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repo
stringlengths 2
152
⌀ | file
stringlengths 15
239
| code
stringlengths 0
58.4M
| file_length
int64 0
58.4M
| avg_line_length
float64 0
1.81M
| max_line_length
int64 0
12.7M
| extension_type
stringclasses 364
values |
|---|---|---|---|---|---|---|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/_lib/_tmpdirs.py
|
''' Contexts for *with* statement providing temporary directories
'''
from __future__ import division, print_function, absolute_import
import os
from contextlib import contextmanager
from shutil import rmtree
from tempfile import mkdtemp
@contextmanager
def tempdir():
"""Create and return a temporary directory. This has the same
behavior as mkdtemp but can be used as a context manager.
Upon exiting the context, the directory and everything contained
in it are removed.
Examples
--------
>>> import os
>>> with tempdir() as tmpdir:
... fname = os.path.join(tmpdir, 'example_file.txt')
... with open(fname, 'wt') as fobj:
... _ = fobj.write('a string\\n')
>>> os.path.exists(tmpdir)
False
"""
d = mkdtemp()
yield d
rmtree(d)
@contextmanager
def in_tempdir():
''' Create, return, and change directory to a temporary directory
Examples
--------
>>> import os
>>> my_cwd = os.getcwd()
>>> with in_tempdir() as tmpdir:
... _ = open('test.txt', 'wt').write('some text')
... assert os.path.isfile('test.txt')
... assert os.path.isfile(os.path.join(tmpdir, 'test.txt'))
>>> os.path.exists(tmpdir)
False
>>> os.getcwd() == my_cwd
True
'''
pwd = os.getcwd()
d = mkdtemp()
os.chdir(d)
yield d
os.chdir(pwd)
rmtree(d)
@contextmanager
def in_dir(dir=None):
""" Change directory to given directory for duration of ``with`` block
Useful when you want to use `in_tempdir` for the final test, but
you are still debugging. For example, you may want to do this in the end:
>>> with in_tempdir() as tmpdir:
... # do something complicated which might break
... pass
But indeed the complicated thing does break, and meanwhile the
``in_tempdir`` context manager wiped out the directory with the
temporary files that you wanted for debugging. So, while debugging, you
replace with something like:
>>> with in_dir() as tmpdir: # Use working directory by default
... # do something complicated which might break
... pass
You can then look at the temporary file outputs to debug what is happening,
fix, and finally replace ``in_dir`` with ``in_tempdir`` again.
"""
cwd = os.getcwd()
if dir is None:
yield cwd
return
os.chdir(dir)
yield dir
os.chdir(cwd)
| 2,439 | 26.727273 | 79 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/_lib/tests/test_tmpdirs.py
|
""" Test tmpdirs module """
from __future__ import division, print_function, absolute_import
from os import getcwd
from os.path import realpath, abspath, dirname, isfile, join as pjoin, exists
from scipy._lib._tmpdirs import tempdir, in_tempdir, in_dir
from numpy.testing import assert_, assert_equal
MY_PATH = abspath(__file__)
MY_DIR = dirname(MY_PATH)
def test_tempdir():
with tempdir() as tmpdir:
fname = pjoin(tmpdir, 'example_file.txt')
with open(fname, 'wt') as fobj:
fobj.write('a string\\n')
assert_(not exists(tmpdir))
def test_in_tempdir():
my_cwd = getcwd()
with in_tempdir() as tmpdir:
with open('test.txt', 'wt') as f:
f.write('some text')
assert_(isfile('test.txt'))
assert_(isfile(pjoin(tmpdir, 'test.txt')))
assert_(not exists(tmpdir))
assert_equal(getcwd(), my_cwd)
def test_given_directory():
# Test InGivenDirectory
cwd = getcwd()
with in_dir() as tmpdir:
assert_equal(tmpdir, abspath(cwd))
assert_equal(tmpdir, abspath(getcwd()))
with in_dir(MY_DIR) as tmpdir:
assert_equal(tmpdir, MY_DIR)
assert_equal(realpath(MY_DIR), realpath(abspath(getcwd())))
# We were deleting the given directory! Check not so now.
assert_(isfile(MY_PATH))
| 1,310 | 27.5 | 77 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/_lib/tests/test__testutils.py
|
from __future__ import division, print_function, absolute_import
import sys
from scipy._lib._testutils import _parse_size, _get_mem_available
import pytest
def test__parse_size():
expected = {
'12': 12e6,
'12 b': 12,
'12k': 12e3,
' 12 M ': 12e6,
' 12 G ': 12e9,
' 12Tb ': 12e12,
'12 Mib ': 12 * 1024.0**2,
'12Tib': 12 * 1024.0**4,
}
for inp, outp in sorted(expected.items()):
if outp is None:
with pytest.raises(ValueError):
_parse_size(inp)
else:
assert _parse_size(inp) == outp
def test__mem_available():
# May return None on non-Linux platforms
available = _get_mem_available()
if sys.platform.startswith('linux'):
assert available >= 0
else:
assert available is None or available >= 0
| 866 | 23.771429 | 65 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/_lib/tests/test_warnings.py
|
"""
Tests which scan for certain occurrences in the code, they may not find
all of these occurrences but should catch almost all. This file was adapted
from numpy.
"""
from __future__ import division, absolute_import, print_function
import sys
import scipy
import pytest
if sys.version_info >= (3, 4):
from pathlib import Path
import ast
import tokenize
class ParseCall(ast.NodeVisitor):
def __init__(self):
self.ls = []
def visit_Attribute(self, node):
ast.NodeVisitor.generic_visit(self, node)
self.ls.append(node.attr)
def visit_Name(self, node):
self.ls.append(node.id)
class FindFuncs(ast.NodeVisitor):
def __init__(self, filename):
super().__init__()
self.__filename = filename
self.bad_filters = []
self.bad_stacklevels = []
def visit_Call(self, node):
p = ParseCall()
p.visit(node.func)
ast.NodeVisitor.generic_visit(self, node)
if p.ls[-1] == 'simplefilter' or p.ls[-1] == 'filterwarnings':
if node.args[0].s == "ignore":
self.bad_filters.append(
"{}:{}".format(self.__filename, node.lineno))
if p.ls[-1] == 'warn' and (
len(p.ls) == 1 or p.ls[-2] == 'warnings'):
if self.__filename == "_lib/tests/test_warnings.py":
# This file
return
# See if stacklevel exists:
if len(node.args) == 3:
return
args = {kw.arg for kw in node.keywords}
if "stacklevel" not in args:
self.bad_stacklevels.append(
"{}:{}".format(self.__filename, node.lineno))
@pytest.fixture(scope="session")
def warning_calls():
# combined "ignore" and stacklevel error
base = Path(scipy.__file__).parent
bad_filters = []
bad_stacklevels = []
for path in base.rglob("*.py"):
# use tokenize to auto-detect encoding on systems where no
# default encoding is defined (e.g. LANG='C')
with tokenize.open(str(path)) as file:
tree = ast.parse(file.read(), filename=str(path))
finder = FindFuncs(path.relative_to(base))
finder.visit(tree)
bad_filters.extend(finder.bad_filters)
bad_stacklevels.extend(finder.bad_stacklevels)
return bad_filters, bad_stacklevels
@pytest.mark.slow
@pytest.mark.skipif(sys.version_info < (3, 4), reason="needs Python >= 3.4")
def test_warning_calls_filters(warning_calls):
bad_filters, bad_stacklevels = warning_calls
# There is still one missing occurrence in optimize.py,
# this is one that should be fixed and this removed then.
bad_filters = [item for item in bad_filters
if 'optimize.py' not in item]
if bad_filters:
raise AssertionError(
"warning ignore filter should not be used, instead, use\n"
"scipy._lib._numpy_compat.suppress_warnings (in tests only);\n"
"found in:\n {}".format(
"\n ".join(bad_filters)))
@pytest.mark.slow
@pytest.mark.skipif(sys.version_info < (3, 4), reason="needs Python >= 3.4")
@pytest.mark.xfail(reason="stacklevels currently missing")
def test_warning_calls_stacklevels(warning_calls):
bad_filters, bad_stacklevels = warning_calls
msg = ""
if bad_filters:
msg += ("warning ignore filter should not be used, instead, use\n"
"scipy._lib._numpy_compat.suppress_warnings (in tests only);\n"
"found in:\n {}".format("\n ".join(bad_filters)))
msg += "\n\n"
if bad_stacklevels:
msg += "warnings should have an appropriate stacklevel:\n {}".format(
"\n ".join(bad_stacklevels))
if msg:
raise AssertionError(msg)
| 3,975 | 31.064516 | 80 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/_lib/tests/test__version.py
|
from __future__ import division, absolute_import, print_function
from numpy.testing import assert_
from pytest import raises as assert_raises
from scipy._lib._version import NumpyVersion
def test_main_versions():
assert_(NumpyVersion('1.8.0') == '1.8.0')
for ver in ['1.9.0', '2.0.0', '1.8.1']:
assert_(NumpyVersion('1.8.0') < ver)
for ver in ['1.7.0', '1.7.1', '0.9.9']:
assert_(NumpyVersion('1.8.0') > ver)
def test_version_1_point_10():
# regression test for gh-2998.
assert_(NumpyVersion('1.9.0') < '1.10.0')
assert_(NumpyVersion('1.11.0') < '1.11.1')
assert_(NumpyVersion('1.11.0') == '1.11.0')
assert_(NumpyVersion('1.99.11') < '1.99.12')
def test_alpha_beta_rc():
assert_(NumpyVersion('1.8.0rc1') == '1.8.0rc1')
for ver in ['1.8.0', '1.8.0rc2']:
assert_(NumpyVersion('1.8.0rc1') < ver)
for ver in ['1.8.0a2', '1.8.0b3', '1.7.2rc4']:
assert_(NumpyVersion('1.8.0rc1') > ver)
assert_(NumpyVersion('1.8.0b1') > '1.8.0a2')
def test_dev_version():
assert_(NumpyVersion('1.9.0.dev-Unknown') < '1.9.0')
for ver in ['1.9.0', '1.9.0a1', '1.9.0b2', '1.9.0b2.dev-ffffffff']:
assert_(NumpyVersion('1.9.0.dev-f16acvda') < ver)
assert_(NumpyVersion('1.9.0.dev-f16acvda') == '1.9.0.dev-11111111')
def test_dev_a_b_rc_mixed():
assert_(NumpyVersion('1.9.0a2.dev-f16acvda') == '1.9.0a2.dev-11111111')
assert_(NumpyVersion('1.9.0a2.dev-6acvda54') < '1.9.0a2')
def test_dev0_version():
assert_(NumpyVersion('1.9.0.dev0+Unknown') < '1.9.0')
for ver in ['1.9.0', '1.9.0a1', '1.9.0b2', '1.9.0b2.dev0+ffffffff']:
assert_(NumpyVersion('1.9.0.dev0+f16acvda') < ver)
assert_(NumpyVersion('1.9.0.dev0+f16acvda') == '1.9.0.dev0+11111111')
def test_dev0_a_b_rc_mixed():
assert_(NumpyVersion('1.9.0a2.dev0+f16acvda') == '1.9.0a2.dev0+11111111')
assert_(NumpyVersion('1.9.0a2.dev0+6acvda54') < '1.9.0a2')
def test_raises():
for ver in ['1.9', '1,9.0', '1.7.x']:
assert_raises(ValueError, NumpyVersion, ver)
| 2,052 | 30.106061 | 77 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/_lib/tests/test__util.py
|
from __future__ import division, print_function, absolute_import
import numpy as np
from numpy.testing import assert_equal, assert_
from pytest import raises as assert_raises
from scipy._lib._util import _aligned_zeros, check_random_state
def test__aligned_zeros():
niter = 10
def check(shape, dtype, order, align):
err_msg = repr((shape, dtype, order, align))
x = _aligned_zeros(shape, dtype, order, align=align)
if align is None:
align = np.dtype(dtype).alignment
assert_equal(x.__array_interface__['data'][0] % align, 0)
if hasattr(shape, '__len__'):
assert_equal(x.shape, shape, err_msg)
else:
assert_equal(x.shape, (shape,), err_msg)
assert_equal(x.dtype, dtype)
if order == "C":
assert_(x.flags.c_contiguous, err_msg)
elif order == "F":
if x.size > 0:
# Size-0 arrays get invalid flags on Numpy 1.5
assert_(x.flags.f_contiguous, err_msg)
elif order is None:
assert_(x.flags.c_contiguous, err_msg)
else:
raise ValueError()
# try various alignments
for align in [1, 2, 3, 4, 8, 16, 32, 64, None]:
for n in [0, 1, 3, 11]:
for order in ["C", "F", None]:
for dtype in [np.uint8, np.float64]:
for shape in [n, (1, 2, 3, n)]:
for j in range(niter):
check(shape, dtype, order, align)
def test_check_random_state():
# If seed is None, return the RandomState singleton used by np.random.
# If seed is an int, return a new RandomState instance seeded with seed.
# If seed is already a RandomState instance, return it.
# Otherwise raise ValueError.
rsi = check_random_state(1)
assert_equal(type(rsi), np.random.RandomState)
rsi = check_random_state(rsi)
assert_equal(type(rsi), np.random.RandomState)
rsi = check_random_state(None)
assert_equal(type(rsi), np.random.RandomState)
assert_raises(ValueError, check_random_state, 'a')
| 2,104 | 35.929825 | 76 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/_lib/tests/test_import_cycles.py
|
from __future__ import division, print_function, absolute_import
import sys
import subprocess
MODULES = [
"scipy.cluster",
"scipy.cluster.vq",
"scipy.cluster.hierarchy",
"scipy.constants",
"scipy.fftpack",
"scipy.integrate",
"scipy.interpolate",
"scipy.io",
"scipy.io.arff",
"scipy.io.harwell_boeing",
"scipy.io.idl",
"scipy.io.matlab",
"scipy.io.netcdf",
"scipy.io.wavfile",
"scipy.linalg",
"scipy.linalg.blas",
"scipy.linalg.cython_blas",
"scipy.linalg.lapack",
"scipy.linalg.cython_lapack",
"scipy.linalg.interpolative",
"scipy.misc",
"scipy.ndimage",
"scipy.odr",
"scipy.optimize",
"scipy.signal",
"scipy.signal.windows",
"scipy.sparse",
"scipy.sparse.linalg",
"scipy.sparse.csgraph",
"scipy.spatial",
"scipy.spatial.distance",
"scipy.special",
"scipy.stats",
"scipy.stats.distributions",
"scipy.stats.mstats",
]
def test_modules_importable():
# Check that all modules are importable in a new Python
# process. This is not necessarily true (esp on Python 2) if there
# are import cycles present.
for module in MODULES:
cmd = 'import {}'.format(module)
subprocess.check_call([sys.executable, '-c', cmd])
| 1,284 | 23.245283 | 70 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/_lib/tests/test_ccallback.py
|
from __future__ import division, print_function, absolute_import
from numpy.testing import assert_equal, assert_
from pytest import raises as assert_raises
import time
import pytest
import ctypes
import threading
from scipy._lib import _ccallback_c as _test_ccallback_cython
from scipy._lib import _test_ccallback
from scipy._lib._ccallback import LowLevelCallable
try:
import cffi
HAVE_CFFI = True
except ImportError:
HAVE_CFFI = False
ERROR_VALUE = 2.0
def callback_python(a, user_data=None):
if a == ERROR_VALUE:
raise ValueError("bad value")
if user_data is None:
return a + 1
else:
return a + user_data
def _get_cffi_func(base, signature):
if not HAVE_CFFI:
pytest.skip("cffi not installed")
# Get function address
voidp = ctypes.cast(base, ctypes.c_void_p)
address = voidp.value
# Create corresponding cffi handle
ffi = cffi.FFI()
func = ffi.cast(signature, address)
return func
def _get_ctypes_data():
value = ctypes.c_double(2.0)
return ctypes.cast(ctypes.pointer(value), ctypes.c_voidp)
def _get_cffi_data():
if not HAVE_CFFI:
pytest.skip("cffi not installed")
ffi = cffi.FFI()
return ffi.new('double *', 2.0)
CALLERS = {
'simple': _test_ccallback.test_call_simple,
'nodata': _test_ccallback.test_call_nodata,
'nonlocal': _test_ccallback.test_call_nonlocal,
'cython': _test_ccallback_cython.test_call_cython,
}
# These functions have signatures known to the callers
FUNCS = {
'python': lambda: callback_python,
'capsule': lambda: _test_ccallback.test_get_plus1_capsule(),
'cython': lambda: LowLevelCallable.from_cython(_test_ccallback_cython, "plus1_cython"),
'ctypes': lambda: _test_ccallback_cython.plus1_ctypes,
'cffi': lambda: _get_cffi_func(_test_ccallback_cython.plus1_ctypes,
'double (*)(double, int *, void *)'),
'capsule_b': lambda: _test_ccallback.test_get_plus1b_capsule(),
'cython_b': lambda: LowLevelCallable.from_cython(_test_ccallback_cython, "plus1b_cython"),
'ctypes_b': lambda: _test_ccallback_cython.plus1b_ctypes,
'cffi_b': lambda: _get_cffi_func(_test_ccallback_cython.plus1b_ctypes,
'double (*)(double, double, int *, void *)'),
}
# These functions have signatures the callers don't know
BAD_FUNCS = {
'capsule_bc': lambda: _test_ccallback.test_get_plus1bc_capsule(),
'cython_bc': lambda: LowLevelCallable.from_cython(_test_ccallback_cython, "plus1bc_cython"),
'ctypes_bc': lambda: _test_ccallback_cython.plus1bc_ctypes,
'cffi_bc': lambda: _get_cffi_func(_test_ccallback_cython.plus1bc_ctypes,
'double (*)(double, double, double, int *, void *)'),
}
USER_DATAS = {
'ctypes': _get_ctypes_data,
'cffi': _get_cffi_data,
'capsule': _test_ccallback.test_get_data_capsule,
}
def test_callbacks():
def check(caller, func, user_data):
caller = CALLERS[caller]
func = FUNCS[func]()
user_data = USER_DATAS[user_data]()
if func is callback_python:
func2 = lambda x: func(x, 2.0)
else:
func2 = LowLevelCallable(func, user_data)
func = LowLevelCallable(func)
# Test basic call
assert_equal(caller(func, 1.0), 2.0)
# Test 'bad' value resulting to an error
assert_raises(ValueError, caller, func, ERROR_VALUE)
# Test passing in user_data
assert_equal(caller(func2, 1.0), 3.0)
for caller in sorted(CALLERS.keys()):
for func in sorted(FUNCS.keys()):
for user_data in sorted(USER_DATAS.keys()):
check(caller, func, user_data)
def test_bad_callbacks():
def check(caller, func, user_data):
caller = CALLERS[caller]
user_data = USER_DATAS[user_data]()
func = BAD_FUNCS[func]()
if func is callback_python:
func2 = lambda x: func(x, 2.0)
else:
func2 = LowLevelCallable(func, user_data)
func = LowLevelCallable(func)
# Test that basic call fails
assert_raises(ValueError, caller, LowLevelCallable(func), 1.0)
# Test that passing in user_data also fails
assert_raises(ValueError, caller, func2, 1.0)
# Test error message
llfunc = LowLevelCallable(func)
try:
caller(llfunc, 1.0)
except ValueError as err:
msg = str(err)
assert_(llfunc.signature in msg, msg)
assert_('double (double, double, int *, void *)' in msg, msg)
for caller in sorted(CALLERS.keys()):
for func in sorted(BAD_FUNCS.keys()):
for user_data in sorted(USER_DATAS.keys()):
check(caller, func, user_data)
def test_signature_override():
caller = _test_ccallback.test_call_simple
func = _test_ccallback.test_get_plus1_capsule()
llcallable = LowLevelCallable(func, signature="bad signature")
assert_equal(llcallable.signature, "bad signature")
assert_raises(ValueError, caller, llcallable, 3)
llcallable = LowLevelCallable(func, signature="double (double, int *, void *)")
assert_equal(llcallable.signature, "double (double, int *, void *)")
assert_equal(caller(llcallable, 3), 4)
def test_threadsafety():
def callback(a, caller):
if a <= 0:
return 1
else:
res = caller(lambda x: callback(x, caller), a - 1)
return 2*res
def check(caller):
caller = CALLERS[caller]
results = []
count = 10
def run():
time.sleep(0.01)
r = caller(lambda x: callback(x, caller), count)
results.append(r)
threads = [threading.Thread(target=run) for j in range(20)]
for thread in threads:
thread.start()
for thread in threads:
thread.join()
assert_equal(results, [2.0**count]*len(threads))
for caller in CALLERS.keys():
check(caller)
| 6,061 | 29.31 | 96 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/_lib/tests/test__threadsafety.py
|
from __future__ import division, print_function, absolute_import
import threading
import time
import traceback
from numpy.testing import assert_
from pytest import raises as assert_raises
from scipy._lib._threadsafety import ReentrancyLock, non_reentrant, ReentrancyError
def test_parallel_threads():
# Check that ReentrancyLock serializes work in parallel threads.
#
# The test is not fully deterministic, and may succeed falsely if
# the timings go wrong.
lock = ReentrancyLock("failure")
failflag = [False]
exceptions_raised = []
def worker(k):
try:
with lock:
assert_(not failflag[0])
failflag[0] = True
time.sleep(0.1 * k)
assert_(failflag[0])
failflag[0] = False
except:
exceptions_raised.append(traceback.format_exc(2))
threads = [threading.Thread(target=lambda k=k: worker(k))
for k in range(3)]
for t in threads:
t.start()
for t in threads:
t.join()
exceptions_raised = "\n".join(exceptions_raised)
assert_(not exceptions_raised, exceptions_raised)
def test_reentering():
# Check that ReentrancyLock prevents re-entering from the same thread.
@non_reentrant()
def func(x):
return func(x)
assert_raises(ReentrancyError, func, 0)
| 1,378 | 24.537037 | 83 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/_lib/tests/test__gcutils.py
|
""" Test for assert_deallocated context manager and gc utilities
"""
from __future__ import division, print_function, absolute_import
import gc
from scipy._lib._gcutils import (set_gc_state, gc_state, assert_deallocated,
ReferenceError, IS_PYPY)
from numpy.testing import assert_equal
import pytest
def test_set_gc_state():
gc_status = gc.isenabled()
try:
for state in (True, False):
gc.enable()
set_gc_state(state)
assert_equal(gc.isenabled(), state)
gc.disable()
set_gc_state(state)
assert_equal(gc.isenabled(), state)
finally:
if gc_status:
gc.enable()
def test_gc_state():
# Test gc_state context manager
gc_status = gc.isenabled()
try:
for pre_state in (True, False):
set_gc_state(pre_state)
for with_state in (True, False):
# Check the gc state is with_state in with block
with gc_state(with_state):
assert_equal(gc.isenabled(), with_state)
# And returns to previous state outside block
assert_equal(gc.isenabled(), pre_state)
# Even if the gc state is set explicitly within the block
with gc_state(with_state):
assert_equal(gc.isenabled(), with_state)
set_gc_state(not with_state)
assert_equal(gc.isenabled(), pre_state)
finally:
if gc_status:
gc.enable()
@pytest.mark.skipif(IS_PYPY, reason="Test not meaningful on PyPy")
def test_assert_deallocated():
# Ordinary use
class C(object):
def __init__(self, arg0, arg1, name='myname'):
self.name = name
for gc_current in (True, False):
with gc_state(gc_current):
# We are deleting from with-block context, so that's OK
with assert_deallocated(C, 0, 2, 'another name') as c:
assert_equal(c.name, 'another name')
del c
# Or not using the thing in with-block context, also OK
with assert_deallocated(C, 0, 2, name='third name'):
pass
assert_equal(gc.isenabled(), gc_current)
@pytest.mark.skipif(IS_PYPY, reason="Test not meaningful on PyPy")
def test_assert_deallocated_nodel():
class C(object):
pass
with pytest.raises(ReferenceError):
# Need to delete after using if in with-block context
with assert_deallocated(C) as c:
pass
@pytest.mark.skipif(IS_PYPY, reason="Test not meaningful on PyPy")
def test_assert_deallocated_circular():
class C(object):
def __init__(self):
self._circular = self
with pytest.raises(ReferenceError):
# Circular reference, no automatic garbage collection
with assert_deallocated(C) as c:
del c
@pytest.mark.skipif(IS_PYPY, reason="Test not meaningful on PyPy")
def test_assert_deallocated_circular2():
class C(object):
def __init__(self):
self._circular = self
with pytest.raises(ReferenceError):
# Still circular reference, no automatic garbage collection
with assert_deallocated(C):
pass
| 3,276 | 31.77 | 76 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/_lib/tests/__init__.py
| 0 | 0 | 0 |
py
|
|
cba-pipeline-public
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cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/cluster/hierarchy.py
|
"""
========================================================
Hierarchical clustering (:mod:`scipy.cluster.hierarchy`)
========================================================
.. currentmodule:: scipy.cluster.hierarchy
These functions cut hierarchical clusterings into flat clusterings
or find the roots of the forest formed by a cut by providing the flat
cluster ids of each observation.
.. autosummary::
:toctree: generated/
fcluster
fclusterdata
leaders
These are routines for agglomerative clustering.
.. autosummary::
:toctree: generated/
linkage
single
complete
average
weighted
centroid
median
ward
These routines compute statistics on hierarchies.
.. autosummary::
:toctree: generated/
cophenet
from_mlab_linkage
inconsistent
maxinconsts
maxdists
maxRstat
to_mlab_linkage
Routines for visualizing flat clusters.
.. autosummary::
:toctree: generated/
dendrogram
These are data structures and routines for representing hierarchies as
tree objects.
.. autosummary::
:toctree: generated/
ClusterNode
leaves_list
to_tree
cut_tree
optimal_leaf_ordering
These are predicates for checking the validity of linkage and
inconsistency matrices as well as for checking isomorphism of two
flat cluster assignments.
.. autosummary::
:toctree: generated/
is_valid_im
is_valid_linkage
is_isomorphic
is_monotonic
correspond
num_obs_linkage
Utility routines for plotting:
.. autosummary::
:toctree: generated/
set_link_color_palette
References
----------
.. [1] "Statistics toolbox." API Reference Documentation. The MathWorks.
http://www.mathworks.com/access/helpdesk/help/toolbox/stats/.
Accessed October 1, 2007.
.. [2] "Hierarchical clustering." API Reference Documentation.
The Wolfram Research, Inc.
https://reference.wolfram.com/language/HierarchicalClustering/tutorial/HierarchicalClustering.html.
Accessed October 1, 2007.
.. [3] Gower, JC and Ross, GJS. "Minimum Spanning Trees and Single Linkage
Cluster Analysis." Applied Statistics. 18(1): pp. 54--64. 1969.
.. [4] Ward Jr, JH. "Hierarchical grouping to optimize an objective
function." Journal of the American Statistical Association. 58(301):
pp. 236--44. 1963.
.. [5] Johnson, SC. "Hierarchical clustering schemes." Psychometrika.
32(2): pp. 241--54. 1966.
.. [6] Sneath, PH and Sokal, RR. "Numerical taxonomy." Nature. 193: pp.
855--60. 1962.
.. [7] Batagelj, V. "Comparing resemblance measures." Journal of
Classification. 12: pp. 73--90. 1995.
.. [8] Sokal, RR and Michener, CD. "A statistical method for evaluating
systematic relationships." Scientific Bulletins. 38(22):
pp. 1409--38. 1958.
.. [9] Edelbrock, C. "Mixture model tests of hierarchical clustering
algorithms: the problem of classifying everybody." Multivariate
Behavioral Research. 14: pp. 367--84. 1979.
.. [10] Jain, A., and Dubes, R., "Algorithms for Clustering Data."
Prentice-Hall. Englewood Cliffs, NJ. 1988.
.. [11] Fisher, RA "The use of multiple measurements in taxonomic
problems." Annals of Eugenics, 7(2): 179-188. 1936
* MATLAB and MathWorks are registered trademarks of The MathWorks, Inc.
* Mathematica is a registered trademark of The Wolfram Research, Inc.
"""
from __future__ import division, print_function, absolute_import
# Copyright (C) Damian Eads, 2007-2008. New BSD License.
# hierarchy.py (derived from cluster.py, http://scipy-cluster.googlecode.com)
#
# Author: Damian Eads
# Date: September 22, 2007
#
# Copyright (c) 2007, 2008, Damian Eads
#
# All rights reserved.
#
# Redistribution and use in source and binary forms, with or without
# modification, are permitted provided that the following conditions
# are met:
# - Redistributions of source code must retain the above
# copyright notice, this list of conditions and the
# following disclaimer.
# - Redistributions in binary form must reproduce the above copyright
# notice, this list of conditions and the following disclaimer
# in the documentation and/or other materials provided with the
# distribution.
# - Neither the name of the author nor the names of its
# contributors may be used to endorse or promote products derived
# from this software without specific prior written permission.
#
# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
# "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
# LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
# A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
# OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
# SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
# LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
# DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
# THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
# (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
# OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
import warnings
import bisect
from collections import deque
import numpy as np
from . import _hierarchy, _optimal_leaf_ordering
import scipy.spatial.distance as distance
from scipy._lib.six import string_types
from scipy._lib.six import xrange
_LINKAGE_METHODS = {'single': 0, 'complete': 1, 'average': 2, 'centroid': 3,
'median': 4, 'ward': 5, 'weighted': 6}
_EUCLIDEAN_METHODS = ('centroid', 'median', 'ward')
__all__ = ['ClusterNode', 'average', 'centroid', 'complete', 'cophenet',
'correspond', 'cut_tree', 'dendrogram', 'fcluster', 'fclusterdata',
'from_mlab_linkage', 'inconsistent', 'is_isomorphic',
'is_monotonic', 'is_valid_im', 'is_valid_linkage', 'leaders',
'leaves_list', 'linkage', 'maxRstat', 'maxdists', 'maxinconsts',
'median', 'num_obs_linkage', 'optimal_leaf_ordering',
'set_link_color_palette', 'single', 'to_mlab_linkage', 'to_tree',
'ward', 'weighted', 'distance']
class ClusterWarning(UserWarning):
pass
def _warning(s):
warnings.warn('scipy.cluster: %s' % s, ClusterWarning, stacklevel=3)
def _copy_array_if_base_present(a):
"""
Copy the array if its base points to a parent array.
"""
if a.base is not None:
return a.copy()
elif np.issubsctype(a, np.float32):
return np.array(a, dtype=np.double)
else:
return a
def _copy_arrays_if_base_present(T):
"""
Accept a tuple of arrays T. Copies the array T[i] if its base array
points to an actual array. Otherwise, the reference is just copied.
This is useful if the arrays are being passed to a C function that
does not do proper striding.
"""
l = [_copy_array_if_base_present(a) for a in T]
return l
def _randdm(pnts):
"""
Generate a random distance matrix stored in condensed form.
Parameters
----------
pnts : int
The number of points in the distance matrix. Has to be at least 2.
Returns
-------
D : ndarray
A ``pnts * (pnts - 1) / 2`` sized vector is returned.
"""
if pnts >= 2:
D = np.random.rand(pnts * (pnts - 1) / 2)
else:
raise ValueError("The number of points in the distance matrix "
"must be at least 2.")
return D
def single(y):
"""
Perform single/min/nearest linkage on the condensed distance matrix ``y``.
Parameters
----------
y : ndarray
The upper triangular of the distance matrix. The result of
``pdist`` is returned in this form.
Returns
-------
Z : ndarray
The linkage matrix.
See Also
--------
linkage: for advanced creation of hierarchical clusterings.
scipy.spatial.distance.pdist : pairwise distance metrics
"""
return linkage(y, method='single', metric='euclidean')
def complete(y):
"""
Perform complete/max/farthest point linkage on a condensed distance matrix.
Parameters
----------
y : ndarray
The upper triangular of the distance matrix. The result of
``pdist`` is returned in this form.
Returns
-------
Z : ndarray
A linkage matrix containing the hierarchical clustering. See
the `linkage` function documentation for more information
on its structure.
See Also
--------
linkage: for advanced creation of hierarchical clusterings.
scipy.spatial.distance.pdist : pairwise distance metrics
"""
return linkage(y, method='complete', metric='euclidean')
def average(y):
"""
Perform average/UPGMA linkage on a condensed distance matrix.
Parameters
----------
y : ndarray
The upper triangular of the distance matrix. The result of
``pdist`` is returned in this form.
Returns
-------
Z : ndarray
A linkage matrix containing the hierarchical clustering. See
`linkage` for more information on its structure.
See Also
--------
linkage: for advanced creation of hierarchical clusterings.
scipy.spatial.distance.pdist : pairwise distance metrics
"""
return linkage(y, method='average', metric='euclidean')
def weighted(y):
"""
Perform weighted/WPGMA linkage on the condensed distance matrix.
See `linkage` for more information on the return
structure and algorithm.
Parameters
----------
y : ndarray
The upper triangular of the distance matrix. The result of
``pdist`` is returned in this form.
Returns
-------
Z : ndarray
A linkage matrix containing the hierarchical clustering. See
`linkage` for more information on its structure.
See Also
--------
linkage : for advanced creation of hierarchical clusterings.
scipy.spatial.distance.pdist : pairwise distance metrics
"""
return linkage(y, method='weighted', metric='euclidean')
def centroid(y):
"""
Perform centroid/UPGMC linkage.
See `linkage` for more information on the input matrix,
return structure, and algorithm.
The following are common calling conventions:
1. ``Z = centroid(y)``
Performs centroid/UPGMC linkage on the condensed distance
matrix ``y``.
2. ``Z = centroid(X)``
Performs centroid/UPGMC linkage on the observation matrix ``X``
using Euclidean distance as the distance metric.
Parameters
----------
y : ndarray
A condensed distance matrix. A condensed
distance matrix is a flat array containing the upper
triangular of the distance matrix. This is the form that
``pdist`` returns. Alternatively, a collection of
m observation vectors in n dimensions may be passed as
a m by n array.
Returns
-------
Z : ndarray
A linkage matrix containing the hierarchical clustering. See
the `linkage` function documentation for more information
on its structure.
See Also
--------
linkage: for advanced creation of hierarchical clusterings.
"""
return linkage(y, method='centroid', metric='euclidean')
def median(y):
"""
Perform median/WPGMC linkage.
See `linkage` for more information on the return structure
and algorithm.
The following are common calling conventions:
1. ``Z = median(y)``
Performs median/WPGMC linkage on the condensed distance matrix
``y``. See ``linkage`` for more information on the return
structure and algorithm.
2. ``Z = median(X)``
Performs median/WPGMC linkage on the observation matrix ``X``
using Euclidean distance as the distance metric. See `linkage`
for more information on the return structure and algorithm.
Parameters
----------
y : ndarray
A condensed distance matrix. A condensed
distance matrix is a flat array containing the upper
triangular of the distance matrix. This is the form that
``pdist`` returns. Alternatively, a collection of
m observation vectors in n dimensions may be passed as
a m by n array.
Returns
-------
Z : ndarray
The hierarchical clustering encoded as a linkage matrix.
See Also
--------
linkage: for advanced creation of hierarchical clusterings.
scipy.spatial.distance.pdist : pairwise distance metrics
"""
return linkage(y, method='median', metric='euclidean')
def ward(y):
"""
Perform Ward's linkage on a condensed distance matrix.
See `linkage` for more information on the return structure
and algorithm.
The following are common calling conventions:
1. ``Z = ward(y)``
Performs Ward's linkage on the condensed distance matrix ``y``.
2. ``Z = ward(X)``
Performs Ward's linkage on the observation matrix ``X`` using
Euclidean distance as the distance metric.
Parameters
----------
y : ndarray
A condensed distance matrix. A condensed
distance matrix is a flat array containing the upper
triangular of the distance matrix. This is the form that
``pdist`` returns. Alternatively, a collection of
m observation vectors in n dimensions may be passed as
a m by n array.
Returns
-------
Z : ndarray
The hierarchical clustering encoded as a linkage matrix. See
`linkage` for more information on the return structure and
algorithm.
See Also
--------
linkage: for advanced creation of hierarchical clusterings.
scipy.spatial.distance.pdist : pairwise distance metrics
"""
return linkage(y, method='ward', metric='euclidean')
def linkage(y, method='single', metric='euclidean', optimal_ordering=False):
"""
Perform hierarchical/agglomerative clustering.
The input y may be either a 1d condensed distance matrix
or a 2d array of observation vectors.
If y is a 1d condensed distance matrix,
then y must be a :math:`\\binom{n}{2}` sized
vector where n is the number of original observations paired
in the distance matrix. The behavior of this function is very
similar to the MATLAB linkage function.
A :math:`(n-1)` by 4 matrix ``Z`` is returned. At the
:math:`i`-th iteration, clusters with indices ``Z[i, 0]`` and
``Z[i, 1]`` are combined to form cluster :math:`n + i`. A
cluster with an index less than :math:`n` corresponds to one of
the :math:`n` original observations. The distance between
clusters ``Z[i, 0]`` and ``Z[i, 1]`` is given by ``Z[i, 2]``. The
fourth value ``Z[i, 3]`` represents the number of original
observations in the newly formed cluster.
The following linkage methods are used to compute the distance
:math:`d(s, t)` between two clusters :math:`s` and
:math:`t`. The algorithm begins with a forest of clusters that
have yet to be used in the hierarchy being formed. When two
clusters :math:`s` and :math:`t` from this forest are combined
into a single cluster :math:`u`, :math:`s` and :math:`t` are
removed from the forest, and :math:`u` is added to the
forest. When only one cluster remains in the forest, the algorithm
stops, and this cluster becomes the root.
A distance matrix is maintained at each iteration. The ``d[i,j]``
entry corresponds to the distance between cluster :math:`i` and
:math:`j` in the original forest.
At each iteration, the algorithm must update the distance matrix
to reflect the distance of the newly formed cluster u with the
remaining clusters in the forest.
Suppose there are :math:`|u|` original observations
:math:`u[0], \\ldots, u[|u|-1]` in cluster :math:`u` and
:math:`|v|` original objects :math:`v[0], \\ldots, v[|v|-1]` in
cluster :math:`v`. Recall :math:`s` and :math:`t` are
combined to form cluster :math:`u`. Let :math:`v` be any
remaining cluster in the forest that is not :math:`u`.
The following are methods for calculating the distance between the
newly formed cluster :math:`u` and each :math:`v`.
* method='single' assigns
.. math::
d(u,v) = \\min(dist(u[i],v[j]))
for all points :math:`i` in cluster :math:`u` and
:math:`j` in cluster :math:`v`. This is also known as the
Nearest Point Algorithm.
* method='complete' assigns
.. math::
d(u, v) = \\max(dist(u[i],v[j]))
for all points :math:`i` in cluster u and :math:`j` in
cluster :math:`v`. This is also known by the Farthest Point
Algorithm or Voor Hees Algorithm.
* method='average' assigns
.. math::
d(u,v) = \\sum_{ij} \\frac{d(u[i], v[j])}
{(|u|*|v|)}
for all points :math:`i` and :math:`j` where :math:`|u|`
and :math:`|v|` are the cardinalities of clusters :math:`u`
and :math:`v`, respectively. This is also called the UPGMA
algorithm.
* method='weighted' assigns
.. math::
d(u,v) = (dist(s,v) + dist(t,v))/2
where cluster u was formed with cluster s and t and v
is a remaining cluster in the forest. (also called WPGMA)
* method='centroid' assigns
.. math::
dist(s,t) = ||c_s-c_t||_2
where :math:`c_s` and :math:`c_t` are the centroids of
clusters :math:`s` and :math:`t`, respectively. When two
clusters :math:`s` and :math:`t` are combined into a new
cluster :math:`u`, the new centroid is computed over all the
original objects in clusters :math:`s` and :math:`t`. The
distance then becomes the Euclidean distance between the
centroid of :math:`u` and the centroid of a remaining cluster
:math:`v` in the forest. This is also known as the UPGMC
algorithm.
* method='median' assigns :math:`d(s,t)` like the ``centroid``
method. When two clusters :math:`s` and :math:`t` are combined
into a new cluster :math:`u`, the average of centroids s and t
give the new centroid :math:`u`. This is also known as the
WPGMC algorithm.
* method='ward' uses the Ward variance minimization algorithm.
The new entry :math:`d(u,v)` is computed as follows,
.. math::
d(u,v) = \\sqrt{\\frac{|v|+|s|}
{T}d(v,s)^2
+ \\frac{|v|+|t|}
{T}d(v,t)^2
- \\frac{|v|}
{T}d(s,t)^2}
where :math:`u` is the newly joined cluster consisting of
clusters :math:`s` and :math:`t`, :math:`v` is an unused
cluster in the forest, :math:`T=|v|+|s|+|t|`, and
:math:`|*|` is the cardinality of its argument. This is also
known as the incremental algorithm.
Warning: When the minimum distance pair in the forest is chosen, there
may be two or more pairs with the same minimum distance. This
implementation may choose a different minimum than the MATLAB
version.
Parameters
----------
y : ndarray
A condensed distance matrix. A condensed distance matrix
is a flat array containing the upper triangular of the distance matrix.
This is the form that ``pdist`` returns. Alternatively, a collection of
:math:`m` observation vectors in :math:`n` dimensions may be passed as
an :math:`m` by :math:`n` array. All elements of the condensed distance
matrix must be finite, i.e. no NaNs or infs.
method : str, optional
The linkage algorithm to use. See the ``Linkage Methods`` section below
for full descriptions.
metric : str or function, optional
The distance metric to use in the case that y is a collection of
observation vectors; ignored otherwise. See the ``pdist``
function for a list of valid distance metrics. A custom distance
function can also be used.
optimal_ordering : bool, optional
If True, the linkage matrix will be reordered so that the distance
between successive leaves is minimal. This results in a more intuitive
tree structure when the data are visualized. defaults to False, because
this algorithm can be slow, particularly on large datasets [2]_. See
also the `optimal_leaf_ordering` function.
.. versionadded:: 1.0.0
Returns
-------
Z : ndarray
The hierarchical clustering encoded as a linkage matrix.
Notes
-----
1. For method 'single' an optimized algorithm based on minimum spanning
tree is implemented. It has time complexity :math:`O(n^2)`.
For methods 'complete', 'average', 'weighted' and 'ward' an algorithm
called nearest-neighbors chain is implemented. It also has time
complexity :math:`O(n^2)`.
For other methods a naive algorithm is implemented with :math:`O(n^3)`
time complexity.
All algorithms use :math:`O(n^2)` memory.
Refer to [1]_ for details about the algorithms.
2. Methods 'centroid', 'median' and 'ward' are correctly defined only if
Euclidean pairwise metric is used. If `y` is passed as precomputed
pairwise distances, then it is a user responsibility to assure that
these distances are in fact Euclidean, otherwise the produced result
will be incorrect.
See Also
--------
scipy.spatial.distance.pdist : pairwise distance metrics
References
----------
.. [1] Daniel Mullner, "Modern hierarchical, agglomerative clustering
algorithms", :arXiv:`1109.2378v1`.
.. [2] Ziv Bar-Joseph, David K. Gifford, Tommi S. Jaakkola, "Fast optimal
leaf ordering for hierarchical clustering", 2001. Bioinformatics
https://doi.org/10.1093/bioinformatics/17.suppl_1.S22
Examples
--------
>>> from scipy.cluster.hierarchy import dendrogram, linkage
>>> from matplotlib import pyplot as plt
>>> X = [[i] for i in [2, 8, 0, 4, 1, 9, 9, 0]]
>>> Z = linkage(X, 'ward')
>>> fig = plt.figure(figsize=(25, 10))
>>> dn = dendrogram(Z)
>>> Z = linkage(X, 'single')
>>> fig = plt.figure(figsize=(25, 10))
>>> dn = dendrogram(Z)
>>> plt.show()
"""
if method not in _LINKAGE_METHODS:
raise ValueError("Invalid method: {0}".format(method))
y = _convert_to_double(np.asarray(y, order='c'))
if y.ndim == 1:
distance.is_valid_y(y, throw=True, name='y')
[y] = _copy_arrays_if_base_present([y])
elif y.ndim == 2:
if method in _EUCLIDEAN_METHODS and metric != 'euclidean':
raise ValueError("Method '{0}' requires the distance metric "
"to be Euclidean".format(method))
if y.shape[0] == y.shape[1] and np.allclose(np.diag(y), 0):
if np.all(y >= 0) and np.allclose(y, y.T):
_warning('The symmetric non-negative hollow observation '
'matrix looks suspiciously like an uncondensed '
'distance matrix')
y = distance.pdist(y, metric)
else:
raise ValueError("`y` must be 1 or 2 dimensional.")
if not np.all(np.isfinite(y)):
raise ValueError("The condensed distance matrix must contain only "
"finite values.")
n = int(distance.num_obs_y(y))
method_code = _LINKAGE_METHODS[method]
if method == 'single':
result = _hierarchy.mst_single_linkage(y, n)
elif method in ['complete', 'average', 'weighted', 'ward']:
result = _hierarchy.nn_chain(y, n, method_code)
else:
result = _hierarchy.fast_linkage(y, n, method_code)
if optimal_ordering:
return optimal_leaf_ordering(result, y)
else:
return result
class ClusterNode:
"""
A tree node class for representing a cluster.
Leaf nodes correspond to original observations, while non-leaf nodes
correspond to non-singleton clusters.
The `to_tree` function converts a matrix returned by the linkage
function into an easy-to-use tree representation.
All parameter names are also attributes.
Parameters
----------
id : int
The node id.
left : ClusterNode instance, optional
The left child tree node.
right : ClusterNode instance, optional
The right child tree node.
dist : float, optional
Distance for this cluster in the linkage matrix.
count : int, optional
The number of samples in this cluster.
See Also
--------
to_tree : for converting a linkage matrix ``Z`` into a tree object.
"""
def __init__(self, id, left=None, right=None, dist=0, count=1):
if id < 0:
raise ValueError('The id must be non-negative.')
if dist < 0:
raise ValueError('The distance must be non-negative.')
if (left is None and right is not None) or \
(left is not None and right is None):
raise ValueError('Only full or proper binary trees are permitted.'
' This node has one child.')
if count < 1:
raise ValueError('A cluster must contain at least one original '
'observation.')
self.id = id
self.left = left
self.right = right
self.dist = dist
if self.left is None:
self.count = count
else:
self.count = left.count + right.count
def __lt__(self, node):
if not isinstance(node, ClusterNode):
raise ValueError("Can't compare ClusterNode "
"to type {}".format(type(node)))
return self.dist < node.dist
def __gt__(self, node):
if not isinstance(node, ClusterNode):
raise ValueError("Can't compare ClusterNode "
"to type {}".format(type(node)))
return self.dist > node.dist
def __eq__(self, node):
if not isinstance(node, ClusterNode):
raise ValueError("Can't compare ClusterNode "
"to type {}".format(type(node)))
return self.dist == node.dist
def get_id(self):
"""
The identifier of the target node.
For ``0 <= i < n``, `i` corresponds to original observation i.
For ``n <= i < 2n-1``, `i` corresponds to non-singleton cluster formed
at iteration ``i-n``.
Returns
-------
id : int
The identifier of the target node.
"""
return self.id
def get_count(self):
"""
The number of leaf nodes (original observations) belonging to
the cluster node nd. If the target node is a leaf, 1 is
returned.
Returns
-------
get_count : int
The number of leaf nodes below the target node.
"""
return self.count
def get_left(self):
"""
Return a reference to the left child tree object.
Returns
-------
left : ClusterNode
The left child of the target node. If the node is a leaf,
None is returned.
"""
return self.left
def get_right(self):
"""
Return a reference to the right child tree object.
Returns
-------
right : ClusterNode
The left child of the target node. If the node is a leaf,
None is returned.
"""
return self.right
def is_leaf(self):
"""
Return True if the target node is a leaf.
Returns
-------
leafness : bool
True if the target node is a leaf node.
"""
return self.left is None
def pre_order(self, func=(lambda x: x.id)):
"""
Perform pre-order traversal without recursive function calls.
When a leaf node is first encountered, ``func`` is called with
the leaf node as its argument, and its result is appended to
the list.
For example, the statement::
ids = root.pre_order(lambda x: x.id)
returns a list of the node ids corresponding to the leaf nodes
of the tree as they appear from left to right.
Parameters
----------
func : function
Applied to each leaf ClusterNode object in the pre-order traversal.
Given the ``i``-th leaf node in the pre-order traversal ``n[i]``,
the result of ``func(n[i])`` is stored in ``L[i]``. If not
provided, the index of the original observation to which the node
corresponds is used.
Returns
-------
L : list
The pre-order traversal.
"""
# Do a preorder traversal, caching the result. To avoid having to do
# recursion, we'll store the previous index we've visited in a vector.
n = self.count
curNode = [None] * (2 * n)
lvisited = set()
rvisited = set()
curNode[0] = self
k = 0
preorder = []
while k >= 0:
nd = curNode[k]
ndid = nd.id
if nd.is_leaf():
preorder.append(func(nd))
k = k - 1
else:
if ndid not in lvisited:
curNode[k + 1] = nd.left
lvisited.add(ndid)
k = k + 1
elif ndid not in rvisited:
curNode[k + 1] = nd.right
rvisited.add(ndid)
k = k + 1
# If we've visited the left and right of this non-leaf
# node already, go up in the tree.
else:
k = k - 1
return preorder
_cnode_bare = ClusterNode(0)
_cnode_type = type(ClusterNode)
def _order_cluster_tree(Z):
"""
Return clustering nodes in bottom-up order by distance.
Parameters
----------
Z : scipy.cluster.linkage array
The linkage matrix.
Returns
-------
nodes : list
A list of ClusterNode objects.
"""
q = deque()
tree = to_tree(Z)
q.append(tree)
nodes = []
while q:
node = q.popleft()
if not node.is_leaf():
bisect.insort_left(nodes, node)
q.append(node.get_right())
q.append(node.get_left())
return nodes
def cut_tree(Z, n_clusters=None, height=None):
"""
Given a linkage matrix Z, return the cut tree.
Parameters
----------
Z : scipy.cluster.linkage array
The linkage matrix.
n_clusters : array_like, optional
Number of clusters in the tree at the cut point.
height : array_like, optional
The height at which to cut the tree. Only possible for ultrametric
trees.
Returns
-------
cutree : array
An array indicating group membership at each agglomeration step. I.e.,
for a full cut tree, in the first column each data point is in its own
cluster. At the next step, two nodes are merged. Finally all
singleton and non-singleton clusters are in one group. If `n_clusters`
or `height` is given, the columns correspond to the columns of
`n_clusters` or `height`.
Examples
--------
>>> from scipy import cluster
>>> np.random.seed(23)
>>> X = np.random.randn(50, 4)
>>> Z = cluster.hierarchy.ward(X)
>>> cutree = cluster.hierarchy.cut_tree(Z, n_clusters=[5, 10])
>>> cutree[:10]
array([[0, 0],
[1, 1],
[2, 2],
[3, 3],
[3, 4],
[2, 2],
[0, 0],
[1, 5],
[3, 6],
[4, 7]])
"""
nobs = num_obs_linkage(Z)
nodes = _order_cluster_tree(Z)
if height is not None and n_clusters is not None:
raise ValueError("At least one of either height or n_clusters "
"must be None")
elif height is None and n_clusters is None: # return the full cut tree
cols_idx = np.arange(nobs)
elif height is not None:
heights = np.array([x.dist for x in nodes])
cols_idx = np.searchsorted(heights, height)
else:
cols_idx = nobs - np.searchsorted(np.arange(nobs), n_clusters)
try:
n_cols = len(cols_idx)
except TypeError: # scalar
n_cols = 1
cols_idx = np.array([cols_idx])
groups = np.zeros((n_cols, nobs), dtype=int)
last_group = np.arange(nobs)
if 0 in cols_idx:
groups[0] = last_group
for i, node in enumerate(nodes):
idx = node.pre_order()
this_group = last_group.copy()
this_group[idx] = last_group[idx].min()
this_group[this_group > last_group[idx].max()] -= 1
if i + 1 in cols_idx:
groups[np.where(i + 1 == cols_idx)[0]] = this_group
last_group = this_group
return groups.T
def to_tree(Z, rd=False):
"""
Convert a linkage matrix into an easy-to-use tree object.
The reference to the root `ClusterNode` object is returned (by default).
Each `ClusterNode` object has a ``left``, ``right``, ``dist``, ``id``,
and ``count`` attribute. The left and right attributes point to
ClusterNode objects that were combined to generate the cluster.
If both are None then the `ClusterNode` object is a leaf node, its count
must be 1, and its distance is meaningless but set to 0.
*Note: This function is provided for the convenience of the library
user. ClusterNodes are not used as input to any of the functions in this
library.*
Parameters
----------
Z : ndarray
The linkage matrix in proper form (see the `linkage`
function documentation).
rd : bool, optional
When False (default), a reference to the root `ClusterNode` object is
returned. Otherwise, a tuple ``(r, d)`` is returned. ``r`` is a
reference to the root node while ``d`` is a list of `ClusterNode`
objects - one per original entry in the linkage matrix plus entries
for all clustering steps. If a cluster id is
less than the number of samples ``n`` in the data that the linkage
matrix describes, then it corresponds to a singleton cluster (leaf
node).
See `linkage` for more information on the assignment of cluster ids
to clusters.
Returns
-------
tree : ClusterNode or tuple (ClusterNode, list of ClusterNode)
If ``rd`` is False, a `ClusterNode`.
If ``rd`` is True, a list of length ``2*n - 1``, with ``n`` the number
of samples. See the description of `rd` above for more details.
See Also
--------
linkage, is_valid_linkage, ClusterNode
Examples
--------
>>> from scipy.cluster import hierarchy
>>> x = np.random.rand(10).reshape(5, 2)
>>> Z = hierarchy.linkage(x)
>>> hierarchy.to_tree(Z)
<scipy.cluster.hierarchy.ClusterNode object at ...
>>> rootnode, nodelist = hierarchy.to_tree(Z, rd=True)
>>> rootnode
<scipy.cluster.hierarchy.ClusterNode object at ...
>>> len(nodelist)
9
"""
Z = np.asarray(Z, order='c')
is_valid_linkage(Z, throw=True, name='Z')
# Number of original objects is equal to the number of rows minus 1.
n = Z.shape[0] + 1
# Create a list full of None's to store the node objects
d = [None] * (n * 2 - 1)
# Create the nodes corresponding to the n original objects.
for i in xrange(0, n):
d[i] = ClusterNode(i)
nd = None
for i in xrange(0, n - 1):
fi = int(Z[i, 0])
fj = int(Z[i, 1])
if fi > i + n:
raise ValueError(('Corrupt matrix Z. Index to derivative cluster '
'is used before it is formed. See row %d, '
'column 0') % fi)
if fj > i + n:
raise ValueError(('Corrupt matrix Z. Index to derivative cluster '
'is used before it is formed. See row %d, '
'column 1') % fj)
nd = ClusterNode(i + n, d[fi], d[fj], Z[i, 2])
# ^ id ^ left ^ right ^ dist
if Z[i, 3] != nd.count:
raise ValueError(('Corrupt matrix Z. The count Z[%d,3] is '
'incorrect.') % i)
d[n + i] = nd
if rd:
return (nd, d)
else:
return nd
def optimal_leaf_ordering(Z, y, metric='euclidean'):
"""
Given a linkage matrix Z and distance, reorder the cut tree.
Parameters
----------
Z : ndarray
The hierarchical clustering encoded as a linkage matrix. See
`linkage` for more information on the return structure and
algorithm.
y : ndarray
The condensed distance matrix from which Z was generated.
Alternatively, a collection of m observation vectors in n
dimensions may be passed as a m by n array.
metric : str or function, optional
The distance metric to use in the case that y is a collection of
observation vectors; ignored otherwise. See the ``pdist``
function for a list of valid distance metrics. A custom distance
function can also be used.
Returns
-------
Z_ordered : ndarray
A copy of the linkage matrix Z, reordered to minimize the distance
between adjacent leaves.
Examples
--------
>>> from scipy.cluster import hierarchy
>>> np.random.seed(23)
>>> X = np.random.randn(10,10)
>>> Z = hierarchy.ward(X)
>>> hierarchy.leaves_list(Z)
array([0, 5, 3, 9, 6, 8, 1, 4, 2, 7], dtype=int32)
>>> hierarchy.leaves_list(hierarchy.optimal_leaf_ordering(Z, X))
array([3, 9, 0, 5, 8, 2, 7, 4, 1, 6], dtype=int32)
"""
Z = np.asarray(Z, order='c')
is_valid_linkage(Z, throw=True, name='Z')
y = _convert_to_double(np.asarray(y, order='c'))
if y.ndim == 1:
distance.is_valid_y(y, throw=True, name='y')
[y] = _copy_arrays_if_base_present([y])
elif y.ndim == 2:
if y.shape[0] == y.shape[1] and np.allclose(np.diag(y), 0):
if np.all(y >= 0) and np.allclose(y, y.T):
_warning('The symmetric non-negative hollow observation '
'matrix looks suspiciously like an uncondensed '
'distance matrix')
y = distance.pdist(y, metric)
else:
raise ValueError("`y` must be 1 or 2 dimensional.")
if not np.all(np.isfinite(y)):
raise ValueError("The condensed distance matrix must contain only "
"finite values.")
return _optimal_leaf_ordering.optimal_leaf_ordering(Z, y)
def _convert_to_bool(X):
if X.dtype != bool:
X = X.astype(bool)
if not X.flags.contiguous:
X = X.copy()
return X
def _convert_to_double(X):
if X.dtype != np.double:
X = X.astype(np.double)
if not X.flags.contiguous:
X = X.copy()
return X
def cophenet(Z, Y=None):
"""
Calculate the cophenetic distances between each observation in
the hierarchical clustering defined by the linkage ``Z``.
Suppose ``p`` and ``q`` are original observations in
disjoint clusters ``s`` and ``t``, respectively and
``s`` and ``t`` are joined by a direct parent cluster
``u``. The cophenetic distance between observations
``i`` and ``j`` is simply the distance between
clusters ``s`` and ``t``.
Parameters
----------
Z : ndarray
The hierarchical clustering encoded as an array
(see `linkage` function).
Y : ndarray (optional)
Calculates the cophenetic correlation coefficient ``c`` of a
hierarchical clustering defined by the linkage matrix `Z`
of a set of :math:`n` observations in :math:`m`
dimensions. `Y` is the condensed distance matrix from which
`Z` was generated.
Returns
-------
c : ndarray
The cophentic correlation distance (if ``Y`` is passed).
d : ndarray
The cophenetic distance matrix in condensed form. The
:math:`ij` th entry is the cophenetic distance between
original observations :math:`i` and :math:`j`.
"""
Z = np.asarray(Z, order='c')
is_valid_linkage(Z, throw=True, name='Z')
Zs = Z.shape
n = Zs[0] + 1
zz = np.zeros((n * (n-1)) // 2, dtype=np.double)
# Since the C code does not support striding using strides.
# The dimensions are used instead.
Z = _convert_to_double(Z)
_hierarchy.cophenetic_distances(Z, zz, int(n))
if Y is None:
return zz
Y = np.asarray(Y, order='c')
distance.is_valid_y(Y, throw=True, name='Y')
z = zz.mean()
y = Y.mean()
Yy = Y - y
Zz = zz - z
numerator = (Yy * Zz)
denomA = Yy**2
denomB = Zz**2
c = numerator.sum() / np.sqrt((denomA.sum() * denomB.sum()))
return (c, zz)
def inconsistent(Z, d=2):
r"""
Calculate inconsistency statistics on a linkage matrix.
Parameters
----------
Z : ndarray
The :math:`(n-1)` by 4 matrix encoding the linkage (hierarchical
clustering). See `linkage` documentation for more information on its
form.
d : int, optional
The number of links up to `d` levels below each non-singleton cluster.
Returns
-------
R : ndarray
A :math:`(n-1)` by 4 matrix where the ``i``'th row contains the link
statistics for the non-singleton cluster ``i``. The link statistics are
computed over the link heights for links :math:`d` levels below the
cluster ``i``. ``R[i,0]`` and ``R[i,1]`` are the mean and standard
deviation of the link heights, respectively; ``R[i,2]`` is the number
of links included in the calculation; and ``R[i,3]`` is the
inconsistency coefficient,
.. math:: \frac{\mathtt{Z[i,2]} - \mathtt{R[i,0]}} {R[i,1]}
Notes
-----
This function behaves similarly to the MATLAB(TM) ``inconsistent``
function.
Examples
--------
>>> from scipy.cluster.hierarchy import inconsistent, linkage
>>> from matplotlib import pyplot as plt
>>> X = [[i] for i in [2, 8, 0, 4, 1, 9, 9, 0]]
>>> Z = linkage(X, 'ward')
>>> print(Z)
[[ 5. 6. 0. 2. ]
[ 2. 7. 0. 2. ]
[ 0. 4. 1. 2. ]
[ 1. 8. 1.15470054 3. ]
[ 9. 10. 2.12132034 4. ]
[ 3. 12. 4.11096096 5. ]
[11. 13. 14.07183949 8. ]]
>>> inconsistent(Z)
array([[ 0. , 0. , 1. , 0. ],
[ 0. , 0. , 1. , 0. ],
[ 1. , 0. , 1. , 0. ],
[ 0.57735027, 0.81649658, 2. , 0.70710678],
[ 1.04044011, 1.06123822, 3. , 1.01850858],
[ 3.11614065, 1.40688837, 2. , 0.70710678],
[ 6.44583366, 6.76770586, 3. , 1.12682288]])
"""
Z = np.asarray(Z, order='c')
Zs = Z.shape
is_valid_linkage(Z, throw=True, name='Z')
if (not d == np.floor(d)) or d < 0:
raise ValueError('The second argument d must be a nonnegative '
'integer value.')
# Since the C code does not support striding using strides.
# The dimensions are used instead.
[Z] = _copy_arrays_if_base_present([Z])
n = Zs[0] + 1
R = np.zeros((n - 1, 4), dtype=np.double)
_hierarchy.inconsistent(Z, R, int(n), int(d))
return R
def from_mlab_linkage(Z):
"""
Convert a linkage matrix generated by MATLAB(TM) to a new
linkage matrix compatible with this module.
The conversion does two things:
* the indices are converted from ``1..N`` to ``0..(N-1)`` form,
and
* a fourth column ``Z[:,3]`` is added where ``Z[i,3]`` represents the
number of original observations (leaves) in the non-singleton
cluster ``i``.
This function is useful when loading in linkages from legacy data
files generated by MATLAB.
Parameters
----------
Z : ndarray
A linkage matrix generated by MATLAB(TM).
Returns
-------
ZS : ndarray
A linkage matrix compatible with ``scipy.cluster.hierarchy``.
"""
Z = np.asarray(Z, dtype=np.double, order='c')
Zs = Z.shape
# If it's empty, return it.
if len(Zs) == 0 or (len(Zs) == 1 and Zs[0] == 0):
return Z.copy()
if len(Zs) != 2:
raise ValueError("The linkage array must be rectangular.")
# If it contains no rows, return it.
if Zs[0] == 0:
return Z.copy()
Zpart = Z.copy()
if Zpart[:, 0:2].min() != 1.0 and Zpart[:, 0:2].max() != 2 * Zs[0]:
raise ValueError('The format of the indices is not 1..N')
Zpart[:, 0:2] -= 1.0
CS = np.zeros((Zs[0],), dtype=np.double)
_hierarchy.calculate_cluster_sizes(Zpart, CS, int(Zs[0]) + 1)
return np.hstack([Zpart, CS.reshape(Zs[0], 1)])
def to_mlab_linkage(Z):
"""
Convert a linkage matrix to a MATLAB(TM) compatible one.
Converts a linkage matrix ``Z`` generated by the linkage function
of this module to a MATLAB(TM) compatible one. The return linkage
matrix has the last column removed and the cluster indices are
converted to ``1..N`` indexing.
Parameters
----------
Z : ndarray
A linkage matrix generated by ``scipy.cluster.hierarchy``.
Returns
-------
to_mlab_linkage : ndarray
A linkage matrix compatible with MATLAB(TM)'s hierarchical
clustering functions.
The return linkage matrix has the last column removed
and the cluster indices are converted to ``1..N`` indexing.
"""
Z = np.asarray(Z, order='c', dtype=np.double)
Zs = Z.shape
if len(Zs) == 0 or (len(Zs) == 1 and Zs[0] == 0):
return Z.copy()
is_valid_linkage(Z, throw=True, name='Z')
ZP = Z[:, 0:3].copy()
ZP[:, 0:2] += 1.0
return ZP
def is_monotonic(Z):
"""
Return True if the linkage passed is monotonic.
The linkage is monotonic if for every cluster :math:`s` and :math:`t`
joined, the distance between them is no less than the distance
between any previously joined clusters.
Parameters
----------
Z : ndarray
The linkage matrix to check for monotonicity.
Returns
-------
b : bool
A boolean indicating whether the linkage is monotonic.
"""
Z = np.asarray(Z, order='c')
is_valid_linkage(Z, throw=True, name='Z')
# We expect the i'th value to be greater than its successor.
return (Z[1:, 2] >= Z[:-1, 2]).all()
def is_valid_im(R, warning=False, throw=False, name=None):
"""Return True if the inconsistency matrix passed is valid.
It must be a :math:`n` by 4 array of doubles. The standard
deviations ``R[:,1]`` must be nonnegative. The link counts
``R[:,2]`` must be positive and no greater than :math:`n-1`.
Parameters
----------
R : ndarray
The inconsistency matrix to check for validity.
warning : bool, optional
When True, issues a Python warning if the linkage
matrix passed is invalid.
throw : bool, optional
When True, throws a Python exception if the linkage
matrix passed is invalid.
name : str, optional
This string refers to the variable name of the invalid
linkage matrix.
Returns
-------
b : bool
True if the inconsistency matrix is valid.
"""
R = np.asarray(R, order='c')
valid = True
name_str = "%r " % name if name else ''
try:
if type(R) != np.ndarray:
raise TypeError('Variable %spassed as inconsistency matrix is not '
'a numpy array.' % name_str)
if R.dtype != np.double:
raise TypeError('Inconsistency matrix %smust contain doubles '
'(double).' % name_str)
if len(R.shape) != 2:
raise ValueError('Inconsistency matrix %smust have shape=2 (i.e. '
'be two-dimensional).' % name_str)
if R.shape[1] != 4:
raise ValueError('Inconsistency matrix %smust have 4 columns.' %
name_str)
if R.shape[0] < 1:
raise ValueError('Inconsistency matrix %smust have at least one '
'row.' % name_str)
if (R[:, 0] < 0).any():
raise ValueError('Inconsistency matrix %scontains negative link '
'height means.' % name_str)
if (R[:, 1] < 0).any():
raise ValueError('Inconsistency matrix %scontains negative link '
'height standard deviations.' % name_str)
if (R[:, 2] < 0).any():
raise ValueError('Inconsistency matrix %scontains negative link '
'counts.' % name_str)
except Exception as e:
if throw:
raise
if warning:
_warning(str(e))
valid = False
return valid
def is_valid_linkage(Z, warning=False, throw=False, name=None):
"""
Check the validity of a linkage matrix.
A linkage matrix is valid if it is a two dimensional array (type double)
with :math:`n` rows and 4 columns. The first two columns must contain
indices between 0 and :math:`2n-1`. For a given row ``i``, the following
two expressions have to hold:
.. math::
0 \\leq \\mathtt{Z[i,0]} \\leq i+n-1
0 \\leq Z[i,1] \\leq i+n-1
I.e. a cluster cannot join another cluster unless the cluster being joined
has been generated.
Parameters
----------
Z : array_like
Linkage matrix.
warning : bool, optional
When True, issues a Python warning if the linkage
matrix passed is invalid.
throw : bool, optional
When True, throws a Python exception if the linkage
matrix passed is invalid.
name : str, optional
This string refers to the variable name of the invalid
linkage matrix.
Returns
-------
b : bool
True if the inconsistency matrix is valid.
"""
Z = np.asarray(Z, order='c')
valid = True
name_str = "%r " % name if name else ''
try:
if type(Z) != np.ndarray:
raise TypeError('Passed linkage argument %sis not a valid array.' %
name_str)
if Z.dtype != np.double:
raise TypeError('Linkage matrix %smust contain doubles.' % name_str)
if len(Z.shape) != 2:
raise ValueError('Linkage matrix %smust have shape=2 (i.e. be '
'two-dimensional).' % name_str)
if Z.shape[1] != 4:
raise ValueError('Linkage matrix %smust have 4 columns.' % name_str)
if Z.shape[0] == 0:
raise ValueError('Linkage must be computed on at least two '
'observations.')
n = Z.shape[0]
if n > 1:
if ((Z[:, 0] < 0).any() or (Z[:, 1] < 0).any()):
raise ValueError('Linkage %scontains negative indices.' %
name_str)
if (Z[:, 2] < 0).any():
raise ValueError('Linkage %scontains negative distances.' %
name_str)
if (Z[:, 3] < 0).any():
raise ValueError('Linkage %scontains negative counts.' %
name_str)
if _check_hierarchy_uses_cluster_before_formed(Z):
raise ValueError('Linkage %suses non-singleton cluster before '
'it is formed.' % name_str)
if _check_hierarchy_uses_cluster_more_than_once(Z):
raise ValueError('Linkage %suses the same cluster more than once.'
% name_str)
except Exception as e:
if throw:
raise
if warning:
_warning(str(e))
valid = False
return valid
def _check_hierarchy_uses_cluster_before_formed(Z):
n = Z.shape[0] + 1
for i in xrange(0, n - 1):
if Z[i, 0] >= n + i or Z[i, 1] >= n + i:
return True
return False
def _check_hierarchy_uses_cluster_more_than_once(Z):
n = Z.shape[0] + 1
chosen = set([])
for i in xrange(0, n - 1):
if (Z[i, 0] in chosen) or (Z[i, 1] in chosen) or Z[i, 0] == Z[i, 1]:
return True
chosen.add(Z[i, 0])
chosen.add(Z[i, 1])
return False
def _check_hierarchy_not_all_clusters_used(Z):
n = Z.shape[0] + 1
chosen = set([])
for i in xrange(0, n - 1):
chosen.add(int(Z[i, 0]))
chosen.add(int(Z[i, 1]))
must_chosen = set(range(0, 2 * n - 2))
return len(must_chosen.difference(chosen)) > 0
def num_obs_linkage(Z):
"""
Return the number of original observations of the linkage matrix passed.
Parameters
----------
Z : ndarray
The linkage matrix on which to perform the operation.
Returns
-------
n : int
The number of original observations in the linkage.
"""
Z = np.asarray(Z, order='c')
is_valid_linkage(Z, throw=True, name='Z')
return (Z.shape[0] + 1)
def correspond(Z, Y):
"""
Check for correspondence between linkage and condensed distance matrices.
They must have the same number of original observations for
the check to succeed.
This function is useful as a sanity check in algorithms that make
extensive use of linkage and distance matrices that must
correspond to the same set of original observations.
Parameters
----------
Z : array_like
The linkage matrix to check for correspondence.
Y : array_like
The condensed distance matrix to check for correspondence.
Returns
-------
b : bool
A boolean indicating whether the linkage matrix and distance
matrix could possibly correspond to one another.
"""
is_valid_linkage(Z, throw=True)
distance.is_valid_y(Y, throw=True)
Z = np.asarray(Z, order='c')
Y = np.asarray(Y, order='c')
return distance.num_obs_y(Y) == num_obs_linkage(Z)
def fcluster(Z, t, criterion='inconsistent', depth=2, R=None, monocrit=None):
"""
Form flat clusters from the hierarchical clustering defined by
the given linkage matrix.
Parameters
----------
Z : ndarray
The hierarchical clustering encoded with the matrix returned
by the `linkage` function.
t : float
The threshold to apply when forming flat clusters.
criterion : str, optional
The criterion to use in forming flat clusters. This can
be any of the following values:
``inconsistent`` :
If a cluster node and all its
descendants have an inconsistent value less than or equal
to `t` then all its leaf descendants belong to the
same flat cluster. When no non-singleton cluster meets
this criterion, every node is assigned to its own
cluster. (Default)
``distance`` :
Forms flat clusters so that the original
observations in each flat cluster have no greater a
cophenetic distance than `t`.
``maxclust`` :
Finds a minimum threshold ``r`` so that
the cophenetic distance between any two original
observations in the same flat cluster is no more than
``r`` and no more than `t` flat clusters are formed.
``monocrit`` :
Forms a flat cluster from a cluster node c
with index i when ``monocrit[j] <= t``.
For example, to threshold on the maximum mean distance
as computed in the inconsistency matrix R with a
threshold of 0.8 do::
MR = maxRstat(Z, R, 3)
cluster(Z, t=0.8, criterion='monocrit', monocrit=MR)
``maxclust_monocrit`` :
Forms a flat cluster from a
non-singleton cluster node ``c`` when ``monocrit[i] <=
r`` for all cluster indices ``i`` below and including
``c``. ``r`` is minimized such that no more than ``t``
flat clusters are formed. monocrit must be
monotonic. For example, to minimize the threshold t on
maximum inconsistency values so that no more than 3 flat
clusters are formed, do::
MI = maxinconsts(Z, R)
cluster(Z, t=3, criterion='maxclust_monocrit', monocrit=MI)
depth : int, optional
The maximum depth to perform the inconsistency calculation.
It has no meaning for the other criteria. Default is 2.
R : ndarray, optional
The inconsistency matrix to use for the 'inconsistent'
criterion. This matrix is computed if not provided.
monocrit : ndarray, optional
An array of length n-1. `monocrit[i]` is the
statistics upon which non-singleton i is thresholded. The
monocrit vector must be monotonic, i.e. given a node c with
index i, for all node indices j corresponding to nodes
below c, ``monocrit[i] >= monocrit[j]``.
Returns
-------
fcluster : ndarray
An array of length ``n``. ``T[i]`` is the flat cluster number to
which original observation ``i`` belongs.
"""
Z = np.asarray(Z, order='c')
is_valid_linkage(Z, throw=True, name='Z')
n = Z.shape[0] + 1
T = np.zeros((n,), dtype='i')
# Since the C code does not support striding using strides.
# The dimensions are used instead.
[Z] = _copy_arrays_if_base_present([Z])
if criterion == 'inconsistent':
if R is None:
R = inconsistent(Z, depth)
else:
R = np.asarray(R, order='c')
is_valid_im(R, throw=True, name='R')
# Since the C code does not support striding using strides.
# The dimensions are used instead.
[R] = _copy_arrays_if_base_present([R])
_hierarchy.cluster_in(Z, R, T, float(t), int(n))
elif criterion == 'distance':
_hierarchy.cluster_dist(Z, T, float(t), int(n))
elif criterion == 'maxclust':
_hierarchy.cluster_maxclust_dist(Z, T, int(n), int(t))
elif criterion == 'monocrit':
[monocrit] = _copy_arrays_if_base_present([monocrit])
_hierarchy.cluster_monocrit(Z, monocrit, T, float(t), int(n))
elif criterion == 'maxclust_monocrit':
[monocrit] = _copy_arrays_if_base_present([monocrit])
_hierarchy.cluster_maxclust_monocrit(Z, monocrit, T, int(n), int(t))
else:
raise ValueError('Invalid cluster formation criterion: %s'
% str(criterion))
return T
def fclusterdata(X, t, criterion='inconsistent',
metric='euclidean', depth=2, method='single', R=None):
"""
Cluster observation data using a given metric.
Clusters the original observations in the n-by-m data
matrix X (n observations in m dimensions), using the euclidean
distance metric to calculate distances between original observations,
performs hierarchical clustering using the single linkage algorithm,
and forms flat clusters using the inconsistency method with `t` as the
cut-off threshold.
A one-dimensional array ``T`` of length ``n`` is returned. ``T[i]`` is
the index of the flat cluster to which the original observation ``i``
belongs.
Parameters
----------
X : (N, M) ndarray
N by M data matrix with N observations in M dimensions.
t : float
The threshold to apply when forming flat clusters.
criterion : str, optional
Specifies the criterion for forming flat clusters. Valid
values are 'inconsistent' (default), 'distance', or 'maxclust'
cluster formation algorithms. See `fcluster` for descriptions.
metric : str, optional
The distance metric for calculating pairwise distances. See
``distance.pdist`` for descriptions and linkage to verify
compatibility with the linkage method.
depth : int, optional
The maximum depth for the inconsistency calculation. See
`inconsistent` for more information.
method : str, optional
The linkage method to use (single, complete, average,
weighted, median centroid, ward). See `linkage` for more
information. Default is "single".
R : ndarray, optional
The inconsistency matrix. It will be computed if necessary
if it is not passed.
Returns
-------
fclusterdata : ndarray
A vector of length n. T[i] is the flat cluster number to
which original observation i belongs.
See Also
--------
scipy.spatial.distance.pdist : pairwise distance metrics
Notes
-----
This function is similar to the MATLAB function ``clusterdata``.
"""
X = np.asarray(X, order='c', dtype=np.double)
if type(X) != np.ndarray or len(X.shape) != 2:
raise TypeError('The observation matrix X must be an n by m numpy '
'array.')
Y = distance.pdist(X, metric=metric)
Z = linkage(Y, method=method)
if R is None:
R = inconsistent(Z, d=depth)
else:
R = np.asarray(R, order='c')
T = fcluster(Z, criterion=criterion, depth=depth, R=R, t=t)
return T
def leaves_list(Z):
"""
Return a list of leaf node ids.
The return corresponds to the observation vector index as it appears
in the tree from left to right. Z is a linkage matrix.
Parameters
----------
Z : ndarray
The hierarchical clustering encoded as a matrix. `Z` is
a linkage matrix. See `linkage` for more information.
Returns
-------
leaves_list : ndarray
The list of leaf node ids.
"""
Z = np.asarray(Z, order='c')
is_valid_linkage(Z, throw=True, name='Z')
n = Z.shape[0] + 1
ML = np.zeros((n,), dtype='i')
[Z] = _copy_arrays_if_base_present([Z])
_hierarchy.prelist(Z, ML, int(n))
return ML
# Maps number of leaves to text size.
#
# p <= 20, size="12"
# 20 < p <= 30, size="10"
# 30 < p <= 50, size="8"
# 50 < p <= np.inf, size="6"
_dtextsizes = {20: 12, 30: 10, 50: 8, 85: 6, np.inf: 5}
_drotation = {20: 0, 40: 45, np.inf: 90}
_dtextsortedkeys = list(_dtextsizes.keys())
_dtextsortedkeys.sort()
_drotationsortedkeys = list(_drotation.keys())
_drotationsortedkeys.sort()
def _remove_dups(L):
"""
Remove duplicates AND preserve the original order of the elements.
The set class is not guaranteed to do this.
"""
seen_before = set([])
L2 = []
for i in L:
if i not in seen_before:
seen_before.add(i)
L2.append(i)
return L2
def _get_tick_text_size(p):
for k in _dtextsortedkeys:
if p <= k:
return _dtextsizes[k]
def _get_tick_rotation(p):
for k in _drotationsortedkeys:
if p <= k:
return _drotation[k]
def _plot_dendrogram(icoords, dcoords, ivl, p, n, mh, orientation,
no_labels, color_list, leaf_font_size=None,
leaf_rotation=None, contraction_marks=None,
ax=None, above_threshold_color='b'):
# Import matplotlib here so that it's not imported unless dendrograms
# are plotted. Raise an informative error if importing fails.
try:
# if an axis is provided, don't use pylab at all
if ax is None:
import matplotlib.pylab
import matplotlib.patches
import matplotlib.collections
except ImportError:
raise ImportError("You must install the matplotlib library to plot "
"the dendrogram. Use no_plot=True to calculate the "
"dendrogram without plotting.")
if ax is None:
ax = matplotlib.pylab.gca()
# if we're using pylab, we want to trigger a draw at the end
trigger_redraw = True
else:
trigger_redraw = False
# Independent variable plot width
ivw = len(ivl) * 10
# Dependent variable plot height
dvw = mh + mh * 0.05
iv_ticks = np.arange(5, len(ivl) * 10 + 5, 10)
if orientation in ('top', 'bottom'):
if orientation == 'top':
ax.set_ylim([0, dvw])
ax.set_xlim([0, ivw])
else:
ax.set_ylim([dvw, 0])
ax.set_xlim([0, ivw])
xlines = icoords
ylines = dcoords
if no_labels:
ax.set_xticks([])
ax.set_xticklabels([])
else:
ax.set_xticks(iv_ticks)
if orientation == 'top':
ax.xaxis.set_ticks_position('bottom')
else:
ax.xaxis.set_ticks_position('top')
# Make the tick marks invisible because they cover up the links
for line in ax.get_xticklines():
line.set_visible(False)
leaf_rot = (float(_get_tick_rotation(len(ivl)))
if (leaf_rotation is None) else leaf_rotation)
leaf_font = (float(_get_tick_text_size(len(ivl)))
if (leaf_font_size is None) else leaf_font_size)
ax.set_xticklabels(ivl, rotation=leaf_rot, size=leaf_font)
elif orientation in ('left', 'right'):
if orientation == 'left':
ax.set_xlim([dvw, 0])
ax.set_ylim([0, ivw])
else:
ax.set_xlim([0, dvw])
ax.set_ylim([0, ivw])
xlines = dcoords
ylines = icoords
if no_labels:
ax.set_yticks([])
ax.set_yticklabels([])
else:
ax.set_yticks(iv_ticks)
if orientation == 'left':
ax.yaxis.set_ticks_position('right')
else:
ax.yaxis.set_ticks_position('left')
# Make the tick marks invisible because they cover up the links
for line in ax.get_yticklines():
line.set_visible(False)
leaf_font = (float(_get_tick_text_size(len(ivl)))
if (leaf_font_size is None) else leaf_font_size)
if leaf_rotation is not None:
ax.set_yticklabels(ivl, rotation=leaf_rotation, size=leaf_font)
else:
ax.set_yticklabels(ivl, size=leaf_font)
# Let's use collections instead. This way there is a separate legend item
# for each tree grouping, rather than stupidly one for each line segment.
colors_used = _remove_dups(color_list)
color_to_lines = {}
for color in colors_used:
color_to_lines[color] = []
for (xline, yline, color) in zip(xlines, ylines, color_list):
color_to_lines[color].append(list(zip(xline, yline)))
colors_to_collections = {}
# Construct the collections.
for color in colors_used:
coll = matplotlib.collections.LineCollection(color_to_lines[color],
colors=(color,))
colors_to_collections[color] = coll
# Add all the groupings below the color threshold.
for color in colors_used:
if color != above_threshold_color:
ax.add_collection(colors_to_collections[color])
# If there's a grouping of links above the color threshold, it goes last.
if above_threshold_color in colors_to_collections:
ax.add_collection(colors_to_collections[above_threshold_color])
if contraction_marks is not None:
Ellipse = matplotlib.patches.Ellipse
for (x, y) in contraction_marks:
if orientation in ('left', 'right'):
e = Ellipse((y, x), width=dvw / 100, height=1.0)
else:
e = Ellipse((x, y), width=1.0, height=dvw / 100)
ax.add_artist(e)
e.set_clip_box(ax.bbox)
e.set_alpha(0.5)
e.set_facecolor('k')
if trigger_redraw:
matplotlib.pylab.draw_if_interactive()
_link_line_colors = ['g', 'r', 'c', 'm', 'y', 'k']
def set_link_color_palette(palette):
"""
Set list of matplotlib color codes for use by dendrogram.
Note that this palette is global (i.e. setting it once changes the colors
for all subsequent calls to `dendrogram`) and that it affects only the
the colors below ``color_threshold``.
Note that `dendrogram` also accepts a custom coloring function through its
``link_color_func`` keyword, which is more flexible and non-global.
Parameters
----------
palette : list of str or None
A list of matplotlib color codes. The order of the color codes is the
order in which the colors are cycled through when color thresholding in
the dendrogram.
If ``None``, resets the palette to its default (which is
``['g', 'r', 'c', 'm', 'y', 'k']``).
Returns
-------
None
See Also
--------
dendrogram
Notes
-----
Ability to reset the palette with ``None`` added in Scipy 0.17.0.
Examples
--------
>>> from scipy.cluster import hierarchy
>>> ytdist = np.array([662., 877., 255., 412., 996., 295., 468., 268.,
... 400., 754., 564., 138., 219., 869., 669.])
>>> Z = hierarchy.linkage(ytdist, 'single')
>>> dn = hierarchy.dendrogram(Z, no_plot=True)
>>> dn['color_list']
['g', 'b', 'b', 'b', 'b']
>>> hierarchy.set_link_color_palette(['c', 'm', 'y', 'k'])
>>> dn = hierarchy.dendrogram(Z, no_plot=True)
>>> dn['color_list']
['c', 'b', 'b', 'b', 'b']
>>> dn = hierarchy.dendrogram(Z, no_plot=True, color_threshold=267,
... above_threshold_color='k')
>>> dn['color_list']
['c', 'm', 'm', 'k', 'k']
Now reset the color palette to its default:
>>> hierarchy.set_link_color_palette(None)
"""
if palette is None:
# reset to its default
palette = ['g', 'r', 'c', 'm', 'y', 'k']
elif type(palette) not in (list, tuple):
raise TypeError("palette must be a list or tuple")
_ptypes = [isinstance(p, string_types) for p in palette]
if False in _ptypes:
raise TypeError("all palette list elements must be color strings")
for i in list(_link_line_colors):
_link_line_colors.remove(i)
_link_line_colors.extend(list(palette))
def dendrogram(Z, p=30, truncate_mode=None, color_threshold=None,
get_leaves=True, orientation='top', labels=None,
count_sort=False, distance_sort=False, show_leaf_counts=True,
no_plot=False, no_labels=False, leaf_font_size=None,
leaf_rotation=None, leaf_label_func=None,
show_contracted=False, link_color_func=None, ax=None,
above_threshold_color='b'):
"""
Plot the hierarchical clustering as a dendrogram.
The dendrogram illustrates how each cluster is
composed by drawing a U-shaped link between a non-singleton
cluster and its children. The top of the U-link indicates a
cluster merge. The two legs of the U-link indicate which clusters
were merged. The length of the two legs of the U-link represents
the distance between the child clusters. It is also the
cophenetic distance between original observations in the two
children clusters.
Parameters
----------
Z : ndarray
The linkage matrix encoding the hierarchical clustering to
render as a dendrogram. See the ``linkage`` function for more
information on the format of ``Z``.
p : int, optional
The ``p`` parameter for ``truncate_mode``.
truncate_mode : str, optional
The dendrogram can be hard to read when the original
observation matrix from which the linkage is derived is
large. Truncation is used to condense the dendrogram. There
are several modes:
``None``
No truncation is performed (default).
Note: ``'none'`` is an alias for ``None`` that's kept for
backward compatibility.
``'lastp'``
The last ``p`` non-singleton clusters formed in the linkage are the
only non-leaf nodes in the linkage; they correspond to rows
``Z[n-p-2:end]`` in ``Z``. All other non-singleton clusters are
contracted into leaf nodes.
``'level'``
No more than ``p`` levels of the dendrogram tree are displayed.
A "level" includes all nodes with ``p`` merges from the last merge.
Note: ``'mtica'`` is an alias for ``'level'`` that's kept for
backward compatibility.
color_threshold : double, optional
For brevity, let :math:`t` be the ``color_threshold``.
Colors all the descendent links below a cluster node
:math:`k` the same color if :math:`k` is the first node below
the cut threshold :math:`t`. All links connecting nodes with
distances greater than or equal to the threshold are colored
blue. If :math:`t` is less than or equal to zero, all nodes
are colored blue. If ``color_threshold`` is None or
'default', corresponding with MATLAB(TM) behavior, the
threshold is set to ``0.7*max(Z[:,2])``.
get_leaves : bool, optional
Includes a list ``R['leaves']=H`` in the result
dictionary. For each :math:`i`, ``H[i] == j``, cluster node
``j`` appears in position ``i`` in the left-to-right traversal
of the leaves, where :math:`j < 2n-1` and :math:`i < n`.
orientation : str, optional
The direction to plot the dendrogram, which can be any
of the following strings:
``'top'``
Plots the root at the top, and plot descendent links going downwards.
(default).
``'bottom'``
Plots the root at the bottom, and plot descendent links going
upwards.
``'left'``
Plots the root at the left, and plot descendent links going right.
``'right'``
Plots the root at the right, and plot descendent links going left.
labels : ndarray, optional
By default ``labels`` is None so the index of the original observation
is used to label the leaf nodes. Otherwise, this is an :math:`n`
-sized list (or tuple). The ``labels[i]`` value is the text to put
under the :math:`i` th leaf node only if it corresponds to an original
observation and not a non-singleton cluster.
count_sort : str or bool, optional
For each node n, the order (visually, from left-to-right) n's
two descendent links are plotted is determined by this
parameter, which can be any of the following values:
``False``
Nothing is done.
``'ascending'`` or ``True``
The child with the minimum number of original objects in its cluster
is plotted first.
``'descendent'``
The child with the maximum number of original objects in its cluster
is plotted first.
Note ``distance_sort`` and ``count_sort`` cannot both be True.
distance_sort : str or bool, optional
For each node n, the order (visually, from left-to-right) n's
two descendent links are plotted is determined by this
parameter, which can be any of the following values:
``False``
Nothing is done.
``'ascending'`` or ``True``
The child with the minimum distance between its direct descendents is
plotted first.
``'descending'``
The child with the maximum distance between its direct descendents is
plotted first.
Note ``distance_sort`` and ``count_sort`` cannot both be True.
show_leaf_counts : bool, optional
When True, leaf nodes representing :math:`k>1` original
observation are labeled with the number of observations they
contain in parentheses.
no_plot : bool, optional
When True, the final rendering is not performed. This is
useful if only the data structures computed for the rendering
are needed or if matplotlib is not available.
no_labels : bool, optional
When True, no labels appear next to the leaf nodes in the
rendering of the dendrogram.
leaf_rotation : double, optional
Specifies the angle (in degrees) to rotate the leaf
labels. When unspecified, the rotation is based on the number of
nodes in the dendrogram (default is 0).
leaf_font_size : int, optional
Specifies the font size (in points) of the leaf labels. When
unspecified, the size based on the number of nodes in the
dendrogram.
leaf_label_func : lambda or function, optional
When leaf_label_func is a callable function, for each
leaf with cluster index :math:`k < 2n-1`. The function
is expected to return a string with the label for the
leaf.
Indices :math:`k < n` correspond to original observations
while indices :math:`k \\geq n` correspond to non-singleton
clusters.
For example, to label singletons with their node id and
non-singletons with their id, count, and inconsistency
coefficient, simply do::
# First define the leaf label function.
def llf(id):
if id < n:
return str(id)
else:
return '[%d %d %1.2f]' % (id, count, R[n-id,3])
# The text for the leaf nodes is going to be big so force
# a rotation of 90 degrees.
dendrogram(Z, leaf_label_func=llf, leaf_rotation=90)
show_contracted : bool, optional
When True the heights of non-singleton nodes contracted
into a leaf node are plotted as crosses along the link
connecting that leaf node. This really is only useful when
truncation is used (see ``truncate_mode`` parameter).
link_color_func : callable, optional
If given, `link_color_function` is called with each non-singleton id
corresponding to each U-shaped link it will paint. The function is
expected to return the color to paint the link, encoded as a matplotlib
color string code. For example::
dendrogram(Z, link_color_func=lambda k: colors[k])
colors the direct links below each untruncated non-singleton node
``k`` using ``colors[k]``.
ax : matplotlib Axes instance, optional
If None and `no_plot` is not True, the dendrogram will be plotted
on the current axes. Otherwise if `no_plot` is not True the
dendrogram will be plotted on the given ``Axes`` instance. This can be
useful if the dendrogram is part of a more complex figure.
above_threshold_color : str, optional
This matplotlib color string sets the color of the links above the
color_threshold. The default is 'b'.
Returns
-------
R : dict
A dictionary of data structures computed to render the
dendrogram. Its has the following keys:
``'color_list'``
A list of color names. The k'th element represents the color of the
k'th link.
``'icoord'`` and ``'dcoord'``
Each of them is a list of lists. Let ``icoord = [I1, I2, ..., Ip]``
where ``Ik = [xk1, xk2, xk3, xk4]`` and ``dcoord = [D1, D2, ..., Dp]``
where ``Dk = [yk1, yk2, yk3, yk4]``, then the k'th link painted is
``(xk1, yk1)`` - ``(xk2, yk2)`` - ``(xk3, yk3)`` - ``(xk4, yk4)``.
``'ivl'``
A list of labels corresponding to the leaf nodes.
``'leaves'``
For each i, ``H[i] == j``, cluster node ``j`` appears in position
``i`` in the left-to-right traversal of the leaves, where
:math:`j < 2n-1` and :math:`i < n`. If ``j`` is less than ``n``, the
``i``-th leaf node corresponds to an original observation.
Otherwise, it corresponds to a non-singleton cluster.
See Also
--------
linkage, set_link_color_palette
Notes
-----
It is expected that the distances in ``Z[:,2]`` be monotonic, otherwise
crossings appear in the dendrogram.
Examples
--------
>>> from scipy.cluster import hierarchy
>>> import matplotlib.pyplot as plt
A very basic example:
>>> ytdist = np.array([662., 877., 255., 412., 996., 295., 468., 268.,
... 400., 754., 564., 138., 219., 869., 669.])
>>> Z = hierarchy.linkage(ytdist, 'single')
>>> plt.figure()
>>> dn = hierarchy.dendrogram(Z)
Now plot in given axes, improve the color scheme and use both vertical and
horizontal orientations:
>>> hierarchy.set_link_color_palette(['m', 'c', 'y', 'k'])
>>> fig, axes = plt.subplots(1, 2, figsize=(8, 3))
>>> dn1 = hierarchy.dendrogram(Z, ax=axes[0], above_threshold_color='y',
... orientation='top')
>>> dn2 = hierarchy.dendrogram(Z, ax=axes[1],
... above_threshold_color='#bcbddc',
... orientation='right')
>>> hierarchy.set_link_color_palette(None) # reset to default after use
>>> plt.show()
"""
# This feature was thought about but never implemented (still useful?):
#
# ... = dendrogram(..., leaves_order=None)
#
# Plots the leaves in the order specified by a vector of
# original observation indices. If the vector contains duplicates
# or results in a crossing, an exception will be thrown. Passing
# None orders leaf nodes based on the order they appear in the
# pre-order traversal.
Z = np.asarray(Z, order='c')
if orientation not in ["top", "left", "bottom", "right"]:
raise ValueError("orientation must be one of 'top', 'left', "
"'bottom', or 'right'")
is_valid_linkage(Z, throw=True, name='Z')
Zs = Z.shape
n = Zs[0] + 1
if type(p) in (int, float):
p = int(p)
else:
raise TypeError('The second argument must be a number')
if truncate_mode not in ('lastp', 'mlab', 'mtica', 'level', 'none', None):
# 'mlab' and 'mtica' are kept working for backwards compat.
raise ValueError('Invalid truncation mode.')
if truncate_mode == 'lastp' or truncate_mode == 'mlab':
if p > n or p == 0:
p = n
if truncate_mode == 'mtica':
# 'mtica' is an alias
truncate_mode = 'level'
if truncate_mode == 'level':
if p <= 0:
p = np.inf
if get_leaves:
lvs = []
else:
lvs = None
icoord_list = []
dcoord_list = []
color_list = []
current_color = [0]
currently_below_threshold = [False]
ivl = [] # list of leaves
if color_threshold is None or (isinstance(color_threshold, string_types) and
color_threshold == 'default'):
color_threshold = max(Z[:, 2]) * 0.7
R = {'icoord': icoord_list, 'dcoord': dcoord_list, 'ivl': ivl,
'leaves': lvs, 'color_list': color_list}
# Empty list will be filled in _dendrogram_calculate_info
contraction_marks = [] if show_contracted else None
_dendrogram_calculate_info(
Z=Z, p=p,
truncate_mode=truncate_mode,
color_threshold=color_threshold,
get_leaves=get_leaves,
orientation=orientation,
labels=labels,
count_sort=count_sort,
distance_sort=distance_sort,
show_leaf_counts=show_leaf_counts,
i=2*n - 2,
iv=0.0,
ivl=ivl,
n=n,
icoord_list=icoord_list,
dcoord_list=dcoord_list,
lvs=lvs,
current_color=current_color,
color_list=color_list,
currently_below_threshold=currently_below_threshold,
leaf_label_func=leaf_label_func,
contraction_marks=contraction_marks,
link_color_func=link_color_func,
above_threshold_color=above_threshold_color)
if not no_plot:
mh = max(Z[:, 2])
_plot_dendrogram(icoord_list, dcoord_list, ivl, p, n, mh, orientation,
no_labels, color_list,
leaf_font_size=leaf_font_size,
leaf_rotation=leaf_rotation,
contraction_marks=contraction_marks,
ax=ax,
above_threshold_color=above_threshold_color)
return R
def _append_singleton_leaf_node(Z, p, n, level, lvs, ivl, leaf_label_func,
i, labels):
# If the leaf id structure is not None and is a list then the caller
# to dendrogram has indicated that cluster id's corresponding to the
# leaf nodes should be recorded.
if lvs is not None:
lvs.append(int(i))
# If leaf node labels are to be displayed...
if ivl is not None:
# If a leaf_label_func has been provided, the label comes from the
# string returned from the leaf_label_func, which is a function
# passed to dendrogram.
if leaf_label_func:
ivl.append(leaf_label_func(int(i)))
else:
# Otherwise, if the dendrogram caller has passed a labels list
# for the leaf nodes, use it.
if labels is not None:
ivl.append(labels[int(i - n)])
else:
# Otherwise, use the id as the label for the leaf.x
ivl.append(str(int(i)))
def _append_nonsingleton_leaf_node(Z, p, n, level, lvs, ivl, leaf_label_func,
i, labels, show_leaf_counts):
# If the leaf id structure is not None and is a list then the caller
# to dendrogram has indicated that cluster id's corresponding to the
# leaf nodes should be recorded.
if lvs is not None:
lvs.append(int(i))
if ivl is not None:
if leaf_label_func:
ivl.append(leaf_label_func(int(i)))
else:
if show_leaf_counts:
ivl.append("(" + str(int(Z[i - n, 3])) + ")")
else:
ivl.append("")
def _append_contraction_marks(Z, iv, i, n, contraction_marks):
_append_contraction_marks_sub(Z, iv, int(Z[i - n, 0]), n, contraction_marks)
_append_contraction_marks_sub(Z, iv, int(Z[i - n, 1]), n, contraction_marks)
def _append_contraction_marks_sub(Z, iv, i, n, contraction_marks):
if i >= n:
contraction_marks.append((iv, Z[i - n, 2]))
_append_contraction_marks_sub(Z, iv, int(Z[i - n, 0]), n, contraction_marks)
_append_contraction_marks_sub(Z, iv, int(Z[i - n, 1]), n, contraction_marks)
def _dendrogram_calculate_info(Z, p, truncate_mode,
color_threshold=np.inf, get_leaves=True,
orientation='top', labels=None,
count_sort=False, distance_sort=False,
show_leaf_counts=False, i=-1, iv=0.0,
ivl=[], n=0, icoord_list=[], dcoord_list=[],
lvs=None, mhr=False,
current_color=[], color_list=[],
currently_below_threshold=[],
leaf_label_func=None, level=0,
contraction_marks=None,
link_color_func=None,
above_threshold_color='b'):
"""
Calculate the endpoints of the links as well as the labels for the
the dendrogram rooted at the node with index i. iv is the independent
variable value to plot the left-most leaf node below the root node i
(if orientation='top', this would be the left-most x value where the
plotting of this root node i and its descendents should begin).
ivl is a list to store the labels of the leaf nodes. The leaf_label_func
is called whenever ivl != None, labels == None, and
leaf_label_func != None. When ivl != None and labels != None, the
labels list is used only for labeling the leaf nodes. When
ivl == None, no labels are generated for leaf nodes.
When get_leaves==True, a list of leaves is built as they are visited
in the dendrogram.
Returns a tuple with l being the independent variable coordinate that
corresponds to the midpoint of cluster to the left of cluster i if
i is non-singleton, otherwise the independent coordinate of the leaf
node if i is a leaf node.
Returns
-------
A tuple (left, w, h, md), where:
* left is the independent variable coordinate of the center of the
the U of the subtree
* w is the amount of space used for the subtree (in independent
variable units)
* h is the height of the subtree in dependent variable units
* md is the ``max(Z[*,2]``) for all nodes ``*`` below and including
the target node.
"""
if n == 0:
raise ValueError("Invalid singleton cluster count n.")
if i == -1:
raise ValueError("Invalid root cluster index i.")
if truncate_mode == 'lastp':
# If the node is a leaf node but corresponds to a non-singleton
# cluster, its label is either the empty string or the number of
# original observations belonging to cluster i.
if 2*n - p > i >= n:
d = Z[i - n, 2]
_append_nonsingleton_leaf_node(Z, p, n, level, lvs, ivl,
leaf_label_func, i, labels,
show_leaf_counts)
if contraction_marks is not None:
_append_contraction_marks(Z, iv + 5.0, i, n, contraction_marks)
return (iv + 5.0, 10.0, 0.0, d)
elif i < n:
_append_singleton_leaf_node(Z, p, n, level, lvs, ivl,
leaf_label_func, i, labels)
return (iv + 5.0, 10.0, 0.0, 0.0)
elif truncate_mode == 'level':
if i > n and level > p:
d = Z[i - n, 2]
_append_nonsingleton_leaf_node(Z, p, n, level, lvs, ivl,
leaf_label_func, i, labels,
show_leaf_counts)
if contraction_marks is not None:
_append_contraction_marks(Z, iv + 5.0, i, n, contraction_marks)
return (iv + 5.0, 10.0, 0.0, d)
elif i < n:
_append_singleton_leaf_node(Z, p, n, level, lvs, ivl,
leaf_label_func, i, labels)
return (iv + 5.0, 10.0, 0.0, 0.0)
elif truncate_mode in ('mlab',):
msg = "Mode 'mlab' is deprecated in scipy 0.19.0 (it never worked)."
warnings.warn(msg, DeprecationWarning)
# Otherwise, only truncate if we have a leaf node.
#
# Only place leaves if they correspond to original observations.
if i < n:
_append_singleton_leaf_node(Z, p, n, level, lvs, ivl,
leaf_label_func, i, labels)
return (iv + 5.0, 10.0, 0.0, 0.0)
# !!! Otherwise, we don't have a leaf node, so work on plotting a
# non-leaf node.
# Actual indices of a and b
aa = int(Z[i - n, 0])
ab = int(Z[i - n, 1])
if aa > n:
# The number of singletons below cluster a
na = Z[aa - n, 3]
# The distance between a's two direct children.
da = Z[aa - n, 2]
else:
na = 1
da = 0.0
if ab > n:
nb = Z[ab - n, 3]
db = Z[ab - n, 2]
else:
nb = 1
db = 0.0
if count_sort == 'ascending' or count_sort:
# If a has a count greater than b, it and its descendents should
# be drawn to the right. Otherwise, to the left.
if na > nb:
# The cluster index to draw to the left (ua) will be ab
# and the one to draw to the right (ub) will be aa
ua = ab
ub = aa
else:
ua = aa
ub = ab
elif count_sort == 'descending':
# If a has a count less than or equal to b, it and its
# descendents should be drawn to the left. Otherwise, to
# the right.
if na > nb:
ua = aa
ub = ab
else:
ua = ab
ub = aa
elif distance_sort == 'ascending' or distance_sort:
# If a has a distance greater than b, it and its descendents should
# be drawn to the right. Otherwise, to the left.
if da > db:
ua = ab
ub = aa
else:
ua = aa
ub = ab
elif distance_sort == 'descending':
# If a has a distance less than or equal to b, it and its
# descendents should be drawn to the left. Otherwise, to
# the right.
if da > db:
ua = aa
ub = ab
else:
ua = ab
ub = aa
else:
ua = aa
ub = ab
# Updated iv variable and the amount of space used.
(uiva, uwa, uah, uamd) = \
_dendrogram_calculate_info(
Z=Z, p=p,
truncate_mode=truncate_mode,
color_threshold=color_threshold,
get_leaves=get_leaves,
orientation=orientation,
labels=labels,
count_sort=count_sort,
distance_sort=distance_sort,
show_leaf_counts=show_leaf_counts,
i=ua, iv=iv, ivl=ivl, n=n,
icoord_list=icoord_list,
dcoord_list=dcoord_list, lvs=lvs,
current_color=current_color,
color_list=color_list,
currently_below_threshold=currently_below_threshold,
leaf_label_func=leaf_label_func,
level=level + 1, contraction_marks=contraction_marks,
link_color_func=link_color_func,
above_threshold_color=above_threshold_color)
h = Z[i - n, 2]
if h >= color_threshold or color_threshold <= 0:
c = above_threshold_color
if currently_below_threshold[0]:
current_color[0] = (current_color[0] + 1) % len(_link_line_colors)
currently_below_threshold[0] = False
else:
currently_below_threshold[0] = True
c = _link_line_colors[current_color[0]]
(uivb, uwb, ubh, ubmd) = \
_dendrogram_calculate_info(
Z=Z, p=p,
truncate_mode=truncate_mode,
color_threshold=color_threshold,
get_leaves=get_leaves,
orientation=orientation,
labels=labels,
count_sort=count_sort,
distance_sort=distance_sort,
show_leaf_counts=show_leaf_counts,
i=ub, iv=iv + uwa, ivl=ivl, n=n,
icoord_list=icoord_list,
dcoord_list=dcoord_list, lvs=lvs,
current_color=current_color,
color_list=color_list,
currently_below_threshold=currently_below_threshold,
leaf_label_func=leaf_label_func,
level=level + 1, contraction_marks=contraction_marks,
link_color_func=link_color_func,
above_threshold_color=above_threshold_color)
max_dist = max(uamd, ubmd, h)
icoord_list.append([uiva, uiva, uivb, uivb])
dcoord_list.append([uah, h, h, ubh])
if link_color_func is not None:
v = link_color_func(int(i))
if not isinstance(v, string_types):
raise TypeError("link_color_func must return a matplotlib "
"color string!")
color_list.append(v)
else:
color_list.append(c)
return (((uiva + uivb) / 2), uwa + uwb, h, max_dist)
def is_isomorphic(T1, T2):
"""
Determine if two different cluster assignments are equivalent.
Parameters
----------
T1 : array_like
An assignment of singleton cluster ids to flat cluster ids.
T2 : array_like
An assignment of singleton cluster ids to flat cluster ids.
Returns
-------
b : bool
Whether the flat cluster assignments `T1` and `T2` are
equivalent.
"""
T1 = np.asarray(T1, order='c')
T2 = np.asarray(T2, order='c')
if type(T1) != np.ndarray:
raise TypeError('T1 must be a numpy array.')
if type(T2) != np.ndarray:
raise TypeError('T2 must be a numpy array.')
T1S = T1.shape
T2S = T2.shape
if len(T1S) != 1:
raise ValueError('T1 must be one-dimensional.')
if len(T2S) != 1:
raise ValueError('T2 must be one-dimensional.')
if T1S[0] != T2S[0]:
raise ValueError('T1 and T2 must have the same number of elements.')
n = T1S[0]
d1 = {}
d2 = {}
for i in xrange(0, n):
if T1[i] in d1:
if not T2[i] in d2:
return False
if d1[T1[i]] != T2[i] or d2[T2[i]] != T1[i]:
return False
elif T2[i] in d2:
return False
else:
d1[T1[i]] = T2[i]
d2[T2[i]] = T1[i]
return True
def maxdists(Z):
"""
Return the maximum distance between any non-singleton cluster.
Parameters
----------
Z : ndarray
The hierarchical clustering encoded as a matrix. See
``linkage`` for more information.
Returns
-------
maxdists : ndarray
A ``(n-1)`` sized numpy array of doubles; ``MD[i]`` represents
the maximum distance between any cluster (including
singletons) below and including the node with index i. More
specifically, ``MD[i] = Z[Q(i)-n, 2].max()`` where ``Q(i)`` is the
set of all node indices below and including node i.
"""
Z = np.asarray(Z, order='c', dtype=np.double)
is_valid_linkage(Z, throw=True, name='Z')
n = Z.shape[0] + 1
MD = np.zeros((n - 1,))
[Z] = _copy_arrays_if_base_present([Z])
_hierarchy.get_max_dist_for_each_cluster(Z, MD, int(n))
return MD
def maxinconsts(Z, R):
"""
Return the maximum inconsistency coefficient for each
non-singleton cluster and its descendents.
Parameters
----------
Z : ndarray
The hierarchical clustering encoded as a matrix. See
`linkage` for more information.
R : ndarray
The inconsistency matrix.
Returns
-------
MI : ndarray
A monotonic ``(n-1)``-sized numpy array of doubles.
"""
Z = np.asarray(Z, order='c')
R = np.asarray(R, order='c')
is_valid_linkage(Z, throw=True, name='Z')
is_valid_im(R, throw=True, name='R')
n = Z.shape[0] + 1
if Z.shape[0] != R.shape[0]:
raise ValueError("The inconsistency matrix and linkage matrix each "
"have a different number of rows.")
MI = np.zeros((n - 1,))
[Z, R] = _copy_arrays_if_base_present([Z, R])
_hierarchy.get_max_Rfield_for_each_cluster(Z, R, MI, int(n), 3)
return MI
def maxRstat(Z, R, i):
"""
Return the maximum statistic for each non-singleton cluster and its
descendents.
Parameters
----------
Z : array_like
The hierarchical clustering encoded as a matrix. See `linkage` for more
information.
R : array_like
The inconsistency matrix.
i : int
The column of `R` to use as the statistic.
Returns
-------
MR : ndarray
Calculates the maximum statistic for the i'th column of the
inconsistency matrix `R` for each non-singleton cluster
node. ``MR[j]`` is the maximum over ``R[Q(j)-n, i]`` where
``Q(j)`` the set of all node ids corresponding to nodes below
and including ``j``.
"""
Z = np.asarray(Z, order='c')
R = np.asarray(R, order='c')
is_valid_linkage(Z, throw=True, name='Z')
is_valid_im(R, throw=True, name='R')
if type(i) is not int:
raise TypeError('The third argument must be an integer.')
if i < 0 or i > 3:
raise ValueError('i must be an integer between 0 and 3 inclusive.')
if Z.shape[0] != R.shape[0]:
raise ValueError("The inconsistency matrix and linkage matrix each "
"have a different number of rows.")
n = Z.shape[0] + 1
MR = np.zeros((n - 1,))
[Z, R] = _copy_arrays_if_base_present([Z, R])
_hierarchy.get_max_Rfield_for_each_cluster(Z, R, MR, int(n), i)
return MR
def leaders(Z, T):
"""
Return the root nodes in a hierarchical clustering.
Returns the root nodes in a hierarchical clustering corresponding
to a cut defined by a flat cluster assignment vector ``T``. See
the ``fcluster`` function for more information on the format of ``T``.
For each flat cluster :math:`j` of the :math:`k` flat clusters
represented in the n-sized flat cluster assignment vector ``T``,
this function finds the lowest cluster node :math:`i` in the linkage
tree Z such that:
* leaf descendents belong only to flat cluster j
(i.e. ``T[p]==j`` for all :math:`p` in :math:`S(i)` where
:math:`S(i)` is the set of leaf ids of leaf nodes descendent
with cluster node :math:`i`)
* there does not exist a leaf that is not descendent with
:math:`i` that also belongs to cluster :math:`j`
(i.e. ``T[q]!=j`` for all :math:`q` not in :math:`S(i)`). If
this condition is violated, ``T`` is not a valid cluster
assignment vector, and an exception will be thrown.
Parameters
----------
Z : ndarray
The hierarchical clustering encoded as a matrix. See
`linkage` for more information.
T : ndarray
The flat cluster assignment vector.
Returns
-------
L : ndarray
The leader linkage node id's stored as a k-element 1-D array
where ``k`` is the number of flat clusters found in ``T``.
``L[j]=i`` is the linkage cluster node id that is the
leader of flat cluster with id M[j]. If ``i < n``, ``i``
corresponds to an original observation, otherwise it
corresponds to a non-singleton cluster.
For example: if ``L[3]=2`` and ``M[3]=8``, the flat cluster with
id 8's leader is linkage node 2.
M : ndarray
The leader linkage node id's stored as a k-element 1-D array where
``k`` is the number of flat clusters found in ``T``. This allows the
set of flat cluster ids to be any arbitrary set of ``k`` integers.
"""
Z = np.asarray(Z, order='c')
T = np.asarray(T, order='c')
if type(T) != np.ndarray or T.dtype != 'i':
raise TypeError('T must be a one-dimensional numpy array of integers.')
is_valid_linkage(Z, throw=True, name='Z')
if len(T) != Z.shape[0] + 1:
raise ValueError('Mismatch: len(T)!=Z.shape[0] + 1.')
Cl = np.unique(T)
kk = len(Cl)
L = np.zeros((kk,), dtype='i')
M = np.zeros((kk,), dtype='i')
n = Z.shape[0] + 1
[Z, T] = _copy_arrays_if_base_present([Z, T])
s = _hierarchy.leaders(Z, T, L, M, int(kk), int(n))
if s >= 0:
raise ValueError(('T is not a valid assignment vector. Error found '
'when examining linkage node %d (< 2n-1).') % s)
return (L, M)
| 103,637 | 33.136364 | 102 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/cluster/vq.py
|
"""
====================================================================
K-means clustering and vector quantization (:mod:`scipy.cluster.vq`)
====================================================================
Provides routines for k-means clustering, generating code books
from k-means models, and quantizing vectors by comparing them with
centroids in a code book.
.. autosummary::
:toctree: generated/
whiten -- Normalize a group of observations so each feature has unit variance
vq -- Calculate code book membership of a set of observation vectors
kmeans -- Performs k-means on a set of observation vectors forming k clusters
kmeans2 -- A different implementation of k-means with more methods
-- for initializing centroids
Background information
======================
The k-means algorithm takes as input the number of clusters to
generate, k, and a set of observation vectors to cluster. It
returns a set of centroids, one for each of the k clusters. An
observation vector is classified with the cluster number or
centroid index of the centroid closest to it.
A vector v belongs to cluster i if it is closer to centroid i than
any other centroids. If v belongs to i, we say centroid i is the
dominating centroid of v. The k-means algorithm tries to
minimize distortion, which is defined as the sum of the squared distances
between each observation vector and its dominating centroid. Each
step of the k-means algorithm refines the choices of centroids to
reduce distortion. The change in distortion is used as a
stopping criterion: when the change is lower than a threshold, the
k-means algorithm is not making sufficient progress and
terminates. One can also define a maximum number of iterations.
Since vector quantization is a natural application for k-means,
information theory terminology is often used. The centroid index
or cluster index is also referred to as a "code" and the table
mapping codes to centroids and vice versa is often referred as a
"code book". The result of k-means, a set of centroids, can be
used to quantize vectors. Quantization aims to find an encoding of
vectors that reduces the expected distortion.
All routines expect obs to be a M by N array where the rows are
the observation vectors. The codebook is a k by N array where the
i'th row is the centroid of code word i. The observation vectors
and centroids have the same feature dimension.
As an example, suppose we wish to compress a 24-bit color image
(each pixel is represented by one byte for red, one for blue, and
one for green) before sending it over the web. By using a smaller
8-bit encoding, we can reduce the amount of data by two
thirds. Ideally, the colors for each of the 256 possible 8-bit
encoding values should be chosen to minimize distortion of the
color. Running k-means with k=256 generates a code book of 256
codes, which fills up all possible 8-bit sequences. Instead of
sending a 3-byte value for each pixel, the 8-bit centroid index
(or code word) of the dominating centroid is transmitted. The code
book is also sent over the wire so each 8-bit code can be
translated back to a 24-bit pixel value representation. If the
image of interest was of an ocean, we would expect many 24-bit
blues to be represented by 8-bit codes. If it was an image of a
human face, more flesh tone colors would be represented in the
code book.
"""
from __future__ import division, print_function, absolute_import
import warnings
import numpy as np
from collections import deque
from scipy._lib._util import _asarray_validated
from scipy._lib.six import xrange
from scipy.spatial.distance import cdist
from . import _vq
__docformat__ = 'restructuredtext'
__all__ = ['whiten', 'vq', 'kmeans', 'kmeans2']
class ClusterError(Exception):
pass
def whiten(obs, check_finite=True):
"""
Normalize a group of observations on a per feature basis.
Before running k-means, it is beneficial to rescale each feature
dimension of the observation set with whitening. Each feature is
divided by its standard deviation across all observations to give
it unit variance.
Parameters
----------
obs : ndarray
Each row of the array is an observation. The
columns are the features seen during each observation.
>>> # f0 f1 f2
>>> obs = [[ 1., 1., 1.], #o0
... [ 2., 2., 2.], #o1
... [ 3., 3., 3.], #o2
... [ 4., 4., 4.]] #o3
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Default: True
Returns
-------
result : ndarray
Contains the values in `obs` scaled by the standard deviation
of each column.
Examples
--------
>>> from scipy.cluster.vq import whiten
>>> features = np.array([[1.9, 2.3, 1.7],
... [1.5, 2.5, 2.2],
... [0.8, 0.6, 1.7,]])
>>> whiten(features)
array([[ 4.17944278, 2.69811351, 7.21248917],
[ 3.29956009, 2.93273208, 9.33380951],
[ 1.75976538, 0.7038557 , 7.21248917]])
"""
obs = _asarray_validated(obs, check_finite=check_finite)
std_dev = obs.std(axis=0)
zero_std_mask = std_dev == 0
if zero_std_mask.any():
std_dev[zero_std_mask] = 1.0
warnings.warn("Some columns have standard deviation zero. "
"The values of these columns will not change.",
RuntimeWarning)
return obs / std_dev
def vq(obs, code_book, check_finite=True):
"""
Assign codes from a code book to observations.
Assigns a code from a code book to each observation. Each
observation vector in the 'M' by 'N' `obs` array is compared with the
centroids in the code book and assigned the code of the closest
centroid.
The features in `obs` should have unit variance, which can be
achieved by passing them through the whiten function. The code
book can be created with the k-means algorithm or a different
encoding algorithm.
Parameters
----------
obs : ndarray
Each row of the 'M' x 'N' array is an observation. The columns are
the "features" seen during each observation. The features must be
whitened first using the whiten function or something equivalent.
code_book : ndarray
The code book is usually generated using the k-means algorithm.
Each row of the array holds a different code, and the columns are
the features of the code.
>>> # f0 f1 f2 f3
>>> code_book = [
... [ 1., 2., 3., 4.], #c0
... [ 1., 2., 3., 4.], #c1
... [ 1., 2., 3., 4.]] #c2
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Default: True
Returns
-------
code : ndarray
A length M array holding the code book index for each observation.
dist : ndarray
The distortion (distance) between the observation and its nearest
code.
Examples
--------
>>> from numpy import array
>>> from scipy.cluster.vq import vq
>>> code_book = array([[1.,1.,1.],
... [2.,2.,2.]])
>>> features = array([[ 1.9,2.3,1.7],
... [ 1.5,2.5,2.2],
... [ 0.8,0.6,1.7]])
>>> vq(features,code_book)
(array([1, 1, 0],'i'), array([ 0.43588989, 0.73484692, 0.83066239]))
"""
obs = _asarray_validated(obs, check_finite=check_finite)
code_book = _asarray_validated(code_book, check_finite=check_finite)
ct = np.common_type(obs, code_book)
c_obs = obs.astype(ct, copy=False)
c_code_book = code_book.astype(ct, copy=False)
if np.issubdtype(ct, np.float64) or np.issubdtype(ct, np.float32):
return _vq.vq(c_obs, c_code_book)
return py_vq(obs, code_book, check_finite=False)
def py_vq(obs, code_book, check_finite=True):
""" Python version of vq algorithm.
The algorithm computes the euclidian distance between each
observation and every frame in the code_book.
Parameters
----------
obs : ndarray
Expects a rank 2 array. Each row is one observation.
code_book : ndarray
Code book to use. Same format than obs. Should have same number of
features (eg columns) than obs.
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Default: True
Returns
-------
code : ndarray
code[i] gives the label of the ith obversation, that its code is
code_book[code[i]].
mind_dist : ndarray
min_dist[i] gives the distance between the ith observation and its
corresponding code.
Notes
-----
This function is slower than the C version but works for
all input types. If the inputs have the wrong types for the
C versions of the function, this one is called as a last resort.
It is about 20 times slower than the C version.
"""
obs = _asarray_validated(obs, check_finite=check_finite)
code_book = _asarray_validated(code_book, check_finite=check_finite)
if obs.ndim != code_book.ndim:
raise ValueError("Observation and code_book should have the same rank")
if obs.ndim == 1:
obs = obs[:, np.newaxis]
code_book = code_book[:, np.newaxis]
dist = cdist(obs, code_book)
code = dist.argmin(axis=1)
min_dist = dist[np.arange(len(code)), code]
return code, min_dist
# py_vq2 was equivalent to py_vq
py_vq2 = np.deprecate(py_vq, old_name='py_vq2', new_name='py_vq')
def _kmeans(obs, guess, thresh=1e-5):
""" "raw" version of k-means.
Returns
-------
code_book
the lowest distortion codebook found.
avg_dist
the average distance a observation is from a code in the book.
Lower means the code_book matches the data better.
See Also
--------
kmeans : wrapper around k-means
Examples
--------
Note: not whitened in this example.
>>> from numpy import array
>>> from scipy.cluster.vq import _kmeans
>>> features = array([[ 1.9,2.3],
... [ 1.5,2.5],
... [ 0.8,0.6],
... [ 0.4,1.8],
... [ 1.0,1.0]])
>>> book = array((features[0],features[2]))
>>> _kmeans(features,book)
(array([[ 1.7 , 2.4 ],
[ 0.73333333, 1.13333333]]), 0.40563916697728591)
"""
code_book = np.asarray(guess)
diff = np.inf
prev_avg_dists = deque([diff], maxlen=2)
while diff > thresh:
# compute membership and distances between obs and code_book
obs_code, distort = vq(obs, code_book, check_finite=False)
prev_avg_dists.append(distort.mean(axis=-1))
# recalc code_book as centroids of associated obs
code_book, has_members = _vq.update_cluster_means(obs, obs_code,
code_book.shape[0])
code_book = code_book[has_members]
diff = prev_avg_dists[0] - prev_avg_dists[1]
return code_book, prev_avg_dists[1]
def kmeans(obs, k_or_guess, iter=20, thresh=1e-5, check_finite=True):
"""
Performs k-means on a set of observation vectors forming k clusters.
The k-means algorithm adjusts the centroids until sufficient
progress cannot be made, i.e. the change in distortion since
the last iteration is less than some threshold. This yields
a code book mapping centroids to codes and vice versa.
Distortion is defined as the sum of the squared differences
between the observations and the corresponding centroid.
Parameters
----------
obs : ndarray
Each row of the M by N array is an observation vector. The
columns are the features seen during each observation.
The features must be whitened first with the `whiten` function.
k_or_guess : int or ndarray
The number of centroids to generate. A code is assigned to
each centroid, which is also the row index of the centroid
in the code_book matrix generated.
The initial k centroids are chosen by randomly selecting
observations from the observation matrix. Alternatively,
passing a k by N array specifies the initial k centroids.
iter : int, optional
The number of times to run k-means, returning the codebook
with the lowest distortion. This argument is ignored if
initial centroids are specified with an array for the
``k_or_guess`` parameter. This parameter does not represent the
number of iterations of the k-means algorithm.
thresh : float, optional
Terminates the k-means algorithm if the change in
distortion since the last k-means iteration is less than
or equal to thresh.
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Default: True
Returns
-------
codebook : ndarray
A k by N array of k centroids. The i'th centroid
codebook[i] is represented with the code i. The centroids
and codes generated represent the lowest distortion seen,
not necessarily the globally minimal distortion.
distortion : float
The distortion between the observations passed and the
centroids generated.
See Also
--------
kmeans2 : a different implementation of k-means clustering
with more methods for generating initial centroids but without
using a distortion change threshold as a stopping criterion.
whiten : must be called prior to passing an observation matrix
to kmeans.
Examples
--------
>>> from numpy import array
>>> from scipy.cluster.vq import vq, kmeans, whiten
>>> import matplotlib.pyplot as plt
>>> features = array([[ 1.9,2.3],
... [ 1.5,2.5],
... [ 0.8,0.6],
... [ 0.4,1.8],
... [ 0.1,0.1],
... [ 0.2,1.8],
... [ 2.0,0.5],
... [ 0.3,1.5],
... [ 1.0,1.0]])
>>> whitened = whiten(features)
>>> book = np.array((whitened[0],whitened[2]))
>>> kmeans(whitened,book)
(array([[ 2.3110306 , 2.86287398], # random
[ 0.93218041, 1.24398691]]), 0.85684700941625547)
>>> from numpy import random
>>> random.seed((1000,2000))
>>> codes = 3
>>> kmeans(whitened,codes)
(array([[ 2.3110306 , 2.86287398], # random
[ 1.32544402, 0.65607529],
[ 0.40782893, 2.02786907]]), 0.5196582527686241)
>>> # Create 50 datapoints in two clusters a and b
>>> pts = 50
>>> a = np.random.multivariate_normal([0, 0], [[4, 1], [1, 4]], size=pts)
>>> b = np.random.multivariate_normal([30, 10],
... [[10, 2], [2, 1]],
... size=pts)
>>> features = np.concatenate((a, b))
>>> # Whiten data
>>> whitened = whiten(features)
>>> # Find 2 clusters in the data
>>> codebook, distortion = kmeans(whitened, 2)
>>> # Plot whitened data and cluster centers in red
>>> plt.scatter(whitened[:, 0], whitened[:, 1])
>>> plt.scatter(codebook[:, 0], codebook[:, 1], c='r')
>>> plt.show()
"""
obs = _asarray_validated(obs, check_finite=check_finite)
if iter < 1:
raise ValueError("iter must be at least 1, got %s" % iter)
# Determine whether a count (scalar) or an initial guess (array) was passed.
if not np.isscalar(k_or_guess):
guess = _asarray_validated(k_or_guess, check_finite=check_finite)
if guess.size < 1:
raise ValueError("Asked for 0 clusters. Initial book was %s" %
guess)
return _kmeans(obs, guess, thresh=thresh)
# k_or_guess is a scalar, now verify that it's an integer
k = int(k_or_guess)
if k != k_or_guess:
raise ValueError("If k_or_guess is a scalar, it must be an integer.")
if k < 1:
raise ValueError("Asked for %d clusters." % k)
# initialize best distance value to a large value
best_dist = np.inf
for i in xrange(iter):
# the initial code book is randomly selected from observations
guess = _kpoints(obs, k)
book, dist = _kmeans(obs, guess, thresh=thresh)
if dist < best_dist:
best_book = book
best_dist = dist
return best_book, best_dist
def _kpoints(data, k):
"""Pick k points at random in data (one row = one observation).
Parameters
----------
data : ndarray
Expect a rank 1 or 2 array. Rank 1 are assumed to describe one
dimensional data, rank 2 multidimensional data, in which case one
row is one observation.
k : int
Number of samples to generate.
"""
idx = np.random.choice(data.shape[0], size=k, replace=False)
return data[idx]
def _krandinit(data, k):
"""Returns k samples of a random variable which parameters depend on data.
More precisely, it returns k observations sampled from a Gaussian random
variable which mean and covariances are the one estimated from data.
Parameters
----------
data : ndarray
Expect a rank 1 or 2 array. Rank 1 are assumed to describe one
dimensional data, rank 2 multidimensional data, in which case one
row is one observation.
k : int
Number of samples to generate.
"""
mu = data.mean(axis=0)
if data.ndim == 1:
cov = np.cov(data)
x = np.random.randn(k)
x *= np.sqrt(cov)
elif data.shape[1] > data.shape[0]:
# initialize when the covariance matrix is rank deficient
_, s, vh = np.linalg.svd(data - mu, full_matrices=False)
x = np.random.randn(k, s.size)
sVh = s[:, None] * vh / np.sqrt(data.shape[0] - 1)
x = x.dot(sVh)
else:
cov = np.atleast_2d(np.cov(data, rowvar=False))
# k rows, d cols (one row = one obs)
# Generate k sample of a random variable ~ Gaussian(mu, cov)
x = np.random.randn(k, mu.size)
x = x.dot(np.linalg.cholesky(cov).T)
x += mu
return x
_valid_init_meth = {'random': _krandinit, 'points': _kpoints}
def _missing_warn():
"""Print a warning when called."""
warnings.warn("One of the clusters is empty. "
"Re-run kmeans with a different initialization.")
def _missing_raise():
"""raise a ClusterError when called."""
raise ClusterError("One of the clusters is empty. "
"Re-run kmeans with a different initialization.")
_valid_miss_meth = {'warn': _missing_warn, 'raise': _missing_raise}
def kmeans2(data, k, iter=10, thresh=1e-5, minit='random',
missing='warn', check_finite=True):
"""
Classify a set of observations into k clusters using the k-means algorithm.
The algorithm attempts to minimize the Euclidian distance between
observations and centroids. Several initialization methods are
included.
Parameters
----------
data : ndarray
A 'M' by 'N' array of 'M' observations in 'N' dimensions or a length
'M' array of 'M' one-dimensional observations.
k : int or ndarray
The number of clusters to form as well as the number of
centroids to generate. If `minit` initialization string is
'matrix', or if a ndarray is given instead, it is
interpreted as initial cluster to use instead.
iter : int, optional
Number of iterations of the k-means algorithm to run. Note
that this differs in meaning from the iters parameter to
the kmeans function.
thresh : float, optional
(not used yet)
minit : str, optional
Method for initialization. Available methods are 'random',
'points', and 'matrix':
'random': generate k centroids from a Gaussian with mean and
variance estimated from the data.
'points': choose k observations (rows) at random from data for
the initial centroids.
'matrix': interpret the k parameter as a k by M (or length k
array for one-dimensional data) array of initial centroids.
missing : str, optional
Method to deal with empty clusters. Available methods are
'warn' and 'raise':
'warn': give a warning and continue.
'raise': raise an ClusterError and terminate the algorithm.
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Default: True
Returns
-------
centroid : ndarray
A 'k' by 'N' array of centroids found at the last iteration of
k-means.
label : ndarray
label[i] is the code or index of the centroid the
i'th observation is closest to.
"""
if int(iter) < 1:
raise ValueError("Invalid iter (%s), "
"must be a positive integer." % iter)
try:
miss_meth = _valid_miss_meth[missing]
except KeyError:
raise ValueError("Unknown missing method %r" % (missing,))
data = _asarray_validated(data, check_finite=check_finite)
if data.ndim == 1:
d = 1
elif data.ndim == 2:
d = data.shape[1]
else:
raise ValueError("Input of rank > 2 is not supported.")
if data.size < 1:
raise ValueError("Empty input is not supported.")
# If k is not a single value it should be compatible with data's shape
if minit == 'matrix' or not np.isscalar(k):
code_book = np.array(k, copy=True)
if data.ndim != code_book.ndim:
raise ValueError("k array doesn't match data rank")
nc = len(code_book)
if data.ndim > 1 and code_book.shape[1] != d:
raise ValueError("k array doesn't match data dimension")
else:
nc = int(k)
if nc < 1:
raise ValueError("Cannot ask kmeans2 for %d clusters"
" (k was %s)" % (nc, k))
elif nc != k:
warnings.warn("k was not an integer, was converted.")
try:
init_meth = _valid_init_meth[minit]
except KeyError:
raise ValueError("Unknown init method %r" % (minit,))
else:
code_book = init_meth(data, k)
for i in xrange(iter):
# Compute the nearest neighbor for each obs using the current code book
label = vq(data, code_book)[0]
# Update the code book by computing centroids
new_code_book, has_members = _vq.update_cluster_means(data, label, nc)
if not has_members.all():
miss_meth()
# Set the empty clusters to their previous positions
new_code_book[~has_members] = code_book[~has_members]
code_book = new_code_book
return code_book, label
| 23,929 | 35.646248 | 80 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/cluster/setup.py
|
from __future__ import division, print_function, absolute_import
import sys
if sys.version_info[0] >= 3:
DEFINE_MACROS = [("SCIPY_PY3K", None)]
else:
DEFINE_MACROS = []
def configuration(parent_package='', top_path=None):
from numpy.distutils.system_info import get_info
from numpy.distutils.misc_util import Configuration, get_numpy_include_dirs
config = Configuration('cluster', parent_package, top_path)
blas_opt = get_info('lapack_opt')
config.add_data_dir('tests')
config.add_extension('_vq',
sources=[('_vq.c')],
include_dirs=[get_numpy_include_dirs()],
extra_info=blas_opt)
config.add_extension('_hierarchy',
sources=[('_hierarchy.c')],
include_dirs=[get_numpy_include_dirs()])
config.add_extension('_optimal_leaf_ordering',
sources=[('_optimal_leaf_ordering.c')],
include_dirs=[get_numpy_include_dirs()])
return config
if __name__ == '__main__':
from numpy.distutils.core import setup
setup(**configuration(top_path='').todict())
| 1,058 | 26.153846 | 79 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/cluster/__init__.py
|
"""
=========================================
Clustering package (:mod:`scipy.cluster`)
=========================================
.. currentmodule:: scipy.cluster
:mod:`scipy.cluster.vq`
Clustering algorithms are useful in information theory, target detection,
communications, compression, and other areas. The `vq` module only
supports vector quantization and the k-means algorithms.
:mod:`scipy.cluster.hierarchy`
The `hierarchy` module provides functions for hierarchical and
agglomerative clustering. Its features include generating hierarchical
clusters from distance matrices,
calculating statistics on clusters, cutting linkages
to generate flat clusters, and visualizing clusters with dendrograms.
"""
from __future__ import division, print_function, absolute_import
__all__ = ['vq', 'hierarchy']
from . import vq, hierarchy
from scipy._lib._testutils import PytestTester
test = PytestTester(__name__)
del PytestTester
| 938 | 28.34375 | 73 |
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cba-pipeline-public
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cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/cluster/tests/test_hierarchy.py
|
#
# Author: Damian Eads
# Date: April 17, 2008
#
# Copyright (C) 2008 Damian Eads
#
# Redistribution and use in source and binary forms, with or without
# modification, are permitted provided that the following conditions
# are met:
#
# 1. Redistributions of source code must retain the above copyright
# notice, this list of conditions and the following disclaimer.
#
# 2. Redistributions in binary form must reproduce the above
# copyright notice, this list of conditions and the following
# disclaimer in the documentation and/or other materials provided
# with the distribution.
#
# 3. The name of the author may not be used to endorse or promote
# products derived from this software without specific prior
# written permission.
#
# THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS
# OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
# WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
# ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY
# DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
# DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE
# GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
# INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY,
# WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
# NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
# SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
from __future__ import division, print_function, absolute_import
import numpy as np
from numpy.testing import assert_allclose, assert_equal, assert_, assert_warns
import pytest
from pytest import raises as assert_raises
from scipy._lib.six import xrange, u
import scipy.cluster.hierarchy
from scipy.cluster.hierarchy import (
ClusterWarning, linkage, from_mlab_linkage, to_mlab_linkage,
num_obs_linkage, inconsistent, cophenet, fclusterdata, fcluster,
is_isomorphic, single, leaders, complete, weighted, centroid,
correspond, is_monotonic, maxdists, maxinconsts, maxRstat,
is_valid_linkage, is_valid_im, to_tree, leaves_list, dendrogram,
set_link_color_palette, cut_tree, optimal_leaf_ordering,
_order_cluster_tree, _hierarchy, _LINKAGE_METHODS)
from scipy.spatial.distance import pdist
from scipy.cluster._hierarchy import Heap
from . import hierarchy_test_data
# Matplotlib is not a scipy dependency but is optionally used in dendrogram, so
# check if it's available
try:
import matplotlib
# and set the backend to be Agg (no gui)
matplotlib.use('Agg')
# before importing pyplot
import matplotlib.pyplot as plt
have_matplotlib = True
except:
have_matplotlib = False
class TestLinkage(object):
def test_linkage_non_finite_elements_in_distance_matrix(self):
# Tests linkage(Y) where Y contains a non-finite element (e.g. NaN or Inf).
# Exception expected.
y = np.zeros((6,))
y[0] = np.nan
assert_raises(ValueError, linkage, y)
def test_linkage_empty_distance_matrix(self):
# Tests linkage(Y) where Y is a 0x4 linkage matrix. Exception expected.
y = np.zeros((0,))
assert_raises(ValueError, linkage, y)
def test_linkage_tdist(self):
for method in ['single', 'complete', 'average', 'weighted', u('single')]:
self.check_linkage_tdist(method)
def check_linkage_tdist(self, method):
# Tests linkage(Y, method) on the tdist data set.
Z = linkage(hierarchy_test_data.ytdist, method)
expectedZ = getattr(hierarchy_test_data, 'linkage_ytdist_' + method)
assert_allclose(Z, expectedZ, atol=1e-10)
def test_linkage_X(self):
for method in ['centroid', 'median', 'ward']:
self.check_linkage_q(method)
def check_linkage_q(self, method):
# Tests linkage(Y, method) on the Q data set.
Z = linkage(hierarchy_test_data.X, method)
expectedZ = getattr(hierarchy_test_data, 'linkage_X_' + method)
assert_allclose(Z, expectedZ, atol=1e-06)
y = scipy.spatial.distance.pdist(hierarchy_test_data.X,
metric="euclidean")
Z = linkage(y, method)
assert_allclose(Z, expectedZ, atol=1e-06)
def test_compare_with_trivial(self):
rng = np.random.RandomState(0)
n = 20
X = rng.rand(n, 2)
d = pdist(X)
for method, code in _LINKAGE_METHODS.items():
Z_trivial = _hierarchy.linkage(d, n, code)
Z = linkage(d, method)
assert_allclose(Z_trivial, Z, rtol=1e-14, atol=1e-15)
def test_optimal_leaf_ordering(self):
Z = linkage(hierarchy_test_data.ytdist, optimal_ordering=True)
expectedZ = getattr(hierarchy_test_data, 'linkage_ytdist_single_olo')
assert_allclose(Z, expectedZ, atol=1e-10)
class TestLinkageTies(object):
_expectations = {
'single': np.array([[0, 1, 1.41421356, 2],
[2, 3, 1.41421356, 3]]),
'complete': np.array([[0, 1, 1.41421356, 2],
[2, 3, 2.82842712, 3]]),
'average': np.array([[0, 1, 1.41421356, 2],
[2, 3, 2.12132034, 3]]),
'weighted': np.array([[0, 1, 1.41421356, 2],
[2, 3, 2.12132034, 3]]),
'centroid': np.array([[0, 1, 1.41421356, 2],
[2, 3, 2.12132034, 3]]),
'median': np.array([[0, 1, 1.41421356, 2],
[2, 3, 2.12132034, 3]]),
'ward': np.array([[0, 1, 1.41421356, 2],
[2, 3, 2.44948974, 3]]),
}
def test_linkage_ties(self):
for method in ['single', 'complete', 'average', 'weighted', 'centroid', 'median', 'ward']:
self.check_linkage_ties(method)
def check_linkage_ties(self, method):
X = np.array([[-1, -1], [0, 0], [1, 1]])
Z = linkage(X, method=method)
expectedZ = self._expectations[method]
assert_allclose(Z, expectedZ, atol=1e-06)
class TestInconsistent(object):
def test_inconsistent_tdist(self):
for depth in hierarchy_test_data.inconsistent_ytdist:
self.check_inconsistent_tdist(depth)
def check_inconsistent_tdist(self, depth):
Z = hierarchy_test_data.linkage_ytdist_single
assert_allclose(inconsistent(Z, depth),
hierarchy_test_data.inconsistent_ytdist[depth])
class TestCopheneticDistance(object):
def test_linkage_cophenet_tdist_Z(self):
# Tests cophenet(Z) on tdist data set.
expectedM = np.array([268, 295, 255, 255, 295, 295, 268, 268, 295, 295,
295, 138, 219, 295, 295])
Z = hierarchy_test_data.linkage_ytdist_single
M = cophenet(Z)
assert_allclose(M, expectedM, atol=1e-10)
def test_linkage_cophenet_tdist_Z_Y(self):
# Tests cophenet(Z, Y) on tdist data set.
Z = hierarchy_test_data.linkage_ytdist_single
(c, M) = cophenet(Z, hierarchy_test_data.ytdist)
expectedM = np.array([268, 295, 255, 255, 295, 295, 268, 268, 295, 295,
295, 138, 219, 295, 295])
expectedc = 0.639931296433393415057366837573
assert_allclose(c, expectedc, atol=1e-10)
assert_allclose(M, expectedM, atol=1e-10)
class TestMLabLinkageConversion(object):
def test_mlab_linkage_conversion_empty(self):
# Tests from/to_mlab_linkage on empty linkage array.
X = np.asarray([])
assert_equal(from_mlab_linkage([]), X)
assert_equal(to_mlab_linkage([]), X)
def test_mlab_linkage_conversion_single_row(self):
# Tests from/to_mlab_linkage on linkage array with single row.
Z = np.asarray([[0., 1., 3., 2.]])
Zm = [[1, 2, 3]]
assert_equal(from_mlab_linkage(Zm), Z)
assert_equal(to_mlab_linkage(Z), Zm)
def test_mlab_linkage_conversion_multiple_rows(self):
# Tests from/to_mlab_linkage on linkage array with multiple rows.
Zm = np.asarray([[3, 6, 138], [4, 5, 219],
[1, 8, 255], [2, 9, 268], [7, 10, 295]])
Z = np.array([[2., 5., 138., 2.],
[3., 4., 219., 2.],
[0., 7., 255., 3.],
[1., 8., 268., 4.],
[6., 9., 295., 6.]],
dtype=np.double)
assert_equal(from_mlab_linkage(Zm), Z)
assert_equal(to_mlab_linkage(Z), Zm)
class TestFcluster(object):
def test_fclusterdata(self):
for t in hierarchy_test_data.fcluster_inconsistent:
self.check_fclusterdata(t, 'inconsistent')
for t in hierarchy_test_data.fcluster_distance:
self.check_fclusterdata(t, 'distance')
for t in hierarchy_test_data.fcluster_maxclust:
self.check_fclusterdata(t, 'maxclust')
def check_fclusterdata(self, t, criterion):
# Tests fclusterdata(X, criterion=criterion, t=t) on a random 3-cluster data set.
expectedT = getattr(hierarchy_test_data, 'fcluster_' + criterion)[t]
X = hierarchy_test_data.Q_X
T = fclusterdata(X, criterion=criterion, t=t)
assert_(is_isomorphic(T, expectedT))
def test_fcluster(self):
for t in hierarchy_test_data.fcluster_inconsistent:
self.check_fcluster(t, 'inconsistent')
for t in hierarchy_test_data.fcluster_distance:
self.check_fcluster(t, 'distance')
for t in hierarchy_test_data.fcluster_maxclust:
self.check_fcluster(t, 'maxclust')
def check_fcluster(self, t, criterion):
# Tests fcluster(Z, criterion=criterion, t=t) on a random 3-cluster data set.
expectedT = getattr(hierarchy_test_data, 'fcluster_' + criterion)[t]
Z = single(hierarchy_test_data.Q_X)
T = fcluster(Z, criterion=criterion, t=t)
assert_(is_isomorphic(T, expectedT))
def test_fcluster_monocrit(self):
for t in hierarchy_test_data.fcluster_distance:
self.check_fcluster_monocrit(t)
for t in hierarchy_test_data.fcluster_maxclust:
self.check_fcluster_maxclust_monocrit(t)
def check_fcluster_monocrit(self, t):
expectedT = hierarchy_test_data.fcluster_distance[t]
Z = single(hierarchy_test_data.Q_X)
T = fcluster(Z, t, criterion='monocrit', monocrit=maxdists(Z))
assert_(is_isomorphic(T, expectedT))
def check_fcluster_maxclust_monocrit(self, t):
expectedT = hierarchy_test_data.fcluster_maxclust[t]
Z = single(hierarchy_test_data.Q_X)
T = fcluster(Z, t, criterion='maxclust_monocrit', monocrit=maxdists(Z))
assert_(is_isomorphic(T, expectedT))
class TestLeaders(object):
def test_leaders_single(self):
# Tests leaders using a flat clustering generated by single linkage.
X = hierarchy_test_data.Q_X
Y = pdist(X)
Z = linkage(Y)
T = fcluster(Z, criterion='maxclust', t=3)
Lright = (np.array([53, 55, 56]), np.array([2, 3, 1]))
L = leaders(Z, T)
assert_equal(L, Lright)
class TestIsIsomorphic(object):
def test_is_isomorphic_1(self):
# Tests is_isomorphic on test case #1 (one flat cluster, different labellings)
a = [1, 1, 1]
b = [2, 2, 2]
assert_(is_isomorphic(a, b))
assert_(is_isomorphic(b, a))
def test_is_isomorphic_2(self):
# Tests is_isomorphic on test case #2 (two flat clusters, different labelings)
a = [1, 7, 1]
b = [2, 3, 2]
assert_(is_isomorphic(a, b))
assert_(is_isomorphic(b, a))
def test_is_isomorphic_3(self):
# Tests is_isomorphic on test case #3 (no flat clusters)
a = []
b = []
assert_(is_isomorphic(a, b))
def test_is_isomorphic_4A(self):
# Tests is_isomorphic on test case #4A (3 flat clusters, different labelings, isomorphic)
a = [1, 2, 3]
b = [1, 3, 2]
assert_(is_isomorphic(a, b))
assert_(is_isomorphic(b, a))
def test_is_isomorphic_4B(self):
# Tests is_isomorphic on test case #4B (3 flat clusters, different labelings, nonisomorphic)
a = [1, 2, 3, 3]
b = [1, 3, 2, 3]
assert_(is_isomorphic(a, b) == False)
assert_(is_isomorphic(b, a) == False)
def test_is_isomorphic_4C(self):
# Tests is_isomorphic on test case #4C (3 flat clusters, different labelings, isomorphic)
a = [7, 2, 3]
b = [6, 3, 2]
assert_(is_isomorphic(a, b))
assert_(is_isomorphic(b, a))
def test_is_isomorphic_5(self):
# Tests is_isomorphic on test case #5 (1000 observations, 2/3/5 random
# clusters, random permutation of the labeling).
for nc in [2, 3, 5]:
self.help_is_isomorphic_randperm(1000, nc)
def test_is_isomorphic_6(self):
# Tests is_isomorphic on test case #5A (1000 observations, 2/3/5 random
# clusters, random permutation of the labeling, slightly
# nonisomorphic.)
for nc in [2, 3, 5]:
self.help_is_isomorphic_randperm(1000, nc, True, 5)
def test_is_isomorphic_7(self):
# Regression test for gh-6271
assert_(not is_isomorphic([1, 2, 3], [1, 1, 1]))
def help_is_isomorphic_randperm(self, nobs, nclusters, noniso=False, nerrors=0):
for k in range(3):
a = np.int_(np.random.rand(nobs) * nclusters)
b = np.zeros(a.size, dtype=np.int_)
P = np.random.permutation(nclusters)
for i in xrange(0, a.shape[0]):
b[i] = P[a[i]]
if noniso:
Q = np.random.permutation(nobs)
b[Q[0:nerrors]] += 1
b[Q[0:nerrors]] %= nclusters
assert_(is_isomorphic(a, b) == (not noniso))
assert_(is_isomorphic(b, a) == (not noniso))
class TestIsValidLinkage(object):
def test_is_valid_linkage_various_size(self):
for nrow, ncol, valid in [(2, 5, False), (2, 3, False),
(1, 4, True), (2, 4, True)]:
self.check_is_valid_linkage_various_size(nrow, ncol, valid)
def check_is_valid_linkage_various_size(self, nrow, ncol, valid):
# Tests is_valid_linkage(Z) with linkage matrics of various sizes
Z = np.asarray([[0, 1, 3.0, 2, 5],
[3, 2, 4.0, 3, 3]], dtype=np.double)
Z = Z[:nrow, :ncol]
assert_(is_valid_linkage(Z) == valid)
if not valid:
assert_raises(ValueError, is_valid_linkage, Z, throw=True)
def test_is_valid_linkage_int_type(self):
# Tests is_valid_linkage(Z) with integer type.
Z = np.asarray([[0, 1, 3.0, 2],
[3, 2, 4.0, 3]], dtype=int)
assert_(is_valid_linkage(Z) == False)
assert_raises(TypeError, is_valid_linkage, Z, throw=True)
def test_is_valid_linkage_empty(self):
# Tests is_valid_linkage(Z) with empty linkage.
Z = np.zeros((0, 4), dtype=np.double)
assert_(is_valid_linkage(Z) == False)
assert_raises(ValueError, is_valid_linkage, Z, throw=True)
def test_is_valid_linkage_4_and_up(self):
# Tests is_valid_linkage(Z) on linkage on observation sets between
# sizes 4 and 15 (step size 3).
for i in xrange(4, 15, 3):
y = np.random.rand(i*(i-1)//2)
Z = linkage(y)
assert_(is_valid_linkage(Z) == True)
def test_is_valid_linkage_4_and_up_neg_index_left(self):
# Tests is_valid_linkage(Z) on linkage on observation sets between
# sizes 4 and 15 (step size 3) with negative indices (left).
for i in xrange(4, 15, 3):
y = np.random.rand(i*(i-1)//2)
Z = linkage(y)
Z[i//2,0] = -2
assert_(is_valid_linkage(Z) == False)
assert_raises(ValueError, is_valid_linkage, Z, throw=True)
def test_is_valid_linkage_4_and_up_neg_index_right(self):
# Tests is_valid_linkage(Z) on linkage on observation sets between
# sizes 4 and 15 (step size 3) with negative indices (right).
for i in xrange(4, 15, 3):
y = np.random.rand(i*(i-1)//2)
Z = linkage(y)
Z[i//2,1] = -2
assert_(is_valid_linkage(Z) == False)
assert_raises(ValueError, is_valid_linkage, Z, throw=True)
def test_is_valid_linkage_4_and_up_neg_dist(self):
# Tests is_valid_linkage(Z) on linkage on observation sets between
# sizes 4 and 15 (step size 3) with negative distances.
for i in xrange(4, 15, 3):
y = np.random.rand(i*(i-1)//2)
Z = linkage(y)
Z[i//2,2] = -0.5
assert_(is_valid_linkage(Z) == False)
assert_raises(ValueError, is_valid_linkage, Z, throw=True)
def test_is_valid_linkage_4_and_up_neg_counts(self):
# Tests is_valid_linkage(Z) on linkage on observation sets between
# sizes 4 and 15 (step size 3) with negative counts.
for i in xrange(4, 15, 3):
y = np.random.rand(i*(i-1)//2)
Z = linkage(y)
Z[i//2,3] = -2
assert_(is_valid_linkage(Z) == False)
assert_raises(ValueError, is_valid_linkage, Z, throw=True)
class TestIsValidInconsistent(object):
def test_is_valid_im_int_type(self):
# Tests is_valid_im(R) with integer type.
R = np.asarray([[0, 1, 3.0, 2],
[3, 2, 4.0, 3]], dtype=int)
assert_(is_valid_im(R) == False)
assert_raises(TypeError, is_valid_im, R, throw=True)
def test_is_valid_im_various_size(self):
for nrow, ncol, valid in [(2, 5, False), (2, 3, False),
(1, 4, True), (2, 4, True)]:
self.check_is_valid_im_various_size(nrow, ncol, valid)
def check_is_valid_im_various_size(self, nrow, ncol, valid):
# Tests is_valid_im(R) with linkage matrics of various sizes
R = np.asarray([[0, 1, 3.0, 2, 5],
[3, 2, 4.0, 3, 3]], dtype=np.double)
R = R[:nrow, :ncol]
assert_(is_valid_im(R) == valid)
if not valid:
assert_raises(ValueError, is_valid_im, R, throw=True)
def test_is_valid_im_empty(self):
# Tests is_valid_im(R) with empty inconsistency matrix.
R = np.zeros((0, 4), dtype=np.double)
assert_(is_valid_im(R) == False)
assert_raises(ValueError, is_valid_im, R, throw=True)
def test_is_valid_im_4_and_up(self):
# Tests is_valid_im(R) on im on observation sets between sizes 4 and 15
# (step size 3).
for i in xrange(4, 15, 3):
y = np.random.rand(i*(i-1)//2)
Z = linkage(y)
R = inconsistent(Z)
assert_(is_valid_im(R) == True)
def test_is_valid_im_4_and_up_neg_index_left(self):
# Tests is_valid_im(R) on im on observation sets between sizes 4 and 15
# (step size 3) with negative link height means.
for i in xrange(4, 15, 3):
y = np.random.rand(i*(i-1)//2)
Z = linkage(y)
R = inconsistent(Z)
R[i//2,0] = -2.0
assert_(is_valid_im(R) == False)
assert_raises(ValueError, is_valid_im, R, throw=True)
def test_is_valid_im_4_and_up_neg_index_right(self):
# Tests is_valid_im(R) on im on observation sets between sizes 4 and 15
# (step size 3) with negative link height standard deviations.
for i in xrange(4, 15, 3):
y = np.random.rand(i*(i-1)//2)
Z = linkage(y)
R = inconsistent(Z)
R[i//2,1] = -2.0
assert_(is_valid_im(R) == False)
assert_raises(ValueError, is_valid_im, R, throw=True)
def test_is_valid_im_4_and_up_neg_dist(self):
# Tests is_valid_im(R) on im on observation sets between sizes 4 and 15
# (step size 3) with negative link counts.
for i in xrange(4, 15, 3):
y = np.random.rand(i*(i-1)//2)
Z = linkage(y)
R = inconsistent(Z)
R[i//2,2] = -0.5
assert_(is_valid_im(R) == False)
assert_raises(ValueError, is_valid_im, R, throw=True)
class TestNumObsLinkage(object):
def test_num_obs_linkage_empty(self):
# Tests num_obs_linkage(Z) with empty linkage.
Z = np.zeros((0, 4), dtype=np.double)
assert_raises(ValueError, num_obs_linkage, Z)
def test_num_obs_linkage_1x4(self):
# Tests num_obs_linkage(Z) on linkage over 2 observations.
Z = np.asarray([[0, 1, 3.0, 2]], dtype=np.double)
assert_equal(num_obs_linkage(Z), 2)
def test_num_obs_linkage_2x4(self):
# Tests num_obs_linkage(Z) on linkage over 3 observations.
Z = np.asarray([[0, 1, 3.0, 2],
[3, 2, 4.0, 3]], dtype=np.double)
assert_equal(num_obs_linkage(Z), 3)
def test_num_obs_linkage_4_and_up(self):
# Tests num_obs_linkage(Z) on linkage on observation sets between sizes
# 4 and 15 (step size 3).
for i in xrange(4, 15, 3):
y = np.random.rand(i*(i-1)//2)
Z = linkage(y)
assert_equal(num_obs_linkage(Z), i)
class TestLeavesList(object):
def test_leaves_list_1x4(self):
# Tests leaves_list(Z) on a 1x4 linkage.
Z = np.asarray([[0, 1, 3.0, 2]], dtype=np.double)
to_tree(Z)
assert_equal(leaves_list(Z), [0, 1])
def test_leaves_list_2x4(self):
# Tests leaves_list(Z) on a 2x4 linkage.
Z = np.asarray([[0, 1, 3.0, 2],
[3, 2, 4.0, 3]], dtype=np.double)
to_tree(Z)
assert_equal(leaves_list(Z), [0, 1, 2])
def test_leaves_list_Q(self):
for method in ['single', 'complete', 'average', 'weighted', 'centroid',
'median', 'ward']:
self.check_leaves_list_Q(method)
def check_leaves_list_Q(self, method):
# Tests leaves_list(Z) on the Q data set
X = hierarchy_test_data.Q_X
Z = linkage(X, method)
node = to_tree(Z)
assert_equal(node.pre_order(), leaves_list(Z))
def test_Q_subtree_pre_order(self):
# Tests that pre_order() works when called on sub-trees.
X = hierarchy_test_data.Q_X
Z = linkage(X, 'single')
node = to_tree(Z)
assert_equal(node.pre_order(), (node.get_left().pre_order()
+ node.get_right().pre_order()))
class TestCorrespond(object):
def test_correspond_empty(self):
# Tests correspond(Z, y) with empty linkage and condensed distance matrix.
y = np.zeros((0,))
Z = np.zeros((0,4))
assert_raises(ValueError, correspond, Z, y)
def test_correspond_2_and_up(self):
# Tests correspond(Z, y) on linkage and CDMs over observation sets of
# different sizes.
for i in xrange(2, 4):
y = np.random.rand(i*(i-1)//2)
Z = linkage(y)
assert_(correspond(Z, y))
for i in xrange(4, 15, 3):
y = np.random.rand(i*(i-1)//2)
Z = linkage(y)
assert_(correspond(Z, y))
def test_correspond_4_and_up(self):
# Tests correspond(Z, y) on linkage and CDMs over observation sets of
# different sizes. Correspondence should be false.
for (i, j) in (list(zip(list(range(2, 4)), list(range(3, 5)))) +
list(zip(list(range(3, 5)), list(range(2, 4))))):
y = np.random.rand(i*(i-1)//2)
y2 = np.random.rand(j*(j-1)//2)
Z = linkage(y)
Z2 = linkage(y2)
assert_equal(correspond(Z, y2), False)
assert_equal(correspond(Z2, y), False)
def test_correspond_4_and_up_2(self):
# Tests correspond(Z, y) on linkage and CDMs over observation sets of
# different sizes. Correspondence should be false.
for (i, j) in (list(zip(list(range(2, 7)), list(range(16, 21)))) +
list(zip(list(range(2, 7)), list(range(16, 21))))):
y = np.random.rand(i*(i-1)//2)
y2 = np.random.rand(j*(j-1)//2)
Z = linkage(y)
Z2 = linkage(y2)
assert_equal(correspond(Z, y2), False)
assert_equal(correspond(Z2, y), False)
def test_num_obs_linkage_multi_matrix(self):
# Tests num_obs_linkage with observation matrices of multiple sizes.
for n in xrange(2, 10):
X = np.random.rand(n, 4)
Y = pdist(X)
Z = linkage(Y)
assert_equal(num_obs_linkage(Z), n)
class TestIsMonotonic(object):
def test_is_monotonic_empty(self):
# Tests is_monotonic(Z) on an empty linkage.
Z = np.zeros((0, 4))
assert_raises(ValueError, is_monotonic, Z)
def test_is_monotonic_1x4(self):
# Tests is_monotonic(Z) on 1x4 linkage. Expecting True.
Z = np.asarray([[0, 1, 0.3, 2]], dtype=np.double)
assert_equal(is_monotonic(Z), True)
def test_is_monotonic_2x4_T(self):
# Tests is_monotonic(Z) on 2x4 linkage. Expecting True.
Z = np.asarray([[0, 1, 0.3, 2],
[2, 3, 0.4, 3]], dtype=np.double)
assert_equal(is_monotonic(Z), True)
def test_is_monotonic_2x4_F(self):
# Tests is_monotonic(Z) on 2x4 linkage. Expecting False.
Z = np.asarray([[0, 1, 0.4, 2],
[2, 3, 0.3, 3]], dtype=np.double)
assert_equal(is_monotonic(Z), False)
def test_is_monotonic_3x4_T(self):
# Tests is_monotonic(Z) on 3x4 linkage. Expecting True.
Z = np.asarray([[0, 1, 0.3, 2],
[2, 3, 0.4, 2],
[4, 5, 0.6, 4]], dtype=np.double)
assert_equal(is_monotonic(Z), True)
def test_is_monotonic_3x4_F1(self):
# Tests is_monotonic(Z) on 3x4 linkage (case 1). Expecting False.
Z = np.asarray([[0, 1, 0.3, 2],
[2, 3, 0.2, 2],
[4, 5, 0.6, 4]], dtype=np.double)
assert_equal(is_monotonic(Z), False)
def test_is_monotonic_3x4_F2(self):
# Tests is_monotonic(Z) on 3x4 linkage (case 2). Expecting False.
Z = np.asarray([[0, 1, 0.8, 2],
[2, 3, 0.4, 2],
[4, 5, 0.6, 4]], dtype=np.double)
assert_equal(is_monotonic(Z), False)
def test_is_monotonic_3x4_F3(self):
# Tests is_monotonic(Z) on 3x4 linkage (case 3). Expecting False
Z = np.asarray([[0, 1, 0.3, 2],
[2, 3, 0.4, 2],
[4, 5, 0.2, 4]], dtype=np.double)
assert_equal(is_monotonic(Z), False)
def test_is_monotonic_tdist_linkage1(self):
# Tests is_monotonic(Z) on clustering generated by single linkage on
# tdist data set. Expecting True.
Z = linkage(hierarchy_test_data.ytdist, 'single')
assert_equal(is_monotonic(Z), True)
def test_is_monotonic_tdist_linkage2(self):
# Tests is_monotonic(Z) on clustering generated by single linkage on
# tdist data set. Perturbing. Expecting False.
Z = linkage(hierarchy_test_data.ytdist, 'single')
Z[2,2] = 0.0
assert_equal(is_monotonic(Z), False)
def test_is_monotonic_Q_linkage(self):
# Tests is_monotonic(Z) on clustering generated by single linkage on
# Q data set. Expecting True.
X = hierarchy_test_data.Q_X
Z = linkage(X, 'single')
assert_equal(is_monotonic(Z), True)
class TestMaxDists(object):
def test_maxdists_empty_linkage(self):
# Tests maxdists(Z) on empty linkage. Expecting exception.
Z = np.zeros((0, 4), dtype=np.double)
assert_raises(ValueError, maxdists, Z)
def test_maxdists_one_cluster_linkage(self):
# Tests maxdists(Z) on linkage with one cluster.
Z = np.asarray([[0, 1, 0.3, 4]], dtype=np.double)
MD = maxdists(Z)
expectedMD = calculate_maximum_distances(Z)
assert_allclose(MD, expectedMD, atol=1e-15)
def test_maxdists_Q_linkage(self):
for method in ['single', 'complete', 'ward', 'centroid', 'median']:
self.check_maxdists_Q_linkage(method)
def check_maxdists_Q_linkage(self, method):
# Tests maxdists(Z) on the Q data set
X = hierarchy_test_data.Q_X
Z = linkage(X, method)
MD = maxdists(Z)
expectedMD = calculate_maximum_distances(Z)
assert_allclose(MD, expectedMD, atol=1e-15)
class TestMaxInconsts(object):
def test_maxinconsts_empty_linkage(self):
# Tests maxinconsts(Z, R) on empty linkage. Expecting exception.
Z = np.zeros((0, 4), dtype=np.double)
R = np.zeros((0, 4), dtype=np.double)
assert_raises(ValueError, maxinconsts, Z, R)
def test_maxinconsts_difrow_linkage(self):
# Tests maxinconsts(Z, R) on linkage and inconsistency matrices with
# different numbers of clusters. Expecting exception.
Z = np.asarray([[0, 1, 0.3, 4]], dtype=np.double)
R = np.random.rand(2, 4)
assert_raises(ValueError, maxinconsts, Z, R)
def test_maxinconsts_one_cluster_linkage(self):
# Tests maxinconsts(Z, R) on linkage with one cluster.
Z = np.asarray([[0, 1, 0.3, 4]], dtype=np.double)
R = np.asarray([[0, 0, 0, 0.3]], dtype=np.double)
MD = maxinconsts(Z, R)
expectedMD = calculate_maximum_inconsistencies(Z, R)
assert_allclose(MD, expectedMD, atol=1e-15)
def test_maxinconsts_Q_linkage(self):
for method in ['single', 'complete', 'ward', 'centroid', 'median']:
self.check_maxinconsts_Q_linkage(method)
def check_maxinconsts_Q_linkage(self, method):
# Tests maxinconsts(Z, R) on the Q data set
X = hierarchy_test_data.Q_X
Z = linkage(X, method)
R = inconsistent(Z)
MD = maxinconsts(Z, R)
expectedMD = calculate_maximum_inconsistencies(Z, R)
assert_allclose(MD, expectedMD, atol=1e-15)
class TestMaxRStat(object):
def test_maxRstat_invalid_index(self):
for i in [3.3, -1, 4]:
self.check_maxRstat_invalid_index(i)
def check_maxRstat_invalid_index(self, i):
# Tests maxRstat(Z, R, i). Expecting exception.
Z = np.asarray([[0, 1, 0.3, 4]], dtype=np.double)
R = np.asarray([[0, 0, 0, 0.3]], dtype=np.double)
if isinstance(i, int):
assert_raises(ValueError, maxRstat, Z, R, i)
else:
assert_raises(TypeError, maxRstat, Z, R, i)
def test_maxRstat_empty_linkage(self):
for i in range(4):
self.check_maxRstat_empty_linkage(i)
def check_maxRstat_empty_linkage(self, i):
# Tests maxRstat(Z, R, i) on empty linkage. Expecting exception.
Z = np.zeros((0, 4), dtype=np.double)
R = np.zeros((0, 4), dtype=np.double)
assert_raises(ValueError, maxRstat, Z, R, i)
def test_maxRstat_difrow_linkage(self):
for i in range(4):
self.check_maxRstat_difrow_linkage(i)
def check_maxRstat_difrow_linkage(self, i):
# Tests maxRstat(Z, R, i) on linkage and inconsistency matrices with
# different numbers of clusters. Expecting exception.
Z = np.asarray([[0, 1, 0.3, 4]], dtype=np.double)
R = np.random.rand(2, 4)
assert_raises(ValueError, maxRstat, Z, R, i)
def test_maxRstat_one_cluster_linkage(self):
for i in range(4):
self.check_maxRstat_one_cluster_linkage(i)
def check_maxRstat_one_cluster_linkage(self, i):
# Tests maxRstat(Z, R, i) on linkage with one cluster.
Z = np.asarray([[0, 1, 0.3, 4]], dtype=np.double)
R = np.asarray([[0, 0, 0, 0.3]], dtype=np.double)
MD = maxRstat(Z, R, 1)
expectedMD = calculate_maximum_inconsistencies(Z, R, 1)
assert_allclose(MD, expectedMD, atol=1e-15)
def test_maxRstat_Q_linkage(self):
for method in ['single', 'complete', 'ward', 'centroid', 'median']:
for i in range(4):
self.check_maxRstat_Q_linkage(method, i)
def check_maxRstat_Q_linkage(self, method, i):
# Tests maxRstat(Z, R, i) on the Q data set
X = hierarchy_test_data.Q_X
Z = linkage(X, method)
R = inconsistent(Z)
MD = maxRstat(Z, R, 1)
expectedMD = calculate_maximum_inconsistencies(Z, R, 1)
assert_allclose(MD, expectedMD, atol=1e-15)
class TestDendrogram(object):
def test_dendrogram_single_linkage_tdist(self):
# Tests dendrogram calculation on single linkage of the tdist data set.
Z = linkage(hierarchy_test_data.ytdist, 'single')
R = dendrogram(Z, no_plot=True)
leaves = R["leaves"]
assert_equal(leaves, [2, 5, 1, 0, 3, 4])
def test_valid_orientation(self):
Z = linkage(hierarchy_test_data.ytdist, 'single')
assert_raises(ValueError, dendrogram, Z, orientation="foo")
@pytest.mark.skipif(not have_matplotlib, reason="no matplotlib")
def test_dendrogram_plot(self):
for orientation in ['top', 'bottom', 'left', 'right']:
self.check_dendrogram_plot(orientation)
def check_dendrogram_plot(self, orientation):
# Tests dendrogram plotting.
Z = linkage(hierarchy_test_data.ytdist, 'single')
expected = {'color_list': ['g', 'b', 'b', 'b', 'b'],
'dcoord': [[0.0, 138.0, 138.0, 0.0],
[0.0, 219.0, 219.0, 0.0],
[0.0, 255.0, 255.0, 219.0],
[0.0, 268.0, 268.0, 255.0],
[138.0, 295.0, 295.0, 268.0]],
'icoord': [[5.0, 5.0, 15.0, 15.0],
[45.0, 45.0, 55.0, 55.0],
[35.0, 35.0, 50.0, 50.0],
[25.0, 25.0, 42.5, 42.5],
[10.0, 10.0, 33.75, 33.75]],
'ivl': ['2', '5', '1', '0', '3', '4'],
'leaves': [2, 5, 1, 0, 3, 4]}
fig = plt.figure()
ax = fig.add_subplot(221)
# test that dendrogram accepts ax keyword
R1 = dendrogram(Z, ax=ax, orientation=orientation)
assert_equal(R1, expected)
# test that dendrogram accepts and handle the leaf_font_size and
# leaf_rotation keywords
R1a = dendrogram(Z, ax=ax, orientation=orientation,
leaf_font_size=20, leaf_rotation=90)
testlabel = (
ax.get_xticklabels()[0]
if orientation in ['top', 'bottom']
else ax.get_yticklabels()[0]
)
assert_equal(testlabel.get_rotation(), 90)
assert_equal(testlabel.get_size(), 20)
R1a = dendrogram(Z, ax=ax, orientation=orientation,
leaf_rotation=90)
testlabel = (
ax.get_xticklabels()[0]
if orientation in ['top', 'bottom']
else ax.get_yticklabels()[0]
)
assert_equal(testlabel.get_rotation(), 90)
R1a = dendrogram(Z, ax=ax, orientation=orientation,
leaf_font_size=20)
testlabel = (
ax.get_xticklabels()[0]
if orientation in ['top', 'bottom']
else ax.get_yticklabels()[0]
)
assert_equal(testlabel.get_size(), 20)
plt.close()
# test plotting to gca (will import pylab)
R2 = dendrogram(Z, orientation=orientation)
plt.close()
assert_equal(R2, expected)
@pytest.mark.skipif(not have_matplotlib, reason="no matplotlib")
def test_dendrogram_truncate_mode(self):
Z = linkage(hierarchy_test_data.ytdist, 'single')
R = dendrogram(Z, 2, 'lastp', show_contracted=True)
plt.close()
assert_equal(R, {'color_list': ['b'],
'dcoord': [[0.0, 295.0, 295.0, 0.0]],
'icoord': [[5.0, 5.0, 15.0, 15.0]],
'ivl': ['(2)', '(4)'],
'leaves': [6, 9]})
R = dendrogram(Z, 2, 'mtica', show_contracted=True)
plt.close()
assert_equal(R, {'color_list': ['g', 'b', 'b', 'b'],
'dcoord': [[0.0, 138.0, 138.0, 0.0],
[0.0, 255.0, 255.0, 0.0],
[0.0, 268.0, 268.0, 255.0],
[138.0, 295.0, 295.0, 268.0]],
'icoord': [[5.0, 5.0, 15.0, 15.0],
[35.0, 35.0, 45.0, 45.0],
[25.0, 25.0, 40.0, 40.0],
[10.0, 10.0, 32.5, 32.5]],
'ivl': ['2', '5', '1', '0', '(2)'],
'leaves': [2, 5, 1, 0, 7]})
def test_dendrogram_colors(self):
# Tests dendrogram plots with alternate colors
Z = linkage(hierarchy_test_data.ytdist, 'single')
set_link_color_palette(['c', 'm', 'y', 'k'])
R = dendrogram(Z, no_plot=True,
above_threshold_color='g', color_threshold=250)
set_link_color_palette(['g', 'r', 'c', 'm', 'y', 'k'])
color_list = R['color_list']
assert_equal(color_list, ['c', 'm', 'g', 'g', 'g'])
# reset color palette (global list)
set_link_color_palette(None)
def calculate_maximum_distances(Z):
# Used for testing correctness of maxdists.
n = Z.shape[0] + 1
B = np.zeros((n-1,))
q = np.zeros((3,))
for i in xrange(0, n - 1):
q[:] = 0.0
left = Z[i, 0]
right = Z[i, 1]
if left >= n:
q[0] = B[int(left) - n]
if right >= n:
q[1] = B[int(right) - n]
q[2] = Z[i, 2]
B[i] = q.max()
return B
def calculate_maximum_inconsistencies(Z, R, k=3):
# Used for testing correctness of maxinconsts.
n = Z.shape[0] + 1
B = np.zeros((n-1,))
q = np.zeros((3,))
for i in xrange(0, n - 1):
q[:] = 0.0
left = Z[i, 0]
right = Z[i, 1]
if left >= n:
q[0] = B[int(left) - n]
if right >= n:
q[1] = B[int(right) - n]
q[2] = R[i, k]
B[i] = q.max()
return B
def within_tol(a, b, tol):
return np.abs(a - b).max() < tol
def test_unsupported_uncondensed_distance_matrix_linkage_warning():
assert_warns(ClusterWarning, linkage, [[0, 1], [1, 0]])
def test_euclidean_linkage_value_error():
for method in scipy.cluster.hierarchy._EUCLIDEAN_METHODS:
assert_raises(ValueError, linkage, [[1, 1], [1, 1]],
method=method, metric='cityblock')
def test_2x2_linkage():
Z1 = linkage([1], method='single', metric='euclidean')
Z2 = linkage([[0, 1], [0, 0]], method='single', metric='euclidean')
assert_allclose(Z1, Z2)
def test_node_compare():
np.random.seed(23)
nobs = 50
X = np.random.randn(nobs, 4)
Z = scipy.cluster.hierarchy.ward(X)
tree = to_tree(Z)
assert_(tree > tree.get_left())
assert_(tree.get_right() > tree.get_left())
assert_(tree.get_right() == tree.get_right())
assert_(tree.get_right() != tree.get_left())
def test_cut_tree():
np.random.seed(23)
nobs = 50
X = np.random.randn(nobs, 4)
Z = scipy.cluster.hierarchy.ward(X)
cutree = cut_tree(Z)
assert_equal(cutree[:, 0], np.arange(nobs))
assert_equal(cutree[:, -1], np.zeros(nobs))
assert_equal(cutree.max(0), np.arange(nobs - 1, -1, -1))
assert_equal(cutree[:, [-5]], cut_tree(Z, n_clusters=5))
assert_equal(cutree[:, [-5, -10]], cut_tree(Z, n_clusters=[5, 10]))
assert_equal(cutree[:, [-10, -5]], cut_tree(Z, n_clusters=[10, 5]))
nodes = _order_cluster_tree(Z)
heights = np.array([node.dist for node in nodes])
assert_equal(cutree[:, np.searchsorted(heights, [5])],
cut_tree(Z, height=5))
assert_equal(cutree[:, np.searchsorted(heights, [5, 10])],
cut_tree(Z, height=[5, 10]))
assert_equal(cutree[:, np.searchsorted(heights, [10, 5])],
cut_tree(Z, height=[10, 5]))
def test_optimal_leaf_ordering():
# test with the distance vector y
Z = optimal_leaf_ordering(linkage(hierarchy_test_data.ytdist),
hierarchy_test_data.ytdist)
expectedZ = hierarchy_test_data.linkage_ytdist_single_olo
assert_allclose(Z, expectedZ, atol=1e-10)
# test with the observation matrix X
Z = optimal_leaf_ordering(linkage(hierarchy_test_data.X, 'ward'),
hierarchy_test_data.X)
expectedZ = hierarchy_test_data.linkage_X_ward_olo
assert_allclose(Z, expectedZ, atol=1e-06)
def test_Heap():
values = np.array([2, -1, 0, -1.5, 3])
heap = Heap(values)
pair = heap.get_min()
assert_equal(pair['key'], 3)
assert_equal(pair['value'], -1.5)
heap.remove_min()
pair = heap.get_min()
assert_equal(pair['key'], 1)
assert_equal(pair['value'], -1)
heap.change_value(1, 2.5)
pair = heap.get_min()
assert_equal(pair['key'], 2)
assert_equal(pair['value'], 0)
heap.remove_min()
heap.remove_min()
heap.change_value(1, 10)
pair = heap.get_min()
assert_equal(pair['key'], 4)
assert_equal(pair['value'], 3)
heap.remove_min()
pair = heap.get_min()
assert_equal(pair['key'], 1)
assert_equal(pair['value'], 10)
| 41,551 | 38.08937 | 100 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/cluster/tests/test_vq.py
|
from __future__ import division, print_function, absolute_import
import warnings
import sys
import numpy as np
from numpy.testing import (assert_array_equal, assert_array_almost_equal,
assert_allclose, assert_equal, assert_)
from scipy._lib._numpy_compat import suppress_warnings
import pytest
from pytest import raises as assert_raises
from scipy.cluster.vq import (kmeans, kmeans2, py_vq, vq, whiten,
ClusterError, _krandinit)
from scipy.cluster import _vq
TESTDATA_2D = np.array([
-2.2, 1.17, -1.63, 1.69, -2.04, 4.38, -3.09, 0.95, -1.7, 4.79, -1.68, 0.68,
-2.26, 3.34, -2.29, 2.55, -1.72, -0.72, -1.99, 2.34, -2.75, 3.43, -2.45,
2.41, -4.26, 3.65, -1.57, 1.87, -1.96, 4.03, -3.01, 3.86, -2.53, 1.28,
-4.0, 3.95, -1.62, 1.25, -3.42, 3.17, -1.17, 0.12, -3.03, -0.27, -2.07,
-0.55, -1.17, 1.34, -2.82, 3.08, -2.44, 0.24, -1.71, 2.48, -5.23, 4.29,
-2.08, 3.69, -1.89, 3.62, -2.09, 0.26, -0.92, 1.07, -2.25, 0.88, -2.25,
2.02, -4.31, 3.86, -2.03, 3.42, -2.76, 0.3, -2.48, -0.29, -3.42, 3.21,
-2.3, 1.73, -2.84, 0.69, -1.81, 2.48, -5.24, 4.52, -2.8, 1.31, -1.67,
-2.34, -1.18, 2.17, -2.17, 2.82, -1.85, 2.25, -2.45, 1.86, -6.79, 3.94,
-2.33, 1.89, -1.55, 2.08, -1.36, 0.93, -2.51, 2.74, -2.39, 3.92, -3.33,
2.99, -2.06, -0.9, -2.83, 3.35, -2.59, 3.05, -2.36, 1.85, -1.69, 1.8,
-1.39, 0.66, -2.06, 0.38, -1.47, 0.44, -4.68, 3.77, -5.58, 3.44, -2.29,
2.24, -1.04, -0.38, -1.85, 4.23, -2.88, 0.73, -2.59, 1.39, -1.34, 1.75,
-1.95, 1.3, -2.45, 3.09, -1.99, 3.41, -5.55, 5.21, -1.73, 2.52, -2.17,
0.85, -2.06, 0.49, -2.54, 2.07, -2.03, 1.3, -3.23, 3.09, -1.55, 1.44,
-0.81, 1.1, -2.99, 2.92, -1.59, 2.18, -2.45, -0.73, -3.12, -1.3, -2.83,
0.2, -2.77, 3.24, -1.98, 1.6, -4.59, 3.39, -4.85, 3.75, -2.25, 1.71, -3.28,
3.38, -1.74, 0.88, -2.41, 1.92, -2.24, 1.19, -2.48, 1.06, -1.68, -0.62,
-1.3, 0.39, -1.78, 2.35, -3.54, 2.44, -1.32, 0.66, -2.38, 2.76, -2.35,
3.95, -1.86, 4.32, -2.01, -1.23, -1.79, 2.76, -2.13, -0.13, -5.25, 3.84,
-2.24, 1.59, -4.85, 2.96, -2.41, 0.01, -0.43, 0.13, -3.92, 2.91, -1.75,
-0.53, -1.69, 1.69, -1.09, 0.15, -2.11, 2.17, -1.53, 1.22, -2.1, -0.86,
-2.56, 2.28, -3.02, 3.33, -1.12, 3.86, -2.18, -1.19, -3.03, 0.79, -0.83,
0.97, -3.19, 1.45, -1.34, 1.28, -2.52, 4.22, -4.53, 3.22, -1.97, 1.75,
-2.36, 3.19, -0.83, 1.53, -1.59, 1.86, -2.17, 2.3, -1.63, 2.71, -2.03,
3.75, -2.57, -0.6, -1.47, 1.33, -1.95, 0.7, -1.65, 1.27, -1.42, 1.09, -3.0,
3.87, -2.51, 3.06, -2.6, 0.74, -1.08, -0.03, -2.44, 1.31, -2.65, 2.99,
-1.84, 1.65, -4.76, 3.75, -2.07, 3.98, -2.4, 2.67, -2.21, 1.49, -1.21,
1.22, -5.29, 2.38, -2.85, 2.28, -5.6, 3.78, -2.7, 0.8, -1.81, 3.5, -3.75,
4.17, -1.29, 2.99, -5.92, 3.43, -1.83, 1.23, -1.24, -1.04, -2.56, 2.37,
-3.26, 0.39, -4.63, 2.51, -4.52, 3.04, -1.7, 0.36, -1.41, 0.04, -2.1, 1.0,
-1.87, 3.78, -4.32, 3.59, -2.24, 1.38, -1.99, -0.22, -1.87, 1.95, -0.84,
2.17, -5.38, 3.56, -1.27, 2.9, -1.79, 3.31, -5.47, 3.85, -1.44, 3.69,
-2.02, 0.37, -1.29, 0.33, -2.34, 2.56, -1.74, -1.27, -1.97, 1.22, -2.51,
-0.16, -1.64, -0.96, -2.99, 1.4, -1.53, 3.31, -2.24, 0.45, -2.46, 1.71,
-2.88, 1.56, -1.63, 1.46, -1.41, 0.68, -1.96, 2.76, -1.61,
2.11]).reshape((200, 2))
# Global data
X = np.array([[3.0, 3], [4, 3], [4, 2],
[9, 2], [5, 1], [6, 2], [9, 4],
[5, 2], [5, 4], [7, 4], [6, 5]])
CODET1 = np.array([[3.0000, 3.0000],
[6.2000, 4.0000],
[5.8000, 1.8000]])
CODET2 = np.array([[11.0/3, 8.0/3],
[6.7500, 4.2500],
[6.2500, 1.7500]])
LABEL1 = np.array([0, 1, 2, 2, 2, 2, 1, 2, 1, 1, 1])
class TestWhiten(object):
def test_whiten(self):
desired = np.array([[5.08738849, 2.97091878],
[3.19909255, 0.69660580],
[4.51041982, 0.02640918],
[4.38567074, 0.95120889],
[2.32191480, 1.63195503]])
for tp in np.array, np.matrix:
obs = tp([[0.98744510, 0.82766775],
[0.62093317, 0.19406729],
[0.87545741, 0.00735733],
[0.85124403, 0.26499712],
[0.45067590, 0.45464607]])
assert_allclose(whiten(obs), desired, rtol=1e-5)
def test_whiten_zero_std(self):
desired = np.array([[0., 1.0, 2.86666544],
[0., 1.0, 1.32460034],
[0., 1.0, 3.74382172]])
for tp in np.array, np.matrix:
obs = tp([[0., 1., 0.74109533],
[0., 1., 0.34243798],
[0., 1., 0.96785929]])
with warnings.catch_warnings(record=True) as w:
warnings.simplefilter('always')
assert_allclose(whiten(obs), desired, rtol=1e-5)
assert_equal(len(w), 1)
assert_(issubclass(w[-1].category, RuntimeWarning))
def test_whiten_not_finite(self):
for tp in np.array, np.matrix:
for bad_value in np.nan, np.inf, -np.inf:
obs = tp([[0.98744510, bad_value],
[0.62093317, 0.19406729],
[0.87545741, 0.00735733],
[0.85124403, 0.26499712],
[0.45067590, 0.45464607]])
assert_raises(ValueError, whiten, obs)
class TestVq(object):
def test_py_vq(self):
initc = np.concatenate(([[X[0]], [X[1]], [X[2]]]))
for tp in np.array, np.matrix:
label1 = py_vq(tp(X), tp(initc))[0]
assert_array_equal(label1, LABEL1)
def test_vq(self):
initc = np.concatenate(([[X[0]], [X[1]], [X[2]]]))
for tp in np.array, np.matrix:
label1, dist = _vq.vq(tp(X), tp(initc))
assert_array_equal(label1, LABEL1)
tlabel1, tdist = vq(tp(X), tp(initc))
def test_vq_1d(self):
# Test special rank 1 vq algo, python implementation.
data = X[:, 0]
initc = data[:3]
a, b = _vq.vq(data, initc)
ta, tb = py_vq(data[:, np.newaxis], initc[:, np.newaxis])
assert_array_equal(a, ta)
assert_array_equal(b, tb)
def test__vq_sametype(self):
a = np.array([1.0, 2.0], dtype=np.float64)
b = a.astype(np.float32)
assert_raises(TypeError, _vq.vq, a, b)
def test__vq_invalid_type(self):
a = np.array([1, 2], dtype=int)
assert_raises(TypeError, _vq.vq, a, a)
def test_vq_large_nfeat(self):
X = np.random.rand(20, 20)
code_book = np.random.rand(3, 20)
codes0, dis0 = _vq.vq(X, code_book)
codes1, dis1 = py_vq(X, code_book)
assert_allclose(dis0, dis1, 1e-5)
assert_array_equal(codes0, codes1)
X = X.astype(np.float32)
code_book = code_book.astype(np.float32)
codes0, dis0 = _vq.vq(X, code_book)
codes1, dis1 = py_vq(X, code_book)
assert_allclose(dis0, dis1, 1e-5)
assert_array_equal(codes0, codes1)
def test_vq_large_features(self):
X = np.random.rand(10, 5) * 1000000
code_book = np.random.rand(2, 5) * 1000000
codes0, dis0 = _vq.vq(X, code_book)
codes1, dis1 = py_vq(X, code_book)
assert_allclose(dis0, dis1, 1e-5)
assert_array_equal(codes0, codes1)
class TestKMean(object):
def test_large_features(self):
# Generate a data set with large values, and run kmeans on it to
# (regression for 1077).
d = 300
n = 100
m1 = np.random.randn(d)
m2 = np.random.randn(d)
x = 10000 * np.random.randn(n, d) - 20000 * m1
y = 10000 * np.random.randn(n, d) + 20000 * m2
data = np.empty((x.shape[0] + y.shape[0], d), np.double)
data[:x.shape[0]] = x
data[x.shape[0]:] = y
kmeans(data, 2)
def test_kmeans_simple(self):
np.random.seed(54321)
initc = np.concatenate(([[X[0]], [X[1]], [X[2]]]))
for tp in np.array, np.matrix:
code1 = kmeans(tp(X), tp(initc), iter=1)[0]
assert_array_almost_equal(code1, CODET2)
def test_kmeans_lost_cluster(self):
# This will cause kmeans to have a cluster with no points.
data = TESTDATA_2D
initk = np.array([[-1.8127404, -0.67128041],
[2.04621601, 0.07401111],
[-2.31149087,-0.05160469]])
kmeans(data, initk)
with suppress_warnings() as sup:
sup.filter(UserWarning,
"One of the clusters is empty. Re-run kmeans with a "
"different initialization")
kmeans2(data, initk, missing='warn')
assert_raises(ClusterError, kmeans2, data, initk, missing='raise')
def test_kmeans2_simple(self):
np.random.seed(12345678)
initc = np.concatenate(([[X[0]], [X[1]], [X[2]]]))
for tp in np.array, np.matrix:
code1 = kmeans2(tp(X), tp(initc), iter=1)[0]
code2 = kmeans2(tp(X), tp(initc), iter=2)[0]
assert_array_almost_equal(code1, CODET1)
assert_array_almost_equal(code2, CODET2)
def test_kmeans2_rank1(self):
data = TESTDATA_2D
data1 = data[:, 0]
initc = data1[:3]
code = initc.copy()
kmeans2(data1, code, iter=1)[0]
kmeans2(data1, code, iter=2)[0]
def test_kmeans2_rank1_2(self):
data = TESTDATA_2D
data1 = data[:, 0]
kmeans2(data1, 2, iter=1)
def test_kmeans2_high_dim(self):
# test kmeans2 when the number of dimensions exceeds the number
# of input points
data = TESTDATA_2D
data = data.reshape((20, 20))[:10]
kmeans2(data, 2)
def test_kmeans2_init(self):
np.random.seed(12345)
data = TESTDATA_2D
kmeans2(data, 3, minit='points')
kmeans2(data[:, :1], 3, minit='points') # special case (1-D)
kmeans2(data, 3, minit='random')
kmeans2(data[:, :1], 3, minit='random') # special case (1-D)
@pytest.mark.skipif(sys.platform == 'win32', reason='Fails with MemoryError in Wine.')
def test_krandinit(self):
data = TESTDATA_2D
datas = [data.reshape((200, 2)), data.reshape((20, 20))[:10]]
k = int(1e6)
for data in datas:
np.random.seed(1234)
init = _krandinit(data, k)
orig_cov = np.cov(data, rowvar=0)
init_cov = np.cov(init, rowvar=0)
assert_allclose(orig_cov, init_cov, atol=1e-2)
def test_kmeans2_empty(self):
# Regression test for gh-1032.
assert_raises(ValueError, kmeans2, [], 2)
def test_kmeans_0k(self):
# Regression test for gh-1073: fail when k arg is 0.
assert_raises(ValueError, kmeans, X, 0)
assert_raises(ValueError, kmeans2, X, 0)
assert_raises(ValueError, kmeans2, X, np.array([]))
def test_kmeans_large_thres(self):
# Regression test for gh-1774
x = np.array([1,2,3,4,10], dtype=float)
res = kmeans(x, 1, thresh=1e16)
assert_allclose(res[0], np.array([4.]))
assert_allclose(res[1], 2.3999999999999999)
| 11,265 | 38.529825 | 90 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/cluster/tests/__init__.py
| 0 | 0 | 0 |
py
|
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/cluster/tests/hierarchy_test_data.py
|
from numpy import array
Q_X = array([[5.26563660e-01, 3.14160190e-01, 8.00656370e-02],
[7.50205180e-01, 4.60299830e-01, 8.98696460e-01],
[6.65461230e-01, 6.94011420e-01, 9.10465700e-01],
[9.64047590e-01, 1.43082200e-03, 7.39874220e-01],
[1.08159060e-01, 5.53028790e-01, 6.63804780e-02],
[9.31359130e-01, 8.25424910e-01, 9.52315440e-01],
[6.78086960e-01, 3.41903970e-01, 5.61481950e-01],
[9.82730940e-01, 7.04605210e-01, 8.70978630e-02],
[6.14691610e-01, 4.69989230e-02, 6.02406450e-01],
[5.80161260e-01, 9.17354970e-01, 5.88163850e-01],
[1.38246310e+00, 1.96358160e+00, 1.94437880e+00],
[2.10675860e+00, 1.67148730e+00, 1.34854480e+00],
[1.39880070e+00, 1.66142050e+00, 1.32224550e+00],
[1.71410460e+00, 1.49176380e+00, 1.45432170e+00],
[1.54102340e+00, 1.84374950e+00, 1.64658950e+00],
[2.08512480e+00, 1.84524350e+00, 2.17340850e+00],
[1.30748740e+00, 1.53801650e+00, 2.16007740e+00],
[1.41447700e+00, 1.99329070e+00, 1.99107420e+00],
[1.61943490e+00, 1.47703280e+00, 1.89788160e+00],
[1.59880600e+00, 1.54988980e+00, 1.57563350e+00],
[3.37247380e+00, 2.69635310e+00, 3.39981700e+00],
[3.13705120e+00, 3.36528090e+00, 3.06089070e+00],
[3.29413250e+00, 3.19619500e+00, 2.90700170e+00],
[2.65510510e+00, 3.06785900e+00, 2.97198540e+00],
[3.30941040e+00, 2.59283970e+00, 2.57714110e+00],
[2.59557220e+00, 3.33477370e+00, 3.08793190e+00],
[2.58206180e+00, 3.41615670e+00, 3.26441990e+00],
[2.71127000e+00, 2.77032450e+00, 2.63466500e+00],
[2.79617850e+00, 3.25473720e+00, 3.41801560e+00],
[2.64741750e+00, 2.54538040e+00, 3.25354110e+00]])
ytdist = array([662., 877., 255., 412., 996., 295., 468., 268., 400., 754.,
564., 138., 219., 869., 669.])
linkage_ytdist_single = array([[2., 5., 138., 2.],
[3., 4., 219., 2.],
[0., 7., 255., 3.],
[1., 8., 268., 4.],
[6., 9., 295., 6.]])
linkage_ytdist_complete = array([[2., 5., 138., 2.],
[3., 4., 219., 2.],
[1., 6., 400., 3.],
[0., 7., 412., 3.],
[8., 9., 996., 6.]])
linkage_ytdist_average = array([[2., 5., 138., 2.],
[3., 4., 219., 2.],
[0., 7., 333.5, 3.],
[1., 6., 347.5, 3.],
[8., 9., 680.77777778, 6.]])
linkage_ytdist_weighted = array([[2., 5., 138., 2.],
[3., 4., 219., 2.],
[0., 7., 333.5, 3.],
[1., 6., 347.5, 3.],
[8., 9., 670.125, 6.]])
# the optimal leaf ordering of linkage_ytdist_single
linkage_ytdist_single_olo = array([[5., 2., 138., 2.],
[4., 3., 219., 2.],
[7., 0., 255., 3.],
[1., 8., 268., 4.],
[6., 9., 295., 6.]])
X = array([[1.43054825, -7.5693489],
[6.95887839, 6.82293382],
[2.87137846, -9.68248579],
[7.87974764, -6.05485803],
[8.24018364, -6.09495602],
[7.39020262, 8.54004355]])
linkage_X_centroid = array([[3., 4., 0.36265956, 2.],
[1., 5., 1.77045373, 2.],
[0., 2., 2.55760419, 2.],
[6., 8., 6.43614494, 4.],
[7., 9., 15.17363237, 6.]])
linkage_X_median = array([[3., 4., 0.36265956, 2.],
[1., 5., 1.77045373, 2.],
[0., 2., 2.55760419, 2.],
[6., 8., 6.43614494, 4.],
[7., 9., 15.17363237, 6.]])
linkage_X_ward = array([[3., 4., 0.36265956, 2.],
[1., 5., 1.77045373, 2.],
[0., 2., 2.55760419, 2.],
[6., 8., 9.10208346, 4.],
[7., 9., 24.7784379, 6.]])
# the optimal leaf ordering of linkage_X_ward
linkage_X_ward_olo = array([[4., 3., 0.36265956, 2.],
[5., 1., 1.77045373, 2.],
[2., 0., 2.55760419, 2.],
[6., 8., 9.10208346, 4.],
[7., 9., 24.7784379, 6.]])
inconsistent_ytdist = {
1: array([[138., 0., 1., 0.],
[219., 0., 1., 0.],
[255., 0., 1., 0.],
[268., 0., 1., 0.],
[295., 0., 1., 0.]]),
2: array([[138., 0., 1., 0.],
[219., 0., 1., 0.],
[237., 25.45584412, 2., 0.70710678],
[261.5, 9.19238816, 2., 0.70710678],
[233.66666667, 83.9424406, 3., 0.7306594]]),
3: array([[138., 0., 1., 0.],
[219., 0., 1., 0.],
[237., 25.45584412, 2., 0.70710678],
[247.33333333, 25.38372182, 3., 0.81417007],
[239., 69.36377537, 4., 0.80733783]]),
4: array([[138., 0., 1., 0.],
[219., 0., 1., 0.],
[237., 25.45584412, 2., 0.70710678],
[247.33333333, 25.38372182, 3., 0.81417007],
[235., 60.73302232, 5., 0.98793042]])}
fcluster_inconsistent = {
0.8: array([6, 2, 2, 4, 6, 2, 3, 7, 3, 5, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1]),
1.0: array([6, 2, 2, 4, 6, 2, 3, 7, 3, 5, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1]),
2.0: array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1])}
fcluster_distance = {
0.6: array([4, 4, 4, 4, 4, 4, 4, 5, 4, 4, 6, 6, 6, 6, 6, 7, 6, 6, 6, 6, 3,
1, 1, 1, 2, 1, 1, 1, 1, 1]),
1.0: array([2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1]),
2.0: array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1])}
fcluster_maxclust = {
8.0: array([5, 5, 5, 5, 5, 5, 5, 6, 5, 5, 7, 7, 7, 7, 7, 8, 7, 7, 7, 7, 4,
1, 1, 1, 3, 1, 1, 1, 1, 2]),
4.0: array([3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2,
1, 1, 1, 1, 1, 1, 1, 1, 1]),
1.0: array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1])}
| 6,850 | 45.924658 | 78 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/odr/setup.py
|
from __future__ import division, print_function, absolute_import
from os.path import join
def configuration(parent_package='', top_path=None):
import warnings
from numpy.distutils.misc_util import Configuration
from numpy.distutils.system_info import get_info, BlasNotFoundError
config = Configuration('odr', parent_package, top_path)
libodr_files = ['d_odr.f',
'd_mprec.f',
'dlunoc.f']
blas_info = get_info('blas_opt')
if blas_info:
libodr_files.append('d_lpk.f')
else:
warnings.warn(BlasNotFoundError.__doc__)
libodr_files.append('d_lpkbls.f')
odrpack_src = [join('odrpack', x) for x in libodr_files]
config.add_library('odrpack', sources=odrpack_src)
sources = ['__odrpack.c']
libraries = ['odrpack'] + blas_info.pop('libraries', [])
include_dirs = ['.'] + blas_info.pop('include_dirs', [])
config.add_extension('__odrpack',
sources=sources,
libraries=libraries,
include_dirs=include_dirs,
depends=(['odrpack.h'] + odrpack_src),
**blas_info
)
config.add_data_dir('tests')
return config
if __name__ == '__main__':
from numpy.distutils.core import setup
setup(**configuration(top_path='').todict())
| 1,290 | 28.340909 | 71 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/odr/odrpack.py
|
"""
Python wrappers for Orthogonal Distance Regression (ODRPACK).
Notes
=====
* Array formats -- FORTRAN stores its arrays in memory column first, i.e. an
array element A(i, j, k) will be next to A(i+1, j, k). In C and, consequently,
NumPy, arrays are stored row first: A[i, j, k] is next to A[i, j, k+1]. For
efficiency and convenience, the input and output arrays of the fitting
function (and its Jacobians) are passed to FORTRAN without transposition.
Therefore, where the ODRPACK documentation says that the X array is of shape
(N, M), it will be passed to the Python function as an array of shape (M, N).
If M==1, the one-dimensional case, then nothing matters; if M>1, then your
Python functions will be dealing with arrays that are indexed in reverse of
the ODRPACK documentation. No real biggie, but watch out for your indexing of
the Jacobians: the i,j'th elements (@f_i/@x_j) evaluated at the n'th
observation will be returned as jacd[j, i, n]. Except for the Jacobians, it
really is easier to deal with x[0] and x[1] than x[:,0] and x[:,1]. Of course,
you can always use the transpose() function from scipy explicitly.
* Examples -- See the accompanying file test/test.py for examples of how to set
up fits of your own. Some are taken from the User's Guide; some are from
other sources.
* Models -- Some common models are instantiated in the accompanying module
models.py . Contributions are welcome.
Credits
=======
* Thanks to Arnold Moene and Gerard Vermeulen for fixing some killer bugs.
Robert Kern
robert.kern@gmail.com
"""
from __future__ import division, print_function, absolute_import
import numpy
from warnings import warn
from scipy.odr import __odrpack
__all__ = ['odr', 'OdrWarning', 'OdrError', 'OdrStop',
'Data', 'RealData', 'Model', 'Output', 'ODR',
'odr_error', 'odr_stop']
odr = __odrpack.odr
class OdrWarning(UserWarning):
"""
Warning indicating that the data passed into
ODR will cause problems when passed into 'odr'
that the user should be aware of.
"""
pass
class OdrError(Exception):
"""
Exception indicating an error in fitting.
This is raised by `scipy.odr` if an error occurs during fitting.
"""
pass
class OdrStop(Exception):
"""
Exception stopping fitting.
You can raise this exception in your objective function to tell
`scipy.odr` to stop fitting.
"""
pass
# Backwards compatibility
odr_error = OdrError
odr_stop = OdrStop
__odrpack._set_exceptions(OdrError, OdrStop)
def _conv(obj, dtype=None):
""" Convert an object to the preferred form for input to the odr routine.
"""
if obj is None:
return obj
else:
if dtype is None:
obj = numpy.asarray(obj)
else:
obj = numpy.asarray(obj, dtype)
if obj.shape == ():
# Scalar.
return obj.dtype.type(obj)
else:
return obj
def _report_error(info):
""" Interprets the return code of the odr routine.
Parameters
----------
info : int
The return code of the odr routine.
Returns
-------
problems : list(str)
A list of messages about why the odr() routine stopped.
"""
stopreason = ('Blank',
'Sum of squares convergence',
'Parameter convergence',
'Both sum of squares and parameter convergence',
'Iteration limit reached')[info % 5]
if info >= 5:
# questionable results or fatal error
I = (info//10000 % 10,
info//1000 % 10,
info//100 % 10,
info//10 % 10,
info % 10)
problems = []
if I[0] == 0:
if I[1] != 0:
problems.append('Derivatives possibly not correct')
if I[2] != 0:
problems.append('Error occurred in callback')
if I[3] != 0:
problems.append('Problem is not full rank at solution')
problems.append(stopreason)
elif I[0] == 1:
if I[1] != 0:
problems.append('N < 1')
if I[2] != 0:
problems.append('M < 1')
if I[3] != 0:
problems.append('NP < 1 or NP > N')
if I[4] != 0:
problems.append('NQ < 1')
elif I[0] == 2:
if I[1] != 0:
problems.append('LDY and/or LDX incorrect')
if I[2] != 0:
problems.append('LDWE, LD2WE, LDWD, and/or LD2WD incorrect')
if I[3] != 0:
problems.append('LDIFX, LDSTPD, and/or LDSCLD incorrect')
if I[4] != 0:
problems.append('LWORK and/or LIWORK too small')
elif I[0] == 3:
if I[1] != 0:
problems.append('STPB and/or STPD incorrect')
if I[2] != 0:
problems.append('SCLB and/or SCLD incorrect')
if I[3] != 0:
problems.append('WE incorrect')
if I[4] != 0:
problems.append('WD incorrect')
elif I[0] == 4:
problems.append('Error in derivatives')
elif I[0] == 5:
problems.append('Error occurred in callback')
elif I[0] == 6:
problems.append('Numerical error detected')
return problems
else:
return [stopreason]
class Data(object):
"""
The data to fit.
Parameters
----------
x : array_like
Observed data for the independent variable of the regression
y : array_like, optional
If array-like, observed data for the dependent variable of the
regression. A scalar input implies that the model to be used on
the data is implicit.
we : array_like, optional
If `we` is a scalar, then that value is used for all data points (and
all dimensions of the response variable).
If `we` is a rank-1 array of length q (the dimensionality of the
response variable), then this vector is the diagonal of the covariant
weighting matrix for all data points.
If `we` is a rank-1 array of length n (the number of data points), then
the i'th element is the weight for the i'th response variable
observation (single-dimensional only).
If `we` is a rank-2 array of shape (q, q), then this is the full
covariant weighting matrix broadcast to each observation.
If `we` is a rank-2 array of shape (q, n), then `we[:,i]` is the
diagonal of the covariant weighting matrix for the i'th observation.
If `we` is a rank-3 array of shape (q, q, n), then `we[:,:,i]` is the
full specification of the covariant weighting matrix for each
observation.
If the fit is implicit, then only a positive scalar value is used.
wd : array_like, optional
If `wd` is a scalar, then that value is used for all data points
(and all dimensions of the input variable). If `wd` = 0, then the
covariant weighting matrix for each observation is set to the identity
matrix (so each dimension of each observation has the same weight).
If `wd` is a rank-1 array of length m (the dimensionality of the input
variable), then this vector is the diagonal of the covariant weighting
matrix for all data points.
If `wd` is a rank-1 array of length n (the number of data points), then
the i'th element is the weight for the i'th input variable observation
(single-dimensional only).
If `wd` is a rank-2 array of shape (m, m), then this is the full
covariant weighting matrix broadcast to each observation.
If `wd` is a rank-2 array of shape (m, n), then `wd[:,i]` is the
diagonal of the covariant weighting matrix for the i'th observation.
If `wd` is a rank-3 array of shape (m, m, n), then `wd[:,:,i]` is the
full specification of the covariant weighting matrix for each
observation.
fix : array_like of ints, optional
The `fix` argument is the same as ifixx in the class ODR. It is an
array of integers with the same shape as data.x that determines which
input observations are treated as fixed. One can use a sequence of
length m (the dimensionality of the input observations) to fix some
dimensions for all observations. A value of 0 fixes the observation,
a value > 0 makes it free.
meta : dict, optional
Free-form dictionary for metadata.
Notes
-----
Each argument is attached to the member of the instance of the same name.
The structures of `x` and `y` are described in the Model class docstring.
If `y` is an integer, then the Data instance can only be used to fit with
implicit models where the dimensionality of the response is equal to the
specified value of `y`.
The `we` argument weights the effect a deviation in the response variable
has on the fit. The `wd` argument weights the effect a deviation in the
input variable has on the fit. To handle multidimensional inputs and
responses easily, the structure of these arguments has the n'th
dimensional axis first. These arguments heavily use the structured
arguments feature of ODRPACK to conveniently and flexibly support all
options. See the ODRPACK User's Guide for a full explanation of how these
weights are used in the algorithm. Basically, a higher value of the weight
for a particular data point makes a deviation at that point more
detrimental to the fit.
"""
def __init__(self, x, y=None, we=None, wd=None, fix=None, meta={}):
self.x = _conv(x)
if not isinstance(self.x, numpy.ndarray):
raise ValueError(("Expected an 'ndarray' of data for 'x', "
"but instead got data of type '{name}'").format(
name=type(self.x).__name__))
self.y = _conv(y)
self.we = _conv(we)
self.wd = _conv(wd)
self.fix = _conv(fix)
self.meta = meta
def set_meta(self, **kwds):
""" Update the metadata dictionary with the keywords and data provided
by keywords.
Examples
--------
::
data.set_meta(lab="Ph 7; Lab 26", title="Ag110 + Ag108 Decay")
"""
self.meta.update(kwds)
def __getattr__(self, attr):
""" Dispatch attribute access to the metadata dictionary.
"""
if attr in self.meta:
return self.meta[attr]
else:
raise AttributeError("'%s' not in metadata" % attr)
class RealData(Data):
"""
The data, with weightings as actual standard deviations and/or
covariances.
Parameters
----------
x : array_like
Observed data for the independent variable of the regression
y : array_like, optional
If array-like, observed data for the dependent variable of the
regression. A scalar input implies that the model to be used on
the data is implicit.
sx : array_like, optional
Standard deviations of `x`.
`sx` are standard deviations of `x` and are converted to weights by
dividing 1.0 by their squares.
sy : array_like, optional
Standard deviations of `y`.
`sy` are standard deviations of `y` and are converted to weights by
dividing 1.0 by their squares.
covx : array_like, optional
Covariance of `x`
`covx` is an array of covariance matrices of `x` and are converted to
weights by performing a matrix inversion on each observation's
covariance matrix.
covy : array_like, optional
Covariance of `y`
`covy` is an array of covariance matrices and are converted to
weights by performing a matrix inversion on each observation's
covariance matrix.
fix : array_like, optional
The argument and member fix is the same as Data.fix and ODR.ifixx:
It is an array of integers with the same shape as `x` that
determines which input observations are treated as fixed. One can
use a sequence of length m (the dimensionality of the input
observations) to fix some dimensions for all observations. A value
of 0 fixes the observation, a value > 0 makes it free.
meta : dict, optional
Free-form dictionary for metadata.
Notes
-----
The weights `wd` and `we` are computed from provided values as follows:
`sx` and `sy` are converted to weights by dividing 1.0 by their squares.
For example, ``wd = 1./numpy.power(`sx`, 2)``.
`covx` and `covy` are arrays of covariance matrices and are converted to
weights by performing a matrix inversion on each observation's covariance
matrix. For example, ``we[i] = numpy.linalg.inv(covy[i])``.
These arguments follow the same structured argument conventions as wd and
we only restricted by their natures: `sx` and `sy` can't be rank-3, but
`covx` and `covy` can be.
Only set *either* `sx` or `covx` (not both). Setting both will raise an
exception. Same with `sy` and `covy`.
"""
def __init__(self, x, y=None, sx=None, sy=None, covx=None, covy=None,
fix=None, meta={}):
if (sx is not None) and (covx is not None):
raise ValueError("cannot set both sx and covx")
if (sy is not None) and (covy is not None):
raise ValueError("cannot set both sy and covy")
# Set flags for __getattr__
self._ga_flags = {}
if sx is not None:
self._ga_flags['wd'] = 'sx'
else:
self._ga_flags['wd'] = 'covx'
if sy is not None:
self._ga_flags['we'] = 'sy'
else:
self._ga_flags['we'] = 'covy'
self.x = _conv(x)
if not isinstance(self.x, numpy.ndarray):
raise ValueError(("Expected an 'ndarray' of data for 'x', "
"but instead got data of type '{name}'").format(
name=type(self.x).__name__))
self.y = _conv(y)
self.sx = _conv(sx)
self.sy = _conv(sy)
self.covx = _conv(covx)
self.covy = _conv(covy)
self.fix = _conv(fix)
self.meta = meta
def _sd2wt(self, sd):
""" Convert standard deviation to weights.
"""
return 1./numpy.power(sd, 2)
def _cov2wt(self, cov):
""" Convert covariance matrix(-ices) to weights.
"""
from numpy.dual import inv
if len(cov.shape) == 2:
return inv(cov)
else:
weights = numpy.zeros(cov.shape, float)
for i in range(cov.shape[-1]): # n
weights[:,:,i] = inv(cov[:,:,i])
return weights
def __getattr__(self, attr):
lookup_tbl = {('wd', 'sx'): (self._sd2wt, self.sx),
('wd', 'covx'): (self._cov2wt, self.covx),
('we', 'sy'): (self._sd2wt, self.sy),
('we', 'covy'): (self._cov2wt, self.covy)}
if attr not in ('wd', 'we'):
if attr in self.meta:
return self.meta[attr]
else:
raise AttributeError("'%s' not in metadata" % attr)
else:
func, arg = lookup_tbl[(attr, self._ga_flags[attr])]
if arg is not None:
return func(*(arg,))
else:
return None
class Model(object):
"""
The Model class stores information about the function you wish to fit.
It stores the function itself, at the least, and optionally stores
functions which compute the Jacobians used during fitting. Also, one
can provide a function that will provide reasonable starting values
for the fit parameters possibly given the set of data.
Parameters
----------
fcn : function
fcn(beta, x) --> y
fjacb : function
Jacobian of fcn wrt the fit parameters beta.
fjacb(beta, x) --> @f_i(x,B)/@B_j
fjacd : function
Jacobian of fcn wrt the (possibly multidimensional) input
variable.
fjacd(beta, x) --> @f_i(x,B)/@x_j
extra_args : tuple, optional
If specified, `extra_args` should be a tuple of extra
arguments to pass to `fcn`, `fjacb`, and `fjacd`. Each will be called
by `apply(fcn, (beta, x) + extra_args)`
estimate : array_like of rank-1
Provides estimates of the fit parameters from the data
estimate(data) --> estbeta
implicit : boolean
If TRUE, specifies that the model
is implicit; i.e `fcn(beta, x)` ~= 0 and there is no y data to fit
against
meta : dict, optional
freeform dictionary of metadata for the model
Notes
-----
Note that the `fcn`, `fjacb`, and `fjacd` operate on NumPy arrays and
return a NumPy array. The `estimate` object takes an instance of the
Data class.
Here are the rules for the shapes of the argument and return
arrays of the callback functions:
`x`
if the input data is single-dimensional, then `x` is rank-1
array; i.e. ``x = array([1, 2, 3, ...]); x.shape = (n,)``
If the input data is multi-dimensional, then `x` is a rank-2 array;
i.e., ``x = array([[1, 2, ...], [2, 4, ...]]); x.shape = (m, n)``.
In all cases, it has the same shape as the input data array passed to
`odr`. `m` is the dimensionality of the input data, `n` is the number
of observations.
`y`
if the response variable is single-dimensional, then `y` is a
rank-1 array, i.e., ``y = array([2, 4, ...]); y.shape = (n,)``.
If the response variable is multi-dimensional, then `y` is a rank-2
array, i.e., ``y = array([[2, 4, ...], [3, 6, ...]]); y.shape =
(q, n)`` where `q` is the dimensionality of the response variable.
`beta`
rank-1 array of length `p` where `p` is the number of parameters;
i.e. ``beta = array([B_1, B_2, ..., B_p])``
`fjacb`
if the response variable is multi-dimensional, then the
return array's shape is `(q, p, n)` such that ``fjacb(x,beta)[l,k,i] =
d f_l(X,B)/d B_k`` evaluated at the i'th data point. If `q == 1`, then
the return array is only rank-2 and with shape `(p, n)`.
`fjacd`
as with fjacb, only the return array's shape is `(q, m, n)`
such that ``fjacd(x,beta)[l,j,i] = d f_l(X,B)/d X_j`` at the i'th data
point. If `q == 1`, then the return array's shape is `(m, n)`. If
`m == 1`, the shape is (q, n). If `m == q == 1`, the shape is `(n,)`.
"""
def __init__(self, fcn, fjacb=None, fjacd=None,
extra_args=None, estimate=None, implicit=0, meta=None):
self.fcn = fcn
self.fjacb = fjacb
self.fjacd = fjacd
if extra_args is not None:
extra_args = tuple(extra_args)
self.extra_args = extra_args
self.estimate = estimate
self.implicit = implicit
self.meta = meta
def set_meta(self, **kwds):
""" Update the metadata dictionary with the keywords and data provided
here.
Examples
--------
set_meta(name="Exponential", equation="y = a exp(b x) + c")
"""
self.meta.update(kwds)
def __getattr__(self, attr):
""" Dispatch attribute access to the metadata.
"""
if attr in self.meta:
return self.meta[attr]
else:
raise AttributeError("'%s' not in metadata" % attr)
class Output(object):
"""
The Output class stores the output of an ODR run.
Attributes
----------
beta : ndarray
Estimated parameter values, of shape (q,).
sd_beta : ndarray
Standard errors of the estimated parameters, of shape (p,).
cov_beta : ndarray
Covariance matrix of the estimated parameters, of shape (p,p).
delta : ndarray, optional
Array of estimated errors in input variables, of same shape as `x`.
eps : ndarray, optional
Array of estimated errors in response variables, of same shape as `y`.
xplus : ndarray, optional
Array of ``x + delta``.
y : ndarray, optional
Array ``y = fcn(x + delta)``.
res_var : float, optional
Residual variance.
sum_square : float, optional
Sum of squares error.
sum_square_delta : float, optional
Sum of squares of delta error.
sum_square_eps : float, optional
Sum of squares of eps error.
inv_condnum : float, optional
Inverse condition number (cf. ODRPACK UG p. 77).
rel_error : float, optional
Relative error in function values computed within fcn.
work : ndarray, optional
Final work array.
work_ind : dict, optional
Indices into work for drawing out values (cf. ODRPACK UG p. 83).
info : int, optional
Reason for returning, as output by ODRPACK (cf. ODRPACK UG p. 38).
stopreason : list of str, optional
`info` interpreted into English.
Notes
-----
Takes one argument for initialization, the return value from the
function `odr`. The attributes listed as "optional" above are only
present if `odr` was run with ``full_output=1``.
"""
def __init__(self, output):
self.beta = output[0]
self.sd_beta = output[1]
self.cov_beta = output[2]
if len(output) == 4:
# full output
self.__dict__.update(output[3])
self.stopreason = _report_error(self.info)
def pprint(self):
""" Pretty-print important results.
"""
print('Beta:', self.beta)
print('Beta Std Error:', self.sd_beta)
print('Beta Covariance:', self.cov_beta)
if hasattr(self, 'info'):
print('Residual Variance:',self.res_var)
print('Inverse Condition #:', self.inv_condnum)
print('Reason(s) for Halting:')
for r in self.stopreason:
print(' %s' % r)
class ODR(object):
"""
The ODR class gathers all information and coordinates the running of the
main fitting routine.
Members of instances of the ODR class have the same names as the arguments
to the initialization routine.
Parameters
----------
data : Data class instance
instance of the Data class
model : Model class instance
instance of the Model class
Other Parameters
----------------
beta0 : array_like of rank-1
a rank-1 sequence of initial parameter values. Optional if
model provides an "estimate" function to estimate these values.
delta0 : array_like of floats of rank-1, optional
a (double-precision) float array to hold the initial values of
the errors in the input variables. Must be same shape as data.x
ifixb : array_like of ints of rank-1, optional
sequence of integers with the same length as beta0 that determines
which parameters are held fixed. A value of 0 fixes the parameter,
a value > 0 makes the parameter free.
ifixx : array_like of ints with same shape as data.x, optional
an array of integers with the same shape as data.x that determines
which input observations are treated as fixed. One can use a sequence
of length m (the dimensionality of the input observations) to fix some
dimensions for all observations. A value of 0 fixes the observation,
a value > 0 makes it free.
job : int, optional
an integer telling ODRPACK what tasks to perform. See p. 31 of the
ODRPACK User's Guide if you absolutely must set the value here. Use the
method set_job post-initialization for a more readable interface.
iprint : int, optional
an integer telling ODRPACK what to print. See pp. 33-34 of the
ODRPACK User's Guide if you absolutely must set the value here. Use the
method set_iprint post-initialization for a more readable interface.
errfile : str, optional
string with the filename to print ODRPACK errors to. *Do Not Open
This File Yourself!*
rptfile : str, optional
string with the filename to print ODRPACK summaries to. *Do Not
Open This File Yourself!*
ndigit : int, optional
integer specifying the number of reliable digits in the computation
of the function.
taufac : float, optional
float specifying the initial trust region. The default value is 1.
The initial trust region is equal to taufac times the length of the
first computed Gauss-Newton step. taufac must be less than 1.
sstol : float, optional
float specifying the tolerance for convergence based on the relative
change in the sum-of-squares. The default value is eps**(1/2) where eps
is the smallest value such that 1 + eps > 1 for double precision
computation on the machine. sstol must be less than 1.
partol : float, optional
float specifying the tolerance for convergence based on the relative
change in the estimated parameters. The default value is eps**(2/3) for
explicit models and ``eps**(1/3)`` for implicit models. partol must be less
than 1.
maxit : int, optional
integer specifying the maximum number of iterations to perform. For
first runs, maxit is the total number of iterations performed and
defaults to 50. For restarts, maxit is the number of additional
iterations to perform and defaults to 10.
stpb : array_like, optional
sequence (``len(stpb) == len(beta0)``) of relative step sizes to compute
finite difference derivatives wrt the parameters.
stpd : optional
array (``stpd.shape == data.x.shape`` or ``stpd.shape == (m,)``) of relative
step sizes to compute finite difference derivatives wrt the input
variable errors. If stpd is a rank-1 array with length m (the
dimensionality of the input variable), then the values are broadcast to
all observations.
sclb : array_like, optional
sequence (``len(stpb) == len(beta0)``) of scaling factors for the
parameters. The purpose of these scaling factors are to scale all of
the parameters to around unity. Normally appropriate scaling factors
are computed if this argument is not specified. Specify them yourself
if the automatic procedure goes awry.
scld : array_like, optional
array (scld.shape == data.x.shape or scld.shape == (m,)) of scaling
factors for the *errors* in the input variables. Again, these factors
are automatically computed if you do not provide them. If scld.shape ==
(m,), then the scaling factors are broadcast to all observations.
work : ndarray, optional
array to hold the double-valued working data for ODRPACK. When
restarting, takes the value of self.output.work.
iwork : ndarray, optional
array to hold the integer-valued working data for ODRPACK. When
restarting, takes the value of self.output.iwork.
Attributes
----------
data : Data
The data for this fit
model : Model
The model used in fit
output : Output
An instance if the Output class containing all of the returned
data from an invocation of ODR.run() or ODR.restart()
"""
def __init__(self, data, model, beta0=None, delta0=None, ifixb=None,
ifixx=None, job=None, iprint=None, errfile=None, rptfile=None,
ndigit=None, taufac=None, sstol=None, partol=None, maxit=None,
stpb=None, stpd=None, sclb=None, scld=None, work=None, iwork=None):
self.data = data
self.model = model
if beta0 is None:
if self.model.estimate is not None:
self.beta0 = _conv(self.model.estimate(self.data))
else:
raise ValueError(
"must specify beta0 or provide an estimater with the model"
)
else:
self.beta0 = _conv(beta0)
self.delta0 = _conv(delta0)
# These really are 32-bit integers in FORTRAN (gfortran), even on 64-bit
# platforms.
# XXX: some other FORTRAN compilers may not agree.
self.ifixx = _conv(ifixx, dtype=numpy.int32)
self.ifixb = _conv(ifixb, dtype=numpy.int32)
self.job = job
self.iprint = iprint
self.errfile = errfile
self.rptfile = rptfile
self.ndigit = ndigit
self.taufac = taufac
self.sstol = sstol
self.partol = partol
self.maxit = maxit
self.stpb = _conv(stpb)
self.stpd = _conv(stpd)
self.sclb = _conv(sclb)
self.scld = _conv(scld)
self.work = _conv(work)
self.iwork = _conv(iwork)
self.output = None
self._check()
def _check(self):
""" Check the inputs for consistency, but don't bother checking things
that the builtin function odr will check.
"""
x_s = list(self.data.x.shape)
if isinstance(self.data.y, numpy.ndarray):
y_s = list(self.data.y.shape)
if self.model.implicit:
raise OdrError("an implicit model cannot use response data")
else:
# implicit model with q == self.data.y
y_s = [self.data.y, x_s[-1]]
if not self.model.implicit:
raise OdrError("an explicit model needs response data")
self.set_job(fit_type=1)
if x_s[-1] != y_s[-1]:
raise OdrError("number of observations do not match")
n = x_s[-1]
if len(x_s) == 2:
m = x_s[0]
else:
m = 1
if len(y_s) == 2:
q = y_s[0]
else:
q = 1
p = len(self.beta0)
# permissible output array shapes
fcn_perms = [(q, n)]
fjacd_perms = [(q, m, n)]
fjacb_perms = [(q, p, n)]
if q == 1:
fcn_perms.append((n,))
fjacd_perms.append((m, n))
fjacb_perms.append((p, n))
if m == 1:
fjacd_perms.append((q, n))
if p == 1:
fjacb_perms.append((q, n))
if m == q == 1:
fjacd_perms.append((n,))
if p == q == 1:
fjacb_perms.append((n,))
# try evaluating the supplied functions to make sure they provide
# sensible outputs
arglist = (self.beta0, self.data.x)
if self.model.extra_args is not None:
arglist = arglist + self.model.extra_args
res = self.model.fcn(*arglist)
if res.shape not in fcn_perms:
print(res.shape)
print(fcn_perms)
raise OdrError("fcn does not output %s-shaped array" % y_s)
if self.model.fjacd is not None:
res = self.model.fjacd(*arglist)
if res.shape not in fjacd_perms:
raise OdrError(
"fjacd does not output %s-shaped array" % repr((q, m, n)))
if self.model.fjacb is not None:
res = self.model.fjacb(*arglist)
if res.shape not in fjacb_perms:
raise OdrError(
"fjacb does not output %s-shaped array" % repr((q, p, n)))
# check shape of delta0
if self.delta0 is not None and self.delta0.shape != self.data.x.shape:
raise OdrError(
"delta0 is not a %s-shaped array" % repr(self.data.x.shape))
if self.data.x.size == 0:
warn(("Empty data detected for ODR instance. "
"Do not expect any fitting to occur"),
OdrWarning)
def _gen_work(self):
""" Generate a suitable work array if one does not already exist.
"""
n = self.data.x.shape[-1]
p = self.beta0.shape[0]
if len(self.data.x.shape) == 2:
m = self.data.x.shape[0]
else:
m = 1
if self.model.implicit:
q = self.data.y
elif len(self.data.y.shape) == 2:
q = self.data.y.shape[0]
else:
q = 1
if self.data.we is None:
ldwe = ld2we = 1
elif len(self.data.we.shape) == 3:
ld2we, ldwe = self.data.we.shape[1:]
else:
# Okay, this isn't precisely right, but for this calculation,
# it's fine
ldwe = 1
ld2we = self.data.we.shape[1]
if self.job % 10 < 2:
# ODR not OLS
lwork = (18 + 11*p + p*p + m + m*m + 4*n*q + 6*n*m + 2*n*q*p +
2*n*q*m + q*q + 5*q + q*(p+m) + ldwe*ld2we*q)
else:
# OLS not ODR
lwork = (18 + 11*p + p*p + m + m*m + 4*n*q + 2*n*m + 2*n*q*p +
5*q + q*(p+m) + ldwe*ld2we*q)
if isinstance(self.work, numpy.ndarray) and self.work.shape == (lwork,)\
and self.work.dtype.str.endswith('f8'):
# the existing array is fine
return
else:
self.work = numpy.zeros((lwork,), float)
def set_job(self, fit_type=None, deriv=None, var_calc=None,
del_init=None, restart=None):
"""
Sets the "job" parameter is a hopefully comprehensible way.
If an argument is not specified, then the value is left as is. The
default value from class initialization is for all of these options set
to 0.
Parameters
----------
fit_type : {0, 1, 2} int
0 -> explicit ODR
1 -> implicit ODR
2 -> ordinary least-squares
deriv : {0, 1, 2, 3} int
0 -> forward finite differences
1 -> central finite differences
2 -> user-supplied derivatives (Jacobians) with results
checked by ODRPACK
3 -> user-supplied derivatives, no checking
var_calc : {0, 1, 2} int
0 -> calculate asymptotic covariance matrix and fit
parameter uncertainties (V_B, s_B) using derivatives
recomputed at the final solution
1 -> calculate V_B and s_B using derivatives from last iteration
2 -> do not calculate V_B and s_B
del_init : {0, 1} int
0 -> initial input variable offsets set to 0
1 -> initial offsets provided by user in variable "work"
restart : {0, 1} int
0 -> fit is not a restart
1 -> fit is a restart
Notes
-----
The permissible values are different from those given on pg. 31 of the
ODRPACK User's Guide only in that one cannot specify numbers greater than
the last value for each variable.
If one does not supply functions to compute the Jacobians, the fitting
procedure will change deriv to 0, finite differences, as a default. To
initialize the input variable offsets by yourself, set del_init to 1 and
put the offsets into the "work" variable correctly.
"""
if self.job is None:
job_l = [0, 0, 0, 0, 0]
else:
job_l = [self.job // 10000 % 10,
self.job // 1000 % 10,
self.job // 100 % 10,
self.job // 10 % 10,
self.job % 10]
if fit_type in (0, 1, 2):
job_l[4] = fit_type
if deriv in (0, 1, 2, 3):
job_l[3] = deriv
if var_calc in (0, 1, 2):
job_l[2] = var_calc
if del_init in (0, 1):
job_l[1] = del_init
if restart in (0, 1):
job_l[0] = restart
self.job = (job_l[0]*10000 + job_l[1]*1000 +
job_l[2]*100 + job_l[3]*10 + job_l[4])
def set_iprint(self, init=None, so_init=None,
iter=None, so_iter=None, iter_step=None, final=None, so_final=None):
""" Set the iprint parameter for the printing of computation reports.
If any of the arguments are specified here, then they are set in the
iprint member. If iprint is not set manually or with this method, then
ODRPACK defaults to no printing. If no filename is specified with the
member rptfile, then ODRPACK prints to stdout. One can tell ODRPACK to
print to stdout in addition to the specified filename by setting the
so_* arguments to this function, but one cannot specify to print to
stdout but not a file since one can do that by not specifying a rptfile
filename.
There are three reports: initialization, iteration, and final reports.
They are represented by the arguments init, iter, and final
respectively. The permissible values are 0, 1, and 2 representing "no
report", "short report", and "long report" respectively.
The argument iter_step (0 <= iter_step <= 9) specifies how often to make
the iteration report; the report will be made for every iter_step'th
iteration starting with iteration one. If iter_step == 0, then no
iteration report is made, regardless of the other arguments.
If the rptfile is None, then any so_* arguments supplied will raise an
exception.
"""
if self.iprint is None:
self.iprint = 0
ip = [self.iprint // 1000 % 10,
self.iprint // 100 % 10,
self.iprint // 10 % 10,
self.iprint % 10]
# make a list to convert iprint digits to/from argument inputs
# rptfile, stdout
ip2arg = [[0, 0], # none, none
[1, 0], # short, none
[2, 0], # long, none
[1, 1], # short, short
[2, 1], # long, short
[1, 2], # short, long
[2, 2]] # long, long
if (self.rptfile is None and
(so_init is not None or
so_iter is not None or
so_final is not None)):
raise OdrError(
"no rptfile specified, cannot output to stdout twice")
iprint_l = ip2arg[ip[0]] + ip2arg[ip[1]] + ip2arg[ip[3]]
if init is not None:
iprint_l[0] = init
if so_init is not None:
iprint_l[1] = so_init
if iter is not None:
iprint_l[2] = iter
if so_iter is not None:
iprint_l[3] = so_iter
if final is not None:
iprint_l[4] = final
if so_final is not None:
iprint_l[5] = so_final
if iter_step in range(10):
# 0..9
ip[2] = iter_step
ip[0] = ip2arg.index(iprint_l[0:2])
ip[1] = ip2arg.index(iprint_l[2:4])
ip[3] = ip2arg.index(iprint_l[4:6])
self.iprint = ip[0]*1000 + ip[1]*100 + ip[2]*10 + ip[3]
def run(self):
""" Run the fitting routine with all of the information given and with ``full_output=1``.
Returns
-------
output : Output instance
This object is also assigned to the attribute .output .
"""
args = (self.model.fcn, self.beta0, self.data.y, self.data.x)
kwds = {'full_output': 1}
kwd_l = ['ifixx', 'ifixb', 'job', 'iprint', 'errfile', 'rptfile',
'ndigit', 'taufac', 'sstol', 'partol', 'maxit', 'stpb',
'stpd', 'sclb', 'scld', 'work', 'iwork']
if self.delta0 is not None and self.job % 1000 // 10 == 1:
# delta0 provided and fit is not a restart
self._gen_work()
d0 = numpy.ravel(self.delta0)
self.work[:len(d0)] = d0
# set the kwds from other objects explicitly
if self.model.fjacb is not None:
kwds['fjacb'] = self.model.fjacb
if self.model.fjacd is not None:
kwds['fjacd'] = self.model.fjacd
if self.data.we is not None:
kwds['we'] = self.data.we
if self.data.wd is not None:
kwds['wd'] = self.data.wd
if self.model.extra_args is not None:
kwds['extra_args'] = self.model.extra_args
# implicitly set kwds from self's members
for attr in kwd_l:
obj = getattr(self, attr)
if obj is not None:
kwds[attr] = obj
self.output = Output(odr(*args, **kwds))
return self.output
def restart(self, iter=None):
""" Restarts the run with iter more iterations.
Parameters
----------
iter : int, optional
ODRPACK's default for the number of new iterations is 10.
Returns
-------
output : Output instance
This object is also assigned to the attribute .output .
"""
if self.output is None:
raise OdrError("cannot restart: run() has not been called before")
self.set_job(restart=1)
self.work = self.output.work
self.iwork = self.output.iwork
self.maxit = iter
return self.run()
| 41,283 | 35.599291 | 97 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/odr/add_newdocs.py
|
from numpy import add_newdoc
add_newdoc('scipy.odr', 'odr',
"""
odr(fcn, beta0, y, x, we=None, wd=None, fjacb=None, fjacd=None, extra_args=None, ifixx=None, ifixb=None, job=0, iprint=0, errfile=None, rptfile=None, ndigit=0, taufac=0.0, sstol=-1.0, partol=-1.0, maxit=-1, stpb=None, stpd=None, sclb=None, scld=None, work=None, iwork=None, full_output=0)
Low-level function for ODR.
See Also
--------
ODR
Model
Data
RealData
Notes
-----
This is a function performing the same operation as the `ODR`,
`Model` and `Data` classes together. The parameters of this
function are explained in the class documentation.
""")
add_newdoc('scipy.odr.__odrpack', '_set_exceptions',
"""
_set_exceptions(odr_error, odr_stop)
Internal function: set exception classes.
""")
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cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/odr/models.py
|
""" Collection of Model instances for use with the odrpack fitting package.
"""
from __future__ import division, print_function, absolute_import
import numpy as np
from scipy.odr.odrpack import Model
__all__ = ['Model', 'exponential', 'multilinear', 'unilinear', 'quadratic',
'polynomial']
def _lin_fcn(B, x):
a, b = B[0], B[1:]
b.shape = (b.shape[0], 1)
return a + (x*b).sum(axis=0)
def _lin_fjb(B, x):
a = np.ones(x.shape[-1], float)
res = np.concatenate((a, x.ravel()))
res.shape = (B.shape[-1], x.shape[-1])
return res
def _lin_fjd(B, x):
b = B[1:]
b = np.repeat(b, (x.shape[-1],)*b.shape[-1],axis=0)
b.shape = x.shape
return b
def _lin_est(data):
# Eh. The answer is analytical, so just return all ones.
# Don't return zeros since that will interfere with
# ODRPACK's auto-scaling procedures.
if len(data.x.shape) == 2:
m = data.x.shape[0]
else:
m = 1
return np.ones((m + 1,), float)
def _poly_fcn(B, x, powers):
a, b = B[0], B[1:]
b.shape = (b.shape[0], 1)
return a + np.sum(b * np.power(x, powers), axis=0)
def _poly_fjacb(B, x, powers):
res = np.concatenate((np.ones(x.shape[-1], float), np.power(x,
powers).flat))
res.shape = (B.shape[-1], x.shape[-1])
return res
def _poly_fjacd(B, x, powers):
b = B[1:]
b.shape = (b.shape[0], 1)
b = b * powers
return np.sum(b * np.power(x, powers-1),axis=0)
def _exp_fcn(B, x):
return B[0] + np.exp(B[1] * x)
def _exp_fjd(B, x):
return B[1] * np.exp(B[1] * x)
def _exp_fjb(B, x):
res = np.concatenate((np.ones(x.shape[-1], float), x * np.exp(B[1] * x)))
res.shape = (2, x.shape[-1])
return res
def _exp_est(data):
# Eh.
return np.array([1., 1.])
multilinear = Model(_lin_fcn, fjacb=_lin_fjb,
fjacd=_lin_fjd, estimate=_lin_est,
meta={'name': 'Arbitrary-dimensional Linear',
'equ':'y = B_0 + Sum[i=1..m, B_i * x_i]',
'TeXequ':r'$y=\beta_0 + \sum_{i=1}^m \beta_i x_i$'})
def polynomial(order):
"""
Factory function for a general polynomial model.
Parameters
----------
order : int or sequence
If an integer, it becomes the order of the polynomial to fit. If
a sequence of numbers, then these are the explicit powers in the
polynomial.
A constant term (power 0) is always included, so don't include 0.
Thus, polynomial(n) is equivalent to polynomial(range(1, n+1)).
Returns
-------
polynomial : Model instance
Model instance.
"""
powers = np.asarray(order)
if powers.shape == ():
# Scalar.
powers = np.arange(1, powers + 1)
powers.shape = (len(powers), 1)
len_beta = len(powers) + 1
def _poly_est(data, len_beta=len_beta):
# Eh. Ignore data and return all ones.
return np.ones((len_beta,), float)
return Model(_poly_fcn, fjacd=_poly_fjacd, fjacb=_poly_fjacb,
estimate=_poly_est, extra_args=(powers,),
meta={'name': 'Sorta-general Polynomial',
'equ': 'y = B_0 + Sum[i=1..%s, B_i * (x**i)]' % (len_beta-1),
'TeXequ': r'$y=\beta_0 + \sum_{i=1}^{%s} \beta_i x^i$' %
(len_beta-1)})
exponential = Model(_exp_fcn, fjacd=_exp_fjd, fjacb=_exp_fjb,
estimate=_exp_est, meta={'name':'Exponential',
'equ': 'y= B_0 + exp(B_1 * x)',
'TeXequ': r'$y=\beta_0 + e^{\beta_1 x}$'})
def _unilin(B, x):
return x*B[0] + B[1]
def _unilin_fjd(B, x):
return np.ones(x.shape, float) * B[0]
def _unilin_fjb(B, x):
_ret = np.concatenate((x, np.ones(x.shape, float)))
_ret.shape = (2,) + x.shape
return _ret
def _unilin_est(data):
return (1., 1.)
def _quadratic(B, x):
return x*(x*B[0] + B[1]) + B[2]
def _quad_fjd(B, x):
return 2*x*B[0] + B[1]
def _quad_fjb(B, x):
_ret = np.concatenate((x*x, x, np.ones(x.shape, float)))
_ret.shape = (3,) + x.shape
return _ret
def _quad_est(data):
return (1.,1.,1.)
unilinear = Model(_unilin, fjacd=_unilin_fjd, fjacb=_unilin_fjb,
estimate=_unilin_est, meta={'name': 'Univariate Linear',
'equ': 'y = B_0 * x + B_1',
'TeXequ': '$y = \\beta_0 x + \\beta_1$'})
quadratic = Model(_quadratic, fjacd=_quad_fjd, fjacb=_quad_fjb,
estimate=_quad_est, meta={'name': 'Quadratic',
'equ': 'y = B_0*x**2 + B_1*x + B_2',
'TeXequ': '$y = \\beta_0 x^2 + \\beta_1 x + \\beta_2'})
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cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/odr/__init__.py
|
"""
=================================================
Orthogonal distance regression (:mod:`scipy.odr`)
=================================================
.. currentmodule:: scipy.odr
Package Content
===============
.. autosummary::
:toctree: generated/
Data -- The data to fit.
RealData -- Data with weights as actual std. dev.s and/or covariances.
Model -- Stores information about the function to be fit.
ODR -- Gathers all info & manages the main fitting routine.
Output -- Result from the fit.
odr -- Low-level function for ODR.
OdrWarning -- Warning about potential problems when running ODR
OdrError -- Error exception.
OdrStop -- Stop exception.
odr_error -- Same as OdrError (for backwards compatibility)
odr_stop -- Same as OdrStop (for backwards compatibility)
Prebuilt models:
.. autosummary::
:toctree: generated/
polynomial
.. data:: exponential
.. data:: multilinear
.. data:: unilinear
.. data:: quadratic
.. data:: polynomial
Usage information
=================
Introduction
------------
Why Orthogonal Distance Regression (ODR)? Sometimes one has
measurement errors in the explanatory (a.k.a., "independent")
variable(s), not just the response (a.k.a., "dependent") variable(s).
Ordinary Least Squares (OLS) fitting procedures treat the data for
explanatory variables as fixed, i.e., not subject to error of any kind.
Furthermore, OLS procedures require that the response variables be an
explicit function of the explanatory variables; sometimes making the
equation explicit is impractical and/or introduces errors. ODR can
handle both of these cases with ease, and can even reduce to the OLS
case if that is sufficient for the problem.
ODRPACK is a FORTRAN-77 library for performing ODR with possibly
non-linear fitting functions. It uses a modified trust-region
Levenberg-Marquardt-type algorithm [1]_ to estimate the function
parameters. The fitting functions are provided by Python functions
operating on NumPy arrays. The required derivatives may be provided
by Python functions as well, or may be estimated numerically. ODRPACK
can do explicit or implicit ODR fits, or it can do OLS. Input and
output variables may be multi-dimensional. Weights can be provided to
account for different variances of the observations, and even
covariances between dimensions of the variables.
The `scipy.odr` package offers an object-oriented interface to
ODRPACK, in addition to the low-level `odr` function.
Additional background information about ODRPACK can be found in the
`ODRPACK User's Guide
<https://docs.scipy.org/doc/external/odrpack_guide.pdf>`_, reading
which is recommended.
Basic usage
-----------
1. Define the function you want to fit against.::
def f(B, x):
'''Linear function y = m*x + b'''
# B is a vector of the parameters.
# x is an array of the current x values.
# x is in the same format as the x passed to Data or RealData.
#
# Return an array in the same format as y passed to Data or RealData.
return B[0]*x + B[1]
2. Create a Model.::
linear = Model(f)
3. Create a Data or RealData instance.::
mydata = Data(x, y, wd=1./power(sx,2), we=1./power(sy,2))
or, when the actual covariances are known::
mydata = RealData(x, y, sx=sx, sy=sy)
4. Instantiate ODR with your data, model and initial parameter estimate.::
myodr = ODR(mydata, linear, beta0=[1., 2.])
5. Run the fit.::
myoutput = myodr.run()
6. Examine output.::
myoutput.pprint()
References
----------
.. [1] P. T. Boggs and J. E. Rogers, "Orthogonal Distance Regression,"
in "Statistical analysis of measurement error models and
applications: proceedings of the AMS-IMS-SIAM joint summer research
conference held June 10-16, 1989," Contemporary Mathematics,
vol. 112, pg. 186, 1990.
"""
# version: 0.7
# author: Robert Kern <robert.kern@gmail.com>
# date: 2006-09-21
from __future__ import division, print_function, absolute_import
from .odrpack import *
from .models import *
from . import add_newdocs
__all__ = [s for s in dir() if not s.startswith('_')]
from scipy._lib._testutils import PytestTester
test = PytestTester(__name__)
del PytestTester
| 4,343 | 29.166667 | 80 |
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|
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cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/odr/tests/test_odr.py
|
from __future__ import division, print_function, absolute_import
# Scipy imports.
import numpy as np
from numpy import pi
from numpy.testing import (assert_array_almost_equal,
assert_equal, assert_warns)
from pytest import raises as assert_raises
from scipy.odr import Data, Model, ODR, RealData, OdrStop, OdrWarning
class TestODR(object):
# Bad Data for 'x'
def test_bad_data(self):
assert_raises(ValueError, Data, 2, 1)
assert_raises(ValueError, RealData, 2, 1)
# Empty Data for 'x'
def empty_data_func(self, B, x):
return B[0]*x + B[1]
def test_empty_data(self):
beta0 = [0.02, 0.0]
linear = Model(self.empty_data_func)
empty_dat = Data([], [])
assert_warns(OdrWarning, ODR,
empty_dat, linear, beta0=beta0)
empty_dat = RealData([], [])
assert_warns(OdrWarning, ODR,
empty_dat, linear, beta0=beta0)
# Explicit Example
def explicit_fcn(self, B, x):
ret = B[0] + B[1] * np.power(np.exp(B[2]*x) - 1.0, 2)
return ret
def explicit_fjd(self, B, x):
eBx = np.exp(B[2]*x)
ret = B[1] * 2.0 * (eBx-1.0) * B[2] * eBx
return ret
def explicit_fjb(self, B, x):
eBx = np.exp(B[2]*x)
res = np.vstack([np.ones(x.shape[-1]),
np.power(eBx-1.0, 2),
B[1]*2.0*(eBx-1.0)*eBx*x])
return res
def test_explicit(self):
explicit_mod = Model(
self.explicit_fcn,
fjacb=self.explicit_fjb,
fjacd=self.explicit_fjd,
meta=dict(name='Sample Explicit Model',
ref='ODRPACK UG, pg. 39'),
)
explicit_dat = Data([0.,0.,5.,7.,7.5,10.,16.,26.,30.,34.,34.5,100.],
[1265.,1263.6,1258.,1254.,1253.,1249.8,1237.,1218.,1220.6,
1213.8,1215.5,1212.])
explicit_odr = ODR(explicit_dat, explicit_mod, beta0=[1500.0, -50.0, -0.1],
ifixx=[0,0,1,1,1,1,1,1,1,1,1,0])
explicit_odr.set_job(deriv=2)
explicit_odr.set_iprint(init=0, iter=0, final=0)
out = explicit_odr.run()
assert_array_almost_equal(
out.beta,
np.array([1.2646548050648876e+03, -5.4018409956678255e+01,
-8.7849712165253724e-02]),
)
assert_array_almost_equal(
out.sd_beta,
np.array([1.0349270280543437, 1.583997785262061, 0.0063321988657267]),
)
assert_array_almost_equal(
out.cov_beta,
np.array([[4.4949592379003039e-01, -3.7421976890364739e-01,
-8.0978217468468912e-04],
[-3.7421976890364739e-01, 1.0529686462751804e+00,
-1.9453521827942002e-03],
[-8.0978217468468912e-04, -1.9453521827942002e-03,
1.6827336938454476e-05]]),
)
# Implicit Example
def implicit_fcn(self, B, x):
return (B[2]*np.power(x[0]-B[0], 2) +
2.0*B[3]*(x[0]-B[0])*(x[1]-B[1]) +
B[4]*np.power(x[1]-B[1], 2) - 1.0)
def test_implicit(self):
implicit_mod = Model(
self.implicit_fcn,
implicit=1,
meta=dict(name='Sample Implicit Model',
ref='ODRPACK UG, pg. 49'),
)
implicit_dat = Data([
[0.5,1.2,1.6,1.86,2.12,2.36,2.44,2.36,2.06,1.74,1.34,0.9,-0.28,
-0.78,-1.36,-1.9,-2.5,-2.88,-3.18,-3.44],
[-0.12,-0.6,-1.,-1.4,-2.54,-3.36,-4.,-4.75,-5.25,-5.64,-5.97,-6.32,
-6.44,-6.44,-6.41,-6.25,-5.88,-5.5,-5.24,-4.86]],
1,
)
implicit_odr = ODR(implicit_dat, implicit_mod,
beta0=[-1.0, -3.0, 0.09, 0.02, 0.08])
out = implicit_odr.run()
assert_array_almost_equal(
out.beta,
np.array([-0.9993809167281279, -2.9310484652026476, 0.0875730502693354,
0.0162299708984738, 0.0797537982976416]),
)
assert_array_almost_equal(
out.sd_beta,
np.array([0.1113840353364371, 0.1097673310686467, 0.0041060738314314,
0.0027500347539902, 0.0034962501532468]),
)
assert_array_almost_equal(
out.cov_beta,
np.array([[2.1089274602333052e+00, -1.9437686411979040e+00,
7.0263550868344446e-02, -4.7175267373474862e-02,
5.2515575927380355e-02],
[-1.9437686411979040e+00, 2.0481509222414456e+00,
-6.1600515853057307e-02, 4.6268827806232933e-02,
-5.8822307501391467e-02],
[7.0263550868344446e-02, -6.1600515853057307e-02,
2.8659542561579308e-03, -1.4628662260014491e-03,
1.4528860663055824e-03],
[-4.7175267373474862e-02, 4.6268827806232933e-02,
-1.4628662260014491e-03, 1.2855592885514335e-03,
-1.2692942951415293e-03],
[5.2515575927380355e-02, -5.8822307501391467e-02,
1.4528860663055824e-03, -1.2692942951415293e-03,
2.0778813389755596e-03]]),
)
# Multi-variable Example
def multi_fcn(self, B, x):
if (x < 0.0).any():
raise OdrStop
theta = pi*B[3]/2.
ctheta = np.cos(theta)
stheta = np.sin(theta)
omega = np.power(2.*pi*x*np.exp(-B[2]), B[3])
phi = np.arctan2((omega*stheta), (1.0 + omega*ctheta))
r = (B[0] - B[1]) * np.power(np.sqrt(np.power(1.0 + omega*ctheta, 2) +
np.power(omega*stheta, 2)), -B[4])
ret = np.vstack([B[1] + r*np.cos(B[4]*phi),
r*np.sin(B[4]*phi)])
return ret
def test_multi(self):
multi_mod = Model(
self.multi_fcn,
meta=dict(name='Sample Multi-Response Model',
ref='ODRPACK UG, pg. 56'),
)
multi_x = np.array([30.0, 50.0, 70.0, 100.0, 150.0, 200.0, 300.0, 500.0,
700.0, 1000.0, 1500.0, 2000.0, 3000.0, 5000.0, 7000.0, 10000.0,
15000.0, 20000.0, 30000.0, 50000.0, 70000.0, 100000.0, 150000.0])
multi_y = np.array([
[4.22, 4.167, 4.132, 4.038, 4.019, 3.956, 3.884, 3.784, 3.713,
3.633, 3.54, 3.433, 3.358, 3.258, 3.193, 3.128, 3.059, 2.984,
2.934, 2.876, 2.838, 2.798, 2.759],
[0.136, 0.167, 0.188, 0.212, 0.236, 0.257, 0.276, 0.297, 0.309,
0.311, 0.314, 0.311, 0.305, 0.289, 0.277, 0.255, 0.24, 0.218,
0.202, 0.182, 0.168, 0.153, 0.139],
])
n = len(multi_x)
multi_we = np.zeros((2, 2, n), dtype=float)
multi_ifixx = np.ones(n, dtype=int)
multi_delta = np.zeros(n, dtype=float)
multi_we[0,0,:] = 559.6
multi_we[1,0,:] = multi_we[0,1,:] = -1634.0
multi_we[1,1,:] = 8397.0
for i in range(n):
if multi_x[i] < 100.0:
multi_ifixx[i] = 0
elif multi_x[i] <= 150.0:
pass # defaults are fine
elif multi_x[i] <= 1000.0:
multi_delta[i] = 25.0
elif multi_x[i] <= 10000.0:
multi_delta[i] = 560.0
elif multi_x[i] <= 100000.0:
multi_delta[i] = 9500.0
else:
multi_delta[i] = 144000.0
if multi_x[i] == 100.0 or multi_x[i] == 150.0:
multi_we[:,:,i] = 0.0
multi_dat = Data(multi_x, multi_y, wd=1e-4/np.power(multi_x, 2),
we=multi_we)
multi_odr = ODR(multi_dat, multi_mod, beta0=[4.,2.,7.,.4,.5],
delta0=multi_delta, ifixx=multi_ifixx)
multi_odr.set_job(deriv=1, del_init=1)
out = multi_odr.run()
assert_array_almost_equal(
out.beta,
np.array([4.3799880305938963, 2.4333057577497703, 8.0028845899503978,
0.5101147161764654, 0.5173902330489161]),
)
assert_array_almost_equal(
out.sd_beta,
np.array([0.0130625231081944, 0.0130499785273277, 0.1167085962217757,
0.0132642749596149, 0.0288529201353984]),
)
assert_array_almost_equal(
out.cov_beta,
np.array([[0.0064918418231375, 0.0036159705923791, 0.0438637051470406,
-0.0058700836512467, 0.011281212888768],
[0.0036159705923791, 0.0064793789429006, 0.0517610978353126,
-0.0051181304940204, 0.0130726943624117],
[0.0438637051470406, 0.0517610978353126, 0.5182263323095322,
-0.0563083340093696, 0.1269490939468611],
[-0.0058700836512467, -0.0051181304940204, -0.0563083340093696,
0.0066939246261263, -0.0140184391377962],
[0.011281212888768, 0.0130726943624117, 0.1269490939468611,
-0.0140184391377962, 0.0316733013820852]]),
)
# Pearson's Data
# K. Pearson, Philosophical Magazine, 2, 559 (1901)
def pearson_fcn(self, B, x):
return B[0] + B[1]*x
def test_pearson(self):
p_x = np.array([0.,.9,1.8,2.6,3.3,4.4,5.2,6.1,6.5,7.4])
p_y = np.array([5.9,5.4,4.4,4.6,3.5,3.7,2.8,2.8,2.4,1.5])
p_sx = np.array([.03,.03,.04,.035,.07,.11,.13,.22,.74,1.])
p_sy = np.array([1.,.74,.5,.35,.22,.22,.12,.12,.1,.04])
p_dat = RealData(p_x, p_y, sx=p_sx, sy=p_sy)
# Reverse the data to test invariance of results
pr_dat = RealData(p_y, p_x, sx=p_sy, sy=p_sx)
p_mod = Model(self.pearson_fcn, meta=dict(name='Uni-linear Fit'))
p_odr = ODR(p_dat, p_mod, beta0=[1.,1.])
pr_odr = ODR(pr_dat, p_mod, beta0=[1.,1.])
out = p_odr.run()
assert_array_almost_equal(
out.beta,
np.array([5.4767400299231674, -0.4796082367610305]),
)
assert_array_almost_equal(
out.sd_beta,
np.array([0.3590121690702467, 0.0706291186037444]),
)
assert_array_almost_equal(
out.cov_beta,
np.array([[0.0854275622946333, -0.0161807025443155],
[-0.0161807025443155, 0.003306337993922]]),
)
rout = pr_odr.run()
assert_array_almost_equal(
rout.beta,
np.array([11.4192022410781231, -2.0850374506165474]),
)
assert_array_almost_equal(
rout.sd_beta,
np.array([0.9820231665657161, 0.3070515616198911]),
)
assert_array_almost_equal(
rout.cov_beta,
np.array([[0.6391799462548782, -0.1955657291119177],
[-0.1955657291119177, 0.0624888159223392]]),
)
# Lorentz Peak
# The data is taken from one of the undergraduate physics labs I performed.
def lorentz(self, beta, x):
return (beta[0]*beta[1]*beta[2] / np.sqrt(np.power(x*x -
beta[2]*beta[2], 2.0) + np.power(beta[1]*x, 2.0)))
def test_lorentz(self):
l_sy = np.array([.29]*18)
l_sx = np.array([.000972971,.000948268,.000707632,.000706679,
.000706074, .000703918,.000698955,.000456856,
.000455207,.000662717,.000654619,.000652694,
.000000859202,.00106589,.00106378,.00125483, .00140818,.00241839])
l_dat = RealData(
[3.9094, 3.85945, 3.84976, 3.84716, 3.84551, 3.83964, 3.82608,
3.78847, 3.78163, 3.72558, 3.70274, 3.6973, 3.67373, 3.65982,
3.6562, 3.62498, 3.55525, 3.41886],
[652, 910.5, 984, 1000, 1007.5, 1053, 1160.5, 1409.5, 1430, 1122,
957.5, 920, 777.5, 709.5, 698, 578.5, 418.5, 275.5],
sx=l_sx,
sy=l_sy,
)
l_mod = Model(self.lorentz, meta=dict(name='Lorentz Peak'))
l_odr = ODR(l_dat, l_mod, beta0=(1000., .1, 3.8))
out = l_odr.run()
assert_array_almost_equal(
out.beta,
np.array([1.4306780846149925e+03, 1.3390509034538309e-01,
3.7798193600109009e+00]),
)
assert_array_almost_equal(
out.sd_beta,
np.array([7.3621186811330963e-01, 3.5068899941471650e-04,
2.4451209281408992e-04]),
)
assert_array_almost_equal(
out.cov_beta,
np.array([[2.4714409064597873e-01, -6.9067261911110836e-05,
-3.1236953270424990e-05],
[-6.9067261911110836e-05, 5.6077531517333009e-08,
3.6133261832722601e-08],
[-3.1236953270424990e-05, 3.6133261832722601e-08,
2.7261220025171730e-08]]),
)
def test_ticket_1253(self):
def linear(c, x):
return c[0]*x+c[1]
c = [2.0, 3.0]
x = np.linspace(0, 10)
y = linear(c, x)
model = Model(linear)
data = Data(x, y, wd=1.0, we=1.0)
job = ODR(data, model, beta0=[1.0, 1.0])
result = job.run()
assert_equal(result.info, 2)
| 13,045 | 36.596542 | 83 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/odr/tests/__init__.py
| 0 | 0 | 0 |
py
|
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/vonmises.py
|
from __future__ import division, print_function, absolute_import
import numpy as np
import scipy.stats
from scipy.special import i0
def von_mises_cdf_series(k,x,p):
x = float(x)
s = np.sin(x)
c = np.cos(x)
sn = np.sin(p*x)
cn = np.cos(p*x)
R = 0
V = 0
for n in range(p-1,0,-1):
sn, cn = sn*c - cn*s, cn*c + sn*s
R = 1./(2*n/k + R)
V = R*(sn/n+V)
return 0.5+x/(2*np.pi) + V/np.pi
def von_mises_cdf_normalapprox(k, x):
b = np.sqrt(2/np.pi)*np.exp(k)/i0(k)
z = b*np.sin(x/2.)
return scipy.stats.norm.cdf(z)
def von_mises_cdf(k,x):
ix = 2*np.pi*np.round(x/(2*np.pi))
x = x-ix
k = float(k)
# These values should give 12 decimal digits
CK = 50
a = [28., 0.5, 100., 5.0]
if k < CK:
p = int(np.ceil(a[0]+a[1]*k-a[2]/(k+a[3])))
F = np.clip(von_mises_cdf_series(k,x,p),0,1)
else:
F = von_mises_cdf_normalapprox(k, x)
return F+ix
| 963 | 19.510638 | 64 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/_continuous_distns.py
|
#
# Author: Travis Oliphant 2002-2011 with contributions from
# SciPy Developers 2004-2011
#
from __future__ import division, print_function, absolute_import
import warnings
import numpy as np
from scipy.misc.doccer import (extend_notes_in_docstring,
replace_notes_in_docstring)
from scipy import optimize
from scipy import integrate
import scipy.special as sc
from scipy._lib._numpy_compat import broadcast_to
from . import _stats
from ._tukeylambda_stats import (tukeylambda_variance as _tlvar,
tukeylambda_kurtosis as _tlkurt)
from ._distn_infrastructure import (get_distribution_names, _kurtosis,
_lazyselect, _lazywhere, _ncx2_cdf,
_ncx2_log_pdf, _ncx2_pdf,
rv_continuous, _skew, valarray)
from ._constants import _XMIN, _EULER, _ZETA3, _XMAX, _LOGXMAX
# In numpy 1.12 and above, np.power refuses to raise integers to negative
# powers, and `np.float_power` is a new replacement.
try:
float_power = np.float_power
except AttributeError:
float_power = np.power
## Kolmogorov-Smirnov one-sided and two-sided test statistics
class ksone_gen(rv_continuous):
"""General Kolmogorov-Smirnov one-sided test.
%(default)s
"""
def _cdf(self, x, n):
return 1.0 - sc.smirnov(n, x)
def _ppf(self, q, n):
return sc.smirnovi(n, 1.0 - q)
ksone = ksone_gen(a=0.0, name='ksone')
class kstwobign_gen(rv_continuous):
"""Kolmogorov-Smirnov two-sided test for large N.
%(default)s
"""
def _cdf(self, x):
return 1.0 - sc.kolmogorov(x)
def _sf(self, x):
return sc.kolmogorov(x)
def _ppf(self, q):
return sc.kolmogi(1.0 - q)
kstwobign = kstwobign_gen(a=0.0, name='kstwobign')
## Normal distribution
# loc = mu, scale = std
# Keep these implementations out of the class definition so they can be reused
# by other distributions.
_norm_pdf_C = np.sqrt(2*np.pi)
_norm_pdf_logC = np.log(_norm_pdf_C)
def _norm_pdf(x):
return np.exp(-x**2/2.0) / _norm_pdf_C
def _norm_logpdf(x):
return -x**2 / 2.0 - _norm_pdf_logC
def _norm_cdf(x):
return sc.ndtr(x)
def _norm_logcdf(x):
return sc.log_ndtr(x)
def _norm_ppf(q):
return sc.ndtri(q)
def _norm_sf(x):
return _norm_cdf(-x)
def _norm_logsf(x):
return _norm_logcdf(-x)
def _norm_isf(q):
return -_norm_ppf(q)
class norm_gen(rv_continuous):
r"""A normal continuous random variable.
The location (loc) keyword specifies the mean.
The scale (scale) keyword specifies the standard deviation.
%(before_notes)s
Notes
-----
The probability density function for `norm` is:
.. math::
f(x) = \frac{\exp(-x^2/2)}{\sqrt{2\pi}}
The survival function, ``norm.sf``, is also referred to as the
Q-function in some contexts (see, e.g.,
`Wikipedia's <https://en.wikipedia.org/wiki/Q-function>`_ definition).
%(after_notes)s
%(example)s
"""
def _rvs(self):
return self._random_state.standard_normal(self._size)
def _pdf(self, x):
# norm.pdf(x) = exp(-x**2/2)/sqrt(2*pi)
return _norm_pdf(x)
def _logpdf(self, x):
return _norm_logpdf(x)
def _cdf(self, x):
return _norm_cdf(x)
def _logcdf(self, x):
return _norm_logcdf(x)
def _sf(self, x):
return _norm_sf(x)
def _logsf(self, x):
return _norm_logsf(x)
def _ppf(self, q):
return _norm_ppf(q)
def _isf(self, q):
return _norm_isf(q)
def _stats(self):
return 0.0, 1.0, 0.0, 0.0
def _entropy(self):
return 0.5*(np.log(2*np.pi)+1)
@replace_notes_in_docstring(rv_continuous, notes="""\
This function uses explicit formulas for the maximum likelihood
estimation of the normal distribution parameters, so the
`optimizer` argument is ignored.\n\n""")
def fit(self, data, **kwds):
floc = kwds.get('floc', None)
fscale = kwds.get('fscale', None)
if floc is not None and fscale is not None:
# This check is for consistency with `rv_continuous.fit`.
# Without this check, this function would just return the
# parameters that were given.
raise ValueError("All parameters fixed. There is nothing to "
"optimize.")
data = np.asarray(data)
if floc is None:
loc = data.mean()
else:
loc = floc
if fscale is None:
scale = np.sqrt(((data - loc)**2).mean())
else:
scale = fscale
return loc, scale
norm = norm_gen(name='norm')
class alpha_gen(rv_continuous):
r"""An alpha continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `alpha` is:
.. math::
f(x, a) = \frac{1}{x^2 \Phi(a) \sqrt{2\pi}} *
\exp(-\frac{1}{2} (a-1/x)^2)
where ``Phi(alpha)`` is the normal CDF, ``x > 0``, and ``a > 0``.
`alpha` takes ``a`` as a shape parameter.
%(after_notes)s
%(example)s
"""
_support_mask = rv_continuous._open_support_mask
def _pdf(self, x, a):
# alpha.pdf(x, a) = 1/(x**2*Phi(a)*sqrt(2*pi)) * exp(-1/2 * (a-1/x)**2)
return 1.0/(x**2)/_norm_cdf(a)*_norm_pdf(a-1.0/x)
def _logpdf(self, x, a):
return -2*np.log(x) + _norm_logpdf(a-1.0/x) - np.log(_norm_cdf(a))
def _cdf(self, x, a):
return _norm_cdf(a-1.0/x) / _norm_cdf(a)
def _ppf(self, q, a):
return 1.0/np.asarray(a-sc.ndtri(q*_norm_cdf(a)))
def _stats(self, a):
return [np.inf]*2 + [np.nan]*2
alpha = alpha_gen(a=0.0, name='alpha')
class anglit_gen(rv_continuous):
r"""An anglit continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `anglit` is:
.. math::
f(x) = \sin(2x + \pi/2) = \cos(2x)
for :math:`-\pi/4 \le x \le \pi/4`.
%(after_notes)s
%(example)s
"""
def _pdf(self, x):
# anglit.pdf(x) = sin(2*x + \pi/2) = cos(2*x)
return np.cos(2*x)
def _cdf(self, x):
return np.sin(x+np.pi/4)**2.0
def _ppf(self, q):
return np.arcsin(np.sqrt(q))-np.pi/4
def _stats(self):
return 0.0, np.pi*np.pi/16-0.5, 0.0, -2*(np.pi**4 - 96)/(np.pi*np.pi-8)**2
def _entropy(self):
return 1-np.log(2)
anglit = anglit_gen(a=-np.pi/4, b=np.pi/4, name='anglit')
class arcsine_gen(rv_continuous):
r"""An arcsine continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `arcsine` is:
.. math::
f(x) = \frac{1}{\pi \sqrt{x (1-x)}}
for :math:`0 \le x \le 1`.
%(after_notes)s
%(example)s
"""
def _pdf(self, x):
# arcsine.pdf(x) = 1/(pi*sqrt(x*(1-x)))
return 1.0/np.pi/np.sqrt(x*(1-x))
def _cdf(self, x):
return 2.0/np.pi*np.arcsin(np.sqrt(x))
def _ppf(self, q):
return np.sin(np.pi/2.0*q)**2.0
def _stats(self):
mu = 0.5
mu2 = 1.0/8
g1 = 0
g2 = -3.0/2.0
return mu, mu2, g1, g2
def _entropy(self):
return -0.24156447527049044468
arcsine = arcsine_gen(a=0.0, b=1.0, name='arcsine')
class FitDataError(ValueError):
# This exception is raised by, for example, beta_gen.fit when both floc
# and fscale are fixed and there are values in the data not in the open
# interval (floc, floc+fscale).
def __init__(self, distr, lower, upper):
self.args = (
"Invalid values in `data`. Maximum likelihood "
"estimation with {distr!r} requires that {lower!r} < x "
"< {upper!r} for each x in `data`.".format(
distr=distr, lower=lower, upper=upper),
)
class FitSolverError(RuntimeError):
# This exception is raised by, for example, beta_gen.fit when
# optimize.fsolve returns with ier != 1.
def __init__(self, mesg):
emsg = "Solver for the MLE equations failed to converge: "
emsg += mesg.replace('\n', '')
self.args = (emsg,)
def _beta_mle_a(a, b, n, s1):
# The zeros of this function give the MLE for `a`, with
# `b`, `n` and `s1` given. `s1` is the sum of the logs of
# the data. `n` is the number of data points.
psiab = sc.psi(a + b)
func = s1 - n * (-psiab + sc.psi(a))
return func
def _beta_mle_ab(theta, n, s1, s2):
# Zeros of this function are critical points of
# the maximum likelihood function. Solving this system
# for theta (which contains a and b) gives the MLE for a and b
# given `n`, `s1` and `s2`. `s1` is the sum of the logs of the data,
# and `s2` is the sum of the logs of 1 - data. `n` is the number
# of data points.
a, b = theta
psiab = sc.psi(a + b)
func = [s1 - n * (-psiab + sc.psi(a)),
s2 - n * (-psiab + sc.psi(b))]
return func
class beta_gen(rv_continuous):
r"""A beta continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `beta` is:
.. math::
f(x, a, b) = \frac{\gamma(a+b) x^{a-1} (1-x)^{b-1}}
{\gamma(a) \gamma(b)}
for :math:`0 < x < 1`, :math:`a > 0`, :math:`b > 0`, where
:math:`\gamma(z)` is the gamma function (`scipy.special.gamma`).
`beta` takes :math:`a` and :math:`b` as shape parameters.
%(after_notes)s
%(example)s
"""
def _rvs(self, a, b):
return self._random_state.beta(a, b, self._size)
def _pdf(self, x, a, b):
# gamma(a+b) * x**(a-1) * (1-x)**(b-1)
# beta.pdf(x, a, b) = ------------------------------------
# gamma(a)*gamma(b)
return np.exp(self._logpdf(x, a, b))
def _logpdf(self, x, a, b):
lPx = sc.xlog1py(b - 1.0, -x) + sc.xlogy(a - 1.0, x)
lPx -= sc.betaln(a, b)
return lPx
def _cdf(self, x, a, b):
return sc.btdtr(a, b, x)
def _ppf(self, q, a, b):
return sc.btdtri(a, b, q)
def _stats(self, a, b):
mn = a*1.0 / (a + b)
var = (a*b*1.0)/(a+b+1.0)/(a+b)**2.0
g1 = 2.0*(b-a)*np.sqrt((1.0+a+b)/(a*b)) / (2+a+b)
g2 = 6.0*(a**3 + a**2*(1-2*b) + b**2*(1+b) - 2*a*b*(2+b))
g2 /= a*b*(a+b+2)*(a+b+3)
return mn, var, g1, g2
def _fitstart(self, data):
g1 = _skew(data)
g2 = _kurtosis(data)
def func(x):
a, b = x
sk = 2*(b-a)*np.sqrt(a + b + 1) / (a + b + 2) / np.sqrt(a*b)
ku = a**3 - a**2*(2*b-1) + b**2*(b+1) - 2*a*b*(b+2)
ku /= a*b*(a+b+2)*(a+b+3)
ku *= 6
return [sk-g1, ku-g2]
a, b = optimize.fsolve(func, (1.0, 1.0))
return super(beta_gen, self)._fitstart(data, args=(a, b))
@extend_notes_in_docstring(rv_continuous, notes="""\
In the special case where both `floc` and `fscale` are given, a
`ValueError` is raised if any value `x` in `data` does not satisfy
`floc < x < floc + fscale`.\n\n""")
def fit(self, data, *args, **kwds):
# Override rv_continuous.fit, so we can more efficiently handle the
# case where floc and fscale are given.
f0 = (kwds.get('f0', None) or kwds.get('fa', None) or
kwds.get('fix_a', None))
f1 = (kwds.get('f1', None) or kwds.get('fb', None) or
kwds.get('fix_b', None))
floc = kwds.get('floc', None)
fscale = kwds.get('fscale', None)
if floc is None or fscale is None:
# do general fit
return super(beta_gen, self).fit(data, *args, **kwds)
if f0 is not None and f1 is not None:
# This check is for consistency with `rv_continuous.fit`.
raise ValueError("All parameters fixed. There is nothing to "
"optimize.")
# Special case: loc and scale are constrained, so we are fitting
# just the shape parameters. This can be done much more efficiently
# than the method used in `rv_continuous.fit`. (See the subsection
# "Two unknown parameters" in the section "Maximum likelihood" of
# the Wikipedia article on the Beta distribution for the formulas.)
# Normalize the data to the interval [0, 1].
data = (np.ravel(data) - floc) / fscale
if np.any(data <= 0) or np.any(data >= 1):
raise FitDataError("beta", lower=floc, upper=floc + fscale)
xbar = data.mean()
if f0 is not None or f1 is not None:
# One of the shape parameters is fixed.
if f0 is not None:
# The shape parameter a is fixed, so swap the parameters
# and flip the data. We always solve for `a`. The result
# will be swapped back before returning.
b = f0
data = 1 - data
xbar = 1 - xbar
else:
b = f1
# Initial guess for a. Use the formula for the mean of the beta
# distribution, E[x] = a / (a + b), to generate a reasonable
# starting point based on the mean of the data and the given
# value of b.
a = b * xbar / (1 - xbar)
# Compute the MLE for `a` by solving _beta_mle_a.
theta, info, ier, mesg = optimize.fsolve(
_beta_mle_a, a,
args=(b, len(data), np.log(data).sum()),
full_output=True
)
if ier != 1:
raise FitSolverError(mesg=mesg)
a = theta[0]
if f0 is not None:
# The shape parameter a was fixed, so swap back the
# parameters.
a, b = b, a
else:
# Neither of the shape parameters is fixed.
# s1 and s2 are used in the extra arguments passed to _beta_mle_ab
# by optimize.fsolve.
s1 = np.log(data).sum()
s2 = sc.log1p(-data).sum()
# Use the "method of moments" to estimate the initial
# guess for a and b.
fac = xbar * (1 - xbar) / data.var(ddof=0) - 1
a = xbar * fac
b = (1 - xbar) * fac
# Compute the MLE for a and b by solving _beta_mle_ab.
theta, info, ier, mesg = optimize.fsolve(
_beta_mle_ab, [a, b],
args=(len(data), s1, s2),
full_output=True
)
if ier != 1:
raise FitSolverError(mesg=mesg)
a, b = theta
return a, b, floc, fscale
beta = beta_gen(a=0.0, b=1.0, name='beta')
class betaprime_gen(rv_continuous):
r"""A beta prime continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `betaprime` is:
.. math::
f(x, a, b) = \frac{x^{a-1} (1+x)^{-a-b}}{\beta(a, b)}
for ``x > 0``, ``a > 0``, ``b > 0``, where ``beta(a, b)`` is the beta
function (see `scipy.special.beta`).
`betaprime` takes ``a`` and ``b`` as shape parameters.
%(after_notes)s
%(example)s
"""
_support_mask = rv_continuous._open_support_mask
def _rvs(self, a, b):
sz, rndm = self._size, self._random_state
u1 = gamma.rvs(a, size=sz, random_state=rndm)
u2 = gamma.rvs(b, size=sz, random_state=rndm)
return u1 / u2
def _pdf(self, x, a, b):
# betaprime.pdf(x, a, b) = x**(a-1) * (1+x)**(-a-b) / beta(a, b)
return np.exp(self._logpdf(x, a, b))
def _logpdf(self, x, a, b):
return sc.xlogy(a - 1.0, x) - sc.xlog1py(a + b, x) - sc.betaln(a, b)
def _cdf(self, x, a, b):
return sc.betainc(a, b, x/(1.+x))
def _munp(self, n, a, b):
if n == 1.0:
return np.where(b > 1,
a/(b-1.0),
np.inf)
elif n == 2.0:
return np.where(b > 2,
a*(a+1.0)/((b-2.0)*(b-1.0)),
np.inf)
elif n == 3.0:
return np.where(b > 3,
a*(a+1.0)*(a+2.0)/((b-3.0)*(b-2.0)*(b-1.0)),
np.inf)
elif n == 4.0:
return np.where(b > 4,
(a*(a + 1.0)*(a + 2.0)*(a + 3.0) /
((b - 4.0)*(b - 3.0)*(b - 2.0)*(b - 1.0))),
np.inf)
else:
raise NotImplementedError
betaprime = betaprime_gen(a=0.0, name='betaprime')
class bradford_gen(rv_continuous):
r"""A Bradford continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `bradford` is:
.. math::
f(x, c) = \frac{c}{k (1+cx)}
for :math:`0 < x < 1`, :math:`c > 0` and :math:`k = \log(1+c)`.
`bradford` takes :math:`c` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _pdf(self, x, c):
# bradford.pdf(x, c) = c / (k * (1+c*x))
return c / (c*x + 1.0) / sc.log1p(c)
def _cdf(self, x, c):
return sc.log1p(c*x) / sc.log1p(c)
def _ppf(self, q, c):
return sc.expm1(q * sc.log1p(c)) / c
def _stats(self, c, moments='mv'):
k = np.log(1.0+c)
mu = (c-k)/(c*k)
mu2 = ((c+2.0)*k-2.0*c)/(2*c*k*k)
g1 = None
g2 = None
if 's' in moments:
g1 = np.sqrt(2)*(12*c*c-9*c*k*(c+2)+2*k*k*(c*(c+3)+3))
g1 /= np.sqrt(c*(c*(k-2)+2*k))*(3*c*(k-2)+6*k)
if 'k' in moments:
g2 = (c**3*(k-3)*(k*(3*k-16)+24)+12*k*c*c*(k-4)*(k-3) +
6*c*k*k*(3*k-14) + 12*k**3)
g2 /= 3*c*(c*(k-2)+2*k)**2
return mu, mu2, g1, g2
def _entropy(self, c):
k = np.log(1+c)
return k/2.0 - np.log(c/k)
bradford = bradford_gen(a=0.0, b=1.0, name='bradford')
class burr_gen(rv_continuous):
r"""A Burr (Type III) continuous random variable.
%(before_notes)s
See Also
--------
fisk : a special case of either `burr` or ``burr12`` with ``d = 1``
burr12 : Burr Type XII distribution
Notes
-----
The probability density function for `burr` is:
.. math::
f(x, c, d) = c d x^{-c-1} (1+x^{-c})^{-d-1}
for :math:`x > 0`.
`burr` takes :math:`c` and :math:`d` as shape parameters.
This is the PDF corresponding to the third CDF given in Burr's list;
specifically, it is equation (11) in Burr's paper [1]_.
%(after_notes)s
References
----------
.. [1] Burr, I. W. "Cumulative frequency functions", Annals of
Mathematical Statistics, 13(2), pp 215-232 (1942).
%(example)s
"""
_support_mask = rv_continuous._open_support_mask
def _pdf(self, x, c, d):
# burr.pdf(x, c, d) = c * d * x**(-c-1) * (1+x**(-c))**(-d-1)
return c * d * (x**(-c - 1.0)) * ((1 + x**(-c))**(-d - 1.0))
def _cdf(self, x, c, d):
return (1 + x**(-c))**(-d)
def _ppf(self, q, c, d):
return (q**(-1.0/d) - 1)**(-1.0/c)
def _munp(self, n, c, d):
nc = 1. * n / c
return d * sc.beta(1.0 - nc, d + nc)
burr = burr_gen(a=0.0, name='burr')
class burr12_gen(rv_continuous):
r"""A Burr (Type XII) continuous random variable.
%(before_notes)s
See Also
--------
fisk : a special case of either `burr` or ``burr12`` with ``d = 1``
burr : Burr Type III distribution
Notes
-----
The probability density function for `burr` is:
.. math::
f(x, c, d) = c d x^{c-1} (1+x^c)^{-d-1}
for :math:`x > 0`.
`burr12` takes :math:`c` and :math:`d` as shape parameters.
This is the PDF corresponding to the twelfth CDF given in Burr's list;
specifically, it is equation (20) in Burr's paper [1]_.
%(after_notes)s
The Burr type 12 distribution is also sometimes referred to as
the Singh-Maddala distribution from NIST [2]_.
References
----------
.. [1] Burr, I. W. "Cumulative frequency functions", Annals of
Mathematical Statistics, 13(2), pp 215-232 (1942).
.. [2] http://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/b12pdf.htm
%(example)s
"""
_support_mask = rv_continuous._open_support_mask
def _pdf(self, x, c, d):
# burr12.pdf(x, c, d) = c * d * x**(c-1) * (1+x**(c))**(-d-1)
return np.exp(self._logpdf(x, c, d))
def _logpdf(self, x, c, d):
return np.log(c) + np.log(d) + sc.xlogy(c - 1, x) + sc.xlog1py(-d-1, x**c)
def _cdf(self, x, c, d):
return -sc.expm1(self._logsf(x, c, d))
def _logcdf(self, x, c, d):
return sc.log1p(-(1 + x**c)**(-d))
def _sf(self, x, c, d):
return np.exp(self._logsf(x, c, d))
def _logsf(self, x, c, d):
return sc.xlog1py(-d, x**c)
def _ppf(self, q, c, d):
# The following is an implementation of
# ((1 - q)**(-1.0/d) - 1)**(1.0/c)
# that does a better job handling small values of q.
return sc.expm1(-1/d * sc.log1p(-q))**(1/c)
def _munp(self, n, c, d):
nc = 1. * n / c
return d * sc.beta(1.0 + nc, d - nc)
burr12 = burr12_gen(a=0.0, name='burr12')
class fisk_gen(burr_gen):
r"""A Fisk continuous random variable.
The Fisk distribution is also known as the log-logistic distribution, and
equals the Burr distribution with ``d == 1``.
`fisk` takes :math:`c` as a shape parameter.
%(before_notes)s
Notes
-----
The probability density function for `fisk` is:
.. math::
f(x, c) = c x^{-c-1} (1 + x^{-c})^{-2}
for :math:`x > 0`.
`fisk` takes :math:`c` as a shape parameters.
%(after_notes)s
See Also
--------
burr
%(example)s
"""
def _pdf(self, x, c):
# fisk.pdf(x, c) = c * x**(-c-1) * (1 + x**(-c))**(-2)
return burr_gen._pdf(self, x, c, 1.0)
def _cdf(self, x, c):
return burr_gen._cdf(self, x, c, 1.0)
def _ppf(self, x, c):
return burr_gen._ppf(self, x, c, 1.0)
def _munp(self, n, c):
return burr_gen._munp(self, n, c, 1.0)
def _entropy(self, c):
return 2 - np.log(c)
fisk = fisk_gen(a=0.0, name='fisk')
# median = loc
class cauchy_gen(rv_continuous):
r"""A Cauchy continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `cauchy` is:
.. math::
f(x) = \frac{1}{\pi (1 + x^2)}
%(after_notes)s
%(example)s
"""
def _pdf(self, x):
# cauchy.pdf(x) = 1 / (pi * (1 + x**2))
return 1.0/np.pi/(1.0+x*x)
def _cdf(self, x):
return 0.5 + 1.0/np.pi*np.arctan(x)
def _ppf(self, q):
return np.tan(np.pi*q-np.pi/2.0)
def _sf(self, x):
return 0.5 - 1.0/np.pi*np.arctan(x)
def _isf(self, q):
return np.tan(np.pi/2.0-np.pi*q)
def _stats(self):
return np.nan, np.nan, np.nan, np.nan
def _entropy(self):
return np.log(4*np.pi)
def _fitstart(self, data, args=None):
# Initialize ML guesses using quartiles instead of moments.
p25, p50, p75 = np.percentile(data, [25, 50, 75])
return p50, (p75 - p25)/2
cauchy = cauchy_gen(name='cauchy')
class chi_gen(rv_continuous):
r"""A chi continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `chi` is:
.. math::
f(x, df) = \frac{x^{df-1} \exp(-x^2/2)}{2^{df/2-1} \gamma(df/2)}
for :math:`x > 0`.
Special cases of `chi` are:
- ``chi(1, loc, scale)`` is equivalent to `halfnorm`
- ``chi(2, 0, scale)`` is equivalent to `rayleigh`
- ``chi(3, 0, scale)`` is equivalent to `maxwell`
`chi` takes ``df`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _rvs(self, df):
sz, rndm = self._size, self._random_state
return np.sqrt(chi2.rvs(df, size=sz, random_state=rndm))
def _pdf(self, x, df):
# x**(df-1) * exp(-x**2/2)
# chi.pdf(x, df) = -------------------------
# 2**(df/2-1) * gamma(df/2)
return np.exp(self._logpdf(x, df))
def _logpdf(self, x, df):
l = np.log(2) - .5*np.log(2)*df - sc.gammaln(.5*df)
return l + sc.xlogy(df - 1., x) - .5*x**2
def _cdf(self, x, df):
return sc.gammainc(.5*df, .5*x**2)
def _ppf(self, q, df):
return np.sqrt(2*sc.gammaincinv(.5*df, q))
def _stats(self, df):
mu = np.sqrt(2)*sc.gamma(df/2.0+0.5)/sc.gamma(df/2.0)
mu2 = df - mu*mu
g1 = (2*mu**3.0 + mu*(1-2*df))/np.asarray(np.power(mu2, 1.5))
g2 = 2*df*(1.0-df)-6*mu**4 + 4*mu**2 * (2*df-1)
g2 /= np.asarray(mu2**2.0)
return mu, mu2, g1, g2
chi = chi_gen(a=0.0, name='chi')
## Chi-squared (gamma-distributed with loc=0 and scale=2 and shape=df/2)
class chi2_gen(rv_continuous):
r"""A chi-squared continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `chi2` is:
.. math::
f(x, df) = \frac{1}{(2 \gamma(df/2)} (x/2)^{df/2-1} \exp(-x/2)
`chi2` takes ``df`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _rvs(self, df):
return self._random_state.chisquare(df, self._size)
def _pdf(self, x, df):
# chi2.pdf(x, df) = 1 / (2*gamma(df/2)) * (x/2)**(df/2-1) * exp(-x/2)
return np.exp(self._logpdf(x, df))
def _logpdf(self, x, df):
return sc.xlogy(df/2.-1, x) - x/2. - sc.gammaln(df/2.) - (np.log(2)*df)/2.
def _cdf(self, x, df):
return sc.chdtr(df, x)
def _sf(self, x, df):
return sc.chdtrc(df, x)
def _isf(self, p, df):
return sc.chdtri(df, p)
def _ppf(self, p, df):
return self._isf(1.0-p, df)
def _stats(self, df):
mu = df
mu2 = 2*df
g1 = 2*np.sqrt(2.0/df)
g2 = 12.0/df
return mu, mu2, g1, g2
chi2 = chi2_gen(a=0.0, name='chi2')
class cosine_gen(rv_continuous):
r"""A cosine continuous random variable.
%(before_notes)s
Notes
-----
The cosine distribution is an approximation to the normal distribution.
The probability density function for `cosine` is:
.. math::
f(x) = \frac{1}{2\pi} (1+\cos(x))
for :math:`-\pi \le x \le \pi`.
%(after_notes)s
%(example)s
"""
def _pdf(self, x):
# cosine.pdf(x) = 1/(2*pi) * (1+cos(x))
return 1.0/2/np.pi*(1+np.cos(x))
def _cdf(self, x):
return 1.0/2/np.pi*(np.pi + x + np.sin(x))
def _stats(self):
return 0.0, np.pi*np.pi/3.0-2.0, 0.0, -6.0*(np.pi**4-90)/(5.0*(np.pi*np.pi-6)**2)
def _entropy(self):
return np.log(4*np.pi)-1.0
cosine = cosine_gen(a=-np.pi, b=np.pi, name='cosine')
class dgamma_gen(rv_continuous):
r"""A double gamma continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `dgamma` is:
.. math::
f(x, a) = \frac{1}{2\gamma(a)} |x|^{a-1} \exp(-|x|)
for :math:`a > 0`.
`dgamma` takes :math:`a` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _rvs(self, a):
sz, rndm = self._size, self._random_state
u = rndm.random_sample(size=sz)
gm = gamma.rvs(a, size=sz, random_state=rndm)
return gm * np.where(u >= 0.5, 1, -1)
def _pdf(self, x, a):
# dgamma.pdf(x, a) = 1 / (2*gamma(a)) * abs(x)**(a-1) * exp(-abs(x))
ax = abs(x)
return 1.0/(2*sc.gamma(a))*ax**(a-1.0) * np.exp(-ax)
def _logpdf(self, x, a):
ax = abs(x)
return sc.xlogy(a - 1.0, ax) - ax - np.log(2) - sc.gammaln(a)
def _cdf(self, x, a):
fac = 0.5*sc.gammainc(a, abs(x))
return np.where(x > 0, 0.5 + fac, 0.5 - fac)
def _sf(self, x, a):
fac = 0.5*sc.gammainc(a, abs(x))
return np.where(x > 0, 0.5-fac, 0.5+fac)
def _ppf(self, q, a):
fac = sc.gammainccinv(a, 1-abs(2*q-1))
return np.where(q > 0.5, fac, -fac)
def _stats(self, a):
mu2 = a*(a+1.0)
return 0.0, mu2, 0.0, (a+2.0)*(a+3.0)/mu2-3.0
dgamma = dgamma_gen(name='dgamma')
class dweibull_gen(rv_continuous):
r"""A double Weibull continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `dweibull` is:
.. math::
f(x, c) = c / 2 |x|^{c-1} \exp(-|x|^c)
`dweibull` takes :math:`d` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _rvs(self, c):
sz, rndm = self._size, self._random_state
u = rndm.random_sample(size=sz)
w = weibull_min.rvs(c, size=sz, random_state=rndm)
return w * (np.where(u >= 0.5, 1, -1))
def _pdf(self, x, c):
# dweibull.pdf(x, c) = c / 2 * abs(x)**(c-1) * exp(-abs(x)**c)
ax = abs(x)
Px = c / 2.0 * ax**(c-1.0) * np.exp(-ax**c)
return Px
def _logpdf(self, x, c):
ax = abs(x)
return np.log(c) - np.log(2.0) + sc.xlogy(c - 1.0, ax) - ax**c
def _cdf(self, x, c):
Cx1 = 0.5 * np.exp(-abs(x)**c)
return np.where(x > 0, 1 - Cx1, Cx1)
def _ppf(self, q, c):
fac = 2. * np.where(q <= 0.5, q, 1. - q)
fac = np.power(-np.log(fac), 1.0 / c)
return np.where(q > 0.5, fac, -fac)
def _munp(self, n, c):
return (1 - (n % 2)) * sc.gamma(1.0 + 1.0 * n / c)
# since we know that all odd moments are zeros, return them at once.
# returning Nones from _stats makes the public stats call _munp
# so overall we're saving one or two gamma function evaluations here.
def _stats(self, c):
return 0, None, 0, None
dweibull = dweibull_gen(name='dweibull')
## Exponential (gamma distributed with a=1.0, loc=loc and scale=scale)
class expon_gen(rv_continuous):
r"""An exponential continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `expon` is:
.. math::
f(x) = \exp(-x)
for :math:`x \ge 0`.
%(after_notes)s
A common parameterization for `expon` is in terms of the rate parameter
``lambda``, such that ``pdf = lambda * exp(-lambda * x)``. This
parameterization corresponds to using ``scale = 1 / lambda``.
%(example)s
"""
def _rvs(self):
return self._random_state.standard_exponential(self._size)
def _pdf(self, x):
# expon.pdf(x) = exp(-x)
return np.exp(-x)
def _logpdf(self, x):
return -x
def _cdf(self, x):
return -sc.expm1(-x)
def _ppf(self, q):
return -sc.log1p(-q)
def _sf(self, x):
return np.exp(-x)
def _logsf(self, x):
return -x
def _isf(self, q):
return -np.log(q)
def _stats(self):
return 1.0, 1.0, 2.0, 6.0
def _entropy(self):
return 1.0
@replace_notes_in_docstring(rv_continuous, notes="""\
This function uses explicit formulas for the maximum likelihood
estimation of the exponential distribution parameters, so the
`optimizer`, `loc` and `scale` keyword arguments are ignored.\n\n""")
def fit(self, data, *args, **kwds):
if len(args) > 0:
raise TypeError("Too many arguments.")
floc = kwds.pop('floc', None)
fscale = kwds.pop('fscale', None)
# Ignore the optimizer-related keyword arguments, if given.
kwds.pop('loc', None)
kwds.pop('scale', None)
kwds.pop('optimizer', None)
if kwds:
raise TypeError("Unknown arguments: %s." % kwds)
if floc is not None and fscale is not None:
# This check is for consistency with `rv_continuous.fit`.
raise ValueError("All parameters fixed. There is nothing to "
"optimize.")
data = np.asarray(data)
data_min = data.min()
if floc is None:
# ML estimate of the location is the minimum of the data.
loc = data_min
else:
loc = floc
if data_min < loc:
# There are values that are less than the specified loc.
raise FitDataError("expon", lower=floc, upper=np.inf)
if fscale is None:
# ML estimate of the scale is the shifted mean.
scale = data.mean() - loc
else:
scale = fscale
# We expect the return values to be floating point, so ensure it
# by explicitly converting to float.
return float(loc), float(scale)
expon = expon_gen(a=0.0, name='expon')
## Exponentially Modified Normal (exponential distribution
## convolved with a Normal).
## This is called an exponentially modified gaussian on wikipedia
class exponnorm_gen(rv_continuous):
r"""An exponentially modified Normal continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `exponnorm` is:
.. math::
f(x, K) = \frac{1}{2K} \exp\left(\frac{1}{2 K^2}\right) \exp(-x / K)
\text{erfc}\left(-\frac{x - 1/K}{\sqrt{2}}\right)
where the shape parameter :math:`K > 0`.
It can be thought of as the sum of a normally distributed random
value with mean ``loc`` and sigma ``scale`` and an exponentially
distributed random number with a pdf proportional to ``exp(-lambda * x)``
where ``lambda = (K * scale)**(-1)``.
%(after_notes)s
An alternative parameterization of this distribution (for example, in
`Wikipedia <http://en.wikipedia.org/wiki/Exponentially_modified_Gaussian_distribution>`_)
involves three parameters, :math:`\mu`, :math:`\lambda` and
:math:`\sigma`.
In the present parameterization this corresponds to having ``loc`` and
``scale`` equal to :math:`\mu` and :math:`\sigma`, respectively, and
shape parameter :math:`K = 1/(\sigma\lambda)`.
.. versionadded:: 0.16.0
%(example)s
"""
def _rvs(self, K):
expval = self._random_state.standard_exponential(self._size) * K
gval = self._random_state.standard_normal(self._size)
return expval + gval
def _pdf(self, x, K):
# exponnorm.pdf(x, K) =
# 1/(2*K) exp(1/(2 * K**2)) exp(-x / K) * erfc-(x - 1/K) / sqrt(2))
invK = 1.0 / K
exparg = 0.5 * invK**2 - invK * x
# Avoid overflows; setting np.exp(exparg) to the max float works
# all right here
expval = _lazywhere(exparg < _LOGXMAX, (exparg,), np.exp, _XMAX)
return 0.5 * invK * expval * sc.erfc(-(x - invK) / np.sqrt(2))
def _logpdf(self, x, K):
invK = 1.0 / K
exparg = 0.5 * invK**2 - invK * x
return exparg + np.log(0.5 * invK * sc.erfc(-(x - invK) / np.sqrt(2)))
def _cdf(self, x, K):
invK = 1.0 / K
expval = invK * (0.5 * invK - x)
return _norm_cdf(x) - np.exp(expval) * _norm_cdf(x - invK)
def _sf(self, x, K):
invK = 1.0 / K
expval = invK * (0.5 * invK - x)
return _norm_cdf(-x) + np.exp(expval) * _norm_cdf(x - invK)
def _stats(self, K):
K2 = K * K
opK2 = 1.0 + K2
skw = 2 * K**3 * opK2**(-1.5)
krt = 6.0 * K2 * K2 * opK2**(-2)
return K, opK2, skw, krt
exponnorm = exponnorm_gen(name='exponnorm')
class exponweib_gen(rv_continuous):
r"""An exponentiated Weibull continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `exponweib` is:
.. math::
f(x, a, c) = a c (1-\exp(-x^c))^{a-1} \exp(-x^c) x^{c-1}
for :math:`x > 0`, :math:`a > 0`, :math:`c > 0`.
`exponweib` takes :math:`a` and :math:`c` as shape parameters.
%(after_notes)s
%(example)s
"""
def _pdf(self, x, a, c):
# exponweib.pdf(x, a, c) =
# a * c * (1-exp(-x**c))**(a-1) * exp(-x**c)*x**(c-1)
return np.exp(self._logpdf(x, a, c))
def _logpdf(self, x, a, c):
negxc = -x**c
exm1c = -sc.expm1(negxc)
logp = (np.log(a) + np.log(c) + sc.xlogy(a - 1.0, exm1c) +
negxc + sc.xlogy(c - 1.0, x))
return logp
def _cdf(self, x, a, c):
exm1c = -sc.expm1(-x**c)
return exm1c**a
def _ppf(self, q, a, c):
return (-sc.log1p(-q**(1.0/a)))**np.asarray(1.0/c)
exponweib = exponweib_gen(a=0.0, name='exponweib')
class exponpow_gen(rv_continuous):
r"""An exponential power continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `exponpow` is:
.. math::
f(x, b) = b x^{b-1} \exp(1 + x^b - \exp(x^b))
for :math:`x \ge 0`, :math:`b > 0``. Note that this is a different
distribution from the exponential power distribution that is also known
under the names "generalized normal" or "generalized Gaussian".
`exponpow` takes :math:`b` as a shape parameter.
%(after_notes)s
References
----------
http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Exponentialpower.pdf
%(example)s
"""
def _pdf(self, x, b):
# exponpow.pdf(x, b) = b * x**(b-1) * exp(1 + x**b - exp(x**b))
return np.exp(self._logpdf(x, b))
def _logpdf(self, x, b):
xb = x**b
f = 1 + np.log(b) + sc.xlogy(b - 1.0, x) + xb - np.exp(xb)
return f
def _cdf(self, x, b):
return -sc.expm1(-sc.expm1(x**b))
def _sf(self, x, b):
return np.exp(-sc.expm1(x**b))
def _isf(self, x, b):
return (sc.log1p(-np.log(x)))**(1./b)
def _ppf(self, q, b):
return pow(sc.log1p(-sc.log1p(-q)), 1.0/b)
exponpow = exponpow_gen(a=0.0, name='exponpow')
class fatiguelife_gen(rv_continuous):
r"""A fatigue-life (Birnbaum-Saunders) continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `fatiguelife` is:
.. math::
f(x, c) = \frac{x+1}{ 2c\sqrt{2\pi x^3} \exp(-\frac{(x-1)^2}{2x c^2}}
for :math:`x > 0`.
`fatiguelife` takes :math:`c` as a shape parameter.
%(after_notes)s
References
----------
.. [1] "Birnbaum-Saunders distribution",
http://en.wikipedia.org/wiki/Birnbaum-Saunders_distribution
%(example)s
"""
_support_mask = rv_continuous._open_support_mask
def _rvs(self, c):
z = self._random_state.standard_normal(self._size)
x = 0.5*c*z
x2 = x*x
t = 1.0 + 2*x2 + 2*x*np.sqrt(1 + x2)
return t
def _pdf(self, x, c):
# fatiguelife.pdf(x, c) =
# (x+1) / (2*c*sqrt(2*pi*x**3)) * exp(-(x-1)**2/(2*x*c**2))
return np.exp(self._logpdf(x, c))
def _logpdf(self, x, c):
return (np.log(x+1) - (x-1)**2 / (2.0*x*c**2) - np.log(2*c) -
0.5*(np.log(2*np.pi) + 3*np.log(x)))
def _cdf(self, x, c):
return _norm_cdf(1.0 / c * (np.sqrt(x) - 1.0/np.sqrt(x)))
def _ppf(self, q, c):
tmp = c*sc.ndtri(q)
return 0.25 * (tmp + np.sqrt(tmp**2 + 4))**2
def _stats(self, c):
# NB: the formula for kurtosis in wikipedia seems to have an error:
# it's 40, not 41. At least it disagrees with the one from Wolfram
# Alpha. And the latter one, below, passes the tests, while the wiki
# one doesn't So far I didn't have the guts to actually check the
# coefficients from the expressions for the raw moments.
c2 = c*c
mu = c2 / 2.0 + 1.0
den = 5.0 * c2 + 4.0
mu2 = c2*den / 4.0
g1 = 4 * c * (11*c2 + 6.0) / np.power(den, 1.5)
g2 = 6 * c2 * (93*c2 + 40.0) / den**2.0
return mu, mu2, g1, g2
fatiguelife = fatiguelife_gen(a=0.0, name='fatiguelife')
class foldcauchy_gen(rv_continuous):
r"""A folded Cauchy continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `foldcauchy` is:
.. math::
f(x, c) = \frac{1}{\pi (1+(x-c)^2)} + \frac{1}{\pi (1+(x+c)^2)}
for :math:`x \ge 0``.
`foldcauchy` takes :math:`c` as a shape parameter.
%(example)s
"""
def _rvs(self, c):
return abs(cauchy.rvs(loc=c, size=self._size,
random_state=self._random_state))
def _pdf(self, x, c):
# foldcauchy.pdf(x, c) = 1/(pi*(1+(x-c)**2)) + 1/(pi*(1+(x+c)**2))
return 1.0/np.pi*(1.0/(1+(x-c)**2) + 1.0/(1+(x+c)**2))
def _cdf(self, x, c):
return 1.0/np.pi*(np.arctan(x-c) + np.arctan(x+c))
def _stats(self, c):
return np.inf, np.inf, np.nan, np.nan
foldcauchy = foldcauchy_gen(a=0.0, name='foldcauchy')
class f_gen(rv_continuous):
r"""An F continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `f` is:
.. math::
f(x, df_1, df_2) = \frac{df_2^{df_2/2} df_1^{df_1/2} x^{df_1 / 2-1}}
{(df_2+df_1 x)^{(df_1+df_2)/2}
B(df_1/2, df_2/2)}
for :math:`x > 0`.
`f` takes ``dfn`` and ``dfd`` as shape parameters.
%(after_notes)s
%(example)s
"""
def _rvs(self, dfn, dfd):
return self._random_state.f(dfn, dfd, self._size)
def _pdf(self, x, dfn, dfd):
# df2**(df2/2) * df1**(df1/2) * x**(df1/2-1)
# F.pdf(x, df1, df2) = --------------------------------------------
# (df2+df1*x)**((df1+df2)/2) * B(df1/2, df2/2)
return np.exp(self._logpdf(x, dfn, dfd))
def _logpdf(self, x, dfn, dfd):
n = 1.0 * dfn
m = 1.0 * dfd
lPx = m/2 * np.log(m) + n/2 * np.log(n) + (n/2 - 1) * np.log(x)
lPx -= ((n+m)/2) * np.log(m + n*x) + sc.betaln(n/2, m/2)
return lPx
def _cdf(self, x, dfn, dfd):
return sc.fdtr(dfn, dfd, x)
def _sf(self, x, dfn, dfd):
return sc.fdtrc(dfn, dfd, x)
def _ppf(self, q, dfn, dfd):
return sc.fdtri(dfn, dfd, q)
def _stats(self, dfn, dfd):
v1, v2 = 1. * dfn, 1. * dfd
v2_2, v2_4, v2_6, v2_8 = v2 - 2., v2 - 4., v2 - 6., v2 - 8.
mu = _lazywhere(
v2 > 2, (v2, v2_2),
lambda v2, v2_2: v2 / v2_2,
np.inf)
mu2 = _lazywhere(
v2 > 4, (v1, v2, v2_2, v2_4),
lambda v1, v2, v2_2, v2_4:
2 * v2 * v2 * (v1 + v2_2) / (v1 * v2_2**2 * v2_4),
np.inf)
g1 = _lazywhere(
v2 > 6, (v1, v2_2, v2_4, v2_6),
lambda v1, v2_2, v2_4, v2_6:
(2 * v1 + v2_2) / v2_6 * np.sqrt(v2_4 / (v1 * (v1 + v2_2))),
np.nan)
g1 *= np.sqrt(8.)
g2 = _lazywhere(
v2 > 8, (g1, v2_6, v2_8),
lambda g1, v2_6, v2_8: (8 + g1 * g1 * v2_6) / v2_8,
np.nan)
g2 *= 3. / 2.
return mu, mu2, g1, g2
f = f_gen(a=0.0, name='f')
## Folded Normal
## abs(Z) where (Z is normal with mu=L and std=S so that c=abs(L)/S)
##
## note: regress docs have scale parameter correct, but first parameter
## he gives is a shape parameter A = c * scale
## Half-normal is folded normal with shape-parameter c=0.
class foldnorm_gen(rv_continuous):
r"""A folded normal continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `foldnorm` is:
.. math::
f(x, c) = \sqrt{2/\pi} cosh(c x) \exp(-\frac{x^2+c^2}{2})
for :math:`c \ge 0`.
`foldnorm` takes :math:`c` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _argcheck(self, c):
return c >= 0
def _rvs(self, c):
return abs(self._random_state.standard_normal(self._size) + c)
def _pdf(self, x, c):
# foldnormal.pdf(x, c) = sqrt(2/pi) * cosh(c*x) * exp(-(x**2+c**2)/2)
return _norm_pdf(x + c) + _norm_pdf(x-c)
def _cdf(self, x, c):
return _norm_cdf(x-c) + _norm_cdf(x+c) - 1.0
def _stats(self, c):
# Regina C. Elandt, Technometrics 3, 551 (1961)
# http://www.jstor.org/stable/1266561
#
c2 = c*c
expfac = np.exp(-0.5*c2) / np.sqrt(2.*np.pi)
mu = 2.*expfac + c * sc.erf(c/np.sqrt(2))
mu2 = c2 + 1 - mu*mu
g1 = 2. * (mu*mu*mu - c2*mu - expfac)
g1 /= np.power(mu2, 1.5)
g2 = c2 * (c2 + 6.) + 3 + 8.*expfac*mu
g2 += (2. * (c2 - 3.) - 3. * mu**2) * mu**2
g2 = g2 / mu2**2.0 - 3.
return mu, mu2, g1, g2
foldnorm = foldnorm_gen(a=0.0, name='foldnorm')
class weibull_min_gen(rv_continuous):
r"""Weibull minimum continuous random variable.
%(before_notes)s
See Also
--------
weibull_max
Notes
-----
The probability density function for `weibull_min` is:
.. math::
f(x, c) = c x^{c-1} \exp(-x^c)
for :math:`x > 0`, :math:`c > 0`.
`weibull_min` takes ``c`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _pdf(self, x, c):
# frechet_r.pdf(x, c) = c * x**(c-1) * exp(-x**c)
return c*pow(x, c-1)*np.exp(-pow(x, c))
def _logpdf(self, x, c):
return np.log(c) + sc.xlogy(c - 1, x) - pow(x, c)
def _cdf(self, x, c):
return -sc.expm1(-pow(x, c))
def _sf(self, x, c):
return np.exp(-pow(x, c))
def _logsf(self, x, c):
return -pow(x, c)
def _ppf(self, q, c):
return pow(-sc.log1p(-q), 1.0/c)
def _munp(self, n, c):
return sc.gamma(1.0+n*1.0/c)
def _entropy(self, c):
return -_EULER / c - np.log(c) + _EULER + 1
weibull_min = weibull_min_gen(a=0.0, name='weibull_min')
class weibull_max_gen(rv_continuous):
r"""Weibull maximum continuous random variable.
%(before_notes)s
See Also
--------
weibull_min
Notes
-----
The probability density function for `weibull_max` is:
.. math::
f(x, c) = c (-x)^{c-1} \exp(-(-x)^c)
for :math:`x < 0`, :math:`c > 0`.
`weibull_max` takes ``c`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _pdf(self, x, c):
# frechet_l.pdf(x, c) = c * (-x)**(c-1) * exp(-(-x)**c)
return c*pow(-x, c-1)*np.exp(-pow(-x, c))
def _logpdf(self, x, c):
return np.log(c) + sc.xlogy(c-1, -x) - pow(-x, c)
def _cdf(self, x, c):
return np.exp(-pow(-x, c))
def _logcdf(self, x, c):
return -pow(-x, c)
def _sf(self, x, c):
return -sc.expm1(-pow(-x, c))
def _ppf(self, q, c):
return -pow(-np.log(q), 1.0/c)
def _munp(self, n, c):
val = sc.gamma(1.0+n*1.0/c)
if int(n) % 2:
sgn = -1
else:
sgn = 1
return sgn * val
def _entropy(self, c):
return -_EULER / c - np.log(c) + _EULER + 1
weibull_max = weibull_max_gen(b=0.0, name='weibull_max')
# Public methods to be deprecated in frechet_r and frechet_l:
# ['__call__', 'cdf', 'entropy', 'expect', 'fit', 'fit_loc_scale', 'freeze',
# 'interval', 'isf', 'logcdf', 'logpdf', 'logsf', 'mean', 'median', 'moment',
# 'nnlf', 'pdf', 'ppf', 'rvs', 'sf', 'stats', 'std', 'var']
_frechet_r_deprec_msg = """\
The distribution `frechet_r` is a synonym for `weibull_min`; this historical
usage is deprecated because of possible confusion with the (quite different)
Frechet distribution. To preserve the existing behavior of the program, use
`scipy.stats.weibull_min`. For the Frechet distribution (i.e. the Type II
extreme value distribution), use `scipy.stats.invweibull`."""
class frechet_r_gen(weibull_min_gen):
@np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg)
def __call__(self, *args, **kwargs):
return weibull_min_gen.__call__(self, *args, **kwargs)
@np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg)
def cdf(self, *args, **kwargs):
return weibull_min_gen.cdf(self, *args, **kwargs)
@np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg)
def entropy(self, *args, **kwargs):
return weibull_min_gen.entropy(self, *args, **kwargs)
@np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg)
def expect(self, *args, **kwargs):
return weibull_min_gen.expect(self, *args, **kwargs)
@np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg)
def fit(self, *args, **kwargs):
return weibull_min_gen.fit(self, *args, **kwargs)
@np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg)
def fit_loc_scale(self, *args, **kwargs):
return weibull_min_gen.fit_loc_scale(self, *args, **kwargs)
@np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg)
def freeze(self, *args, **kwargs):
return weibull_min_gen.freeze(self, *args, **kwargs)
@np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg)
def interval(self, *args, **kwargs):
return weibull_min_gen.interval(self, *args, **kwargs)
@np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg)
def isf(self, *args, **kwargs):
return weibull_min_gen.isf(self, *args, **kwargs)
@np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg)
def logcdf(self, *args, **kwargs):
return weibull_min_gen.logcdf(self, *args, **kwargs)
@np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg)
def logpdf(self, *args, **kwargs):
return weibull_min_gen.logpdf(self, *args, **kwargs)
@np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg)
def logsf(self, *args, **kwargs):
return weibull_min_gen.logsf(self, *args, **kwargs)
@np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg)
def mean(self, *args, **kwargs):
return weibull_min_gen.mean(self, *args, **kwargs)
@np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg)
def median(self, *args, **kwargs):
return weibull_min_gen.median(self, *args, **kwargs)
@np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg)
def moment(self, *args, **kwargs):
return weibull_min_gen.moment(self, *args, **kwargs)
@np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg)
def nnlf(self, *args, **kwargs):
return weibull_min_gen.nnlf(self, *args, **kwargs)
@np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg)
def pdf(self, *args, **kwargs):
return weibull_min_gen.pdf(self, *args, **kwargs)
@np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg)
def ppf(self, *args, **kwargs):
return weibull_min_gen.ppf(self, *args, **kwargs)
@np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg)
def rvs(self, *args, **kwargs):
return weibull_min_gen.rvs(self, *args, **kwargs)
@np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg)
def sf(self, *args, **kwargs):
return weibull_min_gen.sf(self, *args, **kwargs)
@np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg)
def stats(self, *args, **kwargs):
return weibull_min_gen.stats(self, *args, **kwargs)
@np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg)
def std(self, *args, **kwargs):
return weibull_min_gen.std(self, *args, **kwargs)
@np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg)
def var(self, *args, **kwargs):
return weibull_min_gen.var(self, *args, **kwargs)
frechet_r = frechet_r_gen(a=0.0, name='frechet_r')
_frechet_l_deprec_msg = """\
The distribution `frechet_l` is a synonym for `weibull_max`; this historical
usage is deprecated because of possible confusion with the (quite different)
Frechet distribution. To preserve the existing behavior of the program, use
`scipy.stats.weibull_max`. For the Frechet distribution (i.e. the Type II
extreme value distribution), use `scipy.stats.invweibull`."""
class frechet_l_gen(weibull_max_gen):
@np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg)
def __call__(self, *args, **kwargs):
return weibull_max_gen.__call__(self, *args, **kwargs)
@np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg)
def cdf(self, *args, **kwargs):
return weibull_max_gen.cdf(self, *args, **kwargs)
@np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg)
def entropy(self, *args, **kwargs):
return weibull_max_gen.entropy(self, *args, **kwargs)
@np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg)
def expect(self, *args, **kwargs):
return weibull_max_gen.expect(self, *args, **kwargs)
@np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg)
def fit(self, *args, **kwargs):
return weibull_max_gen.fit(self, *args, **kwargs)
@np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg)
def fit_loc_scale(self, *args, **kwargs):
return weibull_max_gen.fit_loc_scale(self, *args, **kwargs)
@np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg)
def freeze(self, *args, **kwargs):
return weibull_max_gen.freeze(self, *args, **kwargs)
@np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg)
def interval(self, *args, **kwargs):
return weibull_max_gen.interval(self, *args, **kwargs)
@np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg)
def isf(self, *args, **kwargs):
return weibull_max_gen.isf(self, *args, **kwargs)
@np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg)
def logcdf(self, *args, **kwargs):
return weibull_max_gen.logcdf(self, *args, **kwargs)
@np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg)
def logpdf(self, *args, **kwargs):
return weibull_max_gen.logpdf(self, *args, **kwargs)
@np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg)
def logsf(self, *args, **kwargs):
return weibull_max_gen.logsf(self, *args, **kwargs)
@np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg)
def mean(self, *args, **kwargs):
return weibull_max_gen.mean(self, *args, **kwargs)
@np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg)
def median(self, *args, **kwargs):
return weibull_max_gen.median(self, *args, **kwargs)
@np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg)
def moment(self, *args, **kwargs):
return weibull_max_gen.moment(self, *args, **kwargs)
@np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg)
def nnlf(self, *args, **kwargs):
return weibull_max_gen.nnlf(self, *args, **kwargs)
@np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg)
def pdf(self, *args, **kwargs):
return weibull_max_gen.pdf(self, *args, **kwargs)
@np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg)
def ppf(self, *args, **kwargs):
return weibull_max_gen.ppf(self, *args, **kwargs)
@np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg)
def rvs(self, *args, **kwargs):
return weibull_max_gen.rvs(self, *args, **kwargs)
@np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg)
def sf(self, *args, **kwargs):
return weibull_max_gen.sf(self, *args, **kwargs)
@np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg)
def stats(self, *args, **kwargs):
return weibull_max_gen.stats(self, *args, **kwargs)
@np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg)
def std(self, *args, **kwargs):
return weibull_max_gen.std(self, *args, **kwargs)
@np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg)
def var(self, *args, **kwargs):
return weibull_max_gen.var(self, *args, **kwargs)
frechet_l = frechet_l_gen(b=0.0, name='frechet_l')
class genlogistic_gen(rv_continuous):
r"""A generalized logistic continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `genlogistic` is:
.. math::
f(x, c) = c \frac{\exp(-x)}
{(1 + \exp(-x))^{c+1}}
for :math:`x > 0`, :math:`c > 0`.
`genlogistic` takes :math:`c` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _pdf(self, x, c):
# genlogistic.pdf(x, c) = c * exp(-x) / (1 + exp(-x))**(c+1)
return np.exp(self._logpdf(x, c))
def _logpdf(self, x, c):
return np.log(c) - x - (c+1.0)*sc.log1p(np.exp(-x))
def _cdf(self, x, c):
Cx = (1+np.exp(-x))**(-c)
return Cx
def _ppf(self, q, c):
vals = -np.log(pow(q, -1.0/c)-1)
return vals
def _stats(self, c):
mu = _EULER + sc.psi(c)
mu2 = np.pi*np.pi/6.0 + sc.zeta(2, c)
g1 = -2*sc.zeta(3, c) + 2*_ZETA3
g1 /= np.power(mu2, 1.5)
g2 = np.pi**4/15.0 + 6*sc.zeta(4, c)
g2 /= mu2**2.0
return mu, mu2, g1, g2
genlogistic = genlogistic_gen(name='genlogistic')
class genpareto_gen(rv_continuous):
r"""A generalized Pareto continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `genpareto` is:
.. math::
f(x, c) = (1 + c x)^{-1 - 1/c}
defined for :math:`x \ge 0` if :math:`c \ge 0`, and for
:math:`0 \le x \le -1/c` if :math:`c < 0`.
`genpareto` takes :math:`c` as a shape parameter.
For ``c == 0``, `genpareto` reduces to the exponential
distribution, `expon`:
.. math::
f(x, c=0) = \exp(-x)
For ``c == -1``, `genpareto` is uniform on ``[0, 1]``:
.. math::
f(x, c=-1) = x
%(after_notes)s
%(example)s
"""
def _argcheck(self, c):
c = np.asarray(c)
self.b = _lazywhere(c < 0, (c,),
lambda c: -1. / c,
np.inf)
return True
def _pdf(self, x, c):
# genpareto.pdf(x, c) = (1 + c * x)**(-1 - 1/c)
return np.exp(self._logpdf(x, c))
def _logpdf(self, x, c):
return _lazywhere((x == x) & (c != 0), (x, c),
lambda x, c: -sc.xlog1py(c + 1., c*x) / c,
-x)
def _cdf(self, x, c):
return -sc.inv_boxcox1p(-x, -c)
def _sf(self, x, c):
return sc.inv_boxcox(-x, -c)
def _logsf(self, x, c):
return _lazywhere((x == x) & (c != 0), (x, c),
lambda x, c: -sc.log1p(c*x) / c,
-x)
def _ppf(self, q, c):
return -sc.boxcox1p(-q, -c)
def _isf(self, q, c):
return -sc.boxcox(q, -c)
def _munp(self, n, c):
def __munp(n, c):
val = 0.0
k = np.arange(0, n + 1)
for ki, cnk in zip(k, sc.comb(n, k)):
val = val + cnk * (-1) ** ki / (1.0 - c * ki)
return np.where(c * n < 1, val * (-1.0 / c) ** n, np.inf)
return _lazywhere(c != 0, (c,),
lambda c: __munp(n, c),
sc.gamma(n + 1))
def _entropy(self, c):
return 1. + c
genpareto = genpareto_gen(a=0.0, name='genpareto')
class genexpon_gen(rv_continuous):
r"""A generalized exponential continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `genexpon` is:
.. math::
f(x, a, b, c) = (a + b (1 - \exp(-c x)))
\exp(-a x - b x + \frac{b}{c} (1-\exp(-c x)))
for :math:`x \ge 0`, :math:`a, b, c > 0`.
`genexpon` takes :math:`a`, :math:`b` and :math:`c` as shape parameters.
%(after_notes)s
References
----------
H.K. Ryu, "An Extension of Marshall and Olkin's Bivariate Exponential
Distribution", Journal of the American Statistical Association, 1993.
N. Balakrishnan, "The Exponential Distribution: Theory, Methods and
Applications", Asit P. Basu.
%(example)s
"""
def _pdf(self, x, a, b, c):
# genexpon.pdf(x, a, b, c) = (a + b * (1 - exp(-c*x))) * \
# exp(-a*x - b*x + b/c * (1-exp(-c*x)))
return (a + b*(-sc.expm1(-c*x)))*np.exp((-a-b)*x +
b*(-sc.expm1(-c*x))/c)
def _cdf(self, x, a, b, c):
return -sc.expm1((-a-b)*x + b*(-sc.expm1(-c*x))/c)
def _logpdf(self, x, a, b, c):
return np.log(a+b*(-sc.expm1(-c*x))) + (-a-b)*x+b*(-sc.expm1(-c*x))/c
genexpon = genexpon_gen(a=0.0, name='genexpon')
class genextreme_gen(rv_continuous):
r"""A generalized extreme value continuous random variable.
%(before_notes)s
See Also
--------
gumbel_r
Notes
-----
For :math:`c=0`, `genextreme` is equal to `gumbel_r`.
The probability density function for `genextreme` is:
.. math::
f(x, c) = \begin{cases}
\exp(-\exp(-x)) \exp(-x) &\text{for } c = 0\\
\exp(-(1-c x)^{1/c}) (1-c x)^{1/c-1} &\text{for }
x \le 1/c, c > 0
\end{cases}
Note that several sources and software packages use the opposite
convention for the sign of the shape parameter :math:`c`.
`genextreme` takes :math:`c` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _argcheck(self, c):
self.b = np.where(c > 0, 1.0 / np.maximum(c, _XMIN), np.inf)
self.a = np.where(c < 0, 1.0 / np.minimum(c, -_XMIN), -np.inf)
return np.where(abs(c) == np.inf, 0, 1)
def _loglogcdf(self, x, c):
return _lazywhere((x == x) & (c != 0), (x, c),
lambda x, c: sc.log1p(-c*x)/c, -x)
def _pdf(self, x, c):
# genextreme.pdf(x, c) =
# exp(-exp(-x))*exp(-x), for c==0
# exp(-(1-c*x)**(1/c))*(1-c*x)**(1/c-1), for x \le 1/c, c > 0
return np.exp(self._logpdf(x, c))
def _logpdf(self, x, c):
cx = _lazywhere((x == x) & (c != 0), (x, c), lambda x, c: c*x, 0.0)
logex2 = sc.log1p(-cx)
logpex2 = self._loglogcdf(x, c)
pex2 = np.exp(logpex2)
# Handle special cases
np.putmask(logpex2, (c == 0) & (x == -np.inf), 0.0)
logpdf = np.where((cx == 1) | (cx == -np.inf),
-np.inf,
-pex2+logpex2-logex2)
np.putmask(logpdf, (c == 1) & (x == 1), 0.0)
return logpdf
def _logcdf(self, x, c):
return -np.exp(self._loglogcdf(x, c))
def _cdf(self, x, c):
return np.exp(self._logcdf(x, c))
def _sf(self, x, c):
return -sc.expm1(self._logcdf(x, c))
def _ppf(self, q, c):
x = -np.log(-np.log(q))
return _lazywhere((x == x) & (c != 0), (x, c),
lambda x, c: -sc.expm1(-c * x) / c, x)
def _isf(self, q, c):
x = -np.log(-sc.log1p(-q))
return _lazywhere((x == x) & (c != 0), (x, c),
lambda x, c: -sc.expm1(-c * x) / c, x)
def _stats(self, c):
g = lambda n: sc.gamma(n*c + 1)
g1 = g(1)
g2 = g(2)
g3 = g(3)
g4 = g(4)
g2mg12 = np.where(abs(c) < 1e-7, (c*np.pi)**2.0/6.0, g2-g1**2.0)
gam2k = np.where(abs(c) < 1e-7, np.pi**2.0/6.0,
sc.expm1(sc.gammaln(2.0*c+1.0)-2*sc.gammaln(c + 1.0))/c**2.0)
eps = 1e-14
gamk = np.where(abs(c) < eps, -_EULER, sc.expm1(sc.gammaln(c + 1))/c)
m = np.where(c < -1.0, np.nan, -gamk)
v = np.where(c < -0.5, np.nan, g1**2.0*gam2k)
# skewness
sk1 = _lazywhere(c >= -1./3,
(c, g1, g2, g3, g2mg12),
lambda c, g1, g2, g3, g2gm12:
np.sign(c)*(-g3 + (g2 + 2*g2mg12)*g1)/g2mg12**1.5,
fillvalue=np.nan)
sk = np.where(abs(c) <= eps**0.29, 12*np.sqrt(6)*_ZETA3/np.pi**3, sk1)
# kurtosis
ku1 = _lazywhere(c >= -1./4,
(g1, g2, g3, g4, g2mg12),
lambda g1, g2, g3, g4, g2mg12:
(g4 + (-4*g3 + 3*(g2 + g2mg12)*g1)*g1)/g2mg12**2,
fillvalue=np.nan)
ku = np.where(abs(c) <= (eps)**0.23, 12.0/5.0, ku1-3.0)
return m, v, sk, ku
def _fitstart(self, data):
# This is better than the default shape of (1,).
g = _skew(data)
if g < 0:
a = 0.5
else:
a = -0.5
return super(genextreme_gen, self)._fitstart(data, args=(a,))
def _munp(self, n, c):
k = np.arange(0, n+1)
vals = 1.0/c**n * np.sum(
sc.comb(n, k) * (-1)**k * sc.gamma(c*k + 1),
axis=0)
return np.where(c*n > -1, vals, np.inf)
def _entropy(self, c):
return _EULER*(1 - c) + 1
genextreme = genextreme_gen(name='genextreme')
def _digammainv(y):
# Inverse of the digamma function (real positive arguments only).
# This function is used in the `fit` method of `gamma_gen`.
# The function uses either optimize.fsolve or optimize.newton
# to solve `sc.digamma(x) - y = 0`. There is probably room for
# improvement, but currently it works over a wide range of y:
# >>> y = 64*np.random.randn(1000000)
# >>> y.min(), y.max()
# (-311.43592651416662, 351.77388222276869)
# x = [_digammainv(t) for t in y]
# np.abs(sc.digamma(x) - y).max()
# 1.1368683772161603e-13
#
_em = 0.5772156649015328606065120
func = lambda x: sc.digamma(x) - y
if y > -0.125:
x0 = np.exp(y) + 0.5
if y < 10:
# Some experimentation shows that newton reliably converges
# must faster than fsolve in this y range. For larger y,
# newton sometimes fails to converge.
value = optimize.newton(func, x0, tol=1e-10)
return value
elif y > -3:
x0 = np.exp(y/2.332) + 0.08661
else:
x0 = 1.0 / (-y - _em)
value, info, ier, mesg = optimize.fsolve(func, x0, xtol=1e-11,
full_output=True)
if ier != 1:
raise RuntimeError("_digammainv: fsolve failed, y = %r" % y)
return value[0]
## Gamma (Use MATLAB and MATHEMATICA (b=theta=scale, a=alpha=shape) definition)
## gamma(a, loc, scale) with a an integer is the Erlang distribution
## gamma(1, loc, scale) is the Exponential distribution
## gamma(df/2, 0, 2) is the chi2 distribution with df degrees of freedom.
class gamma_gen(rv_continuous):
r"""A gamma continuous random variable.
%(before_notes)s
See Also
--------
erlang, expon
Notes
-----
The probability density function for `gamma` is:
.. math::
f(x, a) = \frac{x^{a-1} \exp(-x)}{\Gamma(a)}
for :math:`x \ge 0`, :math:`a > 0`. Here :math:`\Gamma(a)` refers to the
gamma function.
`gamma` has a shape parameter `a` which needs to be set explicitly.
When :math:`a` is an integer, `gamma` reduces to the Erlang
distribution, and when :math:`a=1` to the exponential distribution.
%(after_notes)s
%(example)s
"""
def _rvs(self, a):
return self._random_state.standard_gamma(a, self._size)
def _pdf(self, x, a):
# gamma.pdf(x, a) = x**(a-1) * exp(-x) / gamma(a)
return np.exp(self._logpdf(x, a))
def _logpdf(self, x, a):
return sc.xlogy(a-1.0, x) - x - sc.gammaln(a)
def _cdf(self, x, a):
return sc.gammainc(a, x)
def _sf(self, x, a):
return sc.gammaincc(a, x)
def _ppf(self, q, a):
return sc.gammaincinv(a, q)
def _stats(self, a):
return a, a, 2.0/np.sqrt(a), 6.0/a
def _entropy(self, a):
return sc.psi(a)*(1-a) + a + sc.gammaln(a)
def _fitstart(self, data):
# The skewness of the gamma distribution is `4 / np.sqrt(a)`.
# We invert that to estimate the shape `a` using the skewness
# of the data. The formula is regularized with 1e-8 in the
# denominator to allow for degenerate data where the skewness
# is close to 0.
a = 4 / (1e-8 + _skew(data)**2)
return super(gamma_gen, self)._fitstart(data, args=(a,))
@extend_notes_in_docstring(rv_continuous, notes="""\
When the location is fixed by using the argument `floc`, this
function uses explicit formulas or solves a simpler numerical
problem than the full ML optimization problem. So in that case,
the `optimizer`, `loc` and `scale` arguments are ignored.\n\n""")
def fit(self, data, *args, **kwds):
f0 = (kwds.get('f0', None) or kwds.get('fa', None) or
kwds.get('fix_a', None))
floc = kwds.get('floc', None)
fscale = kwds.get('fscale', None)
if floc is None:
# loc is not fixed. Use the default fit method.
return super(gamma_gen, self).fit(data, *args, **kwds)
# Special case: loc is fixed.
if f0 is not None and fscale is not None:
# This check is for consistency with `rv_continuous.fit`.
# Without this check, this function would just return the
# parameters that were given.
raise ValueError("All parameters fixed. There is nothing to "
"optimize.")
# Fixed location is handled by shifting the data.
data = np.asarray(data)
if np.any(data <= floc):
raise FitDataError("gamma", lower=floc, upper=np.inf)
if floc != 0:
# Don't do the subtraction in-place, because `data` might be a
# view of the input array.
data = data - floc
xbar = data.mean()
# Three cases to handle:
# * shape and scale both free
# * shape fixed, scale free
# * shape free, scale fixed
if fscale is None:
# scale is free
if f0 is not None:
# shape is fixed
a = f0
else:
# shape and scale are both free.
# The MLE for the shape parameter `a` is the solution to:
# np.log(a) - sc.digamma(a) - np.log(xbar) +
# np.log(data.mean) = 0
s = np.log(xbar) - np.log(data).mean()
func = lambda a: np.log(a) - sc.digamma(a) - s
aest = (3-s + np.sqrt((s-3)**2 + 24*s)) / (12*s)
xa = aest*(1-0.4)
xb = aest*(1+0.4)
a = optimize.brentq(func, xa, xb, disp=0)
# The MLE for the scale parameter is just the data mean
# divided by the shape parameter.
scale = xbar / a
else:
# scale is fixed, shape is free
# The MLE for the shape parameter `a` is the solution to:
# sc.digamma(a) - np.log(data).mean() + np.log(fscale) = 0
c = np.log(data).mean() - np.log(fscale)
a = _digammainv(c)
scale = fscale
return a, floc, scale
gamma = gamma_gen(a=0.0, name='gamma')
class erlang_gen(gamma_gen):
"""An Erlang continuous random variable.
%(before_notes)s
See Also
--------
gamma
Notes
-----
The Erlang distribution is a special case of the Gamma distribution, with
the shape parameter `a` an integer. Note that this restriction is not
enforced by `erlang`. It will, however, generate a warning the first time
a non-integer value is used for the shape parameter.
Refer to `gamma` for examples.
"""
def _argcheck(self, a):
allint = np.all(np.floor(a) == a)
allpos = np.all(a > 0)
if not allint:
# An Erlang distribution shouldn't really have a non-integer
# shape parameter, so warn the user.
warnings.warn(
'The shape parameter of the erlang distribution '
'has been given a non-integer value %r.' % (a,),
RuntimeWarning)
return allpos
def _fitstart(self, data):
# Override gamma_gen_fitstart so that an integer initial value is
# used. (Also regularize the division, to avoid issues when
# _skew(data) is 0 or close to 0.)
a = int(4.0 / (1e-8 + _skew(data)**2))
return super(gamma_gen, self)._fitstart(data, args=(a,))
# Trivial override of the fit method, so we can monkey-patch its
# docstring.
def fit(self, data, *args, **kwds):
return super(erlang_gen, self).fit(data, *args, **kwds)
if fit.__doc__ is not None:
fit.__doc__ = (rv_continuous.fit.__doc__ +
"""
Notes
-----
The Erlang distribution is generally defined to have integer values
for the shape parameter. This is not enforced by the `erlang` class.
When fitting the distribution, it will generally return a non-integer
value for the shape parameter. By using the keyword argument
`f0=<integer>`, the fit method can be constrained to fit the data to
a specific integer shape parameter.
""")
erlang = erlang_gen(a=0.0, name='erlang')
class gengamma_gen(rv_continuous):
r"""A generalized gamma continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `gengamma` is:
.. math::
f(x, a, c) = \frac{|c| x^{c a-1} \exp(-x^c)}{\gamma(a)}
for :math:`x \ge 0`, :math:`a > 0`, and :math:`c \ne 0`.
`gengamma` takes :math:`a` and :math:`c` as shape parameters.
%(after_notes)s
%(example)s
"""
def _argcheck(self, a, c):
return (a > 0) & (c != 0)
def _pdf(self, x, a, c):
# gengamma.pdf(x, a, c) = abs(c) * x**(c*a-1) * exp(-x**c) / gamma(a)
return np.exp(self._logpdf(x, a, c))
def _logpdf(self, x, a, c):
return np.log(abs(c)) + sc.xlogy(c*a - 1, x) - x**c - sc.gammaln(a)
def _cdf(self, x, a, c):
xc = x**c
val1 = sc.gammainc(a, xc)
val2 = sc.gammaincc(a, xc)
return np.where(c > 0, val1, val2)
def _sf(self, x, a, c):
xc = x**c
val1 = sc.gammainc(a, xc)
val2 = sc.gammaincc(a, xc)
return np.where(c > 0, val2, val1)
def _ppf(self, q, a, c):
val1 = sc.gammaincinv(a, q)
val2 = sc.gammainccinv(a, q)
return np.where(c > 0, val1, val2)**(1.0/c)
def _isf(self, q, a, c):
val1 = sc.gammaincinv(a, q)
val2 = sc.gammainccinv(a, q)
return np.where(c > 0, val2, val1)**(1.0/c)
def _munp(self, n, a, c):
# Pochhammer symbol: sc.pocha,n) = gamma(a+n)/gamma(a)
return sc.poch(a, n*1.0/c)
def _entropy(self, a, c):
val = sc.psi(a)
return a*(1-val) + 1.0/c*val + sc.gammaln(a) - np.log(abs(c))
gengamma = gengamma_gen(a=0.0, name='gengamma')
class genhalflogistic_gen(rv_continuous):
r"""A generalized half-logistic continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `genhalflogistic` is:
.. math::
f(x, c) = \frac{2 (1 - c x)^{1/(c-1)}}{[1 + (1 - c x)^{1/c}]^2}
for :math:`0 \le x \le 1/c`, and :math:`c > 0`.
`genhalflogistic` takes :math:`c` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _argcheck(self, c):
self.b = 1.0 / c
return c > 0
def _pdf(self, x, c):
# genhalflogistic.pdf(x, c) =
# 2 * (1-c*x)**(1/c-1) / (1+(1-c*x)**(1/c))**2
limit = 1.0/c
tmp = np.asarray(1-c*x)
tmp0 = tmp**(limit-1)
tmp2 = tmp0*tmp
return 2*tmp0 / (1+tmp2)**2
def _cdf(self, x, c):
limit = 1.0/c
tmp = np.asarray(1-c*x)
tmp2 = tmp**(limit)
return (1.0-tmp2) / (1+tmp2)
def _ppf(self, q, c):
return 1.0/c*(1-((1.0-q)/(1.0+q))**c)
def _entropy(self, c):
return 2 - (2*c+1)*np.log(2)
genhalflogistic = genhalflogistic_gen(a=0.0, name='genhalflogistic')
class gompertz_gen(rv_continuous):
r"""A Gompertz (or truncated Gumbel) continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `gompertz` is:
.. math::
f(x, c) = c \exp(x) \exp(-c (e^x-1))
for :math:`x \ge 0`, :math:`c > 0`.
`gompertz` takes :math:`c` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _pdf(self, x, c):
# gompertz.pdf(x, c) = c * exp(x) * exp(-c*(exp(x)-1))
return np.exp(self._logpdf(x, c))
def _logpdf(self, x, c):
return np.log(c) + x - c * sc.expm1(x)
def _cdf(self, x, c):
return -sc.expm1(-c * sc.expm1(x))
def _ppf(self, q, c):
return sc.log1p(-1.0 / c * sc.log1p(-q))
def _entropy(self, c):
return 1.0 - np.log(c) - np.exp(c)*sc.expn(1, c)
gompertz = gompertz_gen(a=0.0, name='gompertz')
class gumbel_r_gen(rv_continuous):
r"""A right-skewed Gumbel continuous random variable.
%(before_notes)s
See Also
--------
gumbel_l, gompertz, genextreme
Notes
-----
The probability density function for `gumbel_r` is:
.. math::
f(x) = \exp(-(x + e^{-x}))
The Gumbel distribution is sometimes referred to as a type I Fisher-Tippett
distribution. It is also related to the extreme value distribution,
log-Weibull and Gompertz distributions.
%(after_notes)s
%(example)s
"""
def _pdf(self, x):
# gumbel_r.pdf(x) = exp(-(x + exp(-x)))
return np.exp(self._logpdf(x))
def _logpdf(self, x):
return -x - np.exp(-x)
def _cdf(self, x):
return np.exp(-np.exp(-x))
def _logcdf(self, x):
return -np.exp(-x)
def _ppf(self, q):
return -np.log(-np.log(q))
def _stats(self):
return _EULER, np.pi*np.pi/6.0, 12*np.sqrt(6)/np.pi**3 * _ZETA3, 12.0/5
def _entropy(self):
# http://en.wikipedia.org/wiki/Gumbel_distribution
return _EULER + 1.
gumbel_r = gumbel_r_gen(name='gumbel_r')
class gumbel_l_gen(rv_continuous):
r"""A left-skewed Gumbel continuous random variable.
%(before_notes)s
See Also
--------
gumbel_r, gompertz, genextreme
Notes
-----
The probability density function for `gumbel_l` is:
.. math::
f(x) = \exp(x - e^x)
The Gumbel distribution is sometimes referred to as a type I Fisher-Tippett
distribution. It is also related to the extreme value distribution,
log-Weibull and Gompertz distributions.
%(after_notes)s
%(example)s
"""
def _pdf(self, x):
# gumbel_l.pdf(x) = exp(x - exp(x))
return np.exp(self._logpdf(x))
def _logpdf(self, x):
return x - np.exp(x)
def _cdf(self, x):
return -sc.expm1(-np.exp(x))
def _ppf(self, q):
return np.log(-sc.log1p(-q))
def _logsf(self, x):
return -np.exp(x)
def _sf(self, x):
return np.exp(-np.exp(x))
def _isf(self, x):
return np.log(-np.log(x))
def _stats(self):
return -_EULER, np.pi*np.pi/6.0, \
-12*np.sqrt(6)/np.pi**3 * _ZETA3, 12.0/5
def _entropy(self):
return _EULER + 1.
gumbel_l = gumbel_l_gen(name='gumbel_l')
class halfcauchy_gen(rv_continuous):
r"""A Half-Cauchy continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `halfcauchy` is:
.. math::
f(x) = \frac{2}{\pi (1 + x^2)}
for :math:`x \ge 0`.
%(after_notes)s
%(example)s
"""
def _pdf(self, x):
# halfcauchy.pdf(x) = 2 / (pi * (1 + x**2))
return 2.0/np.pi/(1.0+x*x)
def _logpdf(self, x):
return np.log(2.0/np.pi) - sc.log1p(x*x)
def _cdf(self, x):
return 2.0/np.pi*np.arctan(x)
def _ppf(self, q):
return np.tan(np.pi/2*q)
def _stats(self):
return np.inf, np.inf, np.nan, np.nan
def _entropy(self):
return np.log(2*np.pi)
halfcauchy = halfcauchy_gen(a=0.0, name='halfcauchy')
class halflogistic_gen(rv_continuous):
r"""A half-logistic continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `halflogistic` is:
.. math::
f(x) = \frac{ 2 e^{-x} }{ (1+e^{-x})^2 } = \frac{1}{2} sech(x/2)^2
for :math:`x \ge 0`.
%(after_notes)s
%(example)s
"""
def _pdf(self, x):
# halflogistic.pdf(x) = 2 * exp(-x) / (1+exp(-x))**2
# = 1/2 * sech(x/2)**2
return np.exp(self._logpdf(x))
def _logpdf(self, x):
return np.log(2) - x - 2. * sc.log1p(np.exp(-x))
def _cdf(self, x):
return np.tanh(x/2.0)
def _ppf(self, q):
return 2*np.arctanh(q)
def _munp(self, n):
if n == 1:
return 2*np.log(2)
if n == 2:
return np.pi*np.pi/3.0
if n == 3:
return 9*_ZETA3
if n == 4:
return 7*np.pi**4 / 15.0
return 2*(1-pow(2.0, 1-n))*sc.gamma(n+1)*sc.zeta(n, 1)
def _entropy(self):
return 2-np.log(2)
halflogistic = halflogistic_gen(a=0.0, name='halflogistic')
class halfnorm_gen(rv_continuous):
r"""A half-normal continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `halfnorm` is:
.. math::
f(x) = \sqrt{2/\pi} e^{-\frac{x^2}{2}}
for :math:`x > 0`.
`halfnorm` is a special case of :math`\chi` with ``df == 1``.
%(after_notes)s
%(example)s
"""
def _rvs(self):
return abs(self._random_state.standard_normal(size=self._size))
def _pdf(self, x):
# halfnorm.pdf(x) = sqrt(2/pi) * exp(-x**2/2)
return np.sqrt(2.0/np.pi)*np.exp(-x*x/2.0)
def _logpdf(self, x):
return 0.5 * np.log(2.0/np.pi) - x*x/2.0
def _cdf(self, x):
return _norm_cdf(x)*2-1.0
def _ppf(self, q):
return sc.ndtri((1+q)/2.0)
def _stats(self):
return (np.sqrt(2.0/np.pi),
1-2.0/np.pi,
np.sqrt(2)*(4-np.pi)/(np.pi-2)**1.5,
8*(np.pi-3)/(np.pi-2)**2)
def _entropy(self):
return 0.5*np.log(np.pi/2.0)+0.5
halfnorm = halfnorm_gen(a=0.0, name='halfnorm')
class hypsecant_gen(rv_continuous):
r"""A hyperbolic secant continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `hypsecant` is:
.. math::
f(x) = \frac{1}{\pi} sech(x)
%(after_notes)s
%(example)s
"""
def _pdf(self, x):
# hypsecant.pdf(x) = 1/pi * sech(x)
return 1.0/(np.pi*np.cosh(x))
def _cdf(self, x):
return 2.0/np.pi*np.arctan(np.exp(x))
def _ppf(self, q):
return np.log(np.tan(np.pi*q/2.0))
def _stats(self):
return 0, np.pi*np.pi/4, 0, 2
def _entropy(self):
return np.log(2*np.pi)
hypsecant = hypsecant_gen(name='hypsecant')
class gausshyper_gen(rv_continuous):
r"""A Gauss hypergeometric continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `gausshyper` is:
.. math::
f(x, a, b, c, z) = C x^{a-1} (1-x)^{b-1} (1+zx)^{-c}
for :math:`0 \le x \le 1`, :math:`a > 0`, :math:`b > 0`, and
:math:`C = \frac{1}{B(a, b) F[2, 1](c, a; a+b; -z)}`
`gausshyper` takes :math:`a`, :math:`b`, :math:`c` and :math:`z` as shape
parameters.
%(after_notes)s
%(example)s
"""
def _argcheck(self, a, b, c, z):
return (a > 0) & (b > 0) & (c == c) & (z == z)
def _pdf(self, x, a, b, c, z):
# gausshyper.pdf(x, a, b, c, z) =
# C * x**(a-1) * (1-x)**(b-1) * (1+z*x)**(-c)
Cinv = sc.gamma(a)*sc.gamma(b)/sc.gamma(a+b)*sc.hyp2f1(c, a, a+b, -z)
return 1.0/Cinv * x**(a-1.0) * (1.0-x)**(b-1.0) / (1.0+z*x)**c
def _munp(self, n, a, b, c, z):
fac = sc.beta(n+a, b) / sc.beta(a, b)
num = sc.hyp2f1(c, a+n, a+b+n, -z)
den = sc.hyp2f1(c, a, a+b, -z)
return fac*num / den
gausshyper = gausshyper_gen(a=0.0, b=1.0, name='gausshyper')
class invgamma_gen(rv_continuous):
r"""An inverted gamma continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `invgamma` is:
.. math::
f(x, a) = \frac{x^{-a-1}}{\gamma(a)} \exp(-\frac{1}{x})
for :math:`x > 0`, :math:`a > 0`.
`invgamma` takes :math:`a` as a shape parameter.
`invgamma` is a special case of `gengamma` with ``c == -1``.
%(after_notes)s
%(example)s
"""
_support_mask = rv_continuous._open_support_mask
def _pdf(self, x, a):
# invgamma.pdf(x, a) = x**(-a-1) / gamma(a) * exp(-1/x)
return np.exp(self._logpdf(x, a))
def _logpdf(self, x, a):
return -(a+1) * np.log(x) - sc.gammaln(a) - 1.0/x
def _cdf(self, x, a):
return sc.gammaincc(a, 1.0 / x)
def _ppf(self, q, a):
return 1.0 / sc.gammainccinv(a, q)
def _sf(self, x, a):
return sc.gammainc(a, 1.0 / x)
def _isf(self, q, a):
return 1.0 / sc.gammaincinv(a, q)
def _stats(self, a, moments='mvsk'):
m1 = _lazywhere(a > 1, (a,), lambda x: 1. / (x - 1.), np.inf)
m2 = _lazywhere(a > 2, (a,), lambda x: 1. / (x - 1.)**2 / (x - 2.),
np.inf)
g1, g2 = None, None
if 's' in moments:
g1 = _lazywhere(
a > 3, (a,),
lambda x: 4. * np.sqrt(x - 2.) / (x - 3.), np.nan)
if 'k' in moments:
g2 = _lazywhere(
a > 4, (a,),
lambda x: 6. * (5. * x - 11.) / (x - 3.) / (x - 4.), np.nan)
return m1, m2, g1, g2
def _entropy(self, a):
return a - (a+1.0) * sc.psi(a) + sc.gammaln(a)
invgamma = invgamma_gen(a=0.0, name='invgamma')
# scale is gamma from DATAPLOT and B from Regress
class invgauss_gen(rv_continuous):
r"""An inverse Gaussian continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `invgauss` is:
.. math::
f(x, \mu) = \frac{1}{\sqrt{2 \pi x^3}}
\exp(-\frac{(x-\mu)^2}{2 x \mu^2})
for :math:`x > 0`.
`invgauss` takes :math:`\mu` as a shape parameter.
%(after_notes)s
When :math:`\mu` is too small, evaluating the cumulative distribution
function will be inaccurate due to ``cdf(mu -> 0) = inf * 0``.
NaNs are returned for :math:`\mu \le 0.0028`.
%(example)s
"""
_support_mask = rv_continuous._open_support_mask
def _rvs(self, mu):
return self._random_state.wald(mu, 1.0, size=self._size)
def _pdf(self, x, mu):
# invgauss.pdf(x, mu) =
# 1 / sqrt(2*pi*x**3) * exp(-(x-mu)**2/(2*x*mu**2))
return 1.0/np.sqrt(2*np.pi*x**3.0)*np.exp(-1.0/(2*x)*((x-mu)/mu)**2)
def _logpdf(self, x, mu):
return -0.5*np.log(2*np.pi) - 1.5*np.log(x) - ((x-mu)/mu)**2/(2*x)
def _cdf(self, x, mu):
fac = np.sqrt(1.0/x)
# Numerical accuracy for small `mu` is bad. See #869.
C1 = _norm_cdf(fac*(x-mu)/mu)
C1 += np.exp(1.0/mu) * _norm_cdf(-fac*(x+mu)/mu) * np.exp(1.0/mu)
return C1
def _stats(self, mu):
return mu, mu**3.0, 3*np.sqrt(mu), 15*mu
invgauss = invgauss_gen(a=0.0, name='invgauss')
class norminvgauss_gen(rv_continuous):
r"""A Normal Inverse Gaussian continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `norminvgauss` is:
.. math::
f(x; a, b) = (a \exp(\sqrt{a^2 - b^2} + b x)) /
(\pi \sqrt{1 + x^2} \, K_1(a * \sqrt{1 + x^2}))
where `x` is a real number, the parameter `a` is the tail heaviness
and `b` is the asymmetry parameter satisfying `a > 0` and `abs(b) <= a`.
`K_1` is the modified Bessel function of second kind (`scipy.special.k1`).
%(after_notes)s
A normal inverse Gaussian random variable with parameters `a` and `b` can
be expressed as `X = b * V + sqrt(V) * X` where `X` is `norm(0,1)`
and `V` is `invgauss(mu=1/sqrt(a**2 - b**2))`. This representation is used
to generate random variates.
References
----------
O. Barndorff-Nielsen, "Hyperbolic Distributions and Distributions on
Hyperbolae", Scandinavian Journal of Statistics, Vol. 5(3),
pp. 151-157, 1978.
O. Barndorff-Nielsen, "Normal Inverse Gaussian Distributions and Stochastic
Volatility Modelling", Scandinavian Journal of Statistics, Vol. 24,
pp. 1–13, 1997.
%(example)s
"""
_support_mask = rv_continuous._open_support_mask
def _argcheck(self, a, b):
return (a > 0) & (np.absolute(b) < a)
def _pdf(self, x, a, b):
gamma = np.sqrt(a**2 - b**2)
fac1 = a / np.pi * np.exp(gamma)
sq = np.hypot(1, x) # reduce overflows
return fac1 * sc.k1e(a * sq) * np.exp(b*x - a*sq) / sq
def _rvs(self, a, b):
# note: X = b * V + sqrt(V) * X is norminvgaus(a,b) if X is standard
# normal and V is invgauss(mu=1/sqrt(a**2 - b**2))
gamma = np.sqrt(a**2 - b**2)
sz, rndm = self._size, self._random_state
ig = invgauss.rvs(mu=1/gamma, size=sz, random_state=rndm)
return b * ig + np.sqrt(ig) * norm.rvs(size=sz, random_state=rndm)
def _stats(self, a, b):
gamma = np.sqrt(a**2 - b**2)
mean = b / gamma
variance = a**2 / gamma**3
skewness = 3.0 * b / (a * np.sqrt(gamma))
kurtosis = 3.0 * (1 + 4 * b**2 / a**2) / gamma
return mean, variance, skewness, kurtosis
norminvgauss = norminvgauss_gen(name="norminvgauss")
class invweibull_gen(rv_continuous):
r"""An inverted Weibull continuous random variable.
This distribution is also known as the Fréchet distribution or the
type II extreme value distribution.
%(before_notes)s
Notes
-----
The probability density function for `invweibull` is:
.. math::
f(x, c) = c x^{-c-1} \exp(-x^{-c})
for :math:`x > 0``, :math:`c > 0``.
`invweibull` takes :math:`c`` as a shape parameter.
%(after_notes)s
References
----------
F.R.S. de Gusmao, E.M.M Ortega and G.M. Cordeiro, "The generalized inverse
Weibull distribution", Stat. Papers, vol. 52, pp. 591-619, 2011.
%(example)s
"""
_support_mask = rv_continuous._open_support_mask
def _pdf(self, x, c):
# invweibull.pdf(x, c) = c * x**(-c-1) * exp(-x**(-c))
xc1 = np.power(x, -c - 1.0)
xc2 = np.power(x, -c)
xc2 = np.exp(-xc2)
return c * xc1 * xc2
def _cdf(self, x, c):
xc1 = np.power(x, -c)
return np.exp(-xc1)
def _ppf(self, q, c):
return np.power(-np.log(q), -1.0/c)
def _munp(self, n, c):
return sc.gamma(1 - n / c)
def _entropy(self, c):
return 1+_EULER + _EULER / c - np.log(c)
invweibull = invweibull_gen(a=0, name='invweibull')
class johnsonsb_gen(rv_continuous):
r"""A Johnson SB continuous random variable.
%(before_notes)s
See Also
--------
johnsonsu
Notes
-----
The probability density function for `johnsonsb` is:
.. math::
f(x, a, b) = \frac{b}{x(1-x)} \phi(a + b \log \frac{x}{1-x} )
for :math:`0 < x < 1` and :math:`a, b > 0`, and :math:`\phi` is the normal
pdf.
`johnsonsb` takes :math:`a` and :math:`b` as shape parameters.
%(after_notes)s
%(example)s
"""
_support_mask = rv_continuous._open_support_mask
def _argcheck(self, a, b):
return (b > 0) & (a == a)
def _pdf(self, x, a, b):
# johnsonsb.pdf(x, a, b) = b / (x*(1-x)) * phi(a + b * log(x/(1-x)))
trm = _norm_pdf(a + b*np.log(x/(1.0-x)))
return b*1.0/(x*(1-x))*trm
def _cdf(self, x, a, b):
return _norm_cdf(a + b*np.log(x/(1.0-x)))
def _ppf(self, q, a, b):
return 1.0 / (1 + np.exp(-1.0 / b * (_norm_ppf(q) - a)))
johnsonsb = johnsonsb_gen(a=0.0, b=1.0, name='johnsonsb')
class johnsonsu_gen(rv_continuous):
r"""A Johnson SU continuous random variable.
%(before_notes)s
See Also
--------
johnsonsb
Notes
-----
The probability density function for `johnsonsu` is:
.. math::
f(x, a, b) = \frac{b}{\sqrt{x^2 + 1}}
\phi(a + b \log(x + \sqrt{x^2 + 1}))
for all :math:`x, a, b > 0`, and :math:`\phi` is the normal pdf.
`johnsonsu` takes :math:`a` and :math:`b` as shape parameters.
%(after_notes)s
%(example)s
"""
def _argcheck(self, a, b):
return (b > 0) & (a == a)
def _pdf(self, x, a, b):
# johnsonsu.pdf(x, a, b) = b / sqrt(x**2 + 1) *
# phi(a + b * log(x + sqrt(x**2 + 1)))
x2 = x*x
trm = _norm_pdf(a + b * np.log(x + np.sqrt(x2+1)))
return b*1.0/np.sqrt(x2+1.0)*trm
def _cdf(self, x, a, b):
return _norm_cdf(a + b * np.log(x + np.sqrt(x*x + 1)))
def _ppf(self, q, a, b):
return np.sinh((_norm_ppf(q) - a) / b)
johnsonsu = johnsonsu_gen(name='johnsonsu')
class laplace_gen(rv_continuous):
r"""A Laplace continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `laplace` is:
.. math::
f(x) = \frac{1}{2} \exp(-|x|)
%(after_notes)s
%(example)s
"""
def _rvs(self):
return self._random_state.laplace(0, 1, size=self._size)
def _pdf(self, x):
# laplace.pdf(x) = 1/2 * exp(-abs(x))
return 0.5*np.exp(-abs(x))
def _cdf(self, x):
return np.where(x > 0, 1.0-0.5*np.exp(-x), 0.5*np.exp(x))
def _ppf(self, q):
return np.where(q > 0.5, -np.log(2*(1-q)), np.log(2*q))
def _stats(self):
return 0, 2, 0, 3
def _entropy(self):
return np.log(2)+1
laplace = laplace_gen(name='laplace')
class levy_gen(rv_continuous):
r"""A Levy continuous random variable.
%(before_notes)s
See Also
--------
levy_stable, levy_l
Notes
-----
The probability density function for `levy` is:
.. math::
f(x) = \frac{1}{x \sqrt{2\pi x}) \exp(-\frac{1}{2x}}
for :math:`x > 0`.
This is the same as the Levy-stable distribution with :math:`a=1/2` and
:math:`b=1`.
%(after_notes)s
%(example)s
"""
_support_mask = rv_continuous._open_support_mask
def _pdf(self, x):
# levy.pdf(x) = 1 / (x * sqrt(2*pi*x)) * exp(-1/(2*x))
return 1 / np.sqrt(2*np.pi*x) / x * np.exp(-1/(2*x))
def _cdf(self, x):
# Equivalent to 2*norm.sf(np.sqrt(1/x))
return sc.erfc(np.sqrt(0.5 / x))
def _ppf(self, q):
# Equivalent to 1.0/(norm.isf(q/2)**2) or 0.5/(erfcinv(q)**2)
val = -sc.ndtri(q/2)
return 1.0 / (val * val)
def _stats(self):
return np.inf, np.inf, np.nan, np.nan
levy = levy_gen(a=0.0, name="levy")
class levy_l_gen(rv_continuous):
r"""A left-skewed Levy continuous random variable.
%(before_notes)s
See Also
--------
levy, levy_stable
Notes
-----
The probability density function for `levy_l` is:
.. math::
f(x) = \frac{1}{|x| \sqrt{2\pi |x|}} \exp(-\frac{1}{2 |x|})
for :math:`x < 0`.
This is the same as the Levy-stable distribution with :math:`a=1/2` and
:math:`b=-1`.
%(after_notes)s
%(example)s
"""
_support_mask = rv_continuous._open_support_mask
def _pdf(self, x):
# levy_l.pdf(x) = 1 / (abs(x) * sqrt(2*pi*abs(x))) * exp(-1/(2*abs(x)))
ax = abs(x)
return 1/np.sqrt(2*np.pi*ax)/ax*np.exp(-1/(2*ax))
def _cdf(self, x):
ax = abs(x)
return 2 * _norm_cdf(1 / np.sqrt(ax)) - 1
def _ppf(self, q):
val = _norm_ppf((q + 1.0) / 2)
return -1.0 / (val * val)
def _stats(self):
return np.inf, np.inf, np.nan, np.nan
levy_l = levy_l_gen(b=0.0, name="levy_l")
class levy_stable_gen(rv_continuous):
r"""A Levy-stable continuous random variable.
%(before_notes)s
See Also
--------
levy, levy_l
Notes
-----
Levy-stable distribution (only random variates available -- ignore other
docs)
"""
def _rvs(self, alpha, beta):
def alpha1func(alpha, beta, TH, aTH, bTH, cosTH, tanTH, W):
return (2/np.pi*(np.pi/2 + bTH)*tanTH -
beta*np.log((np.pi/2*W*cosTH)/(np.pi/2 + bTH)))
def beta0func(alpha, beta, TH, aTH, bTH, cosTH, tanTH, W):
return (W/(cosTH/np.tan(aTH) + np.sin(TH)) *
((np.cos(aTH) + np.sin(aTH)*tanTH)/W)**(1.0/alpha))
def otherwise(alpha, beta, TH, aTH, bTH, cosTH, tanTH, W):
# alpha is not 1 and beta is not 0
val0 = beta*np.tan(np.pi*alpha/2)
th0 = np.arctan(val0)/alpha
val3 = W/(cosTH/np.tan(alpha*(th0 + TH)) + np.sin(TH))
res3 = val3*((np.cos(aTH) + np.sin(aTH)*tanTH -
val0*(np.sin(aTH) - np.cos(aTH)*tanTH))/W)**(1.0/alpha)
return res3
def alphanot1func(alpha, beta, TH, aTH, bTH, cosTH, tanTH, W):
res = _lazywhere(beta == 0,
(alpha, beta, TH, aTH, bTH, cosTH, tanTH, W),
beta0func, f2=otherwise)
return res
sz = self._size
alpha = broadcast_to(alpha, sz)
beta = broadcast_to(beta, sz)
TH = uniform.rvs(loc=-np.pi/2.0, scale=np.pi, size=sz,
random_state=self._random_state)
W = expon.rvs(size=sz, random_state=self._random_state)
aTH = alpha*TH
bTH = beta*TH
cosTH = np.cos(TH)
tanTH = np.tan(TH)
res = _lazywhere(alpha == 1,
(alpha, beta, TH, aTH, bTH, cosTH, tanTH, W),
alpha1func, f2=alphanot1func)
return res
def _argcheck(self, alpha, beta):
return (alpha > 0) & (alpha <= 2) & (beta <= 1) & (beta >= -1)
def _pdf(self, x, alpha, beta):
raise NotImplementedError
levy_stable = levy_stable_gen(name='levy_stable')
class logistic_gen(rv_continuous):
r"""A logistic (or Sech-squared) continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `logistic` is:
.. math::
f(x) = \frac{\exp(-x)}
{(1+exp(-x))^2}
`logistic` is a special case of `genlogistic` with ``c == 1``.
%(after_notes)s
%(example)s
"""
def _rvs(self):
return self._random_state.logistic(size=self._size)
def _pdf(self, x):
# logistic.pdf(x) = exp(-x) / (1+exp(-x))**2
return np.exp(self._logpdf(x))
def _logpdf(self, x):
return -x - 2. * sc.log1p(np.exp(-x))
def _cdf(self, x):
return sc.expit(x)
def _ppf(self, q):
return sc.logit(q)
def _sf(self, x):
return sc.expit(-x)
def _isf(self, q):
return -sc.logit(q)
def _stats(self):
return 0, np.pi*np.pi/3.0, 0, 6.0/5.0
def _entropy(self):
# http://en.wikipedia.org/wiki/Logistic_distribution
return 2.0
logistic = logistic_gen(name='logistic')
class loggamma_gen(rv_continuous):
r"""A log gamma continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `loggamma` is:
.. math::
f(x, c) = \frac{\exp(c x - \exp(x))}
{\gamma(c)}
for all :math:`x, c > 0`.
`loggamma` takes :math:`c` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _rvs(self, c):
return np.log(self._random_state.gamma(c, size=self._size))
def _pdf(self, x, c):
# loggamma.pdf(x, c) = exp(c*x-exp(x)) / gamma(c)
return np.exp(c*x-np.exp(x)-sc.gammaln(c))
def _cdf(self, x, c):
return sc.gammainc(c, np.exp(x))
def _ppf(self, q, c):
return np.log(sc.gammaincinv(c, q))
def _stats(self, c):
# See, for example, "A Statistical Study of Log-Gamma Distribution", by
# Ping Shing Chan (thesis, McMaster University, 1993).
mean = sc.digamma(c)
var = sc.polygamma(1, c)
skewness = sc.polygamma(2, c) / np.power(var, 1.5)
excess_kurtosis = sc.polygamma(3, c) / (var*var)
return mean, var, skewness, excess_kurtosis
loggamma = loggamma_gen(name='loggamma')
class loglaplace_gen(rv_continuous):
r"""A log-Laplace continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `loglaplace` is:
.. math::
f(x, c) = \begin{cases}\frac{c}{2} x^{ c-1} &\text{for } 0 < x < 1\\
\frac{c}{2} x^{-c-1} &\text{for } x \ge 1
\end{cases}
for ``c > 0``.
`loglaplace` takes ``c`` as a shape parameter.
%(after_notes)s
References
----------
T.J. Kozubowski and K. Podgorski, "A log-Laplace growth rate model",
The Mathematical Scientist, vol. 28, pp. 49-60, 2003.
%(example)s
"""
def _pdf(self, x, c):
# loglaplace.pdf(x, c) = c / 2 * x**(c-1), for 0 < x < 1
# = c / 2 * x**(-c-1), for x >= 1
cd2 = c/2.0
c = np.where(x < 1, c, -c)
return cd2*x**(c-1)
def _cdf(self, x, c):
return np.where(x < 1, 0.5*x**c, 1-0.5*x**(-c))
def _ppf(self, q, c):
return np.where(q < 0.5, (2.0*q)**(1.0/c), (2*(1.0-q))**(-1.0/c))
def _munp(self, n, c):
return c**2 / (c**2 - n**2)
def _entropy(self, c):
return np.log(2.0/c) + 1.0
loglaplace = loglaplace_gen(a=0.0, name='loglaplace')
def _lognorm_logpdf(x, s):
return _lazywhere(x != 0, (x, s),
lambda x, s: -np.log(x)**2 / (2*s**2) - np.log(s*x*np.sqrt(2*np.pi)),
-np.inf)
class lognorm_gen(rv_continuous):
r"""A lognormal continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `lognorm` is:
.. math::
f(x, s) = \frac{1}{s x \sqrt{2\pi}}
\exp(-\frac{1}{2} (\frac{\log(x)}{s})^2)
for ``x > 0``, ``s > 0``.
`lognorm` takes ``s`` as a shape parameter.
%(after_notes)s
A common parametrization for a lognormal random variable ``Y`` is in
terms of the mean, ``mu``, and standard deviation, ``sigma``, of the
unique normally distributed random variable ``X`` such that exp(X) = Y.
This parametrization corresponds to setting ``s = sigma`` and ``scale =
exp(mu)``.
%(example)s
"""
_support_mask = rv_continuous._open_support_mask
def _rvs(self, s):
return np.exp(s * self._random_state.standard_normal(self._size))
def _pdf(self, x, s):
# lognorm.pdf(x, s) = 1 / (s*x*sqrt(2*pi)) * exp(-1/2*(log(x)/s)**2)
return np.exp(self._logpdf(x, s))
def _logpdf(self, x, s):
return _lognorm_logpdf(x, s)
def _cdf(self, x, s):
return _norm_cdf(np.log(x) / s)
def _logcdf(self, x, s):
return _norm_logcdf(np.log(x) / s)
def _ppf(self, q, s):
return np.exp(s * _norm_ppf(q))
def _sf(self, x, s):
return _norm_sf(np.log(x) / s)
def _logsf(self, x, s):
return _norm_logsf(np.log(x) / s)
def _stats(self, s):
p = np.exp(s*s)
mu = np.sqrt(p)
mu2 = p*(p-1)
g1 = np.sqrt((p-1))*(2+p)
g2 = np.polyval([1, 2, 3, 0, -6.0], p)
return mu, mu2, g1, g2
def _entropy(self, s):
return 0.5 * (1 + np.log(2*np.pi) + 2 * np.log(s))
@extend_notes_in_docstring(rv_continuous, notes="""\
When the location parameter is fixed by using the `floc` argument,
this function uses explicit formulas for the maximum likelihood
estimation of the log-normal shape and scale parameters, so the
`optimizer`, `loc` and `scale` keyword arguments are ignored.\n\n""")
def fit(self, data, *args, **kwds):
floc = kwds.get('floc', None)
if floc is None:
# loc is not fixed. Use the default fit method.
return super(lognorm_gen, self).fit(data, *args, **kwds)
f0 = (kwds.get('f0', None) or kwds.get('fs', None) or
kwds.get('fix_s', None))
fscale = kwds.get('fscale', None)
if len(args) > 1:
raise TypeError("Too many input arguments.")
for name in ['f0', 'fs', 'fix_s', 'floc', 'fscale', 'loc', 'scale',
'optimizer']:
kwds.pop(name, None)
if kwds:
raise TypeError("Unknown arguments: %s." % kwds)
# Special case: loc is fixed. Use the maximum likelihood formulas
# instead of the numerical solver.
if f0 is not None and fscale is not None:
# This check is for consistency with `rv_continuous.fit`.
raise ValueError("All parameters fixed. There is nothing to "
"optimize.")
data = np.asarray(data)
floc = float(floc)
if floc != 0:
# Shifting the data by floc. Don't do the subtraction in-place,
# because `data` might be a view of the input array.
data = data - floc
if np.any(data <= 0):
raise FitDataError("lognorm", lower=floc, upper=np.inf)
lndata = np.log(data)
# Three cases to handle:
# * shape and scale both free
# * shape fixed, scale free
# * shape free, scale fixed
if fscale is None:
# scale is free.
scale = np.exp(lndata.mean())
if f0 is None:
# shape is free.
shape = lndata.std()
else:
# shape is fixed.
shape = float(f0)
else:
# scale is fixed, shape is free
scale = float(fscale)
shape = np.sqrt(((lndata - np.log(scale))**2).mean())
return shape, floc, scale
lognorm = lognorm_gen(a=0.0, name='lognorm')
class gilbrat_gen(rv_continuous):
r"""A Gilbrat continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `gilbrat` is:
.. math::
f(x) = \frac{1}{x \sqrt{2\pi}} \exp(-\frac{1}{2} (\log(x))^2)
`gilbrat` is a special case of `lognorm` with ``s = 1``.
%(after_notes)s
%(example)s
"""
_support_mask = rv_continuous._open_support_mask
def _rvs(self):
return np.exp(self._random_state.standard_normal(self._size))
def _pdf(self, x):
# gilbrat.pdf(x) = 1/(x*sqrt(2*pi)) * exp(-1/2*(log(x))**2)
return np.exp(self._logpdf(x))
def _logpdf(self, x):
return _lognorm_logpdf(x, 1.0)
def _cdf(self, x):
return _norm_cdf(np.log(x))
def _ppf(self, q):
return np.exp(_norm_ppf(q))
def _stats(self):
p = np.e
mu = np.sqrt(p)
mu2 = p * (p - 1)
g1 = np.sqrt((p - 1)) * (2 + p)
g2 = np.polyval([1, 2, 3, 0, -6.0], p)
return mu, mu2, g1, g2
def _entropy(self):
return 0.5 * np.log(2 * np.pi) + 0.5
gilbrat = gilbrat_gen(a=0.0, name='gilbrat')
class maxwell_gen(rv_continuous):
r"""A Maxwell continuous random variable.
%(before_notes)s
Notes
-----
A special case of a `chi` distribution, with ``df = 3``, ``loc = 0.0``,
and given ``scale = a``, where ``a`` is the parameter used in the
Mathworld description [1]_.
The probability density function for `maxwell` is:
.. math::
f(x) = \sqrt{2/\pi}x^2 \exp(-x^2/2)
for ``x > 0``.
%(after_notes)s
References
----------
.. [1] http://mathworld.wolfram.com/MaxwellDistribution.html
%(example)s
"""
def _rvs(self):
return chi.rvs(3.0, size=self._size, random_state=self._random_state)
def _pdf(self, x):
# maxwell.pdf(x) = sqrt(2/pi)x**2 * exp(-x**2/2)
return np.sqrt(2.0/np.pi)*x*x*np.exp(-x*x/2.0)
def _cdf(self, x):
return sc.gammainc(1.5, x*x/2.0)
def _ppf(self, q):
return np.sqrt(2*sc.gammaincinv(1.5, q))
def _stats(self):
val = 3*np.pi-8
return (2*np.sqrt(2.0/np.pi),
3-8/np.pi,
np.sqrt(2)*(32-10*np.pi)/val**1.5,
(-12*np.pi*np.pi + 160*np.pi - 384) / val**2.0)
def _entropy(self):
return _EULER + 0.5*np.log(2*np.pi)-0.5
maxwell = maxwell_gen(a=0.0, name='maxwell')
class mielke_gen(rv_continuous):
r"""A Mielke's Beta-Kappa continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `mielke` is:
.. math::
f(x, k, s) = \frac{k x^{k-1}}{(1+x^s)^{1+k/s}}
for ``x > 0``.
`mielke` takes ``k`` and ``s`` as shape parameters.
%(after_notes)s
%(example)s
"""
def _pdf(self, x, k, s):
# mielke.pdf(x, k, s) = k * x**(k-1) / (1+x**s)**(1+k/s)
return k*x**(k-1.0) / (1.0+x**s)**(1.0+k*1.0/s)
def _cdf(self, x, k, s):
return x**k / (1.0+x**s)**(k*1.0/s)
def _ppf(self, q, k, s):
qsk = pow(q, s*1.0/k)
return pow(qsk/(1.0-qsk), 1.0/s)
mielke = mielke_gen(a=0.0, name='mielke')
class kappa4_gen(rv_continuous):
r"""Kappa 4 parameter distribution.
%(before_notes)s
Notes
-----
The probability density function for kappa4 is:
.. math::
f(x, h, k) = (1 - k x)^{1/k - 1} (1 - h (1 - k x)^{1/k})^{1/h-1}
if :math:`h` and :math:`k` are not equal to 0.
If :math:`h` or :math:`k` are zero then the pdf can be simplified:
h = 0 and k != 0::
kappa4.pdf(x, h, k) = (1.0 - k*x)**(1.0/k - 1.0)*
exp(-(1.0 - k*x)**(1.0/k))
h != 0 and k = 0::
kappa4.pdf(x, h, k) = exp(-x)*(1.0 - h*exp(-x))**(1.0/h - 1.0)
h = 0 and k = 0::
kappa4.pdf(x, h, k) = exp(-x)*exp(-exp(-x))
kappa4 takes :math:`h` and :math:`k` as shape parameters.
The kappa4 distribution returns other distributions when certain
:math:`h` and :math:`k` values are used.
+------+-------------+----------------+------------------+
| h | k=0.0 | k=1.0 | -inf<=k<=inf |
+======+=============+================+==================+
| -1.0 | Logistic | | Generalized |
| | | | Logistic(1) |
| | | | |
| | logistic(x) | | |
+------+-------------+----------------+------------------+
| 0.0 | Gumbel | Reverse | Generalized |
| | | Exponential(2) | Extreme Value |
| | | | |
| | gumbel_r(x) | | genextreme(x, k) |
+------+-------------+----------------+------------------+
| 1.0 | Exponential | Uniform | Generalized |
| | | | Pareto |
| | | | |
| | expon(x) | uniform(x) | genpareto(x, -k) |
+------+-------------+----------------+------------------+
(1) There are at least five generalized logistic distributions.
Four are described here:
https://en.wikipedia.org/wiki/Generalized_logistic_distribution
The "fifth" one is the one kappa4 should match which currently
isn't implemented in scipy:
https://en.wikipedia.org/wiki/Talk:Generalized_logistic_distribution
http://www.mathwave.com/help/easyfit/html/analyses/distributions/gen_logistic.html
(2) This distribution is currently not in scipy.
References
----------
J.C. Finney, "Optimization of a Skewed Logistic Distribution With Respect
to the Kolmogorov-Smirnov Test", A Dissertation Submitted to the Graduate
Faculty of the Louisiana State University and Agricultural and Mechanical
College, (August, 2004),
http://digitalcommons.lsu.edu/cgi/viewcontent.cgi?article=4671&context=gradschool_dissertations
J.R.M. Hosking, "The four-parameter kappa distribution". IBM J. Res.
Develop. 38 (3), 25 1-258 (1994).
B. Kumphon, A. Kaew-Man, P. Seenoi, "A Rainfall Distribution for the Lampao
Site in the Chi River Basin, Thailand", Journal of Water Resource and
Protection, vol. 4, 866-869, (2012).
http://file.scirp.org/pdf/JWARP20121000009_14676002.pdf
C. Winchester, "On Estimation of the Four-Parameter Kappa Distribution", A
Thesis Submitted to Dalhousie University, Halifax, Nova Scotia, (March
2000).
http://www.nlc-bnc.ca/obj/s4/f2/dsk2/ftp01/MQ57336.pdf
%(after_notes)s
%(example)s
"""
def _argcheck(self, h, k):
condlist = [np.logical_and(h > 0, k > 0),
np.logical_and(h > 0, k == 0),
np.logical_and(h > 0, k < 0),
np.logical_and(h <= 0, k > 0),
np.logical_and(h <= 0, k == 0),
np.logical_and(h <= 0, k < 0)]
def f0(h, k):
return (1.0 - float_power(h, -k))/k
def f1(h, k):
return np.log(h)
def f3(h, k):
a = np.empty(np.shape(h))
a[:] = -np.inf
return a
def f5(h, k):
return 1.0/k
self.a = _lazyselect(condlist,
[f0, f1, f0, f3, f3, f5],
[h, k],
default=np.nan)
def f0(h, k):
return 1.0/k
def f1(h, k):
a = np.empty(np.shape(h))
a[:] = np.inf
return a
self.b = _lazyselect(condlist,
[f0, f1, f1, f0, f1, f1],
[h, k],
default=np.nan)
return h == h
def _pdf(self, x, h, k):
# kappa4.pdf(x, h, k) = (1.0 - k*x)**(1.0/k - 1.0)*
# (1.0 - h*(1.0 - k*x)**(1.0/k))**(1.0/h-1)
return np.exp(self._logpdf(x, h, k))
def _logpdf(self, x, h, k):
condlist = [np.logical_and(h != 0, k != 0),
np.logical_and(h == 0, k != 0),
np.logical_and(h != 0, k == 0),
np.logical_and(h == 0, k == 0)]
def f0(x, h, k):
'''pdf = (1.0 - k*x)**(1.0/k - 1.0)*(
1.0 - h*(1.0 - k*x)**(1.0/k))**(1.0/h-1.0)
logpdf = ...
'''
return (sc.xlog1py(1.0/k - 1.0, -k*x) +
sc.xlog1py(1.0/h - 1.0, -h*(1.0 - k*x)**(1.0/k)))
def f1(x, h, k):
'''pdf = (1.0 - k*x)**(1.0/k - 1.0)*np.exp(-(
1.0 - k*x)**(1.0/k))
logpdf = ...
'''
return sc.xlog1py(1.0/k - 1.0, -k*x) - (1.0 - k*x)**(1.0/k)
def f2(x, h, k):
'''pdf = np.exp(-x)*(1.0 - h*np.exp(-x))**(1.0/h - 1.0)
logpdf = ...
'''
return -x + sc.xlog1py(1.0/h - 1.0, -h*np.exp(-x))
def f3(x, h, k):
'''pdf = np.exp(-x-np.exp(-x))
logpdf = ...
'''
return -x - np.exp(-x)
return _lazyselect(condlist,
[f0, f1, f2, f3],
[x, h, k],
default=np.nan)
def _cdf(self, x, h, k):
return np.exp(self._logcdf(x, h, k))
def _logcdf(self, x, h, k):
condlist = [np.logical_and(h != 0, k != 0),
np.logical_and(h == 0, k != 0),
np.logical_and(h != 0, k == 0),
np.logical_and(h == 0, k == 0)]
def f0(x, h, k):
'''cdf = (1.0 - h*(1.0 - k*x)**(1.0/k))**(1.0/h)
logcdf = ...
'''
return (1.0/h)*sc.log1p(-h*(1.0 - k*x)**(1.0/k))
def f1(x, h, k):
'''cdf = np.exp(-(1.0 - k*x)**(1.0/k))
logcdf = ...
'''
return -(1.0 - k*x)**(1.0/k)
def f2(x, h, k):
'''cdf = (1.0 - h*np.exp(-x))**(1.0/h)
logcdf = ...
'''
return (1.0/h)*sc.log1p(-h*np.exp(-x))
def f3(x, h, k):
'''cdf = np.exp(-np.exp(-x))
logcdf = ...
'''
return -np.exp(-x)
return _lazyselect(condlist,
[f0, f1, f2, f3],
[x, h, k],
default=np.nan)
def _ppf(self, q, h, k):
condlist = [np.logical_and(h != 0, k != 0),
np.logical_and(h == 0, k != 0),
np.logical_and(h != 0, k == 0),
np.logical_and(h == 0, k == 0)]
def f0(q, h, k):
return 1.0/k*(1.0 - ((1.0 - (q**h))/h)**k)
def f1(q, h, k):
return 1.0/k*(1.0 - (-np.log(q))**k)
def f2(q, h, k):
'''ppf = -np.log((1.0 - (q**h))/h)
'''
return -sc.log1p(-(q**h)) + np.log(h)
def f3(q, h, k):
return -np.log(-np.log(q))
return _lazyselect(condlist,
[f0, f1, f2, f3],
[q, h, k],
default=np.nan)
def _stats(self, h, k):
if h >= 0 and k >= 0:
maxr = 5
elif h < 0 and k >= 0:
maxr = int(-1.0/h*k)
elif k < 0:
maxr = int(-1.0/k)
else:
maxr = 5
outputs = [None if r < maxr else np.nan for r in range(1, 5)]
return outputs[:]
kappa4 = kappa4_gen(name='kappa4')
class kappa3_gen(rv_continuous):
r"""Kappa 3 parameter distribution.
%(before_notes)s
Notes
-----
The probability density function for `kappa` is:
.. math::
f(x, a) = \begin{cases}
a [a + x^a]^{-(a + 1)/a}, &\text{for } x > 0\\
0.0, &\text{for } x \le 0
\end{cases}
`kappa3` takes :math:`a` as a shape parameter and :math:`a > 0`.
References
----------
P.W. Mielke and E.S. Johnson, "Three-Parameter Kappa Distribution Maximum
Likelihood and Likelihood Ratio Tests", Methods in Weather Research,
701-707, (September, 1973),
http://docs.lib.noaa.gov/rescue/mwr/101/mwr-101-09-0701.pdf
B. Kumphon, "Maximum Entropy and Maximum Likelihood Estimation for the
Three-Parameter Kappa Distribution", Open Journal of Statistics, vol 2,
415-419 (2012)
http://file.scirp.org/pdf/OJS20120400011_95789012.pdf
%(after_notes)s
%(example)s
"""
def _argcheck(self, a):
return a > 0
def _pdf(self, x, a):
# kappa3.pdf(x, a) =
# a*[a + x**a]**(-(a + 1)/a), for x > 0
# 0.0, for x <= 0
return a*(a + x**a)**(-1.0/a-1)
def _cdf(self, x, a):
return x*(a + x**a)**(-1.0/a)
def _ppf(self, q, a):
return (a/(q**-a - 1.0))**(1.0/a)
def _stats(self, a):
outputs = [None if i < a else np.nan for i in range(1, 5)]
return outputs[:]
kappa3 = kappa3_gen(a=0.0, name='kappa3')
class moyal_gen(rv_continuous):
r"""A Moyal continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `moyal` is:
.. math::
f(x) = \exp(-(x + \exp(-x))/2) / \sqrt{2\pi}
%(after_notes)s
This distribution has utility in high-energy physics and radiation
detection. It describes the energy loss of a charged relativistic
particle due to ionization of the medium [1]_. It also provides an
approximation for the Landau distribution. For an in depth description
see [2]_. For additional description, see [3]_.
References
----------
.. [1] J.E. Moyal, "XXX. Theory of ionization fluctuations",
The London, Edinburgh, and Dublin Philosophical Magazine
and Journal of Science, vol 46, 263-280, (1955).
https://doi.org/10.1080/14786440308521076 (gated)
.. [2] G. Cordeiro et al., "The beta Moyal: a useful skew distribution",
International Journal of Research and Reviews in Applied Sciences,
vol 10, 171-192, (2012).
http://www.arpapress.com/Volumes/Vol10Issue2/IJRRAS_10_2_02.pdf
.. [3] C. Walck, "Handbook on Statistical Distributions for
Experimentalists; International Report SUF-PFY/96-01", Chapter 26,
University of Stockholm: Stockholm, Sweden, (2007).
www.stat.rice.edu/~dobelman/textfiles/DistributionsHandbook.pdf
.. versionadded:: 1.1.0
%(example)s
"""
def _rvs(self):
sz, rndm = self._size, self._random_state
u1 = gamma.rvs(a = 0.5, scale = 2, size=sz, random_state=rndm)
return -np.log(u1)
def _pdf(self, x):
return np.exp(-0.5 * (x + np.exp(-x))) / np.sqrt(2*np.pi)
def _cdf(self, x):
return sc.erfc(np.exp(-0.5 * x) / np.sqrt(2))
def _sf(self, x):
return sc.erf(np.exp(-0.5 * x) / np.sqrt(2))
def _ppf(self, x):
return -np.log(2 * sc.erfcinv(x)**2)
def _stats(self):
mu = np.log(2) + np.euler_gamma
mu2 = np.pi**2 / 2
g1 = 28 * np.sqrt(2) * sc.zeta(3) / np.pi**3
g2 = 4.
return mu, mu2, g1, g2
def _munp(self, n):
if n == 1.0:
return np.log(2) + np.euler_gamma
elif n == 2.0:
return np.pi**2 / 2 + (np.log(2) + np.euler_gamma)**2
elif n == 3.0:
tmp1 = 1.5 * np.pi**2 * (np.log(2)+np.euler_gamma)
tmp2 = (np.log(2)+np.euler_gamma)**3
tmp3 = 14 * sc.zeta(3)
return tmp1 + tmp2 + tmp3
elif n == 4.0:
tmp1 = 4 * 14 * sc.zeta(3) * (np.log(2) + np.euler_gamma)
tmp2 = 3 * np.pi**2 * (np.log(2) + np.euler_gamma)**2
tmp3 = (np.log(2) + np.euler_gamma)**4
tmp4 = 7 * np.pi**4 / 4
return tmp1 + tmp2 + tmp3 + tmp4
else:
# return generic for higher moments
# return rv_continuous._mom1_sc(self, n, b)
return self._mom1_sc(n)
moyal = moyal_gen(name="moyal")
class nakagami_gen(rv_continuous):
r"""A Nakagami continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `nakagami` is:
.. math::
f(x, nu) = \frac{2 \nu^\nu}{\Gamma(\nu)} x^{2\nu-1} \exp(-\nu x^2)
for ``x > 0``, ``nu > 0``.
`nakagami` takes ``nu`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _pdf(self, x, nu):
# nakagami.pdf(x, nu) = 2 * nu**nu / gamma(nu) *
# x**(2*nu-1) * exp(-nu*x**2)
return 2*nu**nu/sc.gamma(nu)*(x**(2*nu-1.0))*np.exp(-nu*x*x)
def _cdf(self, x, nu):
return sc.gammainc(nu, nu*x*x)
def _ppf(self, q, nu):
return np.sqrt(1.0/nu*sc.gammaincinv(nu, q))
def _stats(self, nu):
mu = sc.gamma(nu+0.5)/sc.gamma(nu)/np.sqrt(nu)
mu2 = 1.0-mu*mu
g1 = mu * (1 - 4*nu*mu2) / 2.0 / nu / np.power(mu2, 1.5)
g2 = -6*mu**4*nu + (8*nu-2)*mu**2-2*nu + 1
g2 /= nu*mu2**2.0
return mu, mu2, g1, g2
nakagami = nakagami_gen(a=0.0, name="nakagami")
class ncx2_gen(rv_continuous):
r"""A non-central chi-squared continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `ncx2` is:
.. math::
f(x, df, nc) = \exp(-\frac{nc+x}{2}) \frac{1}{2} (x/nc)^{(df-2)/4}
I[(df-2)/2](\sqrt{nc x})
for :math:`x > 0`.
`ncx2` takes ``df`` and ``nc`` as shape parameters.
%(after_notes)s
%(example)s
"""
def _rvs(self, df, nc):
return self._random_state.noncentral_chisquare(df, nc, self._size)
def _logpdf(self, x, df, nc):
return _ncx2_log_pdf(x, df, nc)
def _pdf(self, x, df, nc):
# ncx2.pdf(x, df, nc) = exp(-(nc+x)/2) * 1/2 * (x/nc)**((df-2)/4)
# * I[(df-2)/2](sqrt(nc*x))
return _ncx2_pdf(x, df, nc)
def _cdf(self, x, df, nc):
return _ncx2_cdf(x, df, nc)
def _ppf(self, q, df, nc):
return sc.chndtrix(q, df, nc)
def _stats(self, df, nc):
val = df + 2.0*nc
return (df + nc,
2*val,
np.sqrt(8)*(val+nc)/val**1.5,
12.0*(val+2*nc)/val**2.0)
ncx2 = ncx2_gen(a=0.0, name='ncx2')
class ncf_gen(rv_continuous):
r"""A non-central F distribution continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `ncf` is:
.. math::
f(x, n_1, n_2, \lambda) =
\exp(\frac{\lambda}{2} + \lambda n_1 \frac{x}{2(n_1 x+n_2)})
n_1^{n_1/2} n_2^{n_2/2} x^{n_1/2 - 1} \\
(n_2+n_1 x)^{-(n_1+n_2)/2}
\gamma(n_1/2) \gamma(1+n_2/2) \\
\frac{L^{\frac{v_1}{2}-1}_{v_2/2}
(-\lambda v_1 \frac{x}{2(v_1 x+v_2)})}
{(B(v_1/2, v_2/2) \gamma(\frac{v_1+v_2}{2})}
for :math:`n_1 > 1`, :math:`n_2, \lambda > 0`. Here :math:`n_1` is the
degrees of freedom in the numerator, :math:`n_2` the degrees of freedom in
the denominator, :math:`\lambda` the non-centrality parameter,
:math:`\gamma` is the logarithm of the Gamma function, :math:`L_n^k` is a
generalized Laguerre polynomial and :math:`B` is the beta function.
`ncf` takes ``df1``, ``df2`` and ``nc`` as shape parameters.
%(after_notes)s
%(example)s
"""
def _rvs(self, dfn, dfd, nc):
return self._random_state.noncentral_f(dfn, dfd, nc, self._size)
def _pdf_skip(self, x, dfn, dfd, nc):
# ncf.pdf(x, df1, df2, nc) = exp(nc/2 + nc*df1*x/(2*(df1*x+df2))) *
# df1**(df1/2) * df2**(df2/2) * x**(df1/2-1) *
# (df2+df1*x)**(-(df1+df2)/2) *
# gamma(df1/2)*gamma(1+df2/2) *
# L^{v1/2-1}^{v2/2}(-nc*v1*x/(2*(v1*x+v2))) /
# (B(v1/2, v2/2) * gamma((v1+v2)/2))
n1, n2 = dfn, dfd
term = -nc/2+nc*n1*x/(2*(n2+n1*x)) + sc.gammaln(n1/2.)+sc.gammaln(1+n2/2.)
term -= sc.gammaln((n1+n2)/2.0)
Px = np.exp(term)
Px *= n1**(n1/2) * n2**(n2/2) * x**(n1/2-1)
Px *= (n2+n1*x)**(-(n1+n2)/2)
Px *= sc.assoc_laguerre(-nc*n1*x/(2.0*(n2+n1*x)), n2/2, n1/2-1)
Px /= sc.beta(n1/2, n2/2)
# This function does not have a return. Drop it for now, the generic
# function seems to work OK.
def _cdf(self, x, dfn, dfd, nc):
return sc.ncfdtr(dfn, dfd, nc, x)
def _ppf(self, q, dfn, dfd, nc):
return sc.ncfdtri(dfn, dfd, nc, q)
def _munp(self, n, dfn, dfd, nc):
val = (dfn * 1.0/dfd)**n
term = sc.gammaln(n+0.5*dfn) + sc.gammaln(0.5*dfd-n) - sc.gammaln(dfd*0.5)
val *= np.exp(-nc / 2.0+term)
val *= sc.hyp1f1(n+0.5*dfn, 0.5*dfn, 0.5*nc)
return val
def _stats(self, dfn, dfd, nc):
mu = np.where(dfd <= 2, np.inf, dfd / (dfd-2.0)*(1+nc*1.0/dfn))
mu2 = np.where(dfd <= 4, np.inf, 2*(dfd*1.0/dfn)**2.0 *
((dfn+nc/2.0)**2.0 + (dfn+nc)*(dfd-2.0)) /
((dfd-2.0)**2.0 * (dfd-4.0)))
return mu, mu2, None, None
ncf = ncf_gen(a=0.0, name='ncf')
class t_gen(rv_continuous):
r"""A Student's T continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `t` is:
.. math::
f(x, df) = \frac{\gamma((df+1)/2)}
{\sqrt{\pi*df} \gamma(df/2) (1+x^2/df)^{(df+1)/2}}
for ``df > 0``.
`t` takes ``df`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _rvs(self, df):
return self._random_state.standard_t(df, size=self._size)
def _pdf(self, x, df):
# gamma((df+1)/2)
# t.pdf(x, df) = ---------------------------------------------------
# sqrt(pi*df) * gamma(df/2) * (1+x**2/df)**((df+1)/2)
r = np.asarray(df*1.0)
Px = np.exp(sc.gammaln((r+1)/2)-sc.gammaln(r/2))
Px /= np.sqrt(r*np.pi)*(1+(x**2)/r)**((r+1)/2)
return Px
def _logpdf(self, x, df):
r = df*1.0
lPx = sc.gammaln((r+1)/2)-sc.gammaln(r/2)
lPx -= 0.5*np.log(r*np.pi) + (r+1)/2*np.log(1+(x**2)/r)
return lPx
def _cdf(self, x, df):
return sc.stdtr(df, x)
def _sf(self, x, df):
return sc.stdtr(df, -x)
def _ppf(self, q, df):
return sc.stdtrit(df, q)
def _isf(self, q, df):
return -sc.stdtrit(df, q)
def _stats(self, df):
mu2 = _lazywhere(df > 2, (df,),
lambda df: df / (df-2.0),
np.inf)
g1 = np.where(df > 3, 0.0, np.nan)
g2 = _lazywhere(df > 4, (df,),
lambda df: 6.0 / (df-4.0),
np.nan)
return 0, mu2, g1, g2
t = t_gen(name='t')
class nct_gen(rv_continuous):
r"""A non-central Student's T continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `nct` is:
.. math::
f(x, df, nc) = \frac{df^{df/2} \gamma(df+1)}{2^{df}
\exp(nc^2 / 2) (df+x^2)^{df/2} \gamma(df/2)}
for ``df > 0``.
`nct` takes ``df`` and ``nc`` as shape parameters.
%(after_notes)s
%(example)s
"""
def _argcheck(self, df, nc):
return (df > 0) & (nc == nc)
def _rvs(self, df, nc):
sz, rndm = self._size, self._random_state
n = norm.rvs(loc=nc, size=sz, random_state=rndm)
c2 = chi2.rvs(df, size=sz, random_state=rndm)
return n * np.sqrt(df) / np.sqrt(c2)
def _pdf(self, x, df, nc):
# nct.pdf(x, df, nc) =
# df**(df/2) * gamma(df+1)
# ----------------------------------------------------
# 2**df*exp(nc**2/2) * (df+x**2)**(df/2) * gamma(df/2)
n = df*1.0
nc = nc*1.0
x2 = x*x
ncx2 = nc*nc*x2
fac1 = n + x2
trm1 = n/2.*np.log(n) + sc.gammaln(n+1)
trm1 -= n*np.log(2)+nc*nc/2.+(n/2.)*np.log(fac1)+sc.gammaln(n/2.)
Px = np.exp(trm1)
valF = ncx2 / (2*fac1)
trm1 = np.sqrt(2)*nc*x*sc.hyp1f1(n/2+1, 1.5, valF)
trm1 /= np.asarray(fac1*sc.gamma((n+1)/2))
trm2 = sc.hyp1f1((n+1)/2, 0.5, valF)
trm2 /= np.asarray(np.sqrt(fac1)*sc.gamma(n/2+1))
Px *= trm1+trm2
return Px
def _cdf(self, x, df, nc):
return sc.nctdtr(df, nc, x)
def _ppf(self, q, df, nc):
return sc.nctdtrit(df, nc, q)
def _stats(self, df, nc, moments='mv'):
#
# See D. Hogben, R.S. Pinkham, and M.B. Wilk,
# 'The moments of the non-central t-distribution'
# Biometrika 48, p. 465 (2961).
# e.g. http://www.jstor.org/stable/2332772 (gated)
#
mu, mu2, g1, g2 = None, None, None, None
gfac = sc.gamma(df/2.-0.5) / sc.gamma(df/2.)
c11 = np.sqrt(df/2.) * gfac
c20 = df / (df-2.)
c22 = c20 - c11*c11
mu = np.where(df > 1, nc*c11, np.inf)
mu2 = np.where(df > 2, c22*nc*nc + c20, np.inf)
if 's' in moments:
c33t = df * (7.-2.*df) / (df-2.) / (df-3.) + 2.*c11*c11
c31t = 3.*df / (df-2.) / (df-3.)
mu3 = (c33t*nc*nc + c31t) * c11*nc
g1 = np.where(df > 3, mu3 / np.power(mu2, 1.5), np.nan)
# kurtosis
if 'k' in moments:
c44 = df*df / (df-2.) / (df-4.)
c44 -= c11*c11 * 2.*df*(5.-df) / (df-2.) / (df-3.)
c44 -= 3.*c11**4
c42 = df / (df-4.) - c11*c11 * (df-1.) / (df-3.)
c42 *= 6.*df / (df-2.)
c40 = 3.*df*df / (df-2.) / (df-4.)
mu4 = c44 * nc**4 + c42*nc**2 + c40
g2 = np.where(df > 4, mu4/mu2**2 - 3., np.nan)
return mu, mu2, g1, g2
nct = nct_gen(name="nct")
class pareto_gen(rv_continuous):
r"""A Pareto continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `pareto` is:
.. math::
f(x, b) = \frac{b}{x^{b+1}}
for :math:`x \ge 1`, :math:`b > 0`.
`pareto` takes :math:`b` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _pdf(self, x, b):
# pareto.pdf(x, b) = b / x**(b+1)
return b * x**(-b-1)
def _cdf(self, x, b):
return 1 - x**(-b)
def _ppf(self, q, b):
return pow(1-q, -1.0/b)
def _sf(self, x, b):
return x**(-b)
def _stats(self, b, moments='mv'):
mu, mu2, g1, g2 = None, None, None, None
if 'm' in moments:
mask = b > 1
bt = np.extract(mask, b)
mu = valarray(np.shape(b), value=np.inf)
np.place(mu, mask, bt / (bt-1.0))
if 'v' in moments:
mask = b > 2
bt = np.extract(mask, b)
mu2 = valarray(np.shape(b), value=np.inf)
np.place(mu2, mask, bt / (bt-2.0) / (bt-1.0)**2)
if 's' in moments:
mask = b > 3
bt = np.extract(mask, b)
g1 = valarray(np.shape(b), value=np.nan)
vals = 2 * (bt + 1.0) * np.sqrt(bt - 2.0) / ((bt - 3.0) * np.sqrt(bt))
np.place(g1, mask, vals)
if 'k' in moments:
mask = b > 4
bt = np.extract(mask, b)
g2 = valarray(np.shape(b), value=np.nan)
vals = (6.0*np.polyval([1.0, 1.0, -6, -2], bt) /
np.polyval([1.0, -7.0, 12.0, 0.0], bt))
np.place(g2, mask, vals)
return mu, mu2, g1, g2
def _entropy(self, c):
return 1 + 1.0/c - np.log(c)
pareto = pareto_gen(a=1.0, name="pareto")
class lomax_gen(rv_continuous):
r"""A Lomax (Pareto of the second kind) continuous random variable.
%(before_notes)s
Notes
-----
The Lomax distribution is a special case of the Pareto distribution, with
(loc=-1.0).
The probability density function for `lomax` is:
.. math::
f(x, c) = \frac{c}{(1+x)^{c+1}}
for :math:`x \ge 0`, ``c > 0``.
`lomax` takes :math:`c` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _pdf(self, x, c):
# lomax.pdf(x, c) = c / (1+x)**(c+1)
return c*1.0/(1.0+x)**(c+1.0)
def _logpdf(self, x, c):
return np.log(c) - (c+1)*sc.log1p(x)
def _cdf(self, x, c):
return -sc.expm1(-c*sc.log1p(x))
def _sf(self, x, c):
return np.exp(-c*sc.log1p(x))
def _logsf(self, x, c):
return -c*sc.log1p(x)
def _ppf(self, q, c):
return sc.expm1(-sc.log1p(-q)/c)
def _stats(self, c):
mu, mu2, g1, g2 = pareto.stats(c, loc=-1.0, moments='mvsk')
return mu, mu2, g1, g2
def _entropy(self, c):
return 1+1.0/c-np.log(c)
lomax = lomax_gen(a=0.0, name="lomax")
class pearson3_gen(rv_continuous):
r"""A pearson type III continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `pearson3` is:
.. math::
f(x, skew) = \frac{|\beta|}{\gamma(\alpha)}
(\beta (x - \zeta))^{alpha - 1} \exp(-\beta (x - \zeta))
where:
.. math::
\beta = \frac{2}{skew stddev}
\alpha = (stddev \beta)^2
\zeta = loc - \frac{\alpha}{\beta}
`pearson3` takes ``skew`` as a shape parameter.
%(after_notes)s
%(example)s
References
----------
R.W. Vogel and D.E. McMartin, "Probability Plot Goodness-of-Fit and
Skewness Estimation Procedures for the Pearson Type 3 Distribution", Water
Resources Research, Vol.27, 3149-3158 (1991).
L.R. Salvosa, "Tables of Pearson's Type III Function", Ann. Math. Statist.,
Vol.1, 191-198 (1930).
"Using Modern Computing Tools to Fit the Pearson Type III Distribution to
Aviation Loads Data", Office of Aviation Research (2003).
"""
def _preprocess(self, x, skew):
# The real 'loc' and 'scale' are handled in the calling pdf(...). The
# local variables 'loc' and 'scale' within pearson3._pdf are set to
# the defaults just to keep them as part of the equations for
# documentation.
loc = 0.0
scale = 1.0
# If skew is small, return _norm_pdf. The divide between pearson3
# and norm was found by brute force and is approximately a skew of
# 0.000016. No one, I hope, would actually use a skew value even
# close to this small.
norm2pearson_transition = 0.000016
ans, x, skew = np.broadcast_arrays([1.0], x, skew)
ans = ans.copy()
# mask is True where skew is small enough to use the normal approx.
mask = np.absolute(skew) < norm2pearson_transition
invmask = ~mask
beta = 2.0 / (skew[invmask] * scale)
alpha = (scale * beta)**2
zeta = loc - alpha / beta
transx = beta * (x[invmask] - zeta)
return ans, x, transx, mask, invmask, beta, alpha, zeta
def _argcheck(self, skew):
# The _argcheck function in rv_continuous only allows positive
# arguments. The skew argument for pearson3 can be zero (which I want
# to handle inside pearson3._pdf) or negative. So just return True
# for all skew args.
return np.ones(np.shape(skew), dtype=bool)
def _stats(self, skew):
_, _, _, _, _, beta, alpha, zeta = (
self._preprocess([1], skew))
m = zeta + alpha / beta
v = alpha / (beta**2)
s = 2.0 / (alpha**0.5) * np.sign(beta)
k = 6.0 / alpha
return m, v, s, k
def _pdf(self, x, skew):
# pearson3.pdf(x, skew) = abs(beta) / gamma(alpha) *
# (beta * (x - zeta))**(alpha - 1) * exp(-beta*(x - zeta))
# Do the calculation in _logpdf since helps to limit
# overflow/underflow problems
ans = np.exp(self._logpdf(x, skew))
if ans.ndim == 0:
if np.isnan(ans):
return 0.0
return ans
ans[np.isnan(ans)] = 0.0
return ans
def _logpdf(self, x, skew):
# PEARSON3 logpdf GAMMA logpdf
# np.log(abs(beta))
# + (alpha - 1)*np.log(beta*(x - zeta)) + (a - 1)*np.log(x)
# - beta*(x - zeta) - x
# - sc.gammalnalpha) - sc.gammalna)
ans, x, transx, mask, invmask, beta, alpha, _ = (
self._preprocess(x, skew))
ans[mask] = np.log(_norm_pdf(x[mask]))
ans[invmask] = np.log(abs(beta)) + gamma._logpdf(transx, alpha)
return ans
def _cdf(self, x, skew):
ans, x, transx, mask, invmask, _, alpha, _ = (
self._preprocess(x, skew))
ans[mask] = _norm_cdf(x[mask])
ans[invmask] = gamma._cdf(transx, alpha)
return ans
def _rvs(self, skew):
skew = broadcast_to(skew, self._size)
ans, _, _, mask, invmask, beta, alpha, zeta = (
self._preprocess([0], skew))
nsmall = mask.sum()
nbig = mask.size - nsmall
ans[mask] = self._random_state.standard_normal(nsmall)
ans[invmask] = (self._random_state.standard_gamma(alpha, nbig)/beta +
zeta)
if self._size == ():
ans = ans[0]
return ans
def _ppf(self, q, skew):
ans, q, _, mask, invmask, beta, alpha, zeta = (
self._preprocess(q, skew))
ans[mask] = _norm_ppf(q[mask])
ans[invmask] = sc.gammaincinv(alpha, q[invmask])/beta + zeta
return ans
pearson3 = pearson3_gen(name="pearson3")
class powerlaw_gen(rv_continuous):
r"""A power-function continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `powerlaw` is:
.. math::
f(x, a) = a x^{a-1}
for :math:`0 \le x \le 1`, :math:`a > 0`.
`powerlaw` takes :math:`a` as a shape parameter.
%(after_notes)s
`powerlaw` is a special case of `beta` with ``b == 1``.
%(example)s
"""
def _pdf(self, x, a):
# powerlaw.pdf(x, a) = a * x**(a-1)
return a*x**(a-1.0)
def _logpdf(self, x, a):
return np.log(a) + sc.xlogy(a - 1, x)
def _cdf(self, x, a):
return x**(a*1.0)
def _logcdf(self, x, a):
return a*np.log(x)
def _ppf(self, q, a):
return pow(q, 1.0/a)
def _stats(self, a):
return (a / (a + 1.0),
a / (a + 2.0) / (a + 1.0) ** 2,
-2.0 * ((a - 1.0) / (a + 3.0)) * np.sqrt((a + 2.0) / a),
6 * np.polyval([1, -1, -6, 2], a) / (a * (a + 3.0) * (a + 4)))
def _entropy(self, a):
return 1 - 1.0/a - np.log(a)
powerlaw = powerlaw_gen(a=0.0, b=1.0, name="powerlaw")
class powerlognorm_gen(rv_continuous):
r"""A power log-normal continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `powerlognorm` is:
.. math::
f(x, c, s) = \frac{c}{x s} \phi(\log(x)/s)
(\Phi(-\log(x)/s))^{c-1}
where :math:`\phi` is the normal pdf, and :math:`\Phi` is the normal cdf,
and :math:`x > 0`, :math:`s, c > 0`.
`powerlognorm` takes :math:`c` and :math:`s` as shape parameters.
%(after_notes)s
%(example)s
"""
_support_mask = rv_continuous._open_support_mask
def _pdf(self, x, c, s):
# powerlognorm.pdf(x, c, s) = c / (x*s) * phi(log(x)/s) *
# (Phi(-log(x)/s))**(c-1),
return (c/(x*s) * _norm_pdf(np.log(x)/s) *
pow(_norm_cdf(-np.log(x)/s), c*1.0-1.0))
def _cdf(self, x, c, s):
return 1.0 - pow(_norm_cdf(-np.log(x)/s), c*1.0)
def _ppf(self, q, c, s):
return np.exp(-s * _norm_ppf(pow(1.0 - q, 1.0 / c)))
powerlognorm = powerlognorm_gen(a=0.0, name="powerlognorm")
class powernorm_gen(rv_continuous):
r"""A power normal continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `powernorm` is:
.. math::
f(x, c) = c \phi(x) (\Phi(-x))^{c-1}
where :math:`\phi` is the normal pdf, and :math:`\Phi` is the normal cdf,
and :math:`x > 0`, :math:`c > 0`.
`powernorm` takes :math:`c` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _pdf(self, x, c):
# powernorm.pdf(x, c) = c * phi(x) * (Phi(-x))**(c-1)
return c*_norm_pdf(x) * (_norm_cdf(-x)**(c-1.0))
def _logpdf(self, x, c):
return np.log(c) + _norm_logpdf(x) + (c-1)*_norm_logcdf(-x)
def _cdf(self, x, c):
return 1.0-_norm_cdf(-x)**(c*1.0)
def _ppf(self, q, c):
return -_norm_ppf(pow(1.0 - q, 1.0 / c))
powernorm = powernorm_gen(name='powernorm')
class rdist_gen(rv_continuous):
r"""An R-distributed continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `rdist` is:
.. math::
f(x, c) = \frac{(1-x^2)^{c/2-1}}{B(1/2, c/2)}
for :math:`-1 \le x \le 1`, :math:`c > 0`.
`rdist` takes :math:`c` as a shape parameter.
This distribution includes the following distribution kernels as
special cases::
c = 2: uniform
c = 4: Epanechnikov (parabolic)
c = 6: quartic (biweight)
c = 8: triweight
%(after_notes)s
%(example)s
"""
def _pdf(self, x, c):
# rdist.pdf(x, c) = (1-x**2)**(c/2-1) / B(1/2, c/2)
return np.power((1.0 - x**2), c / 2.0 - 1) / sc.beta(0.5, c / 2.0)
def _cdf(self, x, c):
term1 = x / sc.beta(0.5, c / 2.0)
res = 0.5 + term1 * sc.hyp2f1(0.5, 1 - c / 2.0, 1.5, x**2)
# There's an issue with hyp2f1, it returns nans near x = +-1, c > 100.
# Use the generic implementation in that case. See gh-1285 for
# background.
if np.any(np.isnan(res)):
return rv_continuous._cdf(self, x, c)
return res
def _munp(self, n, c):
numerator = (1 - (n % 2)) * sc.beta((n + 1.0) / 2, c / 2.0)
return numerator / sc.beta(1. / 2, c / 2.)
rdist = rdist_gen(a=-1.0, b=1.0, name="rdist")
class rayleigh_gen(rv_continuous):
r"""A Rayleigh continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `rayleigh` is:
.. math::
f(r) = r \exp(-r^2/2)
for :math:`x \ge 0`.
`rayleigh` is a special case of `chi` with ``df == 2``.
%(after_notes)s
%(example)s
"""
_support_mask = rv_continuous._open_support_mask
def _rvs(self):
return chi.rvs(2, size=self._size, random_state=self._random_state)
def _pdf(self, r):
# rayleigh.pdf(r) = r * exp(-r**2/2)
return np.exp(self._logpdf(r))
def _logpdf(self, r):
return np.log(r) - 0.5 * r * r
def _cdf(self, r):
return -sc.expm1(-0.5 * r**2)
def _ppf(self, q):
return np.sqrt(-2 * sc.log1p(-q))
def _sf(self, r):
return np.exp(self._logsf(r))
def _logsf(self, r):
return -0.5 * r * r
def _isf(self, q):
return np.sqrt(-2 * np.log(q))
def _stats(self):
val = 4 - np.pi
return (np.sqrt(np.pi/2),
val/2,
2*(np.pi-3)*np.sqrt(np.pi)/val**1.5,
6*np.pi/val-16/val**2)
def _entropy(self):
return _EULER/2.0 + 1 - 0.5*np.log(2)
rayleigh = rayleigh_gen(a=0.0, name="rayleigh")
class reciprocal_gen(rv_continuous):
r"""A reciprocal continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `reciprocal` is:
.. math::
f(x, a, b) = \frac{1}{x \log(b/a)}
for :math:`a \le x \le b`, :math:`a, b > 0`.
`reciprocal` takes :math:`a` and :math:`b` as shape parameters.
%(after_notes)s
%(example)s
"""
def _argcheck(self, a, b):
self.a = a
self.b = b
self.d = np.log(b*1.0 / a)
return (a > 0) & (b > 0) & (b > a)
def _pdf(self, x, a, b):
# reciprocal.pdf(x, a, b) = 1 / (x*log(b/a))
return 1.0 / (x * self.d)
def _logpdf(self, x, a, b):
return -np.log(x) - np.log(self.d)
def _cdf(self, x, a, b):
return (np.log(x)-np.log(a)) / self.d
def _ppf(self, q, a, b):
return a*pow(b*1.0/a, q)
def _munp(self, n, a, b):
return 1.0/self.d / n * (pow(b*1.0, n) - pow(a*1.0, n))
def _entropy(self, a, b):
return 0.5*np.log(a*b)+np.log(np.log(b/a))
reciprocal = reciprocal_gen(name="reciprocal")
class rice_gen(rv_continuous):
r"""A Rice continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `rice` is:
.. math::
f(x, b) = x \exp(- \frac{x^2 + b^2}{2}) I[0](x b)
for :math:`x > 0`, :math:`b > 0`.
`rice` takes :math:`b` as a shape parameter.
%(after_notes)s
The Rice distribution describes the length, :math:`r`, of a 2-D vector with
components :math:`(U+u, V+v)`, where :math:`U, V` are constant, :math:`u,
v` are independent Gaussian random variables with standard deviation
:math:`s`. Let :math:`R = \sqrt{U^2 + V^2}`. Then the pdf of :math:`r` is
``rice.pdf(x, R/s, scale=s)``.
%(example)s
"""
def _argcheck(self, b):
return b >= 0
def _rvs(self, b):
# http://en.wikipedia.org/wiki/Rice_distribution
t = b/np.sqrt(2) + self._random_state.standard_normal(size=(2,) +
self._size)
return np.sqrt((t*t).sum(axis=0))
def _cdf(self, x, b):
return sc.chndtr(np.square(x), 2, np.square(b))
def _ppf(self, q, b):
return np.sqrt(sc.chndtrix(q, 2, np.square(b)))
def _pdf(self, x, b):
# rice.pdf(x, b) = x * exp(-(x**2+b**2)/2) * I[0](x*b)
#
# We use (x**2 + b**2)/2 = ((x-b)**2)/2 + xb.
# The factor of np.exp(-xb) is then included in the i0e function
# in place of the modified Bessel function, i0, improving
# numerical stability for large values of xb.
return x * np.exp(-(x-b)*(x-b)/2.0) * sc.i0e(x*b)
def _munp(self, n, b):
nd2 = n/2.0
n1 = 1 + nd2
b2 = b*b/2.0
return (2.0**(nd2) * np.exp(-b2) * sc.gamma(n1) *
sc.hyp1f1(n1, 1, b2))
rice = rice_gen(a=0.0, name="rice")
# FIXME: PPF does not work.
class recipinvgauss_gen(rv_continuous):
r"""A reciprocal inverse Gaussian continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `recipinvgauss` is:
.. math::
f(x, \mu) = \frac{1}{\sqrt{2\pi x}} \frac{\exp(-(1-\mu x)^2}{2x\mu^2)}
for :math:`x \ge 0`.
`recipinvgauss` takes :math:`\mu` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _pdf(self, x, mu):
# recipinvgauss.pdf(x, mu) =
# 1/sqrt(2*pi*x) * exp(-(1-mu*x)**2/(2*x*mu**2))
return 1.0/np.sqrt(2*np.pi*x)*np.exp(-(1-mu*x)**2.0 / (2*x*mu**2.0))
def _logpdf(self, x, mu):
return -(1-mu*x)**2.0 / (2*x*mu**2.0) - 0.5*np.log(2*np.pi*x)
def _cdf(self, x, mu):
trm1 = 1.0/mu - x
trm2 = 1.0/mu + x
isqx = 1.0/np.sqrt(x)
return 1.0-_norm_cdf(isqx*trm1)-np.exp(2.0/mu)*_norm_cdf(-isqx*trm2)
def _rvs(self, mu):
return 1.0/self._random_state.wald(mu, 1.0, size=self._size)
recipinvgauss = recipinvgauss_gen(a=0.0, name='recipinvgauss')
class semicircular_gen(rv_continuous):
r"""A semicircular continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `semicircular` is:
.. math::
f(x) = \frac{2}{\pi} \sqrt{1-x^2}
for :math:`-1 \le x \le 1`.
%(after_notes)s
%(example)s
"""
def _pdf(self, x):
# semicircular.pdf(x) = 2/pi * sqrt(1-x**2)
return 2.0/np.pi*np.sqrt(1-x*x)
def _cdf(self, x):
return 0.5+1.0/np.pi*(x*np.sqrt(1-x*x) + np.arcsin(x))
def _stats(self):
return 0, 0.25, 0, -1.0
def _entropy(self):
return 0.64472988584940017414
semicircular = semicircular_gen(a=-1.0, b=1.0, name="semicircular")
class skew_norm_gen(rv_continuous):
r"""A skew-normal random variable.
%(before_notes)s
Notes
-----
The pdf is::
skewnorm.pdf(x, a) = 2 * norm.pdf(x) * norm.cdf(a*x)
`skewnorm` takes :math:`a` as a skewness parameter
When ``a = 0`` the distribution is identical to a normal distribution.
rvs implements the method of [1]_.
%(after_notes)s
%(example)s
References
----------
.. [1] A. Azzalini and A. Capitanio (1999). Statistical applications of the
multivariate skew-normal distribution. J. Roy. Statist. Soc., B 61, 579-602.
http://azzalini.stat.unipd.it/SN/faq-r.html
"""
def _argcheck(self, a):
return np.isfinite(a)
def _pdf(self, x, a):
return 2.*_norm_pdf(x)*_norm_cdf(a*x)
def _cdf_single(self, x, *args):
if x <= 0:
cdf = integrate.quad(self._pdf, self.a, x, args=args)[0]
else:
t1 = integrate.quad(self._pdf, self.a, 0, args=args)[0]
t2 = integrate.quad(self._pdf, 0, x, args=args)[0]
cdf = t1 + t2
if cdf > 1:
# Presumably numerical noise, e.g. 1.0000000000000002
cdf = 1.0
return cdf
def _sf(self, x, a):
return self._cdf(-x, -a)
def _rvs(self, a):
u0 = self._random_state.normal(size=self._size)
v = self._random_state.normal(size=self._size)
d = a/np.sqrt(1 + a**2)
u1 = d*u0 + v*np.sqrt(1 - d**2)
return np.where(u0 >= 0, u1, -u1)
def _stats(self, a, moments='mvsk'):
output = [None, None, None, None]
const = np.sqrt(2/np.pi) * a/np.sqrt(1 + a**2)
if 'm' in moments:
output[0] = const
if 'v' in moments:
output[1] = 1 - const**2
if 's' in moments:
output[2] = ((4 - np.pi)/2) * (const/np.sqrt(1 - const**2))**3
if 'k' in moments:
output[3] = (2*(np.pi - 3)) * (const**4/(1 - const**2)**2)
return output
skewnorm = skew_norm_gen(name='skewnorm')
class trapz_gen(rv_continuous):
r"""A trapezoidal continuous random variable.
%(before_notes)s
Notes
-----
The trapezoidal distribution can be represented with an up-sloping line
from ``loc`` to ``(loc + c*scale)``, then constant to ``(loc + d*scale)``
and then downsloping from ``(loc + d*scale)`` to ``(loc+scale)``.
`trapz` takes :math:`c` and :math:`d` as shape parameters.
%(after_notes)s
The standard form is in the range [0, 1] with c the mode.
The location parameter shifts the start to `loc`.
The scale parameter changes the width from 1 to `scale`.
%(example)s
"""
def _argcheck(self, c, d):
return (c >= 0) & (c <= 1) & (d >= 0) & (d <= 1) & (d >= c)
def _pdf(self, x, c, d):
u = 2 / (d-c+1)
return _lazyselect([x < c,
(c <= x) & (x <= d),
x > d],
[lambda x, c, d, u: u * x / c,
lambda x, c, d, u: u,
lambda x, c, d, u: u * (1-x) / (1-d)],
(x, c, d, u))
def _cdf(self, x, c, d):
return _lazyselect([x < c,
(c <= x) & (x <= d),
x > d],
[lambda x, c, d: x**2 / c / (d-c+1),
lambda x, c, d: (c + 2 * (x-c)) / (d-c+1),
lambda x, c, d: 1-((1-x) ** 2
/ (d-c+1) / (1-d))],
(x, c, d))
def _ppf(self, q, c, d):
qc, qd = self._cdf(c, c, d), self._cdf(d, c, d)
condlist = [q < qc, q <= qd, q > qd]
choicelist = [np.sqrt(q * c * (1 + d - c)),
0.5 * q * (1 + d - c) + 0.5 * c,
1 - np.sqrt((1 - q) * (d - c + 1) * (1 - d))]
return np.select(condlist, choicelist)
trapz = trapz_gen(a=0.0, b=1.0, name="trapz")
class triang_gen(rv_continuous):
r"""A triangular continuous random variable.
%(before_notes)s
Notes
-----
The triangular distribution can be represented with an up-sloping line from
``loc`` to ``(loc + c*scale)`` and then downsloping for ``(loc + c*scale)``
to ``(loc+scale)``.
`triang` takes :math:`c` as a shape parameter.
%(after_notes)s
The standard form is in the range [0, 1] with c the mode.
The location parameter shifts the start to `loc`.
The scale parameter changes the width from 1 to `scale`.
%(example)s
"""
def _rvs(self, c):
return self._random_state.triangular(0, c, 1, self._size)
def _argcheck(self, c):
return (c >= 0) & (c <= 1)
def _pdf(self, x, c):
# 0: edge case where c=0
# 1: generalised case for x < c, don't use x <= c, as it doesn't cope
# with c = 0.
# 2: generalised case for x >= c, but doesn't cope with c = 1
# 3: edge case where c=1
r = _lazyselect([c == 0,
x < c,
(x >= c) & (c != 1),
c == 1],
[lambda x, c: 2 - 2 * x,
lambda x, c: 2 * x / c,
lambda x, c: 2 * (1 - x) / (1 - c),
lambda x, c: 2 * x],
(x, c))
return r
def _cdf(self, x, c):
r = _lazyselect([c == 0,
x < c,
(x >= c) & (c != 1),
c == 1],
[lambda x, c: 2*x - x*x,
lambda x, c: x * x / c,
lambda x, c: (x*x - 2*x + c) / (c-1),
lambda x, c: x * x],
(x, c))
return r
def _ppf(self, q, c):
return np.where(q < c, np.sqrt(c * q), 1-np.sqrt((1-c) * (1-q)))
def _stats(self, c):
return ((c+1.0)/3.0,
(1.0-c+c*c)/18,
np.sqrt(2)*(2*c-1)*(c+1)*(c-2) / (5*np.power((1.0-c+c*c), 1.5)),
-3.0/5.0)
def _entropy(self, c):
return 0.5-np.log(2)
triang = triang_gen(a=0.0, b=1.0, name="triang")
class truncexpon_gen(rv_continuous):
r"""A truncated exponential continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `truncexpon` is:
.. math::
f(x, b) = \frac{\exp(-x)}{1 - \exp(-b)}
for :math:`0 < x < b`.
`truncexpon` takes :math:`b` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _argcheck(self, b):
self.b = b
return b > 0
def _pdf(self, x, b):
# truncexpon.pdf(x, b) = exp(-x) / (1-exp(-b))
return np.exp(-x)/(-sc.expm1(-b))
def _logpdf(self, x, b):
return -x - np.log(-sc.expm1(-b))
def _cdf(self, x, b):
return sc.expm1(-x)/sc.expm1(-b)
def _ppf(self, q, b):
return -sc.log1p(q*sc.expm1(-b))
def _munp(self, n, b):
# wrong answer with formula, same as in continuous.pdf
# return sc.gamman+1)-sc.gammainc1+n, b)
if n == 1:
return (1-(b+1)*np.exp(-b))/(-sc.expm1(-b))
elif n == 2:
return 2*(1-0.5*(b*b+2*b+2)*np.exp(-b))/(-sc.expm1(-b))
else:
# return generic for higher moments
# return rv_continuous._mom1_sc(self, n, b)
return self._mom1_sc(n, b)
def _entropy(self, b):
eB = np.exp(b)
return np.log(eB-1)+(1+eB*(b-1.0))/(1.0-eB)
truncexpon = truncexpon_gen(a=0.0, name='truncexpon')
class truncnorm_gen(rv_continuous):
r"""A truncated normal continuous random variable.
%(before_notes)s
Notes
-----
The standard form of this distribution is a standard normal truncated to
the range [a, b] --- notice that a and b are defined over the domain of the
standard normal. To convert clip values for a specific mean and standard
deviation, use::
a, b = (myclip_a - my_mean) / my_std, (myclip_b - my_mean) / my_std
`truncnorm` takes :math:`a` and :math:`b` as shape parameters.
%(after_notes)s
%(example)s
"""
def _argcheck(self, a, b):
self.a = a
self.b = b
self._nb = _norm_cdf(b)
self._na = _norm_cdf(a)
self._sb = _norm_sf(b)
self._sa = _norm_sf(a)
self._delta = np.where(self.a > 0,
-(self._sb - self._sa),
self._nb - self._na)
self._logdelta = np.log(self._delta)
return a != b
def _pdf(self, x, a, b):
return _norm_pdf(x) / self._delta
def _logpdf(self, x, a, b):
return _norm_logpdf(x) - self._logdelta
def _cdf(self, x, a, b):
return (_norm_cdf(x) - self._na) / self._delta
def _ppf(self, q, a, b):
# XXX Use _lazywhere...
ppf = np.where(self.a > 0,
_norm_isf(q*self._sb + self._sa*(1.0-q)),
_norm_ppf(q*self._nb + self._na*(1.0-q)))
return ppf
def _stats(self, a, b):
nA, nB = self._na, self._nb
d = nB - nA
pA, pB = _norm_pdf(a), _norm_pdf(b)
mu = (pA - pB) / d # correction sign
mu2 = 1 + (a*pA - b*pB) / d - mu*mu
return mu, mu2, None, None
truncnorm = truncnorm_gen(name='truncnorm')
# FIXME: RVS does not work.
class tukeylambda_gen(rv_continuous):
r"""A Tukey-Lamdba continuous random variable.
%(before_notes)s
Notes
-----
A flexible distribution, able to represent and interpolate between the
following distributions:
- Cauchy (lam=-1)
- logistic (lam=0.0)
- approx Normal (lam=0.14)
- u-shape (lam = 0.5)
- uniform from -1 to 1 (lam = 1)
`tukeylambda` takes ``lam`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _argcheck(self, lam):
return np.ones(np.shape(lam), dtype=bool)
def _pdf(self, x, lam):
Fx = np.asarray(sc.tklmbda(x, lam))
Px = Fx**(lam-1.0) + (np.asarray(1-Fx))**(lam-1.0)
Px = 1.0/np.asarray(Px)
return np.where((lam <= 0) | (abs(x) < 1.0/np.asarray(lam)), Px, 0.0)
def _cdf(self, x, lam):
return sc.tklmbda(x, lam)
def _ppf(self, q, lam):
return sc.boxcox(q, lam) - sc.boxcox1p(-q, lam)
def _stats(self, lam):
return 0, _tlvar(lam), 0, _tlkurt(lam)
def _entropy(self, lam):
def integ(p):
return np.log(pow(p, lam-1)+pow(1-p, lam-1))
return integrate.quad(integ, 0, 1)[0]
tukeylambda = tukeylambda_gen(name='tukeylambda')
class FitUniformFixedScaleDataError(FitDataError):
def __init__(self, ptp, fscale):
self.args = (
"Invalid values in `data`. Maximum likelihood estimation with "
"the uniform distribution and fixed scale requires that "
"data.ptp() <= fscale, but data.ptp() = %r and fscale = %r." %
(ptp, fscale),
)
class uniform_gen(rv_continuous):
r"""A uniform continuous random variable.
This distribution is constant between `loc` and ``loc + scale``.
%(before_notes)s
%(example)s
"""
def _rvs(self):
return self._random_state.uniform(0.0, 1.0, self._size)
def _pdf(self, x):
return 1.0*(x == x)
def _cdf(self, x):
return x
def _ppf(self, q):
return q
def _stats(self):
return 0.5, 1.0/12, 0, -1.2
def _entropy(self):
return 0.0
def fit(self, data, *args, **kwds):
"""
Maximum likelihood estimate for the location and scale parameters.
`uniform.fit` uses only the following parameters. Because exact
formulas are used, the parameters related to optimization that are
available in the `fit` method of other distributions are ignored
here. The only positional argument accepted is `data`.
Parameters
----------
data : array_like
Data to use in calculating the maximum likelihood estimate.
floc : float, optional
Hold the location parameter fixed to the specified value.
fscale : float, optional
Hold the scale parameter fixed to the specified value.
Returns
-------
loc, scale : float
Maximum likelihood estimates for the location and scale.
Notes
-----
An error is raised if `floc` is given and any values in `data` are
less than `floc`, or if `fscale` is given and `fscale` is less
than ``data.max() - data.min()``. An error is also raised if both
`floc` and `fscale` are given.
Examples
--------
>>> from scipy.stats import uniform
We'll fit the uniform distribution to `x`:
>>> x = np.array([2, 2.5, 3.1, 9.5, 13.0])
For a uniform distribution MLE, the location is the minimum of the
data, and the scale is the maximum minus the minimum.
>>> loc, scale = uniform.fit(x)
>>> loc
2.0
>>> scale
11.0
If we know the data comes from a uniform distribution where the support
starts at 0, we can use `floc=0`:
>>> loc, scale = uniform.fit(x, floc=0)
>>> loc
0.0
>>> scale
13.0
Alternatively, if we know the length of the support is 12, we can use
`fscale=12`:
>>> loc, scale = uniform.fit(x, fscale=12)
>>> loc
1.5
>>> scale
12.0
In that last example, the support interval is [1.5, 13.5]. This
solution is not unique. For example, the distribution with ``loc=2``
and ``scale=12`` has the same likelihood as the one above. When
`fscale` is given and it is larger than ``data.max() - data.min()``,
the parameters returned by the `fit` method center the support over
the interval ``[data.min(), data.max()]``.
"""
if len(args) > 0:
raise TypeError("Too many arguments.")
floc = kwds.pop('floc', None)
fscale = kwds.pop('fscale', None)
# Ignore the optimizer-related keyword arguments, if given.
kwds.pop('loc', None)
kwds.pop('scale', None)
kwds.pop('optimizer', None)
if kwds:
raise TypeError("Unknown arguments: %s." % kwds)
if floc is not None and fscale is not None:
# This check is for consistency with `rv_continuous.fit`.
raise ValueError("All parameters fixed. There is nothing to "
"optimize.")
data = np.asarray(data)
# MLE for the uniform distribution
# --------------------------------
# The PDF is
#
# f(x, loc, scale) = {1/scale for loc <= x <= loc + scale
# {0 otherwise}
#
# The likelihood function is
# L(x, loc, scale) = (1/scale)**n
# where n is len(x), assuming loc <= x <= loc + scale for all x.
# The log-likelihood is
# l(x, loc, scale) = -n*log(scale)
# The log-likelihood is maximized by making scale as small as possible,
# while keeping loc <= x <= loc + scale. So if neither loc nor scale
# are fixed, the log-likelihood is maximized by choosing
# loc = x.min()
# scale = x.ptp()
# If loc is fixed, it must be less than or equal to x.min(), and then
# the scale is
# scale = x.max() - loc
# If scale is fixed, it must not be less than x.ptp(). If scale is
# greater than x.ptp(), the solution is not unique. Note that the
# likelihood does not depend on loc, except for the requirement that
# loc <= x <= loc + scale. All choices of loc for which
# x.max() - scale <= loc <= x.min()
# have the same log-likelihood. In this case, we choose loc such that
# the support is centered over the interval [data.min(), data.max()]:
# loc = x.min() = 0.5*(scale - x.ptp())
if fscale is None:
# scale is not fixed.
if floc is None:
# loc is not fixed, scale is not fixed.
loc = data.min()
scale = data.ptp()
else:
# loc is fixed, scale is not fixed.
loc = floc
scale = data.max() - loc
if data.min() < loc:
raise FitDataError("uniform", lower=loc, upper=loc + scale)
else:
# loc is not fixed, scale is fixed.
ptp = data.ptp()
if ptp > fscale:
raise FitUniformFixedScaleDataError(ptp=ptp, fscale=fscale)
# If ptp < fscale, the ML estimate is not unique; see the comments
# above. We choose the distribution for which the support is
# centered over the interval [data.min(), data.max()].
loc = data.min() - 0.5*(fscale - ptp)
scale = fscale
# We expect the return values to be floating point, so ensure it
# by explicitly converting to float.
return float(loc), float(scale)
uniform = uniform_gen(a=0.0, b=1.0, name='uniform')
class vonmises_gen(rv_continuous):
r"""A Von Mises continuous random variable.
%(before_notes)s
Notes
-----
If `x` is not in range or `loc` is not in range it assumes they are angles
and converts them to [-\pi, \pi] equivalents.
The probability density function for `vonmises` is:
.. math::
f(x, \kappa) = \frac{ \exp(\kappa \cos(x)) }{ 2 \pi I[0](\kappa) }
for :math:`-\pi \le x \le \pi`, :math:`\kappa > 0`.
`vonmises` takes :math:`\kappa` as a shape parameter.
%(after_notes)s
See Also
--------
vonmises_line : The same distribution, defined on a [-\pi, \pi] segment
of the real line.
%(example)s
"""
def _rvs(self, kappa):
return self._random_state.vonmises(0.0, kappa, size=self._size)
def _pdf(self, x, kappa):
# vonmises.pdf(x, \kappa) = exp(\kappa * cos(x)) / (2*pi*I[0](\kappa))
return np.exp(kappa * np.cos(x)) / (2*np.pi*sc.i0(kappa))
def _cdf(self, x, kappa):
return _stats.von_mises_cdf(kappa, x)
def _stats_skip(self, kappa):
return 0, None, 0, None
def _entropy(self, kappa):
return (-kappa * sc.i1(kappa) / sc.i0(kappa) +
np.log(2 * np.pi * sc.i0(kappa)))
vonmises = vonmises_gen(name='vonmises')
vonmises_line = vonmises_gen(a=-np.pi, b=np.pi, name='vonmises_line')
class wald_gen(invgauss_gen):
r"""A Wald continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `wald` is:
.. math::
f(x) = \frac{1}{\sqrt{2\pi x^3}} \exp(- \frac{ (x-1)^2 }{ 2x })
for :math:`x > 0`.
`wald` is a special case of `invgauss` with ``mu == 1``.
%(after_notes)s
%(example)s
"""
_support_mask = rv_continuous._open_support_mask
def _rvs(self):
return self._random_state.wald(1.0, 1.0, size=self._size)
def _pdf(self, x):
# wald.pdf(x) = 1/sqrt(2*pi*x**3) * exp(-(x-1)**2/(2*x))
return invgauss._pdf(x, 1.0)
def _logpdf(self, x):
return invgauss._logpdf(x, 1.0)
def _cdf(self, x):
return invgauss._cdf(x, 1.0)
def _stats(self):
return 1.0, 1.0, 3.0, 15.0
wald = wald_gen(a=0.0, name="wald")
class wrapcauchy_gen(rv_continuous):
r"""A wrapped Cauchy continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `wrapcauchy` is:
.. math::
f(x, c) = \frac{1-c^2}{2\pi (1+c^2 - 2c \cos(x))}
for :math:`0 \le x \le 2\pi`, :math:`0 < c < 1`.
`wrapcauchy` takes :math:`c` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _argcheck(self, c):
return (c > 0) & (c < 1)
def _pdf(self, x, c):
# wrapcauchy.pdf(x, c) = (1-c**2) / (2*pi*(1+c**2-2*c*cos(x)))
return (1.0-c*c)/(2*np.pi*(1+c*c-2*c*np.cos(x)))
def _cdf(self, x, c):
output = np.zeros(x.shape, dtype=x.dtype)
val = (1.0+c)/(1.0-c)
c1 = x < np.pi
c2 = 1-c1
xp = np.extract(c1, x)
xn = np.extract(c2, x)
if np.any(xn):
valn = np.extract(c2, np.ones_like(x)*val)
xn = 2*np.pi - xn
yn = np.tan(xn/2.0)
on = 1.0-1.0/np.pi*np.arctan(valn*yn)
np.place(output, c2, on)
if np.any(xp):
valp = np.extract(c1, np.ones_like(x)*val)
yp = np.tan(xp/2.0)
op = 1.0/np.pi*np.arctan(valp*yp)
np.place(output, c1, op)
return output
def _ppf(self, q, c):
val = (1.0-c)/(1.0+c)
rcq = 2*np.arctan(val*np.tan(np.pi*q))
rcmq = 2*np.pi-2*np.arctan(val*np.tan(np.pi*(1-q)))
return np.where(q < 1.0/2, rcq, rcmq)
def _entropy(self, c):
return np.log(2*np.pi*(1-c*c))
wrapcauchy = wrapcauchy_gen(a=0.0, b=2*np.pi, name='wrapcauchy')
class gennorm_gen(rv_continuous):
r"""A generalized normal continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `gennorm` is [1]_::
beta
gennorm.pdf(x, beta) = --------------- exp(-|x|**beta)
2 gamma(1/beta)
`gennorm` takes :math:`\beta` as a shape parameter.
For :math:`\beta = 1`, it is identical to a Laplace distribution.
For ``\beta = 2``, it is identical to a normal distribution
(with :math:`scale=1/\sqrt{2}`).
See Also
--------
laplace : Laplace distribution
norm : normal distribution
References
----------
.. [1] "Generalized normal distribution, Version 1",
https://en.wikipedia.org/wiki/Generalized_normal_distribution#Version_1
%(example)s
"""
def _pdf(self, x, beta):
return np.exp(self._logpdf(x, beta))
def _logpdf(self, x, beta):
return np.log(0.5*beta) - sc.gammaln(1.0/beta) - abs(x)**beta
def _cdf(self, x, beta):
c = 0.5 * np.sign(x)
# evaluating (.5 + c) first prevents numerical cancellation
return (0.5 + c) - c * sc.gammaincc(1.0/beta, abs(x)**beta)
def _ppf(self, x, beta):
c = np.sign(x - 0.5)
# evaluating (1. + c) first prevents numerical cancellation
return c * sc.gammainccinv(1.0/beta, (1.0 + c) - 2.0*c*x)**(1.0/beta)
def _sf(self, x, beta):
return self._cdf(-x, beta)
def _isf(self, x, beta):
return -self._ppf(x, beta)
def _stats(self, beta):
c1, c3, c5 = sc.gammaln([1.0/beta, 3.0/beta, 5.0/beta])
return 0., np.exp(c3 - c1), 0., np.exp(c5 + c1 - 2.0*c3) - 3.
def _entropy(self, beta):
return 1. / beta - np.log(.5 * beta) + sc.gammaln(1. / beta)
gennorm = gennorm_gen(name='gennorm')
class halfgennorm_gen(rv_continuous):
r"""The upper half of a generalized normal continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `halfgennorm` is:
.. math::
f(x, \beta) = \frac{\beta}{\gamma(1/\beta)} \exp(-|x|^\beta)
`gennorm` takes :math:`\beta` as a shape parameter.
For :math:`\beta = 1`, it is identical to an exponential distribution.
For :math:`\beta = 2`, it is identical to a half normal distribution
(with :math:`scale=1/\sqrt{2}`).
See Also
--------
gennorm : generalized normal distribution
expon : exponential distribution
halfnorm : half normal distribution
References
----------
.. [1] "Generalized normal distribution, Version 1",
https://en.wikipedia.org/wiki/Generalized_normal_distribution#Version_1
%(example)s
"""
def _pdf(self, x, beta):
# beta
# halfgennorm.pdf(x, beta) = ------------- exp(-|x|**beta)
# gamma(1/beta)
return np.exp(self._logpdf(x, beta))
def _logpdf(self, x, beta):
return np.log(beta) - sc.gammaln(1.0/beta) - x**beta
def _cdf(self, x, beta):
return sc.gammainc(1.0/beta, x**beta)
def _ppf(self, x, beta):
return sc.gammaincinv(1.0/beta, x)**(1.0/beta)
def _sf(self, x, beta):
return sc.gammaincc(1.0/beta, x**beta)
def _isf(self, x, beta):
return sc.gammainccinv(1.0/beta, x)**(1.0/beta)
def _entropy(self, beta):
return 1.0/beta - np.log(beta) + sc.gammaln(1.0/beta)
halfgennorm = halfgennorm_gen(a=0, name='halfgennorm')
class crystalball_gen(rv_continuous):
r"""
Crystalball distribution
%(before_notes)s
Notes
-----
The probability density function for `crystalball` is:
.. math::
f(x, \beta, m) = \begin{cases}
N \exp(-x^2 / 2), &\text{for } x > -\beta\\
N A (B - x)^{-m} &\text{for } x \le -\beta
\end{cases}
where :math:`A = (m / |beta|)**n * exp(-beta**2 / 2)`,
:math:`B = m/|beta| - |beta|` and :math:`N` is a normalisation constant.
`crystalball` takes :math:`\beta` and :math:`m` as shape parameters.
:math:`\beta` defines the point where the pdf changes from a power-law to a
gaussian distribution :math:`m` is power of the power-law tail.
References
----------
.. [1] "Crystal Ball Function",
https://en.wikipedia.org/wiki/Crystal_Ball_function
%(after_notes)s
.. versionadded:: 0.19.0
%(example)s
"""
def _pdf(self, x, beta, m):
"""
Return PDF of the crystalball function.
--
| exp(-x**2 / 2), for x > -beta
crystalball.pdf(x, beta, m) = N * |
| A * (B - x)**(-m), for x <= -beta
--
"""
N = 1.0 / (m/beta / (m-1) * np.exp(-beta**2 / 2.0) + _norm_pdf_C * _norm_cdf(beta))
rhs = lambda x, beta, m: np.exp(-x**2 / 2)
lhs = lambda x, beta, m: (m/beta)**m * np.exp(-beta**2 / 2.0) * (m/beta - beta - x)**(-m)
return N * _lazywhere(np.atleast_1d(x > -beta), (x, beta, m), f=rhs, f2=lhs)
def _cdf(self, x, beta, m):
"""
Return CDF of the crystalball function
"""
N = 1.0 / (m/beta / (m-1) * np.exp(-beta**2 / 2.0) + _norm_pdf_C * _norm_cdf(beta))
rhs = lambda x, beta, m: (m/beta) * np.exp(-beta**2 / 2.0) / (m-1) + _norm_pdf_C * (_norm_cdf(x) - _norm_cdf(-beta))
lhs = lambda x, beta, m: (m/beta)**m * np.exp(-beta**2 / 2.0) * (m/beta - beta - x)**(-m+1) / (m-1)
return N * _lazywhere(np.atleast_1d(x > -beta), (x, beta, m), f=rhs, f2=lhs)
def _munp(self, n, beta, m):
"""
Returns the n-th non-central moment of the crystalball function.
"""
N = 1.0 / (m/beta / (m-1) * np.exp(-beta**2 / 2.0) + _norm_pdf_C * _norm_cdf(beta))
def n_th_moment(n, beta, m):
"""
Returns n-th moment. Defined only if n+1 < m
Function cannot broadcast due to the loop over n
"""
A = (m/beta)**m * np.exp(-beta**2 / 2.0)
B = m/beta - beta
rhs = 2**((n-1)/2.0) * sc.gamma((n+1)/2) * (1.0 + (-1)**n * sc.gammainc((n+1)/2, beta**2 / 2))
lhs = np.zeros(rhs.shape)
for k in range(n + 1):
lhs += sc.binom(n, k) * B**(n-k) * (-1)**k / (m - k - 1) * (m/beta)**(-m + k + 1)
return A * lhs + rhs
return N * _lazywhere(np.atleast_1d(n + 1 < m),
(n, beta, m),
np.vectorize(n_th_moment, otypes=[np.float]),
np.inf)
def _argcheck(self, beta, m):
"""
In HEP crystal-ball is also defined for m = 1 (see plot on wikipedia)
But the function doesn't have a finite integral in this corner case,
and isn't a PDF anymore (but can still be used on a finite range).
Here we restrict the function to m > 1.
In addition we restrict beta to be positive
"""
return (m > 1) & (beta > 0)
crystalball = crystalball_gen(name='crystalball', longname="A Crystalball Function")
def _argus_phi(chi):
"""
Utility function for the argus distribution
used in the CDF and norm of the Argus Funktion
"""
return _norm_cdf(chi) - chi * _norm_pdf(chi) - 0.5
class argus_gen(rv_continuous):
r"""
Argus distribution
%(before_notes)s
Notes
-----
The probability density function for `argus` is:
.. math::
f(x, \chi) = \frac{\chi^3}{\sqrt{2\pi} \Psi(\chi)} x \sqrt{1-x^2}
\exp(- 0.5 \chi^2 (1 - x^2))
where:
.. math::
\Psi(\chi) = \Phi(\chi) - \chi \phi(\chi) - 1/2
with :math:`\Phi` and :math:`\phi` being the CDF and PDF of a standard
normal distribution, respectively.
`argus` takes :math:`\chi` as shape a parameter.
References
----------
.. [1] "ARGUS distribution",
https://en.wikipedia.org/wiki/ARGUS_distribution
%(after_notes)s
.. versionadded:: 0.19.0
%(example)s
"""
def _pdf(self, x, chi):
"""
Return PDF of the argus function
argus.pdf(x, chi) = chi**3 / (sqrt(2*pi) * Psi(chi)) * x *
sqrt(1-x**2) * exp(- 0.5 * chi**2 * (1 - x**2))
"""
y = 1.0 - x**2
return chi**3 / (_norm_pdf_C * _argus_phi(chi)) * x * np.sqrt(y) * np.exp(-chi**2 * y / 2)
def _cdf(self, x, chi):
"""
Return CDF of the argus function
"""
return 1.0 - self._sf(x, chi)
def _sf(self, x, chi):
"""
Return survival function of the argus function
"""
return _argus_phi(chi * np.sqrt(1 - x**2)) / _argus_phi(chi)
argus = argus_gen(name='argus', longname="An Argus Function", a=0.0, b=1.0)
class rv_histogram(rv_continuous):
"""
Generates a distribution given by a histogram.
This is useful to generate a template distribution from a binned
datasample.
As a subclass of the `rv_continuous` class, `rv_histogram` inherits from it
a collection of generic methods (see `rv_continuous` for the full list),
and implements them based on the properties of the provided binned
datasample.
Parameters
----------
histogram : tuple of array_like
Tuple containing two array_like objects
The first containing the content of n bins
The second containing the (n+1) bin boundaries
In particular the return value np.histogram is accepted
Notes
-----
There are no additional shape parameters except for the loc and scale.
The pdf is defined as a stepwise function from the provided histogram
The cdf is a linear interpolation of the pdf.
.. versionadded:: 0.19.0
Examples
--------
Create a scipy.stats distribution from a numpy histogram
>>> import scipy.stats
>>> import numpy as np
>>> data = scipy.stats.norm.rvs(size=100000, loc=0, scale=1.5, random_state=123)
>>> hist = np.histogram(data, bins=100)
>>> hist_dist = scipy.stats.rv_histogram(hist)
Behaves like an ordinary scipy rv_continuous distribution
>>> hist_dist.pdf(1.0)
0.20538577847618705
>>> hist_dist.cdf(2.0)
0.90818568543056499
PDF is zero above (below) the highest (lowest) bin of the histogram,
defined by the max (min) of the original dataset
>>> hist_dist.pdf(np.max(data))
0.0
>>> hist_dist.cdf(np.max(data))
1.0
>>> hist_dist.pdf(np.min(data))
7.7591907244498314e-05
>>> hist_dist.cdf(np.min(data))
0.0
PDF and CDF follow the histogram
>>> import matplotlib.pyplot as plt
>>> X = np.linspace(-5.0, 5.0, 100)
>>> plt.title("PDF from Template")
>>> plt.hist(data, density=True, bins=100)
>>> plt.plot(X, hist_dist.pdf(X), label='PDF')
>>> plt.plot(X, hist_dist.cdf(X), label='CDF')
>>> plt.show()
"""
_support_mask = rv_continuous._support_mask
def __init__(self, histogram, *args, **kwargs):
"""
Create a new distribution using the given histogram
Parameters
----------
histogram : tuple of array_like
Tuple containing two array_like objects
The first containing the content of n bins
The second containing the (n+1) bin boundaries
In particular the return value np.histogram is accepted
"""
self._histogram = histogram
if len(histogram) != 2:
raise ValueError("Expected length 2 for parameter histogram")
self._hpdf = np.asarray(histogram[0])
self._hbins = np.asarray(histogram[1])
if len(self._hpdf) + 1 != len(self._hbins):
raise ValueError("Number of elements in histogram content "
"and histogram boundaries do not match, "
"expected n and n+1.")
self._hbin_widths = self._hbins[1:] - self._hbins[:-1]
self._hpdf = self._hpdf / float(np.sum(self._hpdf * self._hbin_widths))
self._hcdf = np.cumsum(self._hpdf * self._hbin_widths)
self._hpdf = np.hstack([0.0, self._hpdf, 0.0])
self._hcdf = np.hstack([0.0, self._hcdf])
# Set support
kwargs['a'] = self._hbins[0]
kwargs['b'] = self._hbins[-1]
super(rv_histogram, self).__init__(*args, **kwargs)
def _pdf(self, x):
"""
PDF of the histogram
"""
return self._hpdf[np.searchsorted(self._hbins, x, side='right')]
def _cdf(self, x):
"""
CDF calculated from the histogram
"""
return np.interp(x, self._hbins, self._hcdf)
def _ppf(self, x):
"""
Percentile function calculated from the histogram
"""
return np.interp(x, self._hcdf, self._hbins)
def _munp(self, n):
"""Compute the n-th non-central moment."""
integrals = (self._hbins[1:]**(n+1) - self._hbins[:-1]**(n+1)) / (n+1)
return np.sum(self._hpdf[1:-1] * integrals)
def _entropy(self):
"""Compute entropy of distribution"""
res = _lazywhere(self._hpdf[1:-1] > 0.0,
(self._hpdf[1:-1],),
np.log,
0.0)
return -np.sum(self._hpdf[1:-1] * res * self._hbin_widths)
def _updated_ctor_param(self):
"""
Set the histogram as additional constructor argument
"""
dct = super(rv_histogram, self)._updated_ctor_param()
dct['histogram'] = self._histogram
return dct
# Collect names of classes and objects in this module.
pairs = list(globals().items())
_distn_names, _distn_gen_names = get_distribution_names(pairs, rv_continuous)
__all__ = _distn_names + _distn_gen_names + ['rv_histogram']
| 186,898 | 26.912037 | 124 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/distributions.py
|
#
# Author: Travis Oliphant 2002-2011 with contributions from
# SciPy Developers 2004-2011
#
# NOTE: To look at history using `git blame`, use `git blame -M -C -C`
# instead of `git blame -Lxxx,+x`.
#
from __future__ import division, print_function, absolute_import
from ._distn_infrastructure import (entropy, rv_discrete, rv_continuous,
rv_frozen)
from . import _continuous_distns
from . import _discrete_distns
from ._continuous_distns import *
from ._discrete_distns import *
# For backwards compatibility e.g. pymc expects distributions.__all__.
__all__ = ['entropy', 'rv_discrete', 'rv_continuous', 'rv_histogram']
# Add only the distribution names, not the *_gen names.
__all__ += _continuous_distns._distn_names
__all__ += _discrete_distns._distn_names
| 819 | 31.8 | 72 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/_binned_statistic.py
|
from __future__ import division, print_function, absolute_import
import numpy as np
from scipy._lib.six import callable, xrange
from scipy._lib._numpy_compat import suppress_warnings
from collections import namedtuple
__all__ = ['binned_statistic',
'binned_statistic_2d',
'binned_statistic_dd']
BinnedStatisticResult = namedtuple('BinnedStatisticResult',
('statistic', 'bin_edges', 'binnumber'))
def binned_statistic(x, values, statistic='mean',
bins=10, range=None):
"""
Compute a binned statistic for one or more sets of data.
This is a generalization of a histogram function. A histogram divides
the space into bins, and returns the count of the number of points in
each bin. This function allows the computation of the sum, mean, median,
or other statistic of the values (or set of values) within each bin.
Parameters
----------
x : (N,) array_like
A sequence of values to be binned.
values : (N,) array_like or list of (N,) array_like
The data on which the statistic will be computed. This must be
the same shape as `x`, or a set of sequences - each the same shape as
`x`. If `values` is a set of sequences, the statistic will be computed
on each independently.
statistic : string or callable, optional
The statistic to compute (default is 'mean').
The following statistics are available:
* 'mean' : compute the mean of values for points within each bin.
Empty bins will be represented by NaN.
* 'median' : compute the median of values for points within each
bin. Empty bins will be represented by NaN.
* 'count' : compute the count of points within each bin. This is
identical to an unweighted histogram. `values` array is not
referenced.
* 'sum' : compute the sum of values for points within each bin.
This is identical to a weighted histogram.
* 'min' : compute the minimum of values for points within each bin.
Empty bins will be represented by NaN.
* 'max' : compute the maximum of values for point within each bin.
Empty bins will be represented by NaN.
* function : a user-defined function which takes a 1D array of
values, and outputs a single numerical statistic. This function
will be called on the values in each bin. Empty bins will be
represented by function([]), or NaN if this returns an error.
bins : int or sequence of scalars, optional
If `bins` is an int, it defines the number of equal-width bins in the
given range (10 by default). If `bins` is a sequence, it defines the
bin edges, including the rightmost edge, allowing for non-uniform bin
widths. Values in `x` that are smaller than lowest bin edge are
assigned to bin number 0, values beyond the highest bin are assigned to
``bins[-1]``. If the bin edges are specified, the number of bins will
be, (nx = len(bins)-1).
range : (float, float) or [(float, float)], optional
The lower and upper range of the bins. If not provided, range
is simply ``(x.min(), x.max())``. Values outside the range are
ignored.
Returns
-------
statistic : array
The values of the selected statistic in each bin.
bin_edges : array of dtype float
Return the bin edges ``(length(statistic)+1)``.
binnumber: 1-D ndarray of ints
Indices of the bins (corresponding to `bin_edges`) in which each value
of `x` belongs. Same length as `values`. A binnumber of `i` means the
corresponding value is between (bin_edges[i-1], bin_edges[i]).
See Also
--------
numpy.digitize, numpy.histogram, binned_statistic_2d, binned_statistic_dd
Notes
-----
All but the last (righthand-most) bin is half-open. In other words, if
`bins` is ``[1, 2, 3, 4]``, then the first bin is ``[1, 2)`` (including 1,
but excluding 2) and the second ``[2, 3)``. The last bin, however, is
``[3, 4]``, which *includes* 4.
.. versionadded:: 0.11.0
Examples
--------
>>> from scipy import stats
>>> import matplotlib.pyplot as plt
First some basic examples:
Create two evenly spaced bins in the range of the given sample, and sum the
corresponding values in each of those bins:
>>> values = [1.0, 1.0, 2.0, 1.5, 3.0]
>>> stats.binned_statistic([1, 1, 2, 5, 7], values, 'sum', bins=2)
(array([ 4. , 4.5]), array([ 1., 4., 7.]), array([1, 1, 1, 2, 2]))
Multiple arrays of values can also be passed. The statistic is calculated
on each set independently:
>>> values = [[1.0, 1.0, 2.0, 1.5, 3.0], [2.0, 2.0, 4.0, 3.0, 6.0]]
>>> stats.binned_statistic([1, 1, 2, 5, 7], values, 'sum', bins=2)
(array([[ 4. , 4.5], [ 8. , 9. ]]), array([ 1., 4., 7.]),
array([1, 1, 1, 2, 2]))
>>> stats.binned_statistic([1, 2, 1, 2, 4], np.arange(5), statistic='mean',
... bins=3)
(array([ 1., 2., 4.]), array([ 1., 2., 3., 4.]),
array([1, 2, 1, 2, 3]))
As a second example, we now generate some random data of sailing boat speed
as a function of wind speed, and then determine how fast our boat is for
certain wind speeds:
>>> windspeed = 8 * np.random.rand(500)
>>> boatspeed = .3 * windspeed**.5 + .2 * np.random.rand(500)
>>> bin_means, bin_edges, binnumber = stats.binned_statistic(windspeed,
... boatspeed, statistic='median', bins=[1,2,3,4,5,6,7])
>>> plt.figure()
>>> plt.plot(windspeed, boatspeed, 'b.', label='raw data')
>>> plt.hlines(bin_means, bin_edges[:-1], bin_edges[1:], colors='g', lw=5,
... label='binned statistic of data')
>>> plt.legend()
Now we can use ``binnumber`` to select all datapoints with a windspeed
below 1:
>>> low_boatspeed = boatspeed[binnumber == 0]
As a final example, we will use ``bin_edges`` and ``binnumber`` to make a
plot of a distribution that shows the mean and distribution around that
mean per bin, on top of a regular histogram and the probability
distribution function:
>>> x = np.linspace(0, 5, num=500)
>>> x_pdf = stats.maxwell.pdf(x)
>>> samples = stats.maxwell.rvs(size=10000)
>>> bin_means, bin_edges, binnumber = stats.binned_statistic(x, x_pdf,
... statistic='mean', bins=25)
>>> bin_width = (bin_edges[1] - bin_edges[0])
>>> bin_centers = bin_edges[1:] - bin_width/2
>>> plt.figure()
>>> plt.hist(samples, bins=50, density=True, histtype='stepfilled',
... alpha=0.2, label='histogram of data')
>>> plt.plot(x, x_pdf, 'r-', label='analytical pdf')
>>> plt.hlines(bin_means, bin_edges[:-1], bin_edges[1:], colors='g', lw=2,
... label='binned statistic of data')
>>> plt.plot((binnumber - 0.5) * bin_width, x_pdf, 'g.', alpha=0.5)
>>> plt.legend(fontsize=10)
>>> plt.show()
"""
try:
N = len(bins)
except TypeError:
N = 1
if N != 1:
bins = [np.asarray(bins, float)]
if range is not None:
if len(range) == 2:
range = [range]
medians, edges, binnumbers = binned_statistic_dd(
[x], values, statistic, bins, range)
return BinnedStatisticResult(medians, edges[0], binnumbers)
BinnedStatistic2dResult = namedtuple('BinnedStatistic2dResult',
('statistic', 'x_edge', 'y_edge',
'binnumber'))
def binned_statistic_2d(x, y, values, statistic='mean',
bins=10, range=None, expand_binnumbers=False):
"""
Compute a bidimensional binned statistic for one or more sets of data.
This is a generalization of a histogram2d function. A histogram divides
the space into bins, and returns the count of the number of points in
each bin. This function allows the computation of the sum, mean, median,
or other statistic of the values (or set of values) within each bin.
Parameters
----------
x : (N,) array_like
A sequence of values to be binned along the first dimension.
y : (N,) array_like
A sequence of values to be binned along the second dimension.
values : (N,) array_like or list of (N,) array_like
The data on which the statistic will be computed. This must be
the same shape as `x`, or a list of sequences - each with the same
shape as `x`. If `values` is such a list, the statistic will be
computed on each independently.
statistic : string or callable, optional
The statistic to compute (default is 'mean').
The following statistics are available:
* 'mean' : compute the mean of values for points within each bin.
Empty bins will be represented by NaN.
* 'median' : compute the median of values for points within each
bin. Empty bins will be represented by NaN.
* 'count' : compute the count of points within each bin. This is
identical to an unweighted histogram. `values` array is not
referenced.
* 'sum' : compute the sum of values for points within each bin.
This is identical to a weighted histogram.
* 'min' : compute the minimum of values for points within each bin.
Empty bins will be represented by NaN.
* 'max' : compute the maximum of values for point within each bin.
Empty bins will be represented by NaN.
* function : a user-defined function which takes a 1D array of
values, and outputs a single numerical statistic. This function
will be called on the values in each bin. Empty bins will be
represented by function([]), or NaN if this returns an error.
bins : int or [int, int] or array_like or [array, array], optional
The bin specification:
* the number of bins for the two dimensions (nx = ny = bins),
* the number of bins in each dimension (nx, ny = bins),
* the bin edges for the two dimensions (x_edge = y_edge = bins),
* the bin edges in each dimension (x_edge, y_edge = bins).
If the bin edges are specified, the number of bins will be,
(nx = len(x_edge)-1, ny = len(y_edge)-1).
range : (2,2) array_like, optional
The leftmost and rightmost edges of the bins along each dimension
(if not specified explicitly in the `bins` parameters):
[[xmin, xmax], [ymin, ymax]]. All values outside of this range will be
considered outliers and not tallied in the histogram.
expand_binnumbers : bool, optional
'False' (default): the returned `binnumber` is a shape (N,) array of
linearized bin indices.
'True': the returned `binnumber` is 'unraveled' into a shape (2,N)
ndarray, where each row gives the bin numbers in the corresponding
dimension.
See the `binnumber` returned value, and the `Examples` section.
.. versionadded:: 0.17.0
Returns
-------
statistic : (nx, ny) ndarray
The values of the selected statistic in each two-dimensional bin.
x_edge : (nx + 1) ndarray
The bin edges along the first dimension.
y_edge : (ny + 1) ndarray
The bin edges along the second dimension.
binnumber : (N,) array of ints or (2,N) ndarray of ints
This assigns to each element of `sample` an integer that represents the
bin in which this observation falls. The representation depends on the
`expand_binnumbers` argument. See `Notes` for details.
See Also
--------
numpy.digitize, numpy.histogram2d, binned_statistic, binned_statistic_dd
Notes
-----
Binedges:
All but the last (righthand-most) bin is half-open. In other words, if
`bins` is ``[1, 2, 3, 4]``, then the first bin is ``[1, 2)`` (including 1,
but excluding 2) and the second ``[2, 3)``. The last bin, however, is
``[3, 4]``, which *includes* 4.
`binnumber`:
This returned argument assigns to each element of `sample` an integer that
represents the bin in which it belongs. The representation depends on the
`expand_binnumbers` argument. If 'False' (default): The returned
`binnumber` is a shape (N,) array of linearized indices mapping each
element of `sample` to its corresponding bin (using row-major ordering).
If 'True': The returned `binnumber` is a shape (2,N) ndarray where
each row indicates bin placements for each dimension respectively. In each
dimension, a binnumber of `i` means the corresponding value is between
(D_edge[i-1], D_edge[i]), where 'D' is either 'x' or 'y'.
.. versionadded:: 0.11.0
Examples
--------
>>> from scipy import stats
Calculate the counts with explicit bin-edges:
>>> x = [0.1, 0.1, 0.1, 0.6]
>>> y = [2.1, 2.6, 2.1, 2.1]
>>> binx = [0.0, 0.5, 1.0]
>>> biny = [2.0, 2.5, 3.0]
>>> ret = stats.binned_statistic_2d(x, y, None, 'count', bins=[binx,biny])
>>> ret.statistic
array([[ 2., 1.],
[ 1., 0.]])
The bin in which each sample is placed is given by the `binnumber`
returned parameter. By default, these are the linearized bin indices:
>>> ret.binnumber
array([5, 6, 5, 9])
The bin indices can also be expanded into separate entries for each
dimension using the `expand_binnumbers` parameter:
>>> ret = stats.binned_statistic_2d(x, y, None, 'count', bins=[binx,biny],
... expand_binnumbers=True)
>>> ret.binnumber
array([[1, 1, 1, 2],
[1, 2, 1, 1]])
Which shows that the first three elements belong in the xbin 1, and the
fourth into xbin 2; and so on for y.
"""
# This code is based on np.histogram2d
try:
N = len(bins)
except TypeError:
N = 1
if N != 1 and N != 2:
xedges = yedges = np.asarray(bins, float)
bins = [xedges, yedges]
medians, edges, binnumbers = binned_statistic_dd(
[x, y], values, statistic, bins, range,
expand_binnumbers=expand_binnumbers)
return BinnedStatistic2dResult(medians, edges[0], edges[1], binnumbers)
BinnedStatisticddResult = namedtuple('BinnedStatisticddResult',
('statistic', 'bin_edges',
'binnumber'))
def binned_statistic_dd(sample, values, statistic='mean',
bins=10, range=None, expand_binnumbers=False):
"""
Compute a multidimensional binned statistic for a set of data.
This is a generalization of a histogramdd function. A histogram divides
the space into bins, and returns the count of the number of points in
each bin. This function allows the computation of the sum, mean, median,
or other statistic of the values within each bin.
Parameters
----------
sample : array_like
Data to histogram passed as a sequence of D arrays of length N, or
as an (N,D) array.
values : (N,) array_like or list of (N,) array_like
The data on which the statistic will be computed. This must be
the same shape as `x`, or a list of sequences - each with the same
shape as `x`. If `values` is such a list, the statistic will be
computed on each independently.
statistic : string or callable, optional
The statistic to compute (default is 'mean').
The following statistics are available:
* 'mean' : compute the mean of values for points within each bin.
Empty bins will be represented by NaN.
* 'median' : compute the median of values for points within each
bin. Empty bins will be represented by NaN.
* 'count' : compute the count of points within each bin. This is
identical to an unweighted histogram. `values` array is not
referenced.
* 'sum' : compute the sum of values for points within each bin.
This is identical to a weighted histogram.
* 'min' : compute the minimum of values for points within each bin.
Empty bins will be represented by NaN.
* 'max' : compute the maximum of values for point within each bin.
Empty bins will be represented by NaN.
* function : a user-defined function which takes a 1D array of
values, and outputs a single numerical statistic. This function
will be called on the values in each bin. Empty bins will be
represented by function([]), or NaN if this returns an error.
bins : sequence or int, optional
The bin specification must be in one of the following forms:
* A sequence of arrays describing the bin edges along each dimension.
* The number of bins for each dimension (nx, ny, ... = bins).
* The number of bins for all dimensions (nx = ny = ... = bins).
range : sequence, optional
A sequence of lower and upper bin edges to be used if the edges are
not given explicitly in `bins`. Defaults to the minimum and maximum
values along each dimension.
expand_binnumbers : bool, optional
'False' (default): the returned `binnumber` is a shape (N,) array of
linearized bin indices.
'True': the returned `binnumber` is 'unraveled' into a shape (D,N)
ndarray, where each row gives the bin numbers in the corresponding
dimension.
See the `binnumber` returned value, and the `Examples` section of
`binned_statistic_2d`.
.. versionadded:: 0.17.0
Returns
-------
statistic : ndarray, shape(nx1, nx2, nx3,...)
The values of the selected statistic in each two-dimensional bin.
bin_edges : list of ndarrays
A list of D arrays describing the (nxi + 1) bin edges for each
dimension.
binnumber : (N,) array of ints or (D,N) ndarray of ints
This assigns to each element of `sample` an integer that represents the
bin in which this observation falls. The representation depends on the
`expand_binnumbers` argument. See `Notes` for details.
See Also
--------
numpy.digitize, numpy.histogramdd, binned_statistic, binned_statistic_2d
Notes
-----
Binedges:
All but the last (righthand-most) bin is half-open in each dimension. In
other words, if `bins` is ``[1, 2, 3, 4]``, then the first bin is
``[1, 2)`` (including 1, but excluding 2) and the second ``[2, 3)``. The
last bin, however, is ``[3, 4]``, which *includes* 4.
`binnumber`:
This returned argument assigns to each element of `sample` an integer that
represents the bin in which it belongs. The representation depends on the
`expand_binnumbers` argument. If 'False' (default): The returned
`binnumber` is a shape (N,) array of linearized indices mapping each
element of `sample` to its corresponding bin (using row-major ordering).
If 'True': The returned `binnumber` is a shape (D,N) ndarray where
each row indicates bin placements for each dimension respectively. In each
dimension, a binnumber of `i` means the corresponding value is between
(bin_edges[D][i-1], bin_edges[D][i]), for each dimension 'D'.
.. versionadded:: 0.11.0
"""
known_stats = ['mean', 'median', 'count', 'sum', 'std','min','max']
if not callable(statistic) and statistic not in known_stats:
raise ValueError('invalid statistic %r' % (statistic,))
# `Ndim` is the number of dimensions (e.g. `2` for `binned_statistic_2d`)
# `Dlen` is the length of elements along each dimension.
# This code is based on np.histogramdd
try:
# `sample` is an ND-array.
Dlen, Ndim = sample.shape
except (AttributeError, ValueError):
# `sample` is a sequence of 1D arrays.
sample = np.atleast_2d(sample).T
Dlen, Ndim = sample.shape
# Store initial shape of `values` to preserve it in the output
values = np.asarray(values)
input_shape = list(values.shape)
# Make sure that `values` is 2D to iterate over rows
values = np.atleast_2d(values)
Vdim, Vlen = values.shape
# Make sure `values` match `sample`
if(statistic != 'count' and Vlen != Dlen):
raise AttributeError('The number of `values` elements must match the '
'length of each `sample` dimension.')
nbin = np.empty(Ndim, int) # Number of bins in each dimension
edges = Ndim * [None] # Bin edges for each dim (will be 2D array)
dedges = Ndim * [None] # Spacing between edges (will be 2D array)
try:
M = len(bins)
if M != Ndim:
raise AttributeError('The dimension of bins must be equal '
'to the dimension of the sample x.')
except TypeError:
bins = Ndim * [bins]
# Select range for each dimension
# Used only if number of bins is given.
if range is None:
smin = np.atleast_1d(np.array(sample.min(axis=0), float))
smax = np.atleast_1d(np.array(sample.max(axis=0), float))
else:
smin = np.zeros(Ndim)
smax = np.zeros(Ndim)
for i in xrange(Ndim):
smin[i], smax[i] = range[i]
# Make sure the bins have a finite width.
for i in xrange(len(smin)):
if smin[i] == smax[i]:
smin[i] = smin[i] - .5
smax[i] = smax[i] + .5
# Create edge arrays
for i in xrange(Ndim):
if np.isscalar(bins[i]):
nbin[i] = bins[i] + 2 # +2 for outlier bins
edges[i] = np.linspace(smin[i], smax[i], nbin[i] - 1)
else:
edges[i] = np.asarray(bins[i], float)
nbin[i] = len(edges[i]) + 1 # +1 for outlier bins
dedges[i] = np.diff(edges[i])
nbin = np.asarray(nbin)
# Compute the bin number each sample falls into, in each dimension
sampBin = [
np.digitize(sample[:, i], edges[i])
for i in xrange(Ndim)
]
# Using `digitize`, values that fall on an edge are put in the right bin.
# For the rightmost bin, we want values equal to the right
# edge to be counted in the last bin, and not as an outlier.
for i in xrange(Ndim):
# Find the rounding precision
decimal = int(-np.log10(dedges[i].min())) + 6
# Find which points are on the rightmost edge.
on_edge = np.where(np.around(sample[:, i], decimal) ==
np.around(edges[i][-1], decimal))[0]
# Shift these points one bin to the left.
sampBin[i][on_edge] -= 1
# Compute the sample indices in the flattened statistic matrix.
binnumbers = np.ravel_multi_index(sampBin, nbin)
result = np.empty([Vdim, nbin.prod()], float)
if statistic == 'mean':
result.fill(np.nan)
flatcount = np.bincount(binnumbers, None)
a = flatcount.nonzero()
for vv in xrange(Vdim):
flatsum = np.bincount(binnumbers, values[vv])
result[vv, a] = flatsum[a] / flatcount[a]
elif statistic == 'std':
result.fill(0)
flatcount = np.bincount(binnumbers, None)
a = flatcount.nonzero()
for vv in xrange(Vdim):
flatsum = np.bincount(binnumbers, values[vv])
flatsum2 = np.bincount(binnumbers, values[vv] ** 2)
result[vv, a] = np.sqrt(flatsum2[a] / flatcount[a] -
(flatsum[a] / flatcount[a]) ** 2)
elif statistic == 'count':
result.fill(0)
flatcount = np.bincount(binnumbers, None)
a = np.arange(len(flatcount))
result[:, a] = flatcount[np.newaxis, :]
elif statistic == 'sum':
result.fill(0)
for vv in xrange(Vdim):
flatsum = np.bincount(binnumbers, values[vv])
a = np.arange(len(flatsum))
result[vv, a] = flatsum
elif statistic == 'median':
result.fill(np.nan)
for i in np.unique(binnumbers):
for vv in xrange(Vdim):
result[vv, i] = np.median(values[vv, binnumbers == i])
elif statistic == 'min':
result.fill(np.nan)
for i in np.unique(binnumbers):
for vv in xrange(Vdim):
result[vv, i] = np.min(values[vv, binnumbers == i])
elif statistic == 'max':
result.fill(np.nan)
for i in np.unique(binnumbers):
for vv in xrange(Vdim):
result[vv, i] = np.max(values[vv, binnumbers == i])
elif callable(statistic):
with np.errstate(invalid='ignore'), suppress_warnings() as sup:
sup.filter(RuntimeWarning)
try:
null = statistic([])
except:
null = np.nan
result.fill(null)
for i in np.unique(binnumbers):
for vv in xrange(Vdim):
result[vv, i] = statistic(values[vv, binnumbers == i])
# Shape into a proper matrix
result = result.reshape(np.append(Vdim, nbin))
# Remove outliers (indices 0 and -1 for each bin-dimension).
core = [slice(None)] + Ndim * [slice(1, -1)]
result = result[core]
# Unravel binnumbers into an ndarray, each row the bins for each dimension
if(expand_binnumbers and Ndim > 1):
binnumbers = np.asarray(np.unravel_index(binnumbers, nbin))
if np.any(result.shape[1:] != nbin - 2):
raise RuntimeError('Internal Shape Error')
# Reshape to have output (`reulst`) match input (`values`) shape
result = result.reshape(input_shape[:-1] + list(nbin-2))
return BinnedStatisticddResult(result, edges, binnumbers)
| 25,912 | 40.795161 | 79 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/_distr_params.py
|
"""
Sane parameters for stats.distributions.
"""
distcont = [
['alpha', (3.5704770516650459,)],
['anglit', ()],
['arcsine', ()],
['argus', (1.0,)],
['beta', (2.3098496451481823, 0.62687954300963677)],
['betaprime', (5, 6)],
['bradford', (0.29891359763170633,)],
['burr', (10.5, 4.3)],
['burr12', (10, 4)],
['cauchy', ()],
['chi', (78,)],
['chi2', (55,)],
['cosine', ()],
['crystalball', (2.0, 3.0)],
['dgamma', (1.1023326088288166,)],
['dweibull', (2.0685080649914673,)],
['erlang', (10,)],
['expon', ()],
['exponnorm', (1.5,)],
['exponpow', (2.697119160358469,)],
['exponweib', (2.8923945291034436, 1.9505288745913174)],
['f', (29, 18)],
['fatiguelife', (29,)], # correction numargs = 1
['fisk', (3.0857548622253179,)],
['foldcauchy', (4.7164673455831894,)],
['foldnorm', (1.9521253373555869,)],
['frechet_l', (3.6279911255583239,)],
['frechet_r', (1.8928171603534227,)],
['gamma', (1.9932305483800778,)],
['gausshyper', (13.763771604130699, 3.1189636648681431,
2.5145980350183019, 5.1811649903971615)], # veryslow
['genexpon', (9.1325976465418908, 16.231956600590632, 3.2819552690843983)],
['genextreme', (-0.1,)],
['gengamma', (4.4162385429431925, 3.1193091679242761)],
['gengamma', (4.4162385429431925, -3.1193091679242761)],
['genhalflogistic', (0.77274727809929322,)],
['genlogistic', (0.41192440799679475,)],
['gennorm', (1.2988442399460265,)],
['halfgennorm', (0.6748054997000371,)],
['genpareto', (0.1,)], # use case with finite moments
['gilbrat', ()],
['gompertz', (0.94743713075105251,)],
['gumbel_l', ()],
['gumbel_r', ()],
['halfcauchy', ()],
['halflogistic', ()],
['halfnorm', ()],
['hypsecant', ()],
['invgamma', (4.0668996136993067,)],
['invgauss', (0.14546264555347513,)],
['invweibull', (10.58,)],
['johnsonsb', (4.3172675099141058, 3.1837781130785063)],
['johnsonsu', (2.554395574161155, 2.2482281679651965)],
['kappa4', (0.0, 0.0)],
['kappa4', (-0.1, 0.1)],
['kappa4', (0.0, 0.1)],
['kappa4', (0.1, 0.0)],
['kappa3', (1.0,)],
['ksone', (1000,)], # replace 22 by 100 to avoid failing range, ticket 956
['kstwobign', ()],
['laplace', ()],
['levy', ()],
['levy_l', ()],
['levy_stable', (0.35667405469844993,
-0.67450531578494011)], # NotImplementedError
# rvs not tested
['loggamma', (0.41411931826052117,)],
['logistic', ()],
['loglaplace', (3.2505926592051435,)],
['lognorm', (0.95368226960575331,)],
['lomax', (1.8771398388773268,)],
['maxwell', ()],
['mielke', (10.4, 3.6)],
['moyal', ()],
['nakagami', (4.9673794866666237,)],
['ncf', (27, 27, 0.41578441799226107)],
['nct', (14, 0.24045031331198066)],
['ncx2', (21, 1.0560465975116415)],
['norm', ()],
['norminvgauss', (1., 0.5)],
['pareto', (2.621716532144454,)],
['pearson3', (0.1,)],
['powerlaw', (1.6591133289905851,)],
['powerlognorm', (2.1413923530064087, 0.44639540782048337)],
['powernorm', (4.4453652254590779,)],
['rayleigh', ()],
['rdist', (0.9,)], # feels also slow
['recipinvgauss', (0.63004267809369119,)],
['reciprocal', (0.0062309367010521255, 1.0062309367010522)],
['rice', (0.7749725210111873,)],
['semicircular', ()],
['skewnorm', (4.0,)],
['t', (2.7433514990818093,)],
['trapz', (0.2, 0.8)],
['triang', (0.15785029824528218,)],
['truncexpon', (4.6907725456810478,)],
['truncnorm', (-1.0978730080013919, 2.7306754109031979)],
['truncnorm', (0.1, 2.)],
['tukeylambda', (3.1321477856738267,)],
['uniform', ()],
['vonmises', (3.9939042581071398,)],
['vonmises_line', (3.9939042581071398,)],
['wald', ()],
['weibull_max', (2.8687961709100187,)],
['weibull_min', (1.7866166930421596,)],
['wrapcauchy', (0.031071279018614728,)]]
distdiscrete = [
['bernoulli',(0.3,)],
['binom', (5, 0.4)],
['boltzmann',(1.4, 19)],
['dlaplace', (0.8,)], # 0.5
['geom', (0.5,)],
['hypergeom',(30, 12, 6)],
['hypergeom',(21,3,12)], # numpy.random (3,18,12) numpy ticket:921
['hypergeom',(21,18,11)], # numpy.random (18,3,11) numpy ticket:921
['logser', (0.6,)], # re-enabled, numpy ticket:921
['nbinom', (5, 0.5)],
['nbinom', (0.4, 0.4)], # from tickets: 583
['planck', (0.51,)], # 4.1
['poisson', (0.6,)],
['randint', (7, 31)],
['skellam', (15, 8)],
['zipf', (6.5,)]
]
| 4,586 | 33.75 | 79 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/setup.py
|
from __future__ import division, print_function, absolute_import
from os.path import join
def configuration(parent_package='',top_path=None):
from numpy.distutils.misc_util import Configuration
config = Configuration('stats', parent_package, top_path)
config.add_data_dir('tests')
statlib_src = [join('statlib', '*.f')]
config.add_library('statlib', sources=statlib_src)
# add statlib module
config.add_extension('statlib',
sources=['statlib.pyf'],
f2py_options=['--no-wrap-functions'],
libraries=['statlib'],
depends=statlib_src
)
# add _stats module
config.add_extension('_stats',
sources=['_stats.c'],
)
# add mvn module
config.add_extension('mvn',
sources=['mvn.pyf','mvndst.f'],
)
return config
if __name__ == '__main__':
from numpy.distutils.core import setup
setup(**configuration(top_path='').todict())
| 938 | 23.076923 | 64 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/kde.py
|
#-------------------------------------------------------------------------------
#
# Define classes for (uni/multi)-variate kernel density estimation.
#
# Currently, only Gaussian kernels are implemented.
#
# Written by: Robert Kern
#
# Date: 2004-08-09
#
# Modified: 2005-02-10 by Robert Kern.
# Contributed to Scipy
# 2005-10-07 by Robert Kern.
# Some fixes to match the new scipy_core
#
# Copyright 2004-2005 by Enthought, Inc.
#
#-------------------------------------------------------------------------------
from __future__ import division, print_function, absolute_import
# Standard library imports.
import warnings
# Scipy imports.
from scipy._lib.six import callable, string_types
from scipy import linalg, special
from scipy.special import logsumexp
from numpy import atleast_2d, reshape, zeros, newaxis, dot, exp, pi, sqrt, \
ravel, power, atleast_1d, squeeze, sum, transpose
import numpy as np
from numpy.random import randint, multivariate_normal
# Local imports.
from . import mvn
__all__ = ['gaussian_kde']
class gaussian_kde(object):
"""Representation of a kernel-density estimate using Gaussian kernels.
Kernel density estimation is a way to estimate the probability density
function (PDF) of a random variable in a non-parametric way.
`gaussian_kde` works for both uni-variate and multi-variate data. It
includes automatic bandwidth determination. The estimation works best for
a unimodal distribution; bimodal or multi-modal distributions tend to be
oversmoothed.
Parameters
----------
dataset : array_like
Datapoints to estimate from. In case of univariate data this is a 1-D
array, otherwise a 2-D array with shape (# of dims, # of data).
bw_method : str, scalar or callable, optional
The method used to calculate the estimator bandwidth. This can be
'scott', 'silverman', a scalar constant or a callable. If a scalar,
this will be used directly as `kde.factor`. If a callable, it should
take a `gaussian_kde` instance as only parameter and return a scalar.
If None (default), 'scott' is used. See Notes for more details.
Attributes
----------
dataset : ndarray
The dataset with which `gaussian_kde` was initialized.
d : int
Number of dimensions.
n : int
Number of datapoints.
factor : float
The bandwidth factor, obtained from `kde.covariance_factor`, with which
the covariance matrix is multiplied.
covariance : ndarray
The covariance matrix of `dataset`, scaled by the calculated bandwidth
(`kde.factor`).
inv_cov : ndarray
The inverse of `covariance`.
Methods
-------
evaluate
__call__
integrate_gaussian
integrate_box_1d
integrate_box
integrate_kde
pdf
logpdf
resample
set_bandwidth
covariance_factor
Notes
-----
Bandwidth selection strongly influences the estimate obtained from the KDE
(much more so than the actual shape of the kernel). Bandwidth selection
can be done by a "rule of thumb", by cross-validation, by "plug-in
methods" or by other means; see [3]_, [4]_ for reviews. `gaussian_kde`
uses a rule of thumb, the default is Scott's Rule.
Scott's Rule [1]_, implemented as `scotts_factor`, is::
n**(-1./(d+4)),
with ``n`` the number of data points and ``d`` the number of dimensions.
Silverman's Rule [2]_, implemented as `silverman_factor`, is::
(n * (d + 2) / 4.)**(-1. / (d + 4)).
Good general descriptions of kernel density estimation can be found in [1]_
and [2]_, the mathematics for this multi-dimensional implementation can be
found in [1]_.
References
----------
.. [1] D.W. Scott, "Multivariate Density Estimation: Theory, Practice, and
Visualization", John Wiley & Sons, New York, Chicester, 1992.
.. [2] B.W. Silverman, "Density Estimation for Statistics and Data
Analysis", Vol. 26, Monographs on Statistics and Applied Probability,
Chapman and Hall, London, 1986.
.. [3] B.A. Turlach, "Bandwidth Selection in Kernel Density Estimation: A
Review", CORE and Institut de Statistique, Vol. 19, pp. 1-33, 1993.
.. [4] D.M. Bashtannyk and R.J. Hyndman, "Bandwidth selection for kernel
conditional density estimation", Computational Statistics & Data
Analysis, Vol. 36, pp. 279-298, 2001.
Examples
--------
Generate some random two-dimensional data:
>>> from scipy import stats
>>> def measure(n):
... "Measurement model, return two coupled measurements."
... m1 = np.random.normal(size=n)
... m2 = np.random.normal(scale=0.5, size=n)
... return m1+m2, m1-m2
>>> m1, m2 = measure(2000)
>>> xmin = m1.min()
>>> xmax = m1.max()
>>> ymin = m2.min()
>>> ymax = m2.max()
Perform a kernel density estimate on the data:
>>> X, Y = np.mgrid[xmin:xmax:100j, ymin:ymax:100j]
>>> positions = np.vstack([X.ravel(), Y.ravel()])
>>> values = np.vstack([m1, m2])
>>> kernel = stats.gaussian_kde(values)
>>> Z = np.reshape(kernel(positions).T, X.shape)
Plot the results:
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> ax.imshow(np.rot90(Z), cmap=plt.cm.gist_earth_r,
... extent=[xmin, xmax, ymin, ymax])
>>> ax.plot(m1, m2, 'k.', markersize=2)
>>> ax.set_xlim([xmin, xmax])
>>> ax.set_ylim([ymin, ymax])
>>> plt.show()
"""
def __init__(self, dataset, bw_method=None):
self.dataset = atleast_2d(dataset)
if not self.dataset.size > 1:
raise ValueError("`dataset` input should have multiple elements.")
self.d, self.n = self.dataset.shape
self.set_bandwidth(bw_method=bw_method)
def evaluate(self, points):
"""Evaluate the estimated pdf on a set of points.
Parameters
----------
points : (# of dimensions, # of points)-array
Alternatively, a (# of dimensions,) vector can be passed in and
treated as a single point.
Returns
-------
values : (# of points,)-array
The values at each point.
Raises
------
ValueError : if the dimensionality of the input points is different than
the dimensionality of the KDE.
"""
points = atleast_2d(points)
d, m = points.shape
if d != self.d:
if d == 1 and m == self.d:
# points was passed in as a row vector
points = reshape(points, (self.d, 1))
m = 1
else:
msg = "points have dimension %s, dataset has dimension %s" % (d,
self.d)
raise ValueError(msg)
result = zeros((m,), dtype=float)
if m >= self.n:
# there are more points than data, so loop over data
for i in range(self.n):
diff = self.dataset[:, i, newaxis] - points
tdiff = dot(self.inv_cov, diff)
energy = sum(diff*tdiff,axis=0) / 2.0
result = result + exp(-energy)
else:
# loop over points
for i in range(m):
diff = self.dataset - points[:, i, newaxis]
tdiff = dot(self.inv_cov, diff)
energy = sum(diff * tdiff, axis=0) / 2.0
result[i] = sum(exp(-energy), axis=0)
result = result / self._norm_factor
return result
__call__ = evaluate
def integrate_gaussian(self, mean, cov):
"""
Multiply estimated density by a multivariate Gaussian and integrate
over the whole space.
Parameters
----------
mean : aray_like
A 1-D array, specifying the mean of the Gaussian.
cov : array_like
A 2-D array, specifying the covariance matrix of the Gaussian.
Returns
-------
result : scalar
The value of the integral.
Raises
------
ValueError
If the mean or covariance of the input Gaussian differs from
the KDE's dimensionality.
"""
mean = atleast_1d(squeeze(mean))
cov = atleast_2d(cov)
if mean.shape != (self.d,):
raise ValueError("mean does not have dimension %s" % self.d)
if cov.shape != (self.d, self.d):
raise ValueError("covariance does not have dimension %s" % self.d)
# make mean a column vector
mean = mean[:, newaxis]
sum_cov = self.covariance + cov
# This will raise LinAlgError if the new cov matrix is not s.p.d
# cho_factor returns (ndarray, bool) where bool is a flag for whether
# or not ndarray is upper or lower triangular
sum_cov_chol = linalg.cho_factor(sum_cov)
diff = self.dataset - mean
tdiff = linalg.cho_solve(sum_cov_chol, diff)
sqrt_det = np.prod(np.diagonal(sum_cov_chol[0]))
norm_const = power(2 * pi, sum_cov.shape[0] / 2.0) * sqrt_det
energies = sum(diff * tdiff, axis=0) / 2.0
result = sum(exp(-energies), axis=0) / norm_const / self.n
return result
def integrate_box_1d(self, low, high):
"""
Computes the integral of a 1D pdf between two bounds.
Parameters
----------
low : scalar
Lower bound of integration.
high : scalar
Upper bound of integration.
Returns
-------
value : scalar
The result of the integral.
Raises
------
ValueError
If the KDE is over more than one dimension.
"""
if self.d != 1:
raise ValueError("integrate_box_1d() only handles 1D pdfs")
stdev = ravel(sqrt(self.covariance))[0]
normalized_low = ravel((low - self.dataset) / stdev)
normalized_high = ravel((high - self.dataset) / stdev)
value = np.mean(special.ndtr(normalized_high) -
special.ndtr(normalized_low))
return value
def integrate_box(self, low_bounds, high_bounds, maxpts=None):
"""Computes the integral of a pdf over a rectangular interval.
Parameters
----------
low_bounds : array_like
A 1-D array containing the lower bounds of integration.
high_bounds : array_like
A 1-D array containing the upper bounds of integration.
maxpts : int, optional
The maximum number of points to use for integration.
Returns
-------
value : scalar
The result of the integral.
"""
if maxpts is not None:
extra_kwds = {'maxpts': maxpts}
else:
extra_kwds = {}
value, inform = mvn.mvnun(low_bounds, high_bounds, self.dataset,
self.covariance, **extra_kwds)
if inform:
msg = ('An integral in mvn.mvnun requires more points than %s' %
(self.d * 1000))
warnings.warn(msg)
return value
def integrate_kde(self, other):
"""
Computes the integral of the product of this kernel density estimate
with another.
Parameters
----------
other : gaussian_kde instance
The other kde.
Returns
-------
value : scalar
The result of the integral.
Raises
------
ValueError
If the KDEs have different dimensionality.
"""
if other.d != self.d:
raise ValueError("KDEs are not the same dimensionality")
# we want to iterate over the smallest number of points
if other.n < self.n:
small = other
large = self
else:
small = self
large = other
sum_cov = small.covariance + large.covariance
sum_cov_chol = linalg.cho_factor(sum_cov)
result = 0.0
for i in range(small.n):
mean = small.dataset[:, i, newaxis]
diff = large.dataset - mean
tdiff = linalg.cho_solve(sum_cov_chol, diff)
energies = sum(diff * tdiff, axis=0) / 2.0
result += sum(exp(-energies), axis=0)
sqrt_det = np.prod(np.diagonal(sum_cov_chol[0]))
norm_const = power(2 * pi, sum_cov.shape[0] / 2.0) * sqrt_det
result /= norm_const * large.n * small.n
return result
def resample(self, size=None):
"""
Randomly sample a dataset from the estimated pdf.
Parameters
----------
size : int, optional
The number of samples to draw. If not provided, then the size is
the same as the underlying dataset.
Returns
-------
resample : (self.d, `size`) ndarray
The sampled dataset.
"""
if size is None:
size = self.n
norm = transpose(multivariate_normal(zeros((self.d,), float),
self.covariance, size=size))
indices = randint(0, self.n, size=size)
means = self.dataset[:, indices]
return means + norm
def scotts_factor(self):
return power(self.n, -1./(self.d+4))
def silverman_factor(self):
return power(self.n*(self.d+2.0)/4.0, -1./(self.d+4))
# Default method to calculate bandwidth, can be overwritten by subclass
covariance_factor = scotts_factor
covariance_factor.__doc__ = """Computes the coefficient (`kde.factor`) that
multiplies the data covariance matrix to obtain the kernel covariance
matrix. The default is `scotts_factor`. A subclass can overwrite this
method to provide a different method, or set it through a call to
`kde.set_bandwidth`."""
def set_bandwidth(self, bw_method=None):
"""Compute the estimator bandwidth with given method.
The new bandwidth calculated after a call to `set_bandwidth` is used
for subsequent evaluations of the estimated density.
Parameters
----------
bw_method : str, scalar or callable, optional
The method used to calculate the estimator bandwidth. This can be
'scott', 'silverman', a scalar constant or a callable. If a
scalar, this will be used directly as `kde.factor`. If a callable,
it should take a `gaussian_kde` instance as only parameter and
return a scalar. If None (default), nothing happens; the current
`kde.covariance_factor` method is kept.
Notes
-----
.. versionadded:: 0.11
Examples
--------
>>> import scipy.stats as stats
>>> x1 = np.array([-7, -5, 1, 4, 5.])
>>> kde = stats.gaussian_kde(x1)
>>> xs = np.linspace(-10, 10, num=50)
>>> y1 = kde(xs)
>>> kde.set_bandwidth(bw_method='silverman')
>>> y2 = kde(xs)
>>> kde.set_bandwidth(bw_method=kde.factor / 3.)
>>> y3 = kde(xs)
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> ax.plot(x1, np.ones(x1.shape) / (4. * x1.size), 'bo',
... label='Data points (rescaled)')
>>> ax.plot(xs, y1, label='Scott (default)')
>>> ax.plot(xs, y2, label='Silverman')
>>> ax.plot(xs, y3, label='Const (1/3 * Silverman)')
>>> ax.legend()
>>> plt.show()
"""
if bw_method is None:
pass
elif bw_method == 'scott':
self.covariance_factor = self.scotts_factor
elif bw_method == 'silverman':
self.covariance_factor = self.silverman_factor
elif np.isscalar(bw_method) and not isinstance(bw_method, string_types):
self._bw_method = 'use constant'
self.covariance_factor = lambda: bw_method
elif callable(bw_method):
self._bw_method = bw_method
self.covariance_factor = lambda: self._bw_method(self)
else:
msg = "`bw_method` should be 'scott', 'silverman', a scalar " \
"or a callable."
raise ValueError(msg)
self._compute_covariance()
def _compute_covariance(self):
"""Computes the covariance matrix for each Gaussian kernel using
covariance_factor().
"""
self.factor = self.covariance_factor()
# Cache covariance and inverse covariance of the data
if not hasattr(self, '_data_inv_cov'):
self._data_covariance = atleast_2d(np.cov(self.dataset, rowvar=1,
bias=False))
self._data_inv_cov = linalg.inv(self._data_covariance)
self.covariance = self._data_covariance * self.factor**2
self.inv_cov = self._data_inv_cov / self.factor**2
self._norm_factor = sqrt(linalg.det(2*pi*self.covariance)) * self.n
def pdf(self, x):
"""
Evaluate the estimated pdf on a provided set of points.
Notes
-----
This is an alias for `gaussian_kde.evaluate`. See the ``evaluate``
docstring for more details.
"""
return self.evaluate(x)
def logpdf(self, x):
"""
Evaluate the log of the estimated pdf on a provided set of points.
"""
points = atleast_2d(x)
d, m = points.shape
if d != self.d:
if d == 1 and m == self.d:
# points was passed in as a row vector
points = reshape(points, (self.d, 1))
m = 1
else:
msg = "points have dimension %s, dataset has dimension %s" % (d,
self.d)
raise ValueError(msg)
result = zeros((m,), dtype=float)
if m >= self.n:
# there are more points than data, so loop over data
energy = zeros((self.n, m), dtype=float)
for i in range(self.n):
diff = self.dataset[:, i, newaxis] - points
tdiff = dot(self.inv_cov, diff)
energy[i] = sum(diff*tdiff,axis=0) / 2.0
result = logsumexp(-energy, b=1/self._norm_factor, axis=0)
else:
# loop over points
for i in range(m):
diff = self.dataset - points[:, i, newaxis]
tdiff = dot(self.inv_cov, diff)
energy = sum(diff * tdiff, axis=0) / 2.0
result[i] = logsumexp(-energy, b=1/self._norm_factor)
return result
| 18,766 | 32.215929 | 80 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/stats.py
|
# Copyright 2002 Gary Strangman. All rights reserved
# Copyright 2002-2016 The SciPy Developers
#
# The original code from Gary Strangman was heavily adapted for
# use in SciPy by Travis Oliphant. The original code came with the
# following disclaimer:
#
# This software is provided "as-is". There are no expressed or implied
# warranties of any kind, including, but not limited to, the warranties
# of merchantability and fitness for a given application. In no event
# shall Gary Strangman be liable for any direct, indirect, incidental,
# special, exemplary or consequential damages (including, but not limited
# to, loss of use, data or profits, or business interruption) however
# caused and on any theory of liability, whether in contract, strict
# liability or tort (including negligence or otherwise) arising in any way
# out of the use of this software, even if advised of the possibility of
# such damage.
"""
A collection of basic statistical functions for Python. The function
names appear below.
Some scalar functions defined here are also available in the scipy.special
package where they work on arbitrary sized arrays.
Disclaimers: The function list is obviously incomplete and, worse, the
functions are not optimized. All functions have been tested (some more
so than others), but they are far from bulletproof. Thus, as with any
free software, no warranty or guarantee is expressed or implied. :-) A
few extra functions that don't appear in the list below can be found by
interested treasure-hunters. These functions don't necessarily have
both list and array versions but were deemed useful.
Central Tendency
----------------
.. autosummary::
:toctree: generated/
gmean
hmean
mode
Moments
-------
.. autosummary::
:toctree: generated/
moment
variation
skew
kurtosis
normaltest
Altered Versions
----------------
.. autosummary::
:toctree: generated/
tmean
tvar
tstd
tsem
describe
Frequency Stats
---------------
.. autosummary::
:toctree: generated/
itemfreq
scoreatpercentile
percentileofscore
cumfreq
relfreq
Variability
-----------
.. autosummary::
:toctree: generated/
obrientransform
sem
zmap
zscore
iqr
Trimming Functions
------------------
.. autosummary::
:toctree: generated/
trimboth
trim1
Correlation Functions
---------------------
.. autosummary::
:toctree: generated/
pearsonr
fisher_exact
spearmanr
pointbiserialr
kendalltau
weightedtau
linregress
theilslopes
Inferential Stats
-----------------
.. autosummary::
:toctree: generated/
ttest_1samp
ttest_ind
ttest_ind_from_stats
ttest_rel
chisquare
power_divergence
ks_2samp
mannwhitneyu
ranksums
wilcoxon
kruskal
friedmanchisquare
combine_pvalues
Statistical Distances
---------------------
.. autosummary::
:toctree: generated/
wasserstein_distance
energy_distance
ANOVA Functions
---------------
.. autosummary::
:toctree: generated/
f_oneway
Support Functions
-----------------
.. autosummary::
:toctree: generated/
rankdata
References
----------
.. [CRCProbStat2000] Zwillinger, D. and Kokoska, S. (2000). CRC Standard
Probability and Statistics Tables and Formulae. Chapman & Hall: New
York. 2000.
"""
from __future__ import division, print_function, absolute_import
import warnings
import math
from collections import namedtuple
import numpy as np
from numpy import array, asarray, ma, zeros
from scipy._lib.six import callable, string_types
from scipy._lib._version import NumpyVersion
import scipy.special as special
import scipy.linalg as linalg
from . import distributions
from . import mstats_basic
from ._distn_infrastructure import _lazywhere
from ._stats_mstats_common import _find_repeats, linregress, theilslopes
from ._stats import _kendall_dis, _toint64, _weightedrankedtau
__all__ = ['find_repeats', 'gmean', 'hmean', 'mode', 'tmean', 'tvar',
'tmin', 'tmax', 'tstd', 'tsem', 'moment', 'variation',
'skew', 'kurtosis', 'describe', 'skewtest', 'kurtosistest',
'normaltest', 'jarque_bera', 'itemfreq',
'scoreatpercentile', 'percentileofscore',
'cumfreq', 'relfreq', 'obrientransform',
'sem', 'zmap', 'zscore', 'iqr',
'sigmaclip', 'trimboth', 'trim1', 'trim_mean', 'f_oneway',
'pearsonr', 'fisher_exact', 'spearmanr', 'pointbiserialr',
'kendalltau', 'weightedtau',
'linregress', 'theilslopes', 'ttest_1samp',
'ttest_ind', 'ttest_ind_from_stats', 'ttest_rel', 'kstest',
'chisquare', 'power_divergence', 'ks_2samp', 'mannwhitneyu',
'tiecorrect', 'ranksums', 'kruskal', 'friedmanchisquare',
'rankdata',
'combine_pvalues', 'wasserstein_distance', 'energy_distance']
def _chk_asarray(a, axis):
if axis is None:
a = np.ravel(a)
outaxis = 0
else:
a = np.asarray(a)
outaxis = axis
if a.ndim == 0:
a = np.atleast_1d(a)
return a, outaxis
def _chk2_asarray(a, b, axis):
if axis is None:
a = np.ravel(a)
b = np.ravel(b)
outaxis = 0
else:
a = np.asarray(a)
b = np.asarray(b)
outaxis = axis
if a.ndim == 0:
a = np.atleast_1d(a)
if b.ndim == 0:
b = np.atleast_1d(b)
return a, b, outaxis
def _contains_nan(a, nan_policy='propagate'):
policies = ['propagate', 'raise', 'omit']
if nan_policy not in policies:
raise ValueError("nan_policy must be one of {%s}" %
', '.join("'%s'" % s for s in policies))
try:
# Calling np.sum to avoid creating a huge array into memory
# e.g. np.isnan(a).any()
with np.errstate(invalid='ignore'):
contains_nan = np.isnan(np.sum(a))
except TypeError:
# If the check cannot be properly performed we fallback to omitting
# nan values and raising a warning. This can happen when attempting to
# sum things that are not numbers (e.g. as in the function `mode`).
contains_nan = False
nan_policy = 'omit'
warnings.warn("The input array could not be properly checked for nan "
"values. nan values will be ignored.", RuntimeWarning)
if contains_nan and nan_policy == 'raise':
raise ValueError("The input contains nan values")
return (contains_nan, nan_policy)
def gmean(a, axis=0, dtype=None):
"""
Compute the geometric mean along the specified axis.
Return the geometric average of the array elements.
That is: n-th root of (x1 * x2 * ... * xn)
Parameters
----------
a : array_like
Input array or object that can be converted to an array.
axis : int or None, optional
Axis along which the geometric mean is computed. Default is 0.
If None, compute over the whole array `a`.
dtype : dtype, optional
Type of the returned array and of the accumulator in which the
elements are summed. If dtype is not specified, it defaults to the
dtype of a, unless a has an integer dtype with a precision less than
that of the default platform integer. In that case, the default
platform integer is used.
Returns
-------
gmean : ndarray
see dtype parameter above
See Also
--------
numpy.mean : Arithmetic average
numpy.average : Weighted average
hmean : Harmonic mean
Notes
-----
The geometric average is computed over a single dimension of the input
array, axis=0 by default, or all values in the array if axis=None.
float64 intermediate and return values are used for integer inputs.
Use masked arrays to ignore any non-finite values in the input or that
arise in the calculations such as Not a Number and infinity because masked
arrays automatically mask any non-finite values.
Examples
--------
>>> from scipy.stats import gmean
>>> gmean([1, 4])
2.0
>>> gmean([1, 2, 3, 4, 5, 6, 7])
3.3800151591412964
"""
if not isinstance(a, np.ndarray):
# if not an ndarray object attempt to convert it
log_a = np.log(np.array(a, dtype=dtype))
elif dtype:
# Must change the default dtype allowing array type
if isinstance(a, np.ma.MaskedArray):
log_a = np.log(np.ma.asarray(a, dtype=dtype))
else:
log_a = np.log(np.asarray(a, dtype=dtype))
else:
log_a = np.log(a)
return np.exp(log_a.mean(axis=axis))
def hmean(a, axis=0, dtype=None):
"""
Calculate the harmonic mean along the specified axis.
That is: n / (1/x1 + 1/x2 + ... + 1/xn)
Parameters
----------
a : array_like
Input array, masked array or object that can be converted to an array.
axis : int or None, optional
Axis along which the harmonic mean is computed. Default is 0.
If None, compute over the whole array `a`.
dtype : dtype, optional
Type of the returned array and of the accumulator in which the
elements are summed. If `dtype` is not specified, it defaults to the
dtype of `a`, unless `a` has an integer `dtype` with a precision less
than that of the default platform integer. In that case, the default
platform integer is used.
Returns
-------
hmean : ndarray
see `dtype` parameter above
See Also
--------
numpy.mean : Arithmetic average
numpy.average : Weighted average
gmean : Geometric mean
Notes
-----
The harmonic mean is computed over a single dimension of the input
array, axis=0 by default, or all values in the array if axis=None.
float64 intermediate and return values are used for integer inputs.
Use masked arrays to ignore any non-finite values in the input or that
arise in the calculations such as Not a Number and infinity.
Examples
--------
>>> from scipy.stats import hmean
>>> hmean([1, 4])
1.6000000000000001
>>> hmean([1, 2, 3, 4, 5, 6, 7])
2.6997245179063363
"""
if not isinstance(a, np.ndarray):
a = np.array(a, dtype=dtype)
if np.all(a > 0):
# Harmonic mean only defined if greater than zero
if isinstance(a, np.ma.MaskedArray):
size = a.count(axis)
else:
if axis is None:
a = a.ravel()
size = a.shape[0]
else:
size = a.shape[axis]
return size / np.sum(1.0 / a, axis=axis, dtype=dtype)
else:
raise ValueError("Harmonic mean only defined if all elements greater "
"than zero")
ModeResult = namedtuple('ModeResult', ('mode', 'count'))
def mode(a, axis=0, nan_policy='propagate'):
"""
Return an array of the modal (most common) value in the passed array.
If there is more than one such value, only the smallest is returned.
The bin-count for the modal bins is also returned.
Parameters
----------
a : array_like
n-dimensional array of which to find mode(s).
axis : int or None, optional
Axis along which to operate. Default is 0. If None, compute over
the whole array `a`.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
mode : ndarray
Array of modal values.
count : ndarray
Array of counts for each mode.
Examples
--------
>>> a = np.array([[6, 8, 3, 0],
... [3, 2, 1, 7],
... [8, 1, 8, 4],
... [5, 3, 0, 5],
... [4, 7, 5, 9]])
>>> from scipy import stats
>>> stats.mode(a)
(array([[3, 1, 0, 0]]), array([[1, 1, 1, 1]]))
To get mode of whole array, specify ``axis=None``:
>>> stats.mode(a, axis=None)
(array([3]), array([3]))
"""
a, axis = _chk_asarray(a, axis)
if a.size == 0:
return ModeResult(np.array([]), np.array([]))
contains_nan, nan_policy = _contains_nan(a, nan_policy)
if contains_nan and nan_policy == 'omit':
a = ma.masked_invalid(a)
return mstats_basic.mode(a, axis)
scores = np.unique(np.ravel(a)) # get ALL unique values
testshape = list(a.shape)
testshape[axis] = 1
oldmostfreq = np.zeros(testshape, dtype=a.dtype)
oldcounts = np.zeros(testshape, dtype=int)
for score in scores:
template = (a == score)
counts = np.expand_dims(np.sum(template, axis), axis)
mostfrequent = np.where(counts > oldcounts, score, oldmostfreq)
oldcounts = np.maximum(counts, oldcounts)
oldmostfreq = mostfrequent
return ModeResult(mostfrequent, oldcounts)
def _mask_to_limits(a, limits, inclusive):
"""Mask an array for values outside of given limits.
This is primarily a utility function.
Parameters
----------
a : array
limits : (float or None, float or None)
A tuple consisting of the (lower limit, upper limit). Values in the
input array less than the lower limit or greater than the upper limit
will be masked out. None implies no limit.
inclusive : (bool, bool)
A tuple consisting of the (lower flag, upper flag). These flags
determine whether values exactly equal to lower or upper are allowed.
Returns
-------
A MaskedArray.
Raises
------
A ValueError if there are no values within the given limits.
"""
lower_limit, upper_limit = limits
lower_include, upper_include = inclusive
am = ma.MaskedArray(a)
if lower_limit is not None:
if lower_include:
am = ma.masked_less(am, lower_limit)
else:
am = ma.masked_less_equal(am, lower_limit)
if upper_limit is not None:
if upper_include:
am = ma.masked_greater(am, upper_limit)
else:
am = ma.masked_greater_equal(am, upper_limit)
if am.count() == 0:
raise ValueError("No array values within given limits")
return am
def tmean(a, limits=None, inclusive=(True, True), axis=None):
"""
Compute the trimmed mean.
This function finds the arithmetic mean of given values, ignoring values
outside the given `limits`.
Parameters
----------
a : array_like
Array of values.
limits : None or (lower limit, upper limit), optional
Values in the input array less than the lower limit or greater than the
upper limit will be ignored. When limits is None (default), then all
values are used. Either of the limit values in the tuple can also be
None representing a half-open interval.
inclusive : (bool, bool), optional
A tuple consisting of the (lower flag, upper flag). These flags
determine whether values exactly equal to the lower or upper limits
are included. The default value is (True, True).
axis : int or None, optional
Axis along which to compute test. Default is None.
Returns
-------
tmean : float
See also
--------
trim_mean : returns mean after trimming a proportion from both tails.
Examples
--------
>>> from scipy import stats
>>> x = np.arange(20)
>>> stats.tmean(x)
9.5
>>> stats.tmean(x, (3,17))
10.0
"""
a = asarray(a)
if limits is None:
return np.mean(a, None)
am = _mask_to_limits(a.ravel(), limits, inclusive)
return am.mean(axis=axis)
def tvar(a, limits=None, inclusive=(True, True), axis=0, ddof=1):
"""
Compute the trimmed variance.
This function computes the sample variance of an array of values,
while ignoring values which are outside of given `limits`.
Parameters
----------
a : array_like
Array of values.
limits : None or (lower limit, upper limit), optional
Values in the input array less than the lower limit or greater than the
upper limit will be ignored. When limits is None, then all values are
used. Either of the limit values in the tuple can also be None
representing a half-open interval. The default value is None.
inclusive : (bool, bool), optional
A tuple consisting of the (lower flag, upper flag). These flags
determine whether values exactly equal to the lower or upper limits
are included. The default value is (True, True).
axis : int or None, optional
Axis along which to operate. Default is 0. If None, compute over the
whole array `a`.
ddof : int, optional
Delta degrees of freedom. Default is 1.
Returns
-------
tvar : float
Trimmed variance.
Notes
-----
`tvar` computes the unbiased sample variance, i.e. it uses a correction
factor ``n / (n - 1)``.
Examples
--------
>>> from scipy import stats
>>> x = np.arange(20)
>>> stats.tvar(x)
35.0
>>> stats.tvar(x, (3,17))
20.0
"""
a = asarray(a)
a = a.astype(float).ravel()
if limits is None:
n = len(a)
return a.var() * n / (n - 1.)
am = _mask_to_limits(a, limits, inclusive)
return np.ma.var(am, ddof=ddof, axis=axis)
def tmin(a, lowerlimit=None, axis=0, inclusive=True, nan_policy='propagate'):
"""
Compute the trimmed minimum.
This function finds the miminum value of an array `a` along the
specified axis, but only considering values greater than a specified
lower limit.
Parameters
----------
a : array_like
array of values
lowerlimit : None or float, optional
Values in the input array less than the given limit will be ignored.
When lowerlimit is None, then all values are used. The default value
is None.
axis : int or None, optional
Axis along which to operate. Default is 0. If None, compute over the
whole array `a`.
inclusive : {True, False}, optional
This flag determines whether values exactly equal to the lower limit
are included. The default value is True.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
tmin : float, int or ndarray
Examples
--------
>>> from scipy import stats
>>> x = np.arange(20)
>>> stats.tmin(x)
0
>>> stats.tmin(x, 13)
13
>>> stats.tmin(x, 13, inclusive=False)
14
"""
a, axis = _chk_asarray(a, axis)
am = _mask_to_limits(a, (lowerlimit, None), (inclusive, False))
contains_nan, nan_policy = _contains_nan(am, nan_policy)
if contains_nan and nan_policy == 'omit':
am = ma.masked_invalid(am)
res = ma.minimum.reduce(am, axis).data
if res.ndim == 0:
return res[()]
return res
def tmax(a, upperlimit=None, axis=0, inclusive=True, nan_policy='propagate'):
"""
Compute the trimmed maximum.
This function computes the maximum value of an array along a given axis,
while ignoring values larger than a specified upper limit.
Parameters
----------
a : array_like
array of values
upperlimit : None or float, optional
Values in the input array greater than the given limit will be ignored.
When upperlimit is None, then all values are used. The default value
is None.
axis : int or None, optional
Axis along which to operate. Default is 0. If None, compute over the
whole array `a`.
inclusive : {True, False}, optional
This flag determines whether values exactly equal to the upper limit
are included. The default value is True.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
tmax : float, int or ndarray
Examples
--------
>>> from scipy import stats
>>> x = np.arange(20)
>>> stats.tmax(x)
19
>>> stats.tmax(x, 13)
13
>>> stats.tmax(x, 13, inclusive=False)
12
"""
a, axis = _chk_asarray(a, axis)
am = _mask_to_limits(a, (None, upperlimit), (False, inclusive))
contains_nan, nan_policy = _contains_nan(am, nan_policy)
if contains_nan and nan_policy == 'omit':
am = ma.masked_invalid(am)
res = ma.maximum.reduce(am, axis).data
if res.ndim == 0:
return res[()]
return res
def tstd(a, limits=None, inclusive=(True, True), axis=0, ddof=1):
"""
Compute the trimmed sample standard deviation.
This function finds the sample standard deviation of given values,
ignoring values outside the given `limits`.
Parameters
----------
a : array_like
array of values
limits : None or (lower limit, upper limit), optional
Values in the input array less than the lower limit or greater than the
upper limit will be ignored. When limits is None, then all values are
used. Either of the limit values in the tuple can also be None
representing a half-open interval. The default value is None.
inclusive : (bool, bool), optional
A tuple consisting of the (lower flag, upper flag). These flags
determine whether values exactly equal to the lower or upper limits
are included. The default value is (True, True).
axis : int or None, optional
Axis along which to operate. Default is 0. If None, compute over the
whole array `a`.
ddof : int, optional
Delta degrees of freedom. Default is 1.
Returns
-------
tstd : float
Notes
-----
`tstd` computes the unbiased sample standard deviation, i.e. it uses a
correction factor ``n / (n - 1)``.
Examples
--------
>>> from scipy import stats
>>> x = np.arange(20)
>>> stats.tstd(x)
5.9160797830996161
>>> stats.tstd(x, (3,17))
4.4721359549995796
"""
return np.sqrt(tvar(a, limits, inclusive, axis, ddof))
def tsem(a, limits=None, inclusive=(True, True), axis=0, ddof=1):
"""
Compute the trimmed standard error of the mean.
This function finds the standard error of the mean for given
values, ignoring values outside the given `limits`.
Parameters
----------
a : array_like
array of values
limits : None or (lower limit, upper limit), optional
Values in the input array less than the lower limit or greater than the
upper limit will be ignored. When limits is None, then all values are
used. Either of the limit values in the tuple can also be None
representing a half-open interval. The default value is None.
inclusive : (bool, bool), optional
A tuple consisting of the (lower flag, upper flag). These flags
determine whether values exactly equal to the lower or upper limits
are included. The default value is (True, True).
axis : int or None, optional
Axis along which to operate. Default is 0. If None, compute over the
whole array `a`.
ddof : int, optional
Delta degrees of freedom. Default is 1.
Returns
-------
tsem : float
Notes
-----
`tsem` uses unbiased sample standard deviation, i.e. it uses a
correction factor ``n / (n - 1)``.
Examples
--------
>>> from scipy import stats
>>> x = np.arange(20)
>>> stats.tsem(x)
1.3228756555322954
>>> stats.tsem(x, (3,17))
1.1547005383792515
"""
a = np.asarray(a).ravel()
if limits is None:
return a.std(ddof=ddof) / np.sqrt(a.size)
am = _mask_to_limits(a, limits, inclusive)
sd = np.sqrt(np.ma.var(am, ddof=ddof, axis=axis))
return sd / np.sqrt(am.count())
#####################################
# MOMENTS #
#####################################
def moment(a, moment=1, axis=0, nan_policy='propagate'):
r"""
Calculate the nth moment about the mean for a sample.
A moment is a specific quantitative measure of the shape of a set of
points. It is often used to calculate coefficients of skewness and kurtosis
due to its close relationship with them.
Parameters
----------
a : array_like
data
moment : int or array_like of ints, optional
order of central moment that is returned. Default is 1.
axis : int or None, optional
Axis along which the central moment is computed. Default is 0.
If None, compute over the whole array `a`.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
n-th central moment : ndarray or float
The appropriate moment along the given axis or over all values if axis
is None. The denominator for the moment calculation is the number of
observations, no degrees of freedom correction is done.
See also
--------
kurtosis, skew, describe
Notes
-----
The k-th central moment of a data sample is:
.. math::
m_k = \frac{1}{n} \sum_{i = 1}^n (x_i - \bar{x})^k
Where n is the number of samples and x-bar is the mean. This function uses
exponentiation by squares [1]_ for efficiency.
References
----------
.. [1] http://eli.thegreenplace.net/2009/03/21/efficient-integer-exponentiation-algorithms
Examples
--------
>>> from scipy.stats import moment
>>> moment([1, 2, 3, 4, 5], moment=1)
0.0
>>> moment([1, 2, 3, 4, 5], moment=2)
2.0
"""
a, axis = _chk_asarray(a, axis)
contains_nan, nan_policy = _contains_nan(a, nan_policy)
if contains_nan and nan_policy == 'omit':
a = ma.masked_invalid(a)
return mstats_basic.moment(a, moment, axis)
if a.size == 0:
# empty array, return nan(s) with shape matching `moment`
if np.isscalar(moment):
return np.nan
else:
return np.ones(np.asarray(moment).shape, dtype=np.float64) * np.nan
# for array_like moment input, return a value for each.
if not np.isscalar(moment):
mmnt = [_moment(a, i, axis) for i in moment]
return np.array(mmnt)
else:
return _moment(a, moment, axis)
def _moment(a, moment, axis):
if np.abs(moment - np.round(moment)) > 0:
raise ValueError("All moment parameters must be integers")
if moment == 0:
# When moment equals 0, the result is 1, by definition.
shape = list(a.shape)
del shape[axis]
if shape:
# return an actual array of the appropriate shape
return np.ones(shape, dtype=float)
else:
# the input was 1D, so return a scalar instead of a rank-0 array
return 1.0
elif moment == 1:
# By definition the first moment about the mean is 0.
shape = list(a.shape)
del shape[axis]
if shape:
# return an actual array of the appropriate shape
return np.zeros(shape, dtype=float)
else:
# the input was 1D, so return a scalar instead of a rank-0 array
return np.float64(0.0)
else:
# Exponentiation by squares: form exponent sequence
n_list = [moment]
current_n = moment
while current_n > 2:
if current_n % 2:
current_n = (current_n - 1) / 2
else:
current_n /= 2
n_list.append(current_n)
# Starting point for exponentiation by squares
a_zero_mean = a - np.expand_dims(np.mean(a, axis), axis)
if n_list[-1] == 1:
s = a_zero_mean.copy()
else:
s = a_zero_mean**2
# Perform multiplications
for n in n_list[-2::-1]:
s = s**2
if n % 2:
s *= a_zero_mean
return np.mean(s, axis)
def variation(a, axis=0, nan_policy='propagate'):
"""
Compute the coefficient of variation, the ratio of the biased standard
deviation to the mean.
Parameters
----------
a : array_like
Input array.
axis : int or None, optional
Axis along which to calculate the coefficient of variation. Default
is 0. If None, compute over the whole array `a`.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
variation : ndarray
The calculated variation along the requested axis.
References
----------
.. [1] Zwillinger, D. and Kokoska, S. (2000). CRC Standard
Probability and Statistics Tables and Formulae. Chapman & Hall: New
York. 2000.
Examples
--------
>>> from scipy.stats import variation
>>> variation([1, 2, 3, 4, 5])
0.47140452079103173
"""
a, axis = _chk_asarray(a, axis)
contains_nan, nan_policy = _contains_nan(a, nan_policy)
if contains_nan and nan_policy == 'omit':
a = ma.masked_invalid(a)
return mstats_basic.variation(a, axis)
return a.std(axis) / a.mean(axis)
def skew(a, axis=0, bias=True, nan_policy='propagate'):
"""
Compute the skewness of a data set.
For normally distributed data, the skewness should be about 0. For
unimodal continuous distributions, a skewness value > 0 means that
there is more weight in the right tail of the distribution. The
function `skewtest` can be used to determine if the skewness value
is close enough to 0, statistically speaking.
Parameters
----------
a : ndarray
data
axis : int or None, optional
Axis along which skewness is calculated. Default is 0.
If None, compute over the whole array `a`.
bias : bool, optional
If False, then the calculations are corrected for statistical bias.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
skewness : ndarray
The skewness of values along an axis, returning 0 where all values are
equal.
References
----------
.. [1] Zwillinger, D. and Kokoska, S. (2000). CRC Standard
Probability and Statistics Tables and Formulae. Chapman & Hall: New
York. 2000.
Section 2.2.24.1
Examples
--------
>>> from scipy.stats import skew
>>> skew([1, 2, 3, 4, 5])
0.0
>>> skew([2, 8, 0, 4, 1, 9, 9, 0])
0.2650554122698573
"""
a, axis = _chk_asarray(a, axis)
n = a.shape[axis]
contains_nan, nan_policy = _contains_nan(a, nan_policy)
if contains_nan and nan_policy == 'omit':
a = ma.masked_invalid(a)
return mstats_basic.skew(a, axis, bias)
m2 = moment(a, 2, axis)
m3 = moment(a, 3, axis)
zero = (m2 == 0)
vals = _lazywhere(~zero, (m2, m3),
lambda m2, m3: m3 / m2**1.5,
0.)
if not bias:
can_correct = (n > 2) & (m2 > 0)
if can_correct.any():
m2 = np.extract(can_correct, m2)
m3 = np.extract(can_correct, m3)
nval = np.sqrt((n - 1.0) * n) / (n - 2.0) * m3 / m2**1.5
np.place(vals, can_correct, nval)
if vals.ndim == 0:
return vals.item()
return vals
def kurtosis(a, axis=0, fisher=True, bias=True, nan_policy='propagate'):
"""
Compute the kurtosis (Fisher or Pearson) of a dataset.
Kurtosis is the fourth central moment divided by the square of the
variance. If Fisher's definition is used, then 3.0 is subtracted from
the result to give 0.0 for a normal distribution.
If bias is False then the kurtosis is calculated using k statistics to
eliminate bias coming from biased moment estimators
Use `kurtosistest` to see if result is close enough to normal.
Parameters
----------
a : array
data for which the kurtosis is calculated
axis : int or None, optional
Axis along which the kurtosis is calculated. Default is 0.
If None, compute over the whole array `a`.
fisher : bool, optional
If True, Fisher's definition is used (normal ==> 0.0). If False,
Pearson's definition is used (normal ==> 3.0).
bias : bool, optional
If False, then the calculations are corrected for statistical bias.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
kurtosis : array
The kurtosis of values along an axis. If all values are equal,
return -3 for Fisher's definition and 0 for Pearson's definition.
References
----------
.. [1] Zwillinger, D. and Kokoska, S. (2000). CRC Standard
Probability and Statistics Tables and Formulae. Chapman & Hall: New
York. 2000.
Examples
--------
>>> from scipy.stats import kurtosis
>>> kurtosis([1, 2, 3, 4, 5])
-1.3
"""
a, axis = _chk_asarray(a, axis)
contains_nan, nan_policy = _contains_nan(a, nan_policy)
if contains_nan and nan_policy == 'omit':
a = ma.masked_invalid(a)
return mstats_basic.kurtosis(a, axis, fisher, bias)
n = a.shape[axis]
m2 = moment(a, 2, axis)
m4 = moment(a, 4, axis)
zero = (m2 == 0)
olderr = np.seterr(all='ignore')
try:
vals = np.where(zero, 0, m4 / m2**2.0)
finally:
np.seterr(**olderr)
if not bias:
can_correct = (n > 3) & (m2 > 0)
if can_correct.any():
m2 = np.extract(can_correct, m2)
m4 = np.extract(can_correct, m4)
nval = 1.0/(n-2)/(n-3) * ((n**2-1.0)*m4/m2**2.0 - 3*(n-1)**2.0)
np.place(vals, can_correct, nval + 3.0)
if vals.ndim == 0:
vals = vals.item() # array scalar
if fisher:
return vals - 3
else:
return vals
DescribeResult = namedtuple('DescribeResult',
('nobs', 'minmax', 'mean', 'variance', 'skewness',
'kurtosis'))
def describe(a, axis=0, ddof=1, bias=True, nan_policy='propagate'):
"""
Compute several descriptive statistics of the passed array.
Parameters
----------
a : array_like
Input data.
axis : int or None, optional
Axis along which statistics are calculated. Default is 0.
If None, compute over the whole array `a`.
ddof : int, optional
Delta degrees of freedom (only for variance). Default is 1.
bias : bool, optional
If False, then the skewness and kurtosis calculations are corrected for
statistical bias.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
nobs : int or ndarray of ints
Number of observations (length of data along `axis`).
When 'omit' is chosen as nan_policy, each column is counted separately.
minmax: tuple of ndarrays or floats
Minimum and maximum value of data array.
mean : ndarray or float
Arithmetic mean of data along axis.
variance : ndarray or float
Unbiased variance of the data along axis, denominator is number of
observations minus one.
skewness : ndarray or float
Skewness, based on moment calculations with denominator equal to
the number of observations, i.e. no degrees of freedom correction.
kurtosis : ndarray or float
Kurtosis (Fisher). The kurtosis is normalized so that it is
zero for the normal distribution. No degrees of freedom are used.
See Also
--------
skew, kurtosis
Examples
--------
>>> from scipy import stats
>>> a = np.arange(10)
>>> stats.describe(a)
DescribeResult(nobs=10, minmax=(0, 9), mean=4.5, variance=9.166666666666666,
skewness=0.0, kurtosis=-1.2242424242424244)
>>> b = [[1, 2], [3, 4]]
>>> stats.describe(b)
DescribeResult(nobs=2, minmax=(array([1, 2]), array([3, 4])),
mean=array([2., 3.]), variance=array([2., 2.]),
skewness=array([0., 0.]), kurtosis=array([-2., -2.]))
"""
a, axis = _chk_asarray(a, axis)
contains_nan, nan_policy = _contains_nan(a, nan_policy)
if contains_nan and nan_policy == 'omit':
a = ma.masked_invalid(a)
return mstats_basic.describe(a, axis, ddof, bias)
if a.size == 0:
raise ValueError("The input must not be empty.")
n = a.shape[axis]
mm = (np.min(a, axis=axis), np.max(a, axis=axis))
m = np.mean(a, axis=axis)
v = np.var(a, axis=axis, ddof=ddof)
sk = skew(a, axis, bias=bias)
kurt = kurtosis(a, axis, bias=bias)
return DescribeResult(n, mm, m, v, sk, kurt)
#####################################
# NORMALITY TESTS #
#####################################
SkewtestResult = namedtuple('SkewtestResult', ('statistic', 'pvalue'))
def skewtest(a, axis=0, nan_policy='propagate'):
"""
Test whether the skew is different from the normal distribution.
This function tests the null hypothesis that the skewness of
the population that the sample was drawn from is the same
as that of a corresponding normal distribution.
Parameters
----------
a : array
The data to be tested
axis : int or None, optional
Axis along which statistics are calculated. Default is 0.
If None, compute over the whole array `a`.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
statistic : float
The computed z-score for this test.
pvalue : float
a 2-sided p-value for the hypothesis test
Notes
-----
The sample size must be at least 8.
References
----------
.. [1] R. B. D'Agostino, A. J. Belanger and R. B. D'Agostino Jr.,
"A suggestion for using powerful and informative tests of
normality", American Statistician 44, pp. 316-321, 1990.
Examples
--------
>>> from scipy.stats import skewtest
>>> skewtest([1, 2, 3, 4, 5, 6, 7, 8])
SkewtestResult(statistic=1.0108048609177787, pvalue=0.3121098361421897)
>>> skewtest([2, 8, 0, 4, 1, 9, 9, 0])
SkewtestResult(statistic=0.44626385374196975, pvalue=0.6554066631275459)
>>> skewtest([1, 2, 3, 4, 5, 6, 7, 8000])
SkewtestResult(statistic=3.571773510360407, pvalue=0.0003545719905823133)
>>> skewtest([100, 100, 100, 100, 100, 100, 100, 101])
SkewtestResult(statistic=3.5717766638478072, pvalue=0.000354567720281634)
"""
a, axis = _chk_asarray(a, axis)
contains_nan, nan_policy = _contains_nan(a, nan_policy)
if contains_nan and nan_policy == 'omit':
a = ma.masked_invalid(a)
return mstats_basic.skewtest(a, axis)
if axis is None:
a = np.ravel(a)
axis = 0
b2 = skew(a, axis)
n = float(a.shape[axis])
if n < 8:
raise ValueError(
"skewtest is not valid with less than 8 samples; %i samples"
" were given." % int(n))
y = b2 * math.sqrt(((n + 1) * (n + 3)) / (6.0 * (n - 2)))
beta2 = (3.0 * (n**2 + 27*n - 70) * (n+1) * (n+3) /
((n-2.0) * (n+5) * (n+7) * (n+9)))
W2 = -1 + math.sqrt(2 * (beta2 - 1))
delta = 1 / math.sqrt(0.5 * math.log(W2))
alpha = math.sqrt(2.0 / (W2 - 1))
y = np.where(y == 0, 1, y)
Z = delta * np.log(y / alpha + np.sqrt((y / alpha)**2 + 1))
return SkewtestResult(Z, 2 * distributions.norm.sf(np.abs(Z)))
KurtosistestResult = namedtuple('KurtosistestResult', ('statistic', 'pvalue'))
def kurtosistest(a, axis=0, nan_policy='propagate'):
"""
Test whether a dataset has normal kurtosis.
This function tests the null hypothesis that the kurtosis
of the population from which the sample was drawn is that
of the normal distribution: ``kurtosis = 3(n-1)/(n+1)``.
Parameters
----------
a : array
array of the sample data
axis : int or None, optional
Axis along which to compute test. Default is 0. If None,
compute over the whole array `a`.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
statistic : float
The computed z-score for this test.
pvalue : float
The 2-sided p-value for the hypothesis test
Notes
-----
Valid only for n>20. The Z-score is set to 0 for bad entries.
This function uses the method described in [1]_.
References
----------
.. [1] see e.g. F. J. Anscombe, W. J. Glynn, "Distribution of the kurtosis
statistic b2 for normal samples", Biometrika, vol. 70, pp. 227-234, 1983.
Examples
--------
>>> from scipy.stats import kurtosistest
>>> kurtosistest(list(range(20)))
KurtosistestResult(statistic=-1.7058104152122062, pvalue=0.08804338332528348)
>>> np.random.seed(28041990)
>>> s = np.random.normal(0, 1, 1000)
>>> kurtosistest(s)
KurtosistestResult(statistic=1.2317590987707365, pvalue=0.21803908613450895)
"""
a, axis = _chk_asarray(a, axis)
contains_nan, nan_policy = _contains_nan(a, nan_policy)
if contains_nan and nan_policy == 'omit':
a = ma.masked_invalid(a)
return mstats_basic.kurtosistest(a, axis)
n = float(a.shape[axis])
if n < 5:
raise ValueError(
"kurtosistest requires at least 5 observations; %i observations"
" were given." % int(n))
if n < 20:
warnings.warn("kurtosistest only valid for n>=20 ... continuing "
"anyway, n=%i" % int(n))
b2 = kurtosis(a, axis, fisher=False)
E = 3.0*(n-1) / (n+1)
varb2 = 24.0*n*(n-2)*(n-3) / ((n+1)*(n+1.)*(n+3)*(n+5)) # [1]_ Eq. 1
x = (b2-E) / np.sqrt(varb2) # [1]_ Eq. 4
# [1]_ Eq. 2:
sqrtbeta1 = 6.0*(n*n-5*n+2)/((n+7)*(n+9)) * np.sqrt((6.0*(n+3)*(n+5)) /
(n*(n-2)*(n-3)))
# [1]_ Eq. 3:
A = 6.0 + 8.0/sqrtbeta1 * (2.0/sqrtbeta1 + np.sqrt(1+4.0/(sqrtbeta1**2)))
term1 = 1 - 2/(9.0*A)
denom = 1 + x*np.sqrt(2/(A-4.0))
denom = np.where(denom < 0, 99, denom)
term2 = np.where(denom < 0, term1, np.power((1-2.0/A)/denom, 1/3.0))
Z = (term1 - term2) / np.sqrt(2/(9.0*A)) # [1]_ Eq. 5
Z = np.where(denom == 99, 0, Z)
if Z.ndim == 0:
Z = Z[()]
# zprob uses upper tail, so Z needs to be positive
return KurtosistestResult(Z, 2 * distributions.norm.sf(np.abs(Z)))
NormaltestResult = namedtuple('NormaltestResult', ('statistic', 'pvalue'))
def normaltest(a, axis=0, nan_policy='propagate'):
"""
Test whether a sample differs from a normal distribution.
This function tests the null hypothesis that a sample comes
from a normal distribution. It is based on D'Agostino and
Pearson's [1]_, [2]_ test that combines skew and kurtosis to
produce an omnibus test of normality.
Parameters
----------
a : array_like
The array containing the sample to be tested.
axis : int or None, optional
Axis along which to compute test. Default is 0. If None,
compute over the whole array `a`.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
statistic : float or array
``s^2 + k^2``, where ``s`` is the z-score returned by `skewtest` and
``k`` is the z-score returned by `kurtosistest`.
pvalue : float or array
A 2-sided chi squared probability for the hypothesis test.
References
----------
.. [1] D'Agostino, R. B. (1971), "An omnibus test of normality for
moderate and large sample size", Biometrika, 58, 341-348
.. [2] D'Agostino, R. and Pearson, E. S. (1973), "Tests for departure from
normality", Biometrika, 60, 613-622
Examples
--------
>>> from scipy import stats
>>> pts = 1000
>>> np.random.seed(28041990)
>>> a = np.random.normal(0, 1, size=pts)
>>> b = np.random.normal(2, 1, size=pts)
>>> x = np.concatenate((a, b))
>>> k2, p = stats.normaltest(x)
>>> alpha = 1e-3
>>> print("p = {:g}".format(p))
p = 3.27207e-11
>>> if p < alpha: # null hypothesis: x comes from a normal distribution
... print("The null hypothesis can be rejected")
... else:
... print("The null hypothesis cannot be rejected")
The null hypothesis can be rejected
"""
a, axis = _chk_asarray(a, axis)
contains_nan, nan_policy = _contains_nan(a, nan_policy)
if contains_nan and nan_policy == 'omit':
a = ma.masked_invalid(a)
return mstats_basic.normaltest(a, axis)
s, _ = skewtest(a, axis)
k, _ = kurtosistest(a, axis)
k2 = s*s + k*k
return NormaltestResult(k2, distributions.chi2.sf(k2, 2))
def jarque_bera(x):
"""
Perform the Jarque-Bera goodness of fit test on sample data.
The Jarque-Bera test tests whether the sample data has the skewness and
kurtosis matching a normal distribution.
Note that this test only works for a large enough number of data samples
(>2000) as the test statistic asymptotically has a Chi-squared distribution
with 2 degrees of freedom.
Parameters
----------
x : array_like
Observations of a random variable.
Returns
-------
jb_value : float
The test statistic.
p : float
The p-value for the hypothesis test.
References
----------
.. [1] Jarque, C. and Bera, A. (1980) "Efficient tests for normality,
homoscedasticity and serial independence of regression residuals",
6 Econometric Letters 255-259.
Examples
--------
>>> from scipy import stats
>>> np.random.seed(987654321)
>>> x = np.random.normal(0, 1, 100000)
>>> y = np.random.rayleigh(1, 100000)
>>> stats.jarque_bera(x)
(4.7165707989581342, 0.09458225503041906)
>>> stats.jarque_bera(y)
(6713.7098548143422, 0.0)
"""
x = np.asarray(x)
n = float(x.size)
if n == 0:
raise ValueError('At least one observation is required.')
mu = x.mean()
diffx = x - mu
skewness = (1 / n * np.sum(diffx**3)) / (1 / n * np.sum(diffx**2))**(3 / 2.)
kurtosis = (1 / n * np.sum(diffx**4)) / (1 / n * np.sum(diffx**2))**2
jb_value = n / 6 * (skewness**2 + (kurtosis - 3)**2 / 4)
p = 1 - distributions.chi2.cdf(jb_value, 2)
return jb_value, p
#####################################
# FREQUENCY FUNCTIONS #
#####################################
@np.deprecate(message="`itemfreq` is deprecated and will be removed in a "
"future version. Use instead `np.unique(..., return_counts=True)`")
def itemfreq(a):
"""
Return a 2-D array of item frequencies.
Parameters
----------
a : (N,) array_like
Input array.
Returns
-------
itemfreq : (K, 2) ndarray
A 2-D frequency table. Column 1 contains sorted, unique values from
`a`, column 2 contains their respective counts.
Examples
--------
>>> from scipy import stats
>>> a = np.array([1, 1, 5, 0, 1, 2, 2, 0, 1, 4])
>>> stats.itemfreq(a)
array([[ 0., 2.],
[ 1., 4.],
[ 2., 2.],
[ 4., 1.],
[ 5., 1.]])
>>> np.bincount(a)
array([2, 4, 2, 0, 1, 1])
>>> stats.itemfreq(a/10.)
array([[ 0. , 2. ],
[ 0.1, 4. ],
[ 0.2, 2. ],
[ 0.4, 1. ],
[ 0.5, 1. ]])
"""
items, inv = np.unique(a, return_inverse=True)
freq = np.bincount(inv)
return np.array([items, freq]).T
def scoreatpercentile(a, per, limit=(), interpolation_method='fraction',
axis=None):
"""
Calculate the score at a given percentile of the input sequence.
For example, the score at `per=50` is the median. If the desired quantile
lies between two data points, we interpolate between them, according to
the value of `interpolation`. If the parameter `limit` is provided, it
should be a tuple (lower, upper) of two values.
Parameters
----------
a : array_like
A 1-D array of values from which to extract score.
per : array_like
Percentile(s) at which to extract score. Values should be in range
[0,100].
limit : tuple, optional
Tuple of two scalars, the lower and upper limits within which to
compute the percentile. Values of `a` outside
this (closed) interval will be ignored.
interpolation_method : {'fraction', 'lower', 'higher'}, optional
This optional parameter specifies the interpolation method to use,
when the desired quantile lies between two data points `i` and `j`
- fraction: ``i + (j - i) * fraction`` where ``fraction`` is the
fractional part of the index surrounded by ``i`` and ``j``.
- lower: ``i``.
- higher: ``j``.
axis : int, optional
Axis along which the percentiles are computed. Default is None. If
None, compute over the whole array `a`.
Returns
-------
score : float or ndarray
Score at percentile(s).
See Also
--------
percentileofscore, numpy.percentile
Notes
-----
This function will become obsolete in the future.
For Numpy 1.9 and higher, `numpy.percentile` provides all the functionality
that `scoreatpercentile` provides. And it's significantly faster.
Therefore it's recommended to use `numpy.percentile` for users that have
numpy >= 1.9.
Examples
--------
>>> from scipy import stats
>>> a = np.arange(100)
>>> stats.scoreatpercentile(a, 50)
49.5
"""
# adapted from NumPy's percentile function. When we require numpy >= 1.8,
# the implementation of this function can be replaced by np.percentile.
a = np.asarray(a)
if a.size == 0:
# empty array, return nan(s) with shape matching `per`
if np.isscalar(per):
return np.nan
else:
return np.ones(np.asarray(per).shape, dtype=np.float64) * np.nan
if limit:
a = a[(limit[0] <= a) & (a <= limit[1])]
sorted = np.sort(a, axis=axis)
if axis is None:
axis = 0
return _compute_qth_percentile(sorted, per, interpolation_method, axis)
# handle sequence of per's without calling sort multiple times
def _compute_qth_percentile(sorted, per, interpolation_method, axis):
if not np.isscalar(per):
score = [_compute_qth_percentile(sorted, i, interpolation_method, axis)
for i in per]
return np.array(score)
if (per < 0) or (per > 100):
raise ValueError("percentile must be in the range [0, 100]")
indexer = [slice(None)] * sorted.ndim
idx = per / 100. * (sorted.shape[axis] - 1)
if int(idx) != idx:
# round fractional indices according to interpolation method
if interpolation_method == 'lower':
idx = int(np.floor(idx))
elif interpolation_method == 'higher':
idx = int(np.ceil(idx))
elif interpolation_method == 'fraction':
pass # keep idx as fraction and interpolate
else:
raise ValueError("interpolation_method can only be 'fraction', "
"'lower' or 'higher'")
i = int(idx)
if i == idx:
indexer[axis] = slice(i, i + 1)
weights = array(1)
sumval = 1.0
else:
indexer[axis] = slice(i, i + 2)
j = i + 1
weights = array([(j - idx), (idx - i)], float)
wshape = [1] * sorted.ndim
wshape[axis] = 2
weights.shape = wshape
sumval = weights.sum()
# Use np.add.reduce (== np.sum but a little faster) to coerce data type
return np.add.reduce(sorted[indexer] * weights, axis=axis) / sumval
def percentileofscore(a, score, kind='rank'):
"""
The percentile rank of a score relative to a list of scores.
A `percentileofscore` of, for example, 80% means that 80% of the
scores in `a` are below the given score. In the case of gaps or
ties, the exact definition depends on the optional keyword, `kind`.
Parameters
----------
a : array_like
Array of scores to which `score` is compared.
score : int or float
Score that is compared to the elements in `a`.
kind : {'rank', 'weak', 'strict', 'mean'}, optional
This optional parameter specifies the interpretation of the
resulting score:
- "rank": Average percentage ranking of score. In case of
multiple matches, average the percentage rankings of
all matching scores.
- "weak": This kind corresponds to the definition of a cumulative
distribution function. A percentileofscore of 80%
means that 80% of values are less than or equal
to the provided score.
- "strict": Similar to "weak", except that only values that are
strictly less than the given score are counted.
- "mean": The average of the "weak" and "strict" scores, often used in
testing. See
http://en.wikipedia.org/wiki/Percentile_rank
Returns
-------
pcos : float
Percentile-position of score (0-100) relative to `a`.
See Also
--------
numpy.percentile
Examples
--------
Three-quarters of the given values lie below a given score:
>>> from scipy import stats
>>> stats.percentileofscore([1, 2, 3, 4], 3)
75.0
With multiple matches, note how the scores of the two matches, 0.6
and 0.8 respectively, are averaged:
>>> stats.percentileofscore([1, 2, 3, 3, 4], 3)
70.0
Only 2/5 values are strictly less than 3:
>>> stats.percentileofscore([1, 2, 3, 3, 4], 3, kind='strict')
40.0
But 4/5 values are less than or equal to 3:
>>> stats.percentileofscore([1, 2, 3, 3, 4], 3, kind='weak')
80.0
The average between the weak and the strict scores is
>>> stats.percentileofscore([1, 2, 3, 3, 4], 3, kind='mean')
60.0
"""
if np.isnan(score):
return np.nan
a = np.asarray(a)
n = len(a)
if n == 0:
return 100.0
if kind == 'rank':
left = np.count_nonzero(a < score)
right = np.count_nonzero(a <= score)
pct = (right + left + (1 if right > left else 0)) * 50.0/n
return pct
elif kind == 'strict':
return np.count_nonzero(a < score) / float(n) * 100
elif kind == 'weak':
return np.count_nonzero(a <= score) / float(n) * 100
elif kind == 'mean':
pct = (np.count_nonzero(a < score) + np.count_nonzero(a <= score)) / float(n) * 50
return pct
else:
raise ValueError("kind can only be 'rank', 'strict', 'weak' or 'mean'")
HistogramResult = namedtuple('HistogramResult',
('count', 'lowerlimit', 'binsize', 'extrapoints'))
def _histogram(a, numbins=10, defaultlimits=None, weights=None, printextras=False):
"""
Separate the range into several bins and return the number of instances
in each bin.
Parameters
----------
a : array_like
Array of scores which will be put into bins.
numbins : int, optional
The number of bins to use for the histogram. Default is 10.
defaultlimits : tuple (lower, upper), optional
The lower and upper values for the range of the histogram.
If no value is given, a range slightly larger than the range of the
values in a is used. Specifically ``(a.min() - s, a.max() + s)``,
where ``s = (1/2)(a.max() - a.min()) / (numbins - 1)``.
weights : array_like, optional
The weights for each value in `a`. Default is None, which gives each
value a weight of 1.0
printextras : bool, optional
If True, if there are extra points (i.e. the points that fall outside
the bin limits) a warning is raised saying how many of those points
there are. Default is False.
Returns
-------
count : ndarray
Number of points (or sum of weights) in each bin.
lowerlimit : float
Lowest value of histogram, the lower limit of the first bin.
binsize : float
The size of the bins (all bins have the same size).
extrapoints : int
The number of points outside the range of the histogram.
See Also
--------
numpy.histogram
Notes
-----
This histogram is based on numpy's histogram but has a larger range by
default if default limits is not set.
"""
a = np.ravel(a)
if defaultlimits is None:
if a.size == 0:
# handle empty arrays. Undetermined range, so use 0-1.
defaultlimits = (0, 1)
else:
# no range given, so use values in `a`
data_min = a.min()
data_max = a.max()
# Have bins extend past min and max values slightly
s = (data_max - data_min) / (2. * (numbins - 1.))
defaultlimits = (data_min - s, data_max + s)
# use numpy's histogram method to compute bins
hist, bin_edges = np.histogram(a, bins=numbins, range=defaultlimits,
weights=weights)
# hist are not always floats, convert to keep with old output
hist = np.array(hist, dtype=float)
# fixed width for bins is assumed, as numpy's histogram gives
# fixed width bins for int values for 'bins'
binsize = bin_edges[1] - bin_edges[0]
# calculate number of extra points
extrapoints = len([v for v in a
if defaultlimits[0] > v or v > defaultlimits[1]])
if extrapoints > 0 and printextras:
warnings.warn("Points outside given histogram range = %s"
% extrapoints)
return HistogramResult(hist, defaultlimits[0], binsize, extrapoints)
CumfreqResult = namedtuple('CumfreqResult',
('cumcount', 'lowerlimit', 'binsize',
'extrapoints'))
def cumfreq(a, numbins=10, defaultreallimits=None, weights=None):
"""
Return a cumulative frequency histogram, using the histogram function.
A cumulative histogram is a mapping that counts the cumulative number of
observations in all of the bins up to the specified bin.
Parameters
----------
a : array_like
Input array.
numbins : int, optional
The number of bins to use for the histogram. Default is 10.
defaultreallimits : tuple (lower, upper), optional
The lower and upper values for the range of the histogram.
If no value is given, a range slightly larger than the range of the
values in `a` is used. Specifically ``(a.min() - s, a.max() + s)``,
where ``s = (1/2)(a.max() - a.min()) / (numbins - 1)``.
weights : array_like, optional
The weights for each value in `a`. Default is None, which gives each
value a weight of 1.0
Returns
-------
cumcount : ndarray
Binned values of cumulative frequency.
lowerlimit : float
Lower real limit
binsize : float
Width of each bin.
extrapoints : int
Extra points.
Examples
--------
>>> import matplotlib.pyplot as plt
>>> from scipy import stats
>>> x = [1, 4, 2, 1, 3, 1]
>>> res = stats.cumfreq(x, numbins=4, defaultreallimits=(1.5, 5))
>>> res.cumcount
array([ 1., 2., 3., 3.])
>>> res.extrapoints
3
Create a normal distribution with 1000 random values
>>> rng = np.random.RandomState(seed=12345)
>>> samples = stats.norm.rvs(size=1000, random_state=rng)
Calculate cumulative frequencies
>>> res = stats.cumfreq(samples, numbins=25)
Calculate space of values for x
>>> x = res.lowerlimit + np.linspace(0, res.binsize*res.cumcount.size,
... res.cumcount.size)
Plot histogram and cumulative histogram
>>> fig = plt.figure(figsize=(10, 4))
>>> ax1 = fig.add_subplot(1, 2, 1)
>>> ax2 = fig.add_subplot(1, 2, 2)
>>> ax1.hist(samples, bins=25)
>>> ax1.set_title('Histogram')
>>> ax2.bar(x, res.cumcount, width=res.binsize)
>>> ax2.set_title('Cumulative histogram')
>>> ax2.set_xlim([x.min(), x.max()])
>>> plt.show()
"""
h, l, b, e = _histogram(a, numbins, defaultreallimits, weights=weights)
cumhist = np.cumsum(h * 1, axis=0)
return CumfreqResult(cumhist, l, b, e)
RelfreqResult = namedtuple('RelfreqResult',
('frequency', 'lowerlimit', 'binsize',
'extrapoints'))
def relfreq(a, numbins=10, defaultreallimits=None, weights=None):
"""
Return a relative frequency histogram, using the histogram function.
A relative frequency histogram is a mapping of the number of
observations in each of the bins relative to the total of observations.
Parameters
----------
a : array_like
Input array.
numbins : int, optional
The number of bins to use for the histogram. Default is 10.
defaultreallimits : tuple (lower, upper), optional
The lower and upper values for the range of the histogram.
If no value is given, a range slightly larger than the range of the
values in a is used. Specifically ``(a.min() - s, a.max() + s)``,
where ``s = (1/2)(a.max() - a.min()) / (numbins - 1)``.
weights : array_like, optional
The weights for each value in `a`. Default is None, which gives each
value a weight of 1.0
Returns
-------
frequency : ndarray
Binned values of relative frequency.
lowerlimit : float
Lower real limit
binsize : float
Width of each bin.
extrapoints : int
Extra points.
Examples
--------
>>> import matplotlib.pyplot as plt
>>> from scipy import stats
>>> a = np.array([2, 4, 1, 2, 3, 2])
>>> res = stats.relfreq(a, numbins=4)
>>> res.frequency
array([ 0.16666667, 0.5 , 0.16666667, 0.16666667])
>>> np.sum(res.frequency) # relative frequencies should add up to 1
1.0
Create a normal distribution with 1000 random values
>>> rng = np.random.RandomState(seed=12345)
>>> samples = stats.norm.rvs(size=1000, random_state=rng)
Calculate relative frequencies
>>> res = stats.relfreq(samples, numbins=25)
Calculate space of values for x
>>> x = res.lowerlimit + np.linspace(0, res.binsize*res.frequency.size,
... res.frequency.size)
Plot relative frequency histogram
>>> fig = plt.figure(figsize=(5, 4))
>>> ax = fig.add_subplot(1, 1, 1)
>>> ax.bar(x, res.frequency, width=res.binsize)
>>> ax.set_title('Relative frequency histogram')
>>> ax.set_xlim([x.min(), x.max()])
>>> plt.show()
"""
a = np.asanyarray(a)
h, l, b, e = _histogram(a, numbins, defaultreallimits, weights=weights)
h = h / float(a.shape[0])
return RelfreqResult(h, l, b, e)
#####################################
# VARIABILITY FUNCTIONS #
#####################################
def obrientransform(*args):
"""
Compute the O'Brien transform on input data (any number of arrays).
Used to test for homogeneity of variance prior to running one-way stats.
Each array in ``*args`` is one level of a factor.
If `f_oneway` is run on the transformed data and found significant,
the variances are unequal. From Maxwell and Delaney [1]_, p.112.
Parameters
----------
args : tuple of array_like
Any number of arrays.
Returns
-------
obrientransform : ndarray
Transformed data for use in an ANOVA. The first dimension
of the result corresponds to the sequence of transformed
arrays. If the arrays given are all 1-D of the same length,
the return value is a 2-D array; otherwise it is a 1-D array
of type object, with each element being an ndarray.
References
----------
.. [1] S. E. Maxwell and H. D. Delaney, "Designing Experiments and
Analyzing Data: A Model Comparison Perspective", Wadsworth, 1990.
Examples
--------
We'll test the following data sets for differences in their variance.
>>> x = [10, 11, 13, 9, 7, 12, 12, 9, 10]
>>> y = [13, 21, 5, 10, 8, 14, 10, 12, 7, 15]
Apply the O'Brien transform to the data.
>>> from scipy.stats import obrientransform
>>> tx, ty = obrientransform(x, y)
Use `scipy.stats.f_oneway` to apply a one-way ANOVA test to the
transformed data.
>>> from scipy.stats import f_oneway
>>> F, p = f_oneway(tx, ty)
>>> p
0.1314139477040335
If we require that ``p < 0.05`` for significance, we cannot conclude
that the variances are different.
"""
TINY = np.sqrt(np.finfo(float).eps)
# `arrays` will hold the transformed arguments.
arrays = []
for arg in args:
a = np.asarray(arg)
n = len(a)
mu = np.mean(a)
sq = (a - mu)**2
sumsq = sq.sum()
# The O'Brien transform.
t = ((n - 1.5) * n * sq - 0.5 * sumsq) / ((n - 1) * (n - 2))
# Check that the mean of the transformed data is equal to the
# original variance.
var = sumsq / (n - 1)
if abs(var - np.mean(t)) > TINY:
raise ValueError('Lack of convergence in obrientransform.')
arrays.append(t)
return np.array(arrays)
def sem(a, axis=0, ddof=1, nan_policy='propagate'):
"""
Calculate the standard error of the mean (or standard error of
measurement) of the values in the input array.
Parameters
----------
a : array_like
An array containing the values for which the standard error is
returned.
axis : int or None, optional
Axis along which to operate. Default is 0. If None, compute over
the whole array `a`.
ddof : int, optional
Delta degrees-of-freedom. How many degrees of freedom to adjust
for bias in limited samples relative to the population estimate
of variance. Defaults to 1.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
s : ndarray or float
The standard error of the mean in the sample(s), along the input axis.
Notes
-----
The default value for `ddof` is different to the default (0) used by other
ddof containing routines, such as np.std and np.nanstd.
Examples
--------
Find standard error along the first axis:
>>> from scipy import stats
>>> a = np.arange(20).reshape(5,4)
>>> stats.sem(a)
array([ 2.8284, 2.8284, 2.8284, 2.8284])
Find standard error across the whole array, using n degrees of freedom:
>>> stats.sem(a, axis=None, ddof=0)
1.2893796958227628
"""
a, axis = _chk_asarray(a, axis)
contains_nan, nan_policy = _contains_nan(a, nan_policy)
if contains_nan and nan_policy == 'omit':
a = ma.masked_invalid(a)
return mstats_basic.sem(a, axis, ddof)
n = a.shape[axis]
s = np.std(a, axis=axis, ddof=ddof) / np.sqrt(n)
return s
def zscore(a, axis=0, ddof=0):
"""
Calculate the z score of each value in the sample, relative to the
sample mean and standard deviation.
Parameters
----------
a : array_like
An array like object containing the sample data.
axis : int or None, optional
Axis along which to operate. Default is 0. If None, compute over
the whole array `a`.
ddof : int, optional
Degrees of freedom correction in the calculation of the
standard deviation. Default is 0.
Returns
-------
zscore : array_like
The z-scores, standardized by mean and standard deviation of
input array `a`.
Notes
-----
This function preserves ndarray subclasses, and works also with
matrices and masked arrays (it uses `asanyarray` instead of
`asarray` for parameters).
Examples
--------
>>> a = np.array([ 0.7972, 0.0767, 0.4383, 0.7866, 0.8091,
... 0.1954, 0.6307, 0.6599, 0.1065, 0.0508])
>>> from scipy import stats
>>> stats.zscore(a)
array([ 1.1273, -1.247 , -0.0552, 1.0923, 1.1664, -0.8559, 0.5786,
0.6748, -1.1488, -1.3324])
Computing along a specified axis, using n-1 degrees of freedom
(``ddof=1``) to calculate the standard deviation:
>>> b = np.array([[ 0.3148, 0.0478, 0.6243, 0.4608],
... [ 0.7149, 0.0775, 0.6072, 0.9656],
... [ 0.6341, 0.1403, 0.9759, 0.4064],
... [ 0.5918, 0.6948, 0.904 , 0.3721],
... [ 0.0921, 0.2481, 0.1188, 0.1366]])
>>> stats.zscore(b, axis=1, ddof=1)
array([[-0.19264823, -1.28415119, 1.07259584, 0.40420358],
[ 0.33048416, -1.37380874, 0.04251374, 1.00081084],
[ 0.26796377, -1.12598418, 1.23283094, -0.37481053],
[-0.22095197, 0.24468594, 1.19042819, -1.21416216],
[-0.82780366, 1.4457416 , -0.43867764, -0.1792603 ]])
"""
a = np.asanyarray(a)
mns = a.mean(axis=axis)
sstd = a.std(axis=axis, ddof=ddof)
if axis and mns.ndim < a.ndim:
return ((a - np.expand_dims(mns, axis=axis)) /
np.expand_dims(sstd, axis=axis))
else:
return (a - mns) / sstd
def zmap(scores, compare, axis=0, ddof=0):
"""
Calculate the relative z-scores.
Return an array of z-scores, i.e., scores that are standardized to
zero mean and unit variance, where mean and variance are calculated
from the comparison array.
Parameters
----------
scores : array_like
The input for which z-scores are calculated.
compare : array_like
The input from which the mean and standard deviation of the
normalization are taken; assumed to have the same dimension as
`scores`.
axis : int or None, optional
Axis over which mean and variance of `compare` are calculated.
Default is 0. If None, compute over the whole array `scores`.
ddof : int, optional
Degrees of freedom correction in the calculation of the
standard deviation. Default is 0.
Returns
-------
zscore : array_like
Z-scores, in the same shape as `scores`.
Notes
-----
This function preserves ndarray subclasses, and works also with
matrices and masked arrays (it uses `asanyarray` instead of
`asarray` for parameters).
Examples
--------
>>> from scipy.stats import zmap
>>> a = [0.5, 2.0, 2.5, 3]
>>> b = [0, 1, 2, 3, 4]
>>> zmap(a, b)
array([-1.06066017, 0. , 0.35355339, 0.70710678])
"""
scores, compare = map(np.asanyarray, [scores, compare])
mns = compare.mean(axis=axis)
sstd = compare.std(axis=axis, ddof=ddof)
if axis and mns.ndim < compare.ndim:
return ((scores - np.expand_dims(mns, axis=axis)) /
np.expand_dims(sstd, axis=axis))
else:
return (scores - mns) / sstd
# Private dictionary initialized only once at module level
# See https://en.wikipedia.org/wiki/Robust_measures_of_scale
_scale_conversions = {'raw': 1.0,
'normal': special.erfinv(0.5) * 2.0 * math.sqrt(2.0)}
def iqr(x, axis=None, rng=(25, 75), scale='raw', nan_policy='propagate',
interpolation='linear', keepdims=False):
"""
Compute the interquartile range of the data along the specified axis.
The interquartile range (IQR) is the difference between the 75th and
25th percentile of the data. It is a measure of the dispersion
similar to standard deviation or variance, but is much more robust
against outliers [2]_.
The ``rng`` parameter allows this function to compute other
percentile ranges than the actual IQR. For example, setting
``rng=(0, 100)`` is equivalent to `numpy.ptp`.
The IQR of an empty array is `np.nan`.
.. versionadded:: 0.18.0
Parameters
----------
x : array_like
Input array or object that can be converted to an array.
axis : int or sequence of int, optional
Axis along which the range is computed. The default is to
compute the IQR for the entire array.
rng : Two-element sequence containing floats in range of [0,100] optional
Percentiles over which to compute the range. Each must be
between 0 and 100, inclusive. The default is the true IQR:
`(25, 75)`. The order of the elements is not important.
scale : scalar or str, optional
The numerical value of scale will be divided out of the final
result. The following string values are recognized:
'raw' : No scaling, just return the raw IQR.
'normal' : Scale by :math:`2 \\sqrt{2} erf^{-1}(\\frac{1}{2}) \\approx 1.349`.
The default is 'raw'. Array-like scale is also allowed, as long
as it broadcasts correctly to the output such that
``out / scale`` is a valid operation. The output dimensions
depend on the input array, `x`, the `axis` argument, and the
`keepdims` flag.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate'
returns nan, 'raise' throws an error, 'omit' performs the
calculations ignoring nan values. Default is 'propagate'.
interpolation : {'linear', 'lower', 'higher', 'midpoint', 'nearest'}, optional
Specifies the interpolation method to use when the percentile
boundaries lie between two data points `i` and `j`:
* 'linear' : `i + (j - i) * fraction`, where `fraction` is the
fractional part of the index surrounded by `i` and `j`.
* 'lower' : `i`.
* 'higher' : `j`.
* 'nearest' : `i` or `j` whichever is nearest.
* 'midpoint' : `(i + j) / 2`.
Default is 'linear'.
keepdims : bool, optional
If this is set to `True`, the reduced axes are left in the
result as dimensions with size one. With this option, the result
will broadcast correctly against the original array `x`.
Returns
-------
iqr : scalar or ndarray
If ``axis=None``, a scalar is returned. If the input contains
integers or floats of smaller precision than ``np.float64``, then the
output data-type is ``np.float64``. Otherwise, the output data-type is
the same as that of the input.
See Also
--------
numpy.std, numpy.var
Examples
--------
>>> from scipy.stats import iqr
>>> x = np.array([[10, 7, 4], [3, 2, 1]])
>>> x
array([[10, 7, 4],
[ 3, 2, 1]])
>>> iqr(x)
4.0
>>> iqr(x, axis=0)
array([ 3.5, 2.5, 1.5])
>>> iqr(x, axis=1)
array([ 3., 1.])
>>> iqr(x, axis=1, keepdims=True)
array([[ 3.],
[ 1.]])
Notes
-----
This function is heavily dependent on the version of `numpy` that is
installed. Versions greater than 1.11.0b3 are highly recommended, as they
include a number of enhancements and fixes to `numpy.percentile` and
`numpy.nanpercentile` that affect the operation of this function. The
following modifications apply:
Below 1.10.0 : `nan_policy` is poorly defined.
The default behavior of `numpy.percentile` is used for 'propagate'. This
is a hybrid of 'omit' and 'propagate' that mostly yields a skewed
version of 'omit' since NaNs are sorted to the end of the data. A
warning is raised if there are NaNs in the data.
Below 1.9.0: `numpy.nanpercentile` does not exist.
This means that `numpy.percentile` is used regardless of `nan_policy`
and a warning is issued. See previous item for a description of the
behavior.
Below 1.9.0: `keepdims` and `interpolation` are not supported.
The keywords get ignored with a warning if supplied with non-default
values. However, multiple axes are still supported.
References
----------
.. [1] "Interquartile range" https://en.wikipedia.org/wiki/Interquartile_range
.. [2] "Robust measures of scale" https://en.wikipedia.org/wiki/Robust_measures_of_scale
.. [3] "Quantile" https://en.wikipedia.org/wiki/Quantile
"""
x = asarray(x)
# This check prevents percentile from raising an error later. Also, it is
# consistent with `np.var` and `np.std`.
if not x.size:
return np.nan
# An error may be raised here, so fail-fast, before doing lengthy
# computations, even though `scale` is not used until later
if isinstance(scale, string_types):
scale_key = scale.lower()
if scale_key not in _scale_conversions:
raise ValueError("{0} not a valid scale for `iqr`".format(scale))
scale = _scale_conversions[scale_key]
# Select the percentile function to use based on nans and policy
contains_nan, nan_policy = _contains_nan(x, nan_policy)
if contains_nan and nan_policy == 'omit':
percentile_func = _iqr_nanpercentile
else:
percentile_func = _iqr_percentile
if len(rng) != 2:
raise TypeError("quantile range must be two element sequence")
rng = sorted(rng)
pct = percentile_func(x, rng, axis=axis, interpolation=interpolation,
keepdims=keepdims, contains_nan=contains_nan)
out = np.subtract(pct[1], pct[0])
if scale != 1.0:
out /= scale
return out
def _iqr_percentile(x, q, axis=None, interpolation='linear', keepdims=False, contains_nan=False):
"""
Private wrapper that works around older versions of `numpy`.
While this function is pretty much necessary for the moment, it
should be removed as soon as the minimum supported numpy version
allows.
"""
if contains_nan and NumpyVersion(np.__version__) < '1.10.0a':
# I see no way to avoid the version check to ensure that the corrected
# NaN behavior has been implemented except to call `percentile` on a
# small array.
msg = "Keyword nan_policy='propagate' not correctly supported for " \
"numpy versions < 1.10.x. The default behavior of " \
"`numpy.percentile` will be used."
warnings.warn(msg, RuntimeWarning)
try:
# For older versions of numpy, there are two things that can cause a
# problem here: missing keywords and non-scalar axis. The former can be
# partially handled with a warning, the latter can be handled fully by
# hacking in an implementation similar to numpy's function for
# providing multi-axis functionality
# (`numpy.lib.function_base._ureduce` for the curious).
result = np.percentile(x, q, axis=axis, keepdims=keepdims,
interpolation=interpolation)
except TypeError:
if interpolation != 'linear' or keepdims:
# At time or writing, this means np.__version__ < 1.9.0
warnings.warn("Keywords interpolation and keepdims not supported "
"for your version of numpy", RuntimeWarning)
try:
# Special processing if axis is an iterable
original_size = len(axis)
except TypeError:
# Axis is a scalar at this point
pass
else:
axis = np.unique(np.asarray(axis) % x.ndim)
if original_size > axis.size:
# mimic numpy if axes are duplicated
raise ValueError("duplicate value in axis")
if axis.size == x.ndim:
# axis includes all axes: revert to None
axis = None
elif axis.size == 1:
# no rolling necessary
axis = axis[0]
else:
# roll multiple axes to the end and flatten that part out
for ax in axis[::-1]:
x = np.rollaxis(x, ax, x.ndim)
x = x.reshape(x.shape[:-axis.size] +
(np.prod(x.shape[-axis.size:]),))
axis = -1
result = np.percentile(x, q, axis=axis)
return result
def _iqr_nanpercentile(x, q, axis=None, interpolation='linear', keepdims=False,
contains_nan=False):
"""
Private wrapper that works around the following:
1. A bug in `np.nanpercentile` that was around until numpy version
1.11.0.
2. A bug in `np.percentile` NaN handling that was fixed in numpy
version 1.10.0.
3. The non-existence of `np.nanpercentile` before numpy version
1.9.0.
While this function is pretty much necessary for the moment, it
should be removed as soon as the minimum supported numpy version
allows.
"""
if hasattr(np, 'nanpercentile'):
# At time or writing, this means np.__version__ < 1.9.0
result = np.nanpercentile(x, q, axis=axis,
interpolation=interpolation,
keepdims=keepdims)
# If non-scalar result and nanpercentile does not do proper axis roll.
# I see no way of avoiding the version test since dimensions may just
# happen to match in the data.
if result.ndim > 1 and NumpyVersion(np.__version__) < '1.11.0a':
axis = np.asarray(axis)
if axis.size == 1:
# If only one axis specified, reduction happens along that dimension
if axis.ndim == 0:
axis = axis[None]
result = np.rollaxis(result, axis[0])
else:
# If multiple axes, reduced dimeision is last
result = np.rollaxis(result, -1)
else:
msg = "Keyword nan_policy='omit' not correctly supported for numpy " \
"versions < 1.9.x. The default behavior of numpy.percentile " \
"will be used."
warnings.warn(msg, RuntimeWarning)
result = _iqr_percentile(x, q, axis=axis)
return result
#####################################
# TRIMMING FUNCTIONS #
#####################################
SigmaclipResult = namedtuple('SigmaclipResult', ('clipped', 'lower', 'upper'))
def sigmaclip(a, low=4., high=4.):
"""
Iterative sigma-clipping of array elements.
Starting from the full sample, all elements outside the critical range are
removed, i.e. all elements of the input array `c` that satisfy either of
the following conditions ::
c < mean(c) - std(c)*low
c > mean(c) + std(c)*high
The iteration continues with the updated sample until no
elements are outside the (updated) range.
Parameters
----------
a : array_like
Data array, will be raveled if not 1-D.
low : float, optional
Lower bound factor of sigma clipping. Default is 4.
high : float, optional
Upper bound factor of sigma clipping. Default is 4.
Returns
-------
clipped : ndarray
Input array with clipped elements removed.
lower : float
Lower threshold value use for clipping.
upper : float
Upper threshold value use for clipping.
Examples
--------
>>> from scipy.stats import sigmaclip
>>> a = np.concatenate((np.linspace(9.5, 10.5, 31),
... np.linspace(0, 20, 5)))
>>> fact = 1.5
>>> c, low, upp = sigmaclip(a, fact, fact)
>>> c
array([ 9.96666667, 10. , 10.03333333, 10. ])
>>> c.var(), c.std()
(0.00055555555555555165, 0.023570226039551501)
>>> low, c.mean() - fact*c.std(), c.min()
(9.9646446609406727, 9.9646446609406727, 9.9666666666666668)
>>> upp, c.mean() + fact*c.std(), c.max()
(10.035355339059327, 10.035355339059327, 10.033333333333333)
>>> a = np.concatenate((np.linspace(9.5, 10.5, 11),
... np.linspace(-100, -50, 3)))
>>> c, low, upp = sigmaclip(a, 1.8, 1.8)
>>> (c == np.linspace(9.5, 10.5, 11)).all()
True
"""
c = np.asarray(a).ravel()
delta = 1
while delta:
c_std = c.std()
c_mean = c.mean()
size = c.size
critlower = c_mean - c_std * low
critupper = c_mean + c_std * high
c = c[(c >= critlower) & (c <= critupper)]
delta = size - c.size
return SigmaclipResult(c, critlower, critupper)
def trimboth(a, proportiontocut, axis=0):
"""
Slices off a proportion of items from both ends of an array.
Slices off the passed proportion of items from both ends of the passed
array (i.e., with `proportiontocut` = 0.1, slices leftmost 10% **and**
rightmost 10% of scores). The trimmed values are the lowest and
highest ones.
Slices off less if proportion results in a non-integer slice index (i.e.,
conservatively slices off`proportiontocut`).
Parameters
----------
a : array_like
Data to trim.
proportiontocut : float
Proportion (in range 0-1) of total data set to trim of each end.
axis : int or None, optional
Axis along which to trim data. Default is 0. If None, compute over
the whole array `a`.
Returns
-------
out : ndarray
Trimmed version of array `a`. The order of the trimmed content
is undefined.
See Also
--------
trim_mean
Examples
--------
>>> from scipy import stats
>>> a = np.arange(20)
>>> b = stats.trimboth(a, 0.1)
>>> b.shape
(16,)
"""
a = np.asarray(a)
if a.size == 0:
return a
if axis is None:
a = a.ravel()
axis = 0
nobs = a.shape[axis]
lowercut = int(proportiontocut * nobs)
uppercut = nobs - lowercut
if (lowercut >= uppercut):
raise ValueError("Proportion too big.")
atmp = np.partition(a, (lowercut, uppercut - 1), axis)
sl = [slice(None)] * atmp.ndim
sl[axis] = slice(lowercut, uppercut)
return atmp[sl]
def trim1(a, proportiontocut, tail='right', axis=0):
"""
Slices off a proportion from ONE end of the passed array distribution.
If `proportiontocut` = 0.1, slices off 'leftmost' or 'rightmost'
10% of scores. The lowest or highest values are trimmed (depending on
the tail).
Slices off less if proportion results in a non-integer slice index
(i.e., conservatively slices off `proportiontocut` ).
Parameters
----------
a : array_like
Input array
proportiontocut : float
Fraction to cut off of 'left' or 'right' of distribution
tail : {'left', 'right'}, optional
Defaults to 'right'.
axis : int or None, optional
Axis along which to trim data. Default is 0. If None, compute over
the whole array `a`.
Returns
-------
trim1 : ndarray
Trimmed version of array `a`. The order of the trimmed content is
undefined.
"""
a = np.asarray(a)
if axis is None:
a = a.ravel()
axis = 0
nobs = a.shape[axis]
# avoid possible corner case
if proportiontocut >= 1:
return []
if tail.lower() == 'right':
lowercut = 0
uppercut = nobs - int(proportiontocut * nobs)
elif tail.lower() == 'left':
lowercut = int(proportiontocut * nobs)
uppercut = nobs
atmp = np.partition(a, (lowercut, uppercut - 1), axis)
return atmp[lowercut:uppercut]
def trim_mean(a, proportiontocut, axis=0):
"""
Return mean of array after trimming distribution from both tails.
If `proportiontocut` = 0.1, slices off 'leftmost' and 'rightmost' 10% of
scores. The input is sorted before slicing. Slices off less if proportion
results in a non-integer slice index (i.e., conservatively slices off
`proportiontocut` ).
Parameters
----------
a : array_like
Input array
proportiontocut : float
Fraction to cut off of both tails of the distribution
axis : int or None, optional
Axis along which the trimmed means are computed. Default is 0.
If None, compute over the whole array `a`.
Returns
-------
trim_mean : ndarray
Mean of trimmed array.
See Also
--------
trimboth
tmean : compute the trimmed mean ignoring values outside given `limits`.
Examples
--------
>>> from scipy import stats
>>> x = np.arange(20)
>>> stats.trim_mean(x, 0.1)
9.5
>>> x2 = x.reshape(5, 4)
>>> x2
array([[ 0, 1, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11],
[12, 13, 14, 15],
[16, 17, 18, 19]])
>>> stats.trim_mean(x2, 0.25)
array([ 8., 9., 10., 11.])
>>> stats.trim_mean(x2, 0.25, axis=1)
array([ 1.5, 5.5, 9.5, 13.5, 17.5])
"""
a = np.asarray(a)
if a.size == 0:
return np.nan
if axis is None:
a = a.ravel()
axis = 0
nobs = a.shape[axis]
lowercut = int(proportiontocut * nobs)
uppercut = nobs - lowercut
if (lowercut > uppercut):
raise ValueError("Proportion too big.")
atmp = np.partition(a, (lowercut, uppercut - 1), axis)
sl = [slice(None)] * atmp.ndim
sl[axis] = slice(lowercut, uppercut)
return np.mean(atmp[sl], axis=axis)
F_onewayResult = namedtuple('F_onewayResult', ('statistic', 'pvalue'))
def f_oneway(*args):
"""
Performs a 1-way ANOVA.
The one-way ANOVA tests the null hypothesis that two or more groups have
the same population mean. The test is applied to samples from two or
more groups, possibly with differing sizes.
Parameters
----------
sample1, sample2, ... : array_like
The sample measurements for each group.
Returns
-------
statistic : float
The computed F-value of the test.
pvalue : float
The associated p-value from the F-distribution.
Notes
-----
The ANOVA test has important assumptions that must be satisfied in order
for the associated p-value to be valid.
1. The samples are independent.
2. Each sample is from a normally distributed population.
3. The population standard deviations of the groups are all equal. This
property is known as homoscedasticity.
If these assumptions are not true for a given set of data, it may still be
possible to use the Kruskal-Wallis H-test (`scipy.stats.kruskal`) although
with some loss of power.
The algorithm is from Heiman[2], pp.394-7.
References
----------
.. [1] Lowry, Richard. "Concepts and Applications of Inferential
Statistics". Chapter 14.
http://faculty.vassar.edu/lowry/ch14pt1.html
.. [2] Heiman, G.W. Research Methods in Statistics. 2002.
.. [3] McDonald, G. H. "Handbook of Biological Statistics", One-way ANOVA.
http://www.biostathandbook.com/onewayanova.html
Examples
--------
>>> import scipy.stats as stats
[3]_ Here are some data on a shell measurement (the length of the anterior
adductor muscle scar, standardized by dividing by length) in the mussel
Mytilus trossulus from five locations: Tillamook, Oregon; Newport, Oregon;
Petersburg, Alaska; Magadan, Russia; and Tvarminne, Finland, taken from a
much larger data set used in McDonald et al. (1991).
>>> tillamook = [0.0571, 0.0813, 0.0831, 0.0976, 0.0817, 0.0859, 0.0735,
... 0.0659, 0.0923, 0.0836]
>>> newport = [0.0873, 0.0662, 0.0672, 0.0819, 0.0749, 0.0649, 0.0835,
... 0.0725]
>>> petersburg = [0.0974, 0.1352, 0.0817, 0.1016, 0.0968, 0.1064, 0.105]
>>> magadan = [0.1033, 0.0915, 0.0781, 0.0685, 0.0677, 0.0697, 0.0764,
... 0.0689]
>>> tvarminne = [0.0703, 0.1026, 0.0956, 0.0973, 0.1039, 0.1045]
>>> stats.f_oneway(tillamook, newport, petersburg, magadan, tvarminne)
(7.1210194716424473, 0.00028122423145345439)
"""
args = [np.asarray(arg, dtype=float) for arg in args]
# ANOVA on N groups, each in its own array
num_groups = len(args)
alldata = np.concatenate(args)
bign = len(alldata)
# Determine the mean of the data, and subtract that from all inputs to a
# variance (via sum_of_sq / sq_of_sum) calculation. Variance is invariance
# to a shift in location, and centering all data around zero vastly
# improves numerical stability.
offset = alldata.mean()
alldata -= offset
sstot = _sum_of_squares(alldata) - (_square_of_sums(alldata) / float(bign))
ssbn = 0
for a in args:
ssbn += _square_of_sums(a - offset) / float(len(a))
# Naming: variables ending in bn/b are for "between treatments", wn/w are
# for "within treatments"
ssbn -= (_square_of_sums(alldata) / float(bign))
sswn = sstot - ssbn
dfbn = num_groups - 1
dfwn = bign - num_groups
msb = ssbn / float(dfbn)
msw = sswn / float(dfwn)
f = msb / msw
prob = special.fdtrc(dfbn, dfwn, f) # equivalent to stats.f.sf
return F_onewayResult(f, prob)
def pearsonr(x, y):
r"""
Calculate a Pearson correlation coefficient and the p-value for testing
non-correlation.
The Pearson correlation coefficient measures the linear relationship
between two datasets. Strictly speaking, Pearson's correlation requires
that each dataset be normally distributed, and not necessarily zero-mean.
Like other correlation coefficients, this one varies between -1 and +1
with 0 implying no correlation. Correlations of -1 or +1 imply an exact
linear relationship. Positive correlations imply that as x increases, so
does y. Negative correlations imply that as x increases, y decreases.
The p-value roughly indicates the probability of an uncorrelated system
producing datasets that have a Pearson correlation at least as extreme
as the one computed from these datasets. The p-values are not entirely
reliable but are probably reasonable for datasets larger than 500 or so.
Parameters
----------
x : (N,) array_like
Input
y : (N,) array_like
Input
Returns
-------
r : float
Pearson's correlation coefficient
p-value : float
2-tailed p-value
Notes
-----
The correlation coefficient is calculated as follows:
.. math::
r_{pb} = \frac{\sum (x - m_x) (y - m_y)
}{\sqrt{\sum (x - m_x)^2 (y - m_y)^2}}
where :math:`m_x` is the mean of the vector :math:`x` and :math:`m_y` is
the mean of the vector :math:`y`.
References
----------
http://www.statsoft.com/textbook/glosp.html#Pearson%20Correlation
Examples
--------
>>> from scipy import stats
>>> a = np.array([0, 0, 0, 1, 1, 1, 1])
>>> b = np.arange(7)
>>> stats.pearsonr(a, b)
(0.8660254037844386, 0.011724811003954654)
>>> stats.pearsonr([1,2,3,4,5], [5,6,7,8,7])
(0.83205029433784372, 0.080509573298498519)
"""
# x and y should have same length.
x = np.asarray(x)
y = np.asarray(y)
n = len(x)
mx = x.mean()
my = y.mean()
xm, ym = x - mx, y - my
r_num = np.add.reduce(xm * ym)
r_den = np.sqrt(_sum_of_squares(xm) * _sum_of_squares(ym))
r = r_num / r_den
# Presumably, if abs(r) > 1, then it is only some small artifact of
# floating point arithmetic.
r = max(min(r, 1.0), -1.0)
df = n - 2
if abs(r) == 1.0:
prob = 0.0
else:
t_squared = r**2 * (df / ((1.0 - r) * (1.0 + r)))
prob = _betai(0.5*df, 0.5, df/(df+t_squared))
return r, prob
def fisher_exact(table, alternative='two-sided'):
"""Performs a Fisher exact test on a 2x2 contingency table.
Parameters
----------
table : array_like of ints
A 2x2 contingency table. Elements should be non-negative integers.
alternative : {'two-sided', 'less', 'greater'}, optional
Which alternative hypothesis to the null hypothesis the test uses.
Default is 'two-sided'.
Returns
-------
oddsratio : float
This is prior odds ratio and not a posterior estimate.
p_value : float
P-value, the probability of obtaining a distribution at least as
extreme as the one that was actually observed, assuming that the
null hypothesis is true.
See Also
--------
chi2_contingency : Chi-square test of independence of variables in a
contingency table.
Notes
-----
The calculated odds ratio is different from the one R uses. This scipy
implementation returns the (more common) "unconditional Maximum
Likelihood Estimate", while R uses the "conditional Maximum Likelihood
Estimate".
For tables with large numbers, the (inexact) chi-square test implemented
in the function `chi2_contingency` can also be used.
Examples
--------
Say we spend a few days counting whales and sharks in the Atlantic and
Indian oceans. In the Atlantic ocean we find 8 whales and 1 shark, in the
Indian ocean 2 whales and 5 sharks. Then our contingency table is::
Atlantic Indian
whales 8 2
sharks 1 5
We use this table to find the p-value:
>>> import scipy.stats as stats
>>> oddsratio, pvalue = stats.fisher_exact([[8, 2], [1, 5]])
>>> pvalue
0.0349...
The probability that we would observe this or an even more imbalanced ratio
by chance is about 3.5%. A commonly used significance level is 5%--if we
adopt that, we can therefore conclude that our observed imbalance is
statistically significant; whales prefer the Atlantic while sharks prefer
the Indian ocean.
"""
hypergeom = distributions.hypergeom
c = np.asarray(table, dtype=np.int64) # int32 is not enough for the algorithm
if not c.shape == (2, 2):
raise ValueError("The input `table` must be of shape (2, 2).")
if np.any(c < 0):
raise ValueError("All values in `table` must be nonnegative.")
if 0 in c.sum(axis=0) or 0 in c.sum(axis=1):
# If both values in a row or column are zero, the p-value is 1 and
# the odds ratio is NaN.
return np.nan, 1.0
if c[1, 0] > 0 and c[0, 1] > 0:
oddsratio = c[0, 0] * c[1, 1] / float(c[1, 0] * c[0, 1])
else:
oddsratio = np.inf
n1 = c[0, 0] + c[0, 1]
n2 = c[1, 0] + c[1, 1]
n = c[0, 0] + c[1, 0]
def binary_search(n, n1, n2, side):
"""Binary search for where to begin lower/upper halves in two-sided
test.
"""
if side == "upper":
minval = mode
maxval = n
else:
minval = 0
maxval = mode
guess = -1
while maxval - minval > 1:
if maxval == minval + 1 and guess == minval:
guess = maxval
else:
guess = (maxval + minval) // 2
pguess = hypergeom.pmf(guess, n1 + n2, n1, n)
if side == "upper":
ng = guess - 1
else:
ng = guess + 1
if pguess <= pexact < hypergeom.pmf(ng, n1 + n2, n1, n):
break
elif pguess < pexact:
maxval = guess
else:
minval = guess
if guess == -1:
guess = minval
if side == "upper":
while guess > 0 and hypergeom.pmf(guess, n1 + n2, n1, n) < pexact * epsilon:
guess -= 1
while hypergeom.pmf(guess, n1 + n2, n1, n) > pexact / epsilon:
guess += 1
else:
while hypergeom.pmf(guess, n1 + n2, n1, n) < pexact * epsilon:
guess += 1
while guess > 0 and hypergeom.pmf(guess, n1 + n2, n1, n) > pexact / epsilon:
guess -= 1
return guess
if alternative == 'less':
pvalue = hypergeom.cdf(c[0, 0], n1 + n2, n1, n)
elif alternative == 'greater':
# Same formula as the 'less' case, but with the second column.
pvalue = hypergeom.cdf(c[0, 1], n1 + n2, n1, c[0, 1] + c[1, 1])
elif alternative == 'two-sided':
mode = int(float((n + 1) * (n1 + 1)) / (n1 + n2 + 2))
pexact = hypergeom.pmf(c[0, 0], n1 + n2, n1, n)
pmode = hypergeom.pmf(mode, n1 + n2, n1, n)
epsilon = 1 - 1e-4
if np.abs(pexact - pmode) / np.maximum(pexact, pmode) <= 1 - epsilon:
return oddsratio, 1.
elif c[0, 0] < mode:
plower = hypergeom.cdf(c[0, 0], n1 + n2, n1, n)
if hypergeom.pmf(n, n1 + n2, n1, n) > pexact / epsilon:
return oddsratio, plower
guess = binary_search(n, n1, n2, "upper")
pvalue = plower + hypergeom.sf(guess - 1, n1 + n2, n1, n)
else:
pupper = hypergeom.sf(c[0, 0] - 1, n1 + n2, n1, n)
if hypergeom.pmf(0, n1 + n2, n1, n) > pexact / epsilon:
return oddsratio, pupper
guess = binary_search(n, n1, n2, "lower")
pvalue = pupper + hypergeom.cdf(guess, n1 + n2, n1, n)
else:
msg = "`alternative` should be one of {'two-sided', 'less', 'greater'}"
raise ValueError(msg)
if pvalue > 1.0:
pvalue = 1.0
return oddsratio, pvalue
SpearmanrResult = namedtuple('SpearmanrResult', ('correlation', 'pvalue'))
def spearmanr(a, b=None, axis=0, nan_policy='propagate'):
"""
Calculate a Spearman rank-order correlation coefficient and the p-value
to test for non-correlation.
The Spearman correlation is a nonparametric measure of the monotonicity
of the relationship between two datasets. Unlike the Pearson correlation,
the Spearman correlation does not assume that both datasets are normally
distributed. Like other correlation coefficients, this one varies
between -1 and +1 with 0 implying no correlation. Correlations of -1 or
+1 imply an exact monotonic relationship. Positive correlations imply that
as x increases, so does y. Negative correlations imply that as x
increases, y decreases.
The p-value roughly indicates the probability of an uncorrelated system
producing datasets that have a Spearman correlation at least as extreme
as the one computed from these datasets. The p-values are not entirely
reliable but are probably reasonable for datasets larger than 500 or so.
Parameters
----------
a, b : 1D or 2D array_like, b is optional
One or two 1-D or 2-D arrays containing multiple variables and
observations. When these are 1-D, each represents a vector of
observations of a single variable. For the behavior in the 2-D case,
see under ``axis``, below.
Both arrays need to have the same length in the ``axis`` dimension.
axis : int or None, optional
If axis=0 (default), then each column represents a variable, with
observations in the rows. If axis=1, the relationship is transposed:
each row represents a variable, while the columns contain observations.
If axis=None, then both arrays will be raveled.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
correlation : float or ndarray (2-D square)
Spearman correlation matrix or correlation coefficient (if only 2
variables are given as parameters. Correlation matrix is square with
length equal to total number of variables (columns or rows) in a and b
combined.
pvalue : float
The two-sided p-value for a hypothesis test whose null hypothesis is
that two sets of data are uncorrelated, has same dimension as rho.
Notes
-----
Changes in scipy 0.8.0: rewrite to add tie-handling, and axis.
References
----------
.. [1] Zwillinger, D. and Kokoska, S. (2000). CRC Standard
Probability and Statistics Tables and Formulae. Chapman & Hall: New
York. 2000.
Section 14.7
Examples
--------
>>> from scipy import stats
>>> stats.spearmanr([1,2,3,4,5], [5,6,7,8,7])
(0.82078268166812329, 0.088587005313543798)
>>> np.random.seed(1234321)
>>> x2n = np.random.randn(100, 2)
>>> y2n = np.random.randn(100, 2)
>>> stats.spearmanr(x2n)
(0.059969996999699973, 0.55338590803773591)
>>> stats.spearmanr(x2n[:,0], x2n[:,1])
(0.059969996999699973, 0.55338590803773591)
>>> rho, pval = stats.spearmanr(x2n, y2n)
>>> rho
array([[ 1. , 0.05997 , 0.18569457, 0.06258626],
[ 0.05997 , 1. , 0.110003 , 0.02534653],
[ 0.18569457, 0.110003 , 1. , 0.03488749],
[ 0.06258626, 0.02534653, 0.03488749, 1. ]])
>>> pval
array([[ 0. , 0.55338591, 0.06435364, 0.53617935],
[ 0.55338591, 0. , 0.27592895, 0.80234077],
[ 0.06435364, 0.27592895, 0. , 0.73039992],
[ 0.53617935, 0.80234077, 0.73039992, 0. ]])
>>> rho, pval = stats.spearmanr(x2n.T, y2n.T, axis=1)
>>> rho
array([[ 1. , 0.05997 , 0.18569457, 0.06258626],
[ 0.05997 , 1. , 0.110003 , 0.02534653],
[ 0.18569457, 0.110003 , 1. , 0.03488749],
[ 0.06258626, 0.02534653, 0.03488749, 1. ]])
>>> stats.spearmanr(x2n, y2n, axis=None)
(0.10816770419260482, 0.1273562188027364)
>>> stats.spearmanr(x2n.ravel(), y2n.ravel())
(0.10816770419260482, 0.1273562188027364)
>>> xint = np.random.randint(10, size=(100, 2))
>>> stats.spearmanr(xint)
(0.052760927029710199, 0.60213045837062351)
"""
a, axisout = _chk_asarray(a, axis)
a_contains_nan, nan_policy = _contains_nan(a, nan_policy)
if a_contains_nan:
a = ma.masked_invalid(a)
if a.size <= 1:
return SpearmanrResult(np.nan, np.nan)
ar = np.apply_along_axis(rankdata, axisout, a)
br = None
if b is not None:
b, axisout = _chk_asarray(b, axis)
b_contains_nan, nan_policy = _contains_nan(b, nan_policy)
if a_contains_nan or b_contains_nan:
b = ma.masked_invalid(b)
if nan_policy == 'propagate':
rho, pval = mstats_basic.spearmanr(a, b, use_ties=True)
return SpearmanrResult(rho * np.nan, pval * np.nan)
if nan_policy == 'omit':
return mstats_basic.spearmanr(a, b, use_ties=True)
br = np.apply_along_axis(rankdata, axisout, b)
n = a.shape[axisout]
rs = np.corrcoef(ar, br, rowvar=axisout)
olderr = np.seterr(divide='ignore') # rs can have elements equal to 1
try:
# clip the small negative values possibly caused by rounding
# errors before taking the square root
t = rs * np.sqrt(((n-2)/((rs+1.0)*(1.0-rs))).clip(0))
finally:
np.seterr(**olderr)
prob = 2 * distributions.t.sf(np.abs(t), n-2)
if rs.shape == (2, 2):
return SpearmanrResult(rs[1, 0], prob[1, 0])
else:
return SpearmanrResult(rs, prob)
PointbiserialrResult = namedtuple('PointbiserialrResult',
('correlation', 'pvalue'))
def pointbiserialr(x, y):
r"""
Calculate a point biserial correlation coefficient and its p-value.
The point biserial correlation is used to measure the relationship
between a binary variable, x, and a continuous variable, y. Like other
correlation coefficients, this one varies between -1 and +1 with 0
implying no correlation. Correlations of -1 or +1 imply a determinative
relationship.
This function uses a shortcut formula but produces the same result as
`pearsonr`.
Parameters
----------
x : array_like of bools
Input array.
y : array_like
Input array.
Returns
-------
correlation : float
R value
pvalue : float
2-tailed p-value
Notes
-----
`pointbiserialr` uses a t-test with ``n-1`` degrees of freedom.
It is equivalent to `pearsonr.`
The value of the point-biserial correlation can be calculated from:
.. math::
r_{pb} = \frac{\overline{Y_{1}} -
\overline{Y_{0}}}{s_{y}}\sqrt{\frac{N_{1} N_{2}}{N (N - 1))}}
Where :math:`Y_{0}` and :math:`Y_{1}` are means of the metric
observations coded 0 and 1 respectively; :math:`N_{0}` and :math:`N_{1}`
are number of observations coded 0 and 1 respectively; :math:`N` is the
total number of observations and :math:`s_{y}` is the standard
deviation of all the metric observations.
A value of :math:`r_{pb}` that is significantly different from zero is
completely equivalent to a significant difference in means between the two
groups. Thus, an independent groups t Test with :math:`N-2` degrees of
freedom may be used to test whether :math:`r_{pb}` is nonzero. The
relation between the t-statistic for comparing two independent groups and
:math:`r_{pb}` is given by:
.. math::
t = \sqrt{N - 2}\frac{r_{pb}}{\sqrt{1 - r^{2}_{pb}}}
References
----------
.. [1] J. Lev, "The Point Biserial Coefficient of Correlation", Ann. Math.
Statist., Vol. 20, no.1, pp. 125-126, 1949.
.. [2] R.F. Tate, "Correlation Between a Discrete and a Continuous
Variable. Point-Biserial Correlation.", Ann. Math. Statist., Vol. 25,
np. 3, pp. 603-607, 1954.
.. [3] http://onlinelibrary.wiley.com/doi/10.1002/9781118445112.stat06227/full
Examples
--------
>>> from scipy import stats
>>> a = np.array([0, 0, 0, 1, 1, 1, 1])
>>> b = np.arange(7)
>>> stats.pointbiserialr(a, b)
(0.8660254037844386, 0.011724811003954652)
>>> stats.pearsonr(a, b)
(0.86602540378443871, 0.011724811003954626)
>>> np.corrcoef(a, b)
array([[ 1. , 0.8660254],
[ 0.8660254, 1. ]])
"""
rpb, prob = pearsonr(x, y)
return PointbiserialrResult(rpb, prob)
KendalltauResult = namedtuple('KendalltauResult', ('correlation', 'pvalue'))
def kendalltau(x, y, initial_lexsort=None, nan_policy='propagate'):
"""
Calculate Kendall's tau, a correlation measure for ordinal data.
Kendall's tau is a measure of the correspondence between two rankings.
Values close to 1 indicate strong agreement, values close to -1 indicate
strong disagreement. This is the 1945 "tau-b" version of Kendall's
tau [2]_, which can account for ties and which reduces to the 1938 "tau-a"
version [1]_ in absence of ties.
Parameters
----------
x, y : array_like
Arrays of rankings, of the same shape. If arrays are not 1-D, they will
be flattened to 1-D.
initial_lexsort : bool, optional
Unused (deprecated).
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'. Note that if the input contains nan
'omit' delegates to mstats_basic.kendalltau(), which has a different
implementation.
Returns
-------
correlation : float
The tau statistic.
pvalue : float
The two-sided p-value for a hypothesis test whose null hypothesis is
an absence of association, tau = 0.
See also
--------
spearmanr : Calculates a Spearman rank-order correlation coefficient.
theilslopes : Computes the Theil-Sen estimator for a set of points (x, y).
weightedtau : Computes a weighted version of Kendall's tau.
Notes
-----
The definition of Kendall's tau that is used is [2]_::
tau = (P - Q) / sqrt((P + Q + T) * (P + Q + U))
where P is the number of concordant pairs, Q the number of discordant
pairs, T the number of ties only in `x`, and U the number of ties only in
`y`. If a tie occurs for the same pair in both `x` and `y`, it is not
added to either T or U.
References
----------
.. [1] Maurice G. Kendall, "A New Measure of Rank Correlation", Biometrika
Vol. 30, No. 1/2, pp. 81-93, 1938.
.. [2] Maurice G. Kendall, "The treatment of ties in ranking problems",
Biometrika Vol. 33, No. 3, pp. 239-251. 1945.
.. [3] Gottfried E. Noether, "Elements of Nonparametric Statistics", John
Wiley & Sons, 1967.
.. [4] Peter M. Fenwick, "A new data structure for cumulative frequency
tables", Software: Practice and Experience, Vol. 24, No. 3,
pp. 327-336, 1994.
Examples
--------
>>> from scipy import stats
>>> x1 = [12, 2, 1, 12, 2]
>>> x2 = [1, 4, 7, 1, 0]
>>> tau, p_value = stats.kendalltau(x1, x2)
>>> tau
-0.47140452079103173
>>> p_value
0.2827454599327748
"""
x = np.asarray(x).ravel()
y = np.asarray(y).ravel()
if x.size != y.size:
raise ValueError("All inputs to `kendalltau` must be of the same size, "
"found x-size %s and y-size %s" % (x.size, y.size))
elif not x.size or not y.size:
return KendalltauResult(np.nan, np.nan) # Return NaN if arrays are empty
# check both x and y
cnx, npx = _contains_nan(x, nan_policy)
cny, npy = _contains_nan(y, nan_policy)
contains_nan = cnx or cny
if npx == 'omit' or npy == 'omit':
nan_policy = 'omit'
if contains_nan and nan_policy == 'propagate':
return KendalltauResult(np.nan, np.nan)
elif contains_nan and nan_policy == 'omit':
x = ma.masked_invalid(x)
y = ma.masked_invalid(y)
return mstats_basic.kendalltau(x, y)
if initial_lexsort is not None: # deprecate to drop!
warnings.warn('"initial_lexsort" is gone!')
def count_rank_tie(ranks):
cnt = np.bincount(ranks).astype('int64', copy=False)
cnt = cnt[cnt > 1]
return ((cnt * (cnt - 1) // 2).sum(),
(cnt * (cnt - 1.) * (cnt - 2)).sum(),
(cnt * (cnt - 1.) * (2*cnt + 5)).sum())
size = x.size
perm = np.argsort(y) # sort on y and convert y to dense ranks
x, y = x[perm], y[perm]
y = np.r_[True, y[1:] != y[:-1]].cumsum(dtype=np.intp)
# stable sort on x and convert x to dense ranks
perm = np.argsort(x, kind='mergesort')
x, y = x[perm], y[perm]
x = np.r_[True, x[1:] != x[:-1]].cumsum(dtype=np.intp)
dis = _kendall_dis(x, y) # discordant pairs
obs = np.r_[True, (x[1:] != x[:-1]) | (y[1:] != y[:-1]), True]
cnt = np.diff(np.where(obs)[0]).astype('int64', copy=False)
ntie = (cnt * (cnt - 1) // 2).sum() # joint ties
xtie, x0, x1 = count_rank_tie(x) # ties in x, stats
ytie, y0, y1 = count_rank_tie(y) # ties in y, stats
tot = (size * (size - 1)) // 2
if xtie == tot or ytie == tot:
return KendalltauResult(np.nan, np.nan)
# Note that tot = con + dis + (xtie - ntie) + (ytie - ntie) + ntie
# = con + dis + xtie + ytie - ntie
con_minus_dis = tot - xtie - ytie + ntie - 2 * dis
tau = con_minus_dis / np.sqrt(tot - xtie) / np.sqrt(tot - ytie)
# Limit range to fix computational errors
tau = min(1., max(-1., tau))
# con_minus_dis is approx normally distributed with this variance [3]_
var = (size * (size - 1) * (2.*size + 5) - x1 - y1) / 18. + (
2. * xtie * ytie) / (size * (size - 1)) + x0 * y0 / (9. *
size * (size - 1) * (size - 2))
pvalue = special.erfc(np.abs(con_minus_dis) / np.sqrt(var) / np.sqrt(2))
# Limit range to fix computational errors
return KendalltauResult(min(1., max(-1., tau)), pvalue)
WeightedTauResult = namedtuple('WeightedTauResult', ('correlation', 'pvalue'))
def weightedtau(x, y, rank=True, weigher=None, additive=True):
r"""
Compute a weighted version of Kendall's :math:`\tau`.
The weighted :math:`\tau` is a weighted version of Kendall's
:math:`\tau` in which exchanges of high weight are more influential than
exchanges of low weight. The default parameters compute the additive
hyperbolic version of the index, :math:`\tau_\mathrm h`, which has
been shown to provide the best balance between important and
unimportant elements [1]_.
The weighting is defined by means of a rank array, which assigns a
nonnegative rank to each element, and a weigher function, which
assigns a weight based from the rank to each element. The weight of an
exchange is then the sum or the product of the weights of the ranks of
the exchanged elements. The default parameters compute
:math:`\tau_\mathrm h`: an exchange between elements with rank
:math:`r` and :math:`s` (starting from zero) has weight
:math:`1/(r+1) + 1/(s+1)`.
Specifying a rank array is meaningful only if you have in mind an
external criterion of importance. If, as it usually happens, you do
not have in mind a specific rank, the weighted :math:`\tau` is
defined by averaging the values obtained using the decreasing
lexicographical rank by (`x`, `y`) and by (`y`, `x`). This is the
behavior with default parameters.
Note that if you are computing the weighted :math:`\tau` on arrays of
ranks, rather than of scores (i.e., a larger value implies a lower
rank) you must negate the ranks, so that elements of higher rank are
associated with a larger value.
Parameters
----------
x, y : array_like
Arrays of scores, of the same shape. If arrays are not 1-D, they will
be flattened to 1-D.
rank: array_like of ints or bool, optional
A nonnegative rank assigned to each element. If it is None, the
decreasing lexicographical rank by (`x`, `y`) will be used: elements of
higher rank will be those with larger `x`-values, using `y`-values to
break ties (in particular, swapping `x` and `y` will give a different
result). If it is False, the element indices will be used
directly as ranks. The default is True, in which case this
function returns the average of the values obtained using the
decreasing lexicographical rank by (`x`, `y`) and by (`y`, `x`).
weigher : callable, optional
The weigher function. Must map nonnegative integers (zero
representing the most important element) to a nonnegative weight.
The default, None, provides hyperbolic weighing, that is,
rank :math:`r` is mapped to weight :math:`1/(r+1)`.
additive : bool, optional
If True, the weight of an exchange is computed by adding the
weights of the ranks of the exchanged elements; otherwise, the weights
are multiplied. The default is True.
Returns
-------
correlation : float
The weighted :math:`\tau` correlation index.
pvalue : float
Presently ``np.nan``, as the null statistics is unknown (even in the
additive hyperbolic case).
See also
--------
kendalltau : Calculates Kendall's tau.
spearmanr : Calculates a Spearman rank-order correlation coefficient.
theilslopes : Computes the Theil-Sen estimator for a set of points (x, y).
Notes
-----
This function uses an :math:`O(n \log n)`, mergesort-based algorithm
[1]_ that is a weighted extension of Knight's algorithm for Kendall's
:math:`\tau` [2]_. It can compute Shieh's weighted :math:`\tau` [3]_
between rankings without ties (i.e., permutations) by setting
`additive` and `rank` to False, as the definition given in [1]_ is a
generalization of Shieh's.
NaNs are considered the smallest possible score.
.. versionadded:: 0.19.0
References
----------
.. [1] Sebastiano Vigna, "A weighted correlation index for rankings with
ties", Proceedings of the 24th international conference on World
Wide Web, pp. 1166-1176, ACM, 2015.
.. [2] W.R. Knight, "A Computer Method for Calculating Kendall's Tau with
Ungrouped Data", Journal of the American Statistical Association,
Vol. 61, No. 314, Part 1, pp. 436-439, 1966.
.. [3] Grace S. Shieh. "A weighted Kendall's tau statistic", Statistics &
Probability Letters, Vol. 39, No. 1, pp. 17-24, 1998.
Examples
--------
>>> from scipy import stats
>>> x = [12, 2, 1, 12, 2]
>>> y = [1, 4, 7, 1, 0]
>>> tau, p_value = stats.weightedtau(x, y)
>>> tau
-0.56694968153682723
>>> p_value
nan
>>> tau, p_value = stats.weightedtau(x, y, additive=False)
>>> tau
-0.62205716951801038
NaNs are considered the smallest possible score:
>>> x = [12, 2, 1, 12, 2]
>>> y = [1, 4, 7, 1, np.nan]
>>> tau, _ = stats.weightedtau(x, y)
>>> tau
-0.56694968153682723
This is exactly Kendall's tau:
>>> x = [12, 2, 1, 12, 2]
>>> y = [1, 4, 7, 1, 0]
>>> tau, _ = stats.weightedtau(x, y, weigher=lambda x: 1)
>>> tau
-0.47140452079103173
>>> x = [12, 2, 1, 12, 2]
>>> y = [1, 4, 7, 1, 0]
>>> stats.weightedtau(x, y, rank=None)
WeightedTauResult(correlation=-0.4157652301037516, pvalue=nan)
>>> stats.weightedtau(y, x, rank=None)
WeightedTauResult(correlation=-0.7181341329699028, pvalue=nan)
"""
x = np.asarray(x).ravel()
y = np.asarray(y).ravel()
if x.size != y.size:
raise ValueError("All inputs to `weightedtau` must be of the same size, "
"found x-size %s and y-size %s" % (x.size, y.size))
if not x.size:
return WeightedTauResult(np.nan, np.nan) # Return NaN if arrays are empty
# If there are NaNs we apply _toint64()
if np.isnan(np.sum(x)):
x = _toint64(x)
if np.isnan(np.sum(x)):
y = _toint64(y)
# Reduce to ranks unsupported types
if x.dtype != y.dtype:
if x.dtype != np.int64:
x = _toint64(x)
if y.dtype != np.int64:
y = _toint64(y)
else:
if x.dtype not in (np.int32, np.int64, np.float32, np.float64):
x = _toint64(x)
y = _toint64(y)
if rank is True:
return WeightedTauResult((
_weightedrankedtau(x, y, None, weigher, additive) +
_weightedrankedtau(y, x, None, weigher, additive)
) / 2, np.nan)
if rank is False:
rank = np.arange(x.size, dtype=np.intp)
elif rank is not None:
rank = np.asarray(rank).ravel()
if rank.size != x.size:
raise ValueError("All inputs to `weightedtau` must be of the same size, "
"found x-size %s and rank-size %s" % (x.size, rank.size))
return WeightedTauResult(_weightedrankedtau(x, y, rank, weigher, additive), np.nan)
#####################################
# INFERENTIAL STATISTICS #
#####################################
Ttest_1sampResult = namedtuple('Ttest_1sampResult', ('statistic', 'pvalue'))
def ttest_1samp(a, popmean, axis=0, nan_policy='propagate'):
"""
Calculate the T-test for the mean of ONE group of scores.
This is a two-sided test for the null hypothesis that the expected value
(mean) of a sample of independent observations `a` is equal to the given
population mean, `popmean`.
Parameters
----------
a : array_like
sample observation
popmean : float or array_like
expected value in null hypothesis. If array_like, then it must have the
same shape as `a` excluding the axis dimension
axis : int or None, optional
Axis along which to compute test. If None, compute over the whole
array `a`.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
statistic : float or array
t-statistic
pvalue : float or array
two-tailed p-value
Examples
--------
>>> from scipy import stats
>>> np.random.seed(7654567) # fix seed to get the same result
>>> rvs = stats.norm.rvs(loc=5, scale=10, size=(50,2))
Test if mean of random sample is equal to true mean, and different mean.
We reject the null hypothesis in the second case and don't reject it in
the first case.
>>> stats.ttest_1samp(rvs,5.0)
(array([-0.68014479, -0.04323899]), array([ 0.49961383, 0.96568674]))
>>> stats.ttest_1samp(rvs,0.0)
(array([ 2.77025808, 4.11038784]), array([ 0.00789095, 0.00014999]))
Examples using axis and non-scalar dimension for population mean.
>>> stats.ttest_1samp(rvs,[5.0,0.0])
(array([-0.68014479, 4.11038784]), array([ 4.99613833e-01, 1.49986458e-04]))
>>> stats.ttest_1samp(rvs.T,[5.0,0.0],axis=1)
(array([-0.68014479, 4.11038784]), array([ 4.99613833e-01, 1.49986458e-04]))
>>> stats.ttest_1samp(rvs,[[5.0],[0.0]])
(array([[-0.68014479, -0.04323899],
[ 2.77025808, 4.11038784]]), array([[ 4.99613833e-01, 9.65686743e-01],
[ 7.89094663e-03, 1.49986458e-04]]))
"""
a, axis = _chk_asarray(a, axis)
contains_nan, nan_policy = _contains_nan(a, nan_policy)
if contains_nan and nan_policy == 'omit':
a = ma.masked_invalid(a)
return mstats_basic.ttest_1samp(a, popmean, axis)
n = a.shape[axis]
df = n - 1
d = np.mean(a, axis) - popmean
v = np.var(a, axis, ddof=1)
denom = np.sqrt(v / float(n))
with np.errstate(divide='ignore', invalid='ignore'):
t = np.divide(d, denom)
t, prob = _ttest_finish(df, t)
return Ttest_1sampResult(t, prob)
def _ttest_finish(df, t):
"""Common code between all 3 t-test functions."""
prob = distributions.t.sf(np.abs(t), df) * 2 # use np.abs to get upper tail
if t.ndim == 0:
t = t[()]
return t, prob
def _ttest_ind_from_stats(mean1, mean2, denom, df):
d = mean1 - mean2
with np.errstate(divide='ignore', invalid='ignore'):
t = np.divide(d, denom)
t, prob = _ttest_finish(df, t)
return (t, prob)
def _unequal_var_ttest_denom(v1, n1, v2, n2):
vn1 = v1 / n1
vn2 = v2 / n2
with np.errstate(divide='ignore', invalid='ignore'):
df = (vn1 + vn2)**2 / (vn1**2 / (n1 - 1) + vn2**2 / (n2 - 1))
# If df is undefined, variances are zero (assumes n1 > 0 & n2 > 0).
# Hence it doesn't matter what df is as long as it's not NaN.
df = np.where(np.isnan(df), 1, df)
denom = np.sqrt(vn1 + vn2)
return df, denom
def _equal_var_ttest_denom(v1, n1, v2, n2):
df = n1 + n2 - 2.0
svar = ((n1 - 1) * v1 + (n2 - 1) * v2) / df
denom = np.sqrt(svar * (1.0 / n1 + 1.0 / n2))
return df, denom
Ttest_indResult = namedtuple('Ttest_indResult', ('statistic', 'pvalue'))
def ttest_ind_from_stats(mean1, std1, nobs1, mean2, std2, nobs2,
equal_var=True):
"""
T-test for means of two independent samples from descriptive statistics.
This is a two-sided test for the null hypothesis that two independent
samples have identical average (expected) values.
Parameters
----------
mean1 : array_like
The mean(s) of sample 1.
std1 : array_like
The standard deviation(s) of sample 1.
nobs1 : array_like
The number(s) of observations of sample 1.
mean2 : array_like
The mean(s) of sample 2
std2 : array_like
The standard deviations(s) of sample 2.
nobs2 : array_like
The number(s) of observations of sample 2.
equal_var : bool, optional
If True (default), perform a standard independent 2 sample test
that assumes equal population variances [1]_.
If False, perform Welch's t-test, which does not assume equal
population variance [2]_.
Returns
-------
statistic : float or array
The calculated t-statistics
pvalue : float or array
The two-tailed p-value.
See Also
--------
scipy.stats.ttest_ind
Notes
-----
.. versionadded:: 0.16.0
References
----------
.. [1] http://en.wikipedia.org/wiki/T-test#Independent_two-sample_t-test
.. [2] http://en.wikipedia.org/wiki/Welch%27s_t_test
Examples
--------
Suppose we have the summary data for two samples, as follows::
Sample Sample
Size Mean Variance
Sample 1 13 15.0 87.5
Sample 2 11 12.0 39.0
Apply the t-test to this data (with the assumption that the population
variances are equal):
>>> from scipy.stats import ttest_ind_from_stats
>>> ttest_ind_from_stats(mean1=15.0, std1=np.sqrt(87.5), nobs1=13,
... mean2=12.0, std2=np.sqrt(39.0), nobs2=11)
Ttest_indResult(statistic=0.9051358093310269, pvalue=0.3751996797581487)
For comparison, here is the data from which those summary statistics
were taken. With this data, we can compute the same result using
`scipy.stats.ttest_ind`:
>>> a = np.array([1, 3, 4, 6, 11, 13, 15, 19, 22, 24, 25, 26, 26])
>>> b = np.array([2, 4, 6, 9, 11, 13, 14, 15, 18, 19, 21])
>>> from scipy.stats import ttest_ind
>>> ttest_ind(a, b)
Ttest_indResult(statistic=0.905135809331027, pvalue=0.3751996797581486)
"""
if equal_var:
df, denom = _equal_var_ttest_denom(std1**2, nobs1, std2**2, nobs2)
else:
df, denom = _unequal_var_ttest_denom(std1**2, nobs1,
std2**2, nobs2)
res = _ttest_ind_from_stats(mean1, mean2, denom, df)
return Ttest_indResult(*res)
def ttest_ind(a, b, axis=0, equal_var=True, nan_policy='propagate'):
"""
Calculate the T-test for the means of *two independent* samples of scores.
This is a two-sided test for the null hypothesis that 2 independent samples
have identical average (expected) values. This test assumes that the
populations have identical variances by default.
Parameters
----------
a, b : array_like
The arrays must have the same shape, except in the dimension
corresponding to `axis` (the first, by default).
axis : int or None, optional
Axis along which to compute test. If None, compute over the whole
arrays, `a`, and `b`.
equal_var : bool, optional
If True (default), perform a standard independent 2 sample test
that assumes equal population variances [1]_.
If False, perform Welch's t-test, which does not assume equal
population variance [2]_.
.. versionadded:: 0.11.0
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
statistic : float or array
The calculated t-statistic.
pvalue : float or array
The two-tailed p-value.
Notes
-----
We can use this test, if we observe two independent samples from
the same or different population, e.g. exam scores of boys and
girls or of two ethnic groups. The test measures whether the
average (expected) value differs significantly across samples. If
we observe a large p-value, for example larger than 0.05 or 0.1,
then we cannot reject the null hypothesis of identical average scores.
If the p-value is smaller than the threshold, e.g. 1%, 5% or 10%,
then we reject the null hypothesis of equal averages.
References
----------
.. [1] http://en.wikipedia.org/wiki/T-test#Independent_two-sample_t-test
.. [2] http://en.wikipedia.org/wiki/Welch%27s_t_test
Examples
--------
>>> from scipy import stats
>>> np.random.seed(12345678)
Test with sample with identical means:
>>> rvs1 = stats.norm.rvs(loc=5,scale=10,size=500)
>>> rvs2 = stats.norm.rvs(loc=5,scale=10,size=500)
>>> stats.ttest_ind(rvs1,rvs2)
(0.26833823296239279, 0.78849443369564776)
>>> stats.ttest_ind(rvs1,rvs2, equal_var = False)
(0.26833823296239279, 0.78849452749500748)
`ttest_ind` underestimates p for unequal variances:
>>> rvs3 = stats.norm.rvs(loc=5, scale=20, size=500)
>>> stats.ttest_ind(rvs1, rvs3)
(-0.46580283298287162, 0.64145827413436174)
>>> stats.ttest_ind(rvs1, rvs3, equal_var = False)
(-0.46580283298287162, 0.64149646246569292)
When n1 != n2, the equal variance t-statistic is no longer equal to the
unequal variance t-statistic:
>>> rvs4 = stats.norm.rvs(loc=5, scale=20, size=100)
>>> stats.ttest_ind(rvs1, rvs4)
(-0.99882539442782481, 0.3182832709103896)
>>> stats.ttest_ind(rvs1, rvs4, equal_var = False)
(-0.69712570584654099, 0.48716927725402048)
T-test with different means, variance, and n:
>>> rvs5 = stats.norm.rvs(loc=8, scale=20, size=100)
>>> stats.ttest_ind(rvs1, rvs5)
(-1.4679669854490653, 0.14263895620529152)
>>> stats.ttest_ind(rvs1, rvs5, equal_var = False)
(-0.94365973617132992, 0.34744170334794122)
"""
a, b, axis = _chk2_asarray(a, b, axis)
# check both a and b
cna, npa = _contains_nan(a, nan_policy)
cnb, npb = _contains_nan(b, nan_policy)
contains_nan = cna or cnb
if npa == 'omit' or npb == 'omit':
nan_policy = 'omit'
if contains_nan and nan_policy == 'omit':
a = ma.masked_invalid(a)
b = ma.masked_invalid(b)
return mstats_basic.ttest_ind(a, b, axis, equal_var)
if a.size == 0 or b.size == 0:
return Ttest_indResult(np.nan, np.nan)
v1 = np.var(a, axis, ddof=1)
v2 = np.var(b, axis, ddof=1)
n1 = a.shape[axis]
n2 = b.shape[axis]
if equal_var:
df, denom = _equal_var_ttest_denom(v1, n1, v2, n2)
else:
df, denom = _unequal_var_ttest_denom(v1, n1, v2, n2)
res = _ttest_ind_from_stats(np.mean(a, axis), np.mean(b, axis), denom, df)
return Ttest_indResult(*res)
Ttest_relResult = namedtuple('Ttest_relResult', ('statistic', 'pvalue'))
def ttest_rel(a, b, axis=0, nan_policy='propagate'):
"""
Calculate the T-test on TWO RELATED samples of scores, a and b.
This is a two-sided test for the null hypothesis that 2 related or
repeated samples have identical average (expected) values.
Parameters
----------
a, b : array_like
The arrays must have the same shape.
axis : int or None, optional
Axis along which to compute test. If None, compute over the whole
arrays, `a`, and `b`.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
statistic : float or array
t-statistic
pvalue : float or array
two-tailed p-value
Notes
-----
Examples for the use are scores of the same set of student in
different exams, or repeated sampling from the same units. The
test measures whether the average score differs significantly
across samples (e.g. exams). If we observe a large p-value, for
example greater than 0.05 or 0.1 then we cannot reject the null
hypothesis of identical average scores. If the p-value is smaller
than the threshold, e.g. 1%, 5% or 10%, then we reject the null
hypothesis of equal averages. Small p-values are associated with
large t-statistics.
References
----------
https://en.wikipedia.org/wiki/T-test#Dependent_t-test_for_paired_samples
Examples
--------
>>> from scipy import stats
>>> np.random.seed(12345678) # fix random seed to get same numbers
>>> rvs1 = stats.norm.rvs(loc=5,scale=10,size=500)
>>> rvs2 = (stats.norm.rvs(loc=5,scale=10,size=500) +
... stats.norm.rvs(scale=0.2,size=500))
>>> stats.ttest_rel(rvs1,rvs2)
(0.24101764965300962, 0.80964043445811562)
>>> rvs3 = (stats.norm.rvs(loc=8,scale=10,size=500) +
... stats.norm.rvs(scale=0.2,size=500))
>>> stats.ttest_rel(rvs1,rvs3)
(-3.9995108708727933, 7.3082402191726459e-005)
"""
a, b, axis = _chk2_asarray(a, b, axis)
cna, npa = _contains_nan(a, nan_policy)
cnb, npb = _contains_nan(b, nan_policy)
contains_nan = cna or cnb
if npa == 'omit' or npb == 'omit':
nan_policy = 'omit'
if contains_nan and nan_policy == 'omit':
a = ma.masked_invalid(a)
b = ma.masked_invalid(b)
m = ma.mask_or(ma.getmask(a), ma.getmask(b))
aa = ma.array(a, mask=m, copy=True)
bb = ma.array(b, mask=m, copy=True)
return mstats_basic.ttest_rel(aa, bb, axis)
if a.shape[axis] != b.shape[axis]:
raise ValueError('unequal length arrays')
if a.size == 0 or b.size == 0:
return np.nan, np.nan
n = a.shape[axis]
df = float(n - 1)
d = (a - b).astype(np.float64)
v = np.var(d, axis, ddof=1)
dm = np.mean(d, axis)
denom = np.sqrt(v / float(n))
with np.errstate(divide='ignore', invalid='ignore'):
t = np.divide(dm, denom)
t, prob = _ttest_finish(df, t)
return Ttest_relResult(t, prob)
KstestResult = namedtuple('KstestResult', ('statistic', 'pvalue'))
def kstest(rvs, cdf, args=(), N=20, alternative='two-sided', mode='approx'):
"""
Perform the Kolmogorov-Smirnov test for goodness of fit.
This performs a test of the distribution G(x) of an observed
random variable against a given distribution F(x). Under the null
hypothesis the two distributions are identical, G(x)=F(x). The
alternative hypothesis can be either 'two-sided' (default), 'less'
or 'greater'. The KS test is only valid for continuous distributions.
Parameters
----------
rvs : str, array or callable
If a string, it should be the name of a distribution in `scipy.stats`.
If an array, it should be a 1-D array of observations of random
variables.
If a callable, it should be a function to generate random variables;
it is required to have a keyword argument `size`.
cdf : str or callable
If a string, it should be the name of a distribution in `scipy.stats`.
If `rvs` is a string then `cdf` can be False or the same as `rvs`.
If a callable, that callable is used to calculate the cdf.
args : tuple, sequence, optional
Distribution parameters, used if `rvs` or `cdf` are strings.
N : int, optional
Sample size if `rvs` is string or callable. Default is 20.
alternative : {'two-sided', 'less','greater'}, optional
Defines the alternative hypothesis (see explanation above).
Default is 'two-sided'.
mode : 'approx' (default) or 'asymp', optional
Defines the distribution used for calculating the p-value.
- 'approx' : use approximation to exact distribution of test statistic
- 'asymp' : use asymptotic distribution of test statistic
Returns
-------
statistic : float
KS test statistic, either D, D+ or D-.
pvalue : float
One-tailed or two-tailed p-value.
Notes
-----
In the one-sided test, the alternative is that the empirical
cumulative distribution function of the random variable is "less"
or "greater" than the cumulative distribution function F(x) of the
hypothesis, ``G(x)<=F(x)``, resp. ``G(x)>=F(x)``.
Examples
--------
>>> from scipy import stats
>>> x = np.linspace(-15, 15, 9)
>>> stats.kstest(x, 'norm')
(0.44435602715924361, 0.038850142705171065)
>>> np.random.seed(987654321) # set random seed to get the same result
>>> stats.kstest('norm', False, N=100)
(0.058352892479417884, 0.88531190944151261)
The above lines are equivalent to:
>>> np.random.seed(987654321)
>>> stats.kstest(stats.norm.rvs(size=100), 'norm')
(0.058352892479417884, 0.88531190944151261)
*Test against one-sided alternative hypothesis*
Shift distribution to larger values, so that ``cdf_dgp(x) < norm.cdf(x)``:
>>> np.random.seed(987654321)
>>> x = stats.norm.rvs(loc=0.2, size=100)
>>> stats.kstest(x,'norm', alternative = 'less')
(0.12464329735846891, 0.040989164077641749)
Reject equal distribution against alternative hypothesis: less
>>> stats.kstest(x,'norm', alternative = 'greater')
(0.0072115233216311081, 0.98531158590396395)
Don't reject equal distribution against alternative hypothesis: greater
>>> stats.kstest(x,'norm', mode='asymp')
(0.12464329735846891, 0.08944488871182088)
*Testing t distributed random variables against normal distribution*
With 100 degrees of freedom the t distribution looks close to the normal
distribution, and the K-S test does not reject the hypothesis that the
sample came from the normal distribution:
>>> np.random.seed(987654321)
>>> stats.kstest(stats.t.rvs(100,size=100),'norm')
(0.072018929165471257, 0.67630062862479168)
With 3 degrees of freedom the t distribution looks sufficiently different
from the normal distribution, that we can reject the hypothesis that the
sample came from the normal distribution at the 10% level:
>>> np.random.seed(987654321)
>>> stats.kstest(stats.t.rvs(3,size=100),'norm')
(0.131016895759829, 0.058826222555312224)
"""
if isinstance(rvs, string_types):
if (not cdf) or (cdf == rvs):
cdf = getattr(distributions, rvs).cdf
rvs = getattr(distributions, rvs).rvs
else:
raise AttributeError("if rvs is string, cdf has to be the "
"same distribution")
if isinstance(cdf, string_types):
cdf = getattr(distributions, cdf).cdf
if callable(rvs):
kwds = {'size': N}
vals = np.sort(rvs(*args, **kwds))
else:
vals = np.sort(rvs)
N = len(vals)
cdfvals = cdf(vals, *args)
# to not break compatibility with existing code
if alternative == 'two_sided':
alternative = 'two-sided'
if alternative in ['two-sided', 'greater']:
Dplus = (np.arange(1.0, N + 1)/N - cdfvals).max()
if alternative == 'greater':
return KstestResult(Dplus, distributions.ksone.sf(Dplus, N))
if alternative in ['two-sided', 'less']:
Dmin = (cdfvals - np.arange(0.0, N)/N).max()
if alternative == 'less':
return KstestResult(Dmin, distributions.ksone.sf(Dmin, N))
if alternative == 'two-sided':
D = np.max([Dplus, Dmin])
if mode == 'asymp':
return KstestResult(D, distributions.kstwobign.sf(D * np.sqrt(N)))
if mode == 'approx':
pval_two = distributions.kstwobign.sf(D * np.sqrt(N))
if N > 2666 or pval_two > 0.80 - N*0.3/1000:
return KstestResult(D, pval_two)
else:
return KstestResult(D, 2 * distributions.ksone.sf(D, N))
# Map from names to lambda_ values used in power_divergence().
_power_div_lambda_names = {
"pearson": 1,
"log-likelihood": 0,
"freeman-tukey": -0.5,
"mod-log-likelihood": -1,
"neyman": -2,
"cressie-read": 2/3,
}
def _count(a, axis=None):
"""
Count the number of non-masked elements of an array.
This function behaves like np.ma.count(), but is much faster
for ndarrays.
"""
if hasattr(a, 'count'):
num = a.count(axis=axis)
if isinstance(num, np.ndarray) and num.ndim == 0:
# In some cases, the `count` method returns a scalar array (e.g.
# np.array(3)), but we want a plain integer.
num = int(num)
else:
if axis is None:
num = a.size
else:
num = a.shape[axis]
return num
Power_divergenceResult = namedtuple('Power_divergenceResult',
('statistic', 'pvalue'))
def power_divergence(f_obs, f_exp=None, ddof=0, axis=0, lambda_=None):
"""
Cressie-Read power divergence statistic and goodness of fit test.
This function tests the null hypothesis that the categorical data
has the given frequencies, using the Cressie-Read power divergence
statistic.
Parameters
----------
f_obs : array_like
Observed frequencies in each category.
f_exp : array_like, optional
Expected frequencies in each category. By default the categories are
assumed to be equally likely.
ddof : int, optional
"Delta degrees of freedom": adjustment to the degrees of freedom
for the p-value. The p-value is computed using a chi-squared
distribution with ``k - 1 - ddof`` degrees of freedom, where `k`
is the number of observed frequencies. The default value of `ddof`
is 0.
axis : int or None, optional
The axis of the broadcast result of `f_obs` and `f_exp` along which to
apply the test. If axis is None, all values in `f_obs` are treated
as a single data set. Default is 0.
lambda_ : float or str, optional
`lambda_` gives the power in the Cressie-Read power divergence
statistic. The default is 1. For convenience, `lambda_` may be
assigned one of the following strings, in which case the
corresponding numerical value is used::
String Value Description
"pearson" 1 Pearson's chi-squared statistic.
In this case, the function is
equivalent to `stats.chisquare`.
"log-likelihood" 0 Log-likelihood ratio. Also known as
the G-test [3]_.
"freeman-tukey" -1/2 Freeman-Tukey statistic.
"mod-log-likelihood" -1 Modified log-likelihood ratio.
"neyman" -2 Neyman's statistic.
"cressie-read" 2/3 The power recommended in [5]_.
Returns
-------
statistic : float or ndarray
The Cressie-Read power divergence test statistic. The value is
a float if `axis` is None or if` `f_obs` and `f_exp` are 1-D.
pvalue : float or ndarray
The p-value of the test. The value is a float if `ddof` and the
return value `stat` are scalars.
See Also
--------
chisquare
Notes
-----
This test is invalid when the observed or expected frequencies in each
category are too small. A typical rule is that all of the observed
and expected frequencies should be at least 5.
When `lambda_` is less than zero, the formula for the statistic involves
dividing by `f_obs`, so a warning or error may be generated if any value
in `f_obs` is 0.
Similarly, a warning or error may be generated if any value in `f_exp` is
zero when `lambda_` >= 0.
The default degrees of freedom, k-1, are for the case when no parameters
of the distribution are estimated. If p parameters are estimated by
efficient maximum likelihood then the correct degrees of freedom are
k-1-p. If the parameters are estimated in a different way, then the
dof can be between k-1-p and k-1. However, it is also possible that
the asymptotic distribution is not a chisquare, in which case this
test is not appropriate.
This function handles masked arrays. If an element of `f_obs` or `f_exp`
is masked, then data at that position is ignored, and does not count
towards the size of the data set.
.. versionadded:: 0.13.0
References
----------
.. [1] Lowry, Richard. "Concepts and Applications of Inferential
Statistics". Chapter 8. http://faculty.vassar.edu/lowry/ch8pt1.html
.. [2] "Chi-squared test", http://en.wikipedia.org/wiki/Chi-squared_test
.. [3] "G-test", http://en.wikipedia.org/wiki/G-test
.. [4] Sokal, R. R. and Rohlf, F. J. "Biometry: the principles and
practice of statistics in biological research", New York: Freeman
(1981)
.. [5] Cressie, N. and Read, T. R. C., "Multinomial Goodness-of-Fit
Tests", J. Royal Stat. Soc. Series B, Vol. 46, No. 3 (1984),
pp. 440-464.
Examples
--------
(See `chisquare` for more examples.)
When just `f_obs` is given, it is assumed that the expected frequencies
are uniform and given by the mean of the observed frequencies. Here we
perform a G-test (i.e. use the log-likelihood ratio statistic):
>>> from scipy.stats import power_divergence
>>> power_divergence([16, 18, 16, 14, 12, 12], lambda_='log-likelihood')
(2.006573162632538, 0.84823476779463769)
The expected frequencies can be given with the `f_exp` argument:
>>> power_divergence([16, 18, 16, 14, 12, 12],
... f_exp=[16, 16, 16, 16, 16, 8],
... lambda_='log-likelihood')
(3.3281031458963746, 0.6495419288047497)
When `f_obs` is 2-D, by default the test is applied to each column.
>>> obs = np.array([[16, 18, 16, 14, 12, 12], [32, 24, 16, 28, 20, 24]]).T
>>> obs.shape
(6, 2)
>>> power_divergence(obs, lambda_="log-likelihood")
(array([ 2.00657316, 6.77634498]), array([ 0.84823477, 0.23781225]))
By setting ``axis=None``, the test is applied to all data in the array,
which is equivalent to applying the test to the flattened array.
>>> power_divergence(obs, axis=None)
(23.31034482758621, 0.015975692534127565)
>>> power_divergence(obs.ravel())
(23.31034482758621, 0.015975692534127565)
`ddof` is the change to make to the default degrees of freedom.
>>> power_divergence([16, 18, 16, 14, 12, 12], ddof=1)
(2.0, 0.73575888234288467)
The calculation of the p-values is done by broadcasting the
test statistic with `ddof`.
>>> power_divergence([16, 18, 16, 14, 12, 12], ddof=[0,1,2])
(2.0, array([ 0.84914504, 0.73575888, 0.5724067 ]))
`f_obs` and `f_exp` are also broadcast. In the following, `f_obs` has
shape (6,) and `f_exp` has shape (2, 6), so the result of broadcasting
`f_obs` and `f_exp` has shape (2, 6). To compute the desired chi-squared
statistics, we must use ``axis=1``:
>>> power_divergence([16, 18, 16, 14, 12, 12],
... f_exp=[[16, 16, 16, 16, 16, 8],
... [8, 20, 20, 16, 12, 12]],
... axis=1)
(array([ 3.5 , 9.25]), array([ 0.62338763, 0.09949846]))
"""
# Convert the input argument `lambda_` to a numerical value.
if isinstance(lambda_, string_types):
if lambda_ not in _power_div_lambda_names:
names = repr(list(_power_div_lambda_names.keys()))[1:-1]
raise ValueError("invalid string for lambda_: {0!r}. Valid strings "
"are {1}".format(lambda_, names))
lambda_ = _power_div_lambda_names[lambda_]
elif lambda_ is None:
lambda_ = 1
f_obs = np.asanyarray(f_obs)
if f_exp is not None:
f_exp = np.atleast_1d(np.asanyarray(f_exp))
else:
# Compute the equivalent of
# f_exp = f_obs.mean(axis=axis, keepdims=True)
# Older versions of numpy do not have the 'keepdims' argument, so
# we have to do a little work to achieve the same result.
# Ignore 'invalid' errors so the edge case of a data set with length 0
# is handled without spurious warnings.
with np.errstate(invalid='ignore'):
f_exp = np.atleast_1d(f_obs.mean(axis=axis))
if axis is not None:
reduced_shape = list(f_obs.shape)
reduced_shape[axis] = 1
f_exp.shape = reduced_shape
# `terms` is the array of terms that are summed along `axis` to create
# the test statistic. We use some specialized code for a few special
# cases of lambda_.
if lambda_ == 1:
# Pearson's chi-squared statistic
terms = (f_obs - f_exp)**2 / f_exp
elif lambda_ == 0:
# Log-likelihood ratio (i.e. G-test)
terms = 2.0 * special.xlogy(f_obs, f_obs / f_exp)
elif lambda_ == -1:
# Modified log-likelihood ratio
terms = 2.0 * special.xlogy(f_exp, f_exp / f_obs)
else:
# General Cressie-Read power divergence.
terms = f_obs * ((f_obs / f_exp)**lambda_ - 1)
terms /= 0.5 * lambda_ * (lambda_ + 1)
stat = terms.sum(axis=axis)
num_obs = _count(terms, axis=axis)
ddof = asarray(ddof)
p = distributions.chi2.sf(stat, num_obs - 1 - ddof)
return Power_divergenceResult(stat, p)
def chisquare(f_obs, f_exp=None, ddof=0, axis=0):
"""
Calculate a one-way chi square test.
The chi square test tests the null hypothesis that the categorical data
has the given frequencies.
Parameters
----------
f_obs : array_like
Observed frequencies in each category.
f_exp : array_like, optional
Expected frequencies in each category. By default the categories are
assumed to be equally likely.
ddof : int, optional
"Delta degrees of freedom": adjustment to the degrees of freedom
for the p-value. The p-value is computed using a chi-squared
distribution with ``k - 1 - ddof`` degrees of freedom, where `k`
is the number of observed frequencies. The default value of `ddof`
is 0.
axis : int or None, optional
The axis of the broadcast result of `f_obs` and `f_exp` along which to
apply the test. If axis is None, all values in `f_obs` are treated
as a single data set. Default is 0.
Returns
-------
chisq : float or ndarray
The chi-squared test statistic. The value is a float if `axis` is
None or `f_obs` and `f_exp` are 1-D.
p : float or ndarray
The p-value of the test. The value is a float if `ddof` and the
return value `chisq` are scalars.
See Also
--------
power_divergence
mstats.chisquare
Notes
-----
This test is invalid when the observed or expected frequencies in each
category are too small. A typical rule is that all of the observed
and expected frequencies should be at least 5.
The default degrees of freedom, k-1, are for the case when no parameters
of the distribution are estimated. If p parameters are estimated by
efficient maximum likelihood then the correct degrees of freedom are
k-1-p. If the parameters are estimated in a different way, then the
dof can be between k-1-p and k-1. However, it is also possible that
the asymptotic distribution is not a chisquare, in which case this
test is not appropriate.
References
----------
.. [1] Lowry, Richard. "Concepts and Applications of Inferential
Statistics". Chapter 8. http://faculty.vassar.edu/lowry/ch8pt1.html
.. [2] "Chi-squared test", http://en.wikipedia.org/wiki/Chi-squared_test
Examples
--------
When just `f_obs` is given, it is assumed that the expected frequencies
are uniform and given by the mean of the observed frequencies.
>>> from scipy.stats import chisquare
>>> chisquare([16, 18, 16, 14, 12, 12])
(2.0, 0.84914503608460956)
With `f_exp` the expected frequencies can be given.
>>> chisquare([16, 18, 16, 14, 12, 12], f_exp=[16, 16, 16, 16, 16, 8])
(3.5, 0.62338762774958223)
When `f_obs` is 2-D, by default the test is applied to each column.
>>> obs = np.array([[16, 18, 16, 14, 12, 12], [32, 24, 16, 28, 20, 24]]).T
>>> obs.shape
(6, 2)
>>> chisquare(obs)
(array([ 2. , 6.66666667]), array([ 0.84914504, 0.24663415]))
By setting ``axis=None``, the test is applied to all data in the array,
which is equivalent to applying the test to the flattened array.
>>> chisquare(obs, axis=None)
(23.31034482758621, 0.015975692534127565)
>>> chisquare(obs.ravel())
(23.31034482758621, 0.015975692534127565)
`ddof` is the change to make to the default degrees of freedom.
>>> chisquare([16, 18, 16, 14, 12, 12], ddof=1)
(2.0, 0.73575888234288467)
The calculation of the p-values is done by broadcasting the
chi-squared statistic with `ddof`.
>>> chisquare([16, 18, 16, 14, 12, 12], ddof=[0,1,2])
(2.0, array([ 0.84914504, 0.73575888, 0.5724067 ]))
`f_obs` and `f_exp` are also broadcast. In the following, `f_obs` has
shape (6,) and `f_exp` has shape (2, 6), so the result of broadcasting
`f_obs` and `f_exp` has shape (2, 6). To compute the desired chi-squared
statistics, we use ``axis=1``:
>>> chisquare([16, 18, 16, 14, 12, 12],
... f_exp=[[16, 16, 16, 16, 16, 8], [8, 20, 20, 16, 12, 12]],
... axis=1)
(array([ 3.5 , 9.25]), array([ 0.62338763, 0.09949846]))
"""
return power_divergence(f_obs, f_exp=f_exp, ddof=ddof, axis=axis,
lambda_="pearson")
Ks_2sampResult = namedtuple('Ks_2sampResult', ('statistic', 'pvalue'))
def ks_2samp(data1, data2):
"""
Compute the Kolmogorov-Smirnov statistic on 2 samples.
This is a two-sided test for the null hypothesis that 2 independent samples
are drawn from the same continuous distribution.
Parameters
----------
data1, data2 : sequence of 1-D ndarrays
two arrays of sample observations assumed to be drawn from a continuous
distribution, sample sizes can be different
Returns
-------
statistic : float
KS statistic
pvalue : float
two-tailed p-value
Notes
-----
This tests whether 2 samples are drawn from the same distribution. Note
that, like in the case of the one-sample K-S test, the distribution is
assumed to be continuous.
This is the two-sided test, one-sided tests are not implemented.
The test uses the two-sided asymptotic Kolmogorov-Smirnov distribution.
If the K-S statistic is small or the p-value is high, then we cannot
reject the hypothesis that the distributions of the two samples
are the same.
Examples
--------
>>> from scipy import stats
>>> np.random.seed(12345678) #fix random seed to get the same result
>>> n1 = 200 # size of first sample
>>> n2 = 300 # size of second sample
For a different distribution, we can reject the null hypothesis since the
pvalue is below 1%:
>>> rvs1 = stats.norm.rvs(size=n1, loc=0., scale=1)
>>> rvs2 = stats.norm.rvs(size=n2, loc=0.5, scale=1.5)
>>> stats.ks_2samp(rvs1, rvs2)
(0.20833333333333337, 4.6674975515806989e-005)
For a slightly different distribution, we cannot reject the null hypothesis
at a 10% or lower alpha since the p-value at 0.144 is higher than 10%
>>> rvs3 = stats.norm.rvs(size=n2, loc=0.01, scale=1.0)
>>> stats.ks_2samp(rvs1, rvs3)
(0.10333333333333333, 0.14498781825751686)
For an identical distribution, we cannot reject the null hypothesis since
the p-value is high, 41%:
>>> rvs4 = stats.norm.rvs(size=n2, loc=0.0, scale=1.0)
>>> stats.ks_2samp(rvs1, rvs4)
(0.07999999999999996, 0.41126949729859719)
"""
data1 = np.sort(data1)
data2 = np.sort(data2)
n1 = data1.shape[0]
n2 = data2.shape[0]
data_all = np.concatenate([data1, data2])
cdf1 = np.searchsorted(data1, data_all, side='right') / (1.0*n1)
cdf2 = np.searchsorted(data2, data_all, side='right') / (1.0*n2)
d = np.max(np.absolute(cdf1 - cdf2))
# Note: d absolute not signed distance
en = np.sqrt(n1 * n2 / float(n1 + n2))
try:
prob = distributions.kstwobign.sf((en + 0.12 + 0.11 / en) * d)
except:
prob = 1.0
return Ks_2sampResult(d, prob)
def tiecorrect(rankvals):
"""
Tie correction factor for ties in the Mann-Whitney U and
Kruskal-Wallis H tests.
Parameters
----------
rankvals : array_like
A 1-D sequence of ranks. Typically this will be the array
returned by `stats.rankdata`.
Returns
-------
factor : float
Correction factor for U or H.
See Also
--------
rankdata : Assign ranks to the data
mannwhitneyu : Mann-Whitney rank test
kruskal : Kruskal-Wallis H test
References
----------
.. [1] Siegel, S. (1956) Nonparametric Statistics for the Behavioral
Sciences. New York: McGraw-Hill.
Examples
--------
>>> from scipy.stats import tiecorrect, rankdata
>>> tiecorrect([1, 2.5, 2.5, 4])
0.9
>>> ranks = rankdata([1, 3, 2, 4, 5, 7, 2, 8, 4])
>>> ranks
array([ 1. , 4. , 2.5, 5.5, 7. , 8. , 2.5, 9. , 5.5])
>>> tiecorrect(ranks)
0.9833333333333333
"""
arr = np.sort(rankvals)
idx = np.nonzero(np.r_[True, arr[1:] != arr[:-1], True])[0]
cnt = np.diff(idx).astype(np.float64)
size = np.float64(arr.size)
return 1.0 if size < 2 else 1.0 - (cnt**3 - cnt).sum() / (size**3 - size)
MannwhitneyuResult = namedtuple('MannwhitneyuResult', ('statistic', 'pvalue'))
def mannwhitneyu(x, y, use_continuity=True, alternative=None):
"""
Compute the Mann-Whitney rank test on samples x and y.
Parameters
----------
x, y : array_like
Array of samples, should be one-dimensional.
use_continuity : bool, optional
Whether a continuity correction (1/2.) should be taken into
account. Default is True.
alternative : None (deprecated), 'less', 'two-sided', or 'greater'
Whether to get the p-value for the one-sided hypothesis ('less'
or 'greater') or for the two-sided hypothesis ('two-sided').
Defaults to None, which results in a p-value half the size of
the 'two-sided' p-value and a different U statistic. The
default behavior is not the same as using 'less' or 'greater':
it only exists for backward compatibility and is deprecated.
Returns
-------
statistic : float
The Mann-Whitney U statistic, equal to min(U for x, U for y) if
`alternative` is equal to None (deprecated; exists for backward
compatibility), and U for y otherwise.
pvalue : float
p-value assuming an asymptotic normal distribution. One-sided or
two-sided, depending on the choice of `alternative`.
Notes
-----
Use only when the number of observation in each sample is > 20 and
you have 2 independent samples of ranks. Mann-Whitney U is
significant if the u-obtained is LESS THAN or equal to the critical
value of U.
This test corrects for ties and by default uses a continuity correction.
References
----------
.. [1] https://en.wikipedia.org/wiki/Mann-Whitney_U_test
.. [2] H.B. Mann and D.R. Whitney, "On a Test of Whether one of Two Random
Variables is Stochastically Larger than the Other," The Annals of
Mathematical Statistics, vol. 18, no. 1, pp. 50-60, 1947.
"""
if alternative is None:
warnings.warn("Calling `mannwhitneyu` without specifying "
"`alternative` is deprecated.", DeprecationWarning)
x = np.asarray(x)
y = np.asarray(y)
n1 = len(x)
n2 = len(y)
ranked = rankdata(np.concatenate((x, y)))
rankx = ranked[0:n1] # get the x-ranks
u1 = n1*n2 + (n1*(n1+1))/2.0 - np.sum(rankx, axis=0) # calc U for x
u2 = n1*n2 - u1 # remainder is U for y
T = tiecorrect(ranked)
if T == 0:
raise ValueError('All numbers are identical in mannwhitneyu')
sd = np.sqrt(T * n1 * n2 * (n1+n2+1) / 12.0)
meanrank = n1*n2/2.0 + 0.5 * use_continuity
if alternative is None or alternative == 'two-sided':
bigu = max(u1, u2)
elif alternative == 'less':
bigu = u1
elif alternative == 'greater':
bigu = u2
else:
raise ValueError("alternative should be None, 'less', 'greater' "
"or 'two-sided'")
z = (bigu - meanrank) / sd
if alternative is None:
# This behavior, equal to half the size of the two-sided
# p-value, is deprecated.
p = distributions.norm.sf(abs(z))
elif alternative == 'two-sided':
p = 2 * distributions.norm.sf(abs(z))
else:
p = distributions.norm.sf(z)
u = u2
# This behavior is deprecated.
if alternative is None:
u = min(u1, u2)
return MannwhitneyuResult(u, p)
RanksumsResult = namedtuple('RanksumsResult', ('statistic', 'pvalue'))
def ranksums(x, y):
"""
Compute the Wilcoxon rank-sum statistic for two samples.
The Wilcoxon rank-sum test tests the null hypothesis that two sets
of measurements are drawn from the same distribution. The alternative
hypothesis is that values in one sample are more likely to be
larger than the values in the other sample.
This test should be used to compare two samples from continuous
distributions. It does not handle ties between measurements
in x and y. For tie-handling and an optional continuity correction
see `scipy.stats.mannwhitneyu`.
Parameters
----------
x,y : array_like
The data from the two samples
Returns
-------
statistic : float
The test statistic under the large-sample approximation that the
rank sum statistic is normally distributed
pvalue : float
The two-sided p-value of the test
References
----------
.. [1] http://en.wikipedia.org/wiki/Wilcoxon_rank-sum_test
"""
x, y = map(np.asarray, (x, y))
n1 = len(x)
n2 = len(y)
alldata = np.concatenate((x, y))
ranked = rankdata(alldata)
x = ranked[:n1]
s = np.sum(x, axis=0)
expected = n1 * (n1+n2+1) / 2.0
z = (s - expected) / np.sqrt(n1*n2*(n1+n2+1)/12.0)
prob = 2 * distributions.norm.sf(abs(z))
return RanksumsResult(z, prob)
KruskalResult = namedtuple('KruskalResult', ('statistic', 'pvalue'))
def kruskal(*args, **kwargs):
"""
Compute the Kruskal-Wallis H-test for independent samples
The Kruskal-Wallis H-test tests the null hypothesis that the population
median of all of the groups are equal. It is a non-parametric version of
ANOVA. The test works on 2 or more independent samples, which may have
different sizes. Note that rejecting the null hypothesis does not
indicate which of the groups differs. Post-hoc comparisons between
groups are required to determine which groups are different.
Parameters
----------
sample1, sample2, ... : array_like
Two or more arrays with the sample measurements can be given as
arguments.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
statistic : float
The Kruskal-Wallis H statistic, corrected for ties
pvalue : float
The p-value for the test using the assumption that H has a chi
square distribution
See Also
--------
f_oneway : 1-way ANOVA
mannwhitneyu : Mann-Whitney rank test on two samples.
friedmanchisquare : Friedman test for repeated measurements
Notes
-----
Due to the assumption that H has a chi square distribution, the number
of samples in each group must not be too small. A typical rule is
that each sample must have at least 5 measurements.
References
----------
.. [1] W. H. Kruskal & W. W. Wallis, "Use of Ranks in
One-Criterion Variance Analysis", Journal of the American Statistical
Association, Vol. 47, Issue 260, pp. 583-621, 1952.
.. [2] http://en.wikipedia.org/wiki/Kruskal-Wallis_one-way_analysis_of_variance
Examples
--------
>>> from scipy import stats
>>> x = [1, 3, 5, 7, 9]
>>> y = [2, 4, 6, 8, 10]
>>> stats.kruskal(x, y)
KruskalResult(statistic=0.2727272727272734, pvalue=0.6015081344405895)
>>> x = [1, 1, 1]
>>> y = [2, 2, 2]
>>> z = [2, 2]
>>> stats.kruskal(x, y, z)
KruskalResult(statistic=7.0, pvalue=0.0301973834223185)
"""
args = list(map(np.asarray, args))
num_groups = len(args)
if num_groups < 2:
raise ValueError("Need at least two groups in stats.kruskal()")
for arg in args:
if arg.size == 0:
return KruskalResult(np.nan, np.nan)
n = np.asarray(list(map(len, args)))
if 'nan_policy' in kwargs.keys():
if kwargs['nan_policy'] not in ('propagate', 'raise', 'omit'):
raise ValueError("nan_policy must be 'propagate', "
"'raise' or'omit'")
else:
nan_policy = kwargs['nan_policy']
else:
nan_policy = 'propagate'
contains_nan = False
for arg in args:
cn = _contains_nan(arg, nan_policy)
if cn[0]:
contains_nan = True
break
if contains_nan and nan_policy == 'omit':
for a in args:
a = ma.masked_invalid(a)
return mstats_basic.kruskal(*args)
if contains_nan and nan_policy == 'propagate':
return KruskalResult(np.nan, np.nan)
alldata = np.concatenate(args)
ranked = rankdata(alldata)
ties = tiecorrect(ranked)
if ties == 0:
raise ValueError('All numbers are identical in kruskal')
# Compute sum^2/n for each group and sum
j = np.insert(np.cumsum(n), 0, 0)
ssbn = 0
for i in range(num_groups):
ssbn += _square_of_sums(ranked[j[i]:j[i+1]]) / float(n[i])
totaln = np.sum(n)
h = 12.0 / (totaln * (totaln + 1)) * ssbn - 3 * (totaln + 1)
df = num_groups - 1
h /= ties
return KruskalResult(h, distributions.chi2.sf(h, df))
FriedmanchisquareResult = namedtuple('FriedmanchisquareResult',
('statistic', 'pvalue'))
def friedmanchisquare(*args):
"""
Compute the Friedman test for repeated measurements
The Friedman test tests the null hypothesis that repeated measurements of
the same individuals have the same distribution. It is often used
to test for consistency among measurements obtained in different ways.
For example, if two measurement techniques are used on the same set of
individuals, the Friedman test can be used to determine if the two
measurement techniques are consistent.
Parameters
----------
measurements1, measurements2, measurements3... : array_like
Arrays of measurements. All of the arrays must have the same number
of elements. At least 3 sets of measurements must be given.
Returns
-------
statistic : float
the test statistic, correcting for ties
pvalue : float
the associated p-value assuming that the test statistic has a chi
squared distribution
Notes
-----
Due to the assumption that the test statistic has a chi squared
distribution, the p-value is only reliable for n > 10 and more than
6 repeated measurements.
References
----------
.. [1] http://en.wikipedia.org/wiki/Friedman_test
"""
k = len(args)
if k < 3:
raise ValueError('Less than 3 levels. Friedman test not appropriate.')
n = len(args[0])
for i in range(1, k):
if len(args[i]) != n:
raise ValueError('Unequal N in friedmanchisquare. Aborting.')
# Rank data
data = np.vstack(args).T
data = data.astype(float)
for i in range(len(data)):
data[i] = rankdata(data[i])
# Handle ties
ties = 0
for i in range(len(data)):
replist, repnum = find_repeats(array(data[i]))
for t in repnum:
ties += t * (t*t - 1)
c = 1 - ties / float(k*(k*k - 1)*n)
ssbn = np.sum(data.sum(axis=0)**2)
chisq = (12.0 / (k*n*(k+1)) * ssbn - 3*n*(k+1)) / c
return FriedmanchisquareResult(chisq, distributions.chi2.sf(chisq, k - 1))
def combine_pvalues(pvalues, method='fisher', weights=None):
"""
Methods for combining the p-values of independent tests bearing upon the
same hypothesis.
Parameters
----------
pvalues : array_like, 1-D
Array of p-values assumed to come from independent tests.
method : {'fisher', 'stouffer'}, optional
Name of method to use to combine p-values. The following methods are
available:
- "fisher": Fisher's method (Fisher's combined probability test),
the default.
- "stouffer": Stouffer's Z-score method.
weights : array_like, 1-D, optional
Optional array of weights used only for Stouffer's Z-score method.
Returns
-------
statistic: float
The statistic calculated by the specified method:
- "fisher": The chi-squared statistic
- "stouffer": The Z-score
pval: float
The combined p-value.
Notes
-----
Fisher's method (also known as Fisher's combined probability test) [1]_ uses
a chi-squared statistic to compute a combined p-value. The closely related
Stouffer's Z-score method [2]_ uses Z-scores rather than p-values. The
advantage of Stouffer's method is that it is straightforward to introduce
weights, which can make Stouffer's method more powerful than Fisher's
method when the p-values are from studies of different size [3]_ [4]_.
Fisher's method may be extended to combine p-values from dependent tests
[5]_. Extensions such as Brown's method and Kost's method are not currently
implemented.
.. versionadded:: 0.15.0
References
----------
.. [1] https://en.wikipedia.org/wiki/Fisher%27s_method
.. [2] http://en.wikipedia.org/wiki/Fisher's_method#Relation_to_Stouffer.27s_Z-score_method
.. [3] Whitlock, M. C. "Combining probability from independent tests: the
weighted Z-method is superior to Fisher's approach." Journal of
Evolutionary Biology 18, no. 5 (2005): 1368-1373.
.. [4] Zaykin, Dmitri V. "Optimally weighted Z-test is a powerful method
for combining probabilities in meta-analysis." Journal of
Evolutionary Biology 24, no. 8 (2011): 1836-1841.
.. [5] https://en.wikipedia.org/wiki/Extensions_of_Fisher%27s_method
"""
pvalues = np.asarray(pvalues)
if pvalues.ndim != 1:
raise ValueError("pvalues is not 1-D")
if method == 'fisher':
Xsq = -2 * np.sum(np.log(pvalues))
pval = distributions.chi2.sf(Xsq, 2 * len(pvalues))
return (Xsq, pval)
elif method == 'stouffer':
if weights is None:
weights = np.ones_like(pvalues)
elif len(weights) != len(pvalues):
raise ValueError("pvalues and weights must be of the same size.")
weights = np.asarray(weights)
if weights.ndim != 1:
raise ValueError("weights is not 1-D")
Zi = distributions.norm.isf(pvalues)
Z = np.dot(weights, Zi) / np.linalg.norm(weights)
pval = distributions.norm.sf(Z)
return (Z, pval)
else:
raise ValueError(
"Invalid method '%s'. Options are 'fisher' or 'stouffer'", method)
#####################################
# PROBABILITY CALCULATIONS #
#####################################
def _betai(a, b, x):
x = np.asarray(x)
x = np.where(x < 1.0, x, 1.0) # if x > 1 then return 1.0
return special.betainc(a, b, x)
#####################################
# STATISTICAL DISTANCES #
#####################################
def wasserstein_distance(u_values, v_values, u_weights=None, v_weights=None):
r"""
Compute the first Wasserstein distance between two 1D distributions.
This distance is also known as the earth mover's distance, since it can be
seen as the minimum amount of "work" required to transform :math:`u` into
:math:`v`, where "work" is measured as the amount of distribution weight
that must be moved, multiplied by the distance it has to be moved.
.. versionadded:: 1.0.0
Parameters
----------
u_values, v_values : array_like
Values observed in the (empirical) distribution.
u_weights, v_weights : array_like, optional
Weight for each value. If unspecified, each value is assigned the same
weight.
`u_weights` (resp. `v_weights`) must have the same length as
`u_values` (resp. `v_values`). If the weight sum differs from 1, it
must still be positive and finite so that the weights can be normalized
to sum to 1.
Returns
-------
distance : float
The computed distance between the distributions.
Notes
-----
The first Wasserstein distance between the distributions :math:`u` and
:math:`v` is:
.. math::
l_1 (u, v) = \inf_{\pi \in \Gamma (u, v)} \int_{\mathbb{R} \times
\mathbb{R}} |x-y| \mathrm{d} \pi (x, y)
where :math:`\Gamma (u, v)` is the set of (probability) distributions on
:math:`\mathbb{R} \times \mathbb{R}` whose marginals are :math:`u` and
:math:`v` on the first and second factors respectively.
If :math:`U` and :math:`V` are the respective CDFs of :math:`u` and
:math:`v`, this distance also equals to:
.. math::
l_1(u, v) = \int_{-\infty}^{+\infty} |U-V|
See [2]_ for a proof of the equivalence of both definitions.
The input distributions can be empirical, therefore coming from samples
whose values are effectively inputs of the function, or they can be seen as
generalized functions, in which case they are weighted sums of Dirac delta
functions located at the specified values.
References
----------
.. [1] "Wasserstein metric", http://en.wikipedia.org/wiki/Wasserstein_metric
.. [2] Ramdas, Garcia, Cuturi "On Wasserstein Two Sample Testing and Related
Families of Nonparametric Tests" (2015). :arXiv:`1509.02237`.
Examples
--------
>>> from scipy.stats import wasserstein_distance
>>> wasserstein_distance([0, 1, 3], [5, 6, 8])
5.0
>>> wasserstein_distance([0, 1], [0, 1], [3, 1], [2, 2])
0.25
>>> wasserstein_distance([3.4, 3.9, 7.5, 7.8], [4.5, 1.4],
... [1.4, 0.9, 3.1, 7.2], [3.2, 3.5])
4.0781331438047861
"""
return _cdf_distance(1, u_values, v_values, u_weights, v_weights)
def energy_distance(u_values, v_values, u_weights=None, v_weights=None):
r"""
Compute the energy distance between two 1D distributions.
.. versionadded:: 1.0.0
Parameters
----------
u_values, v_values : array_like
Values observed in the (empirical) distribution.
u_weights, v_weights : array_like, optional
Weight for each value. If unspecified, each value is assigned the same
weight.
`u_weights` (resp. `v_weights`) must have the same length as
`u_values` (resp. `v_values`). If the weight sum differs from 1, it
must still be positive and finite so that the weights can be normalized
to sum to 1.
Returns
-------
distance : float
The computed distance between the distributions.
Notes
-----
The energy distance between two distributions :math:`u` and :math:`v`, whose
respective CDFs are :math:`U` and :math:`V`, equals to:
.. math::
D(u, v) = \left( 2\mathbb E|X - Y| - \mathbb E|X - X'| -
\mathbb E|Y - Y'| \right)^{1/2}
where :math:`X` and :math:`X'` (resp. :math:`Y` and :math:`Y'`) are
independent random variables whose probability distribution is :math:`u`
(resp. :math:`v`).
As shown in [2]_, for one-dimensional real-valued variables, the energy
distance is linked to the non-distribution-free version of the Cramer-von
Mises distance:
.. math::
D(u, v) = \sqrt{2} l_2(u, v) = \left( 2 \int_{-\infty}^{+\infty} (U-V)^2
\right)^{1/2}
Note that the common Cramer-von Mises criterion uses the distribution-free
version of the distance. See [2]_ (section 2), for more details about both
versions of the distance.
The input distributions can be empirical, therefore coming from samples
whose values are effectively inputs of the function, or they can be seen as
generalized functions, in which case they are weighted sums of Dirac delta
functions located at the specified values.
References
----------
.. [1] "Energy distance", https://en.wikipedia.org/wiki/Energy_distance
.. [2] Szekely "E-statistics: The energy of statistical samples." Bowling
Green State University, Department of Mathematics and Statistics,
Technical Report 02-16 (2002).
.. [3] Rizzo, Szekely "Energy distance." Wiley Interdisciplinary Reviews:
Computational Statistics, 8(1):27-38 (2015).
.. [4] Bellemare, Danihelka, Dabney, Mohamed, Lakshminarayanan, Hoyer,
Munos "The Cramer Distance as a Solution to Biased Wasserstein
Gradients" (2017). :arXiv:`1705.10743`.
Examples
--------
>>> from scipy.stats import energy_distance
>>> energy_distance([0], [2])
2.0000000000000004
>>> energy_distance([0, 8], [0, 8], [3, 1], [2, 2])
1.0000000000000002
>>> energy_distance([0.7, 7.4, 2.4, 6.8], [1.4, 8. ],
... [2.1, 4.2, 7.4, 8. ], [7.6, 8.8])
0.88003340976158217
"""
return np.sqrt(2) * _cdf_distance(2, u_values, v_values,
u_weights, v_weights)
def _cdf_distance(p, u_values, v_values, u_weights=None, v_weights=None):
r"""
Compute, between two one-dimensional distributions :math:`u` and
:math:`v`, whose respective CDFs are :math:`U` and :math:`V`, the
statistical distance that is defined as:
.. math::
l_p(u, v) = \left( \int_{-\infty}^{+\infty} |U-V|^p \right)^{1/p}
p is a positive parameter; p = 1 gives the Wasserstein distance, p = 2
gives the energy distance.
Parameters
----------
u_values, v_values : array_like
Values observed in the (empirical) distribution.
u_weights, v_weights : array_like, optional
Weight for each value. If unspecified, each value is assigned the same
weight.
`u_weights` (resp. `v_weights`) must have the same length as
`u_values` (resp. `v_values`). If the weight sum differs from 1, it
must still be positive and finite so that the weights can be normalized
to sum to 1.
Returns
-------
distance : float
The computed distance between the distributions.
Notes
-----
The input distributions can be empirical, therefore coming from samples
whose values are effectively inputs of the function, or they can be seen as
generalized functions, in which case they are weighted sums of Dirac delta
functions located at the specified values.
References
----------
.. [1] Bellemare, Danihelka, Dabney, Mohamed, Lakshminarayanan, Hoyer,
Munos "The Cramer Distance as a Solution to Biased Wasserstein
Gradients" (2017). :arXiv:`1705.10743`.
"""
u_values, u_weights = _validate_distribution(u_values, u_weights)
v_values, v_weights = _validate_distribution(v_values, v_weights)
u_sorter = np.argsort(u_values)
v_sorter = np.argsort(v_values)
all_values = np.concatenate((u_values, v_values))
all_values.sort(kind='mergesort')
# Compute the differences between pairs of successive values of u and v.
deltas = np.diff(all_values)
# Get the respective positions of the values of u and v among the values of
# both distributions.
u_cdf_indices = u_values[u_sorter].searchsorted(all_values[:-1], 'right')
v_cdf_indices = v_values[v_sorter].searchsorted(all_values[:-1], 'right')
# Calculate the CDFs of u and v using their weights, if specified.
if u_weights is None:
u_cdf = u_cdf_indices / u_values.size
else:
u_sorted_cumweights = np.concatenate(([0],
np.cumsum(u_weights[u_sorter])))
u_cdf = u_sorted_cumweights[u_cdf_indices] / u_sorted_cumweights[-1]
if v_weights is None:
v_cdf = v_cdf_indices / v_values.size
else:
v_sorted_cumweights = np.concatenate(([0],
np.cumsum(v_weights[v_sorter])))
v_cdf = v_sorted_cumweights[v_cdf_indices] / v_sorted_cumweights[-1]
# Compute the value of the integral based on the CDFs.
# If p = 1 or p = 2, we avoid using np.power, which introduces an overhead
# of about 15%.
if p == 1:
return np.sum(np.multiply(np.abs(u_cdf - v_cdf), deltas))
if p == 2:
return np.sqrt(np.sum(np.multiply(np.square(u_cdf - v_cdf), deltas)))
return np.power(np.sum(np.multiply(np.power(np.abs(u_cdf - v_cdf), p),
deltas)), 1/p)
def _validate_distribution(values, weights):
"""
Validate the values and weights from a distribution input of `cdf_distance`
and return them as ndarray objects.
Parameters
----------
values : array_like
Values observed in the (empirical) distribution.
weights : array_like
Weight for each value.
Returns
-------
values : ndarray
Values as ndarray.
weights : ndarray
Weights as ndarray.
"""
# Validate the value array.
values = np.asarray(values, dtype=float)
if len(values) == 0:
raise ValueError("Distribution can't be empty.")
# Validate the weight array, if specified.
if weights is not None:
weights = np.asarray(weights, dtype=float)
if len(weights) != len(values):
raise ValueError('Value and weight array-likes for the same '
'empirical distribution must be of the same size.')
if np.any(weights < 0):
raise ValueError('All weights must be non-negative.')
if not 0 < np.sum(weights) < np.inf:
raise ValueError('Weight array-like sum must be positive and '
'finite. Set as None for an equal distribution of '
'weight.')
return values, weights
return values, None
#####################################
# SUPPORT FUNCTIONS #
#####################################
RepeatedResults = namedtuple('RepeatedResults', ('values', 'counts'))
def find_repeats(arr):
"""
Find repeats and repeat counts.
Parameters
----------
arr : array_like
Input array. This is cast to float64.
Returns
-------
values : ndarray
The unique values from the (flattened) input that are repeated.
counts : ndarray
Number of times the corresponding 'value' is repeated.
Notes
-----
In numpy >= 1.9 `numpy.unique` provides similar functionality. The main
difference is that `find_repeats` only returns repeated values.
Examples
--------
>>> from scipy import stats
>>> stats.find_repeats([2, 1, 2, 3, 2, 2, 5])
RepeatedResults(values=array([2.]), counts=array([4]))
>>> stats.find_repeats([[10, 20, 1, 2], [5, 5, 4, 4]])
RepeatedResults(values=array([4., 5.]), counts=array([2, 2]))
"""
# Note: always copies.
return RepeatedResults(*_find_repeats(np.array(arr, dtype=np.float64)))
def _sum_of_squares(a, axis=0):
"""
Square each element of the input array, and return the sum(s) of that.
Parameters
----------
a : array_like
Input array.
axis : int or None, optional
Axis along which to calculate. Default is 0. If None, compute over
the whole array `a`.
Returns
-------
sum_of_squares : ndarray
The sum along the given axis for (a**2).
See also
--------
_square_of_sums : The square(s) of the sum(s) (the opposite of
`_sum_of_squares`).
"""
a, axis = _chk_asarray(a, axis)
return np.sum(a*a, axis)
def _square_of_sums(a, axis=0):
"""
Sum elements of the input array, and return the square(s) of that sum.
Parameters
----------
a : array_like
Input array.
axis : int or None, optional
Axis along which to calculate. Default is 0. If None, compute over
the whole array `a`.
Returns
-------
square_of_sums : float or ndarray
The square of the sum over `axis`.
See also
--------
_sum_of_squares : The sum of squares (the opposite of `square_of_sums`).
"""
a, axis = _chk_asarray(a, axis)
s = np.sum(a, axis)
if not np.isscalar(s):
return s.astype(float) * s
else:
return float(s) * s
def rankdata(a, method='average'):
"""
Assign ranks to data, dealing with ties appropriately.
Ranks begin at 1. The `method` argument controls how ranks are assigned
to equal values. See [1]_ for further discussion of ranking methods.
Parameters
----------
a : array_like
The array of values to be ranked. The array is first flattened.
method : str, optional
The method used to assign ranks to tied elements.
The options are 'average', 'min', 'max', 'dense' and 'ordinal'.
'average':
The average of the ranks that would have been assigned to
all the tied values is assigned to each value.
'min':
The minimum of the ranks that would have been assigned to all
the tied values is assigned to each value. (This is also
referred to as "competition" ranking.)
'max':
The maximum of the ranks that would have been assigned to all
the tied values is assigned to each value.
'dense':
Like 'min', but the rank of the next highest element is assigned
the rank immediately after those assigned to the tied elements.
'ordinal':
All values are given a distinct rank, corresponding to the order
that the values occur in `a`.
The default is 'average'.
Returns
-------
ranks : ndarray
An array of length equal to the size of `a`, containing rank
scores.
References
----------
.. [1] "Ranking", http://en.wikipedia.org/wiki/Ranking
Examples
--------
>>> from scipy.stats import rankdata
>>> rankdata([0, 2, 3, 2])
array([ 1. , 2.5, 4. , 2.5])
>>> rankdata([0, 2, 3, 2], method='min')
array([ 1, 2, 4, 2])
>>> rankdata([0, 2, 3, 2], method='max')
array([ 1, 3, 4, 3])
>>> rankdata([0, 2, 3, 2], method='dense')
array([ 1, 2, 3, 2])
>>> rankdata([0, 2, 3, 2], method='ordinal')
array([ 1, 2, 4, 3])
"""
if method not in ('average', 'min', 'max', 'dense', 'ordinal'):
raise ValueError('unknown method "{0}"'.format(method))
arr = np.ravel(np.asarray(a))
algo = 'mergesort' if method == 'ordinal' else 'quicksort'
sorter = np.argsort(arr, kind=algo)
inv = np.empty(sorter.size, dtype=np.intp)
inv[sorter] = np.arange(sorter.size, dtype=np.intp)
if method == 'ordinal':
return inv + 1
arr = arr[sorter]
obs = np.r_[True, arr[1:] != arr[:-1]]
dense = obs.cumsum()[inv]
if method == 'dense':
return dense
# cumulative counts of each unique value
count = np.r_[np.nonzero(obs)[0], len(obs)]
if method == 'max':
return count[dense]
if method == 'min':
return count[dense - 1] + 1
# average method
return .5 * (count[dense] + count[dense - 1] + 1)
| 194,013 | 32.818023 | 97 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/_stats_mstats_common.py
|
from collections import namedtuple
import numpy as np
from . import distributions
__all__ = ['_find_repeats', 'linregress', 'theilslopes']
LinregressResult = namedtuple('LinregressResult', ('slope', 'intercept',
'rvalue', 'pvalue',
'stderr'))
def linregress(x, y=None):
"""
Calculate a linear least-squares regression for two sets of measurements.
Parameters
----------
x, y : array_like
Two sets of measurements. Both arrays should have the same length.
If only x is given (and y=None), then it must be a two-dimensional
array where one dimension has length 2. The two sets of measurements
are then found by splitting the array along the length-2 dimension.
Returns
-------
slope : float
slope of the regression line
intercept : float
intercept of the regression line
rvalue : float
correlation coefficient
pvalue : float
two-sided p-value for a hypothesis test whose null hypothesis is
that the slope is zero, using Wald Test with t-distribution of
the test statistic.
stderr : float
Standard error of the estimated gradient.
See also
--------
:func:`scipy.optimize.curve_fit` : Use non-linear
least squares to fit a function to data.
:func:`scipy.optimize.leastsq` : Minimize the sum of
squares of a set of equations.
Examples
--------
>>> import matplotlib.pyplot as plt
>>> from scipy import stats
>>> np.random.seed(12345678)
>>> x = np.random.random(10)
>>> y = np.random.random(10)
>>> slope, intercept, r_value, p_value, std_err = stats.linregress(x, y)
To get coefficient of determination (r_squared)
>>> print("r-squared:", r_value**2)
r-squared: 0.08040226853902833
Plot the data along with the fitted line
>>> plt.plot(x, y, 'o', label='original data')
>>> plt.plot(x, intercept + slope*x, 'r', label='fitted line')
>>> plt.legend()
>>> plt.show()
"""
TINY = 1.0e-20
if y is None: # x is a (2, N) or (N, 2) shaped array_like
x = np.asarray(x)
if x.shape[0] == 2:
x, y = x
elif x.shape[1] == 2:
x, y = x.T
else:
msg = ("If only `x` is given as input, it has to be of shape "
"(2, N) or (N, 2), provided shape was %s" % str(x.shape))
raise ValueError(msg)
else:
x = np.asarray(x)
y = np.asarray(y)
if x.size == 0 or y.size == 0:
raise ValueError("Inputs must not be empty.")
n = len(x)
xmean = np.mean(x, None)
ymean = np.mean(y, None)
# average sum of squares:
ssxm, ssxym, ssyxm, ssym = np.cov(x, y, bias=1).flat
r_num = ssxym
r_den = np.sqrt(ssxm * ssym)
if r_den == 0.0:
r = 0.0
else:
r = r_num / r_den
# test for numerical error propagation
if r > 1.0:
r = 1.0
elif r < -1.0:
r = -1.0
df = n - 2
slope = r_num / ssxm
intercept = ymean - slope*xmean
if n == 2:
# handle case when only two points are passed in
if y[0] == y[1]:
prob = 1.0
else:
prob = 0.0
sterrest = 0.0
else:
t = r * np.sqrt(df / ((1.0 - r + TINY)*(1.0 + r + TINY)))
prob = 2 * distributions.t.sf(np.abs(t), df)
sterrest = np.sqrt((1 - r**2) * ssym / ssxm / df)
return LinregressResult(slope, intercept, r, prob, sterrest)
def theilslopes(y, x=None, alpha=0.95):
r"""
Computes the Theil-Sen estimator for a set of points (x, y).
`theilslopes` implements a method for robust linear regression. It
computes the slope as the median of all slopes between paired values.
Parameters
----------
y : array_like
Dependent variable.
x : array_like or None, optional
Independent variable. If None, use ``arange(len(y))`` instead.
alpha : float, optional
Confidence degree between 0 and 1. Default is 95% confidence.
Note that `alpha` is symmetric around 0.5, i.e. both 0.1 and 0.9 are
interpreted as "find the 90% confidence interval".
Returns
-------
medslope : float
Theil slope.
medintercept : float
Intercept of the Theil line, as ``median(y) - medslope*median(x)``.
lo_slope : float
Lower bound of the confidence interval on `medslope`.
up_slope : float
Upper bound of the confidence interval on `medslope`.
Notes
-----
The implementation of `theilslopes` follows [1]_. The intercept is
not defined in [1]_, and here it is defined as ``median(y) -
medslope*median(x)``, which is given in [3]_. Other definitions of
the intercept exist in the literature. A confidence interval for
the intercept is not given as this question is not addressed in
[1]_.
References
----------
.. [1] P.K. Sen, "Estimates of the regression coefficient based on Kendall's tau",
J. Am. Stat. Assoc., Vol. 63, pp. 1379-1389, 1968.
.. [2] H. Theil, "A rank-invariant method of linear and polynomial
regression analysis I, II and III", Nederl. Akad. Wetensch., Proc.
53:, pp. 386-392, pp. 521-525, pp. 1397-1412, 1950.
.. [3] W.L. Conover, "Practical nonparametric statistics", 2nd ed.,
John Wiley and Sons, New York, pp. 493.
Examples
--------
>>> from scipy import stats
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-5, 5, num=150)
>>> y = x + np.random.normal(size=x.size)
>>> y[11:15] += 10 # add outliers
>>> y[-5:] -= 7
Compute the slope, intercept and 90% confidence interval. For comparison,
also compute the least-squares fit with `linregress`:
>>> res = stats.theilslopes(y, x, 0.90)
>>> lsq_res = stats.linregress(x, y)
Plot the results. The Theil-Sen regression line is shown in red, with the
dashed red lines illustrating the confidence interval of the slope (note
that the dashed red lines are not the confidence interval of the regression
as the confidence interval of the intercept is not included). The green
line shows the least-squares fit for comparison.
>>> fig = plt.figure()
>>> ax = fig.add_subplot(111)
>>> ax.plot(x, y, 'b.')
>>> ax.plot(x, res[1] + res[0] * x, 'r-')
>>> ax.plot(x, res[1] + res[2] * x, 'r--')
>>> ax.plot(x, res[1] + res[3] * x, 'r--')
>>> ax.plot(x, lsq_res[1] + lsq_res[0] * x, 'g-')
>>> plt.show()
"""
# We copy both x and y so we can use _find_repeats.
y = np.array(y).flatten()
if x is None:
x = np.arange(len(y), dtype=float)
else:
x = np.array(x, dtype=float).flatten()
if len(x) != len(y):
raise ValueError("Incompatible lengths ! (%s<>%s)" % (len(y), len(x)))
# Compute sorted slopes only when deltax > 0
deltax = x[:, np.newaxis] - x
deltay = y[:, np.newaxis] - y
slopes = deltay[deltax > 0] / deltax[deltax > 0]
slopes.sort()
medslope = np.median(slopes)
medinter = np.median(y) - medslope * np.median(x)
# Now compute confidence intervals
if alpha > 0.5:
alpha = 1. - alpha
z = distributions.norm.ppf(alpha / 2.)
# This implements (2.6) from Sen (1968)
_, nxreps = _find_repeats(x)
_, nyreps = _find_repeats(y)
nt = len(slopes) # N in Sen (1968)
ny = len(y) # n in Sen (1968)
# Equation 2.6 in Sen (1968):
sigsq = 1/18. * (ny * (ny-1) * (2*ny+5) -
sum(k * (k-1) * (2*k + 5) for k in nxreps) -
sum(k * (k-1) * (2*k + 5) for k in nyreps))
# Find the confidence interval indices in `slopes`
sigma = np.sqrt(sigsq)
Ru = min(int(np.round((nt - z*sigma)/2.)), len(slopes)-1)
Rl = max(int(np.round((nt + z*sigma)/2.)) - 1, 0)
delta = slopes[[Rl, Ru]]
return medslope, medinter, delta[0], delta[1]
def _find_repeats(arr):
# This function assumes it may clobber its input.
if len(arr) == 0:
return np.array(0, np.float64), np.array(0, np.intp)
# XXX This cast was previously needed for the Fortran implementation,
# should we ditch it?
arr = np.asarray(arr, np.float64).ravel()
arr.sort()
# Taken from NumPy 1.9's np.unique.
change = np.concatenate(([True], arr[1:] != arr[:-1]))
unique = arr[change]
change_idx = np.concatenate(np.nonzero(change) + ([arr.size],))
freq = np.diff(change_idx)
atleast2 = freq > 1
return unique[atleast2], freq[atleast2]
| 8,684 | 32.532819 | 86 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/mstats_extras.py
|
"""
Additional statistics functions with support for masked arrays.
"""
# Original author (2007): Pierre GF Gerard-Marchant
from __future__ import division, print_function, absolute_import
__all__ = ['compare_medians_ms',
'hdquantiles', 'hdmedian', 'hdquantiles_sd',
'idealfourths',
'median_cihs','mjci','mquantiles_cimj',
'rsh',
'trimmed_mean_ci',]
import numpy as np
from numpy import float_, int_, ndarray
import numpy.ma as ma
from numpy.ma import MaskedArray
from . import mstats_basic as mstats
from scipy.stats.distributions import norm, beta, t, binom
def hdquantiles(data, prob=list([.25,.5,.75]), axis=None, var=False,):
"""
Computes quantile estimates with the Harrell-Davis method.
The quantile estimates are calculated as a weighted linear combination
of order statistics.
Parameters
----------
data : array_like
Data array.
prob : sequence, optional
Sequence of quantiles to compute.
axis : int or None, optional
Axis along which to compute the quantiles. If None, use a flattened
array.
var : bool, optional
Whether to return the variance of the estimate.
Returns
-------
hdquantiles : MaskedArray
A (p,) array of quantiles (if `var` is False), or a (2,p) array of
quantiles and variances (if `var` is True), where ``p`` is the
number of quantiles.
See Also
--------
hdquantiles_sd
"""
def _hd_1D(data,prob,var):
"Computes the HD quantiles for a 1D array. Returns nan for invalid data."
xsorted = np.squeeze(np.sort(data.compressed().view(ndarray)))
# Don't use length here, in case we have a numpy scalar
n = xsorted.size
hd = np.empty((2,len(prob)), float_)
if n < 2:
hd.flat = np.nan
if var:
return hd
return hd[0]
v = np.arange(n+1) / float(n)
betacdf = beta.cdf
for (i,p) in enumerate(prob):
_w = betacdf(v, (n+1)*p, (n+1)*(1-p))
w = _w[1:] - _w[:-1]
hd_mean = np.dot(w, xsorted)
hd[0,i] = hd_mean
#
hd[1,i] = np.dot(w, (xsorted-hd_mean)**2)
#
hd[0, prob == 0] = xsorted[0]
hd[0, prob == 1] = xsorted[-1]
if var:
hd[1, prob == 0] = hd[1, prob == 1] = np.nan
return hd
return hd[0]
# Initialization & checks
data = ma.array(data, copy=False, dtype=float_)
p = np.array(prob, copy=False, ndmin=1)
# Computes quantiles along axis (or globally)
if (axis is None) or (data.ndim == 1):
result = _hd_1D(data, p, var)
else:
if data.ndim > 2:
raise ValueError("Array 'data' must be at most two dimensional, "
"but got data.ndim = %d" % data.ndim)
result = ma.apply_along_axis(_hd_1D, axis, data, p, var)
return ma.fix_invalid(result, copy=False)
def hdmedian(data, axis=-1, var=False):
"""
Returns the Harrell-Davis estimate of the median along the given axis.
Parameters
----------
data : ndarray
Data array.
axis : int, optional
Axis along which to compute the quantiles. If None, use a flattened
array.
var : bool, optional
Whether to return the variance of the estimate.
Returns
-------
hdmedian : MaskedArray
The median values. If ``var=True``, the variance is returned inside
the masked array. E.g. for a 1-D array the shape change from (1,) to
(2,).
"""
result = hdquantiles(data,[0.5], axis=axis, var=var)
return result.squeeze()
def hdquantiles_sd(data, prob=list([.25,.5,.75]), axis=None):
"""
The standard error of the Harrell-Davis quantile estimates by jackknife.
Parameters
----------
data : array_like
Data array.
prob : sequence, optional
Sequence of quantiles to compute.
axis : int, optional
Axis along which to compute the quantiles. If None, use a flattened
array.
Returns
-------
hdquantiles_sd : MaskedArray
Standard error of the Harrell-Davis quantile estimates.
See Also
--------
hdquantiles
"""
def _hdsd_1D(data, prob):
"Computes the std error for 1D arrays."
xsorted = np.sort(data.compressed())
n = len(xsorted)
hdsd = np.empty(len(prob), float_)
if n < 2:
hdsd.flat = np.nan
vv = np.arange(n) / float(n-1)
betacdf = beta.cdf
for (i,p) in enumerate(prob):
_w = betacdf(vv, (n+1)*p, (n+1)*(1-p))
w = _w[1:] - _w[:-1]
mx_ = np.fromiter([np.dot(w,xsorted[np.r_[list(range(0,k)),
list(range(k+1,n))].astype(int_)])
for k in range(n)], dtype=float_)
mx_var = np.array(mx_.var(), copy=False, ndmin=1) * n / float(n-1)
hdsd[i] = float(n-1) * np.sqrt(np.diag(mx_var).diagonal() / float(n))
return hdsd
# Initialization & checks
data = ma.array(data, copy=False, dtype=float_)
p = np.array(prob, copy=False, ndmin=1)
# Computes quantiles along axis (or globally)
if (axis is None):
result = _hdsd_1D(data, p)
else:
if data.ndim > 2:
raise ValueError("Array 'data' must be at most two dimensional, "
"but got data.ndim = %d" % data.ndim)
result = ma.apply_along_axis(_hdsd_1D, axis, data, p)
return ma.fix_invalid(result, copy=False).ravel()
def trimmed_mean_ci(data, limits=(0.2,0.2), inclusive=(True,True),
alpha=0.05, axis=None):
"""
Selected confidence interval of the trimmed mean along the given axis.
Parameters
----------
data : array_like
Input data.
limits : {None, tuple}, optional
None or a two item tuple.
Tuple of the percentages to cut on each side of the array, with respect
to the number of unmasked data, as floats between 0. and 1. If ``n``
is the number of unmasked data before trimming, then
(``n * limits[0]``)th smallest data and (``n * limits[1]``)th
largest data are masked. The total number of unmasked data after
trimming is ``n * (1. - sum(limits))``.
The value of one limit can be set to None to indicate an open interval.
Defaults to (0.2, 0.2).
inclusive : (2,) tuple of boolean, optional
If relative==False, tuple indicating whether values exactly equal to
the absolute limits are allowed.
If relative==True, tuple indicating whether the number of data being
masked on each side should be rounded (True) or truncated (False).
Defaults to (True, True).
alpha : float, optional
Confidence level of the intervals.
Defaults to 0.05.
axis : int, optional
Axis along which to cut. If None, uses a flattened version of `data`.
Defaults to None.
Returns
-------
trimmed_mean_ci : (2,) ndarray
The lower and upper confidence intervals of the trimmed data.
"""
data = ma.array(data, copy=False)
trimmed = mstats.trimr(data, limits=limits, inclusive=inclusive, axis=axis)
tmean = trimmed.mean(axis)
tstde = mstats.trimmed_stde(data,limits=limits,inclusive=inclusive,axis=axis)
df = trimmed.count(axis) - 1
tppf = t.ppf(1-alpha/2.,df)
return np.array((tmean - tppf*tstde, tmean+tppf*tstde))
def mjci(data, prob=[0.25,0.5,0.75], axis=None):
"""
Returns the Maritz-Jarrett estimators of the standard error of selected
experimental quantiles of the data.
Parameters
----------
data : ndarray
Data array.
prob : sequence, optional
Sequence of quantiles to compute.
axis : int or None, optional
Axis along which to compute the quantiles. If None, use a flattened
array.
"""
def _mjci_1D(data, p):
data = np.sort(data.compressed())
n = data.size
prob = (np.array(p) * n + 0.5).astype(int_)
betacdf = beta.cdf
mj = np.empty(len(prob), float_)
x = np.arange(1,n+1, dtype=float_) / n
y = x - 1./n
for (i,m) in enumerate(prob):
W = betacdf(x,m-1,n-m) - betacdf(y,m-1,n-m)
C1 = np.dot(W,data)
C2 = np.dot(W,data**2)
mj[i] = np.sqrt(C2 - C1**2)
return mj
data = ma.array(data, copy=False)
if data.ndim > 2:
raise ValueError("Array 'data' must be at most two dimensional, "
"but got data.ndim = %d" % data.ndim)
p = np.array(prob, copy=False, ndmin=1)
# Computes quantiles along axis (or globally)
if (axis is None):
return _mjci_1D(data, p)
else:
return ma.apply_along_axis(_mjci_1D, axis, data, p)
def mquantiles_cimj(data, prob=[0.25,0.50,0.75], alpha=0.05, axis=None):
"""
Computes the alpha confidence interval for the selected quantiles of the
data, with Maritz-Jarrett estimators.
Parameters
----------
data : ndarray
Data array.
prob : sequence, optional
Sequence of quantiles to compute.
alpha : float, optional
Confidence level of the intervals.
axis : int or None, optional
Axis along which to compute the quantiles.
If None, use a flattened array.
Returns
-------
ci_lower : ndarray
The lower boundaries of the confidence interval. Of the same length as
`prob`.
ci_upper : ndarray
The upper boundaries of the confidence interval. Of the same length as
`prob`.
"""
alpha = min(alpha, 1 - alpha)
z = norm.ppf(1 - alpha/2.)
xq = mstats.mquantiles(data, prob, alphap=0, betap=0, axis=axis)
smj = mjci(data, prob, axis=axis)
return (xq - z * smj, xq + z * smj)
def median_cihs(data, alpha=0.05, axis=None):
"""
Computes the alpha-level confidence interval for the median of the data.
Uses the Hettmasperger-Sheather method.
Parameters
----------
data : array_like
Input data. Masked values are discarded. The input should be 1D only,
or `axis` should be set to None.
alpha : float, optional
Confidence level of the intervals.
axis : int or None, optional
Axis along which to compute the quantiles. If None, use a flattened
array.
Returns
-------
median_cihs
Alpha level confidence interval.
"""
def _cihs_1D(data, alpha):
data = np.sort(data.compressed())
n = len(data)
alpha = min(alpha, 1-alpha)
k = int(binom._ppf(alpha/2., n, 0.5))
gk = binom.cdf(n-k,n,0.5) - binom.cdf(k-1,n,0.5)
if gk < 1-alpha:
k -= 1
gk = binom.cdf(n-k,n,0.5) - binom.cdf(k-1,n,0.5)
gkk = binom.cdf(n-k-1,n,0.5) - binom.cdf(k,n,0.5)
I = (gk - 1 + alpha)/(gk - gkk)
lambd = (n-k) * I / float(k + (n-2*k)*I)
lims = (lambd*data[k] + (1-lambd)*data[k-1],
lambd*data[n-k-1] + (1-lambd)*data[n-k])
return lims
data = ma.array(data, copy=False)
# Computes quantiles along axis (or globally)
if (axis is None):
result = _cihs_1D(data, alpha)
else:
if data.ndim > 2:
raise ValueError("Array 'data' must be at most two dimensional, "
"but got data.ndim = %d" % data.ndim)
result = ma.apply_along_axis(_cihs_1D, axis, data, alpha)
return result
def compare_medians_ms(group_1, group_2, axis=None):
"""
Compares the medians from two independent groups along the given axis.
The comparison is performed using the McKean-Schrader estimate of the
standard error of the medians.
Parameters
----------
group_1 : array_like
First dataset. Has to be of size >=7.
group_2 : array_like
Second dataset. Has to be of size >=7.
axis : int, optional
Axis along which the medians are estimated. If None, the arrays are
flattened. If `axis` is not None, then `group_1` and `group_2`
should have the same shape.
Returns
-------
compare_medians_ms : {float, ndarray}
If `axis` is None, then returns a float, otherwise returns a 1-D
ndarray of floats with a length equal to the length of `group_1`
along `axis`.
"""
(med_1, med_2) = (ma.median(group_1,axis=axis), ma.median(group_2,axis=axis))
(std_1, std_2) = (mstats.stde_median(group_1, axis=axis),
mstats.stde_median(group_2, axis=axis))
W = np.abs(med_1 - med_2) / ma.sqrt(std_1**2 + std_2**2)
return 1 - norm.cdf(W)
def idealfourths(data, axis=None):
"""
Returns an estimate of the lower and upper quartiles.
Uses the ideal fourths algorithm.
Parameters
----------
data : array_like
Input array.
axis : int, optional
Axis along which the quartiles are estimated. If None, the arrays are
flattened.
Returns
-------
idealfourths : {list of floats, masked array}
Returns the two internal values that divide `data` into four parts
using the ideal fourths algorithm either along the flattened array
(if `axis` is None) or along `axis` of `data`.
"""
def _idf(data):
x = data.compressed()
n = len(x)
if n < 3:
return [np.nan,np.nan]
(j,h) = divmod(n/4. + 5/12.,1)
j = int(j)
qlo = (1-h)*x[j-1] + h*x[j]
k = n - j
qup = (1-h)*x[k] + h*x[k-1]
return [qlo, qup]
data = ma.sort(data, axis=axis).view(MaskedArray)
if (axis is None):
return _idf(data)
else:
return ma.apply_along_axis(_idf, axis, data)
def rsh(data, points=None):
"""
Evaluates Rosenblatt's shifted histogram estimators for each data point.
Rosenblatt's estimator is a centered finite-difference approximation to the
derivative of the empirical cumulative distribution function.
Parameters
----------
data : sequence
Input data, should be 1-D. Masked values are ignored.
points : sequence or None, optional
Sequence of points where to evaluate Rosenblatt shifted histogram.
If None, use the data.
"""
data = ma.array(data, copy=False)
if points is None:
points = data
else:
points = np.array(points, copy=False, ndmin=1)
if data.ndim != 1:
raise AttributeError("The input array should be 1D only !")
n = data.count()
r = idealfourths(data, axis=None)
h = 1.2 * (r[-1]-r[0]) / n**(1./5)
nhi = (data[:,None] <= points[None,:] + h).sum(0)
nlo = (data[:,None] < points[None,:] - h).sum(0)
return (nhi-nlo) / (2.*n*h)
| 14,957 | 30.292887 | 88 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/mstats.py
|
"""
===================================================================
Statistical functions for masked arrays (:mod:`scipy.stats.mstats`)
===================================================================
.. currentmodule:: scipy.stats.mstats
This module contains a large number of statistical functions that can
be used with masked arrays.
Most of these functions are similar to those in scipy.stats but might
have small differences in the API or in the algorithm used. Since this
is a relatively new package, some API changes are still possible.
.. autosummary::
:toctree: generated/
argstoarray
chisquare
count_tied_groups
describe
f_oneway
find_repeats
friedmanchisquare
kendalltau
kendalltau_seasonal
kruskalwallis
ks_twosamp
kurtosis
kurtosistest
linregress
mannwhitneyu
plotting_positions
mode
moment
mquantiles
msign
normaltest
obrientransform
pearsonr
plotting_positions
pointbiserialr
rankdata
scoreatpercentile
sem
skew
skewtest
spearmanr
theilslopes
tmax
tmean
tmin
trim
trima
trimboth
trimmed_stde
trimr
trimtail
tsem
ttest_onesamp
ttest_ind
ttest_onesamp
ttest_rel
tvar
variation
winsorize
zmap
zscore
compare_medians_ms
gmean
hdmedian
hdquantiles
hdquantiles_sd
hmean
idealfourths
kruskal
ks_2samp
median_cihs
meppf
mjci
mquantiles_cimj
rsh
sen_seasonal_slopes
trimmed_mean
trimmed_mean_ci
trimmed_std
trimmed_var
ttest_1samp
"""
from __future__ import division, print_function, absolute_import
from .mstats_basic import *
from .mstats_extras import *
# Functions that support masked array input in stats but need to be kept in the
# mstats namespace for backwards compatibility:
from scipy.stats import gmean, hmean, zmap, zscore, chisquare
| 1,883 | 18.22449 | 79 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/morestats.py
|
from __future__ import division, print_function, absolute_import
import math
import warnings
from collections import namedtuple
import numpy as np
from numpy import (isscalar, r_, log, around, unique, asarray,
zeros, arange, sort, amin, amax, any, atleast_1d,
sqrt, ceil, floor, array, compress,
pi, exp, ravel, count_nonzero, sin, cos, arctan2, hypot)
from scipy._lib.six import string_types
from scipy import optimize
from scipy import special
from . import statlib
from . import stats
from .stats import find_repeats, _contains_nan
from .contingency import chi2_contingency
from . import distributions
from ._distn_infrastructure import rv_generic
__all__ = ['mvsdist',
'bayes_mvs', 'kstat', 'kstatvar', 'probplot', 'ppcc_max', 'ppcc_plot',
'boxcox_llf', 'boxcox', 'boxcox_normmax', 'boxcox_normplot',
'shapiro', 'anderson', 'ansari', 'bartlett', 'levene', 'binom_test',
'fligner', 'mood', 'wilcoxon', 'median_test',
'circmean', 'circvar', 'circstd', 'anderson_ksamp'
]
Mean = namedtuple('Mean', ('statistic', 'minmax'))
Variance = namedtuple('Variance', ('statistic', 'minmax'))
Std_dev = namedtuple('Std_dev', ('statistic', 'minmax'))
def bayes_mvs(data, alpha=0.90):
r"""
Bayesian confidence intervals for the mean, var, and std.
Parameters
----------
data : array_like
Input data, if multi-dimensional it is flattened to 1-D by `bayes_mvs`.
Requires 2 or more data points.
alpha : float, optional
Probability that the returned confidence interval contains
the true parameter.
Returns
-------
mean_cntr, var_cntr, std_cntr : tuple
The three results are for the mean, variance and standard deviation,
respectively. Each result is a tuple of the form::
(center, (lower, upper))
with `center` the mean of the conditional pdf of the value given the
data, and `(lower, upper)` a confidence interval, centered on the
median, containing the estimate to a probability ``alpha``.
See Also
--------
mvsdist
Notes
-----
Each tuple of mean, variance, and standard deviation estimates represent
the (center, (lower, upper)) with center the mean of the conditional pdf
of the value given the data and (lower, upper) is a confidence interval
centered on the median, containing the estimate to a probability
``alpha``.
Converts data to 1-D and assumes all data has the same mean and variance.
Uses Jeffrey's prior for variance and std.
Equivalent to ``tuple((x.mean(), x.interval(alpha)) for x in mvsdist(dat))``
References
----------
T.E. Oliphant, "A Bayesian perspective on estimating mean, variance, and
standard-deviation from data", http://scholarsarchive.byu.edu/facpub/278,
2006.
Examples
--------
First a basic example to demonstrate the outputs:
>>> from scipy import stats
>>> data = [6, 9, 12, 7, 8, 8, 13]
>>> mean, var, std = stats.bayes_mvs(data)
>>> mean
Mean(statistic=9.0, minmax=(7.103650222612533, 10.896349777387467))
>>> var
Variance(statistic=10.0, minmax=(3.176724206..., 24.45910382...))
>>> std
Std_dev(statistic=2.9724954732045084, minmax=(1.7823367265645143, 4.945614605014631))
Now we generate some normally distributed random data, and get estimates of
mean and standard deviation with 95% confidence intervals for those
estimates:
>>> n_samples = 100000
>>> data = stats.norm.rvs(size=n_samples)
>>> res_mean, res_var, res_std = stats.bayes_mvs(data, alpha=0.95)
>>> import matplotlib.pyplot as plt
>>> fig = plt.figure()
>>> ax = fig.add_subplot(111)
>>> ax.hist(data, bins=100, density=True, label='Histogram of data')
>>> ax.vlines(res_mean.statistic, 0, 0.5, colors='r', label='Estimated mean')
>>> ax.axvspan(res_mean.minmax[0],res_mean.minmax[1], facecolor='r',
... alpha=0.2, label=r'Estimated mean (95% limits)')
>>> ax.vlines(res_std.statistic, 0, 0.5, colors='g', label='Estimated scale')
>>> ax.axvspan(res_std.minmax[0],res_std.minmax[1], facecolor='g', alpha=0.2,
... label=r'Estimated scale (95% limits)')
>>> ax.legend(fontsize=10)
>>> ax.set_xlim([-4, 4])
>>> ax.set_ylim([0, 0.5])
>>> plt.show()
"""
m, v, s = mvsdist(data)
if alpha >= 1 or alpha <= 0:
raise ValueError("0 < alpha < 1 is required, but alpha=%s was given."
% alpha)
m_res = Mean(m.mean(), m.interval(alpha))
v_res = Variance(v.mean(), v.interval(alpha))
s_res = Std_dev(s.mean(), s.interval(alpha))
return m_res, v_res, s_res
def mvsdist(data):
"""
'Frozen' distributions for mean, variance, and standard deviation of data.
Parameters
----------
data : array_like
Input array. Converted to 1-D using ravel.
Requires 2 or more data-points.
Returns
-------
mdist : "frozen" distribution object
Distribution object representing the mean of the data
vdist : "frozen" distribution object
Distribution object representing the variance of the data
sdist : "frozen" distribution object
Distribution object representing the standard deviation of the data
See Also
--------
bayes_mvs
Notes
-----
The return values from ``bayes_mvs(data)`` is equivalent to
``tuple((x.mean(), x.interval(0.90)) for x in mvsdist(data))``.
In other words, calling ``<dist>.mean()`` and ``<dist>.interval(0.90)``
on the three distribution objects returned from this function will give
the same results that are returned from `bayes_mvs`.
References
----------
T.E. Oliphant, "A Bayesian perspective on estimating mean, variance, and
standard-deviation from data", http://scholarsarchive.byu.edu/facpub/278,
2006.
Examples
--------
>>> from scipy import stats
>>> data = [6, 9, 12, 7, 8, 8, 13]
>>> mean, var, std = stats.mvsdist(data)
We now have frozen distribution objects "mean", "var" and "std" that we can
examine:
>>> mean.mean()
9.0
>>> mean.interval(0.95)
(6.6120585482655692, 11.387941451734431)
>>> mean.std()
1.1952286093343936
"""
x = ravel(data)
n = len(x)
if n < 2:
raise ValueError("Need at least 2 data-points.")
xbar = x.mean()
C = x.var()
if n > 1000: # gaussian approximations for large n
mdist = distributions.norm(loc=xbar, scale=math.sqrt(C / n))
sdist = distributions.norm(loc=math.sqrt(C), scale=math.sqrt(C / (2. * n)))
vdist = distributions.norm(loc=C, scale=math.sqrt(2.0 / n) * C)
else:
nm1 = n - 1
fac = n * C / 2.
val = nm1 / 2.
mdist = distributions.t(nm1, loc=xbar, scale=math.sqrt(C / nm1))
sdist = distributions.gengamma(val, -2, scale=math.sqrt(fac))
vdist = distributions.invgamma(val, scale=fac)
return mdist, vdist, sdist
def kstat(data, n=2):
r"""
Return the nth k-statistic (1<=n<=4 so far).
The nth k-statistic k_n is the unique symmetric unbiased estimator of the
nth cumulant kappa_n.
Parameters
----------
data : array_like
Input array. Note that n-D input gets flattened.
n : int, {1, 2, 3, 4}, optional
Default is equal to 2.
Returns
-------
kstat : float
The nth k-statistic.
See Also
--------
kstatvar: Returns an unbiased estimator of the variance of the k-statistic.
moment: Returns the n-th central moment about the mean for a sample.
Notes
-----
For a sample size n, the first few k-statistics are given by:
.. math::
k_{1} = \mu
k_{2} = \frac{n}{n-1} m_{2}
k_{3} = \frac{ n^{2} } {(n-1) (n-2)} m_{3}
k_{4} = \frac{ n^{2} [(n + 1)m_{4} - 3(n - 1) m^2_{2}]} {(n-1) (n-2) (n-3)}
where :math:`\mu` is the sample mean, :math:`m_2` is the sample
variance, and :math:`m_i` is the i-th sample central moment.
References
----------
http://mathworld.wolfram.com/k-Statistic.html
http://mathworld.wolfram.com/Cumulant.html
Examples
--------
>>> from scipy import stats
>>> rndm = np.random.RandomState(1234)
As sample size increases, n-th moment and n-th k-statistic converge to the
same number (although they aren't identical). In the case of the normal
distribution, they converge to zero.
>>> for n in [2, 3, 4, 5, 6, 7]:
... x = rndm.normal(size=10**n)
... m, k = stats.moment(x, 3), stats.kstat(x, 3)
... print("%.3g %.3g %.3g" % (m, k, m-k))
-0.631 -0.651 0.0194
0.0282 0.0283 -8.49e-05
-0.0454 -0.0454 1.36e-05
7.53e-05 7.53e-05 -2.26e-09
0.00166 0.00166 -4.99e-09
-2.88e-06 -2.88e-06 8.63e-13
"""
if n > 4 or n < 1:
raise ValueError("k-statistics only supported for 1<=n<=4")
n = int(n)
S = np.zeros(n + 1, np.float64)
data = ravel(data)
N = data.size
# raise ValueError on empty input
if N == 0:
raise ValueError("Data input must not be empty")
# on nan input, return nan without warning
if np.isnan(np.sum(data)):
return np.nan
for k in range(1, n + 1):
S[k] = np.sum(data**k, axis=0)
if n == 1:
return S[1] * 1.0/N
elif n == 2:
return (N*S[2] - S[1]**2.0) / (N*(N - 1.0))
elif n == 3:
return (2*S[1]**3 - 3*N*S[1]*S[2] + N*N*S[3]) / (N*(N - 1.0)*(N - 2.0))
elif n == 4:
return ((-6*S[1]**4 + 12*N*S[1]**2 * S[2] - 3*N*(N-1.0)*S[2]**2 -
4*N*(N+1)*S[1]*S[3] + N*N*(N+1)*S[4]) /
(N*(N-1.0)*(N-2.0)*(N-3.0)))
else:
raise ValueError("Should not be here.")
def kstatvar(data, n=2):
r"""
Returns an unbiased estimator of the variance of the k-statistic.
See `kstat` for more details of the k-statistic.
Parameters
----------
data : array_like
Input array. Note that n-D input gets flattened.
n : int, {1, 2}, optional
Default is equal to 2.
Returns
-------
kstatvar : float
The nth k-statistic variance.
See Also
--------
kstat: Returns the n-th k-statistic.
moment: Returns the n-th central moment about the mean for a sample.
Notes
-----
The variances of the first few k-statistics are given by:
.. math::
var(k_{1}) = \frac{\kappa^2}{n}
var(k_{2}) = \frac{\kappa^4}{n} + \frac{2\kappa^2_{2}}{n - 1}
var(k_{3}) = \frac{\kappa^6}{n} + \frac{9 \kappa_2 \kappa_4}{n - 1} +
\frac{9 \kappa^2_{3}}{n - 1} +
\frac{6 n \kappa^3_{2}}{(n-1) (n-2)}
var(k_{4}) = \frac{\kappa^8}{n} + \frac{16 \kappa_2 \kappa_6}{n - 1} +
\frac{48 \kappa_{3} \kappa_5}{n - 1} +
\frac{34 \kappa^2_{4}}{n-1} + \frac{72 n \kappa^2_{2} \kappa_4}{(n - 1) (n - 2)} +
\frac{144 n \kappa_{2} \kappa^2_{3}}{(n - 1) (n - 2)} +
\frac{24 (n + 1) n \kappa^4_{2}}{(n - 1) (n - 2) (n - 3)}
"""
data = ravel(data)
N = len(data)
if n == 1:
return kstat(data, n=2) * 1.0/N
elif n == 2:
k2 = kstat(data, n=2)
k4 = kstat(data, n=4)
return (2*N*k2**2 + (N-1)*k4) / (N*(N+1))
else:
raise ValueError("Only n=1 or n=2 supported.")
def _calc_uniform_order_statistic_medians(n):
"""
Approximations of uniform order statistic medians.
Parameters
----------
n : int
Sample size.
Returns
-------
v : 1d float array
Approximations of the order statistic medians.
References
----------
.. [1] James J. Filliben, "The Probability Plot Correlation Coefficient
Test for Normality", Technometrics, Vol. 17, pp. 111-117, 1975.
Examples
--------
Order statistics of the uniform distribution on the unit interval
are marginally distributed according to beta distributions.
The expectations of these order statistic are evenly spaced across
the interval, but the distributions are skewed in a way that
pushes the medians slightly towards the endpoints of the unit interval:
>>> n = 4
>>> k = np.arange(1, n+1)
>>> from scipy.stats import beta
>>> a = k
>>> b = n-k+1
>>> beta.mean(a, b)
array([ 0.2, 0.4, 0.6, 0.8])
>>> beta.median(a, b)
array([ 0.15910358, 0.38572757, 0.61427243, 0.84089642])
The Filliben approximation uses the exact medians of the smallest
and greatest order statistics, and the remaining medians are approximated
by points spread evenly across a sub-interval of the unit interval:
>>> from scipy.morestats import _calc_uniform_order_statistic_medians
>>> _calc_uniform_order_statistic_medians(n)
array([ 0.15910358, 0.38545246, 0.61454754, 0.84089642])
This plot shows the skewed distributions of the order statistics
of a sample of size four from a uniform distribution on the unit interval:
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(0.0, 1.0, num=50, endpoint=True)
>>> pdfs = [beta.pdf(x, a[i], b[i]) for i in range(n)]
>>> plt.figure()
>>> plt.plot(x, pdfs[0], x, pdfs[1], x, pdfs[2], x, pdfs[3])
"""
v = np.zeros(n, dtype=np.float64)
v[-1] = 0.5**(1.0 / n)
v[0] = 1 - v[-1]
i = np.arange(2, n)
v[1:-1] = (i - 0.3175) / (n + 0.365)
return v
def _parse_dist_kw(dist, enforce_subclass=True):
"""Parse `dist` keyword.
Parameters
----------
dist : str or stats.distributions instance.
Several functions take `dist` as a keyword, hence this utility
function.
enforce_subclass : bool, optional
If True (default), `dist` needs to be a
`_distn_infrastructure.rv_generic` instance.
It can sometimes be useful to set this keyword to False, if a function
wants to accept objects that just look somewhat like such an instance
(for example, they have a ``ppf`` method).
"""
if isinstance(dist, rv_generic):
pass
elif isinstance(dist, string_types):
try:
dist = getattr(distributions, dist)
except AttributeError:
raise ValueError("%s is not a valid distribution name" % dist)
elif enforce_subclass:
msg = ("`dist` should be a stats.distributions instance or a string "
"with the name of such a distribution.")
raise ValueError(msg)
return dist
def _add_axis_labels_title(plot, xlabel, ylabel, title):
"""Helper function to add axes labels and a title to stats plots"""
try:
if hasattr(plot, 'set_title'):
# Matplotlib Axes instance or something that looks like it
plot.set_title(title)
plot.set_xlabel(xlabel)
plot.set_ylabel(ylabel)
else:
# matplotlib.pyplot module
plot.title(title)
plot.xlabel(xlabel)
plot.ylabel(ylabel)
except:
# Not an MPL object or something that looks (enough) like it.
# Don't crash on adding labels or title
pass
def probplot(x, sparams=(), dist='norm', fit=True, plot=None, rvalue=False):
"""
Calculate quantiles for a probability plot, and optionally show the plot.
Generates a probability plot of sample data against the quantiles of a
specified theoretical distribution (the normal distribution by default).
`probplot` optionally calculates a best-fit line for the data and plots the
results using Matplotlib or a given plot function.
Parameters
----------
x : array_like
Sample/response data from which `probplot` creates the plot.
sparams : tuple, optional
Distribution-specific shape parameters (shape parameters plus location
and scale).
dist : str or stats.distributions instance, optional
Distribution or distribution function name. The default is 'norm' for a
normal probability plot. Objects that look enough like a
stats.distributions instance (i.e. they have a ``ppf`` method) are also
accepted.
fit : bool, optional
Fit a least-squares regression (best-fit) line to the sample data if
True (default).
plot : object, optional
If given, plots the quantiles and least squares fit.
`plot` is an object that has to have methods "plot" and "text".
The `matplotlib.pyplot` module or a Matplotlib Axes object can be used,
or a custom object with the same methods.
Default is None, which means that no plot is created.
Returns
-------
(osm, osr) : tuple of ndarrays
Tuple of theoretical quantiles (osm, or order statistic medians) and
ordered responses (osr). `osr` is simply sorted input `x`.
For details on how `osm` is calculated see the Notes section.
(slope, intercept, r) : tuple of floats, optional
Tuple containing the result of the least-squares fit, if that is
performed by `probplot`. `r` is the square root of the coefficient of
determination. If ``fit=False`` and ``plot=None``, this tuple is not
returned.
Notes
-----
Even if `plot` is given, the figure is not shown or saved by `probplot`;
``plt.show()`` or ``plt.savefig('figname.png')`` should be used after
calling `probplot`.
`probplot` generates a probability plot, which should not be confused with
a Q-Q or a P-P plot. Statsmodels has more extensive functionality of this
type, see ``statsmodels.api.ProbPlot``.
The formula used for the theoretical quantiles (horizontal axis of the
probability plot) is Filliben's estimate::
quantiles = dist.ppf(val), for
0.5**(1/n), for i = n
val = (i - 0.3175) / (n + 0.365), for i = 2, ..., n-1
1 - 0.5**(1/n), for i = 1
where ``i`` indicates the i-th ordered value and ``n`` is the total number
of values.
Examples
--------
>>> from scipy import stats
>>> import matplotlib.pyplot as plt
>>> nsample = 100
>>> np.random.seed(7654321)
A t distribution with small degrees of freedom:
>>> ax1 = plt.subplot(221)
>>> x = stats.t.rvs(3, size=nsample)
>>> res = stats.probplot(x, plot=plt)
A t distribution with larger degrees of freedom:
>>> ax2 = plt.subplot(222)
>>> x = stats.t.rvs(25, size=nsample)
>>> res = stats.probplot(x, plot=plt)
A mixture of two normal distributions with broadcasting:
>>> ax3 = plt.subplot(223)
>>> x = stats.norm.rvs(loc=[0,5], scale=[1,1.5],
... size=(nsample//2,2)).ravel()
>>> res = stats.probplot(x, plot=plt)
A standard normal distribution:
>>> ax4 = plt.subplot(224)
>>> x = stats.norm.rvs(loc=0, scale=1, size=nsample)
>>> res = stats.probplot(x, plot=plt)
Produce a new figure with a loggamma distribution, using the ``dist`` and
``sparams`` keywords:
>>> fig = plt.figure()
>>> ax = fig.add_subplot(111)
>>> x = stats.loggamma.rvs(c=2.5, size=500)
>>> res = stats.probplot(x, dist=stats.loggamma, sparams=(2.5,), plot=ax)
>>> ax.set_title("Probplot for loggamma dist with shape parameter 2.5")
Show the results with Matplotlib:
>>> plt.show()
"""
x = np.asarray(x)
_perform_fit = fit or (plot is not None)
if x.size == 0:
if _perform_fit:
return (x, x), (np.nan, np.nan, 0.0)
else:
return x, x
osm_uniform = _calc_uniform_order_statistic_medians(len(x))
dist = _parse_dist_kw(dist, enforce_subclass=False)
if sparams is None:
sparams = ()
if isscalar(sparams):
sparams = (sparams,)
if not isinstance(sparams, tuple):
sparams = tuple(sparams)
osm = dist.ppf(osm_uniform, *sparams)
osr = sort(x)
if _perform_fit:
# perform a linear least squares fit.
slope, intercept, r, prob, sterrest = stats.linregress(osm, osr)
if plot is not None:
plot.plot(osm, osr, 'bo', osm, slope*osm + intercept, 'r-')
_add_axis_labels_title(plot, xlabel='Theoretical quantiles',
ylabel='Ordered Values',
title='Probability Plot')
# Add R^2 value to the plot as text
if rvalue:
xmin = amin(osm)
xmax = amax(osm)
ymin = amin(x)
ymax = amax(x)
posx = xmin + 0.70 * (xmax - xmin)
posy = ymin + 0.01 * (ymax - ymin)
plot.text(posx, posy, "$R^2=%1.4f$" % r**2)
if fit:
return (osm, osr), (slope, intercept, r)
else:
return osm, osr
def ppcc_max(x, brack=(0.0, 1.0), dist='tukeylambda'):
"""
Calculate the shape parameter that maximizes the PPCC
The probability plot correlation coefficient (PPCC) plot can be used to
determine the optimal shape parameter for a one-parameter family of
distributions. ppcc_max returns the shape parameter that would maximize the
probability plot correlation coefficient for the given data to a
one-parameter family of distributions.
Parameters
----------
x : array_like
Input array.
brack : tuple, optional
Triple (a,b,c) where (a<b<c). If bracket consists of two numbers (a, c)
then they are assumed to be a starting interval for a downhill bracket
search (see `scipy.optimize.brent`).
dist : str or stats.distributions instance, optional
Distribution or distribution function name. Objects that look enough
like a stats.distributions instance (i.e. they have a ``ppf`` method)
are also accepted. The default is ``'tukeylambda'``.
Returns
-------
shape_value : float
The shape parameter at which the probability plot correlation
coefficient reaches its max value.
See also
--------
ppcc_plot, probplot, boxcox
Notes
-----
The brack keyword serves as a starting point which is useful in corner
cases. One can use a plot to obtain a rough visual estimate of the location
for the maximum to start the search near it.
References
----------
.. [1] J.J. Filliben, "The Probability Plot Correlation Coefficient Test for
Normality", Technometrics, Vol. 17, pp. 111-117, 1975.
.. [2] http://www.itl.nist.gov/div898/handbook/eda/section3/ppccplot.htm
Examples
--------
First we generate some random data from a Tukey-Lambda distribution,
with shape parameter -0.7:
>>> from scipy import stats
>>> x = stats.tukeylambda.rvs(-0.7, loc=2, scale=0.5, size=10000,
... random_state=1234567) + 1e4
Now we explore this data with a PPCC plot as well as the related
probability plot and Box-Cox normplot. A red line is drawn where we
expect the PPCC value to be maximal (at the shape parameter -0.7 used
above):
>>> import matplotlib.pyplot as plt
>>> fig = plt.figure(figsize=(8, 6))
>>> ax = fig.add_subplot(111)
>>> res = stats.ppcc_plot(x, -5, 5, plot=ax)
We calculate the value where the shape should reach its maximum and a red
line is drawn there. The line should coincide with the highest point in the
ppcc_plot.
>>> max = stats.ppcc_max(x)
>>> ax.vlines(max, 0, 1, colors='r', label='Expected shape value')
>>> plt.show()
"""
dist = _parse_dist_kw(dist)
osm_uniform = _calc_uniform_order_statistic_medians(len(x))
osr = sort(x)
# this function computes the x-axis values of the probability plot
# and computes a linear regression (including the correlation)
# and returns 1-r so that a minimization function maximizes the
# correlation
def tempfunc(shape, mi, yvals, func):
xvals = func(mi, shape)
r, prob = stats.pearsonr(xvals, yvals)
return 1 - r
return optimize.brent(tempfunc, brack=brack, args=(osm_uniform, osr, dist.ppf))
def ppcc_plot(x, a, b, dist='tukeylambda', plot=None, N=80):
"""
Calculate and optionally plot probability plot correlation coefficient.
The probability plot correlation coefficient (PPCC) plot can be used to
determine the optimal shape parameter for a one-parameter family of
distributions. It cannot be used for distributions without shape parameters
(like the normal distribution) or with multiple shape parameters.
By default a Tukey-Lambda distribution (`stats.tukeylambda`) is used. A
Tukey-Lambda PPCC plot interpolates from long-tailed to short-tailed
distributions via an approximately normal one, and is therefore particularly
useful in practice.
Parameters
----------
x : array_like
Input array.
a, b: scalar
Lower and upper bounds of the shape parameter to use.
dist : str or stats.distributions instance, optional
Distribution or distribution function name. Objects that look enough
like a stats.distributions instance (i.e. they have a ``ppf`` method)
are also accepted. The default is ``'tukeylambda'``.
plot : object, optional
If given, plots PPCC against the shape parameter.
`plot` is an object that has to have methods "plot" and "text".
The `matplotlib.pyplot` module or a Matplotlib Axes object can be used,
or a custom object with the same methods.
Default is None, which means that no plot is created.
N : int, optional
Number of points on the horizontal axis (equally distributed from
`a` to `b`).
Returns
-------
svals : ndarray
The shape values for which `ppcc` was calculated.
ppcc : ndarray
The calculated probability plot correlation coefficient values.
See also
--------
ppcc_max, probplot, boxcox_normplot, tukeylambda
References
----------
J.J. Filliben, "The Probability Plot Correlation Coefficient Test for
Normality", Technometrics, Vol. 17, pp. 111-117, 1975.
Examples
--------
First we generate some random data from a Tukey-Lambda distribution,
with shape parameter -0.7:
>>> from scipy import stats
>>> import matplotlib.pyplot as plt
>>> np.random.seed(1234567)
>>> x = stats.tukeylambda.rvs(-0.7, loc=2, scale=0.5, size=10000) + 1e4
Now we explore this data with a PPCC plot as well as the related
probability plot and Box-Cox normplot. A red line is drawn where we
expect the PPCC value to be maximal (at the shape parameter -0.7 used
above):
>>> fig = plt.figure(figsize=(12, 4))
>>> ax1 = fig.add_subplot(131)
>>> ax2 = fig.add_subplot(132)
>>> ax3 = fig.add_subplot(133)
>>> res = stats.probplot(x, plot=ax1)
>>> res = stats.boxcox_normplot(x, -5, 5, plot=ax2)
>>> res = stats.ppcc_plot(x, -5, 5, plot=ax3)
>>> ax3.vlines(-0.7, 0, 1, colors='r', label='Expected shape value')
>>> plt.show()
"""
if b <= a:
raise ValueError("`b` has to be larger than `a`.")
svals = np.linspace(a, b, num=N)
ppcc = np.empty_like(svals)
for k, sval in enumerate(svals):
_, r2 = probplot(x, sval, dist=dist, fit=True)
ppcc[k] = r2[-1]
if plot is not None:
plot.plot(svals, ppcc, 'x')
_add_axis_labels_title(plot, xlabel='Shape Values',
ylabel='Prob Plot Corr. Coef.',
title='(%s) PPCC Plot' % dist)
return svals, ppcc
def boxcox_llf(lmb, data):
r"""The boxcox log-likelihood function.
Parameters
----------
lmb : scalar
Parameter for Box-Cox transformation. See `boxcox` for details.
data : array_like
Data to calculate Box-Cox log-likelihood for. If `data` is
multi-dimensional, the log-likelihood is calculated along the first
axis.
Returns
-------
llf : float or ndarray
Box-Cox log-likelihood of `data` given `lmb`. A float for 1-D `data`,
an array otherwise.
See Also
--------
boxcox, probplot, boxcox_normplot, boxcox_normmax
Notes
-----
The Box-Cox log-likelihood function is defined here as
.. math::
llf = (\lambda - 1) \sum_i(\log(x_i)) -
N/2 \log(\sum_i (y_i - \bar{y})^2 / N),
where ``y`` is the Box-Cox transformed input data ``x``.
Examples
--------
>>> from scipy import stats
>>> import matplotlib.pyplot as plt
>>> from mpl_toolkits.axes_grid1.inset_locator import inset_axes
>>> np.random.seed(1245)
Generate some random variates and calculate Box-Cox log-likelihood values
for them for a range of ``lmbda`` values:
>>> x = stats.loggamma.rvs(5, loc=10, size=1000)
>>> lmbdas = np.linspace(-2, 10)
>>> llf = np.zeros(lmbdas.shape, dtype=float)
>>> for ii, lmbda in enumerate(lmbdas):
... llf[ii] = stats.boxcox_llf(lmbda, x)
Also find the optimal lmbda value with `boxcox`:
>>> x_most_normal, lmbda_optimal = stats.boxcox(x)
Plot the log-likelihood as function of lmbda. Add the optimal lmbda as a
horizontal line to check that that's really the optimum:
>>> fig = plt.figure()
>>> ax = fig.add_subplot(111)
>>> ax.plot(lmbdas, llf, 'b.-')
>>> ax.axhline(stats.boxcox_llf(lmbda_optimal, x), color='r')
>>> ax.set_xlabel('lmbda parameter')
>>> ax.set_ylabel('Box-Cox log-likelihood')
Now add some probability plots to show that where the log-likelihood is
maximized the data transformed with `boxcox` looks closest to normal:
>>> locs = [3, 10, 4] # 'lower left', 'center', 'lower right'
>>> for lmbda, loc in zip([-1, lmbda_optimal, 9], locs):
... xt = stats.boxcox(x, lmbda=lmbda)
... (osm, osr), (slope, intercept, r_sq) = stats.probplot(xt)
... ax_inset = inset_axes(ax, width="20%", height="20%", loc=loc)
... ax_inset.plot(osm, osr, 'c.', osm, slope*osm + intercept, 'k-')
... ax_inset.set_xticklabels([])
... ax_inset.set_yticklabels([])
... ax_inset.set_title('$\lambda=%1.2f$' % lmbda)
>>> plt.show()
"""
data = np.asarray(data)
N = data.shape[0]
if N == 0:
return np.nan
y = boxcox(data, lmb)
y_mean = np.mean(y, axis=0)
llf = (lmb - 1) * np.sum(np.log(data), axis=0)
llf -= N / 2.0 * np.log(np.sum((y - y_mean)**2. / N, axis=0))
return llf
def _boxcox_conf_interval(x, lmax, alpha):
# Need to find the lambda for which
# f(x,lmbda) >= f(x,lmax) - 0.5*chi^2_alpha;1
fac = 0.5 * distributions.chi2.ppf(1 - alpha, 1)
target = boxcox_llf(lmax, x) - fac
def rootfunc(lmbda, data, target):
return boxcox_llf(lmbda, data) - target
# Find positive endpoint of interval in which answer is to be found
newlm = lmax + 0.5
N = 0
while (rootfunc(newlm, x, target) > 0.0) and (N < 500):
newlm += 0.1
N += 1
if N == 500:
raise RuntimeError("Could not find endpoint.")
lmplus = optimize.brentq(rootfunc, lmax, newlm, args=(x, target))
# Now find negative interval in the same way
newlm = lmax - 0.5
N = 0
while (rootfunc(newlm, x, target) > 0.0) and (N < 500):
newlm -= 0.1
N += 1
if N == 500:
raise RuntimeError("Could not find endpoint.")
lmminus = optimize.brentq(rootfunc, newlm, lmax, args=(x, target))
return lmminus, lmplus
def boxcox(x, lmbda=None, alpha=None):
r"""
Return a positive dataset transformed by a Box-Cox power transformation.
Parameters
----------
x : ndarray
Input array. Should be 1-dimensional.
lmbda : {None, scalar}, optional
If `lmbda` is not None, do the transformation for that value.
If `lmbda` is None, find the lambda that maximizes the log-likelihood
function and return it as the second output argument.
alpha : {None, float}, optional
If ``alpha`` is not None, return the ``100 * (1-alpha)%`` confidence
interval for `lmbda` as the third output argument.
Must be between 0.0 and 1.0.
Returns
-------
boxcox : ndarray
Box-Cox power transformed array.
maxlog : float, optional
If the `lmbda` parameter is None, the second returned argument is
the lambda that maximizes the log-likelihood function.
(min_ci, max_ci) : tuple of float, optional
If `lmbda` parameter is None and ``alpha`` is not None, this returned
tuple of floats represents the minimum and maximum confidence limits
given ``alpha``.
See Also
--------
probplot, boxcox_normplot, boxcox_normmax, boxcox_llf
Notes
-----
The Box-Cox transform is given by::
y = (x**lmbda - 1) / lmbda, for lmbda > 0
log(x), for lmbda = 0
`boxcox` requires the input data to be positive. Sometimes a Box-Cox
transformation provides a shift parameter to achieve this; `boxcox` does
not. Such a shift parameter is equivalent to adding a positive constant to
`x` before calling `boxcox`.
The confidence limits returned when ``alpha`` is provided give the interval
where:
.. math::
llf(\hat{\lambda}) - llf(\lambda) < \frac{1}{2}\chi^2(1 - \alpha, 1),
with ``llf`` the log-likelihood function and :math:`\chi^2` the chi-squared
function.
References
----------
G.E.P. Box and D.R. Cox, "An Analysis of Transformations", Journal of the
Royal Statistical Society B, 26, 211-252 (1964).
Examples
--------
>>> from scipy import stats
>>> import matplotlib.pyplot as plt
We generate some random variates from a non-normal distribution and make a
probability plot for it, to show it is non-normal in the tails:
>>> fig = plt.figure()
>>> ax1 = fig.add_subplot(211)
>>> x = stats.loggamma.rvs(5, size=500) + 5
>>> prob = stats.probplot(x, dist=stats.norm, plot=ax1)
>>> ax1.set_xlabel('')
>>> ax1.set_title('Probplot against normal distribution')
We now use `boxcox` to transform the data so it's closest to normal:
>>> ax2 = fig.add_subplot(212)
>>> xt, _ = stats.boxcox(x)
>>> prob = stats.probplot(xt, dist=stats.norm, plot=ax2)
>>> ax2.set_title('Probplot after Box-Cox transformation')
>>> plt.show()
"""
x = np.asarray(x)
if x.size == 0:
return x
if any(x <= 0):
raise ValueError("Data must be positive.")
if lmbda is not None: # single transformation
return special.boxcox(x, lmbda)
# If lmbda=None, find the lmbda that maximizes the log-likelihood function.
lmax = boxcox_normmax(x, method='mle')
y = boxcox(x, lmax)
if alpha is None:
return y, lmax
else:
# Find confidence interval
interval = _boxcox_conf_interval(x, lmax, alpha)
return y, lmax, interval
def boxcox_normmax(x, brack=(-2.0, 2.0), method='pearsonr'):
"""Compute optimal Box-Cox transform parameter for input data.
Parameters
----------
x : array_like
Input array.
brack : 2-tuple, optional
The starting interval for a downhill bracket search with
`optimize.brent`. Note that this is in most cases not critical; the
final result is allowed to be outside this bracket.
method : str, optional
The method to determine the optimal transform parameter (`boxcox`
``lmbda`` parameter). Options are:
'pearsonr' (default)
Maximizes the Pearson correlation coefficient between
``y = boxcox(x)`` and the expected values for ``y`` if `x` would be
normally-distributed.
'mle'
Minimizes the log-likelihood `boxcox_llf`. This is the method used
in `boxcox`.
'all'
Use all optimization methods available, and return all results.
Useful to compare different methods.
Returns
-------
maxlog : float or ndarray
The optimal transform parameter found. An array instead of a scalar
for ``method='all'``.
See Also
--------
boxcox, boxcox_llf, boxcox_normplot
Examples
--------
>>> from scipy import stats
>>> import matplotlib.pyplot as plt
>>> np.random.seed(1234) # make this example reproducible
Generate some data and determine optimal ``lmbda`` in various ways:
>>> x = stats.loggamma.rvs(5, size=30) + 5
>>> y, lmax_mle = stats.boxcox(x)
>>> lmax_pearsonr = stats.boxcox_normmax(x)
>>> lmax_mle
7.177...
>>> lmax_pearsonr
7.916...
>>> stats.boxcox_normmax(x, method='all')
array([ 7.91667384, 7.17718692])
>>> fig = plt.figure()
>>> ax = fig.add_subplot(111)
>>> prob = stats.boxcox_normplot(x, -10, 10, plot=ax)
>>> ax.axvline(lmax_mle, color='r')
>>> ax.axvline(lmax_pearsonr, color='g', ls='--')
>>> plt.show()
"""
def _pearsonr(x, brack):
osm_uniform = _calc_uniform_order_statistic_medians(len(x))
xvals = distributions.norm.ppf(osm_uniform)
def _eval_pearsonr(lmbda, xvals, samps):
# This function computes the x-axis values of the probability plot
# and computes a linear regression (including the correlation) and
# returns ``1 - r`` so that a minimization function maximizes the
# correlation.
y = boxcox(samps, lmbda)
yvals = np.sort(y)
r, prob = stats.pearsonr(xvals, yvals)
return 1 - r
return optimize.brent(_eval_pearsonr, brack=brack, args=(xvals, x))
def _mle(x, brack):
def _eval_mle(lmb, data):
# function to minimize
return -boxcox_llf(lmb, data)
return optimize.brent(_eval_mle, brack=brack, args=(x,))
def _all(x, brack):
maxlog = np.zeros(2, dtype=float)
maxlog[0] = _pearsonr(x, brack)
maxlog[1] = _mle(x, brack)
return maxlog
methods = {'pearsonr': _pearsonr,
'mle': _mle,
'all': _all}
if method not in methods.keys():
raise ValueError("Method %s not recognized." % method)
optimfunc = methods[method]
return optimfunc(x, brack)
def boxcox_normplot(x, la, lb, plot=None, N=80):
"""Compute parameters for a Box-Cox normality plot, optionally show it.
A Box-Cox normality plot shows graphically what the best transformation
parameter is to use in `boxcox` to obtain a distribution that is close
to normal.
Parameters
----------
x : array_like
Input array.
la, lb : scalar
The lower and upper bounds for the ``lmbda`` values to pass to `boxcox`
for Box-Cox transformations. These are also the limits of the
horizontal axis of the plot if that is generated.
plot : object, optional
If given, plots the quantiles and least squares fit.
`plot` is an object that has to have methods "plot" and "text".
The `matplotlib.pyplot` module or a Matplotlib Axes object can be used,
or a custom object with the same methods.
Default is None, which means that no plot is created.
N : int, optional
Number of points on the horizontal axis (equally distributed from
`la` to `lb`).
Returns
-------
lmbdas : ndarray
The ``lmbda`` values for which a Box-Cox transform was done.
ppcc : ndarray
Probability Plot Correlelation Coefficient, as obtained from `probplot`
when fitting the Box-Cox transformed input `x` against a normal
distribution.
See Also
--------
probplot, boxcox, boxcox_normmax, boxcox_llf, ppcc_max
Notes
-----
Even if `plot` is given, the figure is not shown or saved by
`boxcox_normplot`; ``plt.show()`` or ``plt.savefig('figname.png')``
should be used after calling `probplot`.
Examples
--------
>>> from scipy import stats
>>> import matplotlib.pyplot as plt
Generate some non-normally distributed data, and create a Box-Cox plot:
>>> x = stats.loggamma.rvs(5, size=500) + 5
>>> fig = plt.figure()
>>> ax = fig.add_subplot(111)
>>> prob = stats.boxcox_normplot(x, -20, 20, plot=ax)
Determine and plot the optimal ``lmbda`` to transform ``x`` and plot it in
the same plot:
>>> _, maxlog = stats.boxcox(x)
>>> ax.axvline(maxlog, color='r')
>>> plt.show()
"""
x = np.asarray(x)
if x.size == 0:
return x
if lb <= la:
raise ValueError("`lb` has to be larger than `la`.")
lmbdas = np.linspace(la, lb, num=N)
ppcc = lmbdas * 0.0
for i, val in enumerate(lmbdas):
# Determine for each lmbda the correlation coefficient of transformed x
z = boxcox(x, lmbda=val)
_, r2 = probplot(z, dist='norm', fit=True)
ppcc[i] = r2[-1]
if plot is not None:
plot.plot(lmbdas, ppcc, 'x')
_add_axis_labels_title(plot, xlabel='$\\lambda$',
ylabel='Prob Plot Corr. Coef.',
title='Box-Cox Normality Plot')
return lmbdas, ppcc
def shapiro(x):
"""
Perform the Shapiro-Wilk test for normality.
The Shapiro-Wilk test tests the null hypothesis that the
data was drawn from a normal distribution.
Parameters
----------
x : array_like
Array of sample data.
Returns
-------
W : float
The test statistic.
p-value : float
The p-value for the hypothesis test.
See Also
--------
anderson : The Anderson-Darling test for normality
kstest : The Kolmogorov-Smirnov test for goodness of fit.
Notes
-----
The algorithm used is described in [4]_ but censoring parameters as
described are not implemented. For N > 5000 the W test statistic is accurate
but the p-value may not be.
The chance of rejecting the null hypothesis when it is true is close to 5%
regardless of sample size.
References
----------
.. [1] http://www.itl.nist.gov/div898/handbook/prc/section2/prc213.htm
.. [2] Shapiro, S. S. & Wilk, M.B (1965). An analysis of variance test for
normality (complete samples), Biometrika, Vol. 52, pp. 591-611.
.. [3] Razali, N. M. & Wah, Y. B. (2011) Power comparisons of Shapiro-Wilk,
Kolmogorov-Smirnov, Lilliefors and Anderson-Darling tests, Journal of
Statistical Modeling and Analytics, Vol. 2, pp. 21-33.
.. [4] ALGORITHM AS R94 APPL. STATIST. (1995) VOL. 44, NO. 4.
Examples
--------
>>> from scipy import stats
>>> np.random.seed(12345678)
>>> x = stats.norm.rvs(loc=5, scale=3, size=100)
>>> stats.shapiro(x)
(0.9772805571556091, 0.08144091814756393)
"""
x = np.ravel(x)
N = len(x)
if N < 3:
raise ValueError("Data must be at least length 3.")
a = zeros(N, 'f')
init = 0
y = sort(x)
a, w, pw, ifault = statlib.swilk(y, a[:N//2], init)
if ifault not in [0, 2]:
warnings.warn("Input data for shapiro has range zero. The results "
"may not be accurate.")
if N > 5000:
warnings.warn("p-value may not be accurate for N > 5000.")
return w, pw
# Values from Stephens, M A, "EDF Statistics for Goodness of Fit and
# Some Comparisons", Journal of he American Statistical
# Association, Vol. 69, Issue 347, Sept. 1974, pp 730-737
_Avals_norm = array([0.576, 0.656, 0.787, 0.918, 1.092])
_Avals_expon = array([0.922, 1.078, 1.341, 1.606, 1.957])
# From Stephens, M A, "Goodness of Fit for the Extreme Value Distribution",
# Biometrika, Vol. 64, Issue 3, Dec. 1977, pp 583-588.
_Avals_gumbel = array([0.474, 0.637, 0.757, 0.877, 1.038])
# From Stephens, M A, "Tests of Fit for the Logistic Distribution Based
# on the Empirical Distribution Function.", Biometrika,
# Vol. 66, Issue 3, Dec. 1979, pp 591-595.
_Avals_logistic = array([0.426, 0.563, 0.660, 0.769, 0.906, 1.010])
AndersonResult = namedtuple('AndersonResult', ('statistic',
'critical_values',
'significance_level'))
def anderson(x, dist='norm'):
"""
Anderson-Darling test for data coming from a particular distribution
The Anderson-Darling tests the null hypothesis that a sample is
drawn from a population that follows a particular distribution.
For the Anderson-Darling test, the critical values depend on
which distribution is being tested against. This function works
for normal, exponential, logistic, or Gumbel (Extreme Value
Type I) distributions.
Parameters
----------
x : array_like
array of sample data
dist : {'norm','expon','logistic','gumbel','gumbel_l', gumbel_r',
'extreme1'}, optional
the type of distribution to test against. The default is 'norm'
and 'extreme1', 'gumbel_l' and 'gumbel' are synonyms.
Returns
-------
statistic : float
The Anderson-Darling test statistic
critical_values : list
The critical values for this distribution
significance_level : list
The significance levels for the corresponding critical values
in percents. The function returns critical values for a
differing set of significance levels depending on the
distribution that is being tested against.
See Also
--------
kstest : The Kolmogorov-Smirnov test for goodness-of-fit.
Notes
-----
Critical values provided are for the following significance levels:
normal/exponenential
15%, 10%, 5%, 2.5%, 1%
logistic
25%, 10%, 5%, 2.5%, 1%, 0.5%
Gumbel
25%, 10%, 5%, 2.5%, 1%
If the returned statistic is larger than these critical values then
for the corresponding significance level, the null hypothesis that
the data come from the chosen distribution can be rejected.
The returned statistic is referred to as 'A2' in the references.
References
----------
.. [1] http://www.itl.nist.gov/div898/handbook/prc/section2/prc213.htm
.. [2] Stephens, M. A. (1974). EDF Statistics for Goodness of Fit and
Some Comparisons, Journal of the American Statistical Association,
Vol. 69, pp. 730-737.
.. [3] Stephens, M. A. (1976). Asymptotic Results for Goodness-of-Fit
Statistics with Unknown Parameters, Annals of Statistics, Vol. 4,
pp. 357-369.
.. [4] Stephens, M. A. (1977). Goodness of Fit for the Extreme Value
Distribution, Biometrika, Vol. 64, pp. 583-588.
.. [5] Stephens, M. A. (1977). Goodness of Fit with Special Reference
to Tests for Exponentiality , Technical Report No. 262,
Department of Statistics, Stanford University, Stanford, CA.
.. [6] Stephens, M. A. (1979). Tests of Fit for the Logistic Distribution
Based on the Empirical Distribution Function, Biometrika, Vol. 66,
pp. 591-595.
"""
if dist not in ['norm', 'expon', 'gumbel', 'gumbel_l',
'gumbel_r', 'extreme1', 'logistic']:
raise ValueError("Invalid distribution; dist must be 'norm', "
"'expon', 'gumbel', 'extreme1' or 'logistic'.")
y = sort(x)
xbar = np.mean(x, axis=0)
N = len(y)
if dist == 'norm':
s = np.std(x, ddof=1, axis=0)
w = (y - xbar) / s
logcdf = distributions.norm.logcdf(w)
logsf = distributions.norm.logsf(w)
sig = array([15, 10, 5, 2.5, 1])
critical = around(_Avals_norm / (1.0 + 4.0/N - 25.0/N/N), 3)
elif dist == 'expon':
w = y / xbar
logcdf = distributions.expon.logcdf(w)
logsf = distributions.expon.logsf(w)
sig = array([15, 10, 5, 2.5, 1])
critical = around(_Avals_expon / (1.0 + 0.6/N), 3)
elif dist == 'logistic':
def rootfunc(ab, xj, N):
a, b = ab
tmp = (xj - a) / b
tmp2 = exp(tmp)
val = [np.sum(1.0/(1+tmp2), axis=0) - 0.5*N,
np.sum(tmp*(1.0-tmp2)/(1+tmp2), axis=0) + N]
return array(val)
sol0 = array([xbar, np.std(x, ddof=1, axis=0)])
sol = optimize.fsolve(rootfunc, sol0, args=(x, N), xtol=1e-5)
w = (y - sol[0]) / sol[1]
logcdf = distributions.logistic.logcdf(w)
logsf = distributions.logistic.logsf(w)
sig = array([25, 10, 5, 2.5, 1, 0.5])
critical = around(_Avals_logistic / (1.0 + 0.25/N), 3)
elif dist == 'gumbel_r':
xbar, s = distributions.gumbel_r.fit(x)
w = (y - xbar) / s
logcdf = distributions.gumbel_r.logcdf(w)
logsf = distributions.gumbel_r.logsf(w)
sig = array([25, 10, 5, 2.5, 1])
critical = around(_Avals_gumbel / (1.0 + 0.2/sqrt(N)), 3)
else: # (dist == 'gumbel') or (dist == 'gumbel_l') or (dist == 'extreme1')
xbar, s = distributions.gumbel_l.fit(x)
w = (y - xbar) / s
logcdf = distributions.gumbel_l.logcdf(w)
logsf = distributions.gumbel_l.logsf(w)
sig = array([25, 10, 5, 2.5, 1])
critical = around(_Avals_gumbel / (1.0 + 0.2/sqrt(N)), 3)
i = arange(1, N + 1)
A2 = -N - np.sum((2*i - 1.0) / N * (logcdf + logsf[::-1]), axis=0)
return AndersonResult(A2, critical, sig)
def _anderson_ksamp_midrank(samples, Z, Zstar, k, n, N):
"""
Compute A2akN equation 7 of Scholz and Stephens.
Parameters
----------
samples : sequence of 1-D array_like
Array of sample arrays.
Z : array_like
Sorted array of all observations.
Zstar : array_like
Sorted array of unique observations.
k : int
Number of samples.
n : array_like
Number of observations in each sample.
N : int
Total number of observations.
Returns
-------
A2aKN : float
The A2aKN statistics of Scholz and Stephens 1987.
"""
A2akN = 0.
Z_ssorted_left = Z.searchsorted(Zstar, 'left')
if N == Zstar.size:
lj = 1.
else:
lj = Z.searchsorted(Zstar, 'right') - Z_ssorted_left
Bj = Z_ssorted_left + lj / 2.
for i in arange(0, k):
s = np.sort(samples[i])
s_ssorted_right = s.searchsorted(Zstar, side='right')
Mij = s_ssorted_right.astype(float)
fij = s_ssorted_right - s.searchsorted(Zstar, 'left')
Mij -= fij / 2.
inner = lj / float(N) * (N*Mij - Bj*n[i])**2 / (Bj*(N - Bj) - N*lj/4.)
A2akN += inner.sum() / n[i]
A2akN *= (N - 1.) / N
return A2akN
def _anderson_ksamp_right(samples, Z, Zstar, k, n, N):
"""
Compute A2akN equation 6 of Scholz & Stephens.
Parameters
----------
samples : sequence of 1-D array_like
Array of sample arrays.
Z : array_like
Sorted array of all observations.
Zstar : array_like
Sorted array of unique observations.
k : int
Number of samples.
n : array_like
Number of observations in each sample.
N : int
Total number of observations.
Returns
-------
A2KN : float
The A2KN statistics of Scholz and Stephens 1987.
"""
A2kN = 0.
lj = Z.searchsorted(Zstar[:-1], 'right') - Z.searchsorted(Zstar[:-1],
'left')
Bj = lj.cumsum()
for i in arange(0, k):
s = np.sort(samples[i])
Mij = s.searchsorted(Zstar[:-1], side='right')
inner = lj / float(N) * (N * Mij - Bj * n[i])**2 / (Bj * (N - Bj))
A2kN += inner.sum() / n[i]
return A2kN
Anderson_ksampResult = namedtuple('Anderson_ksampResult',
('statistic', 'critical_values',
'significance_level'))
def anderson_ksamp(samples, midrank=True):
"""The Anderson-Darling test for k-samples.
The k-sample Anderson-Darling test is a modification of the
one-sample Anderson-Darling test. It tests the null hypothesis
that k-samples are drawn from the same population without having
to specify the distribution function of that population. The
critical values depend on the number of samples.
Parameters
----------
samples : sequence of 1-D array_like
Array of sample data in arrays.
midrank : bool, optional
Type of Anderson-Darling test which is computed. Default
(True) is the midrank test applicable to continuous and
discrete populations. If False, the right side empirical
distribution is used.
Returns
-------
statistic : float
Normalized k-sample Anderson-Darling test statistic.
critical_values : array
The critical values for significance levels 25%, 10%, 5%, 2.5%, 1%.
significance_level : float
An approximate significance level at which the null hypothesis for the
provided samples can be rejected.
Raises
------
ValueError
If less than 2 samples are provided, a sample is empty, or no
distinct observations are in the samples.
See Also
--------
ks_2samp : 2 sample Kolmogorov-Smirnov test
anderson : 1 sample Anderson-Darling test
Notes
-----
[1]_ Defines three versions of the k-sample Anderson-Darling test:
one for continuous distributions and two for discrete
distributions, in which ties between samples may occur. The
default of this routine is to compute the version based on the
midrank empirical distribution function. This test is applicable
to continuous and discrete data. If midrank is set to False, the
right side empirical distribution is used for a test for discrete
data. According to [1]_, the two discrete test statistics differ
only slightly if a few collisions due to round-off errors occur in
the test not adjusted for ties between samples.
.. versionadded:: 0.14.0
References
----------
.. [1] Scholz, F. W and Stephens, M. A. (1987), K-Sample
Anderson-Darling Tests, Journal of the American Statistical
Association, Vol. 82, pp. 918-924.
Examples
--------
>>> from scipy import stats
>>> np.random.seed(314159)
The null hypothesis that the two random samples come from the same
distribution can be rejected at the 5% level because the returned
test value is greater than the critical value for 5% (1.961) but
not at the 2.5% level. The interpolation gives an approximate
significance level of 3.1%:
>>> stats.anderson_ksamp([np.random.normal(size=50),
... np.random.normal(loc=0.5, size=30)])
(2.4615796189876105,
array([ 0.325, 1.226, 1.961, 2.718, 3.752]),
0.03134990135800783)
The null hypothesis cannot be rejected for three samples from an
identical distribution. The approximate p-value (87%) has to be
computed by extrapolation and may not be very accurate:
>>> stats.anderson_ksamp([np.random.normal(size=50),
... np.random.normal(size=30), np.random.normal(size=20)])
(-0.73091722665244196,
array([ 0.44925884, 1.3052767 , 1.9434184 , 2.57696569, 3.41634856]),
0.8789283903979661)
"""
k = len(samples)
if (k < 2):
raise ValueError("anderson_ksamp needs at least two samples")
samples = list(map(np.asarray, samples))
Z = np.sort(np.hstack(samples))
N = Z.size
Zstar = np.unique(Z)
if Zstar.size < 2:
raise ValueError("anderson_ksamp needs more than one distinct "
"observation")
n = np.array([sample.size for sample in samples])
if any(n == 0):
raise ValueError("anderson_ksamp encountered sample without "
"observations")
if midrank:
A2kN = _anderson_ksamp_midrank(samples, Z, Zstar, k, n, N)
else:
A2kN = _anderson_ksamp_right(samples, Z, Zstar, k, n, N)
H = (1. / n).sum()
hs_cs = (1. / arange(N - 1, 1, -1)).cumsum()
h = hs_cs[-1] + 1
g = (hs_cs / arange(2, N)).sum()
a = (4*g - 6) * (k - 1) + (10 - 6*g)*H
b = (2*g - 4)*k**2 + 8*h*k + (2*g - 14*h - 4)*H - 8*h + 4*g - 6
c = (6*h + 2*g - 2)*k**2 + (4*h - 4*g + 6)*k + (2*h - 6)*H + 4*h
d = (2*h + 6)*k**2 - 4*h*k
sigmasq = (a*N**3 + b*N**2 + c*N + d) / ((N - 1.) * (N - 2.) * (N - 3.))
m = k - 1
A2 = (A2kN - m) / math.sqrt(sigmasq)
# The b_i values are the interpolation coefficients from Table 2
# of Scholz and Stephens 1987
b0 = np.array([0.675, 1.281, 1.645, 1.96, 2.326])
b1 = np.array([-0.245, 0.25, 0.678, 1.149, 1.822])
b2 = np.array([-0.105, -0.305, -0.362, -0.391, -0.396])
critical = b0 + b1 / math.sqrt(m) + b2 / m
pf = np.polyfit(critical, log(np.array([0.25, 0.1, 0.05, 0.025, 0.01])), 2)
if A2 < critical.min() or A2 > critical.max():
warnings.warn("approximate p-value will be computed by extrapolation")
try:
p = math.exp(np.polyval(pf, A2))
except (OverflowError,):
p = float("inf")
return Anderson_ksampResult(A2, critical, p)
AnsariResult = namedtuple('AnsariResult', ('statistic', 'pvalue'))
def ansari(x, y):
"""
Perform the Ansari-Bradley test for equal scale parameters
The Ansari-Bradley test is a non-parametric test for the equality
of the scale parameter of the distributions from which two
samples were drawn.
Parameters
----------
x, y : array_like
arrays of sample data
Returns
-------
statistic : float
The Ansari-Bradley test statistic
pvalue : float
The p-value of the hypothesis test
See Also
--------
fligner : A non-parametric test for the equality of k variances
mood : A non-parametric test for the equality of two scale parameters
Notes
-----
The p-value given is exact when the sample sizes are both less than
55 and there are no ties, otherwise a normal approximation for the
p-value is used.
References
----------
.. [1] Sprent, Peter and N.C. Smeeton. Applied nonparametric statistical
methods. 3rd ed. Chapman and Hall/CRC. 2001. Section 5.8.2.
"""
x, y = asarray(x), asarray(y)
n = len(x)
m = len(y)
if m < 1:
raise ValueError("Not enough other observations.")
if n < 1:
raise ValueError("Not enough test observations.")
N = m + n
xy = r_[x, y] # combine
rank = stats.rankdata(xy)
symrank = amin(array((rank, N - rank + 1)), 0)
AB = np.sum(symrank[:n], axis=0)
uxy = unique(xy)
repeats = (len(uxy) != len(xy))
exact = ((m < 55) and (n < 55) and not repeats)
if repeats and (m < 55 or n < 55):
warnings.warn("Ties preclude use of exact statistic.")
if exact:
astart, a1, ifault = statlib.gscale(n, m)
ind = AB - astart
total = np.sum(a1, axis=0)
if ind < len(a1)/2.0:
cind = int(ceil(ind))
if ind == cind:
pval = 2.0 * np.sum(a1[:cind+1], axis=0) / total
else:
pval = 2.0 * np.sum(a1[:cind], axis=0) / total
else:
find = int(floor(ind))
if ind == floor(ind):
pval = 2.0 * np.sum(a1[find:], axis=0) / total
else:
pval = 2.0 * np.sum(a1[find+1:], axis=0) / total
return AnsariResult(AB, min(1.0, pval))
# otherwise compute normal approximation
if N % 2: # N odd
mnAB = n * (N+1.0)**2 / 4.0 / N
varAB = n * m * (N+1.0) * (3+N**2) / (48.0 * N**2)
else:
mnAB = n * (N+2.0) / 4.0
varAB = m * n * (N+2) * (N-2.0) / 48 / (N-1.0)
if repeats: # adjust variance estimates
# compute np.sum(tj * rj**2,axis=0)
fac = np.sum(symrank**2, axis=0)
if N % 2: # N odd
varAB = m * n * (16*N*fac - (N+1)**4) / (16.0 * N**2 * (N-1))
else: # N even
varAB = m * n * (16*fac - N*(N+2)**2) / (16.0 * N * (N-1))
z = (AB - mnAB) / sqrt(varAB)
pval = distributions.norm.sf(abs(z)) * 2.0
return AnsariResult(AB, pval)
BartlettResult = namedtuple('BartlettResult', ('statistic', 'pvalue'))
def bartlett(*args):
"""
Perform Bartlett's test for equal variances
Bartlett's test tests the null hypothesis that all input samples
are from populations with equal variances. For samples
from significantly non-normal populations, Levene's test
`levene` is more robust.
Parameters
----------
sample1, sample2,... : array_like
arrays of sample data. May be different lengths.
Returns
-------
statistic : float
The test statistic.
pvalue : float
The p-value of the test.
See Also
--------
fligner : A non-parametric test for the equality of k variances
levene : A robust parametric test for equality of k variances
Notes
-----
Conover et al. (1981) examine many of the existing parametric and
nonparametric tests by extensive simulations and they conclude that the
tests proposed by Fligner and Killeen (1976) and Levene (1960) appear to be
superior in terms of robustness of departures from normality and power [3]_.
References
----------
.. [1] http://www.itl.nist.gov/div898/handbook/eda/section3/eda357.htm
.. [2] Snedecor, George W. and Cochran, William G. (1989), Statistical
Methods, Eighth Edition, Iowa State University Press.
.. [3] Park, C. and Lindsay, B. G. (1999). Robust Scale Estimation and
Hypothesis Testing based on Quadratic Inference Function. Technical
Report #99-03, Center for Likelihood Studies, Pennsylvania State
University.
.. [4] Bartlett, M. S. (1937). Properties of Sufficiency and Statistical
Tests. Proceedings of the Royal Society of London. Series A,
Mathematical and Physical Sciences, Vol. 160, No.901, pp. 268-282.
"""
# Handle empty input
for a in args:
if np.asanyarray(a).size == 0:
return BartlettResult(np.nan, np.nan)
k = len(args)
if k < 2:
raise ValueError("Must enter at least two input sample vectors.")
Ni = zeros(k)
ssq = zeros(k, 'd')
for j in range(k):
Ni[j] = len(args[j])
ssq[j] = np.var(args[j], ddof=1)
Ntot = np.sum(Ni, axis=0)
spsq = np.sum((Ni - 1)*ssq, axis=0) / (1.0*(Ntot - k))
numer = (Ntot*1.0 - k) * log(spsq) - np.sum((Ni - 1.0)*log(ssq), axis=0)
denom = 1.0 + 1.0/(3*(k - 1)) * ((np.sum(1.0/(Ni - 1.0), axis=0)) -
1.0/(Ntot - k))
T = numer / denom
pval = distributions.chi2.sf(T, k - 1) # 1 - cdf
return BartlettResult(T, pval)
LeveneResult = namedtuple('LeveneResult', ('statistic', 'pvalue'))
def levene(*args, **kwds):
"""
Perform Levene test for equal variances.
The Levene test tests the null hypothesis that all input samples
are from populations with equal variances. Levene's test is an
alternative to Bartlett's test `bartlett` in the case where
there are significant deviations from normality.
Parameters
----------
sample1, sample2, ... : array_like
The sample data, possibly with different lengths
center : {'mean', 'median', 'trimmed'}, optional
Which function of the data to use in the test. The default
is 'median'.
proportiontocut : float, optional
When `center` is 'trimmed', this gives the proportion of data points
to cut from each end. (See `scipy.stats.trim_mean`.)
Default is 0.05.
Returns
-------
statistic : float
The test statistic.
pvalue : float
The p-value for the test.
Notes
-----
Three variations of Levene's test are possible. The possibilities
and their recommended usages are:
* 'median' : Recommended for skewed (non-normal) distributions>
* 'mean' : Recommended for symmetric, moderate-tailed distributions.
* 'trimmed' : Recommended for heavy-tailed distributions.
References
----------
.. [1] http://www.itl.nist.gov/div898/handbook/eda/section3/eda35a.htm
.. [2] Levene, H. (1960). In Contributions to Probability and Statistics:
Essays in Honor of Harold Hotelling, I. Olkin et al. eds.,
Stanford University Press, pp. 278-292.
.. [3] Brown, M. B. and Forsythe, A. B. (1974), Journal of the American
Statistical Association, 69, 364-367
"""
# Handle keyword arguments.
center = 'median'
proportiontocut = 0.05
for kw, value in kwds.items():
if kw not in ['center', 'proportiontocut']:
raise TypeError("levene() got an unexpected keyword "
"argument '%s'" % kw)
if kw == 'center':
center = value
else:
proportiontocut = value
k = len(args)
if k < 2:
raise ValueError("Must enter at least two input sample vectors.")
Ni = zeros(k)
Yci = zeros(k, 'd')
if center not in ['mean', 'median', 'trimmed']:
raise ValueError("Keyword argument <center> must be 'mean', 'median'"
" or 'trimmed'.")
if center == 'median':
func = lambda x: np.median(x, axis=0)
elif center == 'mean':
func = lambda x: np.mean(x, axis=0)
else: # center == 'trimmed'
args = tuple(stats.trimboth(np.sort(arg), proportiontocut)
for arg in args)
func = lambda x: np.mean(x, axis=0)
for j in range(k):
Ni[j] = len(args[j])
Yci[j] = func(args[j])
Ntot = np.sum(Ni, axis=0)
# compute Zij's
Zij = [None] * k
for i in range(k):
Zij[i] = abs(asarray(args[i]) - Yci[i])
# compute Zbari
Zbari = zeros(k, 'd')
Zbar = 0.0
for i in range(k):
Zbari[i] = np.mean(Zij[i], axis=0)
Zbar += Zbari[i] * Ni[i]
Zbar /= Ntot
numer = (Ntot - k) * np.sum(Ni * (Zbari - Zbar)**2, axis=0)
# compute denom_variance
dvar = 0.0
for i in range(k):
dvar += np.sum((Zij[i] - Zbari[i])**2, axis=0)
denom = (k - 1.0) * dvar
W = numer / denom
pval = distributions.f.sf(W, k-1, Ntot-k) # 1 - cdf
return LeveneResult(W, pval)
def binom_test(x, n=None, p=0.5, alternative='two-sided'):
"""
Perform a test that the probability of success is p.
This is an exact, two-sided test of the null hypothesis
that the probability of success in a Bernoulli experiment
is `p`.
Parameters
----------
x : integer or array_like
the number of successes, or if x has length 2, it is the
number of successes and the number of failures.
n : integer
the number of trials. This is ignored if x gives both the
number of successes and failures
p : float, optional
The hypothesized probability of success. 0 <= p <= 1. The
default value is p = 0.5
alternative : {'two-sided', 'greater', 'less'}, optional
Indicates the alternative hypothesis. The default value is
'two-sided'.
Returns
-------
p-value : float
The p-value of the hypothesis test
References
----------
.. [1] http://en.wikipedia.org/wiki/Binomial_test
"""
x = atleast_1d(x).astype(np.integer)
if len(x) == 2:
n = x[1] + x[0]
x = x[0]
elif len(x) == 1:
x = x[0]
if n is None or n < x:
raise ValueError("n must be >= x")
n = np.int_(n)
else:
raise ValueError("Incorrect length for x.")
if (p > 1.0) or (p < 0.0):
raise ValueError("p must be in range [0,1]")
if alternative not in ('two-sided', 'less', 'greater'):
raise ValueError("alternative not recognized\n"
"should be 'two-sided', 'less' or 'greater'")
if alternative == 'less':
pval = distributions.binom.cdf(x, n, p)
return pval
if alternative == 'greater':
pval = distributions.binom.sf(x-1, n, p)
return pval
# if alternative was neither 'less' nor 'greater', then it's 'two-sided'
d = distributions.binom.pmf(x, n, p)
rerr = 1 + 1e-7
if x == p * n:
# special case as shortcut, would also be handled by `else` below
pval = 1.
elif x < p * n:
i = np.arange(np.ceil(p * n), n+1)
y = np.sum(distributions.binom.pmf(i, n, p) <= d*rerr, axis=0)
pval = (distributions.binom.cdf(x, n, p) +
distributions.binom.sf(n - y, n, p))
else:
i = np.arange(np.floor(p*n) + 1)
y = np.sum(distributions.binom.pmf(i, n, p) <= d*rerr, axis=0)
pval = (distributions.binom.cdf(y-1, n, p) +
distributions.binom.sf(x-1, n, p))
return min(1.0, pval)
def _apply_func(x, g, func):
# g is list of indices into x
# separating x into different groups
# func should be applied over the groups
g = unique(r_[0, g, len(x)])
output = []
for k in range(len(g) - 1):
output.append(func(x[g[k]:g[k+1]]))
return asarray(output)
FlignerResult = namedtuple('FlignerResult', ('statistic', 'pvalue'))
def fligner(*args, **kwds):
"""
Perform Fligner-Killeen test for equality of variance.
Fligner's test tests the null hypothesis that all input samples
are from populations with equal variances. Fligner-Killeen's test is
distribution free when populations are identical [2]_.
Parameters
----------
sample1, sample2, ... : array_like
Arrays of sample data. Need not be the same length.
center : {'mean', 'median', 'trimmed'}, optional
Keyword argument controlling which function of the data is used in
computing the test statistic. The default is 'median'.
proportiontocut : float, optional
When `center` is 'trimmed', this gives the proportion of data points
to cut from each end. (See `scipy.stats.trim_mean`.)
Default is 0.05.
Returns
-------
statistic : float
The test statistic.
pvalue : float
The p-value for the hypothesis test.
See Also
--------
bartlett : A parametric test for equality of k variances in normal samples
levene : A robust parametric test for equality of k variances
Notes
-----
As with Levene's test there are three variants of Fligner's test that
differ by the measure of central tendency used in the test. See `levene`
for more information.
Conover et al. (1981) examine many of the existing parametric and
nonparametric tests by extensive simulations and they conclude that the
tests proposed by Fligner and Killeen (1976) and Levene (1960) appear to be
superior in terms of robustness of departures from normality and power [3]_.
References
----------
.. [1] Park, C. and Lindsay, B. G. (1999). Robust Scale Estimation and
Hypothesis Testing based on Quadratic Inference Function. Technical
Report #99-03, Center for Likelihood Studies, Pennsylvania State
University.
http://cecas.clemson.edu/~cspark/cv/paper/qif/draftqif2.pdf
.. [2] Fligner, M.A. and Killeen, T.J. (1976). Distribution-free two-sample
tests for scale. 'Journal of the American Statistical Association.'
71(353), 210-213.
.. [3] Park, C. and Lindsay, B. G. (1999). Robust Scale Estimation and
Hypothesis Testing based on Quadratic Inference Function. Technical
Report #99-03, Center for Likelihood Studies, Pennsylvania State
University.
.. [4] Conover, W. J., Johnson, M. E. and Johnson M. M. (1981). A
comparative study of tests for homogeneity of variances, with
applications to the outer continental shelf biding data.
Technometrics, 23(4), 351-361.
"""
# Handle empty input
for a in args:
if np.asanyarray(a).size == 0:
return FlignerResult(np.nan, np.nan)
# Handle keyword arguments.
center = 'median'
proportiontocut = 0.05
for kw, value in kwds.items():
if kw not in ['center', 'proportiontocut']:
raise TypeError("fligner() got an unexpected keyword "
"argument '%s'" % kw)
if kw == 'center':
center = value
else:
proportiontocut = value
k = len(args)
if k < 2:
raise ValueError("Must enter at least two input sample vectors.")
if center not in ['mean', 'median', 'trimmed']:
raise ValueError("Keyword argument <center> must be 'mean', 'median'"
" or 'trimmed'.")
if center == 'median':
func = lambda x: np.median(x, axis=0)
elif center == 'mean':
func = lambda x: np.mean(x, axis=0)
else: # center == 'trimmed'
args = tuple(stats.trimboth(arg, proportiontocut) for arg in args)
func = lambda x: np.mean(x, axis=0)
Ni = asarray([len(args[j]) for j in range(k)])
Yci = asarray([func(args[j]) for j in range(k)])
Ntot = np.sum(Ni, axis=0)
# compute Zij's
Zij = [abs(asarray(args[i]) - Yci[i]) for i in range(k)]
allZij = []
g = [0]
for i in range(k):
allZij.extend(list(Zij[i]))
g.append(len(allZij))
ranks = stats.rankdata(allZij)
a = distributions.norm.ppf(ranks / (2*(Ntot + 1.0)) + 0.5)
# compute Aibar
Aibar = _apply_func(a, g, np.sum) / Ni
anbar = np.mean(a, axis=0)
varsq = np.var(a, axis=0, ddof=1)
Xsq = np.sum(Ni * (asarray(Aibar) - anbar)**2.0, axis=0) / varsq
pval = distributions.chi2.sf(Xsq, k - 1) # 1 - cdf
return FlignerResult(Xsq, pval)
def mood(x, y, axis=0):
"""
Perform Mood's test for equal scale parameters.
Mood's two-sample test for scale parameters is a non-parametric
test for the null hypothesis that two samples are drawn from the
same distribution with the same scale parameter.
Parameters
----------
x, y : array_like
Arrays of sample data.
axis : int, optional
The axis along which the samples are tested. `x` and `y` can be of
different length along `axis`.
If `axis` is None, `x` and `y` are flattened and the test is done on
all values in the flattened arrays.
Returns
-------
z : scalar or ndarray
The z-score for the hypothesis test. For 1-D inputs a scalar is
returned.
p-value : scalar ndarray
The p-value for the hypothesis test.
See Also
--------
fligner : A non-parametric test for the equality of k variances
ansari : A non-parametric test for the equality of 2 variances
bartlett : A parametric test for equality of k variances in normal samples
levene : A parametric test for equality of k variances
Notes
-----
The data are assumed to be drawn from probability distributions ``f(x)``
and ``f(x/s) / s`` respectively, for some probability density function f.
The null hypothesis is that ``s == 1``.
For multi-dimensional arrays, if the inputs are of shapes
``(n0, n1, n2, n3)`` and ``(n0, m1, n2, n3)``, then if ``axis=1``, the
resulting z and p values will have shape ``(n0, n2, n3)``. Note that
``n1`` and ``m1`` don't have to be equal, but the other dimensions do.
Examples
--------
>>> from scipy import stats
>>> np.random.seed(1234)
>>> x2 = np.random.randn(2, 45, 6, 7)
>>> x1 = np.random.randn(2, 30, 6, 7)
>>> z, p = stats.mood(x1, x2, axis=1)
>>> p.shape
(2, 6, 7)
Find the number of points where the difference in scale is not significant:
>>> (p > 0.1).sum()
74
Perform the test with different scales:
>>> x1 = np.random.randn(2, 30)
>>> x2 = np.random.randn(2, 35) * 10.0
>>> stats.mood(x1, x2, axis=1)
(array([-5.7178125 , -5.25342163]), array([ 1.07904114e-08, 1.49299218e-07]))
"""
x = np.asarray(x, dtype=float)
y = np.asarray(y, dtype=float)
if axis is None:
x = x.flatten()
y = y.flatten()
axis = 0
# Determine shape of the result arrays
res_shape = tuple([x.shape[ax] for ax in range(len(x.shape)) if ax != axis])
if not (res_shape == tuple([y.shape[ax] for ax in range(len(y.shape)) if
ax != axis])):
raise ValueError("Dimensions of x and y on all axes except `axis` "
"should match")
n = x.shape[axis]
m = y.shape[axis]
N = m + n
if N < 3:
raise ValueError("Not enough observations.")
xy = np.concatenate((x, y), axis=axis)
if axis != 0:
xy = np.rollaxis(xy, axis)
xy = xy.reshape(xy.shape[0], -1)
# Generalized to the n-dimensional case by adding the axis argument, and
# using for loops, since rankdata is not vectorized. For improving
# performance consider vectorizing rankdata function.
all_ranks = np.zeros_like(xy)
for j in range(xy.shape[1]):
all_ranks[:, j] = stats.rankdata(xy[:, j])
Ri = all_ranks[:n]
M = np.sum((Ri - (N + 1.0) / 2)**2, axis=0)
# Approx stat.
mnM = n * (N * N - 1.0) / 12
varM = m * n * (N + 1.0) * (N + 2) * (N - 2) / 180
z = (M - mnM) / sqrt(varM)
# sf for right tail, cdf for left tail. Factor 2 for two-sidedness
z_pos = z > 0
pval = np.zeros_like(z)
pval[z_pos] = 2 * distributions.norm.sf(z[z_pos])
pval[~z_pos] = 2 * distributions.norm.cdf(z[~z_pos])
if res_shape == ():
# Return scalars, not 0-D arrays
z = z[0]
pval = pval[0]
else:
z.shape = res_shape
pval.shape = res_shape
return z, pval
WilcoxonResult = namedtuple('WilcoxonResult', ('statistic', 'pvalue'))
def wilcoxon(x, y=None, zero_method="wilcox", correction=False):
"""
Calculate the Wilcoxon signed-rank test.
The Wilcoxon signed-rank test tests the null hypothesis that two
related paired samples come from the same distribution. In particular,
it tests whether the distribution of the differences x - y is symmetric
about zero. It is a non-parametric version of the paired T-test.
Parameters
----------
x : array_like
The first set of measurements.
y : array_like, optional
The second set of measurements. If `y` is not given, then the `x`
array is considered to be the differences between the two sets of
measurements.
zero_method : string, {"pratt", "wilcox", "zsplit"}, optional
"pratt":
Pratt treatment: includes zero-differences in the ranking process
(more conservative)
"wilcox":
Wilcox treatment: discards all zero-differences
"zsplit":
Zero rank split: just like Pratt, but spliting the zero rank
between positive and negative ones
correction : bool, optional
If True, apply continuity correction by adjusting the Wilcoxon rank
statistic by 0.5 towards the mean value when computing the
z-statistic. Default is False.
Returns
-------
statistic : float
The sum of the ranks of the differences above or below zero, whichever
is smaller.
pvalue : float
The two-sided p-value for the test.
Notes
-----
Because the normal approximation is used for the calculations, the
samples used should be large. A typical rule is to require that
n > 20.
References
----------
.. [1] http://en.wikipedia.org/wiki/Wilcoxon_signed-rank_test
"""
if zero_method not in ["wilcox", "pratt", "zsplit"]:
raise ValueError("Zero method should be either 'wilcox' "
"or 'pratt' or 'zsplit'")
if y is None:
d = asarray(x)
else:
x, y = map(asarray, (x, y))
if len(x) != len(y):
raise ValueError('Unequal N in wilcoxon. Aborting.')
d = x - y
if zero_method == "wilcox":
# Keep all non-zero differences
d = compress(np.not_equal(d, 0), d, axis=-1)
count = len(d)
if count < 10:
warnings.warn("Warning: sample size too small for normal approximation.")
r = stats.rankdata(abs(d))
r_plus = np.sum((d > 0) * r, axis=0)
r_minus = np.sum((d < 0) * r, axis=0)
if zero_method == "zsplit":
r_zero = np.sum((d == 0) * r, axis=0)
r_plus += r_zero / 2.
r_minus += r_zero / 2.
T = min(r_plus, r_minus)
mn = count * (count + 1.) * 0.25
se = count * (count + 1.) * (2. * count + 1.)
if zero_method == "pratt":
r = r[d != 0]
replist, repnum = find_repeats(r)
if repnum.size != 0:
# Correction for repeated elements.
se -= 0.5 * (repnum * (repnum * repnum - 1)).sum()
se = sqrt(se / 24)
correction = 0.5 * int(bool(correction)) * np.sign(T - mn)
z = (T - mn - correction) / se
prob = 2. * distributions.norm.sf(abs(z))
return WilcoxonResult(T, prob)
def median_test(*args, **kwds):
"""
Mood's median test.
Test that two or more samples come from populations with the same median.
Let ``n = len(args)`` be the number of samples. The "grand median" of
all the data is computed, and a contingency table is formed by
classifying the values in each sample as being above or below the grand
median. The contingency table, along with `correction` and `lambda_`,
are passed to `scipy.stats.chi2_contingency` to compute the test statistic
and p-value.
Parameters
----------
sample1, sample2, ... : array_like
The set of samples. There must be at least two samples.
Each sample must be a one-dimensional sequence containing at least
one value. The samples are not required to have the same length.
ties : str, optional
Determines how values equal to the grand median are classified in
the contingency table. The string must be one of::
"below":
Values equal to the grand median are counted as "below".
"above":
Values equal to the grand median are counted as "above".
"ignore":
Values equal to the grand median are not counted.
The default is "below".
correction : bool, optional
If True, *and* there are just two samples, apply Yates' correction
for continuity when computing the test statistic associated with
the contingency table. Default is True.
lambda_ : float or str, optional.
By default, the statistic computed in this test is Pearson's
chi-squared statistic. `lambda_` allows a statistic from the
Cressie-Read power divergence family to be used instead. See
`power_divergence` for details.
Default is 1 (Pearson's chi-squared statistic).
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
stat : float
The test statistic. The statistic that is returned is determined by
`lambda_`. The default is Pearson's chi-squared statistic.
p : float
The p-value of the test.
m : float
The grand median.
table : ndarray
The contingency table. The shape of the table is (2, n), where
n is the number of samples. The first row holds the counts of the
values above the grand median, and the second row holds the counts
of the values below the grand median. The table allows further
analysis with, for example, `scipy.stats.chi2_contingency`, or with
`scipy.stats.fisher_exact` if there are two samples, without having
to recompute the table. If ``nan_policy`` is "propagate" and there
are nans in the input, the return value for ``table`` is ``None``.
See Also
--------
kruskal : Compute the Kruskal-Wallis H-test for independent samples.
mannwhitneyu : Computes the Mann-Whitney rank test on samples x and y.
Notes
-----
.. versionadded:: 0.15.0
References
----------
.. [1] Mood, A. M., Introduction to the Theory of Statistics. McGraw-Hill
(1950), pp. 394-399.
.. [2] Zar, J. H., Biostatistical Analysis, 5th ed. Prentice Hall (2010).
See Sections 8.12 and 10.15.
Examples
--------
A biologist runs an experiment in which there are three groups of plants.
Group 1 has 16 plants, group 2 has 15 plants, and group 3 has 17 plants.
Each plant produces a number of seeds. The seed counts for each group
are::
Group 1: 10 14 14 18 20 22 24 25 31 31 32 39 43 43 48 49
Group 2: 28 30 31 33 34 35 36 40 44 55 57 61 91 92 99
Group 3: 0 3 9 22 23 25 25 33 34 34 40 45 46 48 62 67 84
The following code applies Mood's median test to these samples.
>>> g1 = [10, 14, 14, 18, 20, 22, 24, 25, 31, 31, 32, 39, 43, 43, 48, 49]
>>> g2 = [28, 30, 31, 33, 34, 35, 36, 40, 44, 55, 57, 61, 91, 92, 99]
>>> g3 = [0, 3, 9, 22, 23, 25, 25, 33, 34, 34, 40, 45, 46, 48, 62, 67, 84]
>>> from scipy.stats import median_test
>>> stat, p, med, tbl = median_test(g1, g2, g3)
The median is
>>> med
34.0
and the contingency table is
>>> tbl
array([[ 5, 10, 7],
[11, 5, 10]])
`p` is too large to conclude that the medians are not the same:
>>> p
0.12609082774093244
The "G-test" can be performed by passing ``lambda_="log-likelihood"`` to
`median_test`.
>>> g, p, med, tbl = median_test(g1, g2, g3, lambda_="log-likelihood")
>>> p
0.12224779737117837
The median occurs several times in the data, so we'll get a different
result if, for example, ``ties="above"`` is used:
>>> stat, p, med, tbl = median_test(g1, g2, g3, ties="above")
>>> p
0.063873276069553273
>>> tbl
array([[ 5, 11, 9],
[11, 4, 8]])
This example demonstrates that if the data set is not large and there
are values equal to the median, the p-value can be sensitive to the
choice of `ties`.
"""
ties = kwds.pop('ties', 'below')
correction = kwds.pop('correction', True)
lambda_ = kwds.pop('lambda_', None)
nan_policy = kwds.pop('nan_policy', 'propagate')
if len(kwds) > 0:
bad_kwd = kwds.keys()[0]
raise TypeError("median_test() got an unexpected keyword "
"argument %r" % bad_kwd)
if len(args) < 2:
raise ValueError('median_test requires two or more samples.')
ties_options = ['below', 'above', 'ignore']
if ties not in ties_options:
raise ValueError("invalid 'ties' option '%s'; 'ties' must be one "
"of: %s" % (ties, str(ties_options)[1:-1]))
data = [np.asarray(arg) for arg in args]
# Validate the sizes and shapes of the arguments.
for k, d in enumerate(data):
if d.size == 0:
raise ValueError("Sample %d is empty. All samples must "
"contain at least one value." % (k + 1))
if d.ndim != 1:
raise ValueError("Sample %d has %d dimensions. All "
"samples must be one-dimensional sequences." %
(k + 1, d.ndim))
cdata = np.concatenate(data)
contains_nan, nan_policy = _contains_nan(cdata, nan_policy)
if contains_nan and nan_policy == 'propagate':
return np.nan, np.nan, np.nan, None
if contains_nan:
grand_median = np.median(cdata[~np.isnan(cdata)])
else:
grand_median = np.median(cdata)
# When the minimum version of numpy supported by scipy is 1.9.0,
# the above if/else statement can be replaced by the single line:
# grand_median = np.nanmedian(cdata)
# Create the contingency table.
table = np.zeros((2, len(data)), dtype=np.int64)
for k, sample in enumerate(data):
sample = sample[~np.isnan(sample)]
nabove = count_nonzero(sample > grand_median)
nbelow = count_nonzero(sample < grand_median)
nequal = sample.size - (nabove + nbelow)
table[0, k] += nabove
table[1, k] += nbelow
if ties == "below":
table[1, k] += nequal
elif ties == "above":
table[0, k] += nequal
# Check that no row or column of the table is all zero.
# Such a table can not be given to chi2_contingency, because it would have
# a zero in the table of expected frequencies.
rowsums = table.sum(axis=1)
if rowsums[0] == 0:
raise ValueError("All values are below the grand median (%r)." %
grand_median)
if rowsums[1] == 0:
raise ValueError("All values are above the grand median (%r)." %
grand_median)
if ties == "ignore":
# We already checked that each sample has at least one value, but it
# is possible that all those values equal the grand median. If `ties`
# is "ignore", that would result in a column of zeros in `table`. We
# check for that case here.
zero_cols = np.where((table == 0).all(axis=0))[0]
if len(zero_cols) > 0:
msg = ("All values in sample %d are equal to the grand "
"median (%r), so they are ignored, resulting in an "
"empty sample." % (zero_cols[0] + 1, grand_median))
raise ValueError(msg)
stat, p, dof, expected = chi2_contingency(table, lambda_=lambda_,
correction=correction)
return stat, p, grand_median, table
def _circfuncs_common(samples, high, low):
samples = np.asarray(samples)
if samples.size == 0:
return np.nan, np.nan
ang = (samples - low)*2.*pi / (high - low)
return samples, ang
def circmean(samples, high=2*pi, low=0, axis=None):
"""
Compute the circular mean for samples in a range.
Parameters
----------
samples : array_like
Input array.
high : float or int, optional
High boundary for circular mean range. Default is ``2*pi``.
low : float or int, optional
Low boundary for circular mean range. Default is 0.
axis : int, optional
Axis along which means are computed. The default is to compute
the mean of the flattened array.
Returns
-------
circmean : float
Circular mean.
Examples
--------
>>> from scipy.stats import circmean
>>> circmean([0.1, 2*np.pi+0.2, 6*np.pi+0.3])
0.2
>>> from scipy.stats import circmean
>>> circmean([0.2, 1.4, 2.6], high = 1, low = 0)
0.4
"""
samples, ang = _circfuncs_common(samples, high, low)
S = sin(ang).sum(axis=axis)
C = cos(ang).sum(axis=axis)
res = arctan2(S, C)
mask = res < 0
if mask.ndim > 0:
res[mask] += 2*pi
elif mask:
res += 2*pi
return res*(high - low)/2.0/pi + low
def circvar(samples, high=2*pi, low=0, axis=None):
"""
Compute the circular variance for samples assumed to be in a range
Parameters
----------
samples : array_like
Input array.
low : float or int, optional
Low boundary for circular variance range. Default is 0.
high : float or int, optional
High boundary for circular variance range. Default is ``2*pi``.
axis : int, optional
Axis along which variances are computed. The default is to compute
the variance of the flattened array.
Returns
-------
circvar : float
Circular variance.
Notes
-----
This uses a definition of circular variance that in the limit of small
angles returns a number close to the 'linear' variance.
Examples
--------
>>> from scipy.stats import circvar
>>> circvar([0, 2*np.pi/3, 5*np.pi/3])
2.19722457734
"""
samples, ang = _circfuncs_common(samples, high, low)
S = sin(ang).mean(axis=axis)
C = cos(ang).mean(axis=axis)
R = hypot(S, C)
return ((high - low)/2.0/pi)**2 * 2 * log(1/R)
def circstd(samples, high=2*pi, low=0, axis=None):
"""
Compute the circular standard deviation for samples assumed to be in the
range [low to high].
Parameters
----------
samples : array_like
Input array.
low : float or int, optional
Low boundary for circular standard deviation range. Default is 0.
high : float or int, optional
High boundary for circular standard deviation range.
Default is ``2*pi``.
axis : int, optional
Axis along which standard deviations are computed. The default is
to compute the standard deviation of the flattened array.
Returns
-------
circstd : float
Circular standard deviation.
Notes
-----
This uses a definition of circular standard deviation that in the limit of
small angles returns a number close to the 'linear' standard deviation.
Examples
--------
>>> from scipy.stats import circstd
>>> circstd([0, 0.1*np.pi/2, 0.001*np.pi, 0.03*np.pi/2])
0.063564063306
"""
samples, ang = _circfuncs_common(samples, high, low)
S = sin(ang).mean(axis=axis)
C = cos(ang).mean(axis=axis)
R = hypot(S, C)
return ((high - low)/2.0/pi) * sqrt(-2*log(R))
| 93,860 | 32.811599 | 103 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/_multivariate.py
|
#
# Author: Joris Vankerschaver 2013
#
from __future__ import division, print_function, absolute_import
import math
import numpy as np
import scipy.linalg
from scipy.misc import doccer
from scipy.special import gammaln, psi, multigammaln, xlogy, entr
from scipy._lib._util import check_random_state
from scipy.linalg.blas import drot
from ._discrete_distns import binom
from . import mvn
__all__ = ['multivariate_normal',
'matrix_normal',
'dirichlet',
'wishart',
'invwishart',
'multinomial',
'special_ortho_group',
'ortho_group',
'random_correlation',
'unitary_group']
_LOG_2PI = np.log(2 * np.pi)
_LOG_2 = np.log(2)
_LOG_PI = np.log(np.pi)
_doc_random_state = """\
random_state : None or int or np.random.RandomState instance, optional
If int or RandomState, use it for drawing the random variates.
If None (or np.random), the global np.random state is used.
Default is None.
"""
def _squeeze_output(out):
"""
Remove single-dimensional entries from array and convert to scalar,
if necessary.
"""
out = out.squeeze()
if out.ndim == 0:
out = out[()]
return out
def _eigvalsh_to_eps(spectrum, cond=None, rcond=None):
"""
Determine which eigenvalues are "small" given the spectrum.
This is for compatibility across various linear algebra functions
that should agree about whether or not a Hermitian matrix is numerically
singular and what is its numerical matrix rank.
This is designed to be compatible with scipy.linalg.pinvh.
Parameters
----------
spectrum : 1d ndarray
Array of eigenvalues of a Hermitian matrix.
cond, rcond : float, optional
Cutoff for small eigenvalues.
Singular values smaller than rcond * largest_eigenvalue are
considered zero.
If None or -1, suitable machine precision is used.
Returns
-------
eps : float
Magnitude cutoff for numerical negligibility.
"""
if rcond is not None:
cond = rcond
if cond in [None, -1]:
t = spectrum.dtype.char.lower()
factor = {'f': 1E3, 'd': 1E6}
cond = factor[t] * np.finfo(t).eps
eps = cond * np.max(abs(spectrum))
return eps
def _pinv_1d(v, eps=1e-5):
"""
A helper function for computing the pseudoinverse.
Parameters
----------
v : iterable of numbers
This may be thought of as a vector of eigenvalues or singular values.
eps : float
Values with magnitude no greater than eps are considered negligible.
Returns
-------
v_pinv : 1d float ndarray
A vector of pseudo-inverted numbers.
"""
return np.array([0 if abs(x) <= eps else 1/x for x in v], dtype=float)
class _PSD(object):
"""
Compute coordinated functions of a symmetric positive semidefinite matrix.
This class addresses two issues. Firstly it allows the pseudoinverse,
the logarithm of the pseudo-determinant, and the rank of the matrix
to be computed using one call to eigh instead of three.
Secondly it allows these functions to be computed in a way
that gives mutually compatible results.
All of the functions are computed with a common understanding as to
which of the eigenvalues are to be considered negligibly small.
The functions are designed to coordinate with scipy.linalg.pinvh()
but not necessarily with np.linalg.det() or with np.linalg.matrix_rank().
Parameters
----------
M : array_like
Symmetric positive semidefinite matrix (2-D).
cond, rcond : float, optional
Cutoff for small eigenvalues.
Singular values smaller than rcond * largest_eigenvalue are
considered zero.
If None or -1, suitable machine precision is used.
lower : bool, optional
Whether the pertinent array data is taken from the lower
or upper triangle of M. (Default: lower)
check_finite : bool, optional
Whether to check that the input matrices contain only finite
numbers. Disabling may give a performance gain, but may result
in problems (crashes, non-termination) if the inputs do contain
infinities or NaNs.
allow_singular : bool, optional
Whether to allow a singular matrix. (Default: True)
Notes
-----
The arguments are similar to those of scipy.linalg.pinvh().
"""
def __init__(self, M, cond=None, rcond=None, lower=True,
check_finite=True, allow_singular=True):
# Compute the symmetric eigendecomposition.
# Note that eigh takes care of array conversion, chkfinite,
# and assertion that the matrix is square.
s, u = scipy.linalg.eigh(M, lower=lower, check_finite=check_finite)
eps = _eigvalsh_to_eps(s, cond, rcond)
if np.min(s) < -eps:
raise ValueError('the input matrix must be positive semidefinite')
d = s[s > eps]
if len(d) < len(s) and not allow_singular:
raise np.linalg.LinAlgError('singular matrix')
s_pinv = _pinv_1d(s, eps)
U = np.multiply(u, np.sqrt(s_pinv))
# Initialize the eagerly precomputed attributes.
self.rank = len(d)
self.U = U
self.log_pdet = np.sum(np.log(d))
# Initialize an attribute to be lazily computed.
self._pinv = None
@property
def pinv(self):
if self._pinv is None:
self._pinv = np.dot(self.U, self.U.T)
return self._pinv
class multi_rv_generic(object):
"""
Class which encapsulates common functionality between all multivariate
distributions.
"""
def __init__(self, seed=None):
super(multi_rv_generic, self).__init__()
self._random_state = check_random_state(seed)
@property
def random_state(self):
""" Get or set the RandomState object for generating random variates.
This can be either None or an existing RandomState object.
If None (or np.random), use the RandomState singleton used by np.random.
If already a RandomState instance, use it.
If an int, use a new RandomState instance seeded with seed.
"""
return self._random_state
@random_state.setter
def random_state(self, seed):
self._random_state = check_random_state(seed)
def _get_random_state(self, random_state):
if random_state is not None:
return check_random_state(random_state)
else:
return self._random_state
class multi_rv_frozen(object):
"""
Class which encapsulates common functionality between all frozen
multivariate distributions.
"""
@property
def random_state(self):
return self._dist._random_state
@random_state.setter
def random_state(self, seed):
self._dist._random_state = check_random_state(seed)
_mvn_doc_default_callparams = """\
mean : array_like, optional
Mean of the distribution (default zero)
cov : array_like, optional
Covariance matrix of the distribution (default one)
allow_singular : bool, optional
Whether to allow a singular covariance matrix. (Default: False)
"""
_mvn_doc_callparams_note = \
"""Setting the parameter `mean` to `None` is equivalent to having `mean`
be the zero-vector. The parameter `cov` can be a scalar, in which case
the covariance matrix is the identity times that value, a vector of
diagonal entries for the covariance matrix, or a two-dimensional
array_like.
"""
_mvn_doc_frozen_callparams = ""
_mvn_doc_frozen_callparams_note = \
"""See class definition for a detailed description of parameters."""
mvn_docdict_params = {
'_mvn_doc_default_callparams': _mvn_doc_default_callparams,
'_mvn_doc_callparams_note': _mvn_doc_callparams_note,
'_doc_random_state': _doc_random_state
}
mvn_docdict_noparams = {
'_mvn_doc_default_callparams': _mvn_doc_frozen_callparams,
'_mvn_doc_callparams_note': _mvn_doc_frozen_callparams_note,
'_doc_random_state': _doc_random_state
}
class multivariate_normal_gen(multi_rv_generic):
r"""
A multivariate normal random variable.
The `mean` keyword specifies the mean. The `cov` keyword specifies the
covariance matrix.
Methods
-------
``pdf(x, mean=None, cov=1, allow_singular=False)``
Probability density function.
``logpdf(x, mean=None, cov=1, allow_singular=False)``
Log of the probability density function.
``cdf(x, mean=None, cov=1, allow_singular=False, maxpts=1000000*dim, abseps=1e-5, releps=1e-5)``
Cumulative distribution function.
``logcdf(x, mean=None, cov=1, allow_singular=False, maxpts=1000000*dim, abseps=1e-5, releps=1e-5)``
Log of the cumulative distribution function.
``rvs(mean=None, cov=1, size=1, random_state=None)``
Draw random samples from a multivariate normal distribution.
``entropy()``
Compute the differential entropy of the multivariate normal.
Parameters
----------
x : array_like
Quantiles, with the last axis of `x` denoting the components.
%(_mvn_doc_default_callparams)s
%(_doc_random_state)s
Alternatively, the object may be called (as a function) to fix the mean
and covariance parameters, returning a "frozen" multivariate normal
random variable:
rv = multivariate_normal(mean=None, cov=1, allow_singular=False)
- Frozen object with the same methods but holding the given
mean and covariance fixed.
Notes
-----
%(_mvn_doc_callparams_note)s
The covariance matrix `cov` must be a (symmetric) positive
semi-definite matrix. The determinant and inverse of `cov` are computed
as the pseudo-determinant and pseudo-inverse, respectively, so
that `cov` does not need to have full rank.
The probability density function for `multivariate_normal` is
.. math::
f(x) = \frac{1}{\sqrt{(2 \pi)^k \det \Sigma}}
\exp\left( -\frac{1}{2} (x - \mu)^T \Sigma^{-1} (x - \mu) \right),
where :math:`\mu` is the mean, :math:`\Sigma` the covariance matrix,
and :math:`k` is the dimension of the space where :math:`x` takes values.
.. versionadded:: 0.14.0
Examples
--------
>>> import matplotlib.pyplot as plt
>>> from scipy.stats import multivariate_normal
>>> x = np.linspace(0, 5, 10, endpoint=False)
>>> y = multivariate_normal.pdf(x, mean=2.5, cov=0.5); y
array([ 0.00108914, 0.01033349, 0.05946514, 0.20755375, 0.43939129,
0.56418958, 0.43939129, 0.20755375, 0.05946514, 0.01033349])
>>> fig1 = plt.figure()
>>> ax = fig1.add_subplot(111)
>>> ax.plot(x, y)
The input quantiles can be any shape of array, as long as the last
axis labels the components. This allows us for instance to
display the frozen pdf for a non-isotropic random variable in 2D as
follows:
>>> x, y = np.mgrid[-1:1:.01, -1:1:.01]
>>> pos = np.dstack((x, y))
>>> rv = multivariate_normal([0.5, -0.2], [[2.0, 0.3], [0.3, 0.5]])
>>> fig2 = plt.figure()
>>> ax2 = fig2.add_subplot(111)
>>> ax2.contourf(x, y, rv.pdf(pos))
"""
def __init__(self, seed=None):
super(multivariate_normal_gen, self).__init__(seed)
self.__doc__ = doccer.docformat(self.__doc__, mvn_docdict_params)
def __call__(self, mean=None, cov=1, allow_singular=False, seed=None):
"""
Create a frozen multivariate normal distribution.
See `multivariate_normal_frozen` for more information.
"""
return multivariate_normal_frozen(mean, cov,
allow_singular=allow_singular,
seed=seed)
def _process_parameters(self, dim, mean, cov):
"""
Infer dimensionality from mean or covariance matrix, ensure that
mean and covariance are full vector resp. matrix.
"""
# Try to infer dimensionality
if dim is None:
if mean is None:
if cov is None:
dim = 1
else:
cov = np.asarray(cov, dtype=float)
if cov.ndim < 2:
dim = 1
else:
dim = cov.shape[0]
else:
mean = np.asarray(mean, dtype=float)
dim = mean.size
else:
if not np.isscalar(dim):
raise ValueError("Dimension of random variable must be a scalar.")
# Check input sizes and return full arrays for mean and cov if necessary
if mean is None:
mean = np.zeros(dim)
mean = np.asarray(mean, dtype=float)
if cov is None:
cov = 1.0
cov = np.asarray(cov, dtype=float)
if dim == 1:
mean.shape = (1,)
cov.shape = (1, 1)
if mean.ndim != 1 or mean.shape[0] != dim:
raise ValueError("Array 'mean' must be a vector of length %d." % dim)
if cov.ndim == 0:
cov = cov * np.eye(dim)
elif cov.ndim == 1:
cov = np.diag(cov)
elif cov.ndim == 2 and cov.shape != (dim, dim):
rows, cols = cov.shape
if rows != cols:
msg = ("Array 'cov' must be square if it is two dimensional,"
" but cov.shape = %s." % str(cov.shape))
else:
msg = ("Dimension mismatch: array 'cov' is of shape %s,"
" but 'mean' is a vector of length %d.")
msg = msg % (str(cov.shape), len(mean))
raise ValueError(msg)
elif cov.ndim > 2:
raise ValueError("Array 'cov' must be at most two-dimensional,"
" but cov.ndim = %d" % cov.ndim)
return dim, mean, cov
def _process_quantiles(self, x, dim):
"""
Adjust quantiles array so that last axis labels the components of
each data point.
"""
x = np.asarray(x, dtype=float)
if x.ndim == 0:
x = x[np.newaxis]
elif x.ndim == 1:
if dim == 1:
x = x[:, np.newaxis]
else:
x = x[np.newaxis, :]
return x
def _logpdf(self, x, mean, prec_U, log_det_cov, rank):
"""
Parameters
----------
x : ndarray
Points at which to evaluate the log of the probability
density function
mean : ndarray
Mean of the distribution
prec_U : ndarray
A decomposition such that np.dot(prec_U, prec_U.T)
is the precision matrix, i.e. inverse of the covariance matrix.
log_det_cov : float
Logarithm of the determinant of the covariance matrix
rank : int
Rank of the covariance matrix.
Notes
-----
As this function does no argument checking, it should not be
called directly; use 'logpdf' instead.
"""
dev = x - mean
maha = np.sum(np.square(np.dot(dev, prec_U)), axis=-1)
return -0.5 * (rank * _LOG_2PI + log_det_cov + maha)
def logpdf(self, x, mean=None, cov=1, allow_singular=False):
"""
Log of the multivariate normal probability density function.
Parameters
----------
x : array_like
Quantiles, with the last axis of `x` denoting the components.
%(_mvn_doc_default_callparams)s
Returns
-------
pdf : ndarray or scalar
Log of the probability density function evaluated at `x`
Notes
-----
%(_mvn_doc_callparams_note)s
"""
dim, mean, cov = self._process_parameters(None, mean, cov)
x = self._process_quantiles(x, dim)
psd = _PSD(cov, allow_singular=allow_singular)
out = self._logpdf(x, mean, psd.U, psd.log_pdet, psd.rank)
return _squeeze_output(out)
def pdf(self, x, mean=None, cov=1, allow_singular=False):
"""
Multivariate normal probability density function.
Parameters
----------
x : array_like
Quantiles, with the last axis of `x` denoting the components.
%(_mvn_doc_default_callparams)s
Returns
-------
pdf : ndarray or scalar
Probability density function evaluated at `x`
Notes
-----
%(_mvn_doc_callparams_note)s
"""
dim, mean, cov = self._process_parameters(None, mean, cov)
x = self._process_quantiles(x, dim)
psd = _PSD(cov, allow_singular=allow_singular)
out = np.exp(self._logpdf(x, mean, psd.U, psd.log_pdet, psd.rank))
return _squeeze_output(out)
def _cdf(self, x, mean, cov, maxpts, abseps, releps):
"""
Parameters
----------
x : ndarray
Points at which to evaluate the cumulative distribution function.
mean : ndarray
Mean of the distribution
cov : array_like
Covariance matrix of the distribution
maxpts: integer
The maximum number of points to use for integration
abseps: float
Absolute error tolerance
releps: float
Relative error tolerance
Notes
-----
As this function does no argument checking, it should not be
called directly; use 'cdf' instead.
.. versionadded:: 1.0.0
"""
lower = np.full(mean.shape, -np.inf)
# mvnun expects 1-d arguments, so process points sequentially
func1d = lambda x_slice: mvn.mvnun(lower, x_slice, mean, cov,
maxpts, abseps, releps)[0]
out = np.apply_along_axis(func1d, -1, x)
return _squeeze_output(out)
def logcdf(self, x, mean=None, cov=1, allow_singular=False, maxpts=None,
abseps=1e-5, releps=1e-5):
"""
Log of the multivariate normal cumulative distribution function.
Parameters
----------
x : array_like
Quantiles, with the last axis of `x` denoting the components.
%(_mvn_doc_default_callparams)s
maxpts: integer, optional
The maximum number of points to use for integration
(default `1000000*dim`)
abseps: float, optional
Absolute error tolerance (default 1e-5)
releps: float, optional
Relative error tolerance (default 1e-5)
Returns
-------
cdf : ndarray or scalar
Log of the cumulative distribution function evaluated at `x`
Notes
-----
%(_mvn_doc_callparams_note)s
.. versionadded:: 1.0.0
"""
dim, mean, cov = self._process_parameters(None, mean, cov)
x = self._process_quantiles(x, dim)
# Use _PSD to check covariance matrix
_PSD(cov, allow_singular=allow_singular)
if not maxpts:
maxpts = 1000000 * dim
out = np.log(self._cdf(x, mean, cov, maxpts, abseps, releps))
return out
def cdf(self, x, mean=None, cov=1, allow_singular=False, maxpts=None,
abseps=1e-5, releps=1e-5):
"""
Multivariate normal cumulative distribution function.
Parameters
----------
x : array_like
Quantiles, with the last axis of `x` denoting the components.
%(_mvn_doc_default_callparams)s
maxpts: integer, optional
The maximum number of points to use for integration
(default `1000000*dim`)
abseps: float, optional
Absolute error tolerance (default 1e-5)
releps: float, optional
Relative error tolerance (default 1e-5)
Returns
-------
cdf : ndarray or scalar
Cumulative distribution function evaluated at `x`
Notes
-----
%(_mvn_doc_callparams_note)s
.. versionadded:: 1.0.0
"""
dim, mean, cov = self._process_parameters(None, mean, cov)
x = self._process_quantiles(x, dim)
# Use _PSD to check covariance matrix
_PSD(cov, allow_singular=allow_singular)
if not maxpts:
maxpts = 1000000 * dim
out = self._cdf(x, mean, cov, maxpts, abseps, releps)
return out
def rvs(self, mean=None, cov=1, size=1, random_state=None):
"""
Draw random samples from a multivariate normal distribution.
Parameters
----------
%(_mvn_doc_default_callparams)s
size : integer, optional
Number of samples to draw (default 1).
%(_doc_random_state)s
Returns
-------
rvs : ndarray or scalar
Random variates of size (`size`, `N`), where `N` is the
dimension of the random variable.
Notes
-----
%(_mvn_doc_callparams_note)s
"""
dim, mean, cov = self._process_parameters(None, mean, cov)
random_state = self._get_random_state(random_state)
out = random_state.multivariate_normal(mean, cov, size)
return _squeeze_output(out)
def entropy(self, mean=None, cov=1):
"""
Compute the differential entropy of the multivariate normal.
Parameters
----------
%(_mvn_doc_default_callparams)s
Returns
-------
h : scalar
Entropy of the multivariate normal distribution
Notes
-----
%(_mvn_doc_callparams_note)s
"""
dim, mean, cov = self._process_parameters(None, mean, cov)
_, logdet = np.linalg.slogdet(2 * np.pi * np.e * cov)
return 0.5 * logdet
multivariate_normal = multivariate_normal_gen()
class multivariate_normal_frozen(multi_rv_frozen):
def __init__(self, mean=None, cov=1, allow_singular=False, seed=None,
maxpts=None, abseps=1e-5, releps=1e-5):
"""
Create a frozen multivariate normal distribution.
Parameters
----------
mean : array_like, optional
Mean of the distribution (default zero)
cov : array_like, optional
Covariance matrix of the distribution (default one)
allow_singular : bool, optional
If this flag is True then tolerate a singular
covariance matrix (default False).
seed : None or int or np.random.RandomState instance, optional
This parameter defines the RandomState object to use for drawing
random variates.
If None (or np.random), the global np.random state is used.
If integer, it is used to seed the local RandomState instance
Default is None.
maxpts: integer, optional
The maximum number of points to use for integration of the
cumulative distribution function (default `1000000*dim`)
abseps: float, optional
Absolute error tolerance for the cumulative distribution function
(default 1e-5)
releps: float, optional
Relative error tolerance for the cumulative distribution function
(default 1e-5)
Examples
--------
When called with the default parameters, this will create a 1D random
variable with mean 0 and covariance 1:
>>> from scipy.stats import multivariate_normal
>>> r = multivariate_normal()
>>> r.mean
array([ 0.])
>>> r.cov
array([[1.]])
"""
self._dist = multivariate_normal_gen(seed)
self.dim, self.mean, self.cov = self._dist._process_parameters(
None, mean, cov)
self.cov_info = _PSD(self.cov, allow_singular=allow_singular)
if not maxpts:
maxpts = 1000000 * self.dim
self.maxpts = maxpts
self.abseps = abseps
self.releps = releps
def logpdf(self, x):
x = self._dist._process_quantiles(x, self.dim)
out = self._dist._logpdf(x, self.mean, self.cov_info.U,
self.cov_info.log_pdet, self.cov_info.rank)
return _squeeze_output(out)
def pdf(self, x):
return np.exp(self.logpdf(x))
def logcdf(self, x):
return np.log(self.cdf(x))
def cdf(self, x):
x = self._dist._process_quantiles(x, self.dim)
out = self._dist._cdf(x, self.mean, self.cov, self.maxpts, self.abseps,
self.releps)
return _squeeze_output(out)
def rvs(self, size=1, random_state=None):
return self._dist.rvs(self.mean, self.cov, size, random_state)
def entropy(self):
"""
Computes the differential entropy of the multivariate normal.
Returns
-------
h : scalar
Entropy of the multivariate normal distribution
"""
log_pdet = self.cov_info.log_pdet
rank = self.cov_info.rank
return 0.5 * (rank * (_LOG_2PI + 1) + log_pdet)
# Set frozen generator docstrings from corresponding docstrings in
# multivariate_normal_gen and fill in default strings in class docstrings
for name in ['logpdf', 'pdf', 'logcdf', 'cdf', 'rvs']:
method = multivariate_normal_gen.__dict__[name]
method_frozen = multivariate_normal_frozen.__dict__[name]
method_frozen.__doc__ = doccer.docformat(method.__doc__, mvn_docdict_noparams)
method.__doc__ = doccer.docformat(method.__doc__, mvn_docdict_params)
_matnorm_doc_default_callparams = """\
mean : array_like, optional
Mean of the distribution (default: `None`)
rowcov : array_like, optional
Among-row covariance matrix of the distribution (default: `1`)
colcov : array_like, optional
Among-column covariance matrix of the distribution (default: `1`)
"""
_matnorm_doc_callparams_note = \
"""If `mean` is set to `None` then a matrix of zeros is used for the mean.
The dimensions of this matrix are inferred from the shape of `rowcov` and
`colcov`, if these are provided, or set to `1` if ambiguous.
`rowcov` and `colcov` can be two-dimensional array_likes specifying the
covariance matrices directly. Alternatively, a one-dimensional array will
be be interpreted as the entries of a diagonal matrix, and a scalar or
zero-dimensional array will be interpreted as this value times the
identity matrix.
"""
_matnorm_doc_frozen_callparams = ""
_matnorm_doc_frozen_callparams_note = \
"""See class definition for a detailed description of parameters."""
matnorm_docdict_params = {
'_matnorm_doc_default_callparams': _matnorm_doc_default_callparams,
'_matnorm_doc_callparams_note': _matnorm_doc_callparams_note,
'_doc_random_state': _doc_random_state
}
matnorm_docdict_noparams = {
'_matnorm_doc_default_callparams': _matnorm_doc_frozen_callparams,
'_matnorm_doc_callparams_note': _matnorm_doc_frozen_callparams_note,
'_doc_random_state': _doc_random_state
}
class matrix_normal_gen(multi_rv_generic):
r"""
A matrix normal random variable.
The `mean` keyword specifies the mean. The `rowcov` keyword specifies the
among-row covariance matrix. The 'colcov' keyword specifies the
among-column covariance matrix.
Methods
-------
``pdf(X, mean=None, rowcov=1, colcov=1)``
Probability density function.
``logpdf(X, mean=None, rowcov=1, colcov=1)``
Log of the probability density function.
``rvs(mean=None, rowcov=1, colcov=1, size=1, random_state=None)``
Draw random samples.
Parameters
----------
X : array_like
Quantiles, with the last two axes of `X` denoting the components.
%(_matnorm_doc_default_callparams)s
%(_doc_random_state)s
Alternatively, the object may be called (as a function) to fix the mean
and covariance parameters, returning a "frozen" matrix normal
random variable:
rv = matrix_normal(mean=None, rowcov=1, colcov=1)
- Frozen object with the same methods but holding the given
mean and covariance fixed.
Notes
-----
%(_matnorm_doc_callparams_note)s
The covariance matrices specified by `rowcov` and `colcov` must be
(symmetric) positive definite. If the samples in `X` are
:math:`m \times n`, then `rowcov` must be :math:`m \times m` and
`colcov` must be :math:`n \times n`. `mean` must be the same shape as `X`.
The probability density function for `matrix_normal` is
.. math::
f(X) = (2 \pi)^{-\frac{mn}{2}}|U|^{-\frac{n}{2}} |V|^{-\frac{m}{2}}
\exp\left( -\frac{1}{2} \mathrm{Tr}\left[ U^{-1} (X-M) V^{-1}
(X-M)^T \right] \right),
where :math:`M` is the mean, :math:`U` the among-row covariance matrix,
:math:`V` the among-column covariance matrix.
The `allow_singular` behaviour of the `multivariate_normal`
distribution is not currently supported. Covariance matrices must be
full rank.
The `matrix_normal` distribution is closely related to the
`multivariate_normal` distribution. Specifically, :math:`\mathrm{Vec}(X)`
(the vector formed by concatenating the columns of :math:`X`) has a
multivariate normal distribution with mean :math:`\mathrm{Vec}(M)`
and covariance :math:`V \otimes U` (where :math:`\otimes` is the Kronecker
product). Sampling and pdf evaluation are
:math:`\mathcal{O}(m^3 + n^3 + m^2 n + m n^2)` for the matrix normal, but
:math:`\mathcal{O}(m^3 n^3)` for the equivalent multivariate normal,
making this equivalent form algorithmically inefficient.
.. versionadded:: 0.17.0
Examples
--------
>>> from scipy.stats import matrix_normal
>>> M = np.arange(6).reshape(3,2); M
array([[0, 1],
[2, 3],
[4, 5]])
>>> U = np.diag([1,2,3]); U
array([[1, 0, 0],
[0, 2, 0],
[0, 0, 3]])
>>> V = 0.3*np.identity(2); V
array([[ 0.3, 0. ],
[ 0. , 0.3]])
>>> X = M + 0.1; X
array([[ 0.1, 1.1],
[ 2.1, 3.1],
[ 4.1, 5.1]])
>>> matrix_normal.pdf(X, mean=M, rowcov=U, colcov=V)
0.023410202050005054
>>> # Equivalent multivariate normal
>>> from scipy.stats import multivariate_normal
>>> vectorised_X = X.T.flatten()
>>> equiv_mean = M.T.flatten()
>>> equiv_cov = np.kron(V,U)
>>> multivariate_normal.pdf(vectorised_X, mean=equiv_mean, cov=equiv_cov)
0.023410202050005054
"""
def __init__(self, seed=None):
super(matrix_normal_gen, self).__init__(seed)
self.__doc__ = doccer.docformat(self.__doc__, matnorm_docdict_params)
def __call__(self, mean=None, rowcov=1, colcov=1, seed=None):
"""
Create a frozen matrix normal distribution.
See `matrix_normal_frozen` for more information.
"""
return matrix_normal_frozen(mean, rowcov, colcov, seed=seed)
def _process_parameters(self, mean, rowcov, colcov):
"""
Infer dimensionality from mean or covariance matrices. Handle
defaults. Ensure compatible dimensions.
"""
# Process mean
if mean is not None:
mean = np.asarray(mean, dtype=float)
meanshape = mean.shape
if len(meanshape) != 2:
raise ValueError("Array `mean` must be two dimensional.")
if np.any(meanshape == 0):
raise ValueError("Array `mean` has invalid shape.")
# Process among-row covariance
rowcov = np.asarray(rowcov, dtype=float)
if rowcov.ndim == 0:
if mean is not None:
rowcov = rowcov * np.identity(meanshape[0])
else:
rowcov = rowcov * np.identity(1)
elif rowcov.ndim == 1:
rowcov = np.diag(rowcov)
rowshape = rowcov.shape
if len(rowshape) != 2:
raise ValueError("`rowcov` must be a scalar or a 2D array.")
if rowshape[0] != rowshape[1]:
raise ValueError("Array `rowcov` must be square.")
if rowshape[0] == 0:
raise ValueError("Array `rowcov` has invalid shape.")
numrows = rowshape[0]
# Process among-column covariance
colcov = np.asarray(colcov, dtype=float)
if colcov.ndim == 0:
if mean is not None:
colcov = colcov * np.identity(meanshape[1])
else:
colcov = colcov * np.identity(1)
elif colcov.ndim == 1:
colcov = np.diag(colcov)
colshape = colcov.shape
if len(colshape) != 2:
raise ValueError("`colcov` must be a scalar or a 2D array.")
if colshape[0] != colshape[1]:
raise ValueError("Array `colcov` must be square.")
if colshape[0] == 0:
raise ValueError("Array `colcov` has invalid shape.")
numcols = colshape[0]
# Ensure mean and covariances compatible
if mean is not None:
if meanshape[0] != numrows:
raise ValueError("Arrays `mean` and `rowcov` must have the"
"same number of rows.")
if meanshape[1] != numcols:
raise ValueError("Arrays `mean` and `colcov` must have the"
"same number of columns.")
else:
mean = np.zeros((numrows,numcols))
dims = (numrows, numcols)
return dims, mean, rowcov, colcov
def _process_quantiles(self, X, dims):
"""
Adjust quantiles array so that last two axes labels the components of
each data point.
"""
X = np.asarray(X, dtype=float)
if X.ndim == 2:
X = X[np.newaxis, :]
if X.shape[-2:] != dims:
raise ValueError("The shape of array `X` is not compatible "
"with the distribution parameters.")
return X
def _logpdf(self, dims, X, mean, row_prec_rt, log_det_rowcov,
col_prec_rt, log_det_colcov):
"""
Parameters
----------
dims : tuple
Dimensions of the matrix variates
X : ndarray
Points at which to evaluate the log of the probability
density function
mean : ndarray
Mean of the distribution
row_prec_rt : ndarray
A decomposition such that np.dot(row_prec_rt, row_prec_rt.T)
is the inverse of the among-row covariance matrix
log_det_rowcov : float
Logarithm of the determinant of the among-row covariance matrix
col_prec_rt : ndarray
A decomposition such that np.dot(col_prec_rt, col_prec_rt.T)
is the inverse of the among-column covariance matrix
log_det_colcov : float
Logarithm of the determinant of the among-column covariance matrix
Notes
-----
As this function does no argument checking, it should not be
called directly; use 'logpdf' instead.
"""
numrows, numcols = dims
roll_dev = np.rollaxis(X-mean, axis=-1, start=0)
scale_dev = np.tensordot(col_prec_rt.T,
np.dot(roll_dev, row_prec_rt), 1)
maha = np.sum(np.sum(np.square(scale_dev), axis=-1), axis=0)
return -0.5 * (numrows*numcols*_LOG_2PI + numcols*log_det_rowcov
+ numrows*log_det_colcov + maha)
def logpdf(self, X, mean=None, rowcov=1, colcov=1):
"""
Log of the matrix normal probability density function.
Parameters
----------
X : array_like
Quantiles, with the last two axes of `X` denoting the components.
%(_matnorm_doc_default_callparams)s
Returns
-------
logpdf : ndarray
Log of the probability density function evaluated at `X`
Notes
-----
%(_matnorm_doc_callparams_note)s
"""
dims, mean, rowcov, colcov = self._process_parameters(mean, rowcov,
colcov)
X = self._process_quantiles(X, dims)
rowpsd = _PSD(rowcov, allow_singular=False)
colpsd = _PSD(colcov, allow_singular=False)
out = self._logpdf(dims, X, mean, rowpsd.U, rowpsd.log_pdet, colpsd.U,
colpsd.log_pdet)
return _squeeze_output(out)
def pdf(self, X, mean=None, rowcov=1, colcov=1):
"""
Matrix normal probability density function.
Parameters
----------
X : array_like
Quantiles, with the last two axes of `X` denoting the components.
%(_matnorm_doc_default_callparams)s
Returns
-------
pdf : ndarray
Probability density function evaluated at `X`
Notes
-----
%(_matnorm_doc_callparams_note)s
"""
return np.exp(self.logpdf(X, mean, rowcov, colcov))
def rvs(self, mean=None, rowcov=1, colcov=1, size=1, random_state=None):
"""
Draw random samples from a matrix normal distribution.
Parameters
----------
%(_matnorm_doc_default_callparams)s
size : integer, optional
Number of samples to draw (default 1).
%(_doc_random_state)s
Returns
-------
rvs : ndarray or scalar
Random variates of size (`size`, `dims`), where `dims` is the
dimension of the random matrices.
Notes
-----
%(_matnorm_doc_callparams_note)s
"""
size = int(size)
dims, mean, rowcov, colcov = self._process_parameters(mean, rowcov,
colcov)
rowchol = scipy.linalg.cholesky(rowcov, lower=True)
colchol = scipy.linalg.cholesky(colcov, lower=True)
random_state = self._get_random_state(random_state)
std_norm = random_state.standard_normal(size=(dims[1],size,dims[0]))
roll_rvs = np.tensordot(colchol, np.dot(std_norm, rowchol.T), 1)
out = np.rollaxis(roll_rvs.T, axis=1, start=0) + mean[np.newaxis,:,:]
if size == 1:
#out = np.squeeze(out, axis=0)
out = out.reshape(mean.shape)
return out
matrix_normal = matrix_normal_gen()
class matrix_normal_frozen(multi_rv_frozen):
def __init__(self, mean=None, rowcov=1, colcov=1, seed=None):
"""
Create a frozen matrix normal distribution.
Parameters
----------
%(_matnorm_doc_default_callparams)s
seed : None or int or np.random.RandomState instance, optional
If int or RandomState, use it for drawing the random variates.
If None (or np.random), the global np.random state is used.
Default is None.
Examples
--------
>>> from scipy.stats import matrix_normal
>>> distn = matrix_normal(mean=np.zeros((3,3)))
>>> X = distn.rvs(); X
array([[-0.02976962, 0.93339138, -0.09663178],
[ 0.67405524, 0.28250467, -0.93308929],
[-0.31144782, 0.74535536, 1.30412916]])
>>> distn.pdf(X)
2.5160642368346784e-05
>>> distn.logpdf(X)
-10.590229595124615
"""
self._dist = matrix_normal_gen(seed)
self.dims, self.mean, self.rowcov, self.colcov = \
self._dist._process_parameters(mean, rowcov, colcov)
self.rowpsd = _PSD(self.rowcov, allow_singular=False)
self.colpsd = _PSD(self.colcov, allow_singular=False)
def logpdf(self, X):
X = self._dist._process_quantiles(X, self.dims)
out = self._dist._logpdf(self.dims, X, self.mean, self.rowpsd.U,
self.rowpsd.log_pdet, self.colpsd.U,
self.colpsd.log_pdet)
return _squeeze_output(out)
def pdf(self, X):
return np.exp(self.logpdf(X))
def rvs(self, size=1, random_state=None):
return self._dist.rvs(self.mean, self.rowcov, self.colcov, size,
random_state)
# Set frozen generator docstrings from corresponding docstrings in
# matrix_normal_gen and fill in default strings in class docstrings
for name in ['logpdf', 'pdf', 'rvs']:
method = matrix_normal_gen.__dict__[name]
method_frozen = matrix_normal_frozen.__dict__[name]
method_frozen.__doc__ = doccer.docformat(method.__doc__, matnorm_docdict_noparams)
method.__doc__ = doccer.docformat(method.__doc__, matnorm_docdict_params)
_dirichlet_doc_default_callparams = """\
alpha : array_like
The concentration parameters. The number of entries determines the
dimensionality of the distribution.
"""
_dirichlet_doc_frozen_callparams = ""
_dirichlet_doc_frozen_callparams_note = \
"""See class definition for a detailed description of parameters."""
dirichlet_docdict_params = {
'_dirichlet_doc_default_callparams': _dirichlet_doc_default_callparams,
'_doc_random_state': _doc_random_state
}
dirichlet_docdict_noparams = {
'_dirichlet_doc_default_callparams': _dirichlet_doc_frozen_callparams,
'_doc_random_state': _doc_random_state
}
def _dirichlet_check_parameters(alpha):
alpha = np.asarray(alpha)
if np.min(alpha) <= 0:
raise ValueError("All parameters must be greater than 0")
elif alpha.ndim != 1:
raise ValueError("Parameter vector 'a' must be one dimensional, "
"but a.shape = %s." % (alpha.shape, ))
return alpha
def _dirichlet_check_input(alpha, x):
x = np.asarray(x)
if x.shape[0] + 1 != alpha.shape[0] and x.shape[0] != alpha.shape[0]:
raise ValueError("Vector 'x' must have either the same number "
"of entries as, or one entry fewer than, "
"parameter vector 'a', but alpha.shape = %s "
"and x.shape = %s." % (alpha.shape, x.shape))
if x.shape[0] != alpha.shape[0]:
xk = np.array([1 - np.sum(x, 0)])
if xk.ndim == 1:
x = np.append(x, xk)
elif xk.ndim == 2:
x = np.vstack((x, xk))
else:
raise ValueError("The input must be one dimensional or a two "
"dimensional matrix containing the entries.")
if np.min(x) < 0:
raise ValueError("Each entry in 'x' must be greater than or equal to zero.")
if np.max(x) > 1:
raise ValueError("Each entry in 'x' must be smaller or equal one.")
# Check x_i > 0 or alpha_i > 1
xeq0 = (x == 0)
alphalt1 = (alpha < 1)
if x.shape != alpha.shape:
alphalt1 = np.repeat(alphalt1, x.shape[-1], axis=-1).reshape(x.shape)
chk = np.logical_and(xeq0, alphalt1)
if np.sum(chk):
raise ValueError("Each entry in 'x' must be greater than zero if its alpha is less than one.")
if (np.abs(np.sum(x, 0) - 1.0) > 10e-10).any():
raise ValueError("The input vector 'x' must lie within the normal "
"simplex. but np.sum(x, 0) = %s." % np.sum(x, 0))
return x
def _lnB(alpha):
r"""
Internal helper function to compute the log of the useful quotient
.. math::
B(\alpha) = \frac{\prod_{i=1}{K}\Gamma(\alpha_i)}{\Gamma\left(\sum_{i=1}^{K}\alpha_i\right)}
Parameters
----------
%(_dirichlet_doc_default_callparams)s
Returns
-------
B : scalar
Helper quotient, internal use only
"""
return np.sum(gammaln(alpha)) - gammaln(np.sum(alpha))
class dirichlet_gen(multi_rv_generic):
r"""
A Dirichlet random variable.
The `alpha` keyword specifies the concentration parameters of the
distribution.
.. versionadded:: 0.15.0
Methods
-------
``pdf(x, alpha)``
Probability density function.
``logpdf(x, alpha)``
Log of the probability density function.
``rvs(alpha, size=1, random_state=None)``
Draw random samples from a Dirichlet distribution.
``mean(alpha)``
The mean of the Dirichlet distribution
``var(alpha)``
The variance of the Dirichlet distribution
``entropy(alpha)``
Compute the differential entropy of the Dirichlet distribution.
Parameters
----------
x : array_like
Quantiles, with the last axis of `x` denoting the components.
%(_dirichlet_doc_default_callparams)s
%(_doc_random_state)s
Alternatively, the object may be called (as a function) to fix
concentration parameters, returning a "frozen" Dirichlet
random variable:
rv = dirichlet(alpha)
- Frozen object with the same methods but holding the given
concentration parameters fixed.
Notes
-----
Each :math:`\alpha` entry must be positive. The distribution has only
support on the simplex defined by
.. math::
\sum_{i=1}^{K} x_i \le 1
The probability density function for `dirichlet` is
.. math::
f(x) = \frac{1}{\mathrm{B}(\boldsymbol\alpha)} \prod_{i=1}^K x_i^{\alpha_i - 1}
where
.. math::
\mathrm{B}(\boldsymbol\alpha) = \frac{\prod_{i=1}^K \Gamma(\alpha_i)}
{\Gamma\bigl(\sum_{i=1}^K \alpha_i\bigr)}
and :math:`\boldsymbol\alpha=(\alpha_1,\ldots,\alpha_K)`, the
concentration parameters and :math:`K` is the dimension of the space
where :math:`x` takes values.
Note that the dirichlet interface is somewhat inconsistent.
The array returned by the rvs function is transposed
with respect to the format expected by the pdf and logpdf.
"""
def __init__(self, seed=None):
super(dirichlet_gen, self).__init__(seed)
self.__doc__ = doccer.docformat(self.__doc__, dirichlet_docdict_params)
def __call__(self, alpha, seed=None):
return dirichlet_frozen(alpha, seed=seed)
def _logpdf(self, x, alpha):
"""
Parameters
----------
x : ndarray
Points at which to evaluate the log of the probability
density function
%(_dirichlet_doc_default_callparams)s
Notes
-----
As this function does no argument checking, it should not be
called directly; use 'logpdf' instead.
"""
lnB = _lnB(alpha)
return - lnB + np.sum((xlogy(alpha - 1, x.T)).T, 0)
def logpdf(self, x, alpha):
"""
Log of the Dirichlet probability density function.
Parameters
----------
x : array_like
Quantiles, with the last axis of `x` denoting the components.
%(_dirichlet_doc_default_callparams)s
Returns
-------
pdf : ndarray or scalar
Log of the probability density function evaluated at `x`.
"""
alpha = _dirichlet_check_parameters(alpha)
x = _dirichlet_check_input(alpha, x)
out = self._logpdf(x, alpha)
return _squeeze_output(out)
def pdf(self, x, alpha):
"""
The Dirichlet probability density function.
Parameters
----------
x : array_like
Quantiles, with the last axis of `x` denoting the components.
%(_dirichlet_doc_default_callparams)s
Returns
-------
pdf : ndarray or scalar
The probability density function evaluated at `x`.
"""
alpha = _dirichlet_check_parameters(alpha)
x = _dirichlet_check_input(alpha, x)
out = np.exp(self._logpdf(x, alpha))
return _squeeze_output(out)
def mean(self, alpha):
"""
Compute the mean of the dirichlet distribution.
Parameters
----------
%(_dirichlet_doc_default_callparams)s
Returns
-------
mu : ndarray or scalar
Mean of the Dirichlet distribution.
"""
alpha = _dirichlet_check_parameters(alpha)
out = alpha / (np.sum(alpha))
return _squeeze_output(out)
def var(self, alpha):
"""
Compute the variance of the dirichlet distribution.
Parameters
----------
%(_dirichlet_doc_default_callparams)s
Returns
-------
v : ndarray or scalar
Variance of the Dirichlet distribution.
"""
alpha = _dirichlet_check_parameters(alpha)
alpha0 = np.sum(alpha)
out = (alpha * (alpha0 - alpha)) / ((alpha0 * alpha0) * (alpha0 + 1))
return _squeeze_output(out)
def entropy(self, alpha):
"""
Compute the differential entropy of the dirichlet distribution.
Parameters
----------
%(_dirichlet_doc_default_callparams)s
Returns
-------
h : scalar
Entropy of the Dirichlet distribution
"""
alpha = _dirichlet_check_parameters(alpha)
alpha0 = np.sum(alpha)
lnB = _lnB(alpha)
K = alpha.shape[0]
out = lnB + (alpha0 - K) * scipy.special.psi(alpha0) - np.sum(
(alpha - 1) * scipy.special.psi(alpha))
return _squeeze_output(out)
def rvs(self, alpha, size=1, random_state=None):
"""
Draw random samples from a Dirichlet distribution.
Parameters
----------
%(_dirichlet_doc_default_callparams)s
size : int, optional
Number of samples to draw (default 1).
%(_doc_random_state)s
Returns
-------
rvs : ndarray or scalar
Random variates of size (`size`, `N`), where `N` is the
dimension of the random variable.
"""
alpha = _dirichlet_check_parameters(alpha)
random_state = self._get_random_state(random_state)
return random_state.dirichlet(alpha, size=size)
dirichlet = dirichlet_gen()
class dirichlet_frozen(multi_rv_frozen):
def __init__(self, alpha, seed=None):
self.alpha = _dirichlet_check_parameters(alpha)
self._dist = dirichlet_gen(seed)
def logpdf(self, x):
return self._dist.logpdf(x, self.alpha)
def pdf(self, x):
return self._dist.pdf(x, self.alpha)
def mean(self):
return self._dist.mean(self.alpha)
def var(self):
return self._dist.var(self.alpha)
def entropy(self):
return self._dist.entropy(self.alpha)
def rvs(self, size=1, random_state=None):
return self._dist.rvs(self.alpha, size, random_state)
# Set frozen generator docstrings from corresponding docstrings in
# multivariate_normal_gen and fill in default strings in class docstrings
for name in ['logpdf', 'pdf', 'rvs', 'mean', 'var', 'entropy']:
method = dirichlet_gen.__dict__[name]
method_frozen = dirichlet_frozen.__dict__[name]
method_frozen.__doc__ = doccer.docformat(
method.__doc__, dirichlet_docdict_noparams)
method.__doc__ = doccer.docformat(method.__doc__, dirichlet_docdict_params)
_wishart_doc_default_callparams = """\
df : int
Degrees of freedom, must be greater than or equal to dimension of the
scale matrix
scale : array_like
Symmetric positive definite scale matrix of the distribution
"""
_wishart_doc_callparams_note = ""
_wishart_doc_frozen_callparams = ""
_wishart_doc_frozen_callparams_note = \
"""See class definition for a detailed description of parameters."""
wishart_docdict_params = {
'_doc_default_callparams': _wishart_doc_default_callparams,
'_doc_callparams_note': _wishart_doc_callparams_note,
'_doc_random_state': _doc_random_state
}
wishart_docdict_noparams = {
'_doc_default_callparams': _wishart_doc_frozen_callparams,
'_doc_callparams_note': _wishart_doc_frozen_callparams_note,
'_doc_random_state': _doc_random_state
}
class wishart_gen(multi_rv_generic):
r"""
A Wishart random variable.
The `df` keyword specifies the degrees of freedom. The `scale` keyword
specifies the scale matrix, which must be symmetric and positive definite.
In this context, the scale matrix is often interpreted in terms of a
multivariate normal precision matrix (the inverse of the covariance
matrix).
Methods
-------
``pdf(x, df, scale)``
Probability density function.
``logpdf(x, df, scale)``
Log of the probability density function.
``rvs(df, scale, size=1, random_state=None)``
Draw random samples from a Wishart distribution.
``entropy()``
Compute the differential entropy of the Wishart distribution.
Parameters
----------
x : array_like
Quantiles, with the last axis of `x` denoting the components.
%(_doc_default_callparams)s
%(_doc_random_state)s
Alternatively, the object may be called (as a function) to fix the degrees
of freedom and scale parameters, returning a "frozen" Wishart random
variable:
rv = wishart(df=1, scale=1)
- Frozen object with the same methods but holding the given
degrees of freedom and scale fixed.
See Also
--------
invwishart, chi2
Notes
-----
%(_doc_callparams_note)s
The scale matrix `scale` must be a symmetric positive definite
matrix. Singular matrices, including the symmetric positive semi-definite
case, are not supported.
The Wishart distribution is often denoted
.. math::
W_p(\nu, \Sigma)
where :math:`\nu` is the degrees of freedom and :math:`\Sigma` is the
:math:`p \times p` scale matrix.
The probability density function for `wishart` has support over positive
definite matrices :math:`S`; if :math:`S \sim W_p(\nu, \Sigma)`, then
its PDF is given by:
.. math::
f(S) = \frac{|S|^{\frac{\nu - p - 1}{2}}}{2^{ \frac{\nu p}{2} }
|\Sigma|^\frac{\nu}{2} \Gamma_p \left ( \frac{\nu}{2} \right )}
\exp\left( -tr(\Sigma^{-1} S) / 2 \right)
If :math:`S \sim W_p(\nu, \Sigma)` (Wishart) then
:math:`S^{-1} \sim W_p^{-1}(\nu, \Sigma^{-1})` (inverse Wishart).
If the scale matrix is 1-dimensional and equal to one, then the Wishart
distribution :math:`W_1(\nu, 1)` collapses to the :math:`\chi^2(\nu)`
distribution.
.. versionadded:: 0.16.0
References
----------
.. [1] M.L. Eaton, "Multivariate Statistics: A Vector Space Approach",
Wiley, 1983.
.. [2] W.B. Smith and R.R. Hocking, "Algorithm AS 53: Wishart Variate
Generator", Applied Statistics, vol. 21, pp. 341-345, 1972.
Examples
--------
>>> import matplotlib.pyplot as plt
>>> from scipy.stats import wishart, chi2
>>> x = np.linspace(1e-5, 8, 100)
>>> w = wishart.pdf(x, df=3, scale=1); w[:5]
array([ 0.00126156, 0.10892176, 0.14793434, 0.17400548, 0.1929669 ])
>>> c = chi2.pdf(x, 3); c[:5]
array([ 0.00126156, 0.10892176, 0.14793434, 0.17400548, 0.1929669 ])
>>> plt.plot(x, w)
The input quantiles can be any shape of array, as long as the last
axis labels the components.
"""
def __init__(self, seed=None):
super(wishart_gen, self).__init__(seed)
self.__doc__ = doccer.docformat(self.__doc__, wishart_docdict_params)
def __call__(self, df=None, scale=None, seed=None):
"""
Create a frozen Wishart distribution.
See `wishart_frozen` for more information.
"""
return wishart_frozen(df, scale, seed)
def _process_parameters(self, df, scale):
if scale is None:
scale = 1.0
scale = np.asarray(scale, dtype=float)
if scale.ndim == 0:
scale = scale[np.newaxis,np.newaxis]
elif scale.ndim == 1:
scale = np.diag(scale)
elif scale.ndim == 2 and not scale.shape[0] == scale.shape[1]:
raise ValueError("Array 'scale' must be square if it is two"
" dimensional, but scale.scale = %s."
% str(scale.shape))
elif scale.ndim > 2:
raise ValueError("Array 'scale' must be at most two-dimensional,"
" but scale.ndim = %d" % scale.ndim)
dim = scale.shape[0]
if df is None:
df = dim
elif not np.isscalar(df):
raise ValueError("Degrees of freedom must be a scalar.")
elif df < dim:
raise ValueError("Degrees of freedom cannot be less than dimension"
" of scale matrix, but df = %d" % df)
return dim, df, scale
def _process_quantiles(self, x, dim):
"""
Adjust quantiles array so that last axis labels the components of
each data point.
"""
x = np.asarray(x, dtype=float)
if x.ndim == 0:
x = x * np.eye(dim)[:, :, np.newaxis]
if x.ndim == 1:
if dim == 1:
x = x[np.newaxis, np.newaxis, :]
else:
x = np.diag(x)[:, :, np.newaxis]
elif x.ndim == 2:
if not x.shape[0] == x.shape[1]:
raise ValueError("Quantiles must be square if they are two"
" dimensional, but x.shape = %s."
% str(x.shape))
x = x[:, :, np.newaxis]
elif x.ndim == 3:
if not x.shape[0] == x.shape[1]:
raise ValueError("Quantiles must be square in the first two"
" dimensions if they are three dimensional"
", but x.shape = %s." % str(x.shape))
elif x.ndim > 3:
raise ValueError("Quantiles must be at most two-dimensional with"
" an additional dimension for multiple"
"components, but x.ndim = %d" % x.ndim)
# Now we have 3-dim array; should have shape [dim, dim, *]
if not x.shape[0:2] == (dim, dim):
raise ValueError('Quantiles have incompatible dimensions: should'
' be %s, got %s.' % ((dim, dim), x.shape[0:2]))
return x
def _process_size(self, size):
size = np.asarray(size)
if size.ndim == 0:
size = size[np.newaxis]
elif size.ndim > 1:
raise ValueError('Size must be an integer or tuple of integers;'
' thus must have dimension <= 1.'
' Got size.ndim = %s' % str(tuple(size)))
n = size.prod()
shape = tuple(size)
return n, shape
def _logpdf(self, x, dim, df, scale, log_det_scale, C):
"""
Parameters
----------
x : ndarray
Points at which to evaluate the log of the probability
density function
dim : int
Dimension of the scale matrix
df : int
Degrees of freedom
scale : ndarray
Scale matrix
log_det_scale : float
Logarithm of the determinant of the scale matrix
C : ndarray
Cholesky factorization of the scale matrix, lower triagular.
Notes
-----
As this function does no argument checking, it should not be
called directly; use 'logpdf' instead.
"""
# log determinant of x
# Note: x has components along the last axis, so that x.T has
# components alone the 0-th axis. Then since det(A) = det(A'), this
# gives us a 1-dim vector of determinants
# Retrieve tr(scale^{-1} x)
log_det_x = np.zeros(x.shape[-1])
scale_inv_x = np.zeros(x.shape)
tr_scale_inv_x = np.zeros(x.shape[-1])
for i in range(x.shape[-1]):
_, log_det_x[i] = self._cholesky_logdet(x[:,:,i])
scale_inv_x[:,:,i] = scipy.linalg.cho_solve((C, True), x[:,:,i])
tr_scale_inv_x[i] = scale_inv_x[:,:,i].trace()
# Log PDF
out = ((0.5 * (df - dim - 1) * log_det_x - 0.5 * tr_scale_inv_x) -
(0.5 * df * dim * _LOG_2 + 0.5 * df * log_det_scale +
multigammaln(0.5*df, dim)))
return out
def logpdf(self, x, df, scale):
"""
Log of the Wishart probability density function.
Parameters
----------
x : array_like
Quantiles, with the last axis of `x` denoting the components.
Each quantile must be a symmetric positive definite matrix.
%(_doc_default_callparams)s
Returns
-------
pdf : ndarray
Log of the probability density function evaluated at `x`
Notes
-----
%(_doc_callparams_note)s
"""
dim, df, scale = self._process_parameters(df, scale)
x = self._process_quantiles(x, dim)
# Cholesky decomposition of scale, get log(det(scale))
C, log_det_scale = self._cholesky_logdet(scale)
out = self._logpdf(x, dim, df, scale, log_det_scale, C)
return _squeeze_output(out)
def pdf(self, x, df, scale):
"""
Wishart probability density function.
Parameters
----------
x : array_like
Quantiles, with the last axis of `x` denoting the components.
Each quantile must be a symmetric positive definite matrix.
%(_doc_default_callparams)s
Returns
-------
pdf : ndarray
Probability density function evaluated at `x`
Notes
-----
%(_doc_callparams_note)s
"""
return np.exp(self.logpdf(x, df, scale))
def _mean(self, dim, df, scale):
"""
Parameters
----------
dim : int
Dimension of the scale matrix
%(_doc_default_callparams)s
Notes
-----
As this function does no argument checking, it should not be
called directly; use 'mean' instead.
"""
return df * scale
def mean(self, df, scale):
"""
Mean of the Wishart distribution
Parameters
----------
%(_doc_default_callparams)s
Returns
-------
mean : float
The mean of the distribution
"""
dim, df, scale = self._process_parameters(df, scale)
out = self._mean(dim, df, scale)
return _squeeze_output(out)
def _mode(self, dim, df, scale):
"""
Parameters
----------
dim : int
Dimension of the scale matrix
%(_doc_default_callparams)s
Notes
-----
As this function does no argument checking, it should not be
called directly; use 'mode' instead.
"""
if df >= dim + 1:
out = (df-dim-1) * scale
else:
out = None
return out
def mode(self, df, scale):
"""
Mode of the Wishart distribution
Only valid if the degrees of freedom are greater than the dimension of
the scale matrix.
Parameters
----------
%(_doc_default_callparams)s
Returns
-------
mode : float or None
The Mode of the distribution
"""
dim, df, scale = self._process_parameters(df, scale)
out = self._mode(dim, df, scale)
return _squeeze_output(out) if out is not None else out
def _var(self, dim, df, scale):
"""
Parameters
----------
dim : int
Dimension of the scale matrix
%(_doc_default_callparams)s
Notes
-----
As this function does no argument checking, it should not be
called directly; use 'var' instead.
"""
var = scale**2
diag = scale.diagonal() # 1 x dim array
var += np.outer(diag, diag)
var *= df
return var
def var(self, df, scale):
"""
Variance of the Wishart distribution
Parameters
----------
%(_doc_default_callparams)s
Returns
-------
var : float
The variance of the distribution
"""
dim, df, scale = self._process_parameters(df, scale)
out = self._var(dim, df, scale)
return _squeeze_output(out)
def _standard_rvs(self, n, shape, dim, df, random_state):
"""
Parameters
----------
n : integer
Number of variates to generate
shape : iterable
Shape of the variates to generate
dim : int
Dimension of the scale matrix
df : int
Degrees of freedom
random_state : np.random.RandomState instance
RandomState used for drawing the random variates.
Notes
-----
As this function does no argument checking, it should not be
called directly; use 'rvs' instead.
"""
# Random normal variates for off-diagonal elements
n_tril = dim * (dim-1) // 2
covariances = random_state.normal(
size=n*n_tril).reshape(shape+(n_tril,))
# Random chi-square variates for diagonal elements
variances = np.r_[[random_state.chisquare(df-(i+1)+1, size=n)**0.5
for i in range(dim)]].reshape((dim,) + shape[::-1]).T
# Create the A matri(ces) - lower triangular
A = np.zeros(shape + (dim, dim))
# Input the covariances
size_idx = tuple([slice(None,None,None)]*len(shape))
tril_idx = np.tril_indices(dim, k=-1)
A[size_idx + tril_idx] = covariances
# Input the variances
diag_idx = np.diag_indices(dim)
A[size_idx + diag_idx] = variances
return A
def _rvs(self, n, shape, dim, df, C, random_state):
"""
Parameters
----------
n : integer
Number of variates to generate
shape : iterable
Shape of the variates to generate
dim : int
Dimension of the scale matrix
df : int
Degrees of freedom
scale : ndarray
Scale matrix
C : ndarray
Cholesky factorization of the scale matrix, lower triangular.
%(_doc_random_state)s
Notes
-----
As this function does no argument checking, it should not be
called directly; use 'rvs' instead.
"""
random_state = self._get_random_state(random_state)
# Calculate the matrices A, which are actually lower triangular
# Cholesky factorizations of a matrix B such that B ~ W(df, I)
A = self._standard_rvs(n, shape, dim, df, random_state)
# Calculate SA = C A A' C', where SA ~ W(df, scale)
# Note: this is the product of a (lower) (lower) (lower)' (lower)'
# or, denoting B = AA', it is C B C' where C is the lower
# triangular Cholesky factorization of the scale matrix.
# this appears to conflict with the instructions in [1]_, which
# suggest that it should be D' B D where D is the lower
# triangular factorization of the scale matrix. However, it is
# meant to refer to the Bartlett (1933) representation of a
# Wishart random variate as L A A' L' where L is lower triangular
# so it appears that understanding D' to be upper triangular
# is either a typo in or misreading of [1]_.
for index in np.ndindex(shape):
CA = np.dot(C, A[index])
A[index] = np.dot(CA, CA.T)
return A
def rvs(self, df, scale, size=1, random_state=None):
"""
Draw random samples from a Wishart distribution.
Parameters
----------
%(_doc_default_callparams)s
size : integer or iterable of integers, optional
Number of samples to draw (default 1).
%(_doc_random_state)s
Returns
-------
rvs : ndarray
Random variates of shape (`size`) + (`dim`, `dim), where `dim` is
the dimension of the scale matrix.
Notes
-----
%(_doc_callparams_note)s
"""
n, shape = self._process_size(size)
dim, df, scale = self._process_parameters(df, scale)
# Cholesky decomposition of scale
C = scipy.linalg.cholesky(scale, lower=True)
out = self._rvs(n, shape, dim, df, C, random_state)
return _squeeze_output(out)
def _entropy(self, dim, df, log_det_scale):
"""
Parameters
----------
dim : int
Dimension of the scale matrix
df : int
Degrees of freedom
log_det_scale : float
Logarithm of the determinant of the scale matrix
Notes
-----
As this function does no argument checking, it should not be
called directly; use 'entropy' instead.
"""
return (
0.5 * (dim+1) * log_det_scale +
0.5 * dim * (dim+1) * _LOG_2 +
multigammaln(0.5*df, dim) -
0.5 * (df - dim - 1) * np.sum(
[psi(0.5*(df + 1 - (i+1))) for i in range(dim)]
) +
0.5 * df * dim
)
def entropy(self, df, scale):
"""
Compute the differential entropy of the Wishart.
Parameters
----------
%(_doc_default_callparams)s
Returns
-------
h : scalar
Entropy of the Wishart distribution
Notes
-----
%(_doc_callparams_note)s
"""
dim, df, scale = self._process_parameters(df, scale)
_, log_det_scale = self._cholesky_logdet(scale)
return self._entropy(dim, df, log_det_scale)
def _cholesky_logdet(self, scale):
"""
Compute Cholesky decomposition and determine (log(det(scale)).
Parameters
----------
scale : ndarray
Scale matrix.
Returns
-------
c_decomp : ndarray
The Cholesky decomposition of `scale`.
logdet : scalar
The log of the determinant of `scale`.
Notes
-----
This computation of ``logdet`` is equivalent to
``np.linalg.slogdet(scale)``. It is ~2x faster though.
"""
c_decomp = scipy.linalg.cholesky(scale, lower=True)
logdet = 2 * np.sum(np.log(c_decomp.diagonal()))
return c_decomp, logdet
wishart = wishart_gen()
class wishart_frozen(multi_rv_frozen):
"""
Create a frozen Wishart distribution.
Parameters
----------
df : array_like
Degrees of freedom of the distribution
scale : array_like
Scale matrix of the distribution
seed : None or int or np.random.RandomState instance, optional
This parameter defines the RandomState object to use for drawing
random variates.
If None (or np.random), the global np.random state is used.
If integer, it is used to seed the local RandomState instance
Default is None.
"""
def __init__(self, df, scale, seed=None):
self._dist = wishart_gen(seed)
self.dim, self.df, self.scale = self._dist._process_parameters(
df, scale)
self.C, self.log_det_scale = self._dist._cholesky_logdet(self.scale)
def logpdf(self, x):
x = self._dist._process_quantiles(x, self.dim)
out = self._dist._logpdf(x, self.dim, self.df, self.scale,
self.log_det_scale, self.C)
return _squeeze_output(out)
def pdf(self, x):
return np.exp(self.logpdf(x))
def mean(self):
out = self._dist._mean(self.dim, self.df, self.scale)
return _squeeze_output(out)
def mode(self):
out = self._dist._mode(self.dim, self.df, self.scale)
return _squeeze_output(out) if out is not None else out
def var(self):
out = self._dist._var(self.dim, self.df, self.scale)
return _squeeze_output(out)
def rvs(self, size=1, random_state=None):
n, shape = self._dist._process_size(size)
out = self._dist._rvs(n, shape, self.dim, self.df,
self.C, random_state)
return _squeeze_output(out)
def entropy(self):
return self._dist._entropy(self.dim, self.df, self.log_det_scale)
# Set frozen generator docstrings from corresponding docstrings in
# Wishart and fill in default strings in class docstrings
for name in ['logpdf', 'pdf', 'mean', 'mode', 'var', 'rvs', 'entropy']:
method = wishart_gen.__dict__[name]
method_frozen = wishart_frozen.__dict__[name]
method_frozen.__doc__ = doccer.docformat(
method.__doc__, wishart_docdict_noparams)
method.__doc__ = doccer.docformat(method.__doc__, wishart_docdict_params)
from numpy import asarray_chkfinite, asarray
from scipy.linalg.misc import LinAlgError
from scipy.linalg.lapack import get_lapack_funcs
def _cho_inv_batch(a, check_finite=True):
"""
Invert the matrices a_i, using a Cholesky factorization of A, where
a_i resides in the last two dimensions of a and the other indices describe
the index i.
Overwrites the data in a.
Parameters
----------
a : array
Array of matrices to invert, where the matrices themselves are stored
in the last two dimensions.
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
x : array
Array of inverses of the matrices ``a_i``.
See also
--------
scipy.linalg.cholesky : Cholesky factorization of a matrix
"""
if check_finite:
a1 = asarray_chkfinite(a)
else:
a1 = asarray(a)
if len(a1.shape) < 2 or a1.shape[-2] != a1.shape[-1]:
raise ValueError('expected square matrix in last two dimensions')
potrf, potri = get_lapack_funcs(('potrf','potri'), (a1,))
tril_idx = np.tril_indices(a.shape[-2], k=-1)
triu_idx = np.triu_indices(a.shape[-2], k=1)
for index in np.ndindex(a1.shape[:-2]):
# Cholesky decomposition
a1[index], info = potrf(a1[index], lower=True, overwrite_a=False,
clean=False)
if info > 0:
raise LinAlgError("%d-th leading minor not positive definite"
% info)
if info < 0:
raise ValueError('illegal value in %d-th argument of internal'
' potrf' % -info)
# Inversion
a1[index], info = potri(a1[index], lower=True, overwrite_c=False)
if info > 0:
raise LinAlgError("the inverse could not be computed")
if info < 0:
raise ValueError('illegal value in %d-th argument of internal'
' potrf' % -info)
# Make symmetric (dpotri only fills in the lower triangle)
a1[index][triu_idx] = a1[index][tril_idx]
return a1
class invwishart_gen(wishart_gen):
r"""
An inverse Wishart random variable.
The `df` keyword specifies the degrees of freedom. The `scale` keyword
specifies the scale matrix, which must be symmetric and positive definite.
In this context, the scale matrix is often interpreted in terms of a
multivariate normal covariance matrix.
Methods
-------
``pdf(x, df, scale)``
Probability density function.
``logpdf(x, df, scale)``
Log of the probability density function.
``rvs(df, scale, size=1, random_state=None)``
Draw random samples from an inverse Wishart distribution.
Parameters
----------
x : array_like
Quantiles, with the last axis of `x` denoting the components.
%(_doc_default_callparams)s
%(_doc_random_state)s
Alternatively, the object may be called (as a function) to fix the degrees
of freedom and scale parameters, returning a "frozen" inverse Wishart
random variable:
rv = invwishart(df=1, scale=1)
- Frozen object with the same methods but holding the given
degrees of freedom and scale fixed.
See Also
--------
wishart
Notes
-----
%(_doc_callparams_note)s
The scale matrix `scale` must be a symmetric positive definite
matrix. Singular matrices, including the symmetric positive semi-definite
case, are not supported.
The inverse Wishart distribution is often denoted
.. math::
W_p^{-1}(\nu, \Psi)
where :math:`\nu` is the degrees of freedom and :math:`\Psi` is the
:math:`p \times p` scale matrix.
The probability density function for `invwishart` has support over positive
definite matrices :math:`S`; if :math:`S \sim W^{-1}_p(\nu, \Sigma)`,
then its PDF is given by:
.. math::
f(S) = \frac{|\Sigma|^\frac{\nu}{2}}{2^{ \frac{\nu p}{2} }
|S|^{\frac{\nu + p + 1}{2}} \Gamma_p \left(\frac{\nu}{2} \right)}
\exp\left( -tr(\Sigma S^{-1}) / 2 \right)
If :math:`S \sim W_p^{-1}(\nu, \Psi)` (inverse Wishart) then
:math:`S^{-1} \sim W_p(\nu, \Psi^{-1})` (Wishart).
If the scale matrix is 1-dimensional and equal to one, then the inverse
Wishart distribution :math:`W_1(\nu, 1)` collapses to the
inverse Gamma distribution with parameters shape = :math:`\frac{\nu}{2}`
and scale = :math:`\frac{1}{2}`.
.. versionadded:: 0.16.0
References
----------
.. [1] M.L. Eaton, "Multivariate Statistics: A Vector Space Approach",
Wiley, 1983.
.. [2] M.C. Jones, "Generating Inverse Wishart Matrices", Communications in
Statistics - Simulation and Computation, vol. 14.2, pp.511-514, 1985.
Examples
--------
>>> import matplotlib.pyplot as plt
>>> from scipy.stats import invwishart, invgamma
>>> x = np.linspace(0.01, 1, 100)
>>> iw = invwishart.pdf(x, df=6, scale=1)
>>> iw[:3]
array([ 1.20546865e-15, 5.42497807e-06, 4.45813929e-03])
>>> ig = invgamma.pdf(x, 6/2., scale=1./2)
>>> ig[:3]
array([ 1.20546865e-15, 5.42497807e-06, 4.45813929e-03])
>>> plt.plot(x, iw)
The input quantiles can be any shape of array, as long as the last
axis labels the components.
"""
def __init__(self, seed=None):
super(invwishart_gen, self).__init__(seed)
self.__doc__ = doccer.docformat(self.__doc__, wishart_docdict_params)
def __call__(self, df=None, scale=None, seed=None):
"""
Create a frozen inverse Wishart distribution.
See `invwishart_frozen` for more information.
"""
return invwishart_frozen(df, scale, seed)
def _logpdf(self, x, dim, df, scale, log_det_scale):
"""
Parameters
----------
x : ndarray
Points at which to evaluate the log of the probability
density function.
dim : int
Dimension of the scale matrix
df : int
Degrees of freedom
scale : ndarray
Scale matrix
log_det_scale : float
Logarithm of the determinant of the scale matrix
Notes
-----
As this function does no argument checking, it should not be
called directly; use 'logpdf' instead.
"""
log_det_x = np.zeros(x.shape[-1])
#scale_x_inv = np.zeros(x.shape)
x_inv = np.copy(x).T
if dim > 1:
_cho_inv_batch(x_inv) # works in-place
else:
x_inv = 1./x_inv
tr_scale_x_inv = np.zeros(x.shape[-1])
for i in range(x.shape[-1]):
C, lower = scipy.linalg.cho_factor(x[:,:,i], lower=True)
log_det_x[i] = 2 * np.sum(np.log(C.diagonal()))
#scale_x_inv[:,:,i] = scipy.linalg.cho_solve((C, True), scale).T
tr_scale_x_inv[i] = np.dot(scale, x_inv[i]).trace()
# Log PDF
out = ((0.5 * df * log_det_scale - 0.5 * tr_scale_x_inv) -
(0.5 * df * dim * _LOG_2 + 0.5 * (df + dim + 1) * log_det_x) -
multigammaln(0.5*df, dim))
return out
def logpdf(self, x, df, scale):
"""
Log of the inverse Wishart probability density function.
Parameters
----------
x : array_like
Quantiles, with the last axis of `x` denoting the components.
Each quantile must be a symmetric positive definite matrix.
%(_doc_default_callparams)s
Returns
-------
pdf : ndarray
Log of the probability density function evaluated at `x`
Notes
-----
%(_doc_callparams_note)s
"""
dim, df, scale = self._process_parameters(df, scale)
x = self._process_quantiles(x, dim)
_, log_det_scale = self._cholesky_logdet(scale)
out = self._logpdf(x, dim, df, scale, log_det_scale)
return _squeeze_output(out)
def pdf(self, x, df, scale):
"""
Inverse Wishart probability density function.
Parameters
----------
x : array_like
Quantiles, with the last axis of `x` denoting the components.
Each quantile must be a symmetric positive definite matrix.
%(_doc_default_callparams)s
Returns
-------
pdf : ndarray
Probability density function evaluated at `x`
Notes
-----
%(_doc_callparams_note)s
"""
return np.exp(self.logpdf(x, df, scale))
def _mean(self, dim, df, scale):
"""
Parameters
----------
dim : int
Dimension of the scale matrix
%(_doc_default_callparams)s
Notes
-----
As this function does no argument checking, it should not be
called directly; use 'mean' instead.
"""
if df > dim + 1:
out = scale / (df - dim - 1)
else:
out = None
return out
def mean(self, df, scale):
"""
Mean of the inverse Wishart distribution
Only valid if the degrees of freedom are greater than the dimension of
the scale matrix plus one.
Parameters
----------
%(_doc_default_callparams)s
Returns
-------
mean : float or None
The mean of the distribution
"""
dim, df, scale = self._process_parameters(df, scale)
out = self._mean(dim, df, scale)
return _squeeze_output(out) if out is not None else out
def _mode(self, dim, df, scale):
"""
Parameters
----------
dim : int
Dimension of the scale matrix
%(_doc_default_callparams)s
Notes
-----
As this function does no argument checking, it should not be
called directly; use 'mode' instead.
"""
return scale / (df + dim + 1)
def mode(self, df, scale):
"""
Mode of the inverse Wishart distribution
Parameters
----------
%(_doc_default_callparams)s
Returns
-------
mode : float
The Mode of the distribution
"""
dim, df, scale = self._process_parameters(df, scale)
out = self._mode(dim, df, scale)
return _squeeze_output(out)
def _var(self, dim, df, scale):
"""
Parameters
----------
dim : int
Dimension of the scale matrix
%(_doc_default_callparams)s
Notes
-----
As this function does no argument checking, it should not be
called directly; use 'var' instead.
"""
if df > dim + 3:
var = (df - dim + 1) * scale**2
diag = scale.diagonal() # 1 x dim array
var += (df - dim - 1) * np.outer(diag, diag)
var /= (df - dim) * (df - dim - 1)**2 * (df - dim - 3)
else:
var = None
return var
def var(self, df, scale):
"""
Variance of the inverse Wishart distribution
Only valid if the degrees of freedom are greater than the dimension of
the scale matrix plus three.
Parameters
----------
%(_doc_default_callparams)s
Returns
-------
var : float
The variance of the distribution
"""
dim, df, scale = self._process_parameters(df, scale)
out = self._var(dim, df, scale)
return _squeeze_output(out) if out is not None else out
def _rvs(self, n, shape, dim, df, C, random_state):
"""
Parameters
----------
n : integer
Number of variates to generate
shape : iterable
Shape of the variates to generate
dim : int
Dimension of the scale matrix
df : int
Degrees of freedom
C : ndarray
Cholesky factorization of the scale matrix, lower triagular.
%(_doc_random_state)s
Notes
-----
As this function does no argument checking, it should not be
called directly; use 'rvs' instead.
"""
random_state = self._get_random_state(random_state)
# Get random draws A such that A ~ W(df, I)
A = super(invwishart_gen, self)._standard_rvs(n, shape, dim,
df, random_state)
# Calculate SA = (CA)'^{-1} (CA)^{-1} ~ iW(df, scale)
eye = np.eye(dim)
trtrs = get_lapack_funcs(('trtrs'), (A,))
for index in np.ndindex(A.shape[:-2]):
# Calculate CA
CA = np.dot(C, A[index])
# Get (C A)^{-1} via triangular solver
if dim > 1:
CA, info = trtrs(CA, eye, lower=True)
if info > 0:
raise LinAlgError("Singular matrix.")
if info < 0:
raise ValueError('Illegal value in %d-th argument of'
' internal trtrs' % -info)
else:
CA = 1. / CA
# Get SA
A[index] = np.dot(CA.T, CA)
return A
def rvs(self, df, scale, size=1, random_state=None):
"""
Draw random samples from an inverse Wishart distribution.
Parameters
----------
%(_doc_default_callparams)s
size : integer or iterable of integers, optional
Number of samples to draw (default 1).
%(_doc_random_state)s
Returns
-------
rvs : ndarray
Random variates of shape (`size`) + (`dim`, `dim), where `dim` is
the dimension of the scale matrix.
Notes
-----
%(_doc_callparams_note)s
"""
n, shape = self._process_size(size)
dim, df, scale = self._process_parameters(df, scale)
# Invert the scale
eye = np.eye(dim)
L, lower = scipy.linalg.cho_factor(scale, lower=True)
inv_scale = scipy.linalg.cho_solve((L, lower), eye)
# Cholesky decomposition of inverted scale
C = scipy.linalg.cholesky(inv_scale, lower=True)
out = self._rvs(n, shape, dim, df, C, random_state)
return _squeeze_output(out)
def entropy(self):
# Need to find reference for inverse Wishart entropy
raise AttributeError
invwishart = invwishart_gen()
class invwishart_frozen(multi_rv_frozen):
def __init__(self, df, scale, seed=None):
"""
Create a frozen inverse Wishart distribution.
Parameters
----------
df : array_like
Degrees of freedom of the distribution
scale : array_like
Scale matrix of the distribution
seed : None or int or np.random.RandomState instance, optional
This parameter defines the RandomState object to use for drawing
random variates.
If None (or np.random), the global np.random state is used.
If integer, it is used to seed the local RandomState instance
Default is None.
"""
self._dist = invwishart_gen(seed)
self.dim, self.df, self.scale = self._dist._process_parameters(
df, scale
)
# Get the determinant via Cholesky factorization
C, lower = scipy.linalg.cho_factor(self.scale, lower=True)
self.log_det_scale = 2 * np.sum(np.log(C.diagonal()))
# Get the inverse using the Cholesky factorization
eye = np.eye(self.dim)
self.inv_scale = scipy.linalg.cho_solve((C, lower), eye)
# Get the Cholesky factorization of the inverse scale
self.C = scipy.linalg.cholesky(self.inv_scale, lower=True)
def logpdf(self, x):
x = self._dist._process_quantiles(x, self.dim)
out = self._dist._logpdf(x, self.dim, self.df, self.scale,
self.log_det_scale)
return _squeeze_output(out)
def pdf(self, x):
return np.exp(self.logpdf(x))
def mean(self):
out = self._dist._mean(self.dim, self.df, self.scale)
return _squeeze_output(out) if out is not None else out
def mode(self):
out = self._dist._mode(self.dim, self.df, self.scale)
return _squeeze_output(out)
def var(self):
out = self._dist._var(self.dim, self.df, self.scale)
return _squeeze_output(out) if out is not None else out
def rvs(self, size=1, random_state=None):
n, shape = self._dist._process_size(size)
out = self._dist._rvs(n, shape, self.dim, self.df,
self.C, random_state)
return _squeeze_output(out)
def entropy(self):
# Need to find reference for inverse Wishart entropy
raise AttributeError
# Set frozen generator docstrings from corresponding docstrings in
# inverse Wishart and fill in default strings in class docstrings
for name in ['logpdf', 'pdf', 'mean', 'mode', 'var', 'rvs']:
method = invwishart_gen.__dict__[name]
method_frozen = wishart_frozen.__dict__[name]
method_frozen.__doc__ = doccer.docformat(
method.__doc__, wishart_docdict_noparams)
method.__doc__ = doccer.docformat(method.__doc__, wishart_docdict_params)
_multinomial_doc_default_callparams = """\
n : int
Number of trials
p : array_like
Probability of a trial falling into each category; should sum to 1
"""
_multinomial_doc_callparams_note = \
"""`n` should be a positive integer. Each element of `p` should be in the
interval :math:`[0,1]` and the elements should sum to 1. If they do not sum to
1, the last element of the `p` array is not used and is replaced with the
remaining probability left over from the earlier elements.
"""
_multinomial_doc_frozen_callparams = ""
_multinomial_doc_frozen_callparams_note = \
"""See class definition for a detailed description of parameters."""
multinomial_docdict_params = {
'_doc_default_callparams': _multinomial_doc_default_callparams,
'_doc_callparams_note': _multinomial_doc_callparams_note,
'_doc_random_state': _doc_random_state
}
multinomial_docdict_noparams = {
'_doc_default_callparams': _multinomial_doc_frozen_callparams,
'_doc_callparams_note': _multinomial_doc_frozen_callparams_note,
'_doc_random_state': _doc_random_state
}
class multinomial_gen(multi_rv_generic):
r"""
A multinomial random variable.
Methods
-------
``pmf(x, n, p)``
Probability mass function.
``logpmf(x, n, p)``
Log of the probability mass function.
``rvs(n, p, size=1, random_state=None)``
Draw random samples from a multinomial distribution.
``entropy(n, p)``
Compute the entropy of the multinomial distribution.
``cov(n, p)``
Compute the covariance matrix of the multinomial distribution.
Parameters
----------
x : array_like
Quantiles, with the last axis of `x` denoting the components.
%(_doc_default_callparams)s
%(_doc_random_state)s
Notes
-----
%(_doc_callparams_note)s
Alternatively, the object may be called (as a function) to fix the `n` and
`p` parameters, returning a "frozen" multinomial random variable:
The probability mass function for `multinomial` is
.. math::
f(x) = \frac{n!}{x_1! \cdots x_k!} p_1^{x_1} \cdots p_k^{x_k},
supported on :math:`x=(x_1, \ldots, x_k)` where each :math:`x_i` is a
nonnegative integer and their sum is :math:`n`.
.. versionadded:: 0.19.0
Examples
--------
>>> from scipy.stats import multinomial
>>> rv = multinomial(8, [0.3, 0.2, 0.5])
>>> rv.pmf([1, 3, 4])
0.042000000000000072
The multinomial distribution for :math:`k=2` is identical to the
corresponding binomial distribution (tiny numerical differences
notwithstanding):
>>> from scipy.stats import binom
>>> multinomial.pmf([3, 4], n=7, p=[0.4, 0.6])
0.29030399999999973
>>> binom.pmf(3, 7, 0.4)
0.29030400000000012
The functions ``pmf``, ``logpmf``, ``entropy``, and ``cov`` support
broadcasting, under the convention that the vector parameters (``x`` and
``p``) are interpreted as if each row along the last axis is a single
object. For instance:
>>> multinomial.pmf([[3, 4], [3, 5]], n=[7, 8], p=[.3, .7])
array([0.2268945, 0.25412184])
Here, ``x.shape == (2, 2)``, ``n.shape == (2,)``, and ``p.shape == (2,)``,
but following the rules mentioned above they behave as if the rows
``[3, 4]`` and ``[3, 5]`` in ``x`` and ``[.3, .7]`` in ``p`` were a single
object, and as if we had ``x.shape = (2,)``, ``n.shape = (2,)``, and
``p.shape = ()``. To obtain the individual elements without broadcasting,
we would do this:
>>> multinomial.pmf([3, 4], n=7, p=[.3, .7])
0.2268945
>>> multinomial.pmf([3, 5], 8, p=[.3, .7])
0.25412184
This broadcasting also works for ``cov``, where the output objects are
square matrices of size ``p.shape[-1]``. For example:
>>> multinomial.cov([4, 5], [[.3, .7], [.4, .6]])
array([[[ 0.84, -0.84],
[-0.84, 0.84]],
[[ 1.2 , -1.2 ],
[-1.2 , 1.2 ]]])
In this example, ``n.shape == (2,)`` and ``p.shape == (2, 2)``, and
following the rules above, these broadcast as if ``p.shape == (2,)``.
Thus the result should also be of shape ``(2,)``, but since each output is
a :math:`2 \times 2` matrix, the result in fact has shape ``(2, 2, 2)``,
where ``result[0]`` is equal to ``multinomial.cov(n=4, p=[.3, .7])`` and
``result[1]`` is equal to ``multinomial.cov(n=5, p=[.4, .6])``.
See also
--------
scipy.stats.binom : The binomial distribution.
numpy.random.multinomial : Sampling from the multinomial distribution.
"""
def __init__(self, seed=None):
super(multinomial_gen, self).__init__(seed)
self.__doc__ = \
doccer.docformat(self.__doc__, multinomial_docdict_params)
def __call__(self, n, p, seed=None):
"""
Create a frozen multinomial distribution.
See `multinomial_frozen` for more information.
"""
return multinomial_frozen(n, p, seed)
def _process_parameters(self, n, p):
"""
Return: n_, p_, npcond.
n_ and p_ are arrays of the correct shape; npcond is a boolean array
flagging values out of the domain.
"""
p = np.array(p, dtype=np.float64, copy=True)
p[...,-1] = 1. - p[...,:-1].sum(axis=-1)
# true for bad p
pcond = np.any(p < 0, axis=-1)
pcond |= np.any(p > 1, axis=-1)
n = np.array(n, dtype=np.int, copy=True)
# true for bad n
ncond = n <= 0
return n, p, ncond | pcond
def _process_quantiles(self, x, n, p):
"""
Return: x_, xcond.
x_ is an int array; xcond is a boolean array flagging values out of the
domain.
"""
xx = np.asarray(x, dtype=np.int)
if xx.ndim == 0:
raise ValueError("x must be an array.")
if xx.size != 0 and not xx.shape[-1] == p.shape[-1]:
raise ValueError("Size of each quantile should be size of p: "
"received %d, but expected %d." % (xx.shape[-1], p.shape[-1]))
# true for x out of the domain
cond = np.any(xx != x, axis=-1)
cond |= np.any(xx < 0, axis=-1)
cond = cond | (np.sum(xx, axis=-1) != n)
return xx, cond
def _checkresult(self, result, cond, bad_value):
result = np.asarray(result)
if cond.ndim != 0:
result[cond] = bad_value
elif cond:
if result.ndim == 0:
return bad_value
result[...] = bad_value
return result
def _logpmf(self, x, n, p):
return gammaln(n+1) + np.sum(xlogy(x, p) - gammaln(x+1), axis=-1)
def logpmf(self, x, n, p):
"""
Log of the Multinomial probability mass function.
Parameters
----------
x : array_like
Quantiles, with the last axis of `x` denoting the components.
Each quantile must be a symmetric positive definite matrix.
%(_doc_default_callparams)s
Returns
-------
logpmf : ndarray or scalar
Log of the probability mass function evaluated at `x`
Notes
-----
%(_doc_callparams_note)s
"""
n, p, npcond = self._process_parameters(n, p)
x, xcond = self._process_quantiles(x, n, p)
result = self._logpmf(x, n, p)
# replace values for which x was out of the domain; broadcast
# xcond to the right shape
xcond_ = xcond | np.zeros(npcond.shape, dtype=np.bool_)
result = self._checkresult(result, xcond_, np.NINF)
# replace values bad for n or p; broadcast npcond to the right shape
npcond_ = npcond | np.zeros(xcond.shape, dtype=np.bool_)
return self._checkresult(result, npcond_, np.NAN)
def pmf(self, x, n, p):
"""
Multinomial probability mass function.
Parameters
----------
x : array_like
Quantiles, with the last axis of `x` denoting the components.
Each quantile must be a symmetric positive definite matrix.
%(_doc_default_callparams)s
Returns
-------
pmf : ndarray or scalar
Probability density function evaluated at `x`
Notes
-----
%(_doc_callparams_note)s
"""
return np.exp(self.logpmf(x, n, p))
def mean(self, n, p):
"""
Mean of the Multinomial distribution
Parameters
----------
%(_doc_default_callparams)s
Returns
-------
mean : float
The mean of the distribution
"""
n, p, npcond = self._process_parameters(n, p)
result = n[..., np.newaxis]*p
return self._checkresult(result, npcond, np.NAN)
def cov(self, n, p):
"""
Covariance matrix of the multinomial distribution.
Parameters
----------
%(_doc_default_callparams)s
Returns
-------
cov : ndarray
The covariance matrix of the distribution
"""
n, p, npcond = self._process_parameters(n, p)
nn = n[..., np.newaxis, np.newaxis]
result = nn * np.einsum('...j,...k->...jk', -p, p)
# change the diagonal
for i in range(p.shape[-1]):
result[...,i, i] += n*p[..., i]
return self._checkresult(result, npcond, np.nan)
def entropy(self, n, p):
r"""
Compute the entropy of the multinomial distribution.
The entropy is computed using this expression:
.. math::
f(x) = - \log n! - n\sum_{i=1}^k p_i \log p_i +
\sum_{i=1}^k \sum_{x=0}^n \binom n x p_i^x(1-p_i)^{n-x} \log x!
Parameters
----------
%(_doc_default_callparams)s
Returns
-------
h : scalar
Entropy of the Multinomial distribution
Notes
-----
%(_doc_callparams_note)s
"""
n, p, npcond = self._process_parameters(n, p)
x = np.r_[1:np.max(n)+1]
term1 = n*np.sum(entr(p), axis=-1)
term1 -= gammaln(n+1)
n = n[..., np.newaxis]
new_axes_needed = max(p.ndim, n.ndim) - x.ndim + 1
x.shape += (1,)*new_axes_needed
term2 = np.sum(binom.pmf(x, n, p)*gammaln(x+1),
axis=(-1, -1-new_axes_needed))
return self._checkresult(term1 + term2, npcond, np.nan)
def rvs(self, n, p, size=None, random_state=None):
"""
Draw random samples from a Multinomial distribution.
Parameters
----------
%(_doc_default_callparams)s
size : integer or iterable of integers, optional
Number of samples to draw (default 1).
%(_doc_random_state)s
Returns
-------
rvs : ndarray or scalar
Random variates of shape (`size`, `len(p)`)
Notes
-----
%(_doc_callparams_note)s
"""
n, p, npcond = self._process_parameters(n, p)
random_state = self._get_random_state(random_state)
return random_state.multinomial(n, p, size)
multinomial = multinomial_gen()
class multinomial_frozen(multi_rv_frozen):
r"""
Create a frozen Multinomial distribution.
Parameters
----------
n : int
number of trials
p: array_like
probability of a trial falling into each category; should sum to 1
seed : None or int or np.random.RandomState instance, optional
This parameter defines the RandomState object to use for drawing
random variates.
If None (or np.random), the global np.random state is used.
If integer, it is used to seed the local RandomState instance
Default is None.
"""
def __init__(self, n, p, seed=None):
self._dist = multinomial_gen(seed)
self.n, self.p, self.npcond = self._dist._process_parameters(n, p)
# monkey patch self._dist
def _process_parameters(n, p):
return self.n, self.p, self.npcond
self._dist._process_parameters = _process_parameters
def logpmf(self, x):
return self._dist.logpmf(x, self.n, self.p)
def pmf(self, x):
return self._dist.pmf(x, self.n, self.p)
def mean(self):
return self._dist.mean(self.n, self.p)
def cov(self):
return self._dist.cov(self.n, self.p)
def entropy(self):
return self._dist.entropy(self.n, self.p)
def rvs(self, size=1, random_state=None):
return self._dist.rvs(self.n, self.p, size, random_state)
# Set frozen generator docstrings from corresponding docstrings in
# multinomial and fill in default strings in class docstrings
for name in ['logpmf', 'pmf', 'mean', 'cov', 'rvs']:
method = multinomial_gen.__dict__[name]
method_frozen = multinomial_frozen.__dict__[name]
method_frozen.__doc__ = doccer.docformat(
method.__doc__, multinomial_docdict_noparams)
method.__doc__ = doccer.docformat(method.__doc__,
multinomial_docdict_params)
class special_ortho_group_gen(multi_rv_generic):
r"""
A matrix-valued SO(N) random variable.
Return a random rotation matrix, drawn from the Haar distribution
(the only uniform distribution on SO(n)).
The `dim` keyword specifies the dimension N.
Methods
-------
``rvs(dim=None, size=1, random_state=None)``
Draw random samples from SO(N).
Parameters
----------
dim : scalar
Dimension of matrices
Notes
----------
This class is wrapping the random_rot code from the MDP Toolkit,
https://github.com/mdp-toolkit/mdp-toolkit
Return a random rotation matrix, drawn from the Haar distribution
(the only uniform distribution on SO(n)).
The algorithm is described in the paper
Stewart, G.W., "The efficient generation of random orthogonal
matrices with an application to condition estimators", SIAM Journal
on Numerical Analysis, 17(3), pp. 403-409, 1980.
For more information see
http://en.wikipedia.org/wiki/Orthogonal_matrix#Randomization
See also the similar `ortho_group`.
Examples
--------
>>> from scipy.stats import special_ortho_group
>>> x = special_ortho_group.rvs(3)
>>> np.dot(x, x.T)
array([[ 1.00000000e+00, 1.13231364e-17, -2.86852790e-16],
[ 1.13231364e-17, 1.00000000e+00, -1.46845020e-16],
[ -2.86852790e-16, -1.46845020e-16, 1.00000000e+00]])
>>> import scipy.linalg
>>> scipy.linalg.det(x)
1.0
This generates one random matrix from SO(3). It is orthogonal and
has a determinant of 1.
"""
def __init__(self, seed=None):
super(special_ortho_group_gen, self).__init__(seed)
self.__doc__ = doccer.docformat(self.__doc__)
def __call__(self, dim=None, seed=None):
"""
Create a frozen SO(N) distribution.
See `special_ortho_group_frozen` for more information.
"""
return special_ortho_group_frozen(dim, seed=seed)
def _process_parameters(self, dim):
"""
Dimension N must be specified; it cannot be inferred.
"""
if dim is None or not np.isscalar(dim) or dim <= 1 or dim != int(dim):
raise ValueError("""Dimension of rotation must be specified,
and must be a scalar greater than 1.""")
return dim
def rvs(self, dim, size=1, random_state=None):
"""
Draw random samples from SO(N).
Parameters
----------
dim : integer
Dimension of rotation space (N).
size : integer, optional
Number of samples to draw (default 1).
Returns
-------
rvs : ndarray or scalar
Random size N-dimensional matrices, dimension (size, dim, dim)
"""
random_state = self._get_random_state(random_state)
size = int(size)
if size > 1:
return np.array([self.rvs(dim, size=1, random_state=random_state)
for i in range(size)])
dim = self._process_parameters(dim)
H = np.eye(dim)
D = np.empty((dim,))
for n in range(dim-1):
x = random_state.normal(size=(dim-n,))
D[n] = np.sign(x[0]) if x[0] != 0 else 1
x[0] += D[n]*np.sqrt((x*x).sum())
# Householder transformation
Hx = (np.eye(dim-n)
- 2.*np.outer(x, x)/(x*x).sum())
mat = np.eye(dim)
mat[n:, n:] = Hx
H = np.dot(H, mat)
D[-1] = (-1)**(dim-1)*D[:-1].prod()
# Equivalent to np.dot(np.diag(D), H) but faster, apparently
H = (D*H.T).T
return H
special_ortho_group = special_ortho_group_gen()
class special_ortho_group_frozen(multi_rv_frozen):
def __init__(self, dim=None, seed=None):
"""
Create a frozen SO(N) distribution.
Parameters
----------
dim : scalar
Dimension of matrices
seed : None or int or np.random.RandomState instance, optional
This parameter defines the RandomState object to use for drawing
random variates.
If None (or np.random), the global np.random state is used.
If integer, it is used to seed the local RandomState instance
Default is None.
Examples
--------
>>> from scipy.stats import special_ortho_group
>>> g = special_ortho_group(5)
>>> x = g.rvs()
"""
self._dist = special_ortho_group_gen(seed)
self.dim = self._dist._process_parameters(dim)
def rvs(self, size=1, random_state=None):
return self._dist.rvs(self.dim, size, random_state)
class ortho_group_gen(multi_rv_generic):
r"""
A matrix-valued O(N) random variable.
Return a random orthogonal matrix, drawn from the O(N) Haar
distribution (the only uniform distribution on O(N)).
The `dim` keyword specifies the dimension N.
Methods
-------
``rvs(dim=None, size=1, random_state=None)``
Draw random samples from O(N).
Parameters
----------
dim : scalar
Dimension of matrices
Notes
----------
This class is closely related to `special_ortho_group`.
Some care is taken to avoid numerical error, as per the paper by Mezzadri.
References
----------
.. [1] F. Mezzadri, "How to generate random matrices from the classical
compact groups", :arXiv:`math-ph/0609050v2`.
Examples
--------
>>> from scipy.stats import ortho_group
>>> x = ortho_group.rvs(3)
>>> np.dot(x, x.T)
array([[ 1.00000000e+00, 1.13231364e-17, -2.86852790e-16],
[ 1.13231364e-17, 1.00000000e+00, -1.46845020e-16],
[ -2.86852790e-16, -1.46845020e-16, 1.00000000e+00]])
>>> import scipy.linalg
>>> np.fabs(scipy.linalg.det(x))
1.0
This generates one random matrix from O(3). It is orthogonal and
has a determinant of +1 or -1.
"""
def __init__(self, seed=None):
super(ortho_group_gen, self).__init__(seed)
self.__doc__ = doccer.docformat(self.__doc__)
def _process_parameters(self, dim):
"""
Dimension N must be specified; it cannot be inferred.
"""
if dim is None or not np.isscalar(dim) or dim <= 1 or dim != int(dim):
raise ValueError("Dimension of rotation must be specified,"
"and must be a scalar greater than 1.")
return dim
def rvs(self, dim, size=1, random_state=None):
"""
Draw random samples from O(N).
Parameters
----------
dim : integer
Dimension of rotation space (N).
size : integer, optional
Number of samples to draw (default 1).
Returns
-------
rvs : ndarray or scalar
Random size N-dimensional matrices, dimension (size, dim, dim)
"""
random_state = self._get_random_state(random_state)
size = int(size)
if size > 1:
return np.array([self.rvs(dim, size=1, random_state=random_state)
for i in range(size)])
dim = self._process_parameters(dim)
H = np.eye(dim)
for n in range(dim):
x = random_state.normal(size=(dim-n,))
# random sign, 50/50, but chosen carefully to avoid roundoff error
D = np.sign(x[0]) if x[0] != 0 else 1
x[0] += D*np.sqrt((x*x).sum())
# Householder transformation
Hx = -D*(np.eye(dim-n)
- 2.*np.outer(x, x)/(x*x).sum())
mat = np.eye(dim)
mat[n:, n:] = Hx
H = np.dot(H, mat)
return H
ortho_group = ortho_group_gen()
class random_correlation_gen(multi_rv_generic):
r"""
A random correlation matrix.
Return a random correlation matrix, given a vector of eigenvalues.
The `eigs` keyword specifies the eigenvalues of the correlation matrix,
and implies the dimension.
Methods
-------
``rvs(eigs=None, random_state=None)``
Draw random correlation matrices, all with eigenvalues eigs.
Parameters
----------
eigs : 1d ndarray
Eigenvalues of correlation matrix.
Notes
----------
Generates a random correlation matrix following a numerically stable
algorithm spelled out by Davies & Higham. This algorithm uses a single O(N)
similarity transformation to construct a symmetric positive semi-definite
matrix, and applies a series of Givens rotations to scale it to have ones
on the diagonal.
References
----------
.. [1] Davies, Philip I; Higham, Nicholas J; "Numerically stable generation
of correlation matrices and their factors", BIT 2000, Vol. 40,
No. 4, pp. 640 651
Examples
--------
>>> from scipy.stats import random_correlation
>>> np.random.seed(514)
>>> x = random_correlation.rvs((.5, .8, 1.2, 1.5))
>>> x
array([[ 1. , -0.20387311, 0.18366501, -0.04953711],
[-0.20387311, 1. , -0.24351129, 0.06703474],
[ 0.18366501, -0.24351129, 1. , 0.38530195],
[-0.04953711, 0.06703474, 0.38530195, 1. ]])
>>> import scipy.linalg
>>> e, v = scipy.linalg.eigh(x)
>>> e
array([ 0.5, 0.8, 1.2, 1.5])
"""
def __init__(self, seed=None):
super(random_correlation_gen, self).__init__(seed)
self.__doc__ = doccer.docformat(self.__doc__)
def _process_parameters(self, eigs, tol):
eigs = np.asarray(eigs, dtype=float)
dim = eigs.size
if eigs.ndim != 1 or eigs.shape[0] != dim or dim <= 1:
raise ValueError("Array 'eigs' must be a vector of length greater than 1.")
if np.fabs(np.sum(eigs) - dim) > tol:
raise ValueError("Sum of eigenvalues must equal dimensionality.")
for x in eigs:
if x < -tol:
raise ValueError("All eigenvalues must be non-negative.")
return dim, eigs
def _givens_to_1(self, aii, ajj, aij):
"""Computes a 2x2 Givens matrix to put 1's on the diagonal for the input matrix.
The input matrix is a 2x2 symmetric matrix M = [ aii aij ; aij ajj ].
The output matrix g is a 2x2 anti-symmetric matrix of the form [ c s ; -s c ];
the elements c and s are returned.
Applying the output matrix to the input matrix (as b=g.T M g)
results in a matrix with bii=1, provided tr(M) - det(M) >= 1
and floating point issues do not occur. Otherwise, some other
valid rotation is returned. When tr(M)==2, also bjj=1.
"""
aiid = aii - 1.
ajjd = ajj - 1.
if ajjd == 0:
# ajj==1, so swap aii and ajj to avoid division by zero
return 0., 1.
dd = math.sqrt(max(aij**2 - aiid*ajjd, 0))
# The choice of t should be chosen to avoid cancellation [1]
t = (aij + math.copysign(dd, aij)) / ajjd
c = 1. / math.sqrt(1. + t*t)
if c == 0:
# Underflow
s = 1.0
else:
s = c*t
return c, s
def _to_corr(self, m):
"""
Given a psd matrix m, rotate to put one's on the diagonal, turning it
into a correlation matrix. This also requires the trace equal the
dimensionality. Note: modifies input matrix
"""
# Check requirements for in-place Givens
if not (m.flags.c_contiguous and m.dtype == np.float64 and m.shape[0] == m.shape[1]):
raise ValueError()
d = m.shape[0]
for i in range(d-1):
if m[i,i] == 1:
continue
elif m[i, i] > 1:
for j in range(i+1, d):
if m[j, j] < 1:
break
else:
for j in range(i+1, d):
if m[j, j] > 1:
break
c, s = self._givens_to_1(m[i,i], m[j,j], m[i,j])
# Use BLAS to apply Givens rotations in-place. Equivalent to:
# g = np.eye(d)
# g[i, i] = g[j,j] = c
# g[j, i] = -s; g[i, j] = s
# m = np.dot(g.T, np.dot(m, g))
mv = m.ravel()
drot(mv, mv, c, -s, n=d,
offx=i*d, incx=1, offy=j*d, incy=1,
overwrite_x=True, overwrite_y=True)
drot(mv, mv, c, -s, n=d,
offx=i, incx=d, offy=j, incy=d,
overwrite_x=True, overwrite_y=True)
return m
def rvs(self, eigs, random_state=None, tol=1e-13, diag_tol=1e-7):
"""
Draw random correlation matrices
Parameters
----------
eigs : 1d ndarray
Eigenvalues of correlation matrix
tol : float, optional
Tolerance for input parameter checks
diag_tol : float, optional
Tolerance for deviation of the diagonal of the resulting
matrix. Default: 1e-7
Raises
------
RuntimeError
Floating point error prevented generating a valid correlation
matrix.
Returns
-------
rvs : ndarray or scalar
Random size N-dimensional matrices, dimension (size, dim, dim),
each having eigenvalues eigs.
"""
dim, eigs = self._process_parameters(eigs, tol=tol)
random_state = self._get_random_state(random_state)
m = ortho_group.rvs(dim, random_state=random_state)
m = np.dot(np.dot(m, np.diag(eigs)), m.T) # Set the trace of m
m = self._to_corr(m) # Carefully rotate to unit diagonal
# Check diagonal
if abs(m.diagonal() - 1).max() > diag_tol:
raise RuntimeError("Failed to generate a valid correlation matrix")
return m
random_correlation = random_correlation_gen()
class unitary_group_gen(multi_rv_generic):
r"""
A matrix-valued U(N) random variable.
Return a random unitary matrix.
The `dim` keyword specifies the dimension N.
Methods
-------
``rvs(dim=None, size=1, random_state=None)``
Draw random samples from U(N).
Parameters
----------
dim : scalar
Dimension of matrices
Notes
----------
This class is similar to `ortho_group`.
References
----------
.. [1] F. Mezzadri, "How to generate random matrices from the classical
compact groups", arXiv:math-ph/0609050v2.
Examples
--------
>>> from scipy.stats import unitary_group
>>> x = unitary_group.rvs(3)
>>> np.dot(x, x.conj().T)
array([[ 1.00000000e+00, 1.13231364e-17, -2.86852790e-16],
[ 1.13231364e-17, 1.00000000e+00, -1.46845020e-16],
[ -2.86852790e-16, -1.46845020e-16, 1.00000000e+00]])
This generates one random matrix from U(3). The dot product confirms that it is unitary up to machine precision.
"""
def __init__(self, seed=None):
super(unitary_group_gen, self).__init__(seed)
self.__doc__ = doccer.docformat(self.__doc__)
def _process_parameters(self, dim):
"""
Dimension N must be specified; it cannot be inferred.
"""
if dim is None or not np.isscalar(dim) or dim <= 1 or dim != int(dim):
raise ValueError("Dimension of rotation must be specified,"
"and must be a scalar greater than 1.")
return dim
def rvs(self, dim, size=1, random_state=None):
"""
Draw random samples from U(N).
Parameters
----------
dim : integer
Dimension of space (N).
size : integer, optional
Number of samples to draw (default 1).
Returns
-------
rvs : ndarray or scalar
Random size N-dimensional matrices, dimension (size, dim, dim)
"""
random_state = self._get_random_state(random_state)
size = int(size)
if size > 1:
return np.array([self.rvs(dim, size=1, random_state=random_state)
for i in range(size)])
dim = self._process_parameters(dim)
z = 1/math.sqrt(2)*(random_state.normal(size=(dim,dim)) +
1j*random_state.normal(size=(dim,dim)))
q, r = scipy.linalg.qr(z)
d = r.diagonal()
q *= d/abs(d)
return q
unitary_group = unitary_group_gen()
| 120,087 | 30.811391 | 116 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/_tukeylambda_stats.py
|
from __future__ import division, print_function, absolute_import
import numpy as np
from numpy import poly1d
from scipy.special import beta
# The following code was used to generate the Pade coefficients for the
# Tukey Lambda variance function. Version 0.17 of mpmath was used.
#---------------------------------------------------------------------------
# import mpmath as mp
#
# mp.mp.dps = 60
#
# one = mp.mpf(1)
# two = mp.mpf(2)
#
# def mpvar(lam):
# if lam == 0:
# v = mp.pi**2 / three
# else:
# v = (two / lam**2) * (one / (one + two*lam) -
# mp.beta(lam + one, lam + one))
# return v
#
# t = mp.taylor(mpvar, 0, 8)
# p, q = mp.pade(t, 4, 4)
# print("p =", [mp.fp.mpf(c) for c in p])
# print("q =", [mp.fp.mpf(c) for c in q])
#---------------------------------------------------------------------------
# Pade coefficients for the Tukey Lambda variance function.
_tukeylambda_var_pc = [3.289868133696453, 0.7306125098871127,
-0.5370742306855439, 0.17292046290190008,
-0.02371146284628187]
_tukeylambda_var_qc = [1.0, 3.683605511659861, 4.184152498888124,
1.7660926747377275, 0.2643989311168465]
# numpy.poly1d instances for the numerator and denominator of the
# Pade approximation to the Tukey Lambda variance.
_tukeylambda_var_p = poly1d(_tukeylambda_var_pc[::-1])
_tukeylambda_var_q = poly1d(_tukeylambda_var_qc[::-1])
def tukeylambda_variance(lam):
"""Variance of the Tukey Lambda distribution.
Parameters
----------
lam : array_like
The lambda values at which to compute the variance.
Returns
-------
v : ndarray
The variance. For lam < -0.5, the variance is not defined, so
np.nan is returned. For lam = 0.5, np.inf is returned.
Notes
-----
In an interval around lambda=0, this function uses the [4,4] Pade
approximation to compute the variance. Otherwise it uses the standard
formula (http://en.wikipedia.org/wiki/Tukey_lambda_distribution). The
Pade approximation is used because the standard formula has a removable
discontinuity at lambda = 0, and does not produce accurate numerical
results near lambda = 0.
"""
lam = np.asarray(lam)
shp = lam.shape
lam = np.atleast_1d(lam).astype(np.float64)
# For absolute values of lam less than threshold, use the Pade
# approximation.
threshold = 0.075
# Play games with masks to implement the conditional evaluation of
# the distribution.
# lambda < -0.5: var = nan
low_mask = lam < -0.5
# lambda == -0.5: var = inf
neghalf_mask = lam == -0.5
# abs(lambda) < threshold: use Pade approximation
small_mask = np.abs(lam) < threshold
# else the "regular" case: use the explicit formula.
reg_mask = ~(low_mask | neghalf_mask | small_mask)
# Get the 'lam' values for the cases where they are needed.
small = lam[small_mask]
reg = lam[reg_mask]
# Compute the function for each case.
v = np.empty_like(lam)
v[low_mask] = np.nan
v[neghalf_mask] = np.inf
if small.size > 0:
# Use the Pade approximation near lambda = 0.
v[small_mask] = _tukeylambda_var_p(small) / _tukeylambda_var_q(small)
if reg.size > 0:
v[reg_mask] = (2.0 / reg**2) * (1.0 / (1.0 + 2 * reg) -
beta(reg + 1, reg + 1))
v.shape = shp
return v
# The following code was used to generate the Pade coefficients for the
# Tukey Lambda kurtosis function. Version 0.17 of mpmath was used.
#---------------------------------------------------------------------------
# import mpmath as mp
#
# mp.mp.dps = 60
#
# one = mp.mpf(1)
# two = mp.mpf(2)
# three = mp.mpf(3)
# four = mp.mpf(4)
#
# def mpkurt(lam):
# if lam == 0:
# k = mp.mpf(6)/5
# else:
# numer = (one/(four*lam+one) - four*mp.beta(three*lam+one, lam+one) +
# three*mp.beta(two*lam+one, two*lam+one))
# denom = two*(one/(two*lam+one) - mp.beta(lam+one,lam+one))**2
# k = numer / denom - three
# return k
#
# # There is a bug in mpmath 0.17: when we use the 'method' keyword of the
# # taylor function and we request a degree 9 Taylor polynomial, we actually
# # get degree 8.
# t = mp.taylor(mpkurt, 0, 9, method='quad', radius=0.01)
# t = [mp.chop(c, tol=1e-15) for c in t]
# p, q = mp.pade(t, 4, 4)
# print("p =", [mp.fp.mpf(c) for c in p])
# print("q =", [mp.fp.mpf(c) for c in q])
#---------------------------------------------------------------------------
# Pade coefficients for the Tukey Lambda kurtosis function.
_tukeylambda_kurt_pc = [1.2, -5.853465139719495, -22.653447381131077,
0.20601184383406815, 4.59796302262789]
_tukeylambda_kurt_qc = [1.0, 7.171149192233599, 12.96663094361842,
0.43075235247853005, -2.789746758009912]
# numpy.poly1d instances for the numerator and denominator of the
# Pade approximation to the Tukey Lambda kurtosis.
_tukeylambda_kurt_p = poly1d(_tukeylambda_kurt_pc[::-1])
_tukeylambda_kurt_q = poly1d(_tukeylambda_kurt_qc[::-1])
def tukeylambda_kurtosis(lam):
"""Kurtosis of the Tukey Lambda distribution.
Parameters
----------
lam : array_like
The lambda values at which to compute the variance.
Returns
-------
v : ndarray
The variance. For lam < -0.25, the variance is not defined, so
np.nan is returned. For lam = 0.25, np.inf is returned.
"""
lam = np.asarray(lam)
shp = lam.shape
lam = np.atleast_1d(lam).astype(np.float64)
# For absolute values of lam less than threshold, use the Pade
# approximation.
threshold = 0.055
# Use masks to implement the conditional evaluation of the kurtosis.
# lambda < -0.25: kurtosis = nan
low_mask = lam < -0.25
# lambda == -0.25: kurtosis = inf
negqrtr_mask = lam == -0.25
# lambda near 0: use Pade approximation
small_mask = np.abs(lam) < threshold
# else the "regular" case: use the explicit formula.
reg_mask = ~(low_mask | negqrtr_mask | small_mask)
# Get the 'lam' values for the cases where they are needed.
small = lam[small_mask]
reg = lam[reg_mask]
# Compute the function for each case.
k = np.empty_like(lam)
k[low_mask] = np.nan
k[negqrtr_mask] = np.inf
if small.size > 0:
k[small_mask] = _tukeylambda_kurt_p(small) / _tukeylambda_kurt_q(small)
if reg.size > 0:
numer = (1.0 / (4 * reg + 1) - 4 * beta(3 * reg + 1, reg + 1) +
3 * beta(2 * reg + 1, 2 * reg + 1))
denom = 2 * (1.0/(2 * reg + 1) - beta(reg + 1, reg + 1))**2
k[reg_mask] = numer / denom - 3
# The return value will be a numpy array; resetting the shape ensures that
# if `lam` was a scalar, the return value is a 0-d array.
k.shape = shp
return k
| 6,934 | 33.331683 | 79 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/mstats_basic.py
|
"""
An extension of scipy.stats.stats to support masked arrays
"""
# Original author (2007): Pierre GF Gerard-Marchant
# TODO : f_value_wilks_lambda looks botched... what are dfnum & dfden for ?
# TODO : ttest_rel looks botched: what are x1,x2,v1,v2 for ?
# TODO : reimplement ksonesamp
from __future__ import division, print_function, absolute_import
__all__ = ['argstoarray',
'count_tied_groups',
'describe',
'f_oneway', 'find_repeats','friedmanchisquare',
'kendalltau','kendalltau_seasonal','kruskal','kruskalwallis',
'ks_twosamp','ks_2samp','kurtosis','kurtosistest',
'linregress',
'mannwhitneyu', 'meppf','mode','moment','mquantiles','msign',
'normaltest',
'obrientransform',
'pearsonr','plotting_positions','pointbiserialr',
'rankdata',
'scoreatpercentile','sem',
'sen_seasonal_slopes','skew','skewtest','spearmanr',
'theilslopes','tmax','tmean','tmin','trim','trimboth',
'trimtail','trima','trimr','trimmed_mean','trimmed_std',
'trimmed_stde','trimmed_var','tsem','ttest_1samp','ttest_onesamp',
'ttest_ind','ttest_rel','tvar',
'variation',
'winsorize',
]
import numpy as np
from numpy import ndarray
import numpy.ma as ma
from numpy.ma import masked, nomask
from scipy._lib.six import iteritems
import itertools
import warnings
from collections import namedtuple
from . import distributions
import scipy.special as special
from ._stats_mstats_common import (
_find_repeats,
linregress as stats_linregress,
theilslopes as stats_theilslopes
)
genmissingvaldoc = """
Notes
-----
Missing values are considered pair-wise: if a value is missing in x,
the corresponding value in y is masked.
"""
def _chk_asarray(a, axis):
# Always returns a masked array, raveled for axis=None
a = ma.asanyarray(a)
if axis is None:
a = ma.ravel(a)
outaxis = 0
else:
outaxis = axis
return a, outaxis
def _chk2_asarray(a, b, axis):
a = ma.asanyarray(a)
b = ma.asanyarray(b)
if axis is None:
a = ma.ravel(a)
b = ma.ravel(b)
outaxis = 0
else:
outaxis = axis
return a, b, outaxis
def _chk_size(a,b):
a = ma.asanyarray(a)
b = ma.asanyarray(b)
(na, nb) = (a.size, b.size)
if na != nb:
raise ValueError("The size of the input array should match!"
" (%s <> %s)" % (na, nb))
return (a, b, na)
def argstoarray(*args):
"""
Constructs a 2D array from a group of sequences.
Sequences are filled with missing values to match the length of the longest
sequence.
Parameters
----------
args : sequences
Group of sequences.
Returns
-------
argstoarray : MaskedArray
A ( `m` x `n` ) masked array, where `m` is the number of arguments and
`n` the length of the longest argument.
Notes
-----
`numpy.ma.row_stack` has identical behavior, but is called with a sequence
of sequences.
"""
if len(args) == 1 and not isinstance(args[0], ndarray):
output = ma.asarray(args[0])
if output.ndim != 2:
raise ValueError("The input should be 2D")
else:
n = len(args)
m = max([len(k) for k in args])
output = ma.array(np.empty((n,m), dtype=float), mask=True)
for (k,v) in enumerate(args):
output[k,:len(v)] = v
output[np.logical_not(np.isfinite(output._data))] = masked
return output
def find_repeats(arr):
"""Find repeats in arr and return a tuple (repeats, repeat_count).
The input is cast to float64. Masked values are discarded.
Parameters
----------
arr : sequence
Input array. The array is flattened if it is not 1D.
Returns
-------
repeats : ndarray
Array of repeated values.
counts : ndarray
Array of counts.
"""
# Make sure we get a copy. ma.compressed promises a "new array", but can
# actually return a reference.
compr = np.asarray(ma.compressed(arr), dtype=np.float64)
try:
need_copy = np.may_share_memory(compr, arr)
except AttributeError:
# numpy < 1.8.2 bug: np.may_share_memory([], []) raises,
# while in numpy 1.8.2 and above it just (correctly) returns False.
need_copy = False
if need_copy:
compr = compr.copy()
return _find_repeats(compr)
def count_tied_groups(x, use_missing=False):
"""
Counts the number of tied values.
Parameters
----------
x : sequence
Sequence of data on which to counts the ties
use_missing : bool, optional
Whether to consider missing values as tied.
Returns
-------
count_tied_groups : dict
Returns a dictionary (nb of ties: nb of groups).
Examples
--------
>>> from scipy.stats import mstats
>>> z = [0, 0, 0, 2, 2, 2, 3, 3, 4, 5, 6]
>>> mstats.count_tied_groups(z)
{2: 1, 3: 2}
In the above example, the ties were 0 (3x), 2 (3x) and 3 (2x).
>>> z = np.ma.array([0, 0, 1, 2, 2, 2, 3, 3, 4, 5, 6])
>>> mstats.count_tied_groups(z)
{2: 2, 3: 1}
>>> z[[1,-1]] = np.ma.masked
>>> mstats.count_tied_groups(z, use_missing=True)
{2: 2, 3: 1}
"""
nmasked = ma.getmask(x).sum()
# We need the copy as find_repeats will overwrite the initial data
data = ma.compressed(x).copy()
(ties, counts) = find_repeats(data)
nties = {}
if len(ties):
nties = dict(zip(np.unique(counts), itertools.repeat(1)))
nties.update(dict(zip(*find_repeats(counts))))
if nmasked and use_missing:
try:
nties[nmasked] += 1
except KeyError:
nties[nmasked] = 1
return nties
def rankdata(data, axis=None, use_missing=False):
"""Returns the rank (also known as order statistics) of each data point
along the given axis.
If some values are tied, their rank is averaged.
If some values are masked, their rank is set to 0 if use_missing is False,
or set to the average rank of the unmasked values if use_missing is True.
Parameters
----------
data : sequence
Input data. The data is transformed to a masked array
axis : {None,int}, optional
Axis along which to perform the ranking.
If None, the array is first flattened. An exception is raised if
the axis is specified for arrays with a dimension larger than 2
use_missing : bool, optional
Whether the masked values have a rank of 0 (False) or equal to the
average rank of the unmasked values (True).
"""
def _rank1d(data, use_missing=False):
n = data.count()
rk = np.empty(data.size, dtype=float)
idx = data.argsort()
rk[idx[:n]] = np.arange(1,n+1)
if use_missing:
rk[idx[n:]] = (n+1)/2.
else:
rk[idx[n:]] = 0
repeats = find_repeats(data.copy())
for r in repeats[0]:
condition = (data == r).filled(False)
rk[condition] = rk[condition].mean()
return rk
data = ma.array(data, copy=False)
if axis is None:
if data.ndim > 1:
return _rank1d(data.ravel(), use_missing).reshape(data.shape)
else:
return _rank1d(data, use_missing)
else:
return ma.apply_along_axis(_rank1d,axis,data,use_missing).view(ndarray)
ModeResult = namedtuple('ModeResult', ('mode', 'count'))
def mode(a, axis=0):
"""
Returns an array of the modal (most common) value in the passed array.
Parameters
----------
a : array_like
n-dimensional array of which to find mode(s).
axis : int or None, optional
Axis along which to operate. Default is 0. If None, compute over
the whole array `a`.
Returns
-------
mode : ndarray
Array of modal values.
count : ndarray
Array of counts for each mode.
Notes
-----
For more details, see `stats.mode`.
"""
a, axis = _chk_asarray(a, axis)
def _mode1D(a):
(rep,cnt) = find_repeats(a)
if not cnt.ndim:
return (0, 0)
elif cnt.size:
return (rep[cnt.argmax()], cnt.max())
else:
not_masked_indices = ma.flatnotmasked_edges(a)
first_not_masked_index = not_masked_indices[0]
return (a[first_not_masked_index], 1)
if axis is None:
output = _mode1D(ma.ravel(a))
output = (ma.array(output[0]), ma.array(output[1]))
else:
output = ma.apply_along_axis(_mode1D, axis, a)
newshape = list(a.shape)
newshape[axis] = 1
slices = [slice(None)] * output.ndim
slices[axis] = 0
modes = output[tuple(slices)].reshape(newshape)
slices[axis] = 1
counts = output[tuple(slices)].reshape(newshape)
output = (modes, counts)
return ModeResult(*output)
def _betai(a, b, x):
x = np.asanyarray(x)
x = ma.where(x < 1.0, x, 1.0) # if x > 1 then return 1.0
return special.betainc(a, b, x)
def msign(x):
"""Returns the sign of x, or 0 if x is masked."""
return ma.filled(np.sign(x), 0)
def pearsonr(x,y):
"""
Calculates a Pearson correlation coefficient and the p-value for testing
non-correlation.
The Pearson correlation coefficient measures the linear relationship
between two datasets. Strictly speaking, Pearson's correlation requires
that each dataset be normally distributed. Like other correlation
coefficients, this one varies between -1 and +1 with 0 implying no
correlation. Correlations of -1 or +1 imply an exact linear
relationship. Positive correlations imply that as `x` increases, so does
`y`. Negative correlations imply that as `x` increases, `y` decreases.
The p-value roughly indicates the probability of an uncorrelated system
producing datasets that have a Pearson correlation at least as extreme
as the one computed from these datasets. The p-values are not entirely
reliable but are probably reasonable for datasets larger than 500 or so.
Parameters
----------
x : 1-D array_like
Input
y : 1-D array_like
Input
Returns
-------
pearsonr : float
Pearson's correlation coefficient, 2-tailed p-value.
References
----------
http://www.statsoft.com/textbook/glosp.html#Pearson%20Correlation
"""
(x, y, n) = _chk_size(x, y)
(x, y) = (x.ravel(), y.ravel())
# Get the common mask and the total nb of unmasked elements
m = ma.mask_or(ma.getmask(x), ma.getmask(y))
n -= m.sum()
df = n-2
if df < 0:
return (masked, masked)
(mx, my) = (x.mean(), y.mean())
(xm, ym) = (x-mx, y-my)
r_num = ma.add.reduce(xm*ym)
r_den = ma.sqrt(ma.dot(xm,xm) * ma.dot(ym,ym))
r = r_num / r_den
# Presumably, if r > 1, then it is only some small artifact of floating
# point arithmetic.
r = min(r, 1.0)
r = max(r, -1.0)
if r is masked or abs(r) == 1.0:
prob = 0.
else:
t_squared = (df / ((1.0 - r) * (1.0 + r))) * r * r
prob = _betai(0.5*df, 0.5, df/(df + t_squared))
return r, prob
SpearmanrResult = namedtuple('SpearmanrResult', ('correlation', 'pvalue'))
def spearmanr(x, y, use_ties=True):
"""
Calculates a Spearman rank-order correlation coefficient and the p-value
to test for non-correlation.
The Spearman correlation is a nonparametric measure of the linear
relationship between two datasets. Unlike the Pearson correlation, the
Spearman correlation does not assume that both datasets are normally
distributed. Like other correlation coefficients, this one varies
between -1 and +1 with 0 implying no correlation. Correlations of -1 or
+1 imply a monotonic relationship. Positive correlations imply that
as `x` increases, so does `y`. Negative correlations imply that as `x`
increases, `y` decreases.
Missing values are discarded pair-wise: if a value is missing in `x`, the
corresponding value in `y` is masked.
The p-value roughly indicates the probability of an uncorrelated system
producing datasets that have a Spearman correlation at least as extreme
as the one computed from these datasets. The p-values are not entirely
reliable but are probably reasonable for datasets larger than 500 or so.
Parameters
----------
x : array_like
The length of `x` must be > 2.
y : array_like
The length of `y` must be > 2.
use_ties : bool, optional
Whether the correction for ties should be computed.
Returns
-------
correlation : float
Spearman correlation coefficient
pvalue : float
2-tailed p-value.
References
----------
[CRCProbStat2000] section 14.7
"""
(x, y, n) = _chk_size(x, y)
(x, y) = (x.ravel(), y.ravel())
m = ma.mask_or(ma.getmask(x), ma.getmask(y))
# need int() here, otherwise numpy defaults to 32 bit
# integer on all Windows architectures, causing overflow.
# int() will keep it infinite precision.
n -= int(m.sum())
if m is not nomask:
x = ma.array(x, mask=m, copy=True)
y = ma.array(y, mask=m, copy=True)
df = n-2
if df < 0:
raise ValueError("The input must have at least 3 entries!")
# Gets the ranks and rank differences
rankx = rankdata(x)
ranky = rankdata(y)
dsq = np.add.reduce((rankx-ranky)**2)
# Tie correction
if use_ties:
xties = count_tied_groups(x)
yties = count_tied_groups(y)
corr_x = sum(v*k*(k**2-1) for (k,v) in iteritems(xties))/12.
corr_y = sum(v*k*(k**2-1) for (k,v) in iteritems(yties))/12.
else:
corr_x = corr_y = 0
denom = n*(n**2 - 1)/6.
if corr_x != 0 or corr_y != 0:
rho = denom - dsq - corr_x - corr_y
rho /= ma.sqrt((denom-2*corr_x)*(denom-2*corr_y))
else:
rho = 1. - dsq/denom
t = ma.sqrt(ma.divide(df,(rho+1.0)*(1.0-rho))) * rho
if t is masked:
prob = 0.
else:
prob = _betai(0.5*df, 0.5, df/(df + t * t))
return SpearmanrResult(rho, prob)
KendalltauResult = namedtuple('KendalltauResult', ('correlation', 'pvalue'))
def kendalltau(x, y, use_ties=True, use_missing=False):
"""
Computes Kendall's rank correlation tau on two variables *x* and *y*.
Parameters
----------
x : sequence
First data list (for example, time).
y : sequence
Second data list.
use_ties : {True, False}, optional
Whether ties correction should be performed.
use_missing : {False, True}, optional
Whether missing data should be allocated a rank of 0 (False) or the
average rank (True)
Returns
-------
correlation : float
Kendall tau
pvalue : float
Approximate 2-side p-value.
"""
(x, y, n) = _chk_size(x, y)
(x, y) = (x.flatten(), y.flatten())
m = ma.mask_or(ma.getmask(x), ma.getmask(y))
if m is not nomask:
x = ma.array(x, mask=m, copy=True)
y = ma.array(y, mask=m, copy=True)
# need int() here, otherwise numpy defaults to 32 bit
# integer on all Windows architectures, causing overflow.
# int() will keep it infinite precision.
n -= int(m.sum())
if n < 2:
return KendalltauResult(np.nan, np.nan)
rx = ma.masked_equal(rankdata(x, use_missing=use_missing), 0)
ry = ma.masked_equal(rankdata(y, use_missing=use_missing), 0)
idx = rx.argsort()
(rx, ry) = (rx[idx], ry[idx])
C = np.sum([((ry[i+1:] > ry[i]) * (rx[i+1:] > rx[i])).filled(0).sum()
for i in range(len(ry)-1)], dtype=float)
D = np.sum([((ry[i+1:] < ry[i])*(rx[i+1:] > rx[i])).filled(0).sum()
for i in range(len(ry)-1)], dtype=float)
if use_ties:
xties = count_tied_groups(x)
yties = count_tied_groups(y)
corr_x = np.sum([v*k*(k-1) for (k,v) in iteritems(xties)], dtype=float)
corr_y = np.sum([v*k*(k-1) for (k,v) in iteritems(yties)], dtype=float)
denom = ma.sqrt((n*(n-1)-corr_x)/2. * (n*(n-1)-corr_y)/2.)
else:
denom = n*(n-1)/2.
tau = (C-D) / denom
var_s = n*(n-1)*(2*n+5)
if use_ties:
var_s -= sum(v*k*(k-1)*(2*k+5)*1. for (k,v) in iteritems(xties))
var_s -= sum(v*k*(k-1)*(2*k+5)*1. for (k,v) in iteritems(yties))
v1 = np.sum([v*k*(k-1) for (k, v) in iteritems(xties)], dtype=float) *\
np.sum([v*k*(k-1) for (k, v) in iteritems(yties)], dtype=float)
v1 /= 2.*n*(n-1)
if n > 2:
v2 = np.sum([v*k*(k-1)*(k-2) for (k,v) in iteritems(xties)],
dtype=float) * \
np.sum([v*k*(k-1)*(k-2) for (k,v) in iteritems(yties)],
dtype=float)
v2 /= 9.*n*(n-1)*(n-2)
else:
v2 = 0
else:
v1 = v2 = 0
var_s /= 18.
var_s += (v1 + v2)
z = (C-D)/np.sqrt(var_s)
prob = special.erfc(abs(z)/np.sqrt(2))
return KendalltauResult(tau, prob)
def kendalltau_seasonal(x):
"""
Computes a multivariate Kendall's rank correlation tau, for seasonal data.
Parameters
----------
x : 2-D ndarray
Array of seasonal data, with seasons in columns.
"""
x = ma.array(x, subok=True, copy=False, ndmin=2)
(n,m) = x.shape
n_p = x.count(0)
S_szn = sum(msign(x[i:]-x[i]).sum(0) for i in range(n))
S_tot = S_szn.sum()
n_tot = x.count()
ties = count_tied_groups(x.compressed())
corr_ties = sum(v*k*(k-1) for (k,v) in iteritems(ties))
denom_tot = ma.sqrt(1.*n_tot*(n_tot-1)*(n_tot*(n_tot-1)-corr_ties))/2.
R = rankdata(x, axis=0, use_missing=True)
K = ma.empty((m,m), dtype=int)
covmat = ma.empty((m,m), dtype=float)
denom_szn = ma.empty(m, dtype=float)
for j in range(m):
ties_j = count_tied_groups(x[:,j].compressed())
corr_j = sum(v*k*(k-1) for (k,v) in iteritems(ties_j))
cmb = n_p[j]*(n_p[j]-1)
for k in range(j,m,1):
K[j,k] = sum(msign((x[i:,j]-x[i,j])*(x[i:,k]-x[i,k])).sum()
for i in range(n))
covmat[j,k] = (K[j,k] + 4*(R[:,j]*R[:,k]).sum() -
n*(n_p[j]+1)*(n_p[k]+1))/3.
K[k,j] = K[j,k]
covmat[k,j] = covmat[j,k]
denom_szn[j] = ma.sqrt(cmb*(cmb-corr_j)) / 2.
var_szn = covmat.diagonal()
z_szn = msign(S_szn) * (abs(S_szn)-1) / ma.sqrt(var_szn)
z_tot_ind = msign(S_tot) * (abs(S_tot)-1) / ma.sqrt(var_szn.sum())
z_tot_dep = msign(S_tot) * (abs(S_tot)-1) / ma.sqrt(covmat.sum())
prob_szn = special.erfc(abs(z_szn)/np.sqrt(2))
prob_tot_ind = special.erfc(abs(z_tot_ind)/np.sqrt(2))
prob_tot_dep = special.erfc(abs(z_tot_dep)/np.sqrt(2))
chi2_tot = (z_szn*z_szn).sum()
chi2_trd = m * z_szn.mean()**2
output = {'seasonal tau': S_szn/denom_szn,
'global tau': S_tot/denom_tot,
'global tau (alt)': S_tot/denom_szn.sum(),
'seasonal p-value': prob_szn,
'global p-value (indep)': prob_tot_ind,
'global p-value (dep)': prob_tot_dep,
'chi2 total': chi2_tot,
'chi2 trend': chi2_trd,
}
return output
PointbiserialrResult = namedtuple('PointbiserialrResult', ('correlation',
'pvalue'))
def pointbiserialr(x, y):
"""Calculates a point biserial correlation coefficient and its p-value.
Parameters
----------
x : array_like of bools
Input array.
y : array_like
Input array.
Returns
-------
correlation : float
R value
pvalue : float
2-tailed p-value
Notes
-----
Missing values are considered pair-wise: if a value is missing in x,
the corresponding value in y is masked.
For more details on `pointbiserialr`, see `stats.pointbiserialr`.
"""
x = ma.fix_invalid(x, copy=True).astype(bool)
y = ma.fix_invalid(y, copy=True).astype(float)
# Get rid of the missing data
m = ma.mask_or(ma.getmask(x), ma.getmask(y))
if m is not nomask:
unmask = np.logical_not(m)
x = x[unmask]
y = y[unmask]
n = len(x)
# phat is the fraction of x values that are True
phat = x.sum() / float(n)
y0 = y[~x] # y-values where x is False
y1 = y[x] # y-values where x is True
y0m = y0.mean()
y1m = y1.mean()
rpb = (y1m - y0m)*np.sqrt(phat * (1-phat)) / y.std()
df = n-2
t = rpb*ma.sqrt(df/(1.0-rpb**2))
prob = _betai(0.5*df, 0.5, df/(df+t*t))
return PointbiserialrResult(rpb, prob)
LinregressResult = namedtuple('LinregressResult', ('slope', 'intercept',
'rvalue', 'pvalue',
'stderr'))
def linregress(x, y=None):
"""
Linear regression calculation
Note that the non-masked version is used, and that this docstring is
replaced by the non-masked docstring + some info on missing data.
"""
if y is None:
x = ma.array(x)
if x.shape[0] == 2:
x, y = x
elif x.shape[1] == 2:
x, y = x.T
else:
msg = ("If only `x` is given as input, it has to be of shape "
"(2, N) or (N, 2), provided shape was %s" % str(x.shape))
raise ValueError(msg)
else:
x = ma.array(x)
y = ma.array(y)
x = x.flatten()
y = y.flatten()
m = ma.mask_or(ma.getmask(x), ma.getmask(y), shrink=False)
if m is not nomask:
x = ma.array(x, mask=m)
y = ma.array(y, mask=m)
if np.any(~m):
slope, intercept, r, prob, sterrest = stats_linregress(x.data[~m],
y.data[~m])
else:
# All data is masked
return None, None, None, None, None
else:
slope, intercept, r, prob, sterrest = stats_linregress(x.data, y.data)
return LinregressResult(slope, intercept, r, prob, sterrest)
if stats_linregress.__doc__:
linregress.__doc__ = stats_linregress.__doc__ + genmissingvaldoc
def theilslopes(y, x=None, alpha=0.95):
r"""
Computes the Theil-Sen estimator for a set of points (x, y).
`theilslopes` implements a method for robust linear regression. It
computes the slope as the median of all slopes between paired values.
Parameters
----------
y : array_like
Dependent variable.
x : array_like or None, optional
Independent variable. If None, use ``arange(len(y))`` instead.
alpha : float, optional
Confidence degree between 0 and 1. Default is 95% confidence.
Note that `alpha` is symmetric around 0.5, i.e. both 0.1 and 0.9 are
interpreted as "find the 90% confidence interval".
Returns
-------
medslope : float
Theil slope.
medintercept : float
Intercept of the Theil line, as ``median(y) - medslope*median(x)``.
lo_slope : float
Lower bound of the confidence interval on `medslope`.
up_slope : float
Upper bound of the confidence interval on `medslope`.
Notes
-----
For more details on `theilslopes`, see `stats.theilslopes`.
"""
y = ma.asarray(y).flatten()
if x is None:
x = ma.arange(len(y), dtype=float)
else:
x = ma.asarray(x).flatten()
if len(x) != len(y):
raise ValueError("Incompatible lengths ! (%s<>%s)" % (len(y),len(x)))
m = ma.mask_or(ma.getmask(x), ma.getmask(y))
y._mask = x._mask = m
# Disregard any masked elements of x or y
y = y.compressed()
x = x.compressed().astype(float)
# We now have unmasked arrays so can use `stats.theilslopes`
return stats_theilslopes(y, x, alpha=alpha)
def sen_seasonal_slopes(x):
x = ma.array(x, subok=True, copy=False, ndmin=2)
(n,_) = x.shape
# Get list of slopes per season
szn_slopes = ma.vstack([(x[i+1:]-x[i])/np.arange(1,n-i)[:,None]
for i in range(n)])
szn_medslopes = ma.median(szn_slopes, axis=0)
medslope = ma.median(szn_slopes, axis=None)
return szn_medslopes, medslope
Ttest_1sampResult = namedtuple('Ttest_1sampResult', ('statistic', 'pvalue'))
def ttest_1samp(a, popmean, axis=0):
"""
Calculates the T-test for the mean of ONE group of scores.
Parameters
----------
a : array_like
sample observation
popmean : float or array_like
expected value in null hypothesis, if array_like than it must have the
same shape as `a` excluding the axis dimension
axis : int or None, optional
Axis along which to compute test. If None, compute over the whole
array `a`.
Returns
-------
statistic : float or array
t-statistic
pvalue : float or array
two-tailed p-value
Notes
-----
For more details on `ttest_1samp`, see `stats.ttest_1samp`.
"""
a, axis = _chk_asarray(a, axis)
if a.size == 0:
return (np.nan, np.nan)
x = a.mean(axis=axis)
v = a.var(axis=axis, ddof=1)
n = a.count(axis=axis)
# force df to be an array for masked division not to throw a warning
df = ma.asanyarray(n - 1.0)
svar = ((n - 1.0) * v) / df
with np.errstate(divide='ignore', invalid='ignore'):
t = (x - popmean) / ma.sqrt(svar / n)
prob = special.betainc(0.5*df, 0.5, df/(df + t*t))
return Ttest_1sampResult(t, prob)
ttest_onesamp = ttest_1samp
Ttest_indResult = namedtuple('Ttest_indResult', ('statistic', 'pvalue'))
def ttest_ind(a, b, axis=0, equal_var=True):
"""
Calculates the T-test for the means of TWO INDEPENDENT samples of scores.
Parameters
----------
a, b : array_like
The arrays must have the same shape, except in the dimension
corresponding to `axis` (the first, by default).
axis : int or None, optional
Axis along which to compute test. If None, compute over the whole
arrays, `a`, and `b`.
equal_var : bool, optional
If True, perform a standard independent 2 sample test that assumes equal
population variances.
If False, perform Welch's t-test, which does not assume equal population
variance.
.. versionadded:: 0.17.0
Returns
-------
statistic : float or array
The calculated t-statistic.
pvalue : float or array
The two-tailed p-value.
Notes
-----
For more details on `ttest_ind`, see `stats.ttest_ind`.
"""
a, b, axis = _chk2_asarray(a, b, axis)
if a.size == 0 or b.size == 0:
return Ttest_indResult(np.nan, np.nan)
(x1, x2) = (a.mean(axis), b.mean(axis))
(v1, v2) = (a.var(axis=axis, ddof=1), b.var(axis=axis, ddof=1))
(n1, n2) = (a.count(axis), b.count(axis))
if equal_var:
# force df to be an array for masked division not to throw a warning
df = ma.asanyarray(n1 + n2 - 2.0)
svar = ((n1-1)*v1+(n2-1)*v2) / df
denom = ma.sqrt(svar*(1.0/n1 + 1.0/n2)) # n-D computation here!
else:
vn1 = v1/n1
vn2 = v2/n2
with np.errstate(divide='ignore', invalid='ignore'):
df = (vn1 + vn2)**2 / (vn1**2 / (n1 - 1) + vn2**2 / (n2 - 1))
# If df is undefined, variances are zero.
# It doesn't matter what df is as long as it is not NaN.
df = np.where(np.isnan(df), 1, df)
denom = ma.sqrt(vn1 + vn2)
with np.errstate(divide='ignore', invalid='ignore'):
t = (x1-x2) / denom
probs = special.betainc(0.5*df, 0.5, df/(df + t*t)).reshape(t.shape)
return Ttest_indResult(t, probs.squeeze())
Ttest_relResult = namedtuple('Ttest_relResult', ('statistic', 'pvalue'))
def ttest_rel(a, b, axis=0):
"""
Calculates the T-test on TWO RELATED samples of scores, a and b.
Parameters
----------
a, b : array_like
The arrays must have the same shape.
axis : int or None, optional
Axis along which to compute test. If None, compute over the whole
arrays, `a`, and `b`.
Returns
-------
statistic : float or array
t-statistic
pvalue : float or array
two-tailed p-value
Notes
-----
For more details on `ttest_rel`, see `stats.ttest_rel`.
"""
a, b, axis = _chk2_asarray(a, b, axis)
if len(a) != len(b):
raise ValueError('unequal length arrays')
if a.size == 0 or b.size == 0:
return Ttest_relResult(np.nan, np.nan)
n = a.count(axis)
df = ma.asanyarray(n-1.0)
d = (a-b).astype('d')
dm = d.mean(axis)
v = d.var(axis=axis, ddof=1)
denom = ma.sqrt(v / n)
with np.errstate(divide='ignore', invalid='ignore'):
t = dm / denom
probs = special.betainc(0.5*df, 0.5, df/(df + t*t)).reshape(t.shape).squeeze()
return Ttest_relResult(t, probs)
MannwhitneyuResult = namedtuple('MannwhitneyuResult', ('statistic',
'pvalue'))
def mannwhitneyu(x,y, use_continuity=True):
"""
Computes the Mann-Whitney statistic
Missing values in `x` and/or `y` are discarded.
Parameters
----------
x : sequence
Input
y : sequence
Input
use_continuity : {True, False}, optional
Whether a continuity correction (1/2.) should be taken into account.
Returns
-------
statistic : float
The Mann-Whitney statistics
pvalue : float
Approximate p-value assuming a normal distribution.
"""
x = ma.asarray(x).compressed().view(ndarray)
y = ma.asarray(y).compressed().view(ndarray)
ranks = rankdata(np.concatenate([x,y]))
(nx, ny) = (len(x), len(y))
nt = nx + ny
U = ranks[:nx].sum() - nx*(nx+1)/2.
U = max(U, nx*ny - U)
u = nx*ny - U
mu = (nx*ny)/2.
sigsq = (nt**3 - nt)/12.
ties = count_tied_groups(ranks)
sigsq -= sum(v*(k**3-k) for (k,v) in iteritems(ties))/12.
sigsq *= nx*ny/float(nt*(nt-1))
if use_continuity:
z = (U - 1/2. - mu) / ma.sqrt(sigsq)
else:
z = (U - mu) / ma.sqrt(sigsq)
prob = special.erfc(abs(z)/np.sqrt(2))
return MannwhitneyuResult(u, prob)
KruskalResult = namedtuple('KruskalResult', ('statistic', 'pvalue'))
def kruskal(*args):
"""
Compute the Kruskal-Wallis H-test for independent samples
Parameters
----------
sample1, sample2, ... : array_like
Two or more arrays with the sample measurements can be given as
arguments.
Returns
-------
statistic : float
The Kruskal-Wallis H statistic, corrected for ties
pvalue : float
The p-value for the test using the assumption that H has a chi
square distribution
Notes
-----
For more details on `kruskal`, see `stats.kruskal`.
"""
output = argstoarray(*args)
ranks = ma.masked_equal(rankdata(output, use_missing=False), 0)
sumrk = ranks.sum(-1)
ngrp = ranks.count(-1)
ntot = ranks.count()
H = 12./(ntot*(ntot+1)) * (sumrk**2/ngrp).sum() - 3*(ntot+1)
# Tie correction
ties = count_tied_groups(ranks)
T = 1. - sum(v*(k**3-k) for (k,v) in iteritems(ties))/float(ntot**3-ntot)
if T == 0:
raise ValueError('All numbers are identical in kruskal')
H /= T
df = len(output) - 1
prob = distributions.chi2.sf(H, df)
return KruskalResult(H, prob)
kruskalwallis = kruskal
def ks_twosamp(data1, data2, alternative="two-sided"):
"""
Computes the Kolmogorov-Smirnov test on two samples.
Missing values are discarded.
Parameters
----------
data1 : array_like
First data set
data2 : array_like
Second data set
alternative : {'two-sided', 'less', 'greater'}, optional
Indicates the alternative hypothesis. Default is 'two-sided'.
Returns
-------
d : float
Value of the Kolmogorov Smirnov test
p : float
Corresponding p-value.
"""
(data1, data2) = (ma.asarray(data1), ma.asarray(data2))
(n1, n2) = (data1.count(), data2.count())
n = (n1*n2/float(n1+n2))
mix = ma.concatenate((data1.compressed(), data2.compressed()))
mixsort = mix.argsort(kind='mergesort')
csum = np.where(mixsort < n1, 1./n1, -1./n2).cumsum()
# Check for ties
if len(np.unique(mix)) < (n1+n2):
csum = csum[np.r_[np.diff(mix[mixsort]).nonzero()[0],-1]]
alternative = str(alternative).lower()[0]
if alternative == 't':
d = ma.abs(csum).max()
prob = special.kolmogorov(np.sqrt(n)*d)
elif alternative == 'l':
d = -csum.min()
prob = np.exp(-2*n*d**2)
elif alternative == 'g':
d = csum.max()
prob = np.exp(-2*n*d**2)
else:
raise ValueError("Invalid value for the alternative hypothesis: "
"should be in 'two-sided', 'less' or 'greater'")
return (d, prob)
ks_2samp = ks_twosamp
def trima(a, limits=None, inclusive=(True,True)):
"""
Trims an array by masking the data outside some given limits.
Returns a masked version of the input array.
Parameters
----------
a : array_like
Input array.
limits : {None, tuple}, optional
Tuple of (lower limit, upper limit) in absolute values.
Values of the input array lower (greater) than the lower (upper) limit
will be masked. A limit is None indicates an open interval.
inclusive : (bool, bool) tuple, optional
Tuple of (lower flag, upper flag), indicating whether values exactly
equal to the lower (upper) limit are allowed.
"""
a = ma.asarray(a)
a.unshare_mask()
if (limits is None) or (limits == (None, None)):
return a
(lower_lim, upper_lim) = limits
(lower_in, upper_in) = inclusive
condition = False
if lower_lim is not None:
if lower_in:
condition |= (a < lower_lim)
else:
condition |= (a <= lower_lim)
if upper_lim is not None:
if upper_in:
condition |= (a > upper_lim)
else:
condition |= (a >= upper_lim)
a[condition.filled(True)] = masked
return a
def trimr(a, limits=None, inclusive=(True, True), axis=None):
"""
Trims an array by masking some proportion of the data on each end.
Returns a masked version of the input array.
Parameters
----------
a : sequence
Input array.
limits : {None, tuple}, optional
Tuple of the percentages to cut on each side of the array, with respect
to the number of unmasked data, as floats between 0. and 1.
Noting n the number of unmasked data before trimming, the
(n*limits[0])th smallest data and the (n*limits[1])th largest data are
masked, and the total number of unmasked data after trimming is
n*(1.-sum(limits)). The value of one limit can be set to None to
indicate an open interval.
inclusive : {(True,True) tuple}, optional
Tuple of flags indicating whether the number of data being masked on
the left (right) end should be truncated (True) or rounded (False) to
integers.
axis : {None,int}, optional
Axis along which to trim. If None, the whole array is trimmed, but its
shape is maintained.
"""
def _trimr1D(a, low_limit, up_limit, low_inclusive, up_inclusive):
n = a.count()
idx = a.argsort()
if low_limit:
if low_inclusive:
lowidx = int(low_limit*n)
else:
lowidx = np.round(low_limit*n)
a[idx[:lowidx]] = masked
if up_limit is not None:
if up_inclusive:
upidx = n - int(n*up_limit)
else:
upidx = n - np.round(n*up_limit)
a[idx[upidx:]] = masked
return a
a = ma.asarray(a)
a.unshare_mask()
if limits is None:
return a
# Check the limits
(lolim, uplim) = limits
errmsg = "The proportion to cut from the %s should be between 0. and 1."
if lolim is not None:
if lolim > 1. or lolim < 0:
raise ValueError(errmsg % 'beginning' + "(got %s)" % lolim)
if uplim is not None:
if uplim > 1. or uplim < 0:
raise ValueError(errmsg % 'end' + "(got %s)" % uplim)
(loinc, upinc) = inclusive
if axis is None:
shp = a.shape
return _trimr1D(a.ravel(),lolim,uplim,loinc,upinc).reshape(shp)
else:
return ma.apply_along_axis(_trimr1D, axis, a, lolim,uplim,loinc,upinc)
trimdoc = """
Parameters
----------
a : sequence
Input array
limits : {None, tuple}, optional
If `relative` is False, tuple (lower limit, upper limit) in absolute values.
Values of the input array lower (greater) than the lower (upper) limit are
masked.
If `relative` is True, tuple (lower percentage, upper percentage) to cut
on each side of the array, with respect to the number of unmasked data.
Noting n the number of unmasked data before trimming, the (n*limits[0])th
smallest data and the (n*limits[1])th largest data are masked, and the
total number of unmasked data after trimming is n*(1.-sum(limits))
In each case, the value of one limit can be set to None to indicate an
open interval.
If limits is None, no trimming is performed
inclusive : {(bool, bool) tuple}, optional
If `relative` is False, tuple indicating whether values exactly equal
to the absolute limits are allowed.
If `relative` is True, tuple indicating whether the number of data
being masked on each side should be rounded (True) or truncated
(False).
relative : bool, optional
Whether to consider the limits as absolute values (False) or proportions
to cut (True).
axis : int, optional
Axis along which to trim.
"""
def trim(a, limits=None, inclusive=(True,True), relative=False, axis=None):
"""
Trims an array by masking the data outside some given limits.
Returns a masked version of the input array.
%s
Examples
--------
>>> from scipy.stats.mstats import trim
>>> z = [ 1, 2, 3, 4, 5, 6, 7, 8, 9,10]
>>> print(trim(z,(3,8)))
[-- -- 3 4 5 6 7 8 -- --]
>>> print(trim(z,(0.1,0.2),relative=True))
[-- 2 3 4 5 6 7 8 -- --]
"""
if relative:
return trimr(a, limits=limits, inclusive=inclusive, axis=axis)
else:
return trima(a, limits=limits, inclusive=inclusive)
if trim.__doc__ is not None:
trim.__doc__ = trim.__doc__ % trimdoc
def trimboth(data, proportiontocut=0.2, inclusive=(True,True), axis=None):
"""
Trims the smallest and largest data values.
Trims the `data` by masking the ``int(proportiontocut * n)`` smallest and
``int(proportiontocut * n)`` largest values of data along the given axis,
where n is the number of unmasked values before trimming.
Parameters
----------
data : ndarray
Data to trim.
proportiontocut : float, optional
Percentage of trimming (as a float between 0 and 1).
If n is the number of unmasked values before trimming, the number of
values after trimming is ``(1 - 2*proportiontocut) * n``.
Default is 0.2.
inclusive : {(bool, bool) tuple}, optional
Tuple indicating whether the number of data being masked on each side
should be rounded (True) or truncated (False).
axis : int, optional
Axis along which to perform the trimming.
If None, the input array is first flattened.
"""
return trimr(data, limits=(proportiontocut,proportiontocut),
inclusive=inclusive, axis=axis)
def trimtail(data, proportiontocut=0.2, tail='left', inclusive=(True,True),
axis=None):
"""
Trims the data by masking values from one tail.
Parameters
----------
data : array_like
Data to trim.
proportiontocut : float, optional
Percentage of trimming. If n is the number of unmasked values
before trimming, the number of values after trimming is
``(1 - proportiontocut) * n``. Default is 0.2.
tail : {'left','right'}, optional
If 'left' the `proportiontocut` lowest values will be masked.
If 'right' the `proportiontocut` highest values will be masked.
Default is 'left'.
inclusive : {(bool, bool) tuple}, optional
Tuple indicating whether the number of data being masked on each side
should be rounded (True) or truncated (False). Default is
(True, True).
axis : int, optional
Axis along which to perform the trimming.
If None, the input array is first flattened. Default is None.
Returns
-------
trimtail : ndarray
Returned array of same shape as `data` with masked tail values.
"""
tail = str(tail).lower()[0]
if tail == 'l':
limits = (proportiontocut,None)
elif tail == 'r':
limits = (None, proportiontocut)
else:
raise TypeError("The tail argument should be in ('left','right')")
return trimr(data, limits=limits, axis=axis, inclusive=inclusive)
trim1 = trimtail
def trimmed_mean(a, limits=(0.1,0.1), inclusive=(1,1), relative=True,
axis=None):
"""Returns the trimmed mean of the data along the given axis.
%s
""" % trimdoc
if (not isinstance(limits,tuple)) and isinstance(limits,float):
limits = (limits, limits)
if relative:
return trimr(a,limits=limits,inclusive=inclusive,axis=axis).mean(axis=axis)
else:
return trima(a,limits=limits,inclusive=inclusive).mean(axis=axis)
def trimmed_var(a, limits=(0.1,0.1), inclusive=(1,1), relative=True,
axis=None, ddof=0):
"""Returns the trimmed variance of the data along the given axis.
%s
ddof : {0,integer}, optional
Means Delta Degrees of Freedom. The denominator used during computations
is (n-ddof). DDOF=0 corresponds to a biased estimate, DDOF=1 to an un-
biased estimate of the variance.
""" % trimdoc
if (not isinstance(limits,tuple)) and isinstance(limits,float):
limits = (limits, limits)
if relative:
out = trimr(a,limits=limits, inclusive=inclusive,axis=axis)
else:
out = trima(a,limits=limits,inclusive=inclusive)
return out.var(axis=axis, ddof=ddof)
def trimmed_std(a, limits=(0.1,0.1), inclusive=(1,1), relative=True,
axis=None, ddof=0):
"""Returns the trimmed standard deviation of the data along the given axis.
%s
ddof : {0,integer}, optional
Means Delta Degrees of Freedom. The denominator used during computations
is (n-ddof). DDOF=0 corresponds to a biased estimate, DDOF=1 to an un-
biased estimate of the variance.
""" % trimdoc
if (not isinstance(limits,tuple)) and isinstance(limits,float):
limits = (limits, limits)
if relative:
out = trimr(a,limits=limits,inclusive=inclusive,axis=axis)
else:
out = trima(a,limits=limits,inclusive=inclusive)
return out.std(axis=axis,ddof=ddof)
def trimmed_stde(a, limits=(0.1,0.1), inclusive=(1,1), axis=None):
"""
Returns the standard error of the trimmed mean along the given axis.
Parameters
----------
a : sequence
Input array
limits : {(0.1,0.1), tuple of float}, optional
tuple (lower percentage, upper percentage) to cut on each side of the
array, with respect to the number of unmasked data.
If n is the number of unmasked data before trimming, the values
smaller than ``n * limits[0]`` and the values larger than
``n * `limits[1]`` are masked, and the total number of unmasked
data after trimming is ``n * (1.-sum(limits))``. In each case,
the value of one limit can be set to None to indicate an open interval.
If `limits` is None, no trimming is performed.
inclusive : {(bool, bool) tuple} optional
Tuple indicating whether the number of data being masked on each side
should be rounded (True) or truncated (False).
axis : int, optional
Axis along which to trim.
Returns
-------
trimmed_stde : scalar or ndarray
"""
def _trimmed_stde_1D(a, low_limit, up_limit, low_inclusive, up_inclusive):
"Returns the standard error of the trimmed mean for a 1D input data."
n = a.count()
idx = a.argsort()
if low_limit:
if low_inclusive:
lowidx = int(low_limit*n)
else:
lowidx = np.round(low_limit*n)
a[idx[:lowidx]] = masked
if up_limit is not None:
if up_inclusive:
upidx = n - int(n*up_limit)
else:
upidx = n - np.round(n*up_limit)
a[idx[upidx:]] = masked
a[idx[:lowidx]] = a[idx[lowidx]]
a[idx[upidx:]] = a[idx[upidx-1]]
winstd = a.std(ddof=1)
return winstd / ((1-low_limit-up_limit)*np.sqrt(len(a)))
a = ma.array(a, copy=True, subok=True)
a.unshare_mask()
if limits is None:
return a.std(axis=axis,ddof=1)/ma.sqrt(a.count(axis))
if (not isinstance(limits,tuple)) and isinstance(limits,float):
limits = (limits, limits)
# Check the limits
(lolim, uplim) = limits
errmsg = "The proportion to cut from the %s should be between 0. and 1."
if lolim is not None:
if lolim > 1. or lolim < 0:
raise ValueError(errmsg % 'beginning' + "(got %s)" % lolim)
if uplim is not None:
if uplim > 1. or uplim < 0:
raise ValueError(errmsg % 'end' + "(got %s)" % uplim)
(loinc, upinc) = inclusive
if (axis is None):
return _trimmed_stde_1D(a.ravel(),lolim,uplim,loinc,upinc)
else:
if a.ndim > 2:
raise ValueError("Array 'a' must be at most two dimensional, but got a.ndim = %d" % a.ndim)
return ma.apply_along_axis(_trimmed_stde_1D, axis, a,
lolim,uplim,loinc,upinc)
def _mask_to_limits(a, limits, inclusive):
"""Mask an array for values outside of given limits.
This is primarily a utility function.
Parameters
----------
a : array
limits : (float or None, float or None)
A tuple consisting of the (lower limit, upper limit). Values in the
input array less than the lower limit or greater than the upper limit
will be masked out. None implies no limit.
inclusive : (bool, bool)
A tuple consisting of the (lower flag, upper flag). These flags
determine whether values exactly equal to lower or upper are allowed.
Returns
-------
A MaskedArray.
Raises
------
A ValueError if there are no values within the given limits.
"""
lower_limit, upper_limit = limits
lower_include, upper_include = inclusive
am = ma.MaskedArray(a)
if lower_limit is not None:
if lower_include:
am = ma.masked_less(am, lower_limit)
else:
am = ma.masked_less_equal(am, lower_limit)
if upper_limit is not None:
if upper_include:
am = ma.masked_greater(am, upper_limit)
else:
am = ma.masked_greater_equal(am, upper_limit)
if am.count() == 0:
raise ValueError("No array values within given limits")
return am
def tmean(a, limits=None, inclusive=(True, True), axis=None):
"""
Compute the trimmed mean.
Parameters
----------
a : array_like
Array of values.
limits : None or (lower limit, upper limit), optional
Values in the input array less than the lower limit or greater than the
upper limit will be ignored. When limits is None (default), then all
values are used. Either of the limit values in the tuple can also be
None representing a half-open interval.
inclusive : (bool, bool), optional
A tuple consisting of the (lower flag, upper flag). These flags
determine whether values exactly equal to the lower or upper limits
are included. The default value is (True, True).
axis : int or None, optional
Axis along which to operate. If None, compute over the
whole array. Default is None.
Returns
-------
tmean : float
Notes
-----
For more details on `tmean`, see `stats.tmean`.
"""
return trima(a, limits=limits, inclusive=inclusive).mean(axis=axis)
def tvar(a, limits=None, inclusive=(True, True), axis=0, ddof=1):
"""
Compute the trimmed variance
This function computes the sample variance of an array of values,
while ignoring values which are outside of given `limits`.
Parameters
----------
a : array_like
Array of values.
limits : None or (lower limit, upper limit), optional
Values in the input array less than the lower limit or greater than the
upper limit will be ignored. When limits is None, then all values are
used. Either of the limit values in the tuple can also be None
representing a half-open interval. The default value is None.
inclusive : (bool, bool), optional
A tuple consisting of the (lower flag, upper flag). These flags
determine whether values exactly equal to the lower or upper limits
are included. The default value is (True, True).
axis : int or None, optional
Axis along which to operate. If None, compute over the
whole array. Default is zero.
ddof : int, optional
Delta degrees of freedom. Default is 1.
Returns
-------
tvar : float
Trimmed variance.
Notes
-----
For more details on `tvar`, see `stats.tvar`.
"""
a = a.astype(float).ravel()
if limits is None:
n = (~a.mask).sum() # todo: better way to do that?
return np.ma.var(a) * n/(n-1.)
am = _mask_to_limits(a, limits=limits, inclusive=inclusive)
return np.ma.var(am, axis=axis, ddof=ddof)
def tmin(a, lowerlimit=None, axis=0, inclusive=True):
"""
Compute the trimmed minimum
Parameters
----------
a : array_like
array of values
lowerlimit : None or float, optional
Values in the input array less than the given limit will be ignored.
When lowerlimit is None, then all values are used. The default value
is None.
axis : int or None, optional
Axis along which to operate. Default is 0. If None, compute over the
whole array `a`.
inclusive : {True, False}, optional
This flag determines whether values exactly equal to the lower limit
are included. The default value is True.
Returns
-------
tmin : float, int or ndarray
Notes
-----
For more details on `tmin`, see `stats.tmin`.
"""
a, axis = _chk_asarray(a, axis)
am = trima(a, (lowerlimit, None), (inclusive, False))
return ma.minimum.reduce(am, axis)
def tmax(a, upperlimit=None, axis=0, inclusive=True):
"""
Compute the trimmed maximum
This function computes the maximum value of an array along a given axis,
while ignoring values larger than a specified upper limit.
Parameters
----------
a : array_like
array of values
upperlimit : None or float, optional
Values in the input array greater than the given limit will be ignored.
When upperlimit is None, then all values are used. The default value
is None.
axis : int or None, optional
Axis along which to operate. Default is 0. If None, compute over the
whole array `a`.
inclusive : {True, False}, optional
This flag determines whether values exactly equal to the upper limit
are included. The default value is True.
Returns
-------
tmax : float, int or ndarray
Notes
-----
For more details on `tmax`, see `stats.tmax`.
"""
a, axis = _chk_asarray(a, axis)
am = trima(a, (None, upperlimit), (False, inclusive))
return ma.maximum.reduce(am, axis)
def tsem(a, limits=None, inclusive=(True, True), axis=0, ddof=1):
"""
Compute the trimmed standard error of the mean.
This function finds the standard error of the mean for given
values, ignoring values outside the given `limits`.
Parameters
----------
a : array_like
array of values
limits : None or (lower limit, upper limit), optional
Values in the input array less than the lower limit or greater than the
upper limit will be ignored. When limits is None, then all values are
used. Either of the limit values in the tuple can also be None
representing a half-open interval. The default value is None.
inclusive : (bool, bool), optional
A tuple consisting of the (lower flag, upper flag). These flags
determine whether values exactly equal to the lower or upper limits
are included. The default value is (True, True).
axis : int or None, optional
Axis along which to operate. If None, compute over the
whole array. Default is zero.
ddof : int, optional
Delta degrees of freedom. Default is 1.
Returns
-------
tsem : float
Notes
-----
For more details on `tsem`, see `stats.tsem`.
"""
a = ma.asarray(a).ravel()
if limits is None:
n = float(a.count())
return a.std(axis=axis, ddof=ddof)/ma.sqrt(n)
am = trima(a.ravel(), limits, inclusive)
sd = np.sqrt(am.var(axis=axis, ddof=ddof))
return sd / np.sqrt(am.count())
def winsorize(a, limits=None, inclusive=(True, True), inplace=False,
axis=None):
"""Returns a Winsorized version of the input array.
The (limits[0])th lowest values are set to the (limits[0])th percentile,
and the (limits[1])th highest values are set to the (1 - limits[1])th
percentile.
Masked values are skipped.
Parameters
----------
a : sequence
Input array.
limits : {None, tuple of float}, optional
Tuple of the percentages to cut on each side of the array, with respect
to the number of unmasked data, as floats between 0. and 1.
Noting n the number of unmasked data before trimming, the
(n*limits[0])th smallest data and the (n*limits[1])th largest data are
masked, and the total number of unmasked data after trimming
is n*(1.-sum(limits)) The value of one limit can be set to None to
indicate an open interval.
inclusive : {(True, True) tuple}, optional
Tuple indicating whether the number of data being masked on each side
should be truncated (True) or rounded (False).
inplace : {False, True}, optional
Whether to winsorize in place (True) or to use a copy (False)
axis : {None, int}, optional
Axis along which to trim. If None, the whole array is trimmed, but its
shape is maintained.
Notes
-----
This function is applied to reduce the effect of possibly spurious outliers
by limiting the extreme values.
"""
def _winsorize1D(a, low_limit, up_limit, low_include, up_include):
n = a.count()
idx = a.argsort()
if low_limit:
if low_include:
lowidx = int(low_limit * n)
else:
lowidx = np.round(low_limit * n).astype(int)
a[idx[:lowidx]] = a[idx[lowidx]]
if up_limit is not None:
if up_include:
upidx = n - int(n * up_limit)
else:
upidx = n - np.round(n * up_limit).astype(int)
a[idx[upidx:]] = a[idx[upidx - 1]]
return a
# We are going to modify a: better make a copy
a = ma.array(a, copy=np.logical_not(inplace))
if limits is None:
return a
if (not isinstance(limits, tuple)) and isinstance(limits, float):
limits = (limits, limits)
# Check the limits
(lolim, uplim) = limits
errmsg = "The proportion to cut from the %s should be between 0. and 1."
if lolim is not None:
if lolim > 1. or lolim < 0:
raise ValueError(errmsg % 'beginning' + "(got %s)" % lolim)
if uplim is not None:
if uplim > 1. or uplim < 0:
raise ValueError(errmsg % 'end' + "(got %s)" % uplim)
(loinc, upinc) = inclusive
if axis is None:
shp = a.shape
return _winsorize1D(a.ravel(), lolim, uplim, loinc, upinc).reshape(shp)
else:
return ma.apply_along_axis(_winsorize1D, axis, a, lolim, uplim, loinc,
upinc)
def moment(a, moment=1, axis=0):
"""
Calculates the nth moment about the mean for a sample.
Parameters
----------
a : array_like
data
moment : int, optional
order of central moment that is returned
axis : int or None, optional
Axis along which the central moment is computed. Default is 0.
If None, compute over the whole array `a`.
Returns
-------
n-th central moment : ndarray or float
The appropriate moment along the given axis or over all values if axis
is None. The denominator for the moment calculation is the number of
observations, no degrees of freedom correction is done.
Notes
-----
For more details about `moment`, see `stats.moment`.
"""
a, axis = _chk_asarray(a, axis)
if moment == 1:
# By definition the first moment about the mean is 0.
shape = list(a.shape)
del shape[axis]
if shape:
# return an actual array of the appropriate shape
return np.zeros(shape, dtype=float)
else:
# the input was 1D, so return a scalar instead of a rank-0 array
return np.float64(0.0)
else:
# Exponentiation by squares: form exponent sequence
n_list = [moment]
current_n = moment
while current_n > 2:
if current_n % 2:
current_n = (current_n-1)/2
else:
current_n /= 2
n_list.append(current_n)
# Starting point for exponentiation by squares
a_zero_mean = a - ma.expand_dims(a.mean(axis), axis)
if n_list[-1] == 1:
s = a_zero_mean.copy()
else:
s = a_zero_mean**2
# Perform multiplications
for n in n_list[-2::-1]:
s = s**2
if n % 2:
s *= a_zero_mean
return s.mean(axis)
def variation(a, axis=0):
"""
Computes the coefficient of variation, the ratio of the biased standard
deviation to the mean.
Parameters
----------
a : array_like
Input array.
axis : int or None, optional
Axis along which to calculate the coefficient of variation. Default
is 0. If None, compute over the whole array `a`.
Returns
-------
variation : ndarray
The calculated variation along the requested axis.
Notes
-----
For more details about `variation`, see `stats.variation`.
"""
a, axis = _chk_asarray(a, axis)
return a.std(axis)/a.mean(axis)
def skew(a, axis=0, bias=True):
"""
Computes the skewness of a data set.
Parameters
----------
a : ndarray
data
axis : int or None, optional
Axis along which skewness is calculated. Default is 0.
If None, compute over the whole array `a`.
bias : bool, optional
If False, then the calculations are corrected for statistical bias.
Returns
-------
skewness : ndarray
The skewness of values along an axis, returning 0 where all values are
equal.
Notes
-----
For more details about `skew`, see `stats.skew`.
"""
a, axis = _chk_asarray(a,axis)
n = a.count(axis)
m2 = moment(a, 2, axis)
m3 = moment(a, 3, axis)
olderr = np.seterr(all='ignore')
try:
vals = ma.where(m2 == 0, 0, m3 / m2**1.5)
finally:
np.seterr(**olderr)
if not bias:
can_correct = (n > 2) & (m2 > 0)
if can_correct.any():
m2 = np.extract(can_correct, m2)
m3 = np.extract(can_correct, m3)
nval = ma.sqrt((n-1.0)*n)/(n-2.0)*m3/m2**1.5
np.place(vals, can_correct, nval)
return vals
def kurtosis(a, axis=0, fisher=True, bias=True):
"""
Computes the kurtosis (Fisher or Pearson) of a dataset.
Kurtosis is the fourth central moment divided by the square of the
variance. If Fisher's definition is used, then 3.0 is subtracted from
the result to give 0.0 for a normal distribution.
If bias is False then the kurtosis is calculated using k statistics to
eliminate bias coming from biased moment estimators
Use `kurtosistest` to see if result is close enough to normal.
Parameters
----------
a : array
data for which the kurtosis is calculated
axis : int or None, optional
Axis along which the kurtosis is calculated. Default is 0.
If None, compute over the whole array `a`.
fisher : bool, optional
If True, Fisher's definition is used (normal ==> 0.0). If False,
Pearson's definition is used (normal ==> 3.0).
bias : bool, optional
If False, then the calculations are corrected for statistical bias.
Returns
-------
kurtosis : array
The kurtosis of values along an axis. If all values are equal,
return -3 for Fisher's definition and 0 for Pearson's definition.
Notes
-----
For more details about `kurtosis`, see `stats.kurtosis`.
"""
a, axis = _chk_asarray(a, axis)
m2 = moment(a, 2, axis)
m4 = moment(a, 4, axis)
olderr = np.seterr(all='ignore')
try:
vals = ma.where(m2 == 0, 0, m4 / m2**2.0)
finally:
np.seterr(**olderr)
if not bias:
n = a.count(axis)
can_correct = (n > 3) & (m2 is not ma.masked and m2 > 0)
if can_correct.any():
n = np.extract(can_correct, n)
m2 = np.extract(can_correct, m2)
m4 = np.extract(can_correct, m4)
nval = 1.0/(n-2)/(n-3)*((n*n-1.0)*m4/m2**2.0-3*(n-1)**2.0)
np.place(vals, can_correct, nval+3.0)
if fisher:
return vals - 3
else:
return vals
DescribeResult = namedtuple('DescribeResult', ('nobs', 'minmax', 'mean',
'variance', 'skewness',
'kurtosis'))
def describe(a, axis=0, ddof=0, bias=True):
"""
Computes several descriptive statistics of the passed array.
Parameters
----------
a : array_like
Data array
axis : int or None, optional
Axis along which to calculate statistics. Default 0. If None,
compute over the whole array `a`.
ddof : int, optional
degree of freedom (default 0); note that default ddof is different
from the same routine in stats.describe
bias : bool, optional
If False, then the skewness and kurtosis calculations are corrected for
statistical bias.
Returns
-------
nobs : int
(size of the data (discarding missing values)
minmax : (int, int)
min, max
mean : float
arithmetic mean
variance : float
unbiased variance
skewness : float
biased skewness
kurtosis : float
biased kurtosis
Examples
--------
>>> from scipy.stats.mstats import describe
>>> ma = np.ma.array(range(6), mask=[0, 0, 0, 1, 1, 1])
>>> describe(ma)
DescribeResult(nobs=3, minmax=(masked_array(data=0,
mask=False,
fill_value=999999), masked_array(data=2,
mask=False,
fill_value=999999)), mean=1.0, variance=0.6666666666666666,
skewness=masked_array(data=0., mask=False, fill_value=1e+20),
kurtosis=-1.5)
"""
a, axis = _chk_asarray(a, axis)
n = a.count(axis)
mm = (ma.minimum.reduce(a), ma.maximum.reduce(a))
m = a.mean(axis)
v = a.var(axis, ddof=ddof)
sk = skew(a, axis, bias=bias)
kurt = kurtosis(a, axis, bias=bias)
return DescribeResult(n, mm, m, v, sk, kurt)
def stde_median(data, axis=None):
"""Returns the McKean-Schrader estimate of the standard error of the sample
median along the given axis. masked values are discarded.
Parameters
----------
data : ndarray
Data to trim.
axis : {None,int}, optional
Axis along which to perform the trimming.
If None, the input array is first flattened.
"""
def _stdemed_1D(data):
data = np.sort(data.compressed())
n = len(data)
z = 2.5758293035489004
k = int(np.round((n+1)/2. - z * np.sqrt(n/4.),0))
return ((data[n-k] - data[k-1])/(2.*z))
data = ma.array(data, copy=False, subok=True)
if (axis is None):
return _stdemed_1D(data)
else:
if data.ndim > 2:
raise ValueError("Array 'data' must be at most two dimensional, "
"but got data.ndim = %d" % data.ndim)
return ma.apply_along_axis(_stdemed_1D, axis, data)
SkewtestResult = namedtuple('SkewtestResult', ('statistic', 'pvalue'))
def skewtest(a, axis=0):
"""
Tests whether the skew is different from the normal distribution.
Parameters
----------
a : array
The data to be tested
axis : int or None, optional
Axis along which statistics are calculated. Default is 0.
If None, compute over the whole array `a`.
Returns
-------
statistic : float
The computed z-score for this test.
pvalue : float
a 2-sided p-value for the hypothesis test
Notes
-----
For more details about `skewtest`, see `stats.skewtest`.
"""
a, axis = _chk_asarray(a, axis)
if axis is None:
a = a.ravel()
axis = 0
b2 = skew(a,axis)
n = a.count(axis)
if np.min(n) < 8:
raise ValueError(
"skewtest is not valid with less than 8 samples; %i samples"
" were given." % np.min(n))
y = b2 * ma.sqrt(((n+1)*(n+3)) / (6.0*(n-2)))
beta2 = (3.0*(n*n+27*n-70)*(n+1)*(n+3)) / ((n-2.0)*(n+5)*(n+7)*(n+9))
W2 = -1 + ma.sqrt(2*(beta2-1))
delta = 1/ma.sqrt(0.5*ma.log(W2))
alpha = ma.sqrt(2.0/(W2-1))
y = ma.where(y == 0, 1, y)
Z = delta*ma.log(y/alpha + ma.sqrt((y/alpha)**2+1))
return SkewtestResult(Z, 2 * distributions.norm.sf(np.abs(Z)))
KurtosistestResult = namedtuple('KurtosistestResult', ('statistic',
'pvalue'))
def kurtosistest(a, axis=0):
"""
Tests whether a dataset has normal kurtosis
Parameters
----------
a : array
array of the sample data
axis : int or None, optional
Axis along which to compute test. Default is 0. If None,
compute over the whole array `a`.
Returns
-------
statistic : float
The computed z-score for this test.
pvalue : float
The 2-sided p-value for the hypothesis test
Notes
-----
For more details about `kurtosistest`, see `stats.kurtosistest`.
"""
a, axis = _chk_asarray(a, axis)
n = a.count(axis=axis)
if np.min(n) < 5:
raise ValueError(
"kurtosistest requires at least 5 observations; %i observations"
" were given." % np.min(n))
if np.min(n) < 20:
warnings.warn(
"kurtosistest only valid for n>=20 ... continuing anyway, n=%i" %
np.min(n))
b2 = kurtosis(a, axis, fisher=False)
E = 3.0*(n-1) / (n+1)
varb2 = 24.0*n*(n-2.)*(n-3) / ((n+1)*(n+1.)*(n+3)*(n+5))
x = (b2-E)/ma.sqrt(varb2)
sqrtbeta1 = 6.0*(n*n-5*n+2)/((n+7)*(n+9)) * np.sqrt((6.0*(n+3)*(n+5)) /
(n*(n-2)*(n-3)))
A = 6.0 + 8.0/sqrtbeta1 * (2.0/sqrtbeta1 + np.sqrt(1+4.0/(sqrtbeta1**2)))
term1 = 1 - 2./(9.0*A)
denom = 1 + x*ma.sqrt(2/(A-4.0))
if np.ma.isMaskedArray(denom):
# For multi-dimensional array input
denom[denom < 0] = masked
elif denom < 0:
denom = masked
term2 = ma.power((1-2.0/A)/denom,1/3.0)
Z = (term1 - term2) / np.sqrt(2/(9.0*A))
return KurtosistestResult(Z, 2 * distributions.norm.sf(np.abs(Z)))
NormaltestResult = namedtuple('NormaltestResult', ('statistic', 'pvalue'))
def normaltest(a, axis=0):
"""
Tests whether a sample differs from a normal distribution.
Parameters
----------
a : array_like
The array containing the data to be tested.
axis : int or None, optional
Axis along which to compute test. Default is 0. If None,
compute over the whole array `a`.
Returns
-------
statistic : float or array
``s^2 + k^2``, where ``s`` is the z-score returned by `skewtest` and
``k`` is the z-score returned by `kurtosistest`.
pvalue : float or array
A 2-sided chi squared probability for the hypothesis test.
Notes
-----
For more details about `normaltest`, see `stats.normaltest`.
"""
a, axis = _chk_asarray(a, axis)
s, _ = skewtest(a, axis)
k, _ = kurtosistest(a, axis)
k2 = s*s + k*k
return NormaltestResult(k2, distributions.chi2.sf(k2, 2))
def mquantiles(a, prob=list([.25,.5,.75]), alphap=.4, betap=.4, axis=None,
limit=()):
"""
Computes empirical quantiles for a data array.
Samples quantile are defined by ``Q(p) = (1-gamma)*x[j] + gamma*x[j+1]``,
where ``x[j]`` is the j-th order statistic, and gamma is a function of
``j = floor(n*p + m)``, ``m = alphap + p*(1 - alphap - betap)`` and
``g = n*p + m - j``.
Reinterpreting the above equations to compare to **R** lead to the
equation: ``p(k) = (k - alphap)/(n + 1 - alphap - betap)``
Typical values of (alphap,betap) are:
- (0,1) : ``p(k) = k/n`` : linear interpolation of cdf
(**R** type 4)
- (.5,.5) : ``p(k) = (k - 1/2.)/n`` : piecewise linear function
(**R** type 5)
- (0,0) : ``p(k) = k/(n+1)`` :
(**R** type 6)
- (1,1) : ``p(k) = (k-1)/(n-1)``: p(k) = mode[F(x[k])].
(**R** type 7, **R** default)
- (1/3,1/3): ``p(k) = (k-1/3)/(n+1/3)``: Then p(k) ~ median[F(x[k])].
The resulting quantile estimates are approximately median-unbiased
regardless of the distribution of x.
(**R** type 8)
- (3/8,3/8): ``p(k) = (k-3/8)/(n+1/4)``: Blom.
The resulting quantile estimates are approximately unbiased
if x is normally distributed
(**R** type 9)
- (.4,.4) : approximately quantile unbiased (Cunnane)
- (.35,.35): APL, used with PWM
Parameters
----------
a : array_like
Input data, as a sequence or array of dimension at most 2.
prob : array_like, optional
List of quantiles to compute.
alphap : float, optional
Plotting positions parameter, default is 0.4.
betap : float, optional
Plotting positions parameter, default is 0.4.
axis : int, optional
Axis along which to perform the trimming.
If None (default), the input array is first flattened.
limit : tuple, optional
Tuple of (lower, upper) values.
Values of `a` outside this open interval are ignored.
Returns
-------
mquantiles : MaskedArray
An array containing the calculated quantiles.
Notes
-----
This formulation is very similar to **R** except the calculation of
``m`` from ``alphap`` and ``betap``, where in **R** ``m`` is defined
with each type.
References
----------
.. [1] *R* statistical software: http://www.r-project.org/
.. [2] *R* ``quantile`` function:
http://stat.ethz.ch/R-manual/R-devel/library/stats/html/quantile.html
Examples
--------
>>> from scipy.stats.mstats import mquantiles
>>> a = np.array([6., 47., 49., 15., 42., 41., 7., 39., 43., 40., 36.])
>>> mquantiles(a)
array([ 19.2, 40. , 42.8])
Using a 2D array, specifying axis and limit.
>>> data = np.array([[ 6., 7., 1.],
... [ 47., 15., 2.],
... [ 49., 36., 3.],
... [ 15., 39., 4.],
... [ 42., 40., -999.],
... [ 41., 41., -999.],
... [ 7., -999., -999.],
... [ 39., -999., -999.],
... [ 43., -999., -999.],
... [ 40., -999., -999.],
... [ 36., -999., -999.]])
>>> print(mquantiles(data, axis=0, limit=(0, 50)))
[[19.2 14.6 1.45]
[40. 37.5 2.5 ]
[42.8 40.05 3.55]]
>>> data[:, 2] = -999.
>>> print(mquantiles(data, axis=0, limit=(0, 50)))
[[19.200000000000003 14.6 --]
[40.0 37.5 --]
[42.800000000000004 40.05 --]]
"""
def _quantiles1D(data,m,p):
x = np.sort(data.compressed())
n = len(x)
if n == 0:
return ma.array(np.empty(len(p), dtype=float), mask=True)
elif n == 1:
return ma.array(np.resize(x, p.shape), mask=nomask)
aleph = (n*p + m)
k = np.floor(aleph.clip(1, n-1)).astype(int)
gamma = (aleph-k).clip(0,1)
return (1.-gamma)*x[(k-1).tolist()] + gamma*x[k.tolist()]
data = ma.array(a, copy=False)
if data.ndim > 2:
raise TypeError("Array should be 2D at most !")
if limit:
condition = (limit[0] < data) & (data < limit[1])
data[~condition.filled(True)] = masked
p = np.array(prob, copy=False, ndmin=1)
m = alphap + p*(1.-alphap-betap)
# Computes quantiles along axis (or globally)
if (axis is None):
return _quantiles1D(data, m, p)
return ma.apply_along_axis(_quantiles1D, axis, data, m, p)
def scoreatpercentile(data, per, limit=(), alphap=.4, betap=.4):
"""Calculate the score at the given 'per' percentile of the
sequence a. For example, the score at per=50 is the median.
This function is a shortcut to mquantile
"""
if (per < 0) or (per > 100.):
raise ValueError("The percentile should be between 0. and 100. !"
" (got %s)" % per)
return mquantiles(data, prob=[per/100.], alphap=alphap, betap=betap,
limit=limit, axis=0).squeeze()
def plotting_positions(data, alpha=0.4, beta=0.4):
"""
Returns plotting positions (or empirical percentile points) for the data.
Plotting positions are defined as ``(i-alpha)/(n+1-alpha-beta)``, where:
- i is the rank order statistics
- n is the number of unmasked values along the given axis
- `alpha` and `beta` are two parameters.
Typical values for `alpha` and `beta` are:
- (0,1) : ``p(k) = k/n``, linear interpolation of cdf (R, type 4)
- (.5,.5) : ``p(k) = (k-1/2.)/n``, piecewise linear function
(R, type 5)
- (0,0) : ``p(k) = k/(n+1)``, Weibull (R type 6)
- (1,1) : ``p(k) = (k-1)/(n-1)``, in this case,
``p(k) = mode[F(x[k])]``. That's R default (R type 7)
- (1/3,1/3): ``p(k) = (k-1/3)/(n+1/3)``, then
``p(k) ~ median[F(x[k])]``.
The resulting quantile estimates are approximately median-unbiased
regardless of the distribution of x. (R type 8)
- (3/8,3/8): ``p(k) = (k-3/8)/(n+1/4)``, Blom.
The resulting quantile estimates are approximately unbiased
if x is normally distributed (R type 9)
- (.4,.4) : approximately quantile unbiased (Cunnane)
- (.35,.35): APL, used with PWM
- (.3175, .3175): used in scipy.stats.probplot
Parameters
----------
data : array_like
Input data, as a sequence or array of dimension at most 2.
alpha : float, optional
Plotting positions parameter. Default is 0.4.
beta : float, optional
Plotting positions parameter. Default is 0.4.
Returns
-------
positions : MaskedArray
The calculated plotting positions.
"""
data = ma.array(data, copy=False).reshape(1,-1)
n = data.count()
plpos = np.empty(data.size, dtype=float)
plpos[n:] = 0
plpos[data.argsort(axis=None)[:n]] = ((np.arange(1, n+1) - alpha) /
(n + 1.0 - alpha - beta))
return ma.array(plpos, mask=data._mask)
meppf = plotting_positions
def obrientransform(*args):
"""
Computes a transform on input data (any number of columns). Used to
test for homogeneity of variance prior to running one-way stats. Each
array in ``*args`` is one level of a factor. If an `f_oneway()` run on
the transformed data and found significant, variances are unequal. From
Maxwell and Delaney, p.112.
Returns: transformed data for use in an ANOVA
"""
data = argstoarray(*args).T
v = data.var(axis=0,ddof=1)
m = data.mean(0)
n = data.count(0).astype(float)
# result = ((N-1.5)*N*(a-m)**2 - 0.5*v*(n-1))/((n-1)*(n-2))
data -= m
data **= 2
data *= (n-1.5)*n
data -= 0.5*v*(n-1)
data /= (n-1.)*(n-2.)
if not ma.allclose(v,data.mean(0)):
raise ValueError("Lack of convergence in obrientransform.")
return data
def sem(a, axis=0, ddof=1):
"""
Calculates the standard error of the mean of the input array.
Also sometimes called standard error of measurement.
Parameters
----------
a : array_like
An array containing the values for which the standard error is
returned.
axis : int or None, optional
If axis is None, ravel `a` first. If axis is an integer, this will be
the axis over which to operate. Defaults to 0.
ddof : int, optional
Delta degrees-of-freedom. How many degrees of freedom to adjust
for bias in limited samples relative to the population estimate
of variance. Defaults to 1.
Returns
-------
s : ndarray or float
The standard error of the mean in the sample(s), along the input axis.
Notes
-----
The default value for `ddof` changed in scipy 0.15.0 to be consistent with
`stats.sem` as well as with the most common definition used (like in the R
documentation).
Examples
--------
Find standard error along the first axis:
>>> from scipy import stats
>>> a = np.arange(20).reshape(5,4)
>>> print(stats.mstats.sem(a))
[2.8284271247461903 2.8284271247461903 2.8284271247461903
2.8284271247461903]
Find standard error across the whole array, using n degrees of freedom:
>>> print(stats.mstats.sem(a, axis=None, ddof=0))
1.2893796958227628
"""
a, axis = _chk_asarray(a, axis)
n = a.count(axis=axis)
s = a.std(axis=axis, ddof=ddof) / ma.sqrt(n)
return s
F_onewayResult = namedtuple('F_onewayResult', ('statistic', 'pvalue'))
def f_oneway(*args):
"""
Performs a 1-way ANOVA, returning an F-value and probability given
any number of groups. From Heiman, pp.394-7.
Usage: ``f_oneway(*args)``, where ``*args`` is 2 or more arrays,
one per treatment group.
Returns
-------
statistic : float
The computed F-value of the test.
pvalue : float
The associated p-value from the F-distribution.
"""
# Construct a single array of arguments: each row is a group
data = argstoarray(*args)
ngroups = len(data)
ntot = data.count()
sstot = (data**2).sum() - (data.sum())**2/float(ntot)
ssbg = (data.count(-1) * (data.mean(-1)-data.mean())**2).sum()
sswg = sstot-ssbg
dfbg = ngroups-1
dfwg = ntot - ngroups
msb = ssbg/float(dfbg)
msw = sswg/float(dfwg)
f = msb/msw
prob = special.fdtrc(dfbg, dfwg, f) # equivalent to stats.f.sf
return F_onewayResult(f, prob)
FriedmanchisquareResult = namedtuple('FriedmanchisquareResult',
('statistic', 'pvalue'))
def friedmanchisquare(*args):
"""Friedman Chi-Square is a non-parametric, one-way within-subjects ANOVA.
This function calculates the Friedman Chi-square test for repeated measures
and returns the result, along with the associated probability value.
Each input is considered a given group. Ideally, the number of treatments
among each group should be equal. If this is not the case, only the first
n treatments are taken into account, where n is the number of treatments
of the smallest group.
If a group has some missing values, the corresponding treatments are masked
in the other groups.
The test statistic is corrected for ties.
Masked values in one group are propagated to the other groups.
Returns
-------
statistic : float
the test statistic.
pvalue : float
the associated p-value.
"""
data = argstoarray(*args).astype(float)
k = len(data)
if k < 3:
raise ValueError("Less than 3 groups (%i): " % k +
"the Friedman test is NOT appropriate.")
ranked = ma.masked_values(rankdata(data, axis=0), 0)
if ranked._mask is not nomask:
ranked = ma.mask_cols(ranked)
ranked = ranked.compressed().reshape(k,-1).view(ndarray)
else:
ranked = ranked._data
(k,n) = ranked.shape
# Ties correction
repeats = [find_repeats(row) for row in ranked.T]
ties = np.array([y for x, y in repeats if x.size > 0])
tie_correction = 1 - (ties**3-ties).sum()/float(n*(k**3-k))
ssbg = np.sum((ranked.sum(-1) - n*(k+1)/2.)**2)
chisq = ssbg * 12./(n*k*(k+1)) * 1./tie_correction
return FriedmanchisquareResult(chisq,
distributions.chi2.sf(chisq, k-1))
| 82,352 | 30.456455 | 103 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/__init__.py
|
"""
==========================================
Statistical functions (:mod:`scipy.stats`)
==========================================
.. module:: scipy.stats
This module contains a large number of probability distributions as
well as a growing library of statistical functions.
Each univariate distribution is an instance of a subclass of `rv_continuous`
(`rv_discrete` for discrete distributions):
.. autosummary::
:toctree: generated/
rv_continuous
rv_discrete
rv_histogram
Continuous distributions
========================
.. autosummary::
:toctree: generated/
alpha -- Alpha
anglit -- Anglit
arcsine -- Arcsine
argus -- Argus
beta -- Beta
betaprime -- Beta Prime
bradford -- Bradford
burr -- Burr (Type III)
burr12 -- Burr (Type XII)
cauchy -- Cauchy
chi -- Chi
chi2 -- Chi-squared
cosine -- Cosine
crystalball -- Crystalball
dgamma -- Double Gamma
dweibull -- Double Weibull
erlang -- Erlang
expon -- Exponential
exponnorm -- Exponentially Modified Normal
exponweib -- Exponentiated Weibull
exponpow -- Exponential Power
f -- F (Snecdor F)
fatiguelife -- Fatigue Life (Birnbaum-Saunders)
fisk -- Fisk
foldcauchy -- Folded Cauchy
foldnorm -- Folded Normal
frechet_r -- Deprecated. Alias for weibull_min
frechet_l -- Deprecated. Alias for weibull_max
genlogistic -- Generalized Logistic
gennorm -- Generalized normal
genpareto -- Generalized Pareto
genexpon -- Generalized Exponential
genextreme -- Generalized Extreme Value
gausshyper -- Gauss Hypergeometric
gamma -- Gamma
gengamma -- Generalized gamma
genhalflogistic -- Generalized Half Logistic
gilbrat -- Gilbrat
gompertz -- Gompertz (Truncated Gumbel)
gumbel_r -- Right Sided Gumbel, Log-Weibull, Fisher-Tippett, Extreme Value Type I
gumbel_l -- Left Sided Gumbel, etc.
halfcauchy -- Half Cauchy
halflogistic -- Half Logistic
halfnorm -- Half Normal
halfgennorm -- Generalized Half Normal
hypsecant -- Hyperbolic Secant
invgamma -- Inverse Gamma
invgauss -- Inverse Gaussian
invweibull -- Inverse Weibull
johnsonsb -- Johnson SB
johnsonsu -- Johnson SU
kappa4 -- Kappa 4 parameter
kappa3 -- Kappa 3 parameter
ksone -- Kolmogorov-Smirnov one-sided (no stats)
kstwobign -- Kolmogorov-Smirnov two-sided test for Large N (no stats)
laplace -- Laplace
levy -- Levy
levy_l
levy_stable
logistic -- Logistic
loggamma -- Log-Gamma
loglaplace -- Log-Laplace (Log Double Exponential)
lognorm -- Log-Normal
lomax -- Lomax (Pareto of the second kind)
maxwell -- Maxwell
mielke -- Mielke's Beta-Kappa
moyal -- Moyal
nakagami -- Nakagami
ncx2 -- Non-central chi-squared
ncf -- Non-central F
nct -- Non-central Student's T
norm -- Normal (Gaussian)
norminvgauss -- Normal Inverse Gaussian
pareto -- Pareto
pearson3 -- Pearson type III
powerlaw -- Power-function
powerlognorm -- Power log normal
powernorm -- Power normal
rdist -- R-distribution
reciprocal -- Reciprocal
rayleigh -- Rayleigh
rice -- Rice
recipinvgauss -- Reciprocal Inverse Gaussian
semicircular -- Semicircular
skewnorm -- Skew normal
t -- Student's T
trapz -- Trapezoidal
triang -- Triangular
truncexpon -- Truncated Exponential
truncnorm -- Truncated Normal
tukeylambda -- Tukey-Lambda
uniform -- Uniform
vonmises -- Von-Mises (Circular)
vonmises_line -- Von-Mises (Line)
wald -- Wald
weibull_min -- Minimum Weibull (see Frechet)
weibull_max -- Maximum Weibull (see Frechet)
wrapcauchy -- Wrapped Cauchy
Multivariate distributions
==========================
.. autosummary::
:toctree: generated/
multivariate_normal -- Multivariate normal distribution
matrix_normal -- Matrix normal distribution
dirichlet -- Dirichlet
wishart -- Wishart
invwishart -- Inverse Wishart
multinomial -- Multinomial distribution
special_ortho_group -- SO(N) group
ortho_group -- O(N) group
unitary_group -- U(N) group
random_correlation -- random correlation matrices
Discrete distributions
======================
.. autosummary::
:toctree: generated/
bernoulli -- Bernoulli
binom -- Binomial
boltzmann -- Boltzmann (Truncated Discrete Exponential)
dlaplace -- Discrete Laplacian
geom -- Geometric
hypergeom -- Hypergeometric
logser -- Logarithmic (Log-Series, Series)
nbinom -- Negative Binomial
planck -- Planck (Discrete Exponential)
poisson -- Poisson
randint -- Discrete Uniform
skellam -- Skellam
zipf -- Zipf
Statistical functions
=====================
Several of these functions have a similar version in scipy.stats.mstats
which work for masked arrays.
.. autosummary::
:toctree: generated/
describe -- Descriptive statistics
gmean -- Geometric mean
hmean -- Harmonic mean
kurtosis -- Fisher or Pearson kurtosis
kurtosistest --
mode -- Modal value
moment -- Central moment
normaltest --
skew -- Skewness
skewtest --
kstat --
kstatvar --
tmean -- Truncated arithmetic mean
tvar -- Truncated variance
tmin --
tmax --
tstd --
tsem --
variation -- Coefficient of variation
find_repeats
trim_mean
.. autosummary::
:toctree: generated/
cumfreq
itemfreq
percentileofscore
scoreatpercentile
relfreq
.. autosummary::
:toctree: generated/
binned_statistic -- Compute a binned statistic for a set of data.
binned_statistic_2d -- Compute a 2-D binned statistic for a set of data.
binned_statistic_dd -- Compute a d-D binned statistic for a set of data.
.. autosummary::
:toctree: generated/
obrientransform
bayes_mvs
mvsdist
sem
zmap
zscore
iqr
.. autosummary::
:toctree: generated/
sigmaclip
trimboth
trim1
.. autosummary::
:toctree: generated/
f_oneway
pearsonr
spearmanr
pointbiserialr
kendalltau
weightedtau
linregress
theilslopes
.. autosummary::
:toctree: generated/
ttest_1samp
ttest_ind
ttest_ind_from_stats
ttest_rel
kstest
chisquare
power_divergence
ks_2samp
mannwhitneyu
tiecorrect
rankdata
ranksums
wilcoxon
kruskal
friedmanchisquare
combine_pvalues
jarque_bera
.. autosummary::
:toctree: generated/
ansari
bartlett
levene
shapiro
anderson
anderson_ksamp
binom_test
fligner
median_test
mood
.. autosummary::
:toctree: generated/
boxcox
boxcox_normmax
boxcox_llf
entropy
.. autosummary::
:toctree: generated/
wasserstein_distance
energy_distance
Circular statistical functions
==============================
.. autosummary::
:toctree: generated/
circmean
circvar
circstd
Contingency table functions
===========================
.. autosummary::
:toctree: generated/
chi2_contingency
contingency.expected_freq
contingency.margins
fisher_exact
Plot-tests
==========
.. autosummary::
:toctree: generated/
ppcc_max
ppcc_plot
probplot
boxcox_normplot
Masked statistics functions
===========================
.. toctree::
stats.mstats
Univariate and multivariate kernel density estimation (:mod:`scipy.stats.kde`)
==============================================================================
.. autosummary::
:toctree: generated/
gaussian_kde
For many more stat related functions install the software R and the
interface package rpy.
"""
from __future__ import division, print_function, absolute_import
from .stats import *
from .distributions import *
from .morestats import *
from ._binned_statistic import *
from .kde import gaussian_kde
from . import mstats
from .contingency import chi2_contingency
from ._multivariate import *
__all__ = [s for s in dir() if not s.startswith("_")] # Remove dunders.
from scipy._lib._testutils import PytestTester
test = PytestTester(__name__)
del PytestTester
| 9,284 | 24.86351 | 93 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/_distn_infrastructure.py
|
#
# Author: Travis Oliphant 2002-2011 with contributions from
# SciPy Developers 2004-2011
#
from __future__ import division, print_function, absolute_import
from scipy._lib.six import string_types, exec_, PY3
from scipy._lib._util import getargspec_no_self as _getargspec
import sys
import keyword
import re
import types
import warnings
from scipy.misc import doccer
from ._distr_params import distcont, distdiscrete
from scipy._lib._util import check_random_state, _lazywhere, _lazyselect
from scipy._lib._util import _valarray as valarray
from scipy.special import (comb, chndtr, entr, rel_entr, kl_div, xlogy, ive)
# for root finding for discrete distribution ppf, and max likelihood estimation
from scipy import optimize
# for functions of continuous distributions (e.g. moments, entropy, cdf)
from scipy import integrate
# to approximate the pdf of a continuous distribution given its cdf
from scipy.misc import derivative
from numpy import (arange, putmask, ravel, take, ones, shape, ndarray,
product, reshape, zeros, floor, logical_and, log, sqrt, exp)
from numpy import (place, argsort, argmax, vectorize,
asarray, nan, inf, isinf, NINF, empty)
import numpy as np
from ._constants import _XMAX
if PY3:
def instancemethod(func, obj, cls):
return types.MethodType(func, obj)
else:
instancemethod = types.MethodType
# These are the docstring parts used for substitution in specific
# distribution docstrings
docheaders = {'methods': """\nMethods\n-------\n""",
'notes': """\nNotes\n-----\n""",
'examples': """\nExamples\n--------\n"""}
_doc_rvs = """\
rvs(%(shapes)s, loc=0, scale=1, size=1, random_state=None)
Random variates.
"""
_doc_pdf = """\
pdf(x, %(shapes)s, loc=0, scale=1)
Probability density function.
"""
_doc_logpdf = """\
logpdf(x, %(shapes)s, loc=0, scale=1)
Log of the probability density function.
"""
_doc_pmf = """\
pmf(k, %(shapes)s, loc=0, scale=1)
Probability mass function.
"""
_doc_logpmf = """\
logpmf(k, %(shapes)s, loc=0, scale=1)
Log of the probability mass function.
"""
_doc_cdf = """\
cdf(x, %(shapes)s, loc=0, scale=1)
Cumulative distribution function.
"""
_doc_logcdf = """\
logcdf(x, %(shapes)s, loc=0, scale=1)
Log of the cumulative distribution function.
"""
_doc_sf = """\
sf(x, %(shapes)s, loc=0, scale=1)
Survival function (also defined as ``1 - cdf``, but `sf` is sometimes more accurate).
"""
_doc_logsf = """\
logsf(x, %(shapes)s, loc=0, scale=1)
Log of the survival function.
"""
_doc_ppf = """\
ppf(q, %(shapes)s, loc=0, scale=1)
Percent point function (inverse of ``cdf`` --- percentiles).
"""
_doc_isf = """\
isf(q, %(shapes)s, loc=0, scale=1)
Inverse survival function (inverse of ``sf``).
"""
_doc_moment = """\
moment(n, %(shapes)s, loc=0, scale=1)
Non-central moment of order n
"""
_doc_stats = """\
stats(%(shapes)s, loc=0, scale=1, moments='mv')
Mean('m'), variance('v'), skew('s'), and/or kurtosis('k').
"""
_doc_entropy = """\
entropy(%(shapes)s, loc=0, scale=1)
(Differential) entropy of the RV.
"""
_doc_fit = """\
fit(data, %(shapes)s, loc=0, scale=1)
Parameter estimates for generic data.
"""
_doc_expect = """\
expect(func, args=(%(shapes_)s), loc=0, scale=1, lb=None, ub=None, conditional=False, **kwds)
Expected value of a function (of one argument) with respect to the distribution.
"""
_doc_expect_discrete = """\
expect(func, args=(%(shapes_)s), loc=0, lb=None, ub=None, conditional=False)
Expected value of a function (of one argument) with respect to the distribution.
"""
_doc_median = """\
median(%(shapes)s, loc=0, scale=1)
Median of the distribution.
"""
_doc_mean = """\
mean(%(shapes)s, loc=0, scale=1)
Mean of the distribution.
"""
_doc_var = """\
var(%(shapes)s, loc=0, scale=1)
Variance of the distribution.
"""
_doc_std = """\
std(%(shapes)s, loc=0, scale=1)
Standard deviation of the distribution.
"""
_doc_interval = """\
interval(alpha, %(shapes)s, loc=0, scale=1)
Endpoints of the range that contains alpha percent of the distribution
"""
_doc_allmethods = ''.join([docheaders['methods'], _doc_rvs, _doc_pdf,
_doc_logpdf, _doc_cdf, _doc_logcdf, _doc_sf,
_doc_logsf, _doc_ppf, _doc_isf, _doc_moment,
_doc_stats, _doc_entropy, _doc_fit,
_doc_expect, _doc_median,
_doc_mean, _doc_var, _doc_std, _doc_interval])
_doc_default_longsummary = """\
As an instance of the `rv_continuous` class, `%(name)s` object inherits from it
a collection of generic methods (see below for the full list),
and completes them with details specific for this particular distribution.
"""
_doc_default_frozen_note = """
Alternatively, the object may be called (as a function) to fix the shape,
location, and scale parameters returning a "frozen" continuous RV object:
rv = %(name)s(%(shapes)s, loc=0, scale=1)
- Frozen RV object with the same methods but holding the given shape,
location, and scale fixed.
"""
_doc_default_example = """\
Examples
--------
>>> from scipy.stats import %(name)s
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)
Calculate a few first moments:
%(set_vals_stmt)s
>>> mean, var, skew, kurt = %(name)s.stats(%(shapes)s, moments='mvsk')
Display the probability density function (``pdf``):
>>> x = np.linspace(%(name)s.ppf(0.01, %(shapes)s),
... %(name)s.ppf(0.99, %(shapes)s), 100)
>>> ax.plot(x, %(name)s.pdf(x, %(shapes)s),
... 'r-', lw=5, alpha=0.6, label='%(name)s pdf')
Alternatively, the distribution object can be called (as a function)
to fix the shape, location and scale parameters. This returns a "frozen"
RV object holding the given parameters fixed.
Freeze the distribution and display the frozen ``pdf``:
>>> rv = %(name)s(%(shapes)s)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
Check accuracy of ``cdf`` and ``ppf``:
>>> vals = %(name)s.ppf([0.001, 0.5, 0.999], %(shapes)s)
>>> np.allclose([0.001, 0.5, 0.999], %(name)s.cdf(vals, %(shapes)s))
True
Generate random numbers:
>>> r = %(name)s.rvs(%(shapes)s, size=1000)
And compare the histogram:
>>> ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()
"""
_doc_default_locscale = """\
The probability density above is defined in the "standardized" form. To shift
and/or scale the distribution use the ``loc`` and ``scale`` parameters.
Specifically, ``%(name)s.pdf(x, %(shapes)s, loc, scale)`` is identically
equivalent to ``%(name)s.pdf(y, %(shapes)s) / scale`` with
``y = (x - loc) / scale``.
"""
_doc_default = ''.join([_doc_default_longsummary,
_doc_allmethods,
'\n',
_doc_default_example])
_doc_default_before_notes = ''.join([_doc_default_longsummary,
_doc_allmethods])
docdict = {
'rvs': _doc_rvs,
'pdf': _doc_pdf,
'logpdf': _doc_logpdf,
'cdf': _doc_cdf,
'logcdf': _doc_logcdf,
'sf': _doc_sf,
'logsf': _doc_logsf,
'ppf': _doc_ppf,
'isf': _doc_isf,
'stats': _doc_stats,
'entropy': _doc_entropy,
'fit': _doc_fit,
'moment': _doc_moment,
'expect': _doc_expect,
'interval': _doc_interval,
'mean': _doc_mean,
'std': _doc_std,
'var': _doc_var,
'median': _doc_median,
'allmethods': _doc_allmethods,
'longsummary': _doc_default_longsummary,
'frozennote': _doc_default_frozen_note,
'example': _doc_default_example,
'default': _doc_default,
'before_notes': _doc_default_before_notes,
'after_notes': _doc_default_locscale
}
# Reuse common content between continuous and discrete docs, change some
# minor bits.
docdict_discrete = docdict.copy()
docdict_discrete['pmf'] = _doc_pmf
docdict_discrete['logpmf'] = _doc_logpmf
docdict_discrete['expect'] = _doc_expect_discrete
_doc_disc_methods = ['rvs', 'pmf', 'logpmf', 'cdf', 'logcdf', 'sf', 'logsf',
'ppf', 'isf', 'stats', 'entropy', 'expect', 'median',
'mean', 'var', 'std', 'interval']
for obj in _doc_disc_methods:
docdict_discrete[obj] = docdict_discrete[obj].replace(', scale=1', '')
_doc_disc_methods_err_varname = ['cdf', 'logcdf', 'sf', 'logsf']
for obj in _doc_disc_methods_err_varname:
docdict_discrete[obj] = docdict_discrete[obj].replace('(x, ', '(k, ')
docdict_discrete.pop('pdf')
docdict_discrete.pop('logpdf')
_doc_allmethods = ''.join([docdict_discrete[obj] for obj in _doc_disc_methods])
docdict_discrete['allmethods'] = docheaders['methods'] + _doc_allmethods
docdict_discrete['longsummary'] = _doc_default_longsummary.replace(
'rv_continuous', 'rv_discrete')
_doc_default_frozen_note = """
Alternatively, the object may be called (as a function) to fix the shape and
location parameters returning a "frozen" discrete RV object:
rv = %(name)s(%(shapes)s, loc=0)
- Frozen RV object with the same methods but holding the given shape and
location fixed.
"""
docdict_discrete['frozennote'] = _doc_default_frozen_note
_doc_default_discrete_example = """\
Examples
--------
>>> from scipy.stats import %(name)s
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)
Calculate a few first moments:
%(set_vals_stmt)s
>>> mean, var, skew, kurt = %(name)s.stats(%(shapes)s, moments='mvsk')
Display the probability mass function (``pmf``):
>>> x = np.arange(%(name)s.ppf(0.01, %(shapes)s),
... %(name)s.ppf(0.99, %(shapes)s))
>>> ax.plot(x, %(name)s.pmf(x, %(shapes)s), 'bo', ms=8, label='%(name)s pmf')
>>> ax.vlines(x, 0, %(name)s.pmf(x, %(shapes)s), colors='b', lw=5, alpha=0.5)
Alternatively, the distribution object can be called (as a function)
to fix the shape and location. This returns a "frozen" RV object holding
the given parameters fixed.
Freeze the distribution and display the frozen ``pmf``:
>>> rv = %(name)s(%(shapes)s)
>>> ax.vlines(x, 0, rv.pmf(x), colors='k', linestyles='-', lw=1,
... label='frozen pmf')
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()
Check accuracy of ``cdf`` and ``ppf``:
>>> prob = %(name)s.cdf(x, %(shapes)s)
>>> np.allclose(x, %(name)s.ppf(prob, %(shapes)s))
True
Generate random numbers:
>>> r = %(name)s.rvs(%(shapes)s, size=1000)
"""
_doc_default_discrete_locscale = """\
The probability mass function above is defined in the "standardized" form.
To shift distribution use the ``loc`` parameter.
Specifically, ``%(name)s.pmf(k, %(shapes)s, loc)`` is identically
equivalent to ``%(name)s.pmf(k - loc, %(shapes)s)``.
"""
docdict_discrete['example'] = _doc_default_discrete_example
docdict_discrete['after_notes'] = _doc_default_discrete_locscale
_doc_default_before_notes = ''.join([docdict_discrete['longsummary'],
docdict_discrete['allmethods']])
docdict_discrete['before_notes'] = _doc_default_before_notes
_doc_default_disc = ''.join([docdict_discrete['longsummary'],
docdict_discrete['allmethods'],
docdict_discrete['frozennote'],
docdict_discrete['example']])
docdict_discrete['default'] = _doc_default_disc
# clean up all the separate docstring elements, we do not need them anymore
for obj in [s for s in dir() if s.startswith('_doc_')]:
exec('del ' + obj)
del obj
try:
del s
except NameError:
# in Python 3, loop variables are not visible after the loop
pass
def _moment(data, n, mu=None):
if mu is None:
mu = data.mean()
return ((data - mu)**n).mean()
def _moment_from_stats(n, mu, mu2, g1, g2, moment_func, args):
if (n == 0):
return 1.0
elif (n == 1):
if mu is None:
val = moment_func(1, *args)
else:
val = mu
elif (n == 2):
if mu2 is None or mu is None:
val = moment_func(2, *args)
else:
val = mu2 + mu*mu
elif (n == 3):
if g1 is None or mu2 is None or mu is None:
val = moment_func(3, *args)
else:
mu3 = g1 * np.power(mu2, 1.5) # 3rd central moment
val = mu3+3*mu*mu2+mu*mu*mu # 3rd non-central moment
elif (n == 4):
if g1 is None or g2 is None or mu2 is None or mu is None:
val = moment_func(4, *args)
else:
mu4 = (g2+3.0)*(mu2**2.0) # 4th central moment
mu3 = g1*np.power(mu2, 1.5) # 3rd central moment
val = mu4+4*mu*mu3+6*mu*mu*mu2+mu*mu*mu*mu
else:
val = moment_func(n, *args)
return val
def _skew(data):
"""
skew is third central moment / variance**(1.5)
"""
data = np.ravel(data)
mu = data.mean()
m2 = ((data - mu)**2).mean()
m3 = ((data - mu)**3).mean()
return m3 / np.power(m2, 1.5)
def _kurtosis(data):
"""
kurtosis is fourth central moment / variance**2 - 3
"""
data = np.ravel(data)
mu = data.mean()
m2 = ((data - mu)**2).mean()
m4 = ((data - mu)**4).mean()
return m4 / m2**2 - 3
# Frozen RV class
class rv_frozen(object):
def __init__(self, dist, *args, **kwds):
self.args = args
self.kwds = kwds
# create a new instance
self.dist = dist.__class__(**dist._updated_ctor_param())
# a, b may be set in _argcheck, depending on *args, **kwds. Ouch.
shapes, _, _ = self.dist._parse_args(*args, **kwds)
self.dist._argcheck(*shapes)
self.a, self.b = self.dist.a, self.dist.b
@property
def random_state(self):
return self.dist._random_state
@random_state.setter
def random_state(self, seed):
self.dist._random_state = check_random_state(seed)
def pdf(self, x): # raises AttributeError in frozen discrete distribution
return self.dist.pdf(x, *self.args, **self.kwds)
def logpdf(self, x):
return self.dist.logpdf(x, *self.args, **self.kwds)
def cdf(self, x):
return self.dist.cdf(x, *self.args, **self.kwds)
def logcdf(self, x):
return self.dist.logcdf(x, *self.args, **self.kwds)
def ppf(self, q):
return self.dist.ppf(q, *self.args, **self.kwds)
def isf(self, q):
return self.dist.isf(q, *self.args, **self.kwds)
def rvs(self, size=None, random_state=None):
kwds = self.kwds.copy()
kwds.update({'size': size, 'random_state': random_state})
return self.dist.rvs(*self.args, **kwds)
def sf(self, x):
return self.dist.sf(x, *self.args, **self.kwds)
def logsf(self, x):
return self.dist.logsf(x, *self.args, **self.kwds)
def stats(self, moments='mv'):
kwds = self.kwds.copy()
kwds.update({'moments': moments})
return self.dist.stats(*self.args, **kwds)
def median(self):
return self.dist.median(*self.args, **self.kwds)
def mean(self):
return self.dist.mean(*self.args, **self.kwds)
def var(self):
return self.dist.var(*self.args, **self.kwds)
def std(self):
return self.dist.std(*self.args, **self.kwds)
def moment(self, n):
return self.dist.moment(n, *self.args, **self.kwds)
def entropy(self):
return self.dist.entropy(*self.args, **self.kwds)
def pmf(self, k):
return self.dist.pmf(k, *self.args, **self.kwds)
def logpmf(self, k):
return self.dist.logpmf(k, *self.args, **self.kwds)
def interval(self, alpha):
return self.dist.interval(alpha, *self.args, **self.kwds)
def expect(self, func=None, lb=None, ub=None, conditional=False, **kwds):
# expect method only accepts shape parameters as positional args
# hence convert self.args, self.kwds, also loc/scale
# See the .expect method docstrings for the meaning of
# other parameters.
a, loc, scale = self.dist._parse_args(*self.args, **self.kwds)
if isinstance(self.dist, rv_discrete):
return self.dist.expect(func, a, loc, lb, ub, conditional, **kwds)
else:
return self.dist.expect(func, a, loc, scale, lb, ub,
conditional, **kwds)
# This should be rewritten
def argsreduce(cond, *args):
"""Return the sequence of ravel(args[i]) where ravel(condition) is
True in 1D.
Examples
--------
>>> import numpy as np
>>> rand = np.random.random_sample
>>> A = rand((4, 5))
>>> B = 2
>>> C = rand((1, 5))
>>> cond = np.ones(A.shape)
>>> [A1, B1, C1] = argsreduce(cond, A, B, C)
>>> B1.shape
(20,)
>>> cond[2,:] = 0
>>> [A2, B2, C2] = argsreduce(cond, A, B, C)
>>> B2.shape
(15,)
"""
newargs = np.atleast_1d(*args)
if not isinstance(newargs, list):
newargs = [newargs, ]
expand_arr = (cond == cond)
return [np.extract(cond, arr1 * expand_arr) for arr1 in newargs]
parse_arg_template = """
def _parse_args(self, %(shape_arg_str)s %(locscale_in)s):
return (%(shape_arg_str)s), %(locscale_out)s
def _parse_args_rvs(self, %(shape_arg_str)s %(locscale_in)s, size=None):
return self._argcheck_rvs(%(shape_arg_str)s %(locscale_out)s, size=size)
def _parse_args_stats(self, %(shape_arg_str)s %(locscale_in)s, moments='mv'):
return (%(shape_arg_str)s), %(locscale_out)s, moments
"""
# Both the continuous and discrete distributions depend on ncx2.
# I think the function name ncx2 is an abbreviation for noncentral chi squared.
def _ncx2_log_pdf(x, df, nc):
# We use (xs**2 + ns**2)/2 = (xs - ns)**2/2 + xs*ns, and include the factor
# of exp(-xs*ns) into the ive function to improve numerical stability
# at large values of xs. See also `rice.pdf`.
df2 = df/2.0 - 1.0
xs, ns = np.sqrt(x), np.sqrt(nc)
res = xlogy(df2/2.0, x/nc) - 0.5*(xs - ns)**2
res += np.log(ive(df2, xs*ns) / 2.0)
return res
def _ncx2_pdf(x, df, nc):
return np.exp(_ncx2_log_pdf(x, df, nc))
def _ncx2_cdf(x, df, nc):
return chndtr(x, df, nc)
class rv_generic(object):
"""Class which encapsulates common functionality between rv_discrete
and rv_continuous.
"""
def __init__(self, seed=None):
super(rv_generic, self).__init__()
# figure out if _stats signature has 'moments' keyword
sign = _getargspec(self._stats)
self._stats_has_moments = ((sign[2] is not None) or
('moments' in sign[0]))
self._random_state = check_random_state(seed)
@property
def random_state(self):
""" Get or set the RandomState object for generating random variates.
This can be either None or an existing RandomState object.
If None (or np.random), use the RandomState singleton used by np.random.
If already a RandomState instance, use it.
If an int, use a new RandomState instance seeded with seed.
"""
return self._random_state
@random_state.setter
def random_state(self, seed):
self._random_state = check_random_state(seed)
def __getstate__(self):
return self._updated_ctor_param(), self._random_state
def __setstate__(self, state):
ctor_param, r = state
self.__init__(**ctor_param)
self._random_state = r
return self
def _construct_argparser(
self, meths_to_inspect, locscale_in, locscale_out):
"""Construct the parser for the shape arguments.
Generates the argument-parsing functions dynamically and attaches
them to the instance.
Is supposed to be called in __init__ of a class for each distribution.
If self.shapes is a non-empty string, interprets it as a
comma-separated list of shape parameters.
Otherwise inspects the call signatures of `meths_to_inspect`
and constructs the argument-parsing functions from these.
In this case also sets `shapes` and `numargs`.
"""
if self.shapes:
# sanitize the user-supplied shapes
if not isinstance(self.shapes, string_types):
raise TypeError('shapes must be a string.')
shapes = self.shapes.replace(',', ' ').split()
for field in shapes:
if keyword.iskeyword(field):
raise SyntaxError('keywords cannot be used as shapes.')
if not re.match('^[_a-zA-Z][_a-zA-Z0-9]*$', field):
raise SyntaxError(
'shapes must be valid python identifiers')
else:
# find out the call signatures (_pdf, _cdf etc), deduce shape
# arguments. Generic methods only have 'self, x', any further args
# are shapes.
shapes_list = []
for meth in meths_to_inspect:
shapes_args = _getargspec(meth) # NB: does not contain self
args = shapes_args.args[1:] # peel off 'x', too
if args:
shapes_list.append(args)
# *args or **kwargs are not allowed w/automatic shapes
if shapes_args.varargs is not None:
raise TypeError(
'*args are not allowed w/out explicit shapes')
if shapes_args.keywords is not None:
raise TypeError(
'**kwds are not allowed w/out explicit shapes')
if shapes_args.defaults is not None:
raise TypeError('defaults are not allowed for shapes')
if shapes_list:
shapes = shapes_list[0]
# make sure the signatures are consistent
for item in shapes_list:
if item != shapes:
raise TypeError('Shape arguments are inconsistent.')
else:
shapes = []
# have the arguments, construct the method from template
shapes_str = ', '.join(shapes) + ', ' if shapes else '' # NB: not None
dct = dict(shape_arg_str=shapes_str,
locscale_in=locscale_in,
locscale_out=locscale_out,
)
ns = {}
exec_(parse_arg_template % dct, ns)
# NB: attach to the instance, not class
for name in ['_parse_args', '_parse_args_stats', '_parse_args_rvs']:
setattr(self, name,
instancemethod(ns[name], self, self.__class__)
)
self.shapes = ', '.join(shapes) if shapes else None
if not hasattr(self, 'numargs'):
# allows more general subclassing with *args
self.numargs = len(shapes)
def _construct_doc(self, docdict, shapes_vals=None):
"""Construct the instance docstring with string substitutions."""
tempdict = docdict.copy()
tempdict['name'] = self.name or 'distname'
tempdict['shapes'] = self.shapes or ''
if shapes_vals is None:
shapes_vals = ()
vals = ', '.join('%.3g' % val for val in shapes_vals)
tempdict['vals'] = vals
tempdict['shapes_'] = self.shapes or ''
if self.shapes and self.numargs == 1:
tempdict['shapes_'] += ','
if self.shapes:
tempdict['set_vals_stmt'] = '>>> %s = %s' % (self.shapes, vals)
else:
tempdict['set_vals_stmt'] = ''
if self.shapes is None:
# remove shapes from call parameters if there are none
for item in ['default', 'before_notes']:
tempdict[item] = tempdict[item].replace(
"\n%(shapes)s : array_like\n shape parameters", "")
for i in range(2):
if self.shapes is None:
# necessary because we use %(shapes)s in two forms (w w/o ", ")
self.__doc__ = self.__doc__.replace("%(shapes)s, ", "")
self.__doc__ = doccer.docformat(self.__doc__, tempdict)
# correct for empty shapes
self.__doc__ = self.__doc__.replace('(, ', '(').replace(', )', ')')
def _construct_default_doc(self, longname=None, extradoc=None,
docdict=None, discrete='continuous'):
"""Construct instance docstring from the default template."""
if longname is None:
longname = 'A'
if extradoc is None:
extradoc = ''
if extradoc.startswith('\n\n'):
extradoc = extradoc[2:]
self.__doc__ = ''.join(['%s %s random variable.' % (longname, discrete),
'\n\n%(before_notes)s\n', docheaders['notes'],
extradoc, '\n%(example)s'])
self._construct_doc(docdict)
def freeze(self, *args, **kwds):
"""Freeze the distribution for the given arguments.
Parameters
----------
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution. Should include all
the non-optional arguments, may include ``loc`` and ``scale``.
Returns
-------
rv_frozen : rv_frozen instance
The frozen distribution.
"""
return rv_frozen(self, *args, **kwds)
def __call__(self, *args, **kwds):
return self.freeze(*args, **kwds)
__call__.__doc__ = freeze.__doc__
# The actual calculation functions (no basic checking need be done)
# If these are defined, the others won't be looked at.
# Otherwise, the other set can be defined.
def _stats(self, *args, **kwds):
return None, None, None, None
# Central moments
def _munp(self, n, *args):
# Silence floating point warnings from integration.
olderr = np.seterr(all='ignore')
vals = self.generic_moment(n, *args)
np.seterr(**olderr)
return vals
def _argcheck_rvs(self, *args, **kwargs):
# Handle broadcasting and size validation of the rvs method.
# Subclasses should not have to override this method.
# The rule is that if `size` is not None, then `size` gives the
# shape of the result (integer values of `size` are treated as
# tuples with length 1; i.e. `size=3` is the same as `size=(3,)`.)
#
# `args` is expected to contain the shape parameters (if any), the
# location and the scale in a flat tuple (e.g. if there are two
# shape parameters `a` and `b`, `args` will be `(a, b, loc, scale)`).
# The only keyword argument expected is 'size'.
size = kwargs.get('size', None)
all_bcast = np.broadcast_arrays(*args)
def squeeze_left(a):
while a.ndim > 0 and a.shape[0] == 1:
a = a[0]
return a
# Eliminate trivial leading dimensions. In the convention
# used by numpy's random variate generators, trivial leading
# dimensions are effectively ignored. In other words, when `size`
# is given, trivial leading dimensions of the broadcast parameters
# in excess of the number of dimensions in size are ignored, e.g.
# >>> np.random.normal([[1, 3, 5]], [[[[0.01]]]], size=3)
# array([ 1.00104267, 3.00422496, 4.99799278])
# If `size` is not given, the exact broadcast shape is preserved:
# >>> np.random.normal([[1, 3, 5]], [[[[0.01]]]])
# array([[[[ 1.00862899, 3.00061431, 4.99867122]]]])
#
all_bcast = [squeeze_left(a) for a in all_bcast]
bcast_shape = all_bcast[0].shape
bcast_ndim = all_bcast[0].ndim
if size is None:
size_ = bcast_shape
else:
size_ = tuple(np.atleast_1d(size))
# Check compatibility of size_ with the broadcast shape of all
# the parameters. This check is intended to be consistent with
# how the numpy random variate generators (e.g. np.random.normal,
# np.random.beta) handle their arguments. The rule is that, if size
# is given, it determines the shape of the output. Broadcasting
# can't change the output size.
# This is the standard broadcasting convention of extending the
# shape with fewer dimensions with enough dimensions of length 1
# so that the two shapes have the same number of dimensions.
ndiff = bcast_ndim - len(size_)
if ndiff < 0:
bcast_shape = (1,)*(-ndiff) + bcast_shape
elif ndiff > 0:
size_ = (1,)*ndiff + size_
# This compatibility test is not standard. In "regular" broadcasting,
# two shapes are compatible if for each dimension, the lengths are the
# same or one of the lengths is 1. Here, the length of a dimension in
# size_ must not be less than the corresponding length in bcast_shape.
ok = all([bcdim == 1 or bcdim == szdim
for (bcdim, szdim) in zip(bcast_shape, size_)])
if not ok:
raise ValueError("size does not match the broadcast shape of "
"the parameters.")
param_bcast = all_bcast[:-2]
loc_bcast = all_bcast[-2]
scale_bcast = all_bcast[-1]
return param_bcast, loc_bcast, scale_bcast, size_
## These are the methods you must define (standard form functions)
## NB: generic _pdf, _logpdf, _cdf are different for
## rv_continuous and rv_discrete hence are defined in there
def _argcheck(self, *args):
"""Default check for correct values on args and keywords.
Returns condition array of 1's where arguments are correct and
0's where they are not.
"""
cond = 1
for arg in args:
cond = logical_and(cond, (asarray(arg) > 0))
return cond
def _support_mask(self, x):
return (self.a <= x) & (x <= self.b)
def _open_support_mask(self, x):
return (self.a < x) & (x < self.b)
def _rvs(self, *args):
# This method must handle self._size being a tuple, and it must
# properly broadcast *args and self._size. self._size might be
# an empty tuple, which means a scalar random variate is to be
# generated.
## Use basic inverse cdf algorithm for RV generation as default.
U = self._random_state.random_sample(self._size)
Y = self._ppf(U, *args)
return Y
def _logcdf(self, x, *args):
return log(self._cdf(x, *args))
def _sf(self, x, *args):
return 1.0-self._cdf(x, *args)
def _logsf(self, x, *args):
return log(self._sf(x, *args))
def _ppf(self, q, *args):
return self._ppfvec(q, *args)
def _isf(self, q, *args):
return self._ppf(1.0-q, *args) # use correct _ppf for subclasses
# These are actually called, and should not be overwritten if you
# want to keep error checking.
def rvs(self, *args, **kwds):
"""
Random variates of given type.
Parameters
----------
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information).
loc : array_like, optional
Location parameter (default=0).
scale : array_like, optional
Scale parameter (default=1).
size : int or tuple of ints, optional
Defining number of random variates (default is 1).
random_state : None or int or ``np.random.RandomState`` instance, optional
If int or RandomState, use it for drawing the random variates.
If None, rely on ``self.random_state``.
Default is None.
Returns
-------
rvs : ndarray or scalar
Random variates of given `size`.
"""
discrete = kwds.pop('discrete', None)
rndm = kwds.pop('random_state', None)
args, loc, scale, size = self._parse_args_rvs(*args, **kwds)
cond = logical_and(self._argcheck(*args), (scale >= 0))
if not np.all(cond):
raise ValueError("Domain error in arguments.")
if np.all(scale == 0):
return loc*ones(size, 'd')
# extra gymnastics needed for a custom random_state
if rndm is not None:
random_state_saved = self._random_state
self._random_state = check_random_state(rndm)
# `size` should just be an argument to _rvs(), but for, um,
# historical reasons, it is made an attribute that is read
# by _rvs().
self._size = size
vals = self._rvs(*args)
vals = vals * scale + loc
# do not forget to restore the _random_state
if rndm is not None:
self._random_state = random_state_saved
# Cast to int if discrete
if discrete:
if size == ():
vals = int(vals)
else:
vals = vals.astype(int)
return vals
def stats(self, *args, **kwds):
"""
Some statistics of the given RV.
Parameters
----------
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information)
loc : array_like, optional
location parameter (default=0)
scale : array_like, optional (continuous RVs only)
scale parameter (default=1)
moments : str, optional
composed of letters ['mvsk'] defining which moments to compute:
'm' = mean,
'v' = variance,
's' = (Fisher's) skew,
'k' = (Fisher's) kurtosis.
(default is 'mv')
Returns
-------
stats : sequence
of requested moments.
"""
args, loc, scale, moments = self._parse_args_stats(*args, **kwds)
# scale = 1 by construction for discrete RVs
loc, scale = map(asarray, (loc, scale))
args = tuple(map(asarray, args))
cond = self._argcheck(*args) & (scale > 0) & (loc == loc)
output = []
default = valarray(shape(cond), self.badvalue)
# Use only entries that are valid in calculation
if np.any(cond):
goodargs = argsreduce(cond, *(args+(scale, loc)))
scale, loc, goodargs = goodargs[-2], goodargs[-1], goodargs[:-2]
if self._stats_has_moments:
mu, mu2, g1, g2 = self._stats(*goodargs,
**{'moments': moments})
else:
mu, mu2, g1, g2 = self._stats(*goodargs)
if g1 is None:
mu3 = None
else:
if mu2 is None:
mu2 = self._munp(2, *goodargs)
if g2 is None:
# (mu2**1.5) breaks down for nan and inf
mu3 = g1 * np.power(mu2, 1.5)
if 'm' in moments:
if mu is None:
mu = self._munp(1, *goodargs)
out0 = default.copy()
place(out0, cond, mu * scale + loc)
output.append(out0)
if 'v' in moments:
if mu2 is None:
mu2p = self._munp(2, *goodargs)
if mu is None:
mu = self._munp(1, *goodargs)
mu2 = mu2p - mu * mu
if np.isinf(mu):
# if mean is inf then var is also inf
mu2 = np.inf
out0 = default.copy()
place(out0, cond, mu2 * scale * scale)
output.append(out0)
if 's' in moments:
if g1 is None:
mu3p = self._munp(3, *goodargs)
if mu is None:
mu = self._munp(1, *goodargs)
if mu2 is None:
mu2p = self._munp(2, *goodargs)
mu2 = mu2p - mu * mu
with np.errstate(invalid='ignore'):
mu3 = mu3p - 3 * mu * mu2 - mu**3
g1 = mu3 / np.power(mu2, 1.5)
out0 = default.copy()
place(out0, cond, g1)
output.append(out0)
if 'k' in moments:
if g2 is None:
mu4p = self._munp(4, *goodargs)
if mu is None:
mu = self._munp(1, *goodargs)
if mu2 is None:
mu2p = self._munp(2, *goodargs)
mu2 = mu2p - mu * mu
if mu3 is None:
mu3p = self._munp(3, *goodargs)
with np.errstate(invalid='ignore'):
mu3 = mu3p - 3 * mu * mu2 - mu**3
with np.errstate(invalid='ignore'):
mu4 = mu4p - 4 * mu * mu3 - 6 * mu * mu * mu2 - mu**4
g2 = mu4 / mu2**2.0 - 3.0
out0 = default.copy()
place(out0, cond, g2)
output.append(out0)
else: # no valid args
output = []
for _ in moments:
out0 = default.copy()
output.append(out0)
if len(output) == 1:
return output[0]
else:
return tuple(output)
def entropy(self, *args, **kwds):
"""
Differential entropy of the RV.
Parameters
----------
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information).
loc : array_like, optional
Location parameter (default=0).
scale : array_like, optional (continuous distributions only).
Scale parameter (default=1).
Notes
-----
Entropy is defined base `e`:
>>> drv = rv_discrete(values=((0, 1), (0.5, 0.5)))
>>> np.allclose(drv.entropy(), np.log(2.0))
True
"""
args, loc, scale = self._parse_args(*args, **kwds)
# NB: for discrete distributions scale=1 by construction in _parse_args
args = tuple(map(asarray, args))
cond0 = self._argcheck(*args) & (scale > 0) & (loc == loc)
output = zeros(shape(cond0), 'd')
place(output, (1-cond0), self.badvalue)
goodargs = argsreduce(cond0, *args)
place(output, cond0, self.vecentropy(*goodargs) + log(scale))
return output
def moment(self, n, *args, **kwds):
"""
n-th order non-central moment of distribution.
Parameters
----------
n : int, n >= 1
Order of moment.
arg1, arg2, arg3,... : float
The shape parameter(s) for the distribution (see docstring of the
instance object for more information).
loc : array_like, optional
location parameter (default=0)
scale : array_like, optional
scale parameter (default=1)
"""
args, loc, scale = self._parse_args(*args, **kwds)
if not (self._argcheck(*args) and (scale > 0)):
return nan
if (floor(n) != n):
raise ValueError("Moment must be an integer.")
if (n < 0):
raise ValueError("Moment must be positive.")
mu, mu2, g1, g2 = None, None, None, None
if (n > 0) and (n < 5):
if self._stats_has_moments:
mdict = {'moments': {1: 'm', 2: 'v', 3: 'vs', 4: 'vk'}[n]}
else:
mdict = {}
mu, mu2, g1, g2 = self._stats(*args, **mdict)
val = _moment_from_stats(n, mu, mu2, g1, g2, self._munp, args)
# Convert to transformed X = L + S*Y
# E[X^n] = E[(L+S*Y)^n] = L^n sum(comb(n, k)*(S/L)^k E[Y^k], k=0...n)
if loc == 0:
return scale**n * val
else:
result = 0
fac = float(scale) / float(loc)
for k in range(n):
valk = _moment_from_stats(k, mu, mu2, g1, g2, self._munp, args)
result += comb(n, k, exact=True)*(fac**k) * valk
result += fac**n * val
return result * loc**n
def median(self, *args, **kwds):
"""
Median of the distribution.
Parameters
----------
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information)
loc : array_like, optional
Location parameter, Default is 0.
scale : array_like, optional
Scale parameter, Default is 1.
Returns
-------
median : float
The median of the distribution.
See Also
--------
stats.distributions.rv_discrete.ppf
Inverse of the CDF
"""
return self.ppf(0.5, *args, **kwds)
def mean(self, *args, **kwds):
"""
Mean of the distribution.
Parameters
----------
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information)
loc : array_like, optional
location parameter (default=0)
scale : array_like, optional
scale parameter (default=1)
Returns
-------
mean : float
the mean of the distribution
"""
kwds['moments'] = 'm'
res = self.stats(*args, **kwds)
if isinstance(res, ndarray) and res.ndim == 0:
return res[()]
return res
def var(self, *args, **kwds):
"""
Variance of the distribution.
Parameters
----------
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information)
loc : array_like, optional
location parameter (default=0)
scale : array_like, optional
scale parameter (default=1)
Returns
-------
var : float
the variance of the distribution
"""
kwds['moments'] = 'v'
res = self.stats(*args, **kwds)
if isinstance(res, ndarray) and res.ndim == 0:
return res[()]
return res
def std(self, *args, **kwds):
"""
Standard deviation of the distribution.
Parameters
----------
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information)
loc : array_like, optional
location parameter (default=0)
scale : array_like, optional
scale parameter (default=1)
Returns
-------
std : float
standard deviation of the distribution
"""
kwds['moments'] = 'v'
res = sqrt(self.stats(*args, **kwds))
return res
def interval(self, alpha, *args, **kwds):
"""
Confidence interval with equal areas around the median.
Parameters
----------
alpha : array_like of float
Probability that an rv will be drawn from the returned range.
Each value should be in the range [0, 1].
arg1, arg2, ... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information).
loc : array_like, optional
location parameter, Default is 0.
scale : array_like, optional
scale parameter, Default is 1.
Returns
-------
a, b : ndarray of float
end-points of range that contain ``100 * alpha %`` of the rv's
possible values.
"""
alpha = asarray(alpha)
if np.any((alpha > 1) | (alpha < 0)):
raise ValueError("alpha must be between 0 and 1 inclusive")
q1 = (1.0-alpha)/2
q2 = (1.0+alpha)/2
a = self.ppf(q1, *args, **kwds)
b = self.ppf(q2, *args, **kwds)
return a, b
## continuous random variables: implement maybe later
##
## hf --- Hazard Function (PDF / SF)
## chf --- Cumulative hazard function (-log(SF))
## psf --- Probability sparsity function (reciprocal of the pdf) in
## units of percent-point-function (as a function of q).
## Also, the derivative of the percent-point function.
class rv_continuous(rv_generic):
"""
A generic continuous random variable class meant for subclassing.
`rv_continuous` is a base class to construct specific distribution classes
and instances for continuous random variables. It cannot be used
directly as a distribution.
Parameters
----------
momtype : int, optional
The type of generic moment calculation to use: 0 for pdf, 1 (default)
for ppf.
a : float, optional
Lower bound of the support of the distribution, default is minus
infinity.
b : float, optional
Upper bound of the support of the distribution, default is plus
infinity.
xtol : float, optional
The tolerance for fixed point calculation for generic ppf.
badvalue : float, optional
The value in a result arrays that indicates a value that for which
some argument restriction is violated, default is np.nan.
name : str, optional
The name of the instance. This string is used to construct the default
example for distributions.
longname : str, optional
This string is used as part of the first line of the docstring returned
when a subclass has no docstring of its own. Note: `longname` exists
for backwards compatibility, do not use for new subclasses.
shapes : str, optional
The shape of the distribution. For example ``"m, n"`` for a
distribution that takes two integers as the two shape arguments for all
its methods. If not provided, shape parameters will be inferred from
the signature of the private methods, ``_pdf`` and ``_cdf`` of the
instance.
extradoc : str, optional, deprecated
This string is used as the last part of the docstring returned when a
subclass has no docstring of its own. Note: `extradoc` exists for
backwards compatibility, do not use for new subclasses.
seed : None or int or ``numpy.random.RandomState`` instance, optional
This parameter defines the RandomState object to use for drawing
random variates.
If None (or np.random), the global np.random state is used.
If integer, it is used to seed the local RandomState instance.
Default is None.
Methods
-------
rvs
pdf
logpdf
cdf
logcdf
sf
logsf
ppf
isf
moment
stats
entropy
expect
median
mean
std
var
interval
__call__
fit
fit_loc_scale
nnlf
Notes
-----
Public methods of an instance of a distribution class (e.g., ``pdf``,
``cdf``) check their arguments and pass valid arguments to private,
computational methods (``_pdf``, ``_cdf``). For ``pdf(x)``, ``x`` is valid
if it is within the support of a distribution, ``self.a <= x <= self.b``.
Whether a shape parameter is valid is decided by an ``_argcheck`` method
(which defaults to checking that its arguments are strictly positive.)
**Subclassing**
New random variables can be defined by subclassing the `rv_continuous` class
and re-defining at least the ``_pdf`` or the ``_cdf`` method (normalized
to location 0 and scale 1).
If positive argument checking is not correct for your RV
then you will also need to re-define the ``_argcheck`` method.
Correct, but potentially slow defaults exist for the remaining
methods but for speed and/or accuracy you can over-ride::
_logpdf, _cdf, _logcdf, _ppf, _rvs, _isf, _sf, _logsf
Rarely would you override ``_isf``, ``_sf`` or ``_logsf``, but you could.
**Methods that can be overwritten by subclasses**
::
_rvs
_pdf
_cdf
_sf
_ppf
_isf
_stats
_munp
_entropy
_argcheck
There are additional (internal and private) generic methods that can
be useful for cross-checking and for debugging, but might work in all
cases when directly called.
A note on ``shapes``: subclasses need not specify them explicitly. In this
case, `shapes` will be automatically deduced from the signatures of the
overridden methods (`pdf`, `cdf` etc).
If, for some reason, you prefer to avoid relying on introspection, you can
specify ``shapes`` explicitly as an argument to the instance constructor.
**Frozen Distributions**
Normally, you must provide shape parameters (and, optionally, location and
scale parameters to each call of a method of a distribution.
Alternatively, the object may be called (as a function) to fix the shape,
location, and scale parameters returning a "frozen" continuous RV object:
rv = generic(<shape(s)>, loc=0, scale=1)
frozen RV object with the same methods but holding the given shape,
location, and scale fixed
**Statistics**
Statistics are computed using numerical integration by default.
For speed you can redefine this using ``_stats``:
- take shape parameters and return mu, mu2, g1, g2
- If you can't compute one of these, return it as None
- Can also be defined with a keyword argument ``moments``, which is a
string composed of "m", "v", "s", and/or "k".
Only the components appearing in string should be computed and
returned in the order "m", "v", "s", or "k" with missing values
returned as None.
Alternatively, you can override ``_munp``, which takes ``n`` and shape
parameters and returns the n-th non-central moment of the distribution.
Examples
--------
To create a new Gaussian distribution, we would do the following:
>>> from scipy.stats import rv_continuous
>>> class gaussian_gen(rv_continuous):
... "Gaussian distribution"
... def _pdf(self, x):
... return np.exp(-x**2 / 2.) / np.sqrt(2.0 * np.pi)
>>> gaussian = gaussian_gen(name='gaussian')
``scipy.stats`` distributions are *instances*, so here we subclass
`rv_continuous` and create an instance. With this, we now have
a fully functional distribution with all relevant methods automagically
generated by the framework.
Note that above we defined a standard normal distribution, with zero mean
and unit variance. Shifting and scaling of the distribution can be done
by using ``loc`` and ``scale`` parameters: ``gaussian.pdf(x, loc, scale)``
essentially computes ``y = (x - loc) / scale`` and
``gaussian._pdf(y) / scale``.
"""
def __init__(self, momtype=1, a=None, b=None, xtol=1e-14,
badvalue=None, name=None, longname=None,
shapes=None, extradoc=None, seed=None):
super(rv_continuous, self).__init__(seed)
# save the ctor parameters, cf generic freeze
self._ctor_param = dict(
momtype=momtype, a=a, b=b, xtol=xtol,
badvalue=badvalue, name=name, longname=longname,
shapes=shapes, extradoc=extradoc, seed=seed)
if badvalue is None:
badvalue = nan
if name is None:
name = 'Distribution'
self.badvalue = badvalue
self.name = name
self.a = a
self.b = b
if a is None:
self.a = -inf
if b is None:
self.b = inf
self.xtol = xtol
self.moment_type = momtype
self.shapes = shapes
self._construct_argparser(meths_to_inspect=[self._pdf, self._cdf],
locscale_in='loc=0, scale=1',
locscale_out='loc, scale')
# nin correction
self._ppfvec = vectorize(self._ppf_single, otypes='d')
self._ppfvec.nin = self.numargs + 1
self.vecentropy = vectorize(self._entropy, otypes='d')
self._cdfvec = vectorize(self._cdf_single, otypes='d')
self._cdfvec.nin = self.numargs + 1
self.extradoc = extradoc
if momtype == 0:
self.generic_moment = vectorize(self._mom0_sc, otypes='d')
else:
self.generic_moment = vectorize(self._mom1_sc, otypes='d')
# Because of the *args argument of _mom0_sc, vectorize cannot count the
# number of arguments correctly.
self.generic_moment.nin = self.numargs + 1
if longname is None:
if name[0] in ['aeiouAEIOU']:
hstr = "An "
else:
hstr = "A "
longname = hstr + name
if sys.flags.optimize < 2:
# Skip adding docstrings if interpreter is run with -OO
if self.__doc__ is None:
self._construct_default_doc(longname=longname,
extradoc=extradoc,
docdict=docdict,
discrete='continuous')
else:
dct = dict(distcont)
self._construct_doc(docdict, dct.get(self.name))
def _updated_ctor_param(self):
""" Return the current version of _ctor_param, possibly updated by user.
Used by freezing and pickling.
Keep this in sync with the signature of __init__.
"""
dct = self._ctor_param.copy()
dct['a'] = self.a
dct['b'] = self.b
dct['xtol'] = self.xtol
dct['badvalue'] = self.badvalue
dct['name'] = self.name
dct['shapes'] = self.shapes
dct['extradoc'] = self.extradoc
return dct
def _ppf_to_solve(self, x, q, *args):
return self.cdf(*(x, )+args)-q
def _ppf_single(self, q, *args):
left = right = None
if self.a > -np.inf:
left = self.a
if self.b < np.inf:
right = self.b
factor = 10.
if not left: # i.e. self.a = -inf
left = -1.*factor
while self._ppf_to_solve(left, q, *args) > 0.:
right = left
left *= factor
# left is now such that cdf(left) < q
if not right: # i.e. self.b = inf
right = factor
while self._ppf_to_solve(right, q, *args) < 0.:
left = right
right *= factor
# right is now such that cdf(right) > q
return optimize.brentq(self._ppf_to_solve,
left, right, args=(q,)+args, xtol=self.xtol)
# moment from definition
def _mom_integ0(self, x, m, *args):
return x**m * self.pdf(x, *args)
def _mom0_sc(self, m, *args):
return integrate.quad(self._mom_integ0, self.a, self.b,
args=(m,)+args)[0]
# moment calculated using ppf
def _mom_integ1(self, q, m, *args):
return (self.ppf(q, *args))**m
def _mom1_sc(self, m, *args):
return integrate.quad(self._mom_integ1, 0, 1, args=(m,)+args)[0]
def _pdf(self, x, *args):
return derivative(self._cdf, x, dx=1e-5, args=args, order=5)
## Could also define any of these
def _logpdf(self, x, *args):
return log(self._pdf(x, *args))
def _cdf_single(self, x, *args):
return integrate.quad(self._pdf, self.a, x, args=args)[0]
def _cdf(self, x, *args):
return self._cdfvec(x, *args)
## generic _argcheck, _logcdf, _sf, _logsf, _ppf, _isf, _rvs are defined
## in rv_generic
def pdf(self, x, *args, **kwds):
"""
Probability density function at x of the given RV.
Parameters
----------
x : array_like
quantiles
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information)
loc : array_like, optional
location parameter (default=0)
scale : array_like, optional
scale parameter (default=1)
Returns
-------
pdf : ndarray
Probability density function evaluated at x
"""
args, loc, scale = self._parse_args(*args, **kwds)
x, loc, scale = map(asarray, (x, loc, scale))
args = tuple(map(asarray, args))
dtyp = np.find_common_type([x.dtype, np.float64], [])
x = np.asarray((x - loc)/scale, dtype=dtyp)
cond0 = self._argcheck(*args) & (scale > 0)
cond1 = self._support_mask(x) & (scale > 0)
cond = cond0 & cond1
output = zeros(shape(cond), dtyp)
putmask(output, (1-cond0)+np.isnan(x), self.badvalue)
if np.any(cond):
goodargs = argsreduce(cond, *((x,)+args+(scale,)))
scale, goodargs = goodargs[-1], goodargs[:-1]
place(output, cond, self._pdf(*goodargs) / scale)
if output.ndim == 0:
return output[()]
return output
def logpdf(self, x, *args, **kwds):
"""
Log of the probability density function at x of the given RV.
This uses a more numerically accurate calculation if available.
Parameters
----------
x : array_like
quantiles
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information)
loc : array_like, optional
location parameter (default=0)
scale : array_like, optional
scale parameter (default=1)
Returns
-------
logpdf : array_like
Log of the probability density function evaluated at x
"""
args, loc, scale = self._parse_args(*args, **kwds)
x, loc, scale = map(asarray, (x, loc, scale))
args = tuple(map(asarray, args))
dtyp = np.find_common_type([x.dtype, np.float64], [])
x = np.asarray((x - loc)/scale, dtype=dtyp)
cond0 = self._argcheck(*args) & (scale > 0)
cond1 = self._support_mask(x) & (scale > 0)
cond = cond0 & cond1
output = empty(shape(cond), dtyp)
output.fill(NINF)
putmask(output, (1-cond0)+np.isnan(x), self.badvalue)
if np.any(cond):
goodargs = argsreduce(cond, *((x,)+args+(scale,)))
scale, goodargs = goodargs[-1], goodargs[:-1]
place(output, cond, self._logpdf(*goodargs) - log(scale))
if output.ndim == 0:
return output[()]
return output
def cdf(self, x, *args, **kwds):
"""
Cumulative distribution function of the given RV.
Parameters
----------
x : array_like
quantiles
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information)
loc : array_like, optional
location parameter (default=0)
scale : array_like, optional
scale parameter (default=1)
Returns
-------
cdf : ndarray
Cumulative distribution function evaluated at `x`
"""
args, loc, scale = self._parse_args(*args, **kwds)
x, loc, scale = map(asarray, (x, loc, scale))
args = tuple(map(asarray, args))
dtyp = np.find_common_type([x.dtype, np.float64], [])
x = np.asarray((x - loc)/scale, dtype=dtyp)
cond0 = self._argcheck(*args) & (scale > 0)
cond1 = self._open_support_mask(x) & (scale > 0)
cond2 = (x >= self.b) & cond0
cond = cond0 & cond1
output = zeros(shape(cond), dtyp)
place(output, (1-cond0)+np.isnan(x), self.badvalue)
place(output, cond2, 1.0)
if np.any(cond): # call only if at least 1 entry
goodargs = argsreduce(cond, *((x,)+args))
place(output, cond, self._cdf(*goodargs))
if output.ndim == 0:
return output[()]
return output
def logcdf(self, x, *args, **kwds):
"""
Log of the cumulative distribution function at x of the given RV.
Parameters
----------
x : array_like
quantiles
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information)
loc : array_like, optional
location parameter (default=0)
scale : array_like, optional
scale parameter (default=1)
Returns
-------
logcdf : array_like
Log of the cumulative distribution function evaluated at x
"""
args, loc, scale = self._parse_args(*args, **kwds)
x, loc, scale = map(asarray, (x, loc, scale))
args = tuple(map(asarray, args))
dtyp = np.find_common_type([x.dtype, np.float64], [])
x = np.asarray((x - loc)/scale, dtype=dtyp)
cond0 = self._argcheck(*args) & (scale > 0)
cond1 = self._open_support_mask(x) & (scale > 0)
cond2 = (x >= self.b) & cond0
cond = cond0 & cond1
output = empty(shape(cond), dtyp)
output.fill(NINF)
place(output, (1-cond0)*(cond1 == cond1)+np.isnan(x), self.badvalue)
place(output, cond2, 0.0)
if np.any(cond): # call only if at least 1 entry
goodargs = argsreduce(cond, *((x,)+args))
place(output, cond, self._logcdf(*goodargs))
if output.ndim == 0:
return output[()]
return output
def sf(self, x, *args, **kwds):
"""
Survival function (1 - `cdf`) at x of the given RV.
Parameters
----------
x : array_like
quantiles
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information)
loc : array_like, optional
location parameter (default=0)
scale : array_like, optional
scale parameter (default=1)
Returns
-------
sf : array_like
Survival function evaluated at x
"""
args, loc, scale = self._parse_args(*args, **kwds)
x, loc, scale = map(asarray, (x, loc, scale))
args = tuple(map(asarray, args))
dtyp = np.find_common_type([x.dtype, np.float64], [])
x = np.asarray((x - loc)/scale, dtype=dtyp)
cond0 = self._argcheck(*args) & (scale > 0)
cond1 = self._open_support_mask(x) & (scale > 0)
cond2 = cond0 & (x <= self.a)
cond = cond0 & cond1
output = zeros(shape(cond), dtyp)
place(output, (1-cond0)+np.isnan(x), self.badvalue)
place(output, cond2, 1.0)
if np.any(cond):
goodargs = argsreduce(cond, *((x,)+args))
place(output, cond, self._sf(*goodargs))
if output.ndim == 0:
return output[()]
return output
def logsf(self, x, *args, **kwds):
"""
Log of the survival function of the given RV.
Returns the log of the "survival function," defined as (1 - `cdf`),
evaluated at `x`.
Parameters
----------
x : array_like
quantiles
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information)
loc : array_like, optional
location parameter (default=0)
scale : array_like, optional
scale parameter (default=1)
Returns
-------
logsf : ndarray
Log of the survival function evaluated at `x`.
"""
args, loc, scale = self._parse_args(*args, **kwds)
x, loc, scale = map(asarray, (x, loc, scale))
args = tuple(map(asarray, args))
dtyp = np.find_common_type([x.dtype, np.float64], [])
x = np.asarray((x - loc)/scale, dtype=dtyp)
cond0 = self._argcheck(*args) & (scale > 0)
cond1 = self._open_support_mask(x) & (scale > 0)
cond2 = cond0 & (x <= self.a)
cond = cond0 & cond1
output = empty(shape(cond), dtyp)
output.fill(NINF)
place(output, (1-cond0)+np.isnan(x), self.badvalue)
place(output, cond2, 0.0)
if np.any(cond):
goodargs = argsreduce(cond, *((x,)+args))
place(output, cond, self._logsf(*goodargs))
if output.ndim == 0:
return output[()]
return output
def ppf(self, q, *args, **kwds):
"""
Percent point function (inverse of `cdf`) at q of the given RV.
Parameters
----------
q : array_like
lower tail probability
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information)
loc : array_like, optional
location parameter (default=0)
scale : array_like, optional
scale parameter (default=1)
Returns
-------
x : array_like
quantile corresponding to the lower tail probability q.
"""
args, loc, scale = self._parse_args(*args, **kwds)
q, loc, scale = map(asarray, (q, loc, scale))
args = tuple(map(asarray, args))
cond0 = self._argcheck(*args) & (scale > 0) & (loc == loc)
cond1 = (0 < q) & (q < 1)
cond2 = cond0 & (q == 0)
cond3 = cond0 & (q == 1)
cond = cond0 & cond1
output = valarray(shape(cond), value=self.badvalue)
lower_bound = self.a * scale + loc
upper_bound = self.b * scale + loc
place(output, cond2, argsreduce(cond2, lower_bound)[0])
place(output, cond3, argsreduce(cond3, upper_bound)[0])
if np.any(cond): # call only if at least 1 entry
goodargs = argsreduce(cond, *((q,)+args+(scale, loc)))
scale, loc, goodargs = goodargs[-2], goodargs[-1], goodargs[:-2]
place(output, cond, self._ppf(*goodargs) * scale + loc)
if output.ndim == 0:
return output[()]
return output
def isf(self, q, *args, **kwds):
"""
Inverse survival function (inverse of `sf`) at q of the given RV.
Parameters
----------
q : array_like
upper tail probability
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information)
loc : array_like, optional
location parameter (default=0)
scale : array_like, optional
scale parameter (default=1)
Returns
-------
x : ndarray or scalar
Quantile corresponding to the upper tail probability q.
"""
args, loc, scale = self._parse_args(*args, **kwds)
q, loc, scale = map(asarray, (q, loc, scale))
args = tuple(map(asarray, args))
cond0 = self._argcheck(*args) & (scale > 0) & (loc == loc)
cond1 = (0 < q) & (q < 1)
cond2 = cond0 & (q == 1)
cond3 = cond0 & (q == 0)
cond = cond0 & cond1
output = valarray(shape(cond), value=self.badvalue)
lower_bound = self.a * scale + loc
upper_bound = self.b * scale + loc
place(output, cond2, argsreduce(cond2, lower_bound)[0])
place(output, cond3, argsreduce(cond3, upper_bound)[0])
if np.any(cond):
goodargs = argsreduce(cond, *((q,)+args+(scale, loc)))
scale, loc, goodargs = goodargs[-2], goodargs[-1], goodargs[:-2]
place(output, cond, self._isf(*goodargs) * scale + loc)
if output.ndim == 0:
return output[()]
return output
def _nnlf(self, x, *args):
return -np.sum(self._logpdf(x, *args), axis=0)
def _unpack_loc_scale(self, theta):
try:
loc = theta[-2]
scale = theta[-1]
args = tuple(theta[:-2])
except IndexError:
raise ValueError("Not enough input arguments.")
return loc, scale, args
def nnlf(self, theta, x):
'''Return negative loglikelihood function.
Notes
-----
This is ``-sum(log pdf(x, theta), axis=0)`` where `theta` are the
parameters (including loc and scale).
'''
loc, scale, args = self._unpack_loc_scale(theta)
if not self._argcheck(*args) or scale <= 0:
return inf
x = asarray((x-loc) / scale)
n_log_scale = len(x) * log(scale)
if np.any(~self._support_mask(x)):
return inf
return self._nnlf(x, *args) + n_log_scale
def _nnlf_and_penalty(self, x, args):
cond0 = ~self._support_mask(x)
n_bad = np.count_nonzero(cond0, axis=0)
if n_bad > 0:
x = argsreduce(~cond0, x)[0]
logpdf = self._logpdf(x, *args)
finite_logpdf = np.isfinite(logpdf)
n_bad += np.sum(~finite_logpdf, axis=0)
if n_bad > 0:
penalty = n_bad * log(_XMAX) * 100
return -np.sum(logpdf[finite_logpdf], axis=0) + penalty
return -np.sum(logpdf, axis=0)
def _penalized_nnlf(self, theta, x):
''' Return penalized negative loglikelihood function,
i.e., - sum (log pdf(x, theta), axis=0) + penalty
where theta are the parameters (including loc and scale)
'''
loc, scale, args = self._unpack_loc_scale(theta)
if not self._argcheck(*args) or scale <= 0:
return inf
x = asarray((x-loc) / scale)
n_log_scale = len(x) * log(scale)
return self._nnlf_and_penalty(x, args) + n_log_scale
# return starting point for fit (shape arguments + loc + scale)
def _fitstart(self, data, args=None):
if args is None:
args = (1.0,)*self.numargs
loc, scale = self._fit_loc_scale_support(data, *args)
return args + (loc, scale)
# Return the (possibly reduced) function to optimize in order to find MLE
# estimates for the .fit method
def _reduce_func(self, args, kwds):
# First of all, convert fshapes params to fnum: eg for stats.beta,
# shapes='a, b'. To fix `a`, can specify either `f1` or `fa`.
# Convert the latter into the former.
if self.shapes:
shapes = self.shapes.replace(',', ' ').split()
for j, s in enumerate(shapes):
val = kwds.pop('f' + s, None) or kwds.pop('fix_' + s, None)
if val is not None:
key = 'f%d' % j
if key in kwds:
raise ValueError("Duplicate entry for %s." % key)
else:
kwds[key] = val
args = list(args)
Nargs = len(args)
fixedn = []
names = ['f%d' % n for n in range(Nargs - 2)] + ['floc', 'fscale']
x0 = []
for n, key in enumerate(names):
if key in kwds:
fixedn.append(n)
args[n] = kwds.pop(key)
else:
x0.append(args[n])
if len(fixedn) == 0:
func = self._penalized_nnlf
restore = None
else:
if len(fixedn) == Nargs:
raise ValueError(
"All parameters fixed. There is nothing to optimize.")
def restore(args, theta):
# Replace with theta for all numbers not in fixedn
# This allows the non-fixed values to vary, but
# we still call self.nnlf with all parameters.
i = 0
for n in range(Nargs):
if n not in fixedn:
args[n] = theta[i]
i += 1
return args
def func(theta, x):
newtheta = restore(args[:], theta)
return self._penalized_nnlf(newtheta, x)
return x0, func, restore, args
def fit(self, data, *args, **kwds):
"""
Return MLEs for shape (if applicable), location, and scale
parameters from data.
MLE stands for Maximum Likelihood Estimate. Starting estimates for
the fit are given by input arguments; for any arguments not provided
with starting estimates, ``self._fitstart(data)`` is called to generate
such.
One can hold some parameters fixed to specific values by passing in
keyword arguments ``f0``, ``f1``, ..., ``fn`` (for shape parameters)
and ``floc`` and ``fscale`` (for location and scale parameters,
respectively).
Parameters
----------
data : array_like
Data to use in calculating the MLEs.
args : floats, optional
Starting value(s) for any shape-characterizing arguments (those not
provided will be determined by a call to ``_fitstart(data)``).
No default value.
kwds : floats, optional
Starting values for the location and scale parameters; no default.
Special keyword arguments are recognized as holding certain
parameters fixed:
- f0...fn : hold respective shape parameters fixed.
Alternatively, shape parameters to fix can be specified by name.
For example, if ``self.shapes == "a, b"``, ``fa``and ``fix_a``
are equivalent to ``f0``, and ``fb`` and ``fix_b`` are
equivalent to ``f1``.
- floc : hold location parameter fixed to specified value.
- fscale : hold scale parameter fixed to specified value.
- optimizer : The optimizer to use. The optimizer must take ``func``,
and starting position as the first two arguments,
plus ``args`` (for extra arguments to pass to the
function to be optimized) and ``disp=0`` to suppress
output as keyword arguments.
Returns
-------
mle_tuple : tuple of floats
MLEs for any shape parameters (if applicable), followed by those
for location and scale. For most random variables, shape statistics
will be returned, but there are exceptions (e.g. ``norm``).
Notes
-----
This fit is computed by maximizing a log-likelihood function, with
penalty applied for samples outside of range of the distribution. The
returned answer is not guaranteed to be the globally optimal MLE, it
may only be locally optimal, or the optimization may fail altogether.
Examples
--------
Generate some data to fit: draw random variates from the `beta`
distribution
>>> from scipy.stats import beta
>>> a, b = 1., 2.
>>> x = beta.rvs(a, b, size=1000)
Now we can fit all four parameters (``a``, ``b``, ``loc`` and ``scale``):
>>> a1, b1, loc1, scale1 = beta.fit(x)
We can also use some prior knowledge about the dataset: let's keep
``loc`` and ``scale`` fixed:
>>> a1, b1, loc1, scale1 = beta.fit(x, floc=0, fscale=1)
>>> loc1, scale1
(0, 1)
We can also keep shape parameters fixed by using ``f``-keywords. To
keep the zero-th shape parameter ``a`` equal 1, use ``f0=1`` or,
equivalently, ``fa=1``:
>>> a1, b1, loc1, scale1 = beta.fit(x, fa=1, floc=0, fscale=1)
>>> a1
1
Not all distributions return estimates for the shape parameters.
``norm`` for example just returns estimates for location and scale:
>>> from scipy.stats import norm
>>> x = norm.rvs(a, b, size=1000, random_state=123)
>>> loc1, scale1 = norm.fit(x)
>>> loc1, scale1
(0.92087172783841631, 2.0015750750324668)
"""
Narg = len(args)
if Narg > self.numargs:
raise TypeError("Too many input arguments.")
start = [None]*2
if (Narg < self.numargs) or not ('loc' in kwds and
'scale' in kwds):
# get distribution specific starting locations
start = self._fitstart(data)
args += start[Narg:-2]
loc = kwds.pop('loc', start[-2])
scale = kwds.pop('scale', start[-1])
args += (loc, scale)
x0, func, restore, args = self._reduce_func(args, kwds)
optimizer = kwds.pop('optimizer', optimize.fmin)
# convert string to function in scipy.optimize
if not callable(optimizer) and isinstance(optimizer, string_types):
if not optimizer.startswith('fmin_'):
optimizer = "fmin_"+optimizer
if optimizer == 'fmin_':
optimizer = 'fmin'
try:
optimizer = getattr(optimize, optimizer)
except AttributeError:
raise ValueError("%s is not a valid optimizer" % optimizer)
# by now kwds must be empty, since everybody took what they needed
if kwds:
raise TypeError("Unknown arguments: %s." % kwds)
vals = optimizer(func, x0, args=(ravel(data),), disp=0)
if restore is not None:
vals = restore(args, vals)
vals = tuple(vals)
return vals
def _fit_loc_scale_support(self, data, *args):
"""
Estimate loc and scale parameters from data accounting for support.
Parameters
----------
data : array_like
Data to fit.
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information).
Returns
-------
Lhat : float
Estimated location parameter for the data.
Shat : float
Estimated scale parameter for the data.
"""
data = np.asarray(data)
# Estimate location and scale according to the method of moments.
loc_hat, scale_hat = self.fit_loc_scale(data, *args)
# Compute the support according to the shape parameters.
self._argcheck(*args)
a, b = self.a, self.b
support_width = b - a
# If the support is empty then return the moment-based estimates.
if support_width <= 0:
return loc_hat, scale_hat
# Compute the proposed support according to the loc and scale estimates.
a_hat = loc_hat + a * scale_hat
b_hat = loc_hat + b * scale_hat
# Use the moment-based estimates if they are compatible with the data.
data_a = np.min(data)
data_b = np.max(data)
if a_hat < data_a and data_b < b_hat:
return loc_hat, scale_hat
# Otherwise find other estimates that are compatible with the data.
data_width = data_b - data_a
rel_margin = 0.1
margin = data_width * rel_margin
# For a finite interval, both the location and scale
# should have interesting values.
if support_width < np.inf:
loc_hat = (data_a - a) - margin
scale_hat = (data_width + 2 * margin) / support_width
return loc_hat, scale_hat
# For a one-sided interval, use only an interesting location parameter.
if a > -np.inf:
return (data_a - a) - margin, 1
elif b < np.inf:
return (data_b - b) + margin, 1
else:
raise RuntimeError
def fit_loc_scale(self, data, *args):
"""
Estimate loc and scale parameters from data using 1st and 2nd moments.
Parameters
----------
data : array_like
Data to fit.
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information).
Returns
-------
Lhat : float
Estimated location parameter for the data.
Shat : float
Estimated scale parameter for the data.
"""
mu, mu2 = self.stats(*args, **{'moments': 'mv'})
tmp = asarray(data)
muhat = tmp.mean()
mu2hat = tmp.var()
Shat = sqrt(mu2hat / mu2)
Lhat = muhat - Shat*mu
if not np.isfinite(Lhat):
Lhat = 0
if not (np.isfinite(Shat) and (0 < Shat)):
Shat = 1
return Lhat, Shat
def _entropy(self, *args):
def integ(x):
val = self._pdf(x, *args)
return entr(val)
# upper limit is often inf, so suppress warnings when integrating
olderr = np.seterr(over='ignore')
h = integrate.quad(integ, self.a, self.b)[0]
np.seterr(**olderr)
if not np.isnan(h):
return h
else:
# try with different limits if integration problems
low, upp = self.ppf([1e-10, 1. - 1e-10], *args)
if np.isinf(self.b):
upper = upp
else:
upper = self.b
if np.isinf(self.a):
lower = low
else:
lower = self.a
return integrate.quad(integ, lower, upper)[0]
def expect(self, func=None, args=(), loc=0, scale=1, lb=None, ub=None,
conditional=False, **kwds):
"""Calculate expected value of a function with respect to the
distribution.
The expected value of a function ``f(x)`` with respect to a
distribution ``dist`` is defined as::
ubound
E[x] = Integral(f(x) * dist.pdf(x))
lbound
Parameters
----------
func : callable, optional
Function for which integral is calculated. Takes only one argument.
The default is the identity mapping f(x) = x.
args : tuple, optional
Shape parameters of the distribution.
loc : float, optional
Location parameter (default=0).
scale : float, optional
Scale parameter (default=1).
lb, ub : scalar, optional
Lower and upper bound for integration. Default is set to the
support of the distribution.
conditional : bool, optional
If True, the integral is corrected by the conditional probability
of the integration interval. The return value is the expectation
of the function, conditional on being in the given interval.
Default is False.
Additional keyword arguments are passed to the integration routine.
Returns
-------
expect : float
The calculated expected value.
Notes
-----
The integration behavior of this function is inherited from
`integrate.quad`.
"""
lockwds = {'loc': loc,
'scale': scale}
self._argcheck(*args)
if func is None:
def fun(x, *args):
return x * self.pdf(x, *args, **lockwds)
else:
def fun(x, *args):
return func(x) * self.pdf(x, *args, **lockwds)
if lb is None:
lb = loc + self.a * scale
if ub is None:
ub = loc + self.b * scale
if conditional:
invfac = (self.sf(lb, *args, **lockwds)
- self.sf(ub, *args, **lockwds))
else:
invfac = 1.0
kwds['args'] = args
# Silence floating point warnings from integration.
olderr = np.seterr(all='ignore')
vals = integrate.quad(fun, lb, ub, **kwds)[0] / invfac
np.seterr(**olderr)
return vals
# Helpers for the discrete distributions
def _drv2_moment(self, n, *args):
"""Non-central moment of discrete distribution."""
def fun(x):
return np.power(x, n) * self._pmf(x, *args)
return _expect(fun, self.a, self.b, self.ppf(0.5, *args), self.inc)
def _drv2_ppfsingle(self, q, *args): # Use basic bisection algorithm
b = self.b
a = self.a
if isinf(b): # Be sure ending point is > q
b = int(max(100*q, 10))
while 1:
if b >= self.b:
qb = 1.0
break
qb = self._cdf(b, *args)
if (qb < q):
b += 10
else:
break
else:
qb = 1.0
if isinf(a): # be sure starting point < q
a = int(min(-100*q, -10))
while 1:
if a <= self.a:
qb = 0.0
break
qa = self._cdf(a, *args)
if (qa > q):
a -= 10
else:
break
else:
qa = self._cdf(a, *args)
while 1:
if (qa == q):
return a
if (qb == q):
return b
if b <= a+1:
# testcase: return wrong number at lower index
# python -c "from scipy.stats import zipf;print zipf.ppf(0.01, 2)" wrong
# python -c "from scipy.stats import zipf;print zipf.ppf([0.01, 0.61, 0.77, 0.83], 2)"
# python -c "from scipy.stats import logser;print logser.ppf([0.1, 0.66, 0.86, 0.93], 0.6)"
if qa > q:
return a
else:
return b
c = int((a+b)/2.0)
qc = self._cdf(c, *args)
if (qc < q):
if a != c:
a = c
else:
raise RuntimeError('updating stopped, endless loop')
qa = qc
elif (qc > q):
if b != c:
b = c
else:
raise RuntimeError('updating stopped, endless loop')
qb = qc
else:
return c
def entropy(pk, qk=None, base=None):
"""Calculate the entropy of a distribution for given probability values.
If only probabilities `pk` are given, the entropy is calculated as
``S = -sum(pk * log(pk), axis=0)``.
If `qk` is not None, then compute the Kullback-Leibler divergence
``S = sum(pk * log(pk / qk), axis=0)``.
This routine will normalize `pk` and `qk` if they don't sum to 1.
Parameters
----------
pk : sequence
Defines the (discrete) distribution. ``pk[i]`` is the (possibly
unnormalized) probability of event ``i``.
qk : sequence, optional
Sequence against which the relative entropy is computed. Should be in
the same format as `pk`.
base : float, optional
The logarithmic base to use, defaults to ``e`` (natural logarithm).
Returns
-------
S : float
The calculated entropy.
"""
pk = asarray(pk)
pk = 1.0*pk / np.sum(pk, axis=0)
if qk is None:
vec = entr(pk)
else:
qk = asarray(qk)
if len(qk) != len(pk):
raise ValueError("qk and pk must have same length.")
qk = 1.0*qk / np.sum(qk, axis=0)
vec = rel_entr(pk, qk)
S = np.sum(vec, axis=0)
if base is not None:
S /= log(base)
return S
# Must over-ride one of _pmf or _cdf or pass in
# x_k, p(x_k) lists in initialization
class rv_discrete(rv_generic):
"""
A generic discrete random variable class meant for subclassing.
`rv_discrete` is a base class to construct specific distribution classes
and instances for discrete random variables. It can also be used
to construct an arbitrary distribution defined by a list of support
points and corresponding probabilities.
Parameters
----------
a : float, optional
Lower bound of the support of the distribution, default: 0
b : float, optional
Upper bound of the support of the distribution, default: plus infinity
moment_tol : float, optional
The tolerance for the generic calculation of moments.
values : tuple of two array_like, optional
``(xk, pk)`` where ``xk`` are integers with non-zero
probabilities ``pk`` with ``sum(pk) = 1``.
inc : integer, optional
Increment for the support of the distribution.
Default is 1. (other values have not been tested)
badvalue : float, optional
The value in a result arrays that indicates a value that for which
some argument restriction is violated, default is np.nan.
name : str, optional
The name of the instance. This string is used to construct the default
example for distributions.
longname : str, optional
This string is used as part of the first line of the docstring returned
when a subclass has no docstring of its own. Note: `longname` exists
for backwards compatibility, do not use for new subclasses.
shapes : str, optional
The shape of the distribution. For example "m, n" for a distribution
that takes two integers as the two shape arguments for all its methods
If not provided, shape parameters will be inferred from
the signatures of the private methods, ``_pmf`` and ``_cdf`` of
the instance.
extradoc : str, optional
This string is used as the last part of the docstring returned when a
subclass has no docstring of its own. Note: `extradoc` exists for
backwards compatibility, do not use for new subclasses.
seed : None or int or ``numpy.random.RandomState`` instance, optional
This parameter defines the RandomState object to use for drawing
random variates.
If None, the global np.random state is used.
If integer, it is used to seed the local RandomState instance.
Default is None.
Methods
-------
rvs
pmf
logpmf
cdf
logcdf
sf
logsf
ppf
isf
moment
stats
entropy
expect
median
mean
std
var
interval
__call__
Notes
-----
This class is similar to `rv_continuous`. Whether a shape parameter is
valid is decided by an ``_argcheck`` method (which defaults to checking
that its arguments are strictly positive.)
The main differences are:
- the support of the distribution is a set of integers
- instead of the probability density function, ``pdf`` (and the
corresponding private ``_pdf``), this class defines the
*probability mass function*, `pmf` (and the corresponding
private ``_pmf``.)
- scale parameter is not defined.
To create a new discrete distribution, we would do the following:
>>> from scipy.stats import rv_discrete
>>> class poisson_gen(rv_discrete):
... "Poisson distribution"
... def _pmf(self, k, mu):
... return exp(-mu) * mu**k / factorial(k)
and create an instance::
>>> poisson = poisson_gen(name="poisson")
Note that above we defined the Poisson distribution in the standard form.
Shifting the distribution can be done by providing the ``loc`` parameter
to the methods of the instance. For example, ``poisson.pmf(x, mu, loc)``
delegates the work to ``poisson._pmf(x-loc, mu)``.
**Discrete distributions from a list of probabilities**
Alternatively, you can construct an arbitrary discrete rv defined
on a finite set of values ``xk`` with ``Prob{X=xk} = pk`` by using the
``values`` keyword argument to the `rv_discrete` constructor.
Examples
--------
Custom made discrete distribution:
>>> from scipy import stats
>>> xk = np.arange(7)
>>> pk = (0.1, 0.2, 0.3, 0.1, 0.1, 0.0, 0.2)
>>> custm = stats.rv_discrete(name='custm', values=(xk, pk))
>>>
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)
>>> ax.plot(xk, custm.pmf(xk), 'ro', ms=12, mec='r')
>>> ax.vlines(xk, 0, custm.pmf(xk), colors='r', lw=4)
>>> plt.show()
Random number generation:
>>> R = custm.rvs(size=100)
"""
def __new__(cls, a=0, b=inf, name=None, badvalue=None,
moment_tol=1e-8, values=None, inc=1, longname=None,
shapes=None, extradoc=None, seed=None):
if values is not None:
# dispatch to a subclass
return super(rv_discrete, cls).__new__(rv_sample)
else:
# business as usual
return super(rv_discrete, cls).__new__(cls)
def __init__(self, a=0, b=inf, name=None, badvalue=None,
moment_tol=1e-8, values=None, inc=1, longname=None,
shapes=None, extradoc=None, seed=None):
super(rv_discrete, self).__init__(seed)
# cf generic freeze
self._ctor_param = dict(
a=a, b=b, name=name, badvalue=badvalue,
moment_tol=moment_tol, values=values, inc=inc,
longname=longname, shapes=shapes, extradoc=extradoc, seed=seed)
if badvalue is None:
badvalue = nan
self.badvalue = badvalue
self.a = a
self.b = b
self.moment_tol = moment_tol
self.inc = inc
self._cdfvec = vectorize(self._cdf_single, otypes='d')
self.vecentropy = vectorize(self._entropy)
self.shapes = shapes
if values is not None:
raise ValueError("rv_discrete.__init__(..., values != None, ...)")
self._construct_argparser(meths_to_inspect=[self._pmf, self._cdf],
locscale_in='loc=0',
# scale=1 for discrete RVs
locscale_out='loc, 1')
# nin correction needs to be after we know numargs
# correct nin for generic moment vectorization
_vec_generic_moment = vectorize(_drv2_moment, otypes='d')
_vec_generic_moment.nin = self.numargs + 2
self.generic_moment = instancemethod(_vec_generic_moment,
self, rv_discrete)
# correct nin for ppf vectorization
_vppf = vectorize(_drv2_ppfsingle, otypes='d')
_vppf.nin = self.numargs + 2
self._ppfvec = instancemethod(_vppf,
self, rv_discrete)
# now that self.numargs is defined, we can adjust nin
self._cdfvec.nin = self.numargs + 1
self._construct_docstrings(name, longname, extradoc)
def _construct_docstrings(self, name, longname, extradoc):
if name is None:
name = 'Distribution'
self.name = name
self.extradoc = extradoc
# generate docstring for subclass instances
if longname is None:
if name[0] in ['aeiouAEIOU']:
hstr = "An "
else:
hstr = "A "
longname = hstr + name
if sys.flags.optimize < 2:
# Skip adding docstrings if interpreter is run with -OO
if self.__doc__ is None:
self._construct_default_doc(longname=longname,
extradoc=extradoc,
docdict=docdict_discrete,
discrete='discrete')
else:
dct = dict(distdiscrete)
self._construct_doc(docdict_discrete, dct.get(self.name))
# discrete RV do not have the scale parameter, remove it
self.__doc__ = self.__doc__.replace(
'\n scale : array_like, '
'optional\n scale parameter (default=1)', '')
def _updated_ctor_param(self):
""" Return the current version of _ctor_param, possibly updated by user.
Used by freezing and pickling.
Keep this in sync with the signature of __init__.
"""
dct = self._ctor_param.copy()
dct['a'] = self.a
dct['b'] = self.b
dct['badvalue'] = self.badvalue
dct['moment_tol'] = self.moment_tol
dct['inc'] = self.inc
dct['name'] = self.name
dct['shapes'] = self.shapes
dct['extradoc'] = self.extradoc
return dct
def _nonzero(self, k, *args):
return floor(k) == k
def _pmf(self, k, *args):
return self._cdf(k, *args) - self._cdf(k-1, *args)
def _logpmf(self, k, *args):
return log(self._pmf(k, *args))
def _cdf_single(self, k, *args):
m = arange(int(self.a), k+1)
return np.sum(self._pmf(m, *args), axis=0)
def _cdf(self, x, *args):
k = floor(x)
return self._cdfvec(k, *args)
# generic _logcdf, _sf, _logsf, _ppf, _isf, _rvs defined in rv_generic
def rvs(self, *args, **kwargs):
"""
Random variates of given type.
Parameters
----------
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information).
loc : array_like, optional
Location parameter (default=0).
size : int or tuple of ints, optional
Defining number of random variates (Default is 1). Note that `size`
has to be given as keyword, not as positional argument.
random_state : None or int or ``np.random.RandomState`` instance, optional
If int or RandomState, use it for drawing the random variates.
If None, rely on ``self.random_state``.
Default is None.
Returns
-------
rvs : ndarray or scalar
Random variates of given `size`.
"""
kwargs['discrete'] = True
return super(rv_discrete, self).rvs(*args, **kwargs)
def pmf(self, k, *args, **kwds):
"""
Probability mass function at k of the given RV.
Parameters
----------
k : array_like
Quantiles.
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information)
loc : array_like, optional
Location parameter (default=0).
Returns
-------
pmf : array_like
Probability mass function evaluated at k
"""
args, loc, _ = self._parse_args(*args, **kwds)
k, loc = map(asarray, (k, loc))
args = tuple(map(asarray, args))
k = asarray((k-loc))
cond0 = self._argcheck(*args)
cond1 = (k >= self.a) & (k <= self.b) & self._nonzero(k, *args)
cond = cond0 & cond1
output = zeros(shape(cond), 'd')
place(output, (1-cond0) + np.isnan(k), self.badvalue)
if np.any(cond):
goodargs = argsreduce(cond, *((k,)+args))
place(output, cond, np.clip(self._pmf(*goodargs), 0, 1))
if output.ndim == 0:
return output[()]
return output
def logpmf(self, k, *args, **kwds):
"""
Log of the probability mass function at k of the given RV.
Parameters
----------
k : array_like
Quantiles.
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information).
loc : array_like, optional
Location parameter. Default is 0.
Returns
-------
logpmf : array_like
Log of the probability mass function evaluated at k.
"""
args, loc, _ = self._parse_args(*args, **kwds)
k, loc = map(asarray, (k, loc))
args = tuple(map(asarray, args))
k = asarray((k-loc))
cond0 = self._argcheck(*args)
cond1 = (k >= self.a) & (k <= self.b) & self._nonzero(k, *args)
cond = cond0 & cond1
output = empty(shape(cond), 'd')
output.fill(NINF)
place(output, (1-cond0) + np.isnan(k), self.badvalue)
if np.any(cond):
goodargs = argsreduce(cond, *((k,)+args))
place(output, cond, self._logpmf(*goodargs))
if output.ndim == 0:
return output[()]
return output
def cdf(self, k, *args, **kwds):
"""
Cumulative distribution function of the given RV.
Parameters
----------
k : array_like, int
Quantiles.
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information).
loc : array_like, optional
Location parameter (default=0).
Returns
-------
cdf : ndarray
Cumulative distribution function evaluated at `k`.
"""
args, loc, _ = self._parse_args(*args, **kwds)
k, loc = map(asarray, (k, loc))
args = tuple(map(asarray, args))
k = asarray((k-loc))
cond0 = self._argcheck(*args)
cond1 = (k >= self.a) & (k < self.b)
cond2 = (k >= self.b)
cond = cond0 & cond1
output = zeros(shape(cond), 'd')
place(output, (1-cond0) + np.isnan(k), self.badvalue)
place(output, cond2*(cond0 == cond0), 1.0)
if np.any(cond):
goodargs = argsreduce(cond, *((k,)+args))
place(output, cond, np.clip(self._cdf(*goodargs), 0, 1))
if output.ndim == 0:
return output[()]
return output
def logcdf(self, k, *args, **kwds):
"""
Log of the cumulative distribution function at k of the given RV.
Parameters
----------
k : array_like, int
Quantiles.
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information).
loc : array_like, optional
Location parameter (default=0).
Returns
-------
logcdf : array_like
Log of the cumulative distribution function evaluated at k.
"""
args, loc, _ = self._parse_args(*args, **kwds)
k, loc = map(asarray, (k, loc))
args = tuple(map(asarray, args))
k = asarray((k-loc))
cond0 = self._argcheck(*args)
cond1 = (k >= self.a) & (k < self.b)
cond2 = (k >= self.b)
cond = cond0 & cond1
output = empty(shape(cond), 'd')
output.fill(NINF)
place(output, (1-cond0) + np.isnan(k), self.badvalue)
place(output, cond2*(cond0 == cond0), 0.0)
if np.any(cond):
goodargs = argsreduce(cond, *((k,)+args))
place(output, cond, self._logcdf(*goodargs))
if output.ndim == 0:
return output[()]
return output
def sf(self, k, *args, **kwds):
"""
Survival function (1 - `cdf`) at k of the given RV.
Parameters
----------
k : array_like
Quantiles.
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information).
loc : array_like, optional
Location parameter (default=0).
Returns
-------
sf : array_like
Survival function evaluated at k.
"""
args, loc, _ = self._parse_args(*args, **kwds)
k, loc = map(asarray, (k, loc))
args = tuple(map(asarray, args))
k = asarray(k-loc)
cond0 = self._argcheck(*args)
cond1 = (k >= self.a) & (k < self.b)
cond2 = (k < self.a) & cond0
cond = cond0 & cond1
output = zeros(shape(cond), 'd')
place(output, (1-cond0) + np.isnan(k), self.badvalue)
place(output, cond2, 1.0)
if np.any(cond):
goodargs = argsreduce(cond, *((k,)+args))
place(output, cond, np.clip(self._sf(*goodargs), 0, 1))
if output.ndim == 0:
return output[()]
return output
def logsf(self, k, *args, **kwds):
"""
Log of the survival function of the given RV.
Returns the log of the "survival function," defined as 1 - `cdf`,
evaluated at `k`.
Parameters
----------
k : array_like
Quantiles.
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information).
loc : array_like, optional
Location parameter (default=0).
Returns
-------
logsf : ndarray
Log of the survival function evaluated at `k`.
"""
args, loc, _ = self._parse_args(*args, **kwds)
k, loc = map(asarray, (k, loc))
args = tuple(map(asarray, args))
k = asarray(k-loc)
cond0 = self._argcheck(*args)
cond1 = (k >= self.a) & (k < self.b)
cond2 = (k < self.a) & cond0
cond = cond0 & cond1
output = empty(shape(cond), 'd')
output.fill(NINF)
place(output, (1-cond0) + np.isnan(k), self.badvalue)
place(output, cond2, 0.0)
if np.any(cond):
goodargs = argsreduce(cond, *((k,)+args))
place(output, cond, self._logsf(*goodargs))
if output.ndim == 0:
return output[()]
return output
def ppf(self, q, *args, **kwds):
"""
Percent point function (inverse of `cdf`) at q of the given RV.
Parameters
----------
q : array_like
Lower tail probability.
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information).
loc : array_like, optional
Location parameter (default=0).
Returns
-------
k : array_like
Quantile corresponding to the lower tail probability, q.
"""
args, loc, _ = self._parse_args(*args, **kwds)
q, loc = map(asarray, (q, loc))
args = tuple(map(asarray, args))
cond0 = self._argcheck(*args) & (loc == loc)
cond1 = (q > 0) & (q < 1)
cond2 = (q == 1) & cond0
cond = cond0 & cond1
output = valarray(shape(cond), value=self.badvalue, typecode='d')
# output type 'd' to handle nin and inf
place(output, (q == 0)*(cond == cond), self.a-1)
place(output, cond2, self.b)
if np.any(cond):
goodargs = argsreduce(cond, *((q,)+args+(loc,)))
loc, goodargs = goodargs[-1], goodargs[:-1]
place(output, cond, self._ppf(*goodargs) + loc)
if output.ndim == 0:
return output[()]
return output
def isf(self, q, *args, **kwds):
"""
Inverse survival function (inverse of `sf`) at q of the given RV.
Parameters
----------
q : array_like
Upper tail probability.
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information).
loc : array_like, optional
Location parameter (default=0).
Returns
-------
k : ndarray or scalar
Quantile corresponding to the upper tail probability, q.
"""
args, loc, _ = self._parse_args(*args, **kwds)
q, loc = map(asarray, (q, loc))
args = tuple(map(asarray, args))
cond0 = self._argcheck(*args) & (loc == loc)
cond1 = (q > 0) & (q < 1)
cond2 = (q == 1) & cond0
cond = cond0 & cond1
# same problem as with ppf; copied from ppf and changed
output = valarray(shape(cond), value=self.badvalue, typecode='d')
# output type 'd' to handle nin and inf
place(output, (q == 0)*(cond == cond), self.b)
place(output, cond2, self.a-1)
# call place only if at least 1 valid argument
if np.any(cond):
goodargs = argsreduce(cond, *((q,)+args+(loc,)))
loc, goodargs = goodargs[-1], goodargs[:-1]
# PB same as ticket 766
place(output, cond, self._isf(*goodargs) + loc)
if output.ndim == 0:
return output[()]
return output
def _entropy(self, *args):
if hasattr(self, 'pk'):
return entropy(self.pk)
else:
return _expect(lambda x: entr(self.pmf(x, *args)),
self.a, self.b, self.ppf(0.5, *args), self.inc)
def expect(self, func=None, args=(), loc=0, lb=None, ub=None,
conditional=False, maxcount=1000, tolerance=1e-10, chunksize=32):
"""
Calculate expected value of a function with respect to the distribution
for discrete distribution.
Parameters
----------
func : callable, optional
Function for which the expectation value is calculated.
Takes only one argument.
The default is the identity mapping f(k) = k.
args : tuple, optional
Shape parameters of the distribution.
loc : float, optional
Location parameter.
Default is 0.
lb, ub : int, optional
Lower and upper bound for the summation, default is set to the
support of the distribution, inclusive (``ul <= k <= ub``).
conditional : bool, optional
If true then the expectation is corrected by the conditional
probability of the summation interval. The return value is the
expectation of the function, `func`, conditional on being in
the given interval (k such that ``ul <= k <= ub``).
Default is False.
maxcount : int, optional
Maximal number of terms to evaluate (to avoid an endless loop for
an infinite sum). Default is 1000.
tolerance : float, optional
Absolute tolerance for the summation. Default is 1e-10.
chunksize : int, optional
Iterate over the support of a distributions in chunks of this size.
Default is 32.
Returns
-------
expect : float
Expected value.
Notes
-----
For heavy-tailed distributions, the expected value may or may not exist,
depending on the function, `func`. If it does exist, but the sum converges
slowly, the accuracy of the result may be rather low. For instance, for
``zipf(4)``, accuracy for mean, variance in example is only 1e-5.
increasing `maxcount` and/or `chunksize` may improve the result, but may also
make zipf very slow.
The function is not vectorized.
"""
if func is None:
def fun(x):
# loc and args from outer scope
return (x+loc)*self._pmf(x, *args)
else:
def fun(x):
# loc and args from outer scope
return func(x+loc)*self._pmf(x, *args)
# used pmf because _pmf does not check support in randint and there
# might be problems(?) with correct self.a, self.b at this stage maybe
# not anymore, seems to work now with _pmf
self._argcheck(*args) # (re)generate scalar self.a and self.b
if lb is None:
lb = self.a
else:
lb = lb - loc # convert bound for standardized distribution
if ub is None:
ub = self.b
else:
ub = ub - loc # convert bound for standardized distribution
if conditional:
invfac = self.sf(lb-1, *args) - self.sf(ub, *args)
else:
invfac = 1.0
# iterate over the support, starting from the median
x0 = self.ppf(0.5, *args)
res = _expect(fun, lb, ub, x0, self.inc, maxcount, tolerance, chunksize)
return res / invfac
def _expect(fun, lb, ub, x0, inc, maxcount=1000, tolerance=1e-10,
chunksize=32):
"""Helper for computing the expectation value of `fun`."""
# short-circuit if the support size is small enough
if (ub - lb) <= chunksize:
supp = np.arange(lb, ub+1, inc)
vals = fun(supp)
return np.sum(vals)
# otherwise, iterate starting from x0
if x0 < lb:
x0 = lb
if x0 > ub:
x0 = ub
count, tot = 0, 0.
# iterate over [x0, ub] inclusive
for x in _iter_chunked(x0, ub+1, chunksize=chunksize, inc=inc):
count += x.size
delta = np.sum(fun(x))
tot += delta
if abs(delta) < tolerance * x.size:
break
if count > maxcount:
warnings.warn('expect(): sum did not converge', RuntimeWarning)
return tot
# iterate over [lb, x0)
for x in _iter_chunked(x0-1, lb-1, chunksize=chunksize, inc=-inc):
count += x.size
delta = np.sum(fun(x))
tot += delta
if abs(delta) < tolerance * x.size:
break
if count > maxcount:
warnings.warn('expect(): sum did not converge', RuntimeWarning)
break
return tot
def _iter_chunked(x0, x1, chunksize=4, inc=1):
"""Iterate from x0 to x1 in chunks of chunksize and steps inc.
x0 must be finite, x1 need not be. In the latter case, the iterator is infinite.
Handles both x0 < x1 and x0 > x1. In the latter case, iterates downwards
(make sure to set inc < 0.)
>>> [x for x in _iter_chunked(2, 5, inc=2)]
[array([2, 4])]
>>> [x for x in _iter_chunked(2, 11, inc=2)]
[array([2, 4, 6, 8]), array([10])]
>>> [x for x in _iter_chunked(2, -5, inc=-2)]
[array([ 2, 0, -2, -4])]
>>> [x for x in _iter_chunked(2, -9, inc=-2)]
[array([ 2, 0, -2, -4]), array([-6, -8])]
"""
if inc == 0:
raise ValueError('Cannot increment by zero.')
if chunksize <= 0:
raise ValueError('Chunk size must be positive; got %s.' % chunksize)
s = 1 if inc > 0 else -1
stepsize = abs(chunksize * inc)
x = x0
while (x - x1) * inc < 0:
delta = min(stepsize, abs(x - x1))
step = delta * s
supp = np.arange(x, x + step, inc)
x += step
yield supp
class rv_sample(rv_discrete):
"""A 'sample' discrete distribution defined by the support and values.
The ctor ignores most of the arguments, only needs the `values` argument.
"""
def __init__(self, a=0, b=inf, name=None, badvalue=None,
moment_tol=1e-8, values=None, inc=1, longname=None,
shapes=None, extradoc=None, seed=None):
super(rv_discrete, self).__init__(seed)
if values is None:
raise ValueError("rv_sample.__init__(..., values=None,...)")
# cf generic freeze
self._ctor_param = dict(
a=a, b=b, name=name, badvalue=badvalue,
moment_tol=moment_tol, values=values, inc=inc,
longname=longname, shapes=shapes, extradoc=extradoc, seed=seed)
if badvalue is None:
badvalue = nan
self.badvalue = badvalue
self.moment_tol = moment_tol
self.inc = inc
self.shapes = shapes
self.vecentropy = self._entropy
xk, pk = values
if len(xk) != len(pk):
raise ValueError("xk and pk need to have the same length.")
if not np.allclose(np.sum(pk), 1):
raise ValueError("The sum of provided pk is not 1.")
indx = np.argsort(np.ravel(xk))
self.xk = np.take(np.ravel(xk), indx, 0)
self.pk = np.take(np.ravel(pk), indx, 0)
self.a = self.xk[0]
self.b = self.xk[-1]
self.qvals = np.cumsum(self.pk, axis=0)
self.shapes = ' ' # bypass inspection
self._construct_argparser(meths_to_inspect=[self._pmf],
locscale_in='loc=0',
# scale=1 for discrete RVs
locscale_out='loc, 1')
self._construct_docstrings(name, longname, extradoc)
def _pmf(self, x):
return np.select([x == k for k in self.xk],
[np.broadcast_arrays(p, x)[0] for p in self.pk], 0)
def _cdf(self, x):
xx, xxk = np.broadcast_arrays(x[:, None], self.xk)
indx = np.argmax(xxk > xx, axis=-1) - 1
return self.qvals[indx]
def _ppf(self, q):
qq, sqq = np.broadcast_arrays(q[..., None], self.qvals)
indx = argmax(sqq >= qq, axis=-1)
return self.xk[indx]
def _rvs(self):
# Need to define it explicitly, otherwise .rvs() with size=None
# fails due to explicit broadcasting in _ppf
U = self._random_state.random_sample(self._size)
if self._size is None:
U = np.array(U, ndmin=1)
Y = self._ppf(U)[0]
else:
Y = self._ppf(U)
return Y
def _entropy(self):
return entropy(self.pk)
def generic_moment(self, n):
n = asarray(n)
return np.sum(self.xk**n[np.newaxis, ...] * self.pk, axis=0)
def get_distribution_names(namespace_pairs, rv_base_class):
"""
Collect names of statistical distributions and their generators.
Parameters
----------
namespace_pairs : sequence
A snapshot of (name, value) pairs in the namespace of a module.
rv_base_class : class
The base class of random variable generator classes in a module.
Returns
-------
distn_names : list of strings
Names of the statistical distributions.
distn_gen_names : list of strings
Names of the generators of the statistical distributions.
Note that these are not simply the names of the statistical
distributions, with a _gen suffix added.
"""
distn_names = []
distn_gen_names = []
for name, value in namespace_pairs:
if name.startswith('_'):
continue
if name.endswith('_gen') and issubclass(value, rv_base_class):
distn_gen_names.append(name)
if isinstance(value, rv_base_class):
distn_names.append(name)
return distn_names, distn_gen_names
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cba-pipeline-public
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cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/_constants.py
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"""
Statistics-related constants.
"""
from __future__ import division, print_function, absolute_import
import numpy as np
# The smallest representable positive number such that 1.0 + _EPS != 1.0.
_EPS = np.finfo(float).eps
# The largest [in magnitude] usable floating value.
_XMAX = np.finfo(float).max
# The log of the largest usable floating value; useful for knowing
# when exp(something) will overflow
_LOGXMAX = np.log(_XMAX)
# The smallest [in magnitude] usable floating value.
_XMIN = np.finfo(float).tiny
# -special.psi(1)
_EULER = 0.577215664901532860606512090082402431042
# special.zeta(3, 1) Apery's constant
_ZETA3 = 1.202056903159594285399738161511449990765
| 681 | 23.357143 | 73 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/_discrete_distns.py
|
#
# Author: Travis Oliphant 2002-2011 with contributions from
# SciPy Developers 2004-2011
#
from __future__ import division, print_function, absolute_import
from scipy import special
from scipy.special import entr, logsumexp, betaln, gammaln as gamln
from scipy._lib._numpy_compat import broadcast_to
from numpy import floor, ceil, log, exp, sqrt, log1p, expm1, tanh, cosh, sinh
import numpy as np
from ._distn_infrastructure import (
rv_discrete, _lazywhere, _ncx2_pdf, _ncx2_cdf, get_distribution_names)
class binom_gen(rv_discrete):
r"""A binomial discrete random variable.
%(before_notes)s
Notes
-----
The probability mass function for `binom` is:
.. math::
f(k) = \binom{n}{k} p^k (1-p)^{n-k}
for ``k`` in ``{0, 1,..., n}``.
`binom` takes ``n`` and ``p`` as shape parameters.
%(after_notes)s
%(example)s
"""
def _rvs(self, n, p):
return self._random_state.binomial(n, p, self._size)
def _argcheck(self, n, p):
self.b = n
return (n >= 0) & (p >= 0) & (p <= 1)
def _logpmf(self, x, n, p):
k = floor(x)
combiln = (gamln(n+1) - (gamln(k+1) + gamln(n-k+1)))
return combiln + special.xlogy(k, p) + special.xlog1py(n-k, -p)
def _pmf(self, x, n, p):
# binom.pmf(k) = choose(n, k) * p**k * (1-p)**(n-k)
return exp(self._logpmf(x, n, p))
def _cdf(self, x, n, p):
k = floor(x)
vals = special.bdtr(k, n, p)
return vals
def _sf(self, x, n, p):
k = floor(x)
return special.bdtrc(k, n, p)
def _ppf(self, q, n, p):
vals = ceil(special.bdtrik(q, n, p))
vals1 = np.maximum(vals - 1, 0)
temp = special.bdtr(vals1, n, p)
return np.where(temp >= q, vals1, vals)
def _stats(self, n, p, moments='mv'):
q = 1.0 - p
mu = n * p
var = n * p * q
g1, g2 = None, None
if 's' in moments:
g1 = (q - p) / sqrt(var)
if 'k' in moments:
g2 = (1.0 - 6*p*q) / var
return mu, var, g1, g2
def _entropy(self, n, p):
k = np.r_[0:n + 1]
vals = self._pmf(k, n, p)
return np.sum(entr(vals), axis=0)
binom = binom_gen(name='binom')
class bernoulli_gen(binom_gen):
r"""A Bernoulli discrete random variable.
%(before_notes)s
Notes
-----
The probability mass function for `bernoulli` is:
.. math::
f(k) = \begin{cases}1-p &\text{if } k = 0\\
p &\text{if } k = 1\end{cases}
for :math:`k` in :math:`\{0, 1\}`.
`bernoulli` takes :math:`p` as shape parameter.
%(after_notes)s
%(example)s
"""
def _rvs(self, p):
return binom_gen._rvs(self, 1, p)
def _argcheck(self, p):
return (p >= 0) & (p <= 1)
def _logpmf(self, x, p):
return binom._logpmf(x, 1, p)
def _pmf(self, x, p):
# bernoulli.pmf(k) = 1-p if k = 0
# = p if k = 1
return binom._pmf(x, 1, p)
def _cdf(self, x, p):
return binom._cdf(x, 1, p)
def _sf(self, x, p):
return binom._sf(x, 1, p)
def _ppf(self, q, p):
return binom._ppf(q, 1, p)
def _stats(self, p):
return binom._stats(1, p)
def _entropy(self, p):
return entr(p) + entr(1-p)
bernoulli = bernoulli_gen(b=1, name='bernoulli')
class nbinom_gen(rv_discrete):
r"""A negative binomial discrete random variable.
%(before_notes)s
Notes
-----
Negative binomial distribution describes a sequence of i.i.d. Bernoulli
trials, repeated until a predefined, non-random number of successes occurs.
The probability mass function of the number of failures for `nbinom` is:
.. math::
f(k) = \binom{k+n-1}{n-1} p^n (1-p)^k
for :math:`k \ge 0`.
`nbinom` takes :math:`n` and :math:`p` as shape parameters where n is the
number of successes, whereas p is the probability of a single success.
%(after_notes)s
%(example)s
"""
def _rvs(self, n, p):
return self._random_state.negative_binomial(n, p, self._size)
def _argcheck(self, n, p):
return (n > 0) & (p >= 0) & (p <= 1)
def _pmf(self, x, n, p):
# nbinom.pmf(k) = choose(k+n-1, n-1) * p**n * (1-p)**k
return exp(self._logpmf(x, n, p))
def _logpmf(self, x, n, p):
coeff = gamln(n+x) - gamln(x+1) - gamln(n)
return coeff + n*log(p) + special.xlog1py(x, -p)
def _cdf(self, x, n, p):
k = floor(x)
return special.betainc(n, k+1, p)
def _sf_skip(self, x, n, p):
# skip because special.nbdtrc doesn't work for 0<n<1
k = floor(x)
return special.nbdtrc(k, n, p)
def _ppf(self, q, n, p):
vals = ceil(special.nbdtrik(q, n, p))
vals1 = (vals-1).clip(0.0, np.inf)
temp = self._cdf(vals1, n, p)
return np.where(temp >= q, vals1, vals)
def _stats(self, n, p):
Q = 1.0 / p
P = Q - 1.0
mu = n*P
var = n*P*Q
g1 = (Q+P)/sqrt(n*P*Q)
g2 = (1.0 + 6*P*Q) / (n*P*Q)
return mu, var, g1, g2
nbinom = nbinom_gen(name='nbinom')
class geom_gen(rv_discrete):
r"""A geometric discrete random variable.
%(before_notes)s
Notes
-----
The probability mass function for `geom` is:
.. math::
f(k) = (1-p)^{k-1} p
for :math:`k \ge 1`.
`geom` takes :math:`p` as shape parameter.
%(after_notes)s
%(example)s
"""
def _rvs(self, p):
return self._random_state.geometric(p, size=self._size)
def _argcheck(self, p):
return (p <= 1) & (p >= 0)
def _pmf(self, k, p):
# geom.pmf(k) = (1-p)**(k-1)*p
return np.power(1-p, k-1) * p
def _logpmf(self, k, p):
return special.xlog1py(k - 1, -p) + log(p)
def _cdf(self, x, p):
k = floor(x)
return -expm1(log1p(-p)*k)
def _sf(self, x, p):
return np.exp(self._logsf(x, p))
def _logsf(self, x, p):
k = floor(x)
return k*log1p(-p)
def _ppf(self, q, p):
vals = ceil(log(1.0-q)/log(1-p))
temp = self._cdf(vals-1, p)
return np.where((temp >= q) & (vals > 0), vals-1, vals)
def _stats(self, p):
mu = 1.0/p
qr = 1.0-p
var = qr / p / p
g1 = (2.0-p) / sqrt(qr)
g2 = np.polyval([1, -6, 6], p)/(1.0-p)
return mu, var, g1, g2
geom = geom_gen(a=1, name='geom', longname="A geometric")
class hypergeom_gen(rv_discrete):
r"""A hypergeometric discrete random variable.
The hypergeometric distribution models drawing objects from a bin.
`M` is the total number of objects, `n` is total number of Type I objects.
The random variate represents the number of Type I objects in `N` drawn
without replacement from the total population.
%(before_notes)s
Notes
-----
The symbols used to denote the shape parameters (`M`, `n`, and `N`) are not
universally accepted. See the Examples for a clarification of the
definitions used here.
The probability mass function is defined as,
.. math:: p(k, M, n, N) = \frac{\binom{n}{k} \binom{M - n}{N - k}}
{\binom{M}{N}}
for :math:`k \in [\max(0, N - M + n), \min(n, N)]`, where the binomial
coefficients are defined as,
.. math:: \binom{n}{k} \equiv \frac{n!}{k! (n - k)!}.
%(after_notes)s
Examples
--------
>>> from scipy.stats import hypergeom
>>> import matplotlib.pyplot as plt
Suppose we have a collection of 20 animals, of which 7 are dogs. Then if
we want to know the probability of finding a given number of dogs if we
choose at random 12 of the 20 animals, we can initialize a frozen
distribution and plot the probability mass function:
>>> [M, n, N] = [20, 7, 12]
>>> rv = hypergeom(M, n, N)
>>> x = np.arange(0, n+1)
>>> pmf_dogs = rv.pmf(x)
>>> fig = plt.figure()
>>> ax = fig.add_subplot(111)
>>> ax.plot(x, pmf_dogs, 'bo')
>>> ax.vlines(x, 0, pmf_dogs, lw=2)
>>> ax.set_xlabel('# of dogs in our group of chosen animals')
>>> ax.set_ylabel('hypergeom PMF')
>>> plt.show()
Instead of using a frozen distribution we can also use `hypergeom`
methods directly. To for example obtain the cumulative distribution
function, use:
>>> prb = hypergeom.cdf(x, M, n, N)
And to generate random numbers:
>>> R = hypergeom.rvs(M, n, N, size=10)
"""
def _rvs(self, M, n, N):
return self._random_state.hypergeometric(n, M-n, N, size=self._size)
def _argcheck(self, M, n, N):
cond = (M > 0) & (n >= 0) & (N >= 0)
cond &= (n <= M) & (N <= M)
self.a = np.maximum(N-(M-n), 0)
self.b = np.minimum(n, N)
return cond
def _logpmf(self, k, M, n, N):
tot, good = M, n
bad = tot - good
return betaln(good+1, 1) + betaln(bad+1,1) + betaln(tot-N+1, N+1)\
- betaln(k+1, good-k+1) - betaln(N-k+1,bad-N+k+1)\
- betaln(tot+1, 1)
def _pmf(self, k, M, n, N):
# same as the following but numerically more precise
# return comb(good, k) * comb(bad, N-k) / comb(tot, N)
return exp(self._logpmf(k, M, n, N))
def _stats(self, M, n, N):
# tot, good, sample_size = M, n, N
# "wikipedia".replace('N', 'M').replace('n', 'N').replace('K', 'n')
M, n, N = 1.*M, 1.*n, 1.*N
m = M - n
p = n/M
mu = N*p
var = m*n*N*(M - N)*1.0/(M*M*(M-1))
g1 = (m - n)*(M-2*N) / (M-2.0) * sqrt((M-1.0) / (m*n*N*(M-N)))
g2 = M*(M+1) - 6.*N*(M-N) - 6.*n*m
g2 *= (M-1)*M*M
g2 += 6.*n*N*(M-N)*m*(5.*M-6)
g2 /= n * N * (M-N) * m * (M-2.) * (M-3.)
return mu, var, g1, g2
def _entropy(self, M, n, N):
k = np.r_[N - (M - n):min(n, N) + 1]
vals = self.pmf(k, M, n, N)
return np.sum(entr(vals), axis=0)
def _sf(self, k, M, n, N):
"""More precise calculation, 1 - cdf doesn't cut it."""
# This for loop is needed because `k` can be an array. If that's the
# case, the sf() method makes M, n and N arrays of the same shape. We
# therefore unpack all inputs args, so we can do the manual
# integration.
res = []
for quant, tot, good, draw in zip(k, M, n, N):
# Manual integration over probability mass function. More accurate
# than integrate.quad.
k2 = np.arange(quant + 1, draw + 1)
res.append(np.sum(self._pmf(k2, tot, good, draw)))
return np.asarray(res)
def _logsf(self, k, M, n, N):
"""
More precise calculation than log(sf)
"""
res = []
for quant, tot, good, draw in zip(k, M, n, N):
# Integration over probability mass function using logsumexp
k2 = np.arange(quant + 1, draw + 1)
res.append(logsumexp(self._logpmf(k2, tot, good, draw)))
return np.asarray(res)
hypergeom = hypergeom_gen(name='hypergeom')
# FIXME: Fails _cdfvec
class logser_gen(rv_discrete):
r"""A Logarithmic (Log-Series, Series) discrete random variable.
%(before_notes)s
Notes
-----
The probability mass function for `logser` is:
.. math::
f(k) = - \frac{p^k}{k \log(1-p)}
for :math:`k \ge 1`.
`logser` takes :math:`p` as shape parameter.
%(after_notes)s
%(example)s
"""
def _rvs(self, p):
# looks wrong for p>0.5, too few k=1
# trying to use generic is worse, no k=1 at all
return self._random_state.logseries(p, size=self._size)
def _argcheck(self, p):
return (p > 0) & (p < 1)
def _pmf(self, k, p):
# logser.pmf(k) = - p**k / (k*log(1-p))
return -np.power(p, k) * 1.0 / k / special.log1p(-p)
def _stats(self, p):
r = special.log1p(-p)
mu = p / (p - 1.0) / r
mu2p = -p / r / (p - 1.0)**2
var = mu2p - mu*mu
mu3p = -p / r * (1.0+p) / (1.0 - p)**3
mu3 = mu3p - 3*mu*mu2p + 2*mu**3
g1 = mu3 / np.power(var, 1.5)
mu4p = -p / r * (
1.0 / (p-1)**2 - 6*p / (p - 1)**3 + 6*p*p / (p-1)**4)
mu4 = mu4p - 4*mu3p*mu + 6*mu2p*mu*mu - 3*mu**4
g2 = mu4 / var**2 - 3.0
return mu, var, g1, g2
logser = logser_gen(a=1, name='logser', longname='A logarithmic')
class poisson_gen(rv_discrete):
r"""A Poisson discrete random variable.
%(before_notes)s
Notes
-----
The probability mass function for `poisson` is:
.. math::
f(k) = \exp(-\mu) \frac{mu^k}{k!}
for :math:`k \ge 0`.
`poisson` takes :math:`\mu` as shape parameter.
%(after_notes)s
%(example)s
"""
# Override rv_discrete._argcheck to allow mu=0.
def _argcheck(self, mu):
return mu >= 0
def _rvs(self, mu):
return self._random_state.poisson(mu, self._size)
def _logpmf(self, k, mu):
Pk = special.xlogy(k, mu) - gamln(k + 1) - mu
return Pk
def _pmf(self, k, mu):
# poisson.pmf(k) = exp(-mu) * mu**k / k!
return exp(self._logpmf(k, mu))
def _cdf(self, x, mu):
k = floor(x)
return special.pdtr(k, mu)
def _sf(self, x, mu):
k = floor(x)
return special.pdtrc(k, mu)
def _ppf(self, q, mu):
vals = ceil(special.pdtrik(q, mu))
vals1 = np.maximum(vals - 1, 0)
temp = special.pdtr(vals1, mu)
return np.where(temp >= q, vals1, vals)
def _stats(self, mu):
var = mu
tmp = np.asarray(mu)
mu_nonzero = tmp > 0
g1 = _lazywhere(mu_nonzero, (tmp,), lambda x: sqrt(1.0/x), np.inf)
g2 = _lazywhere(mu_nonzero, (tmp,), lambda x: 1.0/x, np.inf)
return mu, var, g1, g2
poisson = poisson_gen(name="poisson", longname='A Poisson')
class planck_gen(rv_discrete):
r"""A Planck discrete exponential random variable.
%(before_notes)s
Notes
-----
The probability mass function for `planck` is:
.. math::
f(k) = (1-\exp(-\lambda)) \exp(-\lambda k)
for :math:`k \lambda \ge 0`.
`planck` takes :math:`\lambda` as shape parameter.
%(after_notes)s
%(example)s
"""
def _argcheck(self, lambda_):
self.a = np.where(lambda_ > 0, 0, -np.inf)
self.b = np.where(lambda_ > 0, np.inf, 0)
return lambda_ != 0
def _pmf(self, k, lambda_):
# planck.pmf(k) = (1-exp(-lambda_))*exp(-lambda_*k)
fact = (1-exp(-lambda_))
return fact*exp(-lambda_*k)
def _cdf(self, x, lambda_):
k = floor(x)
return 1-exp(-lambda_*(k+1))
def _sf(self, x, lambda_):
return np.exp(self._logsf(x, lambda_))
def _logsf(self, x, lambda_):
k = floor(x)
return -lambda_*(k+1)
def _ppf(self, q, lambda_):
vals = ceil(-1.0/lambda_ * log1p(-q)-1)
vals1 = (vals-1).clip(self.a, np.inf)
temp = self._cdf(vals1, lambda_)
return np.where(temp >= q, vals1, vals)
def _stats(self, lambda_):
mu = 1/(exp(lambda_)-1)
var = exp(-lambda_)/(expm1(-lambda_))**2
g1 = 2*cosh(lambda_/2.0)
g2 = 4+2*cosh(lambda_)
return mu, var, g1, g2
def _entropy(self, lambda_):
l = lambda_
C = (1-exp(-l))
return l*exp(-l)/C - log(C)
planck = planck_gen(name='planck', longname='A discrete exponential ')
class boltzmann_gen(rv_discrete):
r"""A Boltzmann (Truncated Discrete Exponential) random variable.
%(before_notes)s
Notes
-----
The probability mass function for `boltzmann` is:
.. math::
f(k) = (1-\exp(-\lambda) \exp(-\lambda k)/(1-\exp(-\lambda N))
for :math:`k = 0,..., N-1`.
`boltzmann` takes :math:`\lambda` and :math:`N` as shape parameters.
%(after_notes)s
%(example)s
"""
def _pmf(self, k, lambda_, N):
# boltzmann.pmf(k) =
# (1-exp(-lambda_)*exp(-lambda_*k)/(1-exp(-lambda_*N))
fact = (1-exp(-lambda_))/(1-exp(-lambda_*N))
return fact*exp(-lambda_*k)
def _cdf(self, x, lambda_, N):
k = floor(x)
return (1-exp(-lambda_*(k+1)))/(1-exp(-lambda_*N))
def _ppf(self, q, lambda_, N):
qnew = q*(1-exp(-lambda_*N))
vals = ceil(-1.0/lambda_ * log(1-qnew)-1)
vals1 = (vals-1).clip(0.0, np.inf)
temp = self._cdf(vals1, lambda_, N)
return np.where(temp >= q, vals1, vals)
def _stats(self, lambda_, N):
z = exp(-lambda_)
zN = exp(-lambda_*N)
mu = z/(1.0-z)-N*zN/(1-zN)
var = z/(1.0-z)**2 - N*N*zN/(1-zN)**2
trm = (1-zN)/(1-z)
trm2 = (z*trm**2 - N*N*zN)
g1 = z*(1+z)*trm**3 - N**3*zN*(1+zN)
g1 = g1 / trm2**(1.5)
g2 = z*(1+4*z+z*z)*trm**4 - N**4 * zN*(1+4*zN+zN*zN)
g2 = g2 / trm2 / trm2
return mu, var, g1, g2
boltzmann = boltzmann_gen(name='boltzmann',
longname='A truncated discrete exponential ')
class randint_gen(rv_discrete):
r"""A uniform discrete random variable.
%(before_notes)s
Notes
-----
The probability mass function for `randint` is:
.. math::
f(k) = \frac{1}{high - low}
for ``k = low, ..., high - 1``.
`randint` takes ``low`` and ``high`` as shape parameters.
%(after_notes)s
%(example)s
"""
def _argcheck(self, low, high):
self.a = low
self.b = high - 1
return (high > low)
def _pmf(self, k, low, high):
# randint.pmf(k) = 1./(high - low)
p = np.ones_like(k) / (high - low)
return np.where((k >= low) & (k < high), p, 0.)
def _cdf(self, x, low, high):
k = floor(x)
return (k - low + 1.) / (high - low)
def _ppf(self, q, low, high):
vals = ceil(q * (high - low) + low) - 1
vals1 = (vals - 1).clip(low, high)
temp = self._cdf(vals1, low, high)
return np.where(temp >= q, vals1, vals)
def _stats(self, low, high):
m2, m1 = np.asarray(high), np.asarray(low)
mu = (m2 + m1 - 1.0) / 2
d = m2 - m1
var = (d*d - 1) / 12.0
g1 = 0.0
g2 = -6.0/5.0 * (d*d + 1.0) / (d*d - 1.0)
return mu, var, g1, g2
def _rvs(self, low, high):
"""An array of *size* random integers >= ``low`` and < ``high``."""
if self._size is not None:
# Numpy's RandomState.randint() doesn't broadcast its arguments.
# Use `broadcast_to()` to extend the shapes of low and high
# up to self._size. Then we can use the numpy.vectorize'd
# randint without needing to pass it a `size` argument.
low = broadcast_to(low, self._size)
high = broadcast_to(high, self._size)
randint = np.vectorize(self._random_state.randint, otypes=[np.int_])
return randint(low, high)
def _entropy(self, low, high):
return log(high - low)
randint = randint_gen(name='randint', longname='A discrete uniform '
'(random integer)')
# FIXME: problems sampling.
class zipf_gen(rv_discrete):
r"""A Zipf discrete random variable.
%(before_notes)s
Notes
-----
The probability mass function for `zipf` is:
.. math::
f(k, a) = \frac{1}{\zeta(a) k^a}
for :math:`k \ge 1`.
`zipf` takes :math:`a` as shape parameter.
%(after_notes)s
%(example)s
"""
def _rvs(self, a):
return self._random_state.zipf(a, size=self._size)
def _argcheck(self, a):
return a > 1
def _pmf(self, k, a):
# zipf.pmf(k, a) = 1/(zeta(a) * k**a)
Pk = 1.0 / special.zeta(a, 1) / k**a
return Pk
def _munp(self, n, a):
return _lazywhere(
a > n + 1, (a, n),
lambda a, n: special.zeta(a - n, 1) / special.zeta(a, 1),
np.inf)
zipf = zipf_gen(a=1, name='zipf', longname='A Zipf')
class dlaplace_gen(rv_discrete):
r"""A Laplacian discrete random variable.
%(before_notes)s
Notes
-----
The probability mass function for `dlaplace` is:
.. math::
f(k) = \tanh(a/2) \exp(-a |k|)
for :math:`a > 0`.
`dlaplace` takes :math:`a` as shape parameter.
%(after_notes)s
%(example)s
"""
def _pmf(self, k, a):
# dlaplace.pmf(k) = tanh(a/2) * exp(-a*abs(k))
return tanh(a/2.0) * exp(-a * abs(k))
def _cdf(self, x, a):
k = floor(x)
f = lambda k, a: 1.0 - exp(-a * k) / (exp(a) + 1)
f2 = lambda k, a: exp(a * (k+1)) / (exp(a) + 1)
return _lazywhere(k >= 0, (k, a), f=f, f2=f2)
def _ppf(self, q, a):
const = 1 + exp(a)
vals = ceil(np.where(q < 1.0 / (1 + exp(-a)), log(q*const) / a - 1,
-log((1-q) * const) / a))
vals1 = vals - 1
return np.where(self._cdf(vals1, a) >= q, vals1, vals)
def _stats(self, a):
ea = exp(a)
mu2 = 2.*ea/(ea-1.)**2
mu4 = 2.*ea*(ea**2+10.*ea+1.) / (ea-1.)**4
return 0., mu2, 0., mu4/mu2**2 - 3.
def _entropy(self, a):
return a / sinh(a) - log(tanh(a/2.0))
dlaplace = dlaplace_gen(a=-np.inf,
name='dlaplace', longname='A discrete Laplacian')
class skellam_gen(rv_discrete):
r"""A Skellam discrete random variable.
%(before_notes)s
Notes
-----
Probability distribution of the difference of two correlated or
uncorrelated Poisson random variables.
Let :math:`k_1` and :math:`k_2` be two Poisson-distributed r.v. with
expected values lam1 and lam2. Then, :math:`k_1 - k_2` follows a Skellam
distribution with parameters
:math:`\mu_1 = \lambda_1 - \rho \sqrt{\lambda_1 \lambda_2}` and
:math:`\mu_2 = \lambda_2 - \rho \sqrt{\lambda_1 \lambda_2}`, where
:math:`\rho` is the correlation coefficient between :math:`k_1` and
:math:`k_2`. If the two Poisson-distributed r.v. are independent then
:math:`\rho = 0`.
Parameters :math:`\mu_1` and :math:`\mu_2` must be strictly positive.
For details see: http://en.wikipedia.org/wiki/Skellam_distribution
`skellam` takes :math:`\mu_1` and :math:`\mu_2` as shape parameters.
%(after_notes)s
%(example)s
"""
def _rvs(self, mu1, mu2):
n = self._size
return (self._random_state.poisson(mu1, n) -
self._random_state.poisson(mu2, n))
def _pmf(self, x, mu1, mu2):
px = np.where(x < 0,
_ncx2_pdf(2*mu2, 2*(1-x), 2*mu1)*2,
_ncx2_pdf(2*mu1, 2*(1+x), 2*mu2)*2)
# ncx2.pdf() returns nan's for extremely low probabilities
return px
def _cdf(self, x, mu1, mu2):
x = floor(x)
px = np.where(x < 0,
_ncx2_cdf(2*mu2, -2*x, 2*mu1),
1-_ncx2_cdf(2*mu1, 2*(x+1), 2*mu2))
return px
def _stats(self, mu1, mu2):
mean = mu1 - mu2
var = mu1 + mu2
g1 = mean / sqrt((var)**3)
g2 = 1 / var
return mean, var, g1, g2
skellam = skellam_gen(a=-np.inf, name="skellam", longname='A Skellam')
# Collect names of classes and objects in this module.
pairs = list(globals().items())
_distn_names, _distn_gen_names = get_distribution_names(pairs, rv_discrete)
__all__ = _distn_names + _distn_gen_names
| 23,447 | 25.435175 | 79 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/contingency.py
|
"""Some functions for working with contingency tables (i.e. cross tabulations).
"""
from __future__ import division, print_function, absolute_import
from functools import reduce
import numpy as np
from .stats import power_divergence
__all__ = ['margins', 'expected_freq', 'chi2_contingency']
def margins(a):
"""Return a list of the marginal sums of the array `a`.
Parameters
----------
a : ndarray
The array for which to compute the marginal sums.
Returns
-------
margsums : list of ndarrays
A list of length `a.ndim`. `margsums[k]` is the result
of summing `a` over all axes except `k`; it has the same
number of dimensions as `a`, but the length of each axis
except axis `k` will be 1.
Examples
--------
>>> a = np.arange(12).reshape(2, 6)
>>> a
array([[ 0, 1, 2, 3, 4, 5],
[ 6, 7, 8, 9, 10, 11]])
>>> m0, m1 = margins(a)
>>> m0
array([[15],
[51]])
>>> m1
array([[ 6, 8, 10, 12, 14, 16]])
>>> b = np.arange(24).reshape(2,3,4)
>>> m0, m1, m2 = margins(b)
>>> m0
array([[[ 66]],
[[210]]])
>>> m1
array([[[ 60],
[ 92],
[124]]])
>>> m2
array([[[60, 66, 72, 78]]])
"""
margsums = []
ranged = list(range(a.ndim))
for k in ranged:
marg = np.apply_over_axes(np.sum, a, [j for j in ranged if j != k])
margsums.append(marg)
return margsums
def expected_freq(observed):
"""
Compute the expected frequencies from a contingency table.
Given an n-dimensional contingency table of observed frequencies,
compute the expected frequencies for the table based on the marginal
sums under the assumption that the groups associated with each
dimension are independent.
Parameters
----------
observed : array_like
The table of observed frequencies. (While this function can handle
a 1-D array, that case is trivial. Generally `observed` is at
least 2-D.)
Returns
-------
expected : ndarray of float64
The expected frequencies, based on the marginal sums of the table.
Same shape as `observed`.
Examples
--------
>>> observed = np.array([[10, 10, 20],[20, 20, 20]])
>>> from scipy.stats import expected_freq
>>> expected_freq(observed)
array([[ 12., 12., 16.],
[ 18., 18., 24.]])
"""
# Typically `observed` is an integer array. If `observed` has a large
# number of dimensions or holds large values, some of the following
# computations may overflow, so we first switch to floating point.
observed = np.asarray(observed, dtype=np.float64)
# Create a list of the marginal sums.
margsums = margins(observed)
# Create the array of expected frequencies. The shapes of the
# marginal sums returned by apply_over_axes() are just what we
# need for broadcasting in the following product.
d = observed.ndim
expected = reduce(np.multiply, margsums) / observed.sum() ** (d - 1)
return expected
def chi2_contingency(observed, correction=True, lambda_=None):
"""Chi-square test of independence of variables in a contingency table.
This function computes the chi-square statistic and p-value for the
hypothesis test of independence of the observed frequencies in the
contingency table [1]_ `observed`. The expected frequencies are computed
based on the marginal sums under the assumption of independence; see
`scipy.stats.contingency.expected_freq`. The number of degrees of
freedom is (expressed using numpy functions and attributes)::
dof = observed.size - sum(observed.shape) + observed.ndim - 1
Parameters
----------
observed : array_like
The contingency table. The table contains the observed frequencies
(i.e. number of occurrences) in each category. In the two-dimensional
case, the table is often described as an "R x C table".
correction : bool, optional
If True, *and* the degrees of freedom is 1, apply Yates' correction
for continuity. The effect of the correction is to adjust each
observed value by 0.5 towards the corresponding expected value.
lambda_ : float or str, optional.
By default, the statistic computed in this test is Pearson's
chi-squared statistic [2]_. `lambda_` allows a statistic from the
Cressie-Read power divergence family [3]_ to be used instead. See
`power_divergence` for details.
Returns
-------
chi2 : float
The test statistic.
p : float
The p-value of the test
dof : int
Degrees of freedom
expected : ndarray, same shape as `observed`
The expected frequencies, based on the marginal sums of the table.
See Also
--------
contingency.expected_freq
fisher_exact
chisquare
power_divergence
Notes
-----
An often quoted guideline for the validity of this calculation is that
the test should be used only if the observed and expected frequencies
in each cell are at least 5.
This is a test for the independence of different categories of a
population. The test is only meaningful when the dimension of
`observed` is two or more. Applying the test to a one-dimensional
table will always result in `expected` equal to `observed` and a
chi-square statistic equal to 0.
This function does not handle masked arrays, because the calculation
does not make sense with missing values.
Like stats.chisquare, this function computes a chi-square statistic;
the convenience this function provides is to figure out the expected
frequencies and degrees of freedom from the given contingency table.
If these were already known, and if the Yates' correction was not
required, one could use stats.chisquare. That is, if one calls::
chi2, p, dof, ex = chi2_contingency(obs, correction=False)
then the following is true::
(chi2, p) == stats.chisquare(obs.ravel(), f_exp=ex.ravel(),
ddof=obs.size - 1 - dof)
The `lambda_` argument was added in version 0.13.0 of scipy.
References
----------
.. [1] "Contingency table", http://en.wikipedia.org/wiki/Contingency_table
.. [2] "Pearson's chi-squared test",
http://en.wikipedia.org/wiki/Pearson%27s_chi-squared_test
.. [3] Cressie, N. and Read, T. R. C., "Multinomial Goodness-of-Fit
Tests", J. Royal Stat. Soc. Series B, Vol. 46, No. 3 (1984),
pp. 440-464.
Examples
--------
A two-way example (2 x 3):
>>> from scipy.stats import chi2_contingency
>>> obs = np.array([[10, 10, 20], [20, 20, 20]])
>>> chi2_contingency(obs)
(2.7777777777777777,
0.24935220877729619,
2,
array([[ 12., 12., 16.],
[ 18., 18., 24.]]))
Perform the test using the log-likelihood ratio (i.e. the "G-test")
instead of Pearson's chi-squared statistic.
>>> g, p, dof, expctd = chi2_contingency(obs, lambda_="log-likelihood")
>>> g, p
(2.7688587616781319, 0.25046668010954165)
A four-way example (2 x 2 x 2 x 2):
>>> obs = np.array(
... [[[[12, 17],
... [11, 16]],
... [[11, 12],
... [15, 16]]],
... [[[23, 15],
... [30, 22]],
... [[14, 17],
... [15, 16]]]])
>>> chi2_contingency(obs)
(8.7584514426741897,
0.64417725029295503,
11,
array([[[[ 14.15462386, 14.15462386],
[ 16.49423111, 16.49423111]],
[[ 11.2461395 , 11.2461395 ],
[ 13.10500554, 13.10500554]]],
[[[ 19.5591166 , 19.5591166 ],
[ 22.79202844, 22.79202844]],
[[ 15.54012004, 15.54012004],
[ 18.10873492, 18.10873492]]]]))
"""
observed = np.asarray(observed)
if np.any(observed < 0):
raise ValueError("All values in `observed` must be nonnegative.")
if observed.size == 0:
raise ValueError("No data; `observed` has size 0.")
expected = expected_freq(observed)
if np.any(expected == 0):
# Include one of the positions where expected is zero in
# the exception message.
zeropos = list(zip(*np.where(expected == 0)))[0]
raise ValueError("The internally computed table of expected "
"frequencies has a zero element at %s." % (zeropos,))
# The degrees of freedom
dof = expected.size - sum(expected.shape) + expected.ndim - 1
if dof == 0:
# Degenerate case; this occurs when `observed` is 1D (or, more
# generally, when it has only one nontrivial dimension). In this
# case, we also have observed == expected, so chi2 is 0.
chi2 = 0.0
p = 1.0
else:
if dof == 1 and correction:
# Adjust `observed` according to Yates' correction for continuity.
observed = observed + 0.5 * np.sign(expected - observed)
chi2, p = power_divergence(observed, expected,
ddof=observed.size - 1 - dof, axis=None,
lambda_=lambda_)
return chi2, p, dof, expected
| 9,324 | 33.032847 | 79 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/tests/test_binned_statistic.py
|
from __future__ import division, print_function, absolute_import
import numpy as np
from numpy.testing import assert_allclose
from scipy.stats import (binned_statistic, binned_statistic_2d,
binned_statistic_dd)
from scipy._lib.six import u
from .common_tests import check_named_results
class TestBinnedStatistic(object):
@classmethod
def setup_class(cls):
np.random.seed(9865)
cls.x = np.random.random(100)
cls.y = np.random.random(100)
cls.v = np.random.random(100)
cls.X = np.random.random((100, 3))
cls.w = np.random.random(100)
def test_1d_count(self):
x = self.x
v = self.v
count1, edges1, bc = binned_statistic(x, v, 'count', bins=10)
count2, edges2 = np.histogram(x, bins=10)
assert_allclose(count1, count2)
assert_allclose(edges1, edges2)
def test_gh5927(self):
# smoke test for gh5927 - binned_statistic was using `is` for string
# comparison
x = self.x
v = self.v
statistics = [u'mean', u'median', u'count', u'sum']
for statistic in statistics:
res = binned_statistic(x, v, statistic, bins=10)
def test_1d_result_attributes(self):
x = self.x
v = self.v
res = binned_statistic(x, v, 'count', bins=10)
attributes = ('statistic', 'bin_edges', 'binnumber')
check_named_results(res, attributes)
def test_1d_sum(self):
x = self.x
v = self.v
sum1, edges1, bc = binned_statistic(x, v, 'sum', bins=10)
sum2, edges2 = np.histogram(x, bins=10, weights=v)
assert_allclose(sum1, sum2)
assert_allclose(edges1, edges2)
def test_1d_mean(self):
x = self.x
v = self.v
stat1, edges1, bc = binned_statistic(x, v, 'mean', bins=10)
stat2, edges2, bc = binned_statistic(x, v, np.mean, bins=10)
assert_allclose(stat1, stat2)
assert_allclose(edges1, edges2)
def test_1d_std(self):
x = self.x
v = self.v
stat1, edges1, bc = binned_statistic(x, v, 'std', bins=10)
stat2, edges2, bc = binned_statistic(x, v, np.std, bins=10)
assert_allclose(stat1, stat2)
assert_allclose(edges1, edges2)
def test_1d_min(self):
x = self.x
v = self.v
stat1, edges1, bc = binned_statistic(x, v, 'min', bins=10)
stat2, edges2, bc = binned_statistic(x, v, np.min, bins=10)
assert_allclose(stat1, stat2)
assert_allclose(edges1, edges2)
def test_1d_max(self):
x = self.x
v = self.v
stat1, edges1, bc = binned_statistic(x, v, 'max', bins=10)
stat2, edges2, bc = binned_statistic(x, v, np.max, bins=10)
assert_allclose(stat1, stat2)
assert_allclose(edges1, edges2)
def test_1d_median(self):
x = self.x
v = self.v
stat1, edges1, bc = binned_statistic(x, v, 'median', bins=10)
stat2, edges2, bc = binned_statistic(x, v, np.median, bins=10)
assert_allclose(stat1, stat2)
assert_allclose(edges1, edges2)
def test_1d_bincode(self):
x = self.x[:20]
v = self.v[:20]
count1, edges1, bc = binned_statistic(x, v, 'count', bins=3)
bc2 = np.array([3, 2, 1, 3, 2, 3, 3, 3, 3, 1, 1, 3, 3, 1, 2, 3, 1,
1, 2, 1])
bcount = [(bc == i).sum() for i in np.unique(bc)]
assert_allclose(bc, bc2)
assert_allclose(bcount, count1)
def test_1d_range_keyword(self):
# Regression test for gh-3063, range can be (min, max) or [(min, max)]
np.random.seed(9865)
x = np.arange(30)
data = np.random.random(30)
mean, bins, _ = binned_statistic(x[:15], data[:15])
mean_range, bins_range, _ = binned_statistic(x, data, range=[(0, 14)])
mean_range2, bins_range2, _ = binned_statistic(x, data, range=(0, 14))
assert_allclose(mean, mean_range)
assert_allclose(bins, bins_range)
assert_allclose(mean, mean_range2)
assert_allclose(bins, bins_range2)
def test_1d_multi_values(self):
x = self.x
v = self.v
w = self.w
stat1v, edges1v, bc1v = binned_statistic(x, v, 'mean', bins=10)
stat1w, edges1w, bc1w = binned_statistic(x, w, 'mean', bins=10)
stat2, edges2, bc2 = binned_statistic(x, [v, w], 'mean', bins=10)
assert_allclose(stat2[0], stat1v)
assert_allclose(stat2[1], stat1w)
assert_allclose(edges1v, edges2)
assert_allclose(bc1v, bc2)
def test_2d_count(self):
x = self.x
y = self.y
v = self.v
count1, binx1, biny1, bc = binned_statistic_2d(
x, y, v, 'count', bins=5)
count2, binx2, biny2 = np.histogram2d(x, y, bins=5)
assert_allclose(count1, count2)
assert_allclose(binx1, binx2)
assert_allclose(biny1, biny2)
def test_2d_result_attributes(self):
x = self.x
y = self.y
v = self.v
res = binned_statistic_2d(x, y, v, 'count', bins=5)
attributes = ('statistic', 'x_edge', 'y_edge', 'binnumber')
check_named_results(res, attributes)
def test_2d_sum(self):
x = self.x
y = self.y
v = self.v
sum1, binx1, biny1, bc = binned_statistic_2d(x, y, v, 'sum', bins=5)
sum2, binx2, biny2 = np.histogram2d(x, y, bins=5, weights=v)
assert_allclose(sum1, sum2)
assert_allclose(binx1, binx2)
assert_allclose(biny1, biny2)
def test_2d_mean(self):
x = self.x
y = self.y
v = self.v
stat1, binx1, biny1, bc = binned_statistic_2d(x, y, v, 'mean', bins=5)
stat2, binx2, biny2, bc = binned_statistic_2d(x, y, v, np.mean, bins=5)
assert_allclose(stat1, stat2)
assert_allclose(binx1, binx2)
assert_allclose(biny1, biny2)
def test_2d_mean_unicode(self):
x = self.x
y = self.y
v = self.v
stat1, binx1, biny1, bc = binned_statistic_2d(
x, y, v, u('mean'), bins=5)
stat2, binx2, biny2, bc = binned_statistic_2d(x, y, v, np.mean, bins=5)
assert_allclose(stat1, stat2)
assert_allclose(binx1, binx2)
assert_allclose(biny1, biny2)
def test_2d_std(self):
x = self.x
y = self.y
v = self.v
stat1, binx1, biny1, bc = binned_statistic_2d(x, y, v, 'std', bins=5)
stat2, binx2, biny2, bc = binned_statistic_2d(x, y, v, np.std, bins=5)
assert_allclose(stat1, stat2)
assert_allclose(binx1, binx2)
assert_allclose(biny1, biny2)
def test_2d_min(self):
x = self.x
y = self.y
v = self.v
stat1, binx1, biny1, bc = binned_statistic_2d(x, y, v, 'min', bins=5)
stat2, binx2, biny2, bc = binned_statistic_2d(x, y, v, np.min, bins=5)
assert_allclose(stat1, stat2)
assert_allclose(binx1, binx2)
assert_allclose(biny1, biny2)
def test_2d_max(self):
x = self.x
y = self.y
v = self.v
stat1, binx1, biny1, bc = binned_statistic_2d(x, y, v, 'max', bins=5)
stat2, binx2, biny2, bc = binned_statistic_2d(x, y, v, np.max, bins=5)
assert_allclose(stat1, stat2)
assert_allclose(binx1, binx2)
assert_allclose(biny1, biny2)
def test_2d_median(self):
x = self.x
y = self.y
v = self.v
stat1, binx1, biny1, bc = binned_statistic_2d(
x, y, v, 'median', bins=5)
stat2, binx2, biny2, bc = binned_statistic_2d(
x, y, v, np.median, bins=5)
assert_allclose(stat1, stat2)
assert_allclose(binx1, binx2)
assert_allclose(biny1, biny2)
def test_2d_bincode(self):
x = self.x[:20]
y = self.y[:20]
v = self.v[:20]
count1, binx1, biny1, bc = binned_statistic_2d(
x, y, v, 'count', bins=3)
bc2 = np.array([17, 11, 6, 16, 11, 17, 18, 17, 17, 7, 6, 18, 16,
6, 11, 16, 6, 6, 11, 8])
bcount = [(bc == i).sum() for i in np.unique(bc)]
assert_allclose(bc, bc2)
count1adj = count1[count1.nonzero()]
assert_allclose(bcount, count1adj)
def test_2d_multi_values(self):
x = self.x
y = self.y
v = self.v
w = self.w
stat1v, binx1v, biny1v, bc1v = binned_statistic_2d(
x, y, v, 'mean', bins=8)
stat1w, binx1w, biny1w, bc1w = binned_statistic_2d(
x, y, w, 'mean', bins=8)
stat2, binx2, biny2, bc2 = binned_statistic_2d(
x, y, [v, w], 'mean', bins=8)
assert_allclose(stat2[0], stat1v)
assert_allclose(stat2[1], stat1w)
assert_allclose(binx1v, binx2)
assert_allclose(biny1w, biny2)
assert_allclose(bc1v, bc2)
def test_2d_binnumbers_unraveled(self):
x = self.x
y = self.y
v = self.v
stat, edgesx, bcx = binned_statistic(x, v, 'mean', bins=20)
stat, edgesy, bcy = binned_statistic(y, v, 'mean', bins=10)
stat2, edgesx2, edgesy2, bc2 = binned_statistic_2d(
x, y, v, 'mean', bins=(20, 10), expand_binnumbers=True)
bcx3 = np.searchsorted(edgesx, x, side='right')
bcy3 = np.searchsorted(edgesy, y, side='right')
# `numpy.searchsorted` is non-inclusive on right-edge, compensate
bcx3[x == x.max()] -= 1
bcy3[y == y.max()] -= 1
assert_allclose(bcx, bc2[0])
assert_allclose(bcy, bc2[1])
assert_allclose(bcx3, bc2[0])
assert_allclose(bcy3, bc2[1])
def test_dd_count(self):
X = self.X
v = self.v
count1, edges1, bc = binned_statistic_dd(X, v, 'count', bins=3)
count2, edges2 = np.histogramdd(X, bins=3)
assert_allclose(count1, count2)
assert_allclose(edges1, edges2)
def test_dd_result_attributes(self):
X = self.X
v = self.v
res = binned_statistic_dd(X, v, 'count', bins=3)
attributes = ('statistic', 'bin_edges', 'binnumber')
check_named_results(res, attributes)
def test_dd_sum(self):
X = self.X
v = self.v
sum1, edges1, bc = binned_statistic_dd(X, v, 'sum', bins=3)
sum2, edges2 = np.histogramdd(X, bins=3, weights=v)
assert_allclose(sum1, sum2)
assert_allclose(edges1, edges2)
def test_dd_mean(self):
X = self.X
v = self.v
stat1, edges1, bc = binned_statistic_dd(X, v, 'mean', bins=3)
stat2, edges2, bc = binned_statistic_dd(X, v, np.mean, bins=3)
assert_allclose(stat1, stat2)
assert_allclose(edges1, edges2)
def test_dd_std(self):
X = self.X
v = self.v
stat1, edges1, bc = binned_statistic_dd(X, v, 'std', bins=3)
stat2, edges2, bc = binned_statistic_dd(X, v, np.std, bins=3)
assert_allclose(stat1, stat2)
assert_allclose(edges1, edges2)
def test_dd_min(self):
X = self.X
v = self.v
stat1, edges1, bc = binned_statistic_dd(X, v, 'min', bins=3)
stat2, edges2, bc = binned_statistic_dd(X, v, np.min, bins=3)
assert_allclose(stat1, stat2)
assert_allclose(edges1, edges2)
def test_dd_max(self):
X = self.X
v = self.v
stat1, edges1, bc = binned_statistic_dd(X, v, 'max', bins=3)
stat2, edges2, bc = binned_statistic_dd(X, v, np.max, bins=3)
assert_allclose(stat1, stat2)
assert_allclose(edges1, edges2)
def test_dd_median(self):
X = self.X
v = self.v
stat1, edges1, bc = binned_statistic_dd(X, v, 'median', bins=3)
stat2, edges2, bc = binned_statistic_dd(X, v, np.median, bins=3)
assert_allclose(stat1, stat2)
assert_allclose(edges1, edges2)
def test_dd_bincode(self):
X = self.X[:20]
v = self.v[:20]
count1, edges1, bc = binned_statistic_dd(X, v, 'count', bins=3)
bc2 = np.array([63, 33, 86, 83, 88, 67, 57, 33, 42, 41, 82, 83, 92,
32, 36, 91, 43, 87, 81, 81])
bcount = [(bc == i).sum() for i in np.unique(bc)]
assert_allclose(bc, bc2)
count1adj = count1[count1.nonzero()]
assert_allclose(bcount, count1adj)
def test_dd_multi_values(self):
X = self.X
v = self.v
w = self.w
stat1v, edges1v, bc1v = binned_statistic_dd(X, v, np.std, bins=8)
stat1w, edges1w, bc1w = binned_statistic_dd(X, w, np.std, bins=8)
stat2, edges2, bc2 = binned_statistic_dd(X, [v, w], np.std, bins=8)
assert_allclose(stat2[0], stat1v)
assert_allclose(stat2[1], stat1w)
assert_allclose(edges1v, edges2)
assert_allclose(edges1w, edges2)
assert_allclose(bc1v, bc2)
def test_dd_binnumbers_unraveled(self):
X = self.X
v = self.v
stat, edgesx, bcx = binned_statistic(X[:, 0], v, 'mean', bins=15)
stat, edgesy, bcy = binned_statistic(X[:, 1], v, 'mean', bins=20)
stat, edgesz, bcz = binned_statistic(X[:, 2], v, 'mean', bins=10)
stat2, edges2, bc2 = binned_statistic_dd(
X, v, 'mean', bins=(15, 20, 10), expand_binnumbers=True)
assert_allclose(bcx, bc2[0])
assert_allclose(bcy, bc2[1])
assert_allclose(bcz, bc2[2])
| 13,447 | 29.703196 | 79 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/tests/test_morestats.py
|
# Author: Travis Oliphant, 2002
#
# Further enhancements and tests added by numerous SciPy developers.
#
from __future__ import division, print_function, absolute_import
import warnings
import numpy as np
from numpy.random import RandomState
from numpy.testing import (assert_array_equal,
assert_almost_equal, assert_array_less, assert_array_almost_equal,
assert_, assert_allclose, assert_equal, assert_warns)
import pytest
from pytest import raises as assert_raises
from scipy._lib._numpy_compat import suppress_warnings
from scipy import stats
from .common_tests import check_named_results
# Matplotlib is not a scipy dependency but is optionally used in probplot, so
# check if it's available
try:
import matplotlib.pyplot as plt
have_matplotlib = True
except:
have_matplotlib = False
g1 = [1.006, 0.996, 0.998, 1.000, 0.992, 0.993, 1.002, 0.999, 0.994, 1.000]
g2 = [0.998, 1.006, 1.000, 1.002, 0.997, 0.998, 0.996, 1.000, 1.006, 0.988]
g3 = [0.991, 0.987, 0.997, 0.999, 0.995, 0.994, 1.000, 0.999, 0.996, 0.996]
g4 = [1.005, 1.002, 0.994, 1.000, 0.995, 0.994, 0.998, 0.996, 1.002, 0.996]
g5 = [0.998, 0.998, 0.982, 0.990, 1.002, 0.984, 0.996, 0.993, 0.980, 0.996]
g6 = [1.009, 1.013, 1.009, 0.997, 0.988, 1.002, 0.995, 0.998, 0.981, 0.996]
g7 = [0.990, 1.004, 0.996, 1.001, 0.998, 1.000, 1.018, 1.010, 0.996, 1.002]
g8 = [0.998, 1.000, 1.006, 1.000, 1.002, 0.996, 0.998, 0.996, 1.002, 1.006]
g9 = [1.002, 0.998, 0.996, 0.995, 0.996, 1.004, 1.004, 0.998, 0.999, 0.991]
g10 = [0.991, 0.995, 0.984, 0.994, 0.997, 0.997, 0.991, 0.998, 1.004, 0.997]
class TestBayes_mvs(object):
def test_basic(self):
# Expected values in this test simply taken from the function. For
# some checks regarding correctness of implementation, see review in
# gh-674
data = [6, 9, 12, 7, 8, 8, 13]
mean, var, std = stats.bayes_mvs(data)
assert_almost_equal(mean.statistic, 9.0)
assert_allclose(mean.minmax, (7.1036502226125329, 10.896349777387467),
rtol=1e-14)
assert_almost_equal(var.statistic, 10.0)
assert_allclose(var.minmax, (3.1767242068607087, 24.45910381334018),
rtol=1e-09)
assert_almost_equal(std.statistic, 2.9724954732045084, decimal=14)
assert_allclose(std.minmax, (1.7823367265645145, 4.9456146050146312),
rtol=1e-14)
def test_empty_input(self):
assert_raises(ValueError, stats.bayes_mvs, [])
def test_result_attributes(self):
x = np.arange(15)
attributes = ('statistic', 'minmax')
res = stats.bayes_mvs(x)
for i in res:
check_named_results(i, attributes)
class TestMvsdist(object):
def test_basic(self):
data = [6, 9, 12, 7, 8, 8, 13]
mean, var, std = stats.mvsdist(data)
assert_almost_equal(mean.mean(), 9.0)
assert_allclose(mean.interval(0.9), (7.1036502226125329,
10.896349777387467), rtol=1e-14)
assert_almost_equal(var.mean(), 10.0)
assert_allclose(var.interval(0.9), (3.1767242068607087,
24.45910381334018), rtol=1e-09)
assert_almost_equal(std.mean(), 2.9724954732045084, decimal=14)
assert_allclose(std.interval(0.9), (1.7823367265645145,
4.9456146050146312), rtol=1e-14)
def test_empty_input(self):
assert_raises(ValueError, stats.mvsdist, [])
def test_bad_arg(self):
# Raise ValueError if fewer than two data points are given.
data = [1]
assert_raises(ValueError, stats.mvsdist, data)
def test_warns(self):
# regression test for gh-5270
# make sure there are no spurious divide-by-zero warnings
with warnings.catch_warnings():
warnings.simplefilter('error', RuntimeWarning)
[x.mean() for x in stats.mvsdist([1, 2, 3])]
[x.mean() for x in stats.mvsdist([1, 2, 3, 4, 5])]
class TestShapiro(object):
def test_basic(self):
x1 = [0.11, 7.87, 4.61, 10.14, 7.95, 3.14, 0.46,
4.43, 0.21, 4.75, 0.71, 1.52, 3.24,
0.93, 0.42, 4.97, 9.53, 4.55, 0.47, 6.66]
w, pw = stats.shapiro(x1)
assert_almost_equal(w, 0.90047299861907959, 6)
assert_almost_equal(pw, 0.042089745402336121, 6)
x2 = [1.36, 1.14, 2.92, 2.55, 1.46, 1.06, 5.27, -1.11,
3.48, 1.10, 0.88, -0.51, 1.46, 0.52, 6.20, 1.69,
0.08, 3.67, 2.81, 3.49]
w, pw = stats.shapiro(x2)
assert_almost_equal(w, 0.9590270, 6)
assert_almost_equal(pw, 0.52460, 3)
# Verified against R
np.random.seed(12345678)
x3 = stats.norm.rvs(loc=5, scale=3, size=100)
w, pw = stats.shapiro(x3)
assert_almost_equal(w, 0.9772805571556091, decimal=6)
assert_almost_equal(pw, 0.08144091814756393, decimal=3)
# Extracted from original paper
x4 = [0.139, 0.157, 0.175, 0.256, 0.344, 0.413, 0.503, 0.577, 0.614,
0.655, 0.954, 1.392, 1.557, 1.648, 1.690, 1.994, 2.174, 2.206,
3.245, 3.510, 3.571, 4.354, 4.980, 6.084, 8.351]
W_expected = 0.83467
p_expected = 0.000914
w, pw = stats.shapiro(x4)
assert_almost_equal(w, W_expected, decimal=4)
assert_almost_equal(pw, p_expected, decimal=5)
def test_2d(self):
x1 = [[0.11, 7.87, 4.61, 10.14, 7.95, 3.14, 0.46,
4.43, 0.21, 4.75], [0.71, 1.52, 3.24,
0.93, 0.42, 4.97, 9.53, 4.55, 0.47, 6.66]]
w, pw = stats.shapiro(x1)
assert_almost_equal(w, 0.90047299861907959, 6)
assert_almost_equal(pw, 0.042089745402336121, 6)
x2 = [[1.36, 1.14, 2.92, 2.55, 1.46, 1.06, 5.27, -1.11,
3.48, 1.10], [0.88, -0.51, 1.46, 0.52, 6.20, 1.69,
0.08, 3.67, 2.81, 3.49]]
w, pw = stats.shapiro(x2)
assert_almost_equal(w, 0.9590270, 6)
assert_almost_equal(pw, 0.52460, 3)
def test_empty_input(self):
assert_raises(ValueError, stats.shapiro, [])
assert_raises(ValueError, stats.shapiro, [[], [], []])
def test_not_enough_values(self):
assert_raises(ValueError, stats.shapiro, [1, 2])
assert_raises(ValueError, stats.shapiro, [[], [2]])
def test_bad_arg(self):
# Length of x is less than 3.
x = [1]
assert_raises(ValueError, stats.shapiro, x)
def test_nan_input(self):
x = np.arange(10.)
x[9] = np.nan
w, pw = stats.shapiro(x)
assert_equal(w, np.nan)
assert_almost_equal(pw, 1.0)
class TestAnderson(object):
def test_normal(self):
rs = RandomState(1234567890)
x1 = rs.standard_exponential(size=50)
x2 = rs.standard_normal(size=50)
A, crit, sig = stats.anderson(x1)
assert_array_less(crit[:-1], A)
A, crit, sig = stats.anderson(x2)
assert_array_less(A, crit[-2:])
v = np.ones(10)
v[0] = 0
A, crit, sig = stats.anderson(v)
# The expected statistic 3.208057 was computed independently of scipy.
# For example, in R:
# > library(nortest)
# > v <- rep(1, 10)
# > v[1] <- 0
# > result <- ad.test(v)
# > result$statistic
# A
# 3.208057
assert_allclose(A, 3.208057)
def test_expon(self):
rs = RandomState(1234567890)
x1 = rs.standard_exponential(size=50)
x2 = rs.standard_normal(size=50)
A, crit, sig = stats.anderson(x1, 'expon')
assert_array_less(A, crit[-2:])
olderr = np.seterr(all='ignore')
try:
A, crit, sig = stats.anderson(x2, 'expon')
finally:
np.seterr(**olderr)
assert_(A > crit[-1])
def test_gumbel(self):
# Regression test for gh-6306. Before that issue was fixed,
# this case would return a2=inf.
v = np.ones(100)
v[0] = 0.0
a2, crit, sig = stats.anderson(v, 'gumbel')
# A brief reimplementation of the calculation of the statistic.
n = len(v)
xbar, s = stats.gumbel_l.fit(v)
logcdf = stats.gumbel_l.logcdf(v, xbar, s)
logsf = stats.gumbel_l.logsf(v, xbar, s)
i = np.arange(1, n+1)
expected_a2 = -n - np.mean((2*i - 1) * (logcdf + logsf[::-1]))
assert_allclose(a2, expected_a2)
def test_bad_arg(self):
assert_raises(ValueError, stats.anderson, [1], dist='plate_of_shrimp')
def test_result_attributes(self):
rs = RandomState(1234567890)
x = rs.standard_exponential(size=50)
res = stats.anderson(x)
attributes = ('statistic', 'critical_values', 'significance_level')
check_named_results(res, attributes)
def test_gumbel_l(self):
# gh-2592, gh-6337
# Adds support to 'gumbel_r' and 'gumbel_l' as valid inputs for dist.
rs = RandomState(1234567890)
x = rs.gumbel(size=100)
A1, crit1, sig1 = stats.anderson(x, 'gumbel')
A2, crit2, sig2 = stats.anderson(x, 'gumbel_l')
assert_allclose(A2, A1)
def test_gumbel_r(self):
# gh-2592, gh-6337
# Adds support to 'gumbel_r' and 'gumbel_l' as valid inputs for dist.
rs = RandomState(1234567890)
x1 = rs.gumbel(size=100)
x2 = np.ones(100)
A1, crit1, sig1 = stats.anderson(x1, 'gumbel_r')
A2, crit2, sig2 = stats.anderson(x2, 'gumbel_r')
assert_array_less(A1, crit1[-2:])
assert_(A2 > crit2[-1])
class TestAndersonKSamp(object):
def test_example1a(self):
# Example data from Scholz & Stephens (1987), originally
# published in Lehmann (1995, Nonparametrics, Statistical
# Methods Based on Ranks, p. 309)
# Pass a mixture of lists and arrays
t1 = [38.7, 41.5, 43.8, 44.5, 45.5, 46.0, 47.7, 58.0]
t2 = np.array([39.2, 39.3, 39.7, 41.4, 41.8, 42.9, 43.3, 45.8])
t3 = np.array([34.0, 35.0, 39.0, 40.0, 43.0, 43.0, 44.0, 45.0])
t4 = np.array([34.0, 34.8, 34.8, 35.4, 37.2, 37.8, 41.2, 42.8])
assert_warns(UserWarning, stats.anderson_ksamp, (t1, t2, t3, t4),
midrank=False)
with suppress_warnings() as sup:
sup.filter(UserWarning, message='approximate p-value')
Tk, tm, p = stats.anderson_ksamp((t1, t2, t3, t4), midrank=False)
assert_almost_equal(Tk, 4.449, 3)
assert_array_almost_equal([0.4985, 1.3237, 1.9158, 2.4930, 3.2459],
tm, 4)
assert_almost_equal(p, 0.0021, 4)
def test_example1b(self):
# Example data from Scholz & Stephens (1987), originally
# published in Lehmann (1995, Nonparametrics, Statistical
# Methods Based on Ranks, p. 309)
# Pass arrays
t1 = np.array([38.7, 41.5, 43.8, 44.5, 45.5, 46.0, 47.7, 58.0])
t2 = np.array([39.2, 39.3, 39.7, 41.4, 41.8, 42.9, 43.3, 45.8])
t3 = np.array([34.0, 35.0, 39.0, 40.0, 43.0, 43.0, 44.0, 45.0])
t4 = np.array([34.0, 34.8, 34.8, 35.4, 37.2, 37.8, 41.2, 42.8])
with suppress_warnings() as sup:
sup.filter(UserWarning, message='approximate p-value')
Tk, tm, p = stats.anderson_ksamp((t1, t2, t3, t4), midrank=True)
assert_almost_equal(Tk, 4.480, 3)
assert_array_almost_equal([0.4985, 1.3237, 1.9158, 2.4930, 3.2459],
tm, 4)
assert_almost_equal(p, 0.0020, 4)
def test_example2a(self):
# Example data taken from an earlier technical report of
# Scholz and Stephens
# Pass lists instead of arrays
t1 = [194, 15, 41, 29, 33, 181]
t2 = [413, 14, 58, 37, 100, 65, 9, 169, 447, 184, 36, 201, 118]
t3 = [34, 31, 18, 18, 67, 57, 62, 7, 22, 34]
t4 = [90, 10, 60, 186, 61, 49, 14, 24, 56, 20, 79, 84, 44, 59, 29,
118, 25, 156, 310, 76, 26, 44, 23, 62]
t5 = [130, 208, 70, 101, 208]
t6 = [74, 57, 48, 29, 502, 12, 70, 21, 29, 386, 59, 27]
t7 = [55, 320, 56, 104, 220, 239, 47, 246, 176, 182, 33]
t8 = [23, 261, 87, 7, 120, 14, 62, 47, 225, 71, 246, 21, 42, 20, 5,
12, 120, 11, 3, 14, 71, 11, 14, 11, 16, 90, 1, 16, 52, 95]
t9 = [97, 51, 11, 4, 141, 18, 142, 68, 77, 80, 1, 16, 106, 206, 82,
54, 31, 216, 46, 111, 39, 63, 18, 191, 18, 163, 24]
t10 = [50, 44, 102, 72, 22, 39, 3, 15, 197, 188, 79, 88, 46, 5, 5, 36,
22, 139, 210, 97, 30, 23, 13, 14]
t11 = [359, 9, 12, 270, 603, 3, 104, 2, 438]
t12 = [50, 254, 5, 283, 35, 12]
t13 = [487, 18, 100, 7, 98, 5, 85, 91, 43, 230, 3, 130]
t14 = [102, 209, 14, 57, 54, 32, 67, 59, 134, 152, 27, 14, 230, 66,
61, 34]
with suppress_warnings() as sup:
sup.filter(UserWarning, message='approximate p-value')
Tk, tm, p = stats.anderson_ksamp((t1, t2, t3, t4, t5, t6, t7, t8,
t9, t10, t11, t12, t13, t14),
midrank=False)
assert_almost_equal(Tk, 3.288, 3)
assert_array_almost_equal([0.5990, 1.3269, 1.8052, 2.2486, 2.8009],
tm, 4)
assert_almost_equal(p, 0.0041, 4)
def test_example2b(self):
# Example data taken from an earlier technical report of
# Scholz and Stephens
t1 = [194, 15, 41, 29, 33, 181]
t2 = [413, 14, 58, 37, 100, 65, 9, 169, 447, 184, 36, 201, 118]
t3 = [34, 31, 18, 18, 67, 57, 62, 7, 22, 34]
t4 = [90, 10, 60, 186, 61, 49, 14, 24, 56, 20, 79, 84, 44, 59, 29,
118, 25, 156, 310, 76, 26, 44, 23, 62]
t5 = [130, 208, 70, 101, 208]
t6 = [74, 57, 48, 29, 502, 12, 70, 21, 29, 386, 59, 27]
t7 = [55, 320, 56, 104, 220, 239, 47, 246, 176, 182, 33]
t8 = [23, 261, 87, 7, 120, 14, 62, 47, 225, 71, 246, 21, 42, 20, 5,
12, 120, 11, 3, 14, 71, 11, 14, 11, 16, 90, 1, 16, 52, 95]
t9 = [97, 51, 11, 4, 141, 18, 142, 68, 77, 80, 1, 16, 106, 206, 82,
54, 31, 216, 46, 111, 39, 63, 18, 191, 18, 163, 24]
t10 = [50, 44, 102, 72, 22, 39, 3, 15, 197, 188, 79, 88, 46, 5, 5, 36,
22, 139, 210, 97, 30, 23, 13, 14]
t11 = [359, 9, 12, 270, 603, 3, 104, 2, 438]
t12 = [50, 254, 5, 283, 35, 12]
t13 = [487, 18, 100, 7, 98, 5, 85, 91, 43, 230, 3, 130]
t14 = [102, 209, 14, 57, 54, 32, 67, 59, 134, 152, 27, 14, 230, 66,
61, 34]
with suppress_warnings() as sup:
sup.filter(UserWarning, message='approximate p-value')
Tk, tm, p = stats.anderson_ksamp((t1, t2, t3, t4, t5, t6, t7, t8,
t9, t10, t11, t12, t13, t14),
midrank=True)
assert_almost_equal(Tk, 3.294, 3)
assert_array_almost_equal([0.5990, 1.3269, 1.8052, 2.2486, 2.8009],
tm, 4)
assert_almost_equal(p, 0.0041, 4)
def test_not_enough_samples(self):
assert_raises(ValueError, stats.anderson_ksamp, np.ones(5))
def test_no_distinct_observations(self):
assert_raises(ValueError, stats.anderson_ksamp,
(np.ones(5), np.ones(5)))
def test_empty_sample(self):
assert_raises(ValueError, stats.anderson_ksamp, (np.ones(5), []))
def test_result_attributes(self):
# Example data from Scholz & Stephens (1987), originally
# published in Lehmann (1995, Nonparametrics, Statistical
# Methods Based on Ranks, p. 309)
# Pass a mixture of lists and arrays
t1 = [38.7, 41.5, 43.8, 44.5, 45.5, 46.0, 47.7, 58.0]
t2 = np.array([39.2, 39.3, 39.7, 41.4, 41.8, 42.9, 43.3, 45.8])
t3 = np.array([34.0, 35.0, 39.0, 40.0, 43.0, 43.0, 44.0, 45.0])
t4 = np.array([34.0, 34.8, 34.8, 35.4, 37.2, 37.8, 41.2, 42.8])
with suppress_warnings() as sup:
sup.filter(UserWarning, message='approximate p-value')
res = stats.anderson_ksamp((t1, t2, t3, t4), midrank=False)
attributes = ('statistic', 'critical_values', 'significance_level')
check_named_results(res, attributes)
def test_overflow(self):
# when significance_level approximation overflows, should still return
with suppress_warnings() as sup:
sup.filter(UserWarning, message='approximate p-value')
res = stats.anderson_ksamp([[-20, -10] * 100, [-10, 40, 12] * 100])
assert_almost_equal(res[0], 272.796, 3)
class TestAnsari(object):
def test_small(self):
x = [1, 2, 3, 3, 4]
y = [3, 2, 6, 1, 6, 1, 4, 1]
with suppress_warnings() as sup:
sup.filter(UserWarning, "Ties preclude use of exact statistic.")
W, pval = stats.ansari(x, y)
assert_almost_equal(W, 23.5, 11)
assert_almost_equal(pval, 0.13499256881897437, 11)
def test_approx(self):
ramsay = np.array((111, 107, 100, 99, 102, 106, 109, 108, 104, 99,
101, 96, 97, 102, 107, 113, 116, 113, 110, 98))
parekh = np.array((107, 108, 106, 98, 105, 103, 110, 105, 104,
100, 96, 108, 103, 104, 114, 114, 113, 108,
106, 99))
with suppress_warnings() as sup:
sup.filter(UserWarning, "Ties preclude use of exact statistic.")
W, pval = stats.ansari(ramsay, parekh)
assert_almost_equal(W, 185.5, 11)
assert_almost_equal(pval, 0.18145819972867083, 11)
def test_exact(self):
W, pval = stats.ansari([1, 2, 3, 4], [15, 5, 20, 8, 10, 12])
assert_almost_equal(W, 10.0, 11)
assert_almost_equal(pval, 0.533333333333333333, 7)
def test_bad_arg(self):
assert_raises(ValueError, stats.ansari, [], [1])
assert_raises(ValueError, stats.ansari, [1], [])
def test_result_attributes(self):
x = [1, 2, 3, 3, 4]
y = [3, 2, 6, 1, 6, 1, 4, 1]
with suppress_warnings() as sup:
sup.filter(UserWarning, "Ties preclude use of exact statistic.")
res = stats.ansari(x, y)
attributes = ('statistic', 'pvalue')
check_named_results(res, attributes)
class TestBartlett(object):
def test_data(self):
args = [g1, g2, g3, g4, g5, g6, g7, g8, g9, g10]
T, pval = stats.bartlett(*args)
assert_almost_equal(T, 20.78587342806484, 7)
assert_almost_equal(pval, 0.0136358632781, 7)
def test_bad_arg(self):
# Too few args raises ValueError.
assert_raises(ValueError, stats.bartlett, [1])
def test_result_attributes(self):
args = [g1, g2, g3, g4, g5, g6, g7, g8, g9, g10]
res = stats.bartlett(*args)
attributes = ('statistic', 'pvalue')
check_named_results(res, attributes)
def test_empty_arg(self):
args = (g1, g2, g3, g4, g5, g6, g7, g8, g9, g10, [])
assert_equal((np.nan, np.nan), stats.bartlett(*args))
class TestLevene(object):
def test_data(self):
args = [g1, g2, g3, g4, g5, g6, g7, g8, g9, g10]
W, pval = stats.levene(*args)
assert_almost_equal(W, 1.7059176930008939, 7)
assert_almost_equal(pval, 0.0990829755522, 7)
def test_trimmed1(self):
# Test that center='trimmed' gives the same result as center='mean'
# when proportiontocut=0.
W1, pval1 = stats.levene(g1, g2, g3, center='mean')
W2, pval2 = stats.levene(g1, g2, g3, center='trimmed',
proportiontocut=0.0)
assert_almost_equal(W1, W2)
assert_almost_equal(pval1, pval2)
def test_trimmed2(self):
x = [1.2, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 100.0]
y = [0.0, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 200.0]
np.random.seed(1234)
x2 = np.random.permutation(x)
# Use center='trimmed'
W0, pval0 = stats.levene(x, y, center='trimmed',
proportiontocut=0.125)
W1, pval1 = stats.levene(x2, y, center='trimmed',
proportiontocut=0.125)
# Trim the data here, and use center='mean'
W2, pval2 = stats.levene(x[1:-1], y[1:-1], center='mean')
# Result should be the same.
assert_almost_equal(W0, W2)
assert_almost_equal(W1, W2)
assert_almost_equal(pval1, pval2)
def test_equal_mean_median(self):
x = np.linspace(-1, 1, 21)
np.random.seed(1234)
x2 = np.random.permutation(x)
y = x**3
W1, pval1 = stats.levene(x, y, center='mean')
W2, pval2 = stats.levene(x2, y, center='median')
assert_almost_equal(W1, W2)
assert_almost_equal(pval1, pval2)
def test_bad_keyword(self):
x = np.linspace(-1, 1, 21)
assert_raises(TypeError, stats.levene, x, x, portiontocut=0.1)
def test_bad_center_value(self):
x = np.linspace(-1, 1, 21)
assert_raises(ValueError, stats.levene, x, x, center='trim')
def test_too_few_args(self):
assert_raises(ValueError, stats.levene, [1])
def test_result_attributes(self):
args = [g1, g2, g3, g4, g5, g6, g7, g8, g9, g10]
res = stats.levene(*args)
attributes = ('statistic', 'pvalue')
check_named_results(res, attributes)
class TestBinomP(object):
def test_data(self):
pval = stats.binom_test(100, 250)
assert_almost_equal(pval, 0.0018833009350757682, 11)
pval = stats.binom_test(201, 405)
assert_almost_equal(pval, 0.92085205962670713, 11)
pval = stats.binom_test([682, 243], p=3.0/4)
assert_almost_equal(pval, 0.38249155957481695, 11)
def test_bad_len_x(self):
# Length of x must be 1 or 2.
assert_raises(ValueError, stats.binom_test, [1, 2, 3])
def test_bad_n(self):
# len(x) is 1, but n is invalid.
# Missing n
assert_raises(ValueError, stats.binom_test, [100])
# n less than x[0]
assert_raises(ValueError, stats.binom_test, [100], n=50)
def test_bad_p(self):
assert_raises(ValueError, stats.binom_test, [50, 50], p=2.0)
def test_alternatives(self):
res = stats.binom_test(51, 235, p=1./6, alternative='less')
assert_almost_equal(res, 0.982022657605858)
res = stats.binom_test(51, 235, p=1./6, alternative='greater')
assert_almost_equal(res, 0.02654424571169085)
res = stats.binom_test(51, 235, p=1./6, alternative='two-sided')
assert_almost_equal(res, 0.0437479701823997)
class TestFligner(object):
def test_data(self):
# numbers from R: fligner.test in package stats
x1 = np.arange(5)
assert_array_almost_equal(stats.fligner(x1, x1**2),
(3.2282229927203536, 0.072379187848207877),
11)
def test_trimmed1(self):
# Perturb input to break ties in the transformed data
# See https://github.com/scipy/scipy/pull/8042 for more details
rs = np.random.RandomState(123)
_perturb = lambda g: (np.asarray(g) + 1e-10*rs.randn(len(g))).tolist()
g1_ = _perturb(g1)
g2_ = _perturb(g2)
g3_ = _perturb(g3)
# Test that center='trimmed' gives the same result as center='mean'
# when proportiontocut=0.
Xsq1, pval1 = stats.fligner(g1_, g2_, g3_, center='mean')
Xsq2, pval2 = stats.fligner(g1_, g2_, g3_, center='trimmed',
proportiontocut=0.0)
assert_almost_equal(Xsq1, Xsq2)
assert_almost_equal(pval1, pval2)
def test_trimmed2(self):
x = [1.2, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 100.0]
y = [0.0, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 200.0]
# Use center='trimmed'
Xsq1, pval1 = stats.fligner(x, y, center='trimmed',
proportiontocut=0.125)
# Trim the data here, and use center='mean'
Xsq2, pval2 = stats.fligner(x[1:-1], y[1:-1], center='mean')
# Result should be the same.
assert_almost_equal(Xsq1, Xsq2)
assert_almost_equal(pval1, pval2)
# The following test looks reasonable at first, but fligner() uses the
# function stats.rankdata(), and in one of the cases in this test,
# there are ties, while in the other (because of normal rounding
# errors) there are not. This difference leads to differences in the
# third significant digit of W.
#
#def test_equal_mean_median(self):
# x = np.linspace(-1,1,21)
# y = x**3
# W1, pval1 = stats.fligner(x, y, center='mean')
# W2, pval2 = stats.fligner(x, y, center='median')
# assert_almost_equal(W1, W2)
# assert_almost_equal(pval1, pval2)
def test_bad_keyword(self):
x = np.linspace(-1, 1, 21)
assert_raises(TypeError, stats.fligner, x, x, portiontocut=0.1)
def test_bad_center_value(self):
x = np.linspace(-1, 1, 21)
assert_raises(ValueError, stats.fligner, x, x, center='trim')
def test_bad_num_args(self):
# Too few args raises ValueError.
assert_raises(ValueError, stats.fligner, [1])
def test_empty_arg(self):
x = np.arange(5)
assert_equal((np.nan, np.nan), stats.fligner(x, x**2, []))
class TestMood(object):
def test_mood(self):
# numbers from R: mood.test in package stats
x1 = np.arange(5)
assert_array_almost_equal(stats.mood(x1, x1**2),
(-1.3830857299399906, 0.16663858066771478),
11)
def test_mood_order_of_args(self):
# z should change sign when the order of arguments changes, pvalue
# should not change
np.random.seed(1234)
x1 = np.random.randn(10, 1)
x2 = np.random.randn(15, 1)
z1, p1 = stats.mood(x1, x2)
z2, p2 = stats.mood(x2, x1)
assert_array_almost_equal([z1, p1], [-z2, p2])
def test_mood_with_axis_none(self):
# Test with axis = None, compare with results from R
x1 = [-0.626453810742332, 0.183643324222082, -0.835628612410047,
1.59528080213779, 0.329507771815361, -0.820468384118015,
0.487429052428485, 0.738324705129217, 0.575781351653492,
-0.305388387156356, 1.51178116845085, 0.389843236411431,
-0.621240580541804, -2.2146998871775, 1.12493091814311,
-0.0449336090152309, -0.0161902630989461, 0.943836210685299,
0.821221195098089, 0.593901321217509]
x2 = [-0.896914546624981, 0.184849184646742, 1.58784533120882,
-1.13037567424629, -0.0802517565509893, 0.132420284381094,
0.707954729271733, -0.23969802417184, 1.98447393665293,
-0.138787012119665, 0.417650750792556, 0.981752777463662,
-0.392695355503813, -1.03966897694891, 1.78222896030858,
-2.31106908460517, 0.878604580921265, 0.035806718015226,
1.01282869212708, 0.432265154539617, 2.09081920524915,
-1.19992581964387, 1.58963820029007, 1.95465164222325,
0.00493777682814261, -2.45170638784613, 0.477237302613617,
-0.596558168631403, 0.792203270299649, 0.289636710177348]
x1 = np.array(x1)
x2 = np.array(x2)
x1.shape = (10, 2)
x2.shape = (15, 2)
assert_array_almost_equal(stats.mood(x1, x2, axis=None),
[-1.31716607555, 0.18778296257])
def test_mood_2d(self):
# Test if the results of mood test in 2-D case are consistent with the
# R result for the same inputs. Numbers from R mood.test().
ny = 5
np.random.seed(1234)
x1 = np.random.randn(10, ny)
x2 = np.random.randn(15, ny)
z_vectest, pval_vectest = stats.mood(x1, x2)
for j in range(ny):
assert_array_almost_equal([z_vectest[j], pval_vectest[j]],
stats.mood(x1[:, j], x2[:, j]))
# inverse order of dimensions
x1 = x1.transpose()
x2 = x2.transpose()
z_vectest, pval_vectest = stats.mood(x1, x2, axis=1)
for i in range(ny):
# check axis handling is self consistent
assert_array_almost_equal([z_vectest[i], pval_vectest[i]],
stats.mood(x1[i, :], x2[i, :]))
def test_mood_3d(self):
shape = (10, 5, 6)
np.random.seed(1234)
x1 = np.random.randn(*shape)
x2 = np.random.randn(*shape)
for axis in range(3):
z_vectest, pval_vectest = stats.mood(x1, x2, axis=axis)
# Tests that result for 3-D arrays is equal to that for the
# same calculation on a set of 1-D arrays taken from the
# 3-D array
axes_idx = ([1, 2], [0, 2], [0, 1]) # the two axes != axis
for i in range(shape[axes_idx[axis][0]]):
for j in range(shape[axes_idx[axis][1]]):
if axis == 0:
slice1 = x1[:, i, j]
slice2 = x2[:, i, j]
elif axis == 1:
slice1 = x1[i, :, j]
slice2 = x2[i, :, j]
else:
slice1 = x1[i, j, :]
slice2 = x2[i, j, :]
assert_array_almost_equal([z_vectest[i, j],
pval_vectest[i, j]],
stats.mood(slice1, slice2))
def test_mood_bad_arg(self):
# Raise ValueError when the sum of the lengths of the args is
# less than 3
assert_raises(ValueError, stats.mood, [1], [])
class TestProbplot(object):
def test_basic(self):
np.random.seed(12345)
x = stats.norm.rvs(size=20)
osm, osr = stats.probplot(x, fit=False)
osm_expected = [-1.8241636, -1.38768012, -1.11829229, -0.91222575,
-0.73908135, -0.5857176, -0.44506467, -0.31273668,
-0.18568928, -0.06158146, 0.06158146, 0.18568928,
0.31273668, 0.44506467, 0.5857176, 0.73908135,
0.91222575, 1.11829229, 1.38768012, 1.8241636]
assert_allclose(osr, np.sort(x))
assert_allclose(osm, osm_expected)
res, res_fit = stats.probplot(x, fit=True)
res_fit_expected = [1.05361841, 0.31297795, 0.98741609]
assert_allclose(res_fit, res_fit_expected)
def test_sparams_keyword(self):
np.random.seed(123456)
x = stats.norm.rvs(size=100)
# Check that None, () and 0 (loc=0, for normal distribution) all work
# and give the same results
osm1, osr1 = stats.probplot(x, sparams=None, fit=False)
osm2, osr2 = stats.probplot(x, sparams=0, fit=False)
osm3, osr3 = stats.probplot(x, sparams=(), fit=False)
assert_allclose(osm1, osm2)
assert_allclose(osm1, osm3)
assert_allclose(osr1, osr2)
assert_allclose(osr1, osr3)
# Check giving (loc, scale) params for normal distribution
osm, osr = stats.probplot(x, sparams=(), fit=False)
def test_dist_keyword(self):
np.random.seed(12345)
x = stats.norm.rvs(size=20)
osm1, osr1 = stats.probplot(x, fit=False, dist='t', sparams=(3,))
osm2, osr2 = stats.probplot(x, fit=False, dist=stats.t, sparams=(3,))
assert_allclose(osm1, osm2)
assert_allclose(osr1, osr2)
assert_raises(ValueError, stats.probplot, x, dist='wrong-dist-name')
assert_raises(AttributeError, stats.probplot, x, dist=[])
class custom_dist(object):
"""Some class that looks just enough like a distribution."""
def ppf(self, q):
return stats.norm.ppf(q, loc=2)
osm1, osr1 = stats.probplot(x, sparams=(2,), fit=False)
osm2, osr2 = stats.probplot(x, dist=custom_dist(), fit=False)
assert_allclose(osm1, osm2)
assert_allclose(osr1, osr2)
@pytest.mark.skipif(not have_matplotlib, reason="no matplotlib")
def test_plot_kwarg(self):
np.random.seed(7654321)
fig = plt.figure()
fig.add_subplot(111)
x = stats.t.rvs(3, size=100)
res1, fitres1 = stats.probplot(x, plot=plt)
plt.close()
res2, fitres2 = stats.probplot(x, plot=None)
res3 = stats.probplot(x, fit=False, plot=plt)
plt.close()
res4 = stats.probplot(x, fit=False, plot=None)
# Check that results are consistent between combinations of `fit` and
# `plot` keywords.
assert_(len(res1) == len(res2) == len(res3) == len(res4) == 2)
assert_allclose(res1, res2)
assert_allclose(res1, res3)
assert_allclose(res1, res4)
assert_allclose(fitres1, fitres2)
# Check that a Matplotlib Axes object is accepted
fig = plt.figure()
ax = fig.add_subplot(111)
stats.probplot(x, fit=False, plot=ax)
plt.close()
def test_probplot_bad_args(self):
# Raise ValueError when given an invalid distribution.
assert_raises(ValueError, stats.probplot, [1], dist="plate_of_shrimp")
def test_empty(self):
assert_equal(stats.probplot([], fit=False),
(np.array([]), np.array([])))
assert_equal(stats.probplot([], fit=True),
((np.array([]), np.array([])),
(np.nan, np.nan, 0.0)))
def test_array_of_size_one(self):
with np.errstate(invalid='ignore'):
assert_equal(stats.probplot([1], fit=True),
((np.array([0.]), np.array([1])),
(np.nan, np.nan, 0.0)))
def test_wilcoxon_bad_arg():
# Raise ValueError when two args of different lengths are given or
# zero_method is unknown.
assert_raises(ValueError, stats.wilcoxon, [1], [1, 2])
assert_raises(ValueError, stats.wilcoxon, [1, 2], [1, 2], "dummy")
def test_wilcoxon_arg_type():
# Should be able to accept list as arguments.
# Address issue 6070.
arr = [1, 2, 3, 0, -1, 3, 1, 2, 1, 1, 2]
_ = stats.wilcoxon(arr, zero_method="pratt")
_ = stats.wilcoxon(arr, zero_method="zsplit")
_ = stats.wilcoxon(arr, zero_method="wilcox")
class TestKstat(object):
def test_moments_normal_distribution(self):
np.random.seed(32149)
data = np.random.randn(12345)
moments = []
for n in [1, 2, 3, 4]:
moments.append(stats.kstat(data, n))
expected = [0.011315, 1.017931, 0.05811052, 0.0754134]
assert_allclose(moments, expected, rtol=1e-4)
# test equivalence with `stats.moment`
m1 = stats.moment(data, moment=1)
m2 = stats.moment(data, moment=2)
m3 = stats.moment(data, moment=3)
assert_allclose((m1, m2, m3), expected[:-1], atol=0.02, rtol=1e-2)
def test_empty_input(self):
assert_raises(ValueError, stats.kstat, [])
def test_nan_input(self):
data = np.arange(10.)
data[6] = np.nan
assert_equal(stats.kstat(data), np.nan)
def test_kstat_bad_arg(self):
# Raise ValueError if n > 4 or n < 1.
data = np.arange(10)
for n in [0, 4.001]:
assert_raises(ValueError, stats.kstat, data, n=n)
class TestKstatVar(object):
def test_empty_input(self):
assert_raises(ValueError, stats.kstatvar, [])
def test_nan_input(self):
data = np.arange(10.)
data[6] = np.nan
assert_equal(stats.kstat(data), np.nan)
def test_bad_arg(self):
# Raise ValueError is n is not 1 or 2.
data = [1]
n = 10
assert_raises(ValueError, stats.kstatvar, data, n=n)
class TestPpccPlot(object):
def setup_method(self):
np.random.seed(7654321)
self.x = stats.loggamma.rvs(5, size=500) + 5
def test_basic(self):
N = 5
svals, ppcc = stats.ppcc_plot(self.x, -10, 10, N=N)
ppcc_expected = [0.21139644, 0.21384059, 0.98766719, 0.97980182,
0.93519298]
assert_allclose(svals, np.linspace(-10, 10, num=N))
assert_allclose(ppcc, ppcc_expected)
def test_dist(self):
# Test that we can specify distributions both by name and as objects.
svals1, ppcc1 = stats.ppcc_plot(self.x, -10, 10, dist='tukeylambda')
svals2, ppcc2 = stats.ppcc_plot(self.x, -10, 10,
dist=stats.tukeylambda)
assert_allclose(svals1, svals2, rtol=1e-20)
assert_allclose(ppcc1, ppcc2, rtol=1e-20)
# Test that 'tukeylambda' is the default dist
svals3, ppcc3 = stats.ppcc_plot(self.x, -10, 10)
assert_allclose(svals1, svals3, rtol=1e-20)
assert_allclose(ppcc1, ppcc3, rtol=1e-20)
@pytest.mark.skipif(not have_matplotlib, reason="no matplotlib")
def test_plot_kwarg(self):
# Check with the matplotlib.pyplot module
fig = plt.figure()
ax = fig.add_subplot(111)
stats.ppcc_plot(self.x, -20, 20, plot=plt)
fig.delaxes(ax)
# Check that a Matplotlib Axes object is accepted
ax = fig.add_subplot(111)
stats.ppcc_plot(self.x, -20, 20, plot=ax)
plt.close()
def test_invalid_inputs(self):
# `b` has to be larger than `a`
assert_raises(ValueError, stats.ppcc_plot, self.x, 1, 0)
# Raise ValueError when given an invalid distribution.
assert_raises(ValueError, stats.ppcc_plot, [1, 2, 3], 0, 1,
dist="plate_of_shrimp")
def test_empty(self):
# For consistency with probplot return for one empty array,
# ppcc contains all zeros and svals is the same as for normal array
# input.
svals, ppcc = stats.ppcc_plot([], 0, 1)
assert_allclose(svals, np.linspace(0, 1, num=80))
assert_allclose(ppcc, np.zeros(80, dtype=float))
class TestPpccMax(object):
def test_ppcc_max_bad_arg(self):
# Raise ValueError when given an invalid distribution.
data = [1]
assert_raises(ValueError, stats.ppcc_max, data, dist="plate_of_shrimp")
def test_ppcc_max_basic(self):
np.random.seed(1234567)
x = stats.tukeylambda.rvs(-0.7, loc=2, scale=0.5, size=10000) + 1e4
# On Python 2.6 the result is accurate to 5 decimals. On Python >= 2.7
# it is accurate up to 16 decimals
assert_almost_equal(stats.ppcc_max(x), -0.71215366521264145, decimal=5)
def test_dist(self):
np.random.seed(1234567)
x = stats.tukeylambda.rvs(-0.7, loc=2, scale=0.5, size=10000) + 1e4
# Test that we can specify distributions both by name and as objects.
max1 = stats.ppcc_max(x, dist='tukeylambda')
max2 = stats.ppcc_max(x, dist=stats.tukeylambda)
assert_almost_equal(max1, -0.71215366521264145, decimal=5)
assert_almost_equal(max2, -0.71215366521264145, decimal=5)
# Test that 'tukeylambda' is the default dist
max3 = stats.ppcc_max(x)
assert_almost_equal(max3, -0.71215366521264145, decimal=5)
def test_brack(self):
np.random.seed(1234567)
x = stats.tukeylambda.rvs(-0.7, loc=2, scale=0.5, size=10000) + 1e4
assert_raises(ValueError, stats.ppcc_max, x, brack=(0.0, 1.0, 0.5))
# On Python 2.6 the result is accurate to 5 decimals. On Python >= 2.7
# it is accurate up to 16 decimals
assert_almost_equal(stats.ppcc_max(x, brack=(0, 1)),
-0.71215366521264145, decimal=5)
# On Python 2.6 the result is accurate to 5 decimals. On Python >= 2.7
# it is accurate up to 16 decimals
assert_almost_equal(stats.ppcc_max(x, brack=(-2, 2)),
-0.71215366521264145, decimal=5)
class TestBoxcox_llf(object):
def test_basic(self):
np.random.seed(54321)
x = stats.norm.rvs(size=10000, loc=10)
lmbda = 1
llf = stats.boxcox_llf(lmbda, x)
llf_expected = -x.size / 2. * np.log(np.sum(x.std()**2))
assert_allclose(llf, llf_expected)
def test_array_like(self):
np.random.seed(54321)
x = stats.norm.rvs(size=100, loc=10)
lmbda = 1
llf = stats.boxcox_llf(lmbda, x)
llf2 = stats.boxcox_llf(lmbda, list(x))
assert_allclose(llf, llf2, rtol=1e-12)
def test_2d_input(self):
# Note: boxcox_llf() was already working with 2-D input (sort of), so
# keep it like that. boxcox() doesn't work with 2-D input though, due
# to brent() returning a scalar.
np.random.seed(54321)
x = stats.norm.rvs(size=100, loc=10)
lmbda = 1
llf = stats.boxcox_llf(lmbda, x)
llf2 = stats.boxcox_llf(lmbda, np.vstack([x, x]).T)
assert_allclose([llf, llf], llf2, rtol=1e-12)
def test_empty(self):
assert_(np.isnan(stats.boxcox_llf(1, [])))
class TestBoxcox(object):
def test_fixed_lmbda(self):
np.random.seed(12345)
x = stats.loggamma.rvs(5, size=50) + 5
xt = stats.boxcox(x, lmbda=1)
assert_allclose(xt, x - 1)
xt = stats.boxcox(x, lmbda=-1)
assert_allclose(xt, 1 - 1/x)
xt = stats.boxcox(x, lmbda=0)
assert_allclose(xt, np.log(x))
# Also test that array_like input works
xt = stats.boxcox(list(x), lmbda=0)
assert_allclose(xt, np.log(x))
def test_lmbda_None(self):
np.random.seed(1234567)
# Start from normal rv's, do inverse transform to check that
# optimization function gets close to the right answer.
np.random.seed(1245)
lmbda = 2.5
x = stats.norm.rvs(loc=10, size=50000)
x_inv = (x * lmbda + 1)**(-lmbda)
xt, maxlog = stats.boxcox(x_inv)
assert_almost_equal(maxlog, -1 / lmbda, decimal=2)
def test_alpha(self):
np.random.seed(1234)
x = stats.loggamma.rvs(5, size=50) + 5
# Some regular values for alpha, on a small sample size
_, _, interval = stats.boxcox(x, alpha=0.75)
assert_allclose(interval, [4.004485780226041, 5.138756355035744])
_, _, interval = stats.boxcox(x, alpha=0.05)
assert_allclose(interval, [1.2138178554857557, 8.209033272375663])
# Try some extreme values, see we don't hit the N=500 limit
x = stats.loggamma.rvs(7, size=500) + 15
_, _, interval = stats.boxcox(x, alpha=0.001)
assert_allclose(interval, [0.3988867, 11.40553131])
_, _, interval = stats.boxcox(x, alpha=0.999)
assert_allclose(interval, [5.83316246, 5.83735292])
def test_boxcox_bad_arg(self):
# Raise ValueError if any data value is negative.
x = np.array([-1])
assert_raises(ValueError, stats.boxcox, x)
def test_empty(self):
assert_(stats.boxcox([]).shape == (0,))
class TestBoxcoxNormmax(object):
def setup_method(self):
np.random.seed(12345)
self.x = stats.loggamma.rvs(5, size=50) + 5
def test_pearsonr(self):
maxlog = stats.boxcox_normmax(self.x)
assert_allclose(maxlog, 1.804465, rtol=1e-6)
def test_mle(self):
maxlog = stats.boxcox_normmax(self.x, method='mle')
assert_allclose(maxlog, 1.758101, rtol=1e-6)
# Check that boxcox() uses 'mle'
_, maxlog_boxcox = stats.boxcox(self.x)
assert_allclose(maxlog_boxcox, maxlog)
def test_all(self):
maxlog_all = stats.boxcox_normmax(self.x, method='all')
assert_allclose(maxlog_all, [1.804465, 1.758101], rtol=1e-6)
class TestBoxcoxNormplot(object):
def setup_method(self):
np.random.seed(7654321)
self.x = stats.loggamma.rvs(5, size=500) + 5
def test_basic(self):
N = 5
lmbdas, ppcc = stats.boxcox_normplot(self.x, -10, 10, N=N)
ppcc_expected = [0.57783375, 0.83610988, 0.97524311, 0.99756057,
0.95843297]
assert_allclose(lmbdas, np.linspace(-10, 10, num=N))
assert_allclose(ppcc, ppcc_expected)
@pytest.mark.skipif(not have_matplotlib, reason="no matplotlib")
def test_plot_kwarg(self):
# Check with the matplotlib.pyplot module
fig = plt.figure()
ax = fig.add_subplot(111)
stats.boxcox_normplot(self.x, -20, 20, plot=plt)
fig.delaxes(ax)
# Check that a Matplotlib Axes object is accepted
ax = fig.add_subplot(111)
stats.boxcox_normplot(self.x, -20, 20, plot=ax)
plt.close()
def test_invalid_inputs(self):
# `lb` has to be larger than `la`
assert_raises(ValueError, stats.boxcox_normplot, self.x, 1, 0)
# `x` can not contain negative values
assert_raises(ValueError, stats.boxcox_normplot, [-1, 1], 0, 1)
def test_empty(self):
assert_(stats.boxcox_normplot([], 0, 1).size == 0)
class TestCircFuncs(object):
def test_circfuncs(self):
x = np.array([355, 5, 2, 359, 10, 350])
M = stats.circmean(x, high=360)
Mval = 0.167690146
assert_allclose(M, Mval, rtol=1e-7)
V = stats.circvar(x, high=360)
Vval = 42.51955609
assert_allclose(V, Vval, rtol=1e-7)
S = stats.circstd(x, high=360)
Sval = 6.520702116
assert_allclose(S, Sval, rtol=1e-7)
def test_circfuncs_small(self):
x = np.array([20, 21, 22, 18, 19, 20.5, 19.2])
M1 = x.mean()
M2 = stats.circmean(x, high=360)
assert_allclose(M2, M1, rtol=1e-5)
V1 = x.var()
V2 = stats.circvar(x, high=360)
assert_allclose(V2, V1, rtol=1e-4)
S1 = x.std()
S2 = stats.circstd(x, high=360)
assert_allclose(S2, S1, rtol=1e-4)
def test_circmean_axis(self):
x = np.array([[355, 5, 2, 359, 10, 350],
[351, 7, 4, 352, 9, 349],
[357, 9, 8, 358, 4, 356]])
M1 = stats.circmean(x, high=360)
M2 = stats.circmean(x.ravel(), high=360)
assert_allclose(M1, M2, rtol=1e-14)
M1 = stats.circmean(x, high=360, axis=1)
M2 = [stats.circmean(x[i], high=360) for i in range(x.shape[0])]
assert_allclose(M1, M2, rtol=1e-14)
M1 = stats.circmean(x, high=360, axis=0)
M2 = [stats.circmean(x[:, i], high=360) for i in range(x.shape[1])]
assert_allclose(M1, M2, rtol=1e-14)
def test_circvar_axis(self):
x = np.array([[355, 5, 2, 359, 10, 350],
[351, 7, 4, 352, 9, 349],
[357, 9, 8, 358, 4, 356]])
V1 = stats.circvar(x, high=360)
V2 = stats.circvar(x.ravel(), high=360)
assert_allclose(V1, V2, rtol=1e-11)
V1 = stats.circvar(x, high=360, axis=1)
V2 = [stats.circvar(x[i], high=360) for i in range(x.shape[0])]
assert_allclose(V1, V2, rtol=1e-11)
V1 = stats.circvar(x, high=360, axis=0)
V2 = [stats.circvar(x[:, i], high=360) for i in range(x.shape[1])]
assert_allclose(V1, V2, rtol=1e-11)
def test_circstd_axis(self):
x = np.array([[355, 5, 2, 359, 10, 350],
[351, 7, 4, 352, 9, 349],
[357, 9, 8, 358, 4, 356]])
S1 = stats.circstd(x, high=360)
S2 = stats.circstd(x.ravel(), high=360)
assert_allclose(S1, S2, rtol=1e-11)
S1 = stats.circstd(x, high=360, axis=1)
S2 = [stats.circstd(x[i], high=360) for i in range(x.shape[0])]
assert_allclose(S1, S2, rtol=1e-11)
S1 = stats.circstd(x, high=360, axis=0)
S2 = [stats.circstd(x[:, i], high=360) for i in range(x.shape[1])]
assert_allclose(S1, S2, rtol=1e-11)
def test_circfuncs_array_like(self):
x = [355, 5, 2, 359, 10, 350]
assert_allclose(stats.circmean(x, high=360), 0.167690146, rtol=1e-7)
assert_allclose(stats.circvar(x, high=360), 42.51955609, rtol=1e-7)
assert_allclose(stats.circstd(x, high=360), 6.520702116, rtol=1e-7)
def test_empty(self):
assert_(np.isnan(stats.circmean([])))
assert_(np.isnan(stats.circstd([])))
assert_(np.isnan(stats.circvar([])))
def test_circmean_scalar(self):
x = 1.
M1 = x
M2 = stats.circmean(x)
assert_allclose(M2, M1, rtol=1e-5)
def test_circmean_range(self):
# regression test for gh-6420: circmean(..., high, low) must be
# between `high` and `low`
m = stats.circmean(np.arange(0, 2, 0.1), np.pi, -np.pi)
assert_(m < np.pi)
assert_(m > -np.pi)
def test_circfuncs_unit8(self):
# regression test for gh-7255: overflow when working with
# numpy uint8 data type
x = np.array([150, 10], dtype='uint8')
assert_equal(stats.circmean(x, high=180), 170.0)
assert_allclose(stats.circvar(x, high=180), 437.45871686, rtol=1e-7)
assert_allclose(stats.circstd(x, high=180), 20.91551378, rtol=1e-7)
def test_accuracy_wilcoxon():
freq = [1, 4, 16, 15, 8, 4, 5, 1, 2]
nums = range(-4, 5)
x = np.concatenate([[u] * v for u, v in zip(nums, freq)])
y = np.zeros(x.size)
T, p = stats.wilcoxon(x, y, "pratt")
assert_allclose(T, 423)
assert_allclose(p, 0.00197547303533107)
T, p = stats.wilcoxon(x, y, "zsplit")
assert_allclose(T, 441)
assert_allclose(p, 0.0032145343172473055)
T, p = stats.wilcoxon(x, y, "wilcox")
assert_allclose(T, 327)
assert_allclose(p, 0.00641346115861)
# Test the 'correction' option, using values computed in R with:
# > wilcox.test(x, y, paired=TRUE, exact=FALSE, correct={FALSE,TRUE})
x = np.array([120, 114, 181, 188, 180, 146, 121, 191, 132, 113, 127, 112])
y = np.array([133, 143, 119, 189, 112, 199, 198, 113, 115, 121, 142, 187])
T, p = stats.wilcoxon(x, y, correction=False)
assert_equal(T, 34)
assert_allclose(p, 0.6948866, rtol=1e-6)
T, p = stats.wilcoxon(x, y, correction=True)
assert_equal(T, 34)
assert_allclose(p, 0.7240817, rtol=1e-6)
def test_wilcoxon_result_attributes():
x = np.array([120, 114, 181, 188, 180, 146, 121, 191, 132, 113, 127, 112])
y = np.array([133, 143, 119, 189, 112, 199, 198, 113, 115, 121, 142, 187])
res = stats.wilcoxon(x, y, correction=False)
attributes = ('statistic', 'pvalue')
check_named_results(res, attributes)
def test_wilcoxon_tie():
# Regression test for gh-2391.
# Corresponding R code is:
# > result = wilcox.test(rep(0.1, 10), exact=FALSE, correct=FALSE)
# > result$p.value
# [1] 0.001565402
# > result = wilcox.test(rep(0.1, 10), exact=FALSE, correct=TRUE)
# > result$p.value
# [1] 0.001904195
stat, p = stats.wilcoxon([0.1] * 10)
expected_p = 0.001565402
assert_equal(stat, 0)
assert_allclose(p, expected_p, rtol=1e-6)
stat, p = stats.wilcoxon([0.1] * 10, correction=True)
expected_p = 0.001904195
assert_equal(stat, 0)
assert_allclose(p, expected_p, rtol=1e-6)
class TestMedianTest(object):
def test_bad_n_samples(self):
# median_test requires at least two samples.
assert_raises(ValueError, stats.median_test, [1, 2, 3])
def test_empty_sample(self):
# Each sample must contain at least one value.
assert_raises(ValueError, stats.median_test, [], [1, 2, 3])
def test_empty_when_ties_ignored(self):
# The grand median is 1, and all values in the first argument are
# equal to the grand median. With ties="ignore", those values are
# ignored, which results in the first sample being (in effect) empty.
# This should raise a ValueError.
assert_raises(ValueError, stats.median_test,
[1, 1, 1, 1], [2, 0, 1], [2, 0], ties="ignore")
def test_empty_contingency_row(self):
# The grand median is 1, and with the default ties="below", all the
# values in the samples are counted as being below the grand median.
# This would result a row of zeros in the contingency table, which is
# an error.
assert_raises(ValueError, stats.median_test, [1, 1, 1], [1, 1, 1])
# With ties="above", all the values are counted as above the
# grand median.
assert_raises(ValueError, stats.median_test, [1, 1, 1], [1, 1, 1],
ties="above")
def test_bad_ties(self):
assert_raises(ValueError, stats.median_test, [1, 2, 3], [4, 5],
ties="foo")
def test_bad_nan_policy(self):
assert_raises(ValueError, stats.median_test, [1, 2, 3], [4, 5], nan_policy='foobar')
def test_bad_keyword(self):
assert_raises(TypeError, stats.median_test, [1, 2, 3], [4, 5],
foo="foo")
def test_simple(self):
x = [1, 2, 3]
y = [1, 2, 3]
stat, p, med, tbl = stats.median_test(x, y)
# The median is floating point, but this equality test should be safe.
assert_equal(med, 2.0)
assert_array_equal(tbl, [[1, 1], [2, 2]])
# The expected values of the contingency table equal the contingency
# table, so the statistic should be 0 and the p-value should be 1.
assert_equal(stat, 0)
assert_equal(p, 1)
def test_ties_options(self):
# Test the contingency table calculation.
x = [1, 2, 3, 4]
y = [5, 6]
z = [7, 8, 9]
# grand median is 5.
# Default 'ties' option is "below".
stat, p, m, tbl = stats.median_test(x, y, z)
assert_equal(m, 5)
assert_equal(tbl, [[0, 1, 3], [4, 1, 0]])
stat, p, m, tbl = stats.median_test(x, y, z, ties="ignore")
assert_equal(m, 5)
assert_equal(tbl, [[0, 1, 3], [4, 0, 0]])
stat, p, m, tbl = stats.median_test(x, y, z, ties="above")
assert_equal(m, 5)
assert_equal(tbl, [[0, 2, 3], [4, 0, 0]])
def test_nan_policy_options(self):
x = [1, 2, np.nan]
y = [4, 5, 6]
mt1 = stats.median_test(x, y, nan_policy='propagate')
s, p, m, t = stats.median_test(x, y, nan_policy='omit')
assert_equal(mt1, (np.nan, np.nan, np.nan, None))
assert_allclose(s, 0.31250000000000006)
assert_allclose(p, 0.57615012203057869)
assert_equal(m, 4.0)
assert_equal(t, np.array([[0, 2],[2, 1]]))
assert_raises(ValueError, stats.median_test, x, y, nan_policy='raise')
def test_basic(self):
# median_test calls chi2_contingency to compute the test statistic
# and p-value. Make sure it hasn't screwed up the call...
x = [1, 2, 3, 4, 5]
y = [2, 4, 6, 8]
stat, p, m, tbl = stats.median_test(x, y)
assert_equal(m, 4)
assert_equal(tbl, [[1, 2], [4, 2]])
exp_stat, exp_p, dof, e = stats.chi2_contingency(tbl)
assert_allclose(stat, exp_stat)
assert_allclose(p, exp_p)
stat, p, m, tbl = stats.median_test(x, y, lambda_=0)
assert_equal(m, 4)
assert_equal(tbl, [[1, 2], [4, 2]])
exp_stat, exp_p, dof, e = stats.chi2_contingency(tbl, lambda_=0)
assert_allclose(stat, exp_stat)
assert_allclose(p, exp_p)
stat, p, m, tbl = stats.median_test(x, y, correction=False)
assert_equal(m, 4)
assert_equal(tbl, [[1, 2], [4, 2]])
exp_stat, exp_p, dof, e = stats.chi2_contingency(tbl, correction=False)
assert_allclose(stat, exp_stat)
assert_allclose(p, exp_p)
| 55,766 | 37.889121 | 92 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/tests/test_tukeylambda_stats.py
|
from __future__ import division, print_function, absolute_import
import numpy as np
from numpy.testing import assert_allclose, assert_equal
from scipy.stats._tukeylambda_stats import (tukeylambda_variance,
tukeylambda_kurtosis)
def test_tukeylambda_stats_known_exact():
"""Compare results with some known exact formulas."""
# Some exact values of the Tukey Lambda variance and kurtosis:
# lambda var kurtosis
# 0 pi**2/3 6/5 (logistic distribution)
# 0.5 4 - pi (5/3 - pi/2)/(pi/4 - 1)**2 - 3
# 1 1/3 -6/5 (uniform distribution on (-1,1))
# 2 1/12 -6/5 (uniform distribution on (-1/2, 1/2))
# lambda = 0
var = tukeylambda_variance(0)
assert_allclose(var, np.pi**2 / 3, atol=1e-12)
kurt = tukeylambda_kurtosis(0)
assert_allclose(kurt, 1.2, atol=1e-10)
# lambda = 0.5
var = tukeylambda_variance(0.5)
assert_allclose(var, 4 - np.pi, atol=1e-12)
kurt = tukeylambda_kurtosis(0.5)
desired = (5./3 - np.pi/2) / (np.pi/4 - 1)**2 - 3
assert_allclose(kurt, desired, atol=1e-10)
# lambda = 1
var = tukeylambda_variance(1)
assert_allclose(var, 1.0 / 3, atol=1e-12)
kurt = tukeylambda_kurtosis(1)
assert_allclose(kurt, -1.2, atol=1e-10)
# lambda = 2
var = tukeylambda_variance(2)
assert_allclose(var, 1.0 / 12, atol=1e-12)
kurt = tukeylambda_kurtosis(2)
assert_allclose(kurt, -1.2, atol=1e-10)
def test_tukeylambda_stats_mpmath():
"""Compare results with some values that were computed using mpmath."""
a10 = dict(atol=1e-10, rtol=0)
a12 = dict(atol=1e-12, rtol=0)
data = [
# lambda variance kurtosis
[-0.1, 4.78050217874253547, 3.78559520346454510],
[-0.0649, 4.16428023599895777, 2.52019675947435718],
[-0.05, 3.93672267890775277, 2.13129793057777277],
[-0.001, 3.30128380390964882, 1.21452460083542988],
[0.001, 3.27850775649572176, 1.18560634779287585],
[0.03125, 2.95927803254615800, 0.804487555161819980],
[0.05, 2.78281053405464501, 0.611604043886644327],
[0.0649, 2.65282386754100551, 0.476834119532774540],
[1.2, 0.242153920578588346, -1.23428047169049726],
[10.0, 0.00095237579757703597, 2.37810697355144933],
[20.0, 0.00012195121951131043, 7.37654321002709531],
]
for lam, var_expected, kurt_expected in data:
var = tukeylambda_variance(lam)
assert_allclose(var, var_expected, **a12)
kurt = tukeylambda_kurtosis(lam)
assert_allclose(kurt, kurt_expected, **a10)
# Test with vector arguments (most of the other tests are for single
# values).
lam, var_expected, kurt_expected = zip(*data)
var = tukeylambda_variance(lam)
assert_allclose(var, var_expected, **a12)
kurt = tukeylambda_kurtosis(lam)
assert_allclose(kurt, kurt_expected, **a10)
def test_tukeylambda_stats_invalid():
"""Test values of lambda outside the domains of the functions."""
lam = [-1.0, -0.5]
var = tukeylambda_variance(lam)
assert_equal(var, np.array([np.nan, np.inf]))
lam = [-1.0, -0.25]
kurt = tukeylambda_kurtosis(lam)
assert_equal(kurt, np.array([np.nan, np.inf]))
| 3,298 | 36.067416 | 75 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/tests/test_contingency.py
|
from __future__ import division, print_function, absolute_import
import numpy as np
from numpy.testing import (assert_equal, assert_array_equal,
assert_array_almost_equal, assert_approx_equal, assert_allclose)
from pytest import raises as assert_raises
from scipy.special import xlogy
from scipy.stats.contingency import margins, expected_freq, chi2_contingency
def test_margins():
a = np.array([1])
m = margins(a)
assert_equal(len(m), 1)
m0 = m[0]
assert_array_equal(m0, np.array([1]))
a = np.array([[1]])
m0, m1 = margins(a)
expected0 = np.array([[1]])
expected1 = np.array([[1]])
assert_array_equal(m0, expected0)
assert_array_equal(m1, expected1)
a = np.arange(12).reshape(2, 6)
m0, m1 = margins(a)
expected0 = np.array([[15], [51]])
expected1 = np.array([[6, 8, 10, 12, 14, 16]])
assert_array_equal(m0, expected0)
assert_array_equal(m1, expected1)
a = np.arange(24).reshape(2, 3, 4)
m0, m1, m2 = margins(a)
expected0 = np.array([[[66]], [[210]]])
expected1 = np.array([[[60], [92], [124]]])
expected2 = np.array([[[60, 66, 72, 78]]])
assert_array_equal(m0, expected0)
assert_array_equal(m1, expected1)
assert_array_equal(m2, expected2)
def test_expected_freq():
assert_array_equal(expected_freq([1]), np.array([1.0]))
observed = np.array([[[2, 0], [0, 2]], [[0, 2], [2, 0]], [[1, 1], [1, 1]]])
e = expected_freq(observed)
assert_array_equal(e, np.ones_like(observed))
observed = np.array([[10, 10, 20], [20, 20, 20]])
e = expected_freq(observed)
correct = np.array([[12., 12., 16.], [18., 18., 24.]])
assert_array_almost_equal(e, correct)
def test_chi2_contingency_trivial():
# Some very simple tests for chi2_contingency.
# A trivial case
obs = np.array([[1, 2], [1, 2]])
chi2, p, dof, expected = chi2_contingency(obs, correction=False)
assert_equal(chi2, 0.0)
assert_equal(p, 1.0)
assert_equal(dof, 1)
assert_array_equal(obs, expected)
# A *really* trivial case: 1-D data.
obs = np.array([1, 2, 3])
chi2, p, dof, expected = chi2_contingency(obs, correction=False)
assert_equal(chi2, 0.0)
assert_equal(p, 1.0)
assert_equal(dof, 0)
assert_array_equal(obs, expected)
def test_chi2_contingency_R():
# Some test cases that were computed independently, using R.
Rcode = \
"""
# Data vector.
data <- c(
12, 34, 23, 4, 47, 11,
35, 31, 11, 34, 10, 18,
12, 32, 9, 18, 13, 19,
12, 12, 14, 9, 33, 25
)
# Create factor tags:r=rows, c=columns, t=tiers
r <- factor(gl(4, 2*3, 2*3*4, labels=c("r1", "r2", "r3", "r4")))
c <- factor(gl(3, 1, 2*3*4, labels=c("c1", "c2", "c3")))
t <- factor(gl(2, 3, 2*3*4, labels=c("t1", "t2")))
# 3-way Chi squared test of independence
s = summary(xtabs(data~r+c+t))
print(s)
"""
Routput = \
"""
Call: xtabs(formula = data ~ r + c + t)
Number of cases in table: 478
Number of factors: 3
Test for independence of all factors:
Chisq = 102.17, df = 17, p-value = 3.514e-14
"""
obs = np.array(
[[[12, 34, 23],
[35, 31, 11],
[12, 32, 9],
[12, 12, 14]],
[[4, 47, 11],
[34, 10, 18],
[18, 13, 19],
[9, 33, 25]]])
chi2, p, dof, expected = chi2_contingency(obs)
assert_approx_equal(chi2, 102.17, significant=5)
assert_approx_equal(p, 3.514e-14, significant=4)
assert_equal(dof, 17)
Rcode = \
"""
# Data vector.
data <- c(
#
12, 17,
11, 16,
#
11, 12,
15, 16,
#
23, 15,
30, 22,
#
14, 17,
15, 16
)
# Create factor tags:r=rows, c=columns, d=depths(?), t=tiers
r <- factor(gl(2, 2, 2*2*2*2, labels=c("r1", "r2")))
c <- factor(gl(2, 1, 2*2*2*2, labels=c("c1", "c2")))
d <- factor(gl(2, 4, 2*2*2*2, labels=c("d1", "d2")))
t <- factor(gl(2, 8, 2*2*2*2, labels=c("t1", "t2")))
# 4-way Chi squared test of independence
s = summary(xtabs(data~r+c+d+t))
print(s)
"""
Routput = \
"""
Call: xtabs(formula = data ~ r + c + d + t)
Number of cases in table: 262
Number of factors: 4
Test for independence of all factors:
Chisq = 8.758, df = 11, p-value = 0.6442
"""
obs = np.array(
[[[[12, 17],
[11, 16]],
[[11, 12],
[15, 16]]],
[[[23, 15],
[30, 22]],
[[14, 17],
[15, 16]]]])
chi2, p, dof, expected = chi2_contingency(obs)
assert_approx_equal(chi2, 8.758, significant=4)
assert_approx_equal(p, 0.6442, significant=4)
assert_equal(dof, 11)
def test_chi2_contingency_g():
c = np.array([[15, 60], [15, 90]])
g, p, dof, e = chi2_contingency(c, lambda_='log-likelihood', correction=False)
assert_allclose(g, 2*xlogy(c, c/e).sum())
g, p, dof, e = chi2_contingency(c, lambda_='log-likelihood', correction=True)
c_corr = c + np.array([[-0.5, 0.5], [0.5, -0.5]])
assert_allclose(g, 2*xlogy(c_corr, c_corr/e).sum())
c = np.array([[10, 12, 10], [12, 10, 10]])
g, p, dof, e = chi2_contingency(c, lambda_='log-likelihood')
assert_allclose(g, 2*xlogy(c, c/e).sum())
def test_chi2_contingency_bad_args():
# Test that "bad" inputs raise a ValueError.
# Negative value in the array of observed frequencies.
obs = np.array([[-1, 10], [1, 2]])
assert_raises(ValueError, chi2_contingency, obs)
# The zeros in this will result in zeros in the array
# of expected frequencies.
obs = np.array([[0, 1], [0, 1]])
assert_raises(ValueError, chi2_contingency, obs)
# A degenerate case: `observed` has size 0.
obs = np.empty((0, 8))
assert_raises(ValueError, chi2_contingency, obs)
| 5,910 | 28.40796 | 82 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/tests/test_continuous_basic.py
|
from __future__ import division, print_function, absolute_import
import numpy as np
import numpy.testing as npt
import pytest
from pytest import raises as assert_raises
from scipy._lib._numpy_compat import suppress_warnings
from scipy.integrate import IntegrationWarning
from scipy import stats
from scipy.special import betainc
from. common_tests import (check_normalization, check_moment, check_mean_expect,
check_var_expect, check_skew_expect,
check_kurt_expect, check_entropy,
check_private_entropy,
check_edge_support, check_named_args,
check_random_state_property,
check_meth_dtype, check_ppf_dtype, check_cmplx_deriv,
check_pickling, check_rvs_broadcast)
from scipy.stats._distr_params import distcont
"""
Test all continuous distributions.
Parameters were chosen for those distributions that pass the
Kolmogorov-Smirnov test. This provides safe parameters for each
distributions so that we can perform further testing of class methods.
These tests currently check only/mostly for serious errors and exceptions,
not for numerically exact results.
"""
# Note that you need to add new distributions you want tested
# to _distr_params
DECIMAL = 5 # specify the precision of the tests # increased from 0 to 5
# Last four of these fail all around. Need to be checked
distcont_extra = [
['betaprime', (100, 86)],
['fatiguelife', (5,)],
['mielke', (4.6420495492121487, 0.59707419545516938)],
['invweibull', (0.58847112119264788,)],
# burr: sample mean test fails still for c<1
['burr', (0.94839838075366045, 4.3820284068855795)],
# genextreme: sample mean test, sf-logsf test fail
['genextreme', (3.3184017469423535,)],
]
distslow = ['rdist', 'gausshyper', 'recipinvgauss', 'ksone', 'genexpon',
'vonmises', 'vonmises_line', 'mielke', 'semicircular',
'cosine', 'invweibull', 'powerlognorm', 'johnsonsu', 'kstwobign']
# distslow are sorted by speed (very slow to slow)
# These distributions fail the complex derivative test below.
# Here 'fail' mean produce wrong results and/or raise exceptions, depending
# on the implementation details of corresponding special functions.
# cf https://github.com/scipy/scipy/pull/4979 for a discussion.
fails_cmplx = set(['beta', 'betaprime', 'chi', 'chi2', 'dgamma', 'dweibull',
'erlang', 'f', 'gamma', 'gausshyper', 'gengamma',
'gennorm', 'genpareto', 'halfgennorm', 'invgamma',
'ksone', 'kstwobign', 'levy_l', 'loggamma', 'logistic',
'maxwell', 'nakagami', 'ncf', 'nct', 'ncx2', 'norminvgauss',
'pearson3', 'rice', 't', 'skewnorm', 'tukeylambda',
'vonmises', 'vonmises_line', 'rv_histogram_instance'])
_h = np.histogram([1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6,
6, 6, 6, 7, 7, 7, 8, 8, 9], bins=8)
histogram_test_instance = stats.rv_histogram(_h)
def cases_test_cont_basic():
for distname, arg in distcont[:] + [(histogram_test_instance, tuple())]:
if distname == 'levy_stable':
continue
elif distname in distslow:
yield pytest.param(distname, arg, marks=pytest.mark.slow)
else:
yield distname, arg
@pytest.mark.parametrize('distname,arg', cases_test_cont_basic())
def test_cont_basic(distname, arg):
# this test skips slow distributions
if distname == 'truncnorm':
pytest.xfail(reason=distname)
try:
distfn = getattr(stats, distname)
except TypeError:
distfn = distname
distname = 'rv_histogram_instance'
np.random.seed(765456)
sn = 500
with suppress_warnings() as sup:
# frechet_l and frechet_r are deprecated, so all their
# methods generate DeprecationWarnings.
sup.filter(category=DeprecationWarning, message=".*frechet_")
rvs = distfn.rvs(size=sn, *arg)
sm = rvs.mean()
sv = rvs.var()
m, v = distfn.stats(*arg)
check_sample_meanvar_(distfn, arg, m, v, sm, sv, sn, distname + 'sample mean test')
check_cdf_ppf(distfn, arg, distname)
check_sf_isf(distfn, arg, distname)
check_pdf(distfn, arg, distname)
check_pdf_logpdf(distfn, arg, distname)
check_cdf_logcdf(distfn, arg, distname)
check_sf_logsf(distfn, arg, distname)
alpha = 0.01
if distname == 'rv_histogram_instance':
check_distribution_rvs(distfn.cdf, arg, alpha, rvs)
else:
check_distribution_rvs(distname, arg, alpha, rvs)
locscale_defaults = (0, 1)
meths = [distfn.pdf, distfn.logpdf, distfn.cdf, distfn.logcdf,
distfn.logsf]
# make sure arguments are within support
spec_x = {'frechet_l': -0.5, 'weibull_max': -0.5, 'levy_l': -0.5,
'pareto': 1.5, 'tukeylambda': 0.3,
'rv_histogram_instance': 5.0}
x = spec_x.get(distname, 0.5)
if distname == 'invweibull':
arg = (1,)
elif distname == 'ksone':
arg = (3,)
check_named_args(distfn, x, arg, locscale_defaults, meths)
check_random_state_property(distfn, arg)
check_pickling(distfn, arg)
# Entropy
if distname not in ['ksone', 'kstwobign']:
check_entropy(distfn, arg, distname)
if distfn.numargs == 0:
check_vecentropy(distfn, arg)
if (distfn.__class__._entropy != stats.rv_continuous._entropy
and distname != 'vonmises'):
check_private_entropy(distfn, arg, stats.rv_continuous)
check_edge_support(distfn, arg)
check_meth_dtype(distfn, arg, meths)
check_ppf_dtype(distfn, arg)
if distname not in fails_cmplx:
check_cmplx_deriv(distfn, arg)
if distname != 'truncnorm':
check_ppf_private(distfn, arg, distname)
def test_levy_stable_random_state_property():
# levy_stable only implements rvs(), so it is skipped in the
# main loop in test_cont_basic(). Here we apply just the test
# check_random_state_property to levy_stable.
check_random_state_property(stats.levy_stable, (0.5, 0.1))
def cases_test_moments():
fail_normalization = set(['vonmises', 'ksone'])
fail_higher = set(['vonmises', 'ksone', 'ncf'])
for distname, arg in distcont[:] + [(histogram_test_instance, tuple())]:
if distname == 'levy_stable':
continue
cond1 = distname not in fail_normalization
cond2 = distname not in fail_higher
yield distname, arg, cond1, cond2, False
if not cond1 or not cond2:
# Run the distributions that have issues twice, once skipping the
# not_ok parts, once with the not_ok parts but marked as knownfail
yield pytest.param(distname, arg, True, True, True,
marks=pytest.mark.xfail)
@pytest.mark.slow
@pytest.mark.parametrize('distname,arg,normalization_ok,higher_ok,is_xfailing',
cases_test_moments())
def test_moments(distname, arg, normalization_ok, higher_ok, is_xfailing):
try:
distfn = getattr(stats, distname)
except TypeError:
distfn = distname
distname = 'rv_histogram_instance'
with suppress_warnings() as sup:
sup.filter(IntegrationWarning,
"The integral is probably divergent, or slowly convergent.")
sup.filter(category=DeprecationWarning, message=".*frechet_")
if is_xfailing:
sup.filter(IntegrationWarning)
m, v, s, k = distfn.stats(*arg, moments='mvsk')
if normalization_ok:
check_normalization(distfn, arg, distname)
if higher_ok:
check_mean_expect(distfn, arg, m, distname)
check_skew_expect(distfn, arg, m, v, s, distname)
check_var_expect(distfn, arg, m, v, distname)
check_kurt_expect(distfn, arg, m, v, k, distname)
check_loc_scale(distfn, arg, m, v, distname)
check_moment(distfn, arg, m, v, distname)
@pytest.mark.parametrize('dist,shape_args', distcont)
def test_rvs_broadcast(dist, shape_args):
if dist in ['gausshyper', 'genexpon']:
pytest.skip("too slow")
# If shape_only is True, it means the _rvs method of the
# distribution uses more than one random number to generate a random
# variate. That means the result of using rvs with broadcasting or
# with a nontrivial size will not necessarily be the same as using the
# numpy.vectorize'd version of rvs(), so we can only compare the shapes
# of the results, not the values.
# Whether or not a distribution is in the following list is an
# implementation detail of the distribution, not a requirement. If
# the implementation the rvs() method of a distribution changes, this
# test might also have to be changed.
shape_only = dist in ['betaprime', 'dgamma', 'exponnorm', 'norminvgauss',
'nct', 'dweibull', 'rice', 'levy_stable', 'skewnorm']
distfunc = getattr(stats, dist)
loc = np.zeros(2)
scale = np.ones((3, 1))
nargs = distfunc.numargs
allargs = []
bshape = [3, 2]
# Generate shape parameter arguments...
for k in range(nargs):
shp = (k + 4,) + (1,)*(k + 2)
allargs.append(shape_args[k]*np.ones(shp))
bshape.insert(0, k + 4)
allargs.extend([loc, scale])
# bshape holds the expected shape when loc, scale, and the shape
# parameters are all broadcast together.
check_rvs_broadcast(distfunc, dist, allargs, bshape, shape_only, 'd')
def test_rvs_gh2069_regression():
# Regression tests for gh-2069. In scipy 0.17 and earlier,
# these tests would fail.
#
# A typical example of the broken behavior:
# >>> norm.rvs(loc=np.zeros(5), scale=np.ones(5))
# array([-2.49613705, -2.49613705, -2.49613705, -2.49613705, -2.49613705])
np.random.seed(123)
vals = stats.norm.rvs(loc=np.zeros(5), scale=1)
d = np.diff(vals)
npt.assert_(np.all(d != 0), "All the values are equal, but they shouldn't be!")
vals = stats.norm.rvs(loc=0, scale=np.ones(5))
d = np.diff(vals)
npt.assert_(np.all(d != 0), "All the values are equal, but they shouldn't be!")
vals = stats.norm.rvs(loc=np.zeros(5), scale=np.ones(5))
d = np.diff(vals)
npt.assert_(np.all(d != 0), "All the values are equal, but they shouldn't be!")
vals = stats.norm.rvs(loc=np.array([[0], [0]]), scale=np.ones(5))
d = np.diff(vals.ravel())
npt.assert_(np.all(d != 0), "All the values are equal, but they shouldn't be!")
assert_raises(ValueError, stats.norm.rvs, [[0, 0], [0, 0]],
[[1, 1], [1, 1]], 1)
assert_raises(ValueError, stats.gamma.rvs, [2, 3, 4, 5], 0, 1, (2, 2))
assert_raises(ValueError, stats.gamma.rvs, [1, 1, 1, 1], [0, 0, 0, 0],
[[1], [2]], (4,))
def check_sample_meanvar_(distfn, arg, m, v, sm, sv, sn, msg):
# this did not work, skipped silently by nose
if np.isfinite(m):
check_sample_mean(sm, sv, sn, m)
if np.isfinite(v):
check_sample_var(sv, sn, v)
def check_sample_mean(sm, v, n, popmean):
# from stats.stats.ttest_1samp(a, popmean):
# Calculates the t-obtained for the independent samples T-test on ONE group
# of scores a, given a population mean.
#
# Returns: t-value, two-tailed prob
df = n-1
svar = ((n-1)*v) / float(df) # looks redundant
t = (sm-popmean) / np.sqrt(svar*(1.0/n))
prob = betainc(0.5*df, 0.5, df/(df + t*t))
# return t,prob
npt.assert_(prob > 0.01, 'mean fail, t,prob = %f, %f, m, sm=%f,%f' %
(t, prob, popmean, sm))
def check_sample_var(sv, n, popvar):
# two-sided chisquare test for sample variance equal to
# hypothesized variance
df = n-1
chi2 = (n-1)*popvar/float(popvar)
pval = stats.distributions.chi2.sf(chi2, df) * 2
npt.assert_(pval > 0.01, 'var fail, t, pval = %f, %f, v, sv=%f, %f' %
(chi2, pval, popvar, sv))
def check_cdf_ppf(distfn, arg, msg):
values = [0.001, 0.5, 0.999]
npt.assert_almost_equal(distfn.cdf(distfn.ppf(values, *arg), *arg),
values, decimal=DECIMAL, err_msg=msg +
' - cdf-ppf roundtrip')
def check_sf_isf(distfn, arg, msg):
npt.assert_almost_equal(distfn.sf(distfn.isf([0.1, 0.5, 0.9], *arg), *arg),
[0.1, 0.5, 0.9], decimal=DECIMAL, err_msg=msg +
' - sf-isf roundtrip')
npt.assert_almost_equal(distfn.cdf([0.1, 0.9], *arg),
1.0 - distfn.sf([0.1, 0.9], *arg),
decimal=DECIMAL, err_msg=msg +
' - cdf-sf relationship')
def check_pdf(distfn, arg, msg):
# compares pdf at median with numerical derivative of cdf
median = distfn.ppf(0.5, *arg)
eps = 1e-6
pdfv = distfn.pdf(median, *arg)
if (pdfv < 1e-4) or (pdfv > 1e4):
# avoid checking a case where pdf is close to zero or
# huge (singularity)
median = median + 0.1
pdfv = distfn.pdf(median, *arg)
cdfdiff = (distfn.cdf(median + eps, *arg) -
distfn.cdf(median - eps, *arg))/eps/2.0
# replace with better diff and better test (more points),
# actually, this works pretty well
msg += ' - cdf-pdf relationship'
npt.assert_almost_equal(pdfv, cdfdiff, decimal=DECIMAL, err_msg=msg)
def check_pdf_logpdf(distfn, args, msg):
# compares pdf at several points with the log of the pdf
points = np.array([0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8])
vals = distfn.ppf(points, *args)
pdf = distfn.pdf(vals, *args)
logpdf = distfn.logpdf(vals, *args)
pdf = pdf[pdf != 0]
logpdf = logpdf[np.isfinite(logpdf)]
msg += " - logpdf-log(pdf) relationship"
npt.assert_almost_equal(np.log(pdf), logpdf, decimal=7, err_msg=msg)
def check_sf_logsf(distfn, args, msg):
# compares sf at several points with the log of the sf
points = np.array([0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8])
vals = distfn.ppf(points, *args)
sf = distfn.sf(vals, *args)
logsf = distfn.logsf(vals, *args)
sf = sf[sf != 0]
logsf = logsf[np.isfinite(logsf)]
msg += " - logsf-log(sf) relationship"
npt.assert_almost_equal(np.log(sf), logsf, decimal=7, err_msg=msg)
def check_cdf_logcdf(distfn, args, msg):
# compares cdf at several points with the log of the cdf
points = np.array([0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8])
vals = distfn.ppf(points, *args)
cdf = distfn.cdf(vals, *args)
logcdf = distfn.logcdf(vals, *args)
cdf = cdf[cdf != 0]
logcdf = logcdf[np.isfinite(logcdf)]
msg += " - logcdf-log(cdf) relationship"
npt.assert_almost_equal(np.log(cdf), logcdf, decimal=7, err_msg=msg)
def check_distribution_rvs(dist, args, alpha, rvs):
# test from scipy.stats.tests
# this version reuses existing random variables
D, pval = stats.kstest(rvs, dist, args=args, N=1000)
if (pval < alpha):
D, pval = stats.kstest(dist, '', args=args, N=1000)
npt.assert_(pval > alpha, "D = " + str(D) + "; pval = " + str(pval) +
"; alpha = " + str(alpha) + "\nargs = " + str(args))
def check_vecentropy(distfn, args):
npt.assert_equal(distfn.vecentropy(*args), distfn._entropy(*args))
def check_loc_scale(distfn, arg, m, v, msg):
loc, scale = 10.0, 10.0
mt, vt = distfn.stats(loc=loc, scale=scale, *arg)
npt.assert_allclose(m*scale + loc, mt)
npt.assert_allclose(v*scale*scale, vt)
def check_ppf_private(distfn, arg, msg):
# fails by design for truncnorm self.nb not defined
ppfs = distfn._ppf(np.array([0.1, 0.5, 0.9]), *arg)
npt.assert_(not np.any(np.isnan(ppfs)), msg + 'ppf private is nan')
| 16,009 | 37.671498 | 91 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/tests/common_tests.py
|
from __future__ import division, print_function, absolute_import
import pickle
import numpy as np
import numpy.testing as npt
from numpy.testing import assert_allclose, assert_equal
from scipy._lib._numpy_compat import suppress_warnings
from pytest import raises as assert_raises
import numpy.ma.testutils as ma_npt
from scipy._lib._util import getargspec_no_self as _getargspec
from scipy import stats
def check_named_results(res, attributes, ma=False):
for i, attr in enumerate(attributes):
if ma:
ma_npt.assert_equal(res[i], getattr(res, attr))
else:
npt.assert_equal(res[i], getattr(res, attr))
def check_normalization(distfn, args, distname):
norm_moment = distfn.moment(0, *args)
npt.assert_allclose(norm_moment, 1.0)
# this is a temporary plug: either ncf or expect is problematic;
# best be marked as a knownfail, but I've no clue how to do it.
if distname == "ncf":
atol, rtol = 1e-5, 0
else:
atol, rtol = 1e-7, 1e-7
normalization_expect = distfn.expect(lambda x: 1, args=args)
npt.assert_allclose(normalization_expect, 1.0, atol=atol, rtol=rtol,
err_msg=distname, verbose=True)
normalization_cdf = distfn.cdf(distfn.b, *args)
npt.assert_allclose(normalization_cdf, 1.0)
def check_moment(distfn, arg, m, v, msg):
m1 = distfn.moment(1, *arg)
m2 = distfn.moment(2, *arg)
if not np.isinf(m):
npt.assert_almost_equal(m1, m, decimal=10, err_msg=msg +
' - 1st moment')
else: # or np.isnan(m1),
npt.assert_(np.isinf(m1),
msg + ' - 1st moment -infinite, m1=%s' % str(m1))
if not np.isinf(v):
npt.assert_almost_equal(m2 - m1 * m1, v, decimal=10, err_msg=msg +
' - 2ndt moment')
else: # or np.isnan(m2),
npt.assert_(np.isinf(m2),
msg + ' - 2nd moment -infinite, m2=%s' % str(m2))
def check_mean_expect(distfn, arg, m, msg):
if np.isfinite(m):
m1 = distfn.expect(lambda x: x, arg)
npt.assert_almost_equal(m1, m, decimal=5, err_msg=msg +
' - 1st moment (expect)')
def check_var_expect(distfn, arg, m, v, msg):
if np.isfinite(v):
m2 = distfn.expect(lambda x: x*x, arg)
npt.assert_almost_equal(m2, v + m*m, decimal=5, err_msg=msg +
' - 2st moment (expect)')
def check_skew_expect(distfn, arg, m, v, s, msg):
if np.isfinite(s):
m3e = distfn.expect(lambda x: np.power(x-m, 3), arg)
npt.assert_almost_equal(m3e, s * np.power(v, 1.5),
decimal=5, err_msg=msg + ' - skew')
else:
npt.assert_(np.isnan(s))
def check_kurt_expect(distfn, arg, m, v, k, msg):
if np.isfinite(k):
m4e = distfn.expect(lambda x: np.power(x-m, 4), arg)
npt.assert_allclose(m4e, (k + 3.) * np.power(v, 2), atol=1e-5, rtol=1e-5,
err_msg=msg + ' - kurtosis')
else:
npt.assert_(np.isnan(k))
def check_entropy(distfn, arg, msg):
ent = distfn.entropy(*arg)
npt.assert_(not np.isnan(ent), msg + 'test Entropy is nan')
def check_private_entropy(distfn, args, superclass):
# compare a generic _entropy with the distribution-specific implementation
npt.assert_allclose(distfn._entropy(*args),
superclass._entropy(distfn, *args))
def check_edge_support(distfn, args):
# Make sure that x=self.a and self.b are handled correctly.
x = [distfn.a, distfn.b]
if isinstance(distfn, stats.rv_discrete):
x = [distfn.a - 1, distfn.b]
npt.assert_equal(distfn.cdf(x, *args), [0.0, 1.0])
npt.assert_equal(distfn.sf(x, *args), [1.0, 0.0])
if distfn.name not in ('skellam', 'dlaplace'):
# with a = -inf, log(0) generates warnings
npt.assert_equal(distfn.logcdf(x, *args), [-np.inf, 0.0])
npt.assert_equal(distfn.logsf(x, *args), [0.0, -np.inf])
npt.assert_equal(distfn.ppf([0.0, 1.0], *args), x)
npt.assert_equal(distfn.isf([0.0, 1.0], *args), x[::-1])
# out-of-bounds for isf & ppf
npt.assert_(np.isnan(distfn.isf([-1, 2], *args)).all())
npt.assert_(np.isnan(distfn.ppf([-1, 2], *args)).all())
def check_named_args(distfn, x, shape_args, defaults, meths):
## Check calling w/ named arguments.
# check consistency of shapes, numargs and _parse signature
signature = _getargspec(distfn._parse_args)
npt.assert_(signature.varargs is None)
npt.assert_(signature.keywords is None)
npt.assert_(list(signature.defaults) == list(defaults))
shape_argnames = signature.args[:-len(defaults)] # a, b, loc=0, scale=1
if distfn.shapes:
shapes_ = distfn.shapes.replace(',', ' ').split()
else:
shapes_ = ''
npt.assert_(len(shapes_) == distfn.numargs)
npt.assert_(len(shapes_) == len(shape_argnames))
# check calling w/ named arguments
shape_args = list(shape_args)
vals = [meth(x, *shape_args) for meth in meths]
npt.assert_(np.all(np.isfinite(vals)))
names, a, k = shape_argnames[:], shape_args[:], {}
while names:
k.update({names.pop(): a.pop()})
v = [meth(x, *a, **k) for meth in meths]
npt.assert_array_equal(vals, v)
if 'n' not in k.keys():
# `n` is first parameter of moment(), so can't be used as named arg
npt.assert_equal(distfn.moment(1, *a, **k),
distfn.moment(1, *shape_args))
# unknown arguments should not go through:
k.update({'kaboom': 42})
assert_raises(TypeError, distfn.cdf, x, **k)
def check_random_state_property(distfn, args):
# check the random_state attribute of a distribution *instance*
# This test fiddles with distfn.random_state. This breaks other tests,
# hence need to save it and then restore.
rndm = distfn.random_state
# baseline: this relies on the global state
np.random.seed(1234)
distfn.random_state = None
r0 = distfn.rvs(*args, size=8)
# use an explicit instance-level random_state
distfn.random_state = 1234
r1 = distfn.rvs(*args, size=8)
npt.assert_equal(r0, r1)
distfn.random_state = np.random.RandomState(1234)
r2 = distfn.rvs(*args, size=8)
npt.assert_equal(r0, r2)
# can override the instance-level random_state for an individual .rvs call
distfn.random_state = 2
orig_state = distfn.random_state.get_state()
r3 = distfn.rvs(*args, size=8, random_state=np.random.RandomState(1234))
npt.assert_equal(r0, r3)
# ... and that does not alter the instance-level random_state!
npt.assert_equal(distfn.random_state.get_state(), orig_state)
# finally, restore the random_state
distfn.random_state = rndm
def check_meth_dtype(distfn, arg, meths):
q0 = [0.25, 0.5, 0.75]
x0 = distfn.ppf(q0, *arg)
x_cast = [x0.astype(tp) for tp in
(np.int_, np.float16, np.float32, np.float64)]
for x in x_cast:
# casting may have clipped the values, exclude those
distfn._argcheck(*arg)
x = x[(distfn.a < x) & (x < distfn.b)]
for meth in meths:
val = meth(x, *arg)
npt.assert_(val.dtype == np.float_)
def check_ppf_dtype(distfn, arg):
q0 = np.asarray([0.25, 0.5, 0.75])
q_cast = [q0.astype(tp) for tp in (np.float16, np.float32, np.float64)]
for q in q_cast:
for meth in [distfn.ppf, distfn.isf]:
val = meth(q, *arg)
npt.assert_(val.dtype == np.float_)
def check_cmplx_deriv(distfn, arg):
# Distributions allow complex arguments.
def deriv(f, x, *arg):
x = np.asarray(x)
h = 1e-10
return (f(x + h*1j, *arg)/h).imag
x0 = distfn.ppf([0.25, 0.51, 0.75], *arg)
x_cast = [x0.astype(tp) for tp in
(np.int_, np.float16, np.float32, np.float64)]
for x in x_cast:
# casting may have clipped the values, exclude those
distfn._argcheck(*arg)
x = x[(distfn.a < x) & (x < distfn.b)]
pdf, cdf, sf = distfn.pdf(x, *arg), distfn.cdf(x, *arg), distfn.sf(x, *arg)
assert_allclose(deriv(distfn.cdf, x, *arg), pdf, rtol=1e-5)
assert_allclose(deriv(distfn.logcdf, x, *arg), pdf/cdf, rtol=1e-5)
assert_allclose(deriv(distfn.sf, x, *arg), -pdf, rtol=1e-5)
assert_allclose(deriv(distfn.logsf, x, *arg), -pdf/sf, rtol=1e-5)
assert_allclose(deriv(distfn.logpdf, x, *arg),
deriv(distfn.pdf, x, *arg) / distfn.pdf(x, *arg),
rtol=1e-5)
def check_pickling(distfn, args):
# check that a distribution instance pickles and unpickles
# pay special attention to the random_state property
# save the random_state (restore later)
rndm = distfn.random_state
distfn.random_state = 1234
distfn.rvs(*args, size=8)
s = pickle.dumps(distfn)
r0 = distfn.rvs(*args, size=8)
unpickled = pickle.loads(s)
r1 = unpickled.rvs(*args, size=8)
npt.assert_equal(r0, r1)
# also smoke test some methods
medians = [distfn.ppf(0.5, *args), unpickled.ppf(0.5, *args)]
npt.assert_equal(medians[0], medians[1])
npt.assert_equal(distfn.cdf(medians[0], *args),
unpickled.cdf(medians[1], *args))
# restore the random_state
distfn.random_state = rndm
def check_rvs_broadcast(distfunc, distname, allargs, shape, shape_only, otype):
np.random.seed(123)
with suppress_warnings() as sup:
# frechet_l and frechet_r are deprecated, so all their
# methods generate DeprecationWarnings.
sup.filter(category=DeprecationWarning, message=".*frechet_")
sample = distfunc.rvs(*allargs)
assert_equal(sample.shape, shape, "%s: rvs failed to broadcast" % distname)
if not shape_only:
rvs = np.vectorize(lambda *allargs: distfunc.rvs(*allargs), otypes=otype)
np.random.seed(123)
expected = rvs(*allargs)
assert_allclose(sample, expected, rtol=1e-15)
| 10,081 | 33.646048 | 85 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/tests/test_discrete_basic.py
|
from __future__ import division, print_function, absolute_import
import numpy.testing as npt
import numpy as np
from scipy._lib.six import xrange
import pytest
from scipy import stats
from .common_tests import (check_normalization, check_moment, check_mean_expect,
check_var_expect, check_skew_expect,
check_kurt_expect, check_entropy,
check_private_entropy, check_edge_support,
check_named_args, check_random_state_property,
check_pickling, check_rvs_broadcast)
from scipy.stats._distr_params import distdiscrete
vals = ([1, 2, 3, 4], [0.1, 0.2, 0.3, 0.4])
distdiscrete += [[stats.rv_discrete(values=vals), ()]]
def cases_test_discrete_basic():
seen = set()
for distname, arg in distdiscrete:
yield distname, arg, distname not in seen
seen.add(distname)
@pytest.mark.parametrize('distname,arg,first_case', cases_test_discrete_basic())
def test_discrete_basic(distname, arg, first_case):
try:
distfn = getattr(stats, distname)
except TypeError:
distfn = distname
distname = 'sample distribution'
np.random.seed(9765456)
rvs = distfn.rvs(size=2000, *arg)
supp = np.unique(rvs)
m, v = distfn.stats(*arg)
check_cdf_ppf(distfn, arg, supp, distname + ' cdf_ppf')
check_pmf_cdf(distfn, arg, distname)
check_oth(distfn, arg, supp, distname + ' oth')
check_edge_support(distfn, arg)
alpha = 0.01
check_discrete_chisquare(distfn, arg, rvs, alpha,
distname + ' chisquare')
if first_case:
locscale_defaults = (0,)
meths = [distfn.pmf, distfn.logpmf, distfn.cdf, distfn.logcdf,
distfn.logsf]
# make sure arguments are within support
spec_k = {'randint': 11, 'hypergeom': 4, 'bernoulli': 0, }
k = spec_k.get(distname, 1)
check_named_args(distfn, k, arg, locscale_defaults, meths)
if distname != 'sample distribution':
check_scale_docstring(distfn)
check_random_state_property(distfn, arg)
check_pickling(distfn, arg)
# Entropy
check_entropy(distfn, arg, distname)
if distfn.__class__._entropy != stats.rv_discrete._entropy:
check_private_entropy(distfn, arg, stats.rv_discrete)
@pytest.mark.parametrize('distname,arg', distdiscrete)
def test_moments(distname, arg):
try:
distfn = getattr(stats, distname)
except TypeError:
distfn = distname
distname = 'sample distribution'
m, v, s, k = distfn.stats(*arg, moments='mvsk')
check_normalization(distfn, arg, distname)
# compare `stats` and `moment` methods
check_moment(distfn, arg, m, v, distname)
check_mean_expect(distfn, arg, m, distname)
check_var_expect(distfn, arg, m, v, distname)
check_skew_expect(distfn, arg, m, v, s, distname)
if distname not in ['zipf']:
check_kurt_expect(distfn, arg, m, v, k, distname)
# frozen distr moments
check_moment_frozen(distfn, arg, m, 1)
check_moment_frozen(distfn, arg, v+m*m, 2)
@pytest.mark.parametrize('dist,shape_args', distdiscrete)
def test_rvs_broadcast(dist, shape_args):
# If shape_only is True, it means the _rvs method of the
# distribution uses more than one random number to generate a random
# variate. That means the result of using rvs with broadcasting or
# with a nontrivial size will not necessarily be the same as using the
# numpy.vectorize'd version of rvs(), so we can only compare the shapes
# of the results, not the values.
# Whether or not a distribution is in the following list is an
# implementation detail of the distribution, not a requirement. If
# the implementation the rvs() method of a distribution changes, this
# test might also have to be changed.
shape_only = dist in ['skellam']
try:
distfunc = getattr(stats, dist)
except TypeError:
distfunc = dist
dist = 'rv_discrete(values=(%r, %r))' % (dist.xk, dist.pk)
loc = np.zeros(2)
nargs = distfunc.numargs
allargs = []
bshape = []
# Generate shape parameter arguments...
for k in range(nargs):
shp = (k + 3,) + (1,)*(k + 1)
param_val = shape_args[k]
allargs.append(param_val*np.ones(shp, dtype=np.array(param_val).dtype))
bshape.insert(0, shp[0])
allargs.append(loc)
bshape.append(loc.size)
# bshape holds the expected shape when loc, scale, and the shape
# parameters are all broadcast together.
check_rvs_broadcast(distfunc, dist, allargs, bshape, shape_only, [np.int_])
def check_cdf_ppf(distfn, arg, supp, msg):
# cdf is a step function, and ppf(q) = min{k : cdf(k) >= q, k integer}
npt.assert_array_equal(distfn.ppf(distfn.cdf(supp, *arg), *arg),
supp, msg + '-roundtrip')
npt.assert_array_equal(distfn.ppf(distfn.cdf(supp, *arg) - 1e-8, *arg),
supp, msg + '-roundtrip')
if not hasattr(distfn, 'xk'):
supp1 = supp[supp < distfn.b]
npt.assert_array_equal(distfn.ppf(distfn.cdf(supp1, *arg) + 1e-8, *arg),
supp1 + distfn.inc, msg + ' ppf-cdf-next')
# -1e-8 could cause an error if pmf < 1e-8
def check_pmf_cdf(distfn, arg, distname):
if hasattr(distfn, 'xk'):
index = distfn.xk
else:
startind = int(distfn.ppf(0.01, *arg) - 1)
index = list(range(startind, startind + 10))
cdfs = distfn.cdf(index, *arg)
pmfs_cum = distfn.pmf(index, *arg).cumsum()
atol, rtol = 1e-10, 1e-10
if distname == 'skellam': # ncx2 accuracy
atol, rtol = 1e-5, 1e-5
npt.assert_allclose(cdfs - cdfs[0], pmfs_cum - pmfs_cum[0],
atol=atol, rtol=rtol)
def check_moment_frozen(distfn, arg, m, k):
npt.assert_allclose(distfn(*arg).moment(k), m,
atol=1e-10, rtol=1e-10)
def check_oth(distfn, arg, supp, msg):
# checking other methods of distfn
npt.assert_allclose(distfn.sf(supp, *arg), 1. - distfn.cdf(supp, *arg),
atol=1e-10, rtol=1e-10)
q = np.linspace(0.01, 0.99, 20)
npt.assert_allclose(distfn.isf(q, *arg), distfn.ppf(1. - q, *arg),
atol=1e-10, rtol=1e-10)
median_sf = distfn.isf(0.5, *arg)
npt.assert_(distfn.sf(median_sf - 1, *arg) > 0.5)
npt.assert_(distfn.cdf(median_sf + 1, *arg) > 0.5)
def check_discrete_chisquare(distfn, arg, rvs, alpha, msg):
"""Perform chisquare test for random sample of a discrete distribution
Parameters
----------
distname : string
name of distribution function
arg : sequence
parameters of distribution
alpha : float
significance level, threshold for p-value
Returns
-------
result : bool
0 if test passes, 1 if test fails
"""
wsupp = 0.05
# construct intervals with minimum mass `wsupp`.
# intervals are left-half-open as in a cdf difference
lo = int(max(distfn.a, -1000))
distsupport = xrange(lo, int(min(distfn.b, 1000)) + 1)
last = 0
distsupp = [lo]
distmass = []
for ii in distsupport:
current = distfn.cdf(ii, *arg)
if current - last >= wsupp - 1e-14:
distsupp.append(ii)
distmass.append(current - last)
last = current
if current > (1 - wsupp):
break
if distsupp[-1] < distfn.b:
distsupp.append(distfn.b)
distmass.append(1 - last)
distsupp = np.array(distsupp)
distmass = np.array(distmass)
# convert intervals to right-half-open as required by histogram
histsupp = distsupp + 1e-8
histsupp[0] = distfn.a
# find sample frequencies and perform chisquare test
freq, hsupp = np.histogram(rvs, histsupp)
chis, pval = stats.chisquare(np.array(freq), len(rvs)*distmass)
npt.assert_(pval > alpha,
'chisquare - test for %s at arg = %s with pval = %s' %
(msg, str(arg), str(pval)))
def check_scale_docstring(distfn):
if distfn.__doc__ is not None:
# Docstrings can be stripped if interpreter is run with -OO
npt.assert_('scale' not in distfn.__doc__)
| 8,295 | 34.302128 | 80 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/tests/test_kdeoth.py
|
from __future__ import division, print_function, absolute_import
from scipy import stats
import numpy as np
from numpy.testing import (assert_almost_equal, assert_,
assert_array_almost_equal, assert_array_almost_equal_nulp)
from pytest import raises as assert_raises
def test_kde_1d():
#some basic tests comparing to normal distribution
np.random.seed(8765678)
n_basesample = 500
xn = np.random.randn(n_basesample)
xnmean = xn.mean()
xnstd = xn.std(ddof=1)
# get kde for original sample
gkde = stats.gaussian_kde(xn)
# evaluate the density function for the kde for some points
xs = np.linspace(-7,7,501)
kdepdf = gkde.evaluate(xs)
normpdf = stats.norm.pdf(xs, loc=xnmean, scale=xnstd)
intervall = xs[1] - xs[0]
assert_(np.sum((kdepdf - normpdf)**2)*intervall < 0.01)
prob1 = gkde.integrate_box_1d(xnmean, np.inf)
prob2 = gkde.integrate_box_1d(-np.inf, xnmean)
assert_almost_equal(prob1, 0.5, decimal=1)
assert_almost_equal(prob2, 0.5, decimal=1)
assert_almost_equal(gkde.integrate_box(xnmean, np.inf), prob1, decimal=13)
assert_almost_equal(gkde.integrate_box(-np.inf, xnmean), prob2, decimal=13)
assert_almost_equal(gkde.integrate_kde(gkde),
(kdepdf**2).sum()*intervall, decimal=2)
assert_almost_equal(gkde.integrate_gaussian(xnmean, xnstd**2),
(kdepdf*normpdf).sum()*intervall, decimal=2)
def test_kde_2d():
#some basic tests comparing to normal distribution
np.random.seed(8765678)
n_basesample = 500
mean = np.array([1.0, 3.0])
covariance = np.array([[1.0, 2.0], [2.0, 6.0]])
# Need transpose (shape (2, 500)) for kde
xn = np.random.multivariate_normal(mean, covariance, size=n_basesample).T
# get kde for original sample
gkde = stats.gaussian_kde(xn)
# evaluate the density function for the kde for some points
x, y = np.mgrid[-7:7:500j, -7:7:500j]
grid_coords = np.vstack([x.ravel(), y.ravel()])
kdepdf = gkde.evaluate(grid_coords)
kdepdf = kdepdf.reshape(500, 500)
normpdf = stats.multivariate_normal.pdf(np.dstack([x, y]), mean=mean, cov=covariance)
intervall = y.ravel()[1] - y.ravel()[0]
assert_(np.sum((kdepdf - normpdf)**2) * (intervall**2) < 0.01)
small = -1e100
large = 1e100
prob1 = gkde.integrate_box([small, mean[1]], [large, large])
prob2 = gkde.integrate_box([small, small], [large, mean[1]])
assert_almost_equal(prob1, 0.5, decimal=1)
assert_almost_equal(prob2, 0.5, decimal=1)
assert_almost_equal(gkde.integrate_kde(gkde),
(kdepdf**2).sum()*(intervall**2), decimal=2)
assert_almost_equal(gkde.integrate_gaussian(mean, covariance),
(kdepdf*normpdf).sum()*(intervall**2), decimal=2)
def test_kde_bandwidth_method():
def scotts_factor(kde_obj):
"""Same as default, just check that it works."""
return np.power(kde_obj.n, -1./(kde_obj.d+4))
np.random.seed(8765678)
n_basesample = 50
xn = np.random.randn(n_basesample)
# Default
gkde = stats.gaussian_kde(xn)
# Supply a callable
gkde2 = stats.gaussian_kde(xn, bw_method=scotts_factor)
# Supply a scalar
gkde3 = stats.gaussian_kde(xn, bw_method=gkde.factor)
xs = np.linspace(-7,7,51)
kdepdf = gkde.evaluate(xs)
kdepdf2 = gkde2.evaluate(xs)
assert_almost_equal(kdepdf, kdepdf2)
kdepdf3 = gkde3.evaluate(xs)
assert_almost_equal(kdepdf, kdepdf3)
assert_raises(ValueError, stats.gaussian_kde, xn, bw_method='wrongstring')
# Subclasses that should stay working (extracted from various sources).
# Unfortunately the earlier design of gaussian_kde made it necessary for users
# to create these kinds of subclasses, or call _compute_covariance() directly.
class _kde_subclass1(stats.gaussian_kde):
def __init__(self, dataset):
self.dataset = np.atleast_2d(dataset)
self.d, self.n = self.dataset.shape
self.covariance_factor = self.scotts_factor
self._compute_covariance()
class _kde_subclass2(stats.gaussian_kde):
def __init__(self, dataset):
self.covariance_factor = self.scotts_factor
super(_kde_subclass2, self).__init__(dataset)
class _kde_subclass3(stats.gaussian_kde):
def __init__(self, dataset, covariance):
self.covariance = covariance
stats.gaussian_kde.__init__(self, dataset)
def _compute_covariance(self):
self.inv_cov = np.linalg.inv(self.covariance)
self._norm_factor = np.sqrt(np.linalg.det(2*np.pi * self.covariance)) \
* self.n
class _kde_subclass4(stats.gaussian_kde):
def covariance_factor(self):
return 0.5 * self.silverman_factor()
def test_gaussian_kde_subclassing():
x1 = np.array([-7, -5, 1, 4, 5], dtype=float)
xs = np.linspace(-10, 10, num=50)
# gaussian_kde itself
kde = stats.gaussian_kde(x1)
ys = kde(xs)
# subclass 1
kde1 = _kde_subclass1(x1)
y1 = kde1(xs)
assert_array_almost_equal_nulp(ys, y1, nulp=10)
# subclass 2
kde2 = _kde_subclass2(x1)
y2 = kde2(xs)
assert_array_almost_equal_nulp(ys, y2, nulp=10)
# subclass 3
kde3 = _kde_subclass3(x1, kde.covariance)
y3 = kde3(xs)
assert_array_almost_equal_nulp(ys, y3, nulp=10)
# subclass 4
kde4 = _kde_subclass4(x1)
y4 = kde4(x1)
y_expected = [0.06292987, 0.06346938, 0.05860291, 0.08657652, 0.07904017]
assert_array_almost_equal(y_expected, y4, decimal=6)
# Not a subclass, but check for use of _compute_covariance()
kde5 = kde
kde5.covariance_factor = lambda: kde.factor
kde5._compute_covariance()
y5 = kde5(xs)
assert_array_almost_equal_nulp(ys, y5, nulp=10)
def test_gaussian_kde_covariance_caching():
x1 = np.array([-7, -5, 1, 4, 5], dtype=float)
xs = np.linspace(-10, 10, num=5)
# These expected values are from scipy 0.10, before some changes to
# gaussian_kde. They were not compared with any external reference.
y_expected = [0.02463386, 0.04689208, 0.05395444, 0.05337754, 0.01664475]
# Set the bandwidth, then reset it to the default.
kde = stats.gaussian_kde(x1)
kde.set_bandwidth(bw_method=0.5)
kde.set_bandwidth(bw_method='scott')
y2 = kde(xs)
assert_array_almost_equal(y_expected, y2, decimal=7)
def test_gaussian_kde_monkeypatch():
"""Ugly, but people may rely on this. See scipy pull request 123,
specifically the linked ML thread "Width of the Gaussian in stats.kde".
If it is necessary to break this later on, that is to be discussed on ML.
"""
x1 = np.array([-7, -5, 1, 4, 5], dtype=float)
xs = np.linspace(-10, 10, num=50)
# The old monkeypatched version to get at Silverman's Rule.
kde = stats.gaussian_kde(x1)
kde.covariance_factor = kde.silverman_factor
kde._compute_covariance()
y1 = kde(xs)
# The new saner version.
kde2 = stats.gaussian_kde(x1, bw_method='silverman')
y2 = kde2(xs)
assert_array_almost_equal_nulp(y1, y2, nulp=10)
def test_kde_integer_input():
"""Regression test for #1181."""
x1 = np.arange(5)
kde = stats.gaussian_kde(x1)
y_expected = [0.13480721, 0.18222869, 0.19514935, 0.18222869, 0.13480721]
assert_array_almost_equal(kde(x1), y_expected, decimal=6)
def test_pdf_logpdf():
np.random.seed(1)
n_basesample = 50
xn = np.random.randn(n_basesample)
# Default
gkde = stats.gaussian_kde(xn)
xs = np.linspace(-15, 12, 25)
pdf = gkde.evaluate(xs)
pdf2 = gkde.pdf(xs)
assert_almost_equal(pdf, pdf2, decimal=12)
logpdf = np.log(pdf)
logpdf2 = gkde.logpdf(xs)
assert_almost_equal(logpdf, logpdf2, decimal=12)
# There are more points than data
gkde = stats.gaussian_kde(xs)
pdf = np.log(gkde.evaluate(xn))
pdf2 = gkde.logpdf(xn)
assert_almost_equal(pdf, pdf2, decimal=12)
| 7,923 | 31.342857 | 89 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/tests/test_distributions.py
|
""" Test functions for stats module
"""
from __future__ import division, print_function, absolute_import
import warnings
import re
import sys
import pickle
from numpy.testing import (assert_equal,
assert_array_equal, assert_almost_equal, assert_array_almost_equal,
assert_allclose, assert_, assert_warns)
import pytest
from pytest import raises as assert_raises
from scipy._lib._numpy_compat import suppress_warnings
import numpy
import numpy as np
from numpy import typecodes, array
from scipy import special
from scipy.integrate import IntegrationWarning
import scipy.stats as stats
from scipy.stats._distn_infrastructure import argsreduce
import scipy.stats.distributions
from scipy.special import xlogy
from .test_continuous_basic import distcont
# python -OO strips docstrings
DOCSTRINGS_STRIPPED = sys.flags.optimize > 1
# Generate test cases to test cdf and distribution consistency.
# Note that this list does not include all distributions.
dists = ['uniform', 'norm', 'lognorm', 'expon', 'beta',
'powerlaw', 'bradford', 'burr', 'fisk', 'cauchy', 'halfcauchy',
'foldcauchy', 'gamma', 'gengamma', 'loggamma',
'alpha', 'anglit', 'arcsine', 'betaprime', 'dgamma', 'moyal',
'exponnorm', 'exponweib', 'exponpow', 'frechet_l', 'frechet_r',
'gilbrat', 'f', 'ncf', 'chi2', 'chi', 'nakagami', 'genpareto',
'genextreme', 'genhalflogistic', 'pareto', 'lomax', 'halfnorm',
'halflogistic', 'fatiguelife', 'foldnorm', 'ncx2', 't', 'nct',
'weibull_min', 'weibull_max', 'dweibull', 'maxwell', 'rayleigh',
'genlogistic', 'logistic', 'gumbel_l', 'gumbel_r', 'gompertz',
'hypsecant', 'laplace', 'reciprocal', 'trapz', 'triang', 'tukeylambda',
'vonmises', 'vonmises_line', 'pearson3', 'gennorm', 'halfgennorm',
'rice', 'kappa4', 'kappa3', 'truncnorm', 'argus', 'crystalball']
def _assert_hasattr(a, b, msg=None):
if msg is None:
msg = '%s does not have attribute %s' % (a, b)
assert_(hasattr(a, b), msg=msg)
def test_api_regression():
# https://github.com/scipy/scipy/issues/3802
_assert_hasattr(scipy.stats.distributions, 'f_gen')
# check function for test generator
def check_distribution(dist, args, alpha):
with suppress_warnings() as sup:
# frechet_l and frechet_r are deprecated, so all their
# methods generate DeprecationWarnings.
sup.filter(category=DeprecationWarning, message=".*frechet_")
D, pval = stats.kstest(dist, '', args=args, N=1000)
if (pval < alpha):
D, pval = stats.kstest(dist, '', args=args, N=1000)
assert_(pval > alpha,
msg="D = {}; pval = {}; alpha = {}; args = {}".format(
D, pval, alpha, args))
def cases_test_all_distributions():
np.random.seed(1234)
for dist in dists:
distfunc = getattr(stats, dist)
nargs = distfunc.numargs
alpha = 0.01
if dist == 'fatiguelife':
alpha = 0.001
if dist == 'trapz':
args = tuple(np.sort(np.random.random(nargs)))
elif dist == 'triang':
args = tuple(np.random.random(nargs))
elif dist == 'reciprocal' or dist == 'truncnorm':
vals = np.random.random(nargs)
vals[1] = vals[0] + 1.0
args = tuple(vals)
elif dist == 'vonmises':
yield dist, (10,), alpha
yield dist, (101,), alpha
args = tuple(1.0 + np.random.random(nargs))
else:
args = tuple(1.0 + np.random.random(nargs))
yield dist, args, alpha
@pytest.mark.parametrize('dist,args,alpha', cases_test_all_distributions())
def test_all_distributions(dist, args, alpha):
check_distribution(dist, args, alpha)
def check_vonmises_pdf_periodic(k, l, s, x):
vm = stats.vonmises(k, loc=l, scale=s)
assert_almost_equal(vm.pdf(x), vm.pdf(x % (2*numpy.pi*s)))
def check_vonmises_cdf_periodic(k, l, s, x):
vm = stats.vonmises(k, loc=l, scale=s)
assert_almost_equal(vm.cdf(x) % 1, vm.cdf(x % (2*numpy.pi*s)) % 1)
def test_vonmises_pdf_periodic():
for k in [0.1, 1, 101]:
for x in [0, 1, numpy.pi, 10, 100]:
check_vonmises_pdf_periodic(k, 0, 1, x)
check_vonmises_pdf_periodic(k, 1, 1, x)
check_vonmises_pdf_periodic(k, 0, 10, x)
check_vonmises_cdf_periodic(k, 0, 1, x)
check_vonmises_cdf_periodic(k, 1, 1, x)
check_vonmises_cdf_periodic(k, 0, 10, x)
def test_vonmises_line_support():
assert_equal(stats.vonmises_line.a, -np.pi)
assert_equal(stats.vonmises_line.b, np.pi)
def test_vonmises_numerical():
vm = stats.vonmises(800)
assert_almost_equal(vm.cdf(0), 0.5)
@pytest.mark.parametrize('dist', ['alpha', 'betaprime', 'burr', 'burr12',
'fatiguelife', 'invgamma', 'invgauss', 'invweibull',
'johnsonsb', 'levy', 'levy_l', 'lognorm', 'gilbrat',
'powerlognorm', 'rayleigh', 'wald'])
def test_support(dist):
"""gh-6235"""
dct = dict(distcont)
args = dct[dist]
dist = getattr(stats, dist)
assert_almost_equal(dist.pdf(dist.a, *args), 0)
assert_equal(dist.logpdf(dist.a, *args), -np.inf)
assert_almost_equal(dist.pdf(dist.b, *args), 0)
assert_equal(dist.logpdf(dist.b, *args), -np.inf)
class TestRandInt(object):
def setup_method(self):
np.random.seed(1234)
def test_rvs(self):
vals = stats.randint.rvs(5, 30, size=100)
assert_(numpy.all(vals < 30) & numpy.all(vals >= 5))
assert_(len(vals) == 100)
vals = stats.randint.rvs(5, 30, size=(2, 50))
assert_(numpy.shape(vals) == (2, 50))
assert_(vals.dtype.char in typecodes['AllInteger'])
val = stats.randint.rvs(15, 46)
assert_((val >= 15) & (val < 46))
assert_(isinstance(val, numpy.ScalarType), msg=repr(type(val)))
val = stats.randint(15, 46).rvs(3)
assert_(val.dtype.char in typecodes['AllInteger'])
def test_pdf(self):
k = numpy.r_[0:36]
out = numpy.where((k >= 5) & (k < 30), 1.0/(30-5), 0)
vals = stats.randint.pmf(k, 5, 30)
assert_array_almost_equal(vals, out)
def test_cdf(self):
x = numpy.r_[0:36:100j]
k = numpy.floor(x)
out = numpy.select([k >= 30, k >= 5], [1.0, (k-5.0+1)/(30-5.0)], 0)
vals = stats.randint.cdf(x, 5, 30)
assert_array_almost_equal(vals, out, decimal=12)
class TestBinom(object):
def setup_method(self):
np.random.seed(1234)
def test_rvs(self):
vals = stats.binom.rvs(10, 0.75, size=(2, 50))
assert_(numpy.all(vals >= 0) & numpy.all(vals <= 10))
assert_(numpy.shape(vals) == (2, 50))
assert_(vals.dtype.char in typecodes['AllInteger'])
val = stats.binom.rvs(10, 0.75)
assert_(isinstance(val, int))
val = stats.binom(10, 0.75).rvs(3)
assert_(isinstance(val, numpy.ndarray))
assert_(val.dtype.char in typecodes['AllInteger'])
def test_pmf(self):
# regression test for Ticket #1842
vals1 = stats.binom.pmf(100, 100, 1)
vals2 = stats.binom.pmf(0, 100, 0)
assert_allclose(vals1, 1.0, rtol=1e-15, atol=0)
assert_allclose(vals2, 1.0, rtol=1e-15, atol=0)
def test_entropy(self):
# Basic entropy tests.
b = stats.binom(2, 0.5)
expected_p = np.array([0.25, 0.5, 0.25])
expected_h = -sum(xlogy(expected_p, expected_p))
h = b.entropy()
assert_allclose(h, expected_h)
b = stats.binom(2, 0.0)
h = b.entropy()
assert_equal(h, 0.0)
b = stats.binom(2, 1.0)
h = b.entropy()
assert_equal(h, 0.0)
def test_warns_p0(self):
# no spurious warnigns are generated for p=0; gh-3817
with warnings.catch_warnings():
warnings.simplefilter("error", RuntimeWarning)
assert_equal(stats.binom(n=2, p=0).mean(), 0)
assert_equal(stats.binom(n=2, p=0).std(), 0)
class TestBernoulli(object):
def setup_method(self):
np.random.seed(1234)
def test_rvs(self):
vals = stats.bernoulli.rvs(0.75, size=(2, 50))
assert_(numpy.all(vals >= 0) & numpy.all(vals <= 1))
assert_(numpy.shape(vals) == (2, 50))
assert_(vals.dtype.char in typecodes['AllInteger'])
val = stats.bernoulli.rvs(0.75)
assert_(isinstance(val, int))
val = stats.bernoulli(0.75).rvs(3)
assert_(isinstance(val, numpy.ndarray))
assert_(val.dtype.char in typecodes['AllInteger'])
def test_entropy(self):
# Simple tests of entropy.
b = stats.bernoulli(0.25)
expected_h = -0.25*np.log(0.25) - 0.75*np.log(0.75)
h = b.entropy()
assert_allclose(h, expected_h)
b = stats.bernoulli(0.0)
h = b.entropy()
assert_equal(h, 0.0)
b = stats.bernoulli(1.0)
h = b.entropy()
assert_equal(h, 0.0)
class TestBradford(object):
# gh-6216
def test_cdf_ppf(self):
c = 0.1
x = np.logspace(-20, -4)
q = stats.bradford.cdf(x, c)
xx = stats.bradford.ppf(q, c)
assert_allclose(x, xx)
class TestNBinom(object):
def setup_method(self):
np.random.seed(1234)
def test_rvs(self):
vals = stats.nbinom.rvs(10, 0.75, size=(2, 50))
assert_(numpy.all(vals >= 0))
assert_(numpy.shape(vals) == (2, 50))
assert_(vals.dtype.char in typecodes['AllInteger'])
val = stats.nbinom.rvs(10, 0.75)
assert_(isinstance(val, int))
val = stats.nbinom(10, 0.75).rvs(3)
assert_(isinstance(val, numpy.ndarray))
assert_(val.dtype.char in typecodes['AllInteger'])
def test_pmf(self):
# regression test for ticket 1779
assert_allclose(np.exp(stats.nbinom.logpmf(700, 721, 0.52)),
stats.nbinom.pmf(700, 721, 0.52))
# logpmf(0,1,1) shouldn't return nan (regression test for gh-4029)
val = scipy.stats.nbinom.logpmf(0, 1, 1)
assert_equal(val, 0)
class TestNormInvGauss(object):
def setup_method(self):
np.random.seed(1234)
def test_cdf_R(self):
# test pdf and cdf vals against R
# require("GeneralizedHyperbolic")
# x_test <- c(-7, -5, 0, 8, 15)
# r_cdf <- GeneralizedHyperbolic::pnig(x_test, mu = 0, a = 1, b = 0.5)
# r_pdf <- GeneralizedHyperbolic::dnig(x_test, mu = 0, a = 1, b = 0.5)
r_cdf = np.array([8.034920282e-07, 2.512671945e-05, 3.186661051e-01,
9.988650664e-01, 9.999848769e-01])
x_test = np.array([-7, -5, 0, 8, 15])
vals_cdf = stats.norminvgauss.cdf(x_test, a=1, b=0.5)
assert_allclose(vals_cdf, r_cdf, atol=1e-9)
def test_pdf_R(self):
# values from R as defined in test_cdf_R
r_pdf = np.array([1.359600783e-06, 4.413878805e-05, 4.555014266e-01,
7.450485342e-04, 8.917889931e-06])
x_test = np.array([-7, -5, 0, 8, 15])
vals_pdf = stats.norminvgauss.pdf(x_test, a=1, b=0.5)
assert_allclose(vals_pdf, r_pdf, atol=1e-9)
def test_stats(self):
a, b = 1, 0.5
gamma = np.sqrt(a**2 - b**2)
v_stats = (b / gamma, a**2 / gamma**3, 3.0 * b / (a * np.sqrt(gamma)),
3.0 * (1 + 4 * b**2 / a**2) / gamma)
assert_equal(v_stats, stats.norminvgauss.stats(a, b, moments='mvsk'))
def test_ppf(self):
a, b = 1, 0.5
x_test = np.array([0.001, 0.5, 0.999])
vals = stats.norminvgauss.ppf(x_test, a, b)
assert_allclose(x_test, stats.norminvgauss.cdf(vals, a, b))
class TestGeom(object):
def setup_method(self):
np.random.seed(1234)
def test_rvs(self):
vals = stats.geom.rvs(0.75, size=(2, 50))
assert_(numpy.all(vals >= 0))
assert_(numpy.shape(vals) == (2, 50))
assert_(vals.dtype.char in typecodes['AllInteger'])
val = stats.geom.rvs(0.75)
assert_(isinstance(val, int))
val = stats.geom(0.75).rvs(3)
assert_(isinstance(val, numpy.ndarray))
assert_(val.dtype.char in typecodes['AllInteger'])
def test_pmf(self):
vals = stats.geom.pmf([1, 2, 3], 0.5)
assert_array_almost_equal(vals, [0.5, 0.25, 0.125])
def test_logpmf(self):
# regression test for ticket 1793
vals1 = np.log(stats.geom.pmf([1, 2, 3], 0.5))
vals2 = stats.geom.logpmf([1, 2, 3], 0.5)
assert_allclose(vals1, vals2, rtol=1e-15, atol=0)
# regression test for gh-4028
val = stats.geom.logpmf(1, 1)
assert_equal(val, 0.0)
def test_cdf_sf(self):
vals = stats.geom.cdf([1, 2, 3], 0.5)
vals_sf = stats.geom.sf([1, 2, 3], 0.5)
expected = array([0.5, 0.75, 0.875])
assert_array_almost_equal(vals, expected)
assert_array_almost_equal(vals_sf, 1-expected)
def test_logcdf_logsf(self):
vals = stats.geom.logcdf([1, 2, 3], 0.5)
vals_sf = stats.geom.logsf([1, 2, 3], 0.5)
expected = array([0.5, 0.75, 0.875])
assert_array_almost_equal(vals, np.log(expected))
assert_array_almost_equal(vals_sf, np.log1p(-expected))
def test_ppf(self):
vals = stats.geom.ppf([0.5, 0.75, 0.875], 0.5)
expected = array([1.0, 2.0, 3.0])
assert_array_almost_equal(vals, expected)
class TestPlanck(object):
def setup_method(self):
np.random.seed(1234)
def test_sf(self):
vals = stats.planck.sf([1, 2, 3], 5.)
expected = array([4.5399929762484854e-05, 3.0590232050182579e-07, 2.0611536224385579e-09])
assert_array_almost_equal(vals, expected)
def test_logsf(self):
vals = stats.planck.logsf([1000., 2000., 3000.], 1000.)
expected = array([-1001000., -2001000., -3001000.])
assert_array_almost_equal(vals, expected)
class TestGennorm(object):
def test_laplace(self):
# test against Laplace (special case for beta=1)
points = [1, 2, 3]
pdf1 = stats.gennorm.pdf(points, 1)
pdf2 = stats.laplace.pdf(points)
assert_almost_equal(pdf1, pdf2)
def test_norm(self):
# test against normal (special case for beta=2)
points = [1, 2, 3]
pdf1 = stats.gennorm.pdf(points, 2)
pdf2 = stats.norm.pdf(points, scale=2**-.5)
assert_almost_equal(pdf1, pdf2)
class TestHalfgennorm(object):
def test_expon(self):
# test against exponential (special case for beta=1)
points = [1, 2, 3]
pdf1 = stats.halfgennorm.pdf(points, 1)
pdf2 = stats.expon.pdf(points)
assert_almost_equal(pdf1, pdf2)
def test_halfnorm(self):
# test against half normal (special case for beta=2)
points = [1, 2, 3]
pdf1 = stats.halfgennorm.pdf(points, 2)
pdf2 = stats.halfnorm.pdf(points, scale=2**-.5)
assert_almost_equal(pdf1, pdf2)
def test_gennorm(self):
# test against generalized normal
points = [1, 2, 3]
pdf1 = stats.halfgennorm.pdf(points, .497324)
pdf2 = stats.gennorm.pdf(points, .497324)
assert_almost_equal(pdf1, 2*pdf2)
class TestTruncnorm(object):
def setup_method(self):
np.random.seed(1234)
def test_ppf_ticket1131(self):
vals = stats.truncnorm.ppf([-0.5, 0, 1e-4, 0.5, 1-1e-4, 1, 2], -1., 1.,
loc=[3]*7, scale=2)
expected = np.array([np.nan, 1, 1.00056419, 3, 4.99943581, 5, np.nan])
assert_array_almost_equal(vals, expected)
def test_isf_ticket1131(self):
vals = stats.truncnorm.isf([-0.5, 0, 1e-4, 0.5, 1-1e-4, 1, 2], -1., 1.,
loc=[3]*7, scale=2)
expected = np.array([np.nan, 5, 4.99943581, 3, 1.00056419, 1, np.nan])
assert_array_almost_equal(vals, expected)
def test_gh_2477_small_values(self):
# Check a case that worked in the original issue.
low, high = -11, -10
x = stats.truncnorm.rvs(low, high, 0, 1, size=10)
assert_(low < x.min() < x.max() < high)
# Check a case that failed in the original issue.
low, high = 10, 11
x = stats.truncnorm.rvs(low, high, 0, 1, size=10)
assert_(low < x.min() < x.max() < high)
@pytest.mark.xfail(reason="truncnorm rvs is know to fail at extreme tails")
def test_gh_2477_large_values(self):
# Check a case that fails because of extreme tailness.
low, high = 100, 101
with np.errstate(divide='ignore'):
x = stats.truncnorm.rvs(low, high, 0, 1, size=10)
assert_(low < x.min() < x.max() < high)
def test_gh_1489_trac_962_rvs(self):
# Check the original example.
low, high = 10, 15
x = stats.truncnorm.rvs(low, high, 0, 1, size=10)
assert_(low < x.min() < x.max() < high)
class TestHypergeom(object):
def setup_method(self):
np.random.seed(1234)
def test_rvs(self):
vals = stats.hypergeom.rvs(20, 10, 3, size=(2, 50))
assert_(numpy.all(vals >= 0) &
numpy.all(vals <= 3))
assert_(numpy.shape(vals) == (2, 50))
assert_(vals.dtype.char in typecodes['AllInteger'])
val = stats.hypergeom.rvs(20, 3, 10)
assert_(isinstance(val, int))
val = stats.hypergeom(20, 3, 10).rvs(3)
assert_(isinstance(val, numpy.ndarray))
assert_(val.dtype.char in typecodes['AllInteger'])
def test_precision(self):
# comparison number from mpmath
M = 2500
n = 50
N = 500
tot = M
good = n
hgpmf = stats.hypergeom.pmf(2, tot, good, N)
assert_almost_equal(hgpmf, 0.0010114963068932233, 11)
def test_args(self):
# test correct output for corner cases of arguments
# see gh-2325
assert_almost_equal(stats.hypergeom.pmf(0, 2, 1, 0), 1.0, 11)
assert_almost_equal(stats.hypergeom.pmf(1, 2, 1, 0), 0.0, 11)
assert_almost_equal(stats.hypergeom.pmf(0, 2, 0, 2), 1.0, 11)
assert_almost_equal(stats.hypergeom.pmf(1, 2, 1, 0), 0.0, 11)
def test_cdf_above_one(self):
# for some values of parameters, hypergeom cdf was >1, see gh-2238
assert_(0 <= stats.hypergeom.cdf(30, 13397950, 4363, 12390) <= 1.0)
def test_precision2(self):
# Test hypergeom precision for large numbers. See #1218.
# Results compared with those from R.
oranges = 9.9e4
pears = 1.1e5
fruits_eaten = np.array([3, 3.8, 3.9, 4, 4.1, 4.2, 5]) * 1e4
quantile = 2e4
res = []
for eaten in fruits_eaten:
res.append(stats.hypergeom.sf(quantile, oranges + pears, oranges,
eaten))
expected = np.array([0, 1.904153e-114, 2.752693e-66, 4.931217e-32,
8.265601e-11, 0.1237904, 1])
assert_allclose(res, expected, atol=0, rtol=5e-7)
# Test with array_like first argument
quantiles = [1.9e4, 2e4, 2.1e4, 2.15e4]
res2 = stats.hypergeom.sf(quantiles, oranges + pears, oranges, 4.2e4)
expected2 = [1, 0.1237904, 6.511452e-34, 3.277667e-69]
assert_allclose(res2, expected2, atol=0, rtol=5e-7)
def test_entropy(self):
# Simple tests of entropy.
hg = stats.hypergeom(4, 1, 1)
h = hg.entropy()
expected_p = np.array([0.75, 0.25])
expected_h = -np.sum(xlogy(expected_p, expected_p))
assert_allclose(h, expected_h)
hg = stats.hypergeom(1, 1, 1)
h = hg.entropy()
assert_equal(h, 0.0)
def test_logsf(self):
# Test logsf for very large numbers. See issue #4982
# Results compare with those from R (v3.2.0):
# phyper(k, n, M-n, N, lower.tail=FALSE, log.p=TRUE)
# -2239.771
k = 1e4
M = 1e7
n = 1e6
N = 5e4
result = stats.hypergeom.logsf(k, M, n, N)
exspected = -2239.771 # From R
assert_almost_equal(result, exspected, decimal=3)
class TestLoggamma(object):
def test_stats(self):
# The following precomputed values are from the table in section 2.2
# of "A Statistical Study of Log-Gamma Distribution", by Ping Shing
# Chan (thesis, McMaster University, 1993).
table = np.array([
# c, mean, var, skew, exc. kurt.
0.5, -1.9635, 4.9348, -1.5351, 4.0000,
1.0, -0.5772, 1.6449, -1.1395, 2.4000,
12.0, 2.4427, 0.0869, -0.2946, 0.1735,
]).reshape(-1, 5)
for c, mean, var, skew, kurt in table:
computed = stats.loggamma.stats(c, moments='msvk')
assert_array_almost_equal(computed, [mean, var, skew, kurt],
decimal=4)
class TestLogistic(object):
# gh-6226
def test_cdf_ppf(self):
x = np.linspace(-20, 20)
y = stats.logistic.cdf(x)
xx = stats.logistic.ppf(y)
assert_allclose(x, xx)
def test_sf_isf(self):
x = np.linspace(-20, 20)
y = stats.logistic.sf(x)
xx = stats.logistic.isf(y)
assert_allclose(x, xx)
def test_extreme_values(self):
# p is chosen so that 1 - (1 - p) == p in double precision
p = 9.992007221626409e-16
desired = 34.53957599234088
assert_allclose(stats.logistic.ppf(1 - p), desired)
assert_allclose(stats.logistic.isf(p), desired)
class TestLogser(object):
def setup_method(self):
np.random.seed(1234)
def test_rvs(self):
vals = stats.logser.rvs(0.75, size=(2, 50))
assert_(numpy.all(vals >= 1))
assert_(numpy.shape(vals) == (2, 50))
assert_(vals.dtype.char in typecodes['AllInteger'])
val = stats.logser.rvs(0.75)
assert_(isinstance(val, int))
val = stats.logser(0.75).rvs(3)
assert_(isinstance(val, numpy.ndarray))
assert_(val.dtype.char in typecodes['AllInteger'])
def test_pmf_small_p(self):
m = stats.logser.pmf(4, 1e-20)
# The expected value was computed using mpmath:
# >>> import mpmath
# >>> mpmath.mp.dps = 64
# >>> k = 4
# >>> p = mpmath.mpf('1e-20')
# >>> float(-(p**k)/k/mpmath.log(1-p))
# 2.5e-61
# It is also clear from noticing that for very small p,
# log(1-p) is approximately -p, and the formula becomes
# p**(k-1) / k
assert_allclose(m, 2.5e-61)
def test_mean_small_p(self):
m = stats.logser.mean(1e-8)
# The expected mean was computed using mpmath:
# >>> import mpmath
# >>> mpmath.dps = 60
# >>> p = mpmath.mpf('1e-8')
# >>> float(-p / ((1 - p)*mpmath.log(1 - p)))
# 1.000000005
assert_allclose(m, 1.000000005)
class TestPareto(object):
def test_stats(self):
# Check the stats() method with some simple values. Also check
# that the calculations do not trigger RuntimeWarnings.
with warnings.catch_warnings():
warnings.simplefilter("error", RuntimeWarning)
m, v, s, k = stats.pareto.stats(0.5, moments='mvsk')
assert_equal(m, np.inf)
assert_equal(v, np.inf)
assert_equal(s, np.nan)
assert_equal(k, np.nan)
m, v, s, k = stats.pareto.stats(1.0, moments='mvsk')
assert_equal(m, np.inf)
assert_equal(v, np.inf)
assert_equal(s, np.nan)
assert_equal(k, np.nan)
m, v, s, k = stats.pareto.stats(1.5, moments='mvsk')
assert_equal(m, 3.0)
assert_equal(v, np.inf)
assert_equal(s, np.nan)
assert_equal(k, np.nan)
m, v, s, k = stats.pareto.stats(2.0, moments='mvsk')
assert_equal(m, 2.0)
assert_equal(v, np.inf)
assert_equal(s, np.nan)
assert_equal(k, np.nan)
m, v, s, k = stats.pareto.stats(2.5, moments='mvsk')
assert_allclose(m, 2.5 / 1.5)
assert_allclose(v, 2.5 / (1.5*1.5*0.5))
assert_equal(s, np.nan)
assert_equal(k, np.nan)
m, v, s, k = stats.pareto.stats(3.0, moments='mvsk')
assert_allclose(m, 1.5)
assert_allclose(v, 0.75)
assert_equal(s, np.nan)
assert_equal(k, np.nan)
m, v, s, k = stats.pareto.stats(3.5, moments='mvsk')
assert_allclose(m, 3.5 / 2.5)
assert_allclose(v, 3.5 / (2.5*2.5*1.5))
assert_allclose(s, (2*4.5/0.5)*np.sqrt(1.5/3.5))
assert_equal(k, np.nan)
m, v, s, k = stats.pareto.stats(4.0, moments='mvsk')
assert_allclose(m, 4.0 / 3.0)
assert_allclose(v, 4.0 / 18.0)
assert_allclose(s, 2*(1+4.0)/(4.0-3) * np.sqrt((4.0-2)/4.0))
assert_equal(k, np.nan)
m, v, s, k = stats.pareto.stats(4.5, moments='mvsk')
assert_allclose(m, 4.5 / 3.5)
assert_allclose(v, 4.5 / (3.5*3.5*2.5))
assert_allclose(s, (2*5.5/1.5) * np.sqrt(2.5/4.5))
assert_allclose(k, 6*(4.5**3 + 4.5**2 - 6*4.5 - 2)/(4.5*1.5*0.5))
def test_sf(self):
x = 1e9
b = 2
scale = 1.5
p = stats.pareto.sf(x, b, loc=0, scale=scale)
expected = (scale/x)**b # 2.25e-18
assert_allclose(p, expected)
class TestGenpareto(object):
def test_ab(self):
# c >= 0: a, b = [0, inf]
for c in [1., 0.]:
c = np.asarray(c)
stats.genpareto._argcheck(c) # ugh
assert_equal(stats.genpareto.a, 0.)
assert_(np.isposinf(stats.genpareto.b))
# c < 0: a=0, b=1/|c|
c = np.asarray(-2.)
stats.genpareto._argcheck(c)
assert_allclose([stats.genpareto.a, stats.genpareto.b], [0., 0.5])
def test_c0(self):
# with c=0, genpareto reduces to the exponential distribution
rv = stats.genpareto(c=0.)
x = np.linspace(0, 10., 30)
assert_allclose(rv.pdf(x), stats.expon.pdf(x))
assert_allclose(rv.cdf(x), stats.expon.cdf(x))
assert_allclose(rv.sf(x), stats.expon.sf(x))
q = np.linspace(0., 1., 10)
assert_allclose(rv.ppf(q), stats.expon.ppf(q))
def test_cm1(self):
# with c=-1, genpareto reduces to the uniform distr on [0, 1]
rv = stats.genpareto(c=-1.)
x = np.linspace(0, 10., 30)
assert_allclose(rv.pdf(x), stats.uniform.pdf(x))
assert_allclose(rv.cdf(x), stats.uniform.cdf(x))
assert_allclose(rv.sf(x), stats.uniform.sf(x))
q = np.linspace(0., 1., 10)
assert_allclose(rv.ppf(q), stats.uniform.ppf(q))
# logpdf(1., c=-1) should be zero
assert_allclose(rv.logpdf(1), 0)
def test_x_inf(self):
# make sure x=inf is handled gracefully
rv = stats.genpareto(c=0.1)
assert_allclose([rv.pdf(np.inf), rv.cdf(np.inf)], [0., 1.])
assert_(np.isneginf(rv.logpdf(np.inf)))
rv = stats.genpareto(c=0.)
assert_allclose([rv.pdf(np.inf), rv.cdf(np.inf)], [0., 1.])
assert_(np.isneginf(rv.logpdf(np.inf)))
rv = stats.genpareto(c=-1.)
assert_allclose([rv.pdf(np.inf), rv.cdf(np.inf)], [0., 1.])
assert_(np.isneginf(rv.logpdf(np.inf)))
def test_c_continuity(self):
# pdf is continuous at c=0, -1
x = np.linspace(0, 10, 30)
for c in [0, -1]:
pdf0 = stats.genpareto.pdf(x, c)
for dc in [1e-14, -1e-14]:
pdfc = stats.genpareto.pdf(x, c + dc)
assert_allclose(pdf0, pdfc, atol=1e-12)
cdf0 = stats.genpareto.cdf(x, c)
for dc in [1e-14, 1e-14]:
cdfc = stats.genpareto.cdf(x, c + dc)
assert_allclose(cdf0, cdfc, atol=1e-12)
def test_c_continuity_ppf(self):
q = np.r_[np.logspace(1e-12, 0.01, base=0.1),
np.linspace(0.01, 1, 30, endpoint=False),
1. - np.logspace(1e-12, 0.01, base=0.1)]
for c in [0., -1.]:
ppf0 = stats.genpareto.ppf(q, c)
for dc in [1e-14, -1e-14]:
ppfc = stats.genpareto.ppf(q, c + dc)
assert_allclose(ppf0, ppfc, atol=1e-12)
def test_c_continuity_isf(self):
q = np.r_[np.logspace(1e-12, 0.01, base=0.1),
np.linspace(0.01, 1, 30, endpoint=False),
1. - np.logspace(1e-12, 0.01, base=0.1)]
for c in [0., -1.]:
isf0 = stats.genpareto.isf(q, c)
for dc in [1e-14, -1e-14]:
isfc = stats.genpareto.isf(q, c + dc)
assert_allclose(isf0, isfc, atol=1e-12)
def test_cdf_ppf_roundtrip(self):
# this should pass with machine precision. hat tip @pbrod
q = np.r_[np.logspace(1e-12, 0.01, base=0.1),
np.linspace(0.01, 1, 30, endpoint=False),
1. - np.logspace(1e-12, 0.01, base=0.1)]
for c in [1e-8, -1e-18, 1e-15, -1e-15]:
assert_allclose(stats.genpareto.cdf(stats.genpareto.ppf(q, c), c),
q, atol=1e-15)
def test_logsf(self):
logp = stats.genpareto.logsf(1e10, .01, 0, 1)
assert_allclose(logp, -1842.0680753952365)
class TestPearson3(object):
def setup_method(self):
np.random.seed(1234)
def test_rvs(self):
vals = stats.pearson3.rvs(0.1, size=(2, 50))
assert_(numpy.shape(vals) == (2, 50))
assert_(vals.dtype.char in typecodes['AllFloat'])
val = stats.pearson3.rvs(0.5)
assert_(isinstance(val, float))
val = stats.pearson3(0.5).rvs(3)
assert_(isinstance(val, numpy.ndarray))
assert_(val.dtype.char in typecodes['AllFloat'])
assert_(len(val) == 3)
def test_pdf(self):
vals = stats.pearson3.pdf(2, [0.0, 0.1, 0.2])
assert_allclose(vals, np.array([0.05399097, 0.05555481, 0.05670246]),
atol=1e-6)
vals = stats.pearson3.pdf(-3, 0.1)
assert_allclose(vals, np.array([0.00313791]), atol=1e-6)
vals = stats.pearson3.pdf([-3, -2, -1, 0, 1], 0.1)
assert_allclose(vals, np.array([0.00313791, 0.05192304, 0.25028092,
0.39885918, 0.23413173]), atol=1e-6)
def test_cdf(self):
vals = stats.pearson3.cdf(2, [0.0, 0.1, 0.2])
assert_allclose(vals, np.array([0.97724987, 0.97462004, 0.97213626]),
atol=1e-6)
vals = stats.pearson3.cdf(-3, 0.1)
assert_allclose(vals, [0.00082256], atol=1e-6)
vals = stats.pearson3.cdf([-3, -2, -1, 0, 1], 0.1)
assert_allclose(vals, [8.22563821e-04, 1.99860448e-02, 1.58550710e-01,
5.06649130e-01, 8.41442111e-01], atol=1e-6)
class TestKappa4(object):
def test_cdf_genpareto(self):
# h = 1 and k != 0 is generalized Pareto
x = [0.0, 0.1, 0.2, 0.5]
h = 1.0
for k in [-1.9, -1.0, -0.5, -0.2, -0.1, 0.1, 0.2, 0.5, 1.0,
1.9]:
vals = stats.kappa4.cdf(x, h, k)
# shape parameter is opposite what is expected
vals_comp = stats.genpareto.cdf(x, -k)
assert_allclose(vals, vals_comp)
def test_cdf_genextreme(self):
# h = 0 and k != 0 is generalized extreme value
x = np.linspace(-5, 5, 10)
h = 0.0
k = np.linspace(-3, 3, 10)
vals = stats.kappa4.cdf(x, h, k)
vals_comp = stats.genextreme.cdf(x, k)
assert_allclose(vals, vals_comp)
def test_cdf_expon(self):
# h = 1 and k = 0 is exponential
x = np.linspace(0, 10, 10)
h = 1.0
k = 0.0
vals = stats.kappa4.cdf(x, h, k)
vals_comp = stats.expon.cdf(x)
assert_allclose(vals, vals_comp)
def test_cdf_gumbel_r(self):
# h = 0 and k = 0 is gumbel_r
x = np.linspace(-5, 5, 10)
h = 0.0
k = 0.0
vals = stats.kappa4.cdf(x, h, k)
vals_comp = stats.gumbel_r.cdf(x)
assert_allclose(vals, vals_comp)
def test_cdf_logistic(self):
# h = -1 and k = 0 is logistic
x = np.linspace(-5, 5, 10)
h = -1.0
k = 0.0
vals = stats.kappa4.cdf(x, h, k)
vals_comp = stats.logistic.cdf(x)
assert_allclose(vals, vals_comp)
def test_cdf_uniform(self):
# h = 1 and k = 1 is uniform
x = np.linspace(-5, 5, 10)
h = 1.0
k = 1.0
vals = stats.kappa4.cdf(x, h, k)
vals_comp = stats.uniform.cdf(x)
assert_allclose(vals, vals_comp)
def test_integers_ctor(self):
# regression test for gh-7416: _argcheck fails for integer h and k
# in numpy 1.12
stats.kappa4(1, 2)
class TestPoisson(object):
def setup_method(self):
np.random.seed(1234)
def test_pmf_basic(self):
# Basic case
ln2 = np.log(2)
vals = stats.poisson.pmf([0, 1, 2], ln2)
expected = [0.5, ln2/2, ln2**2/4]
assert_allclose(vals, expected)
def test_mu0(self):
# Edge case: mu=0
vals = stats.poisson.pmf([0, 1, 2], 0)
expected = [1, 0, 0]
assert_array_equal(vals, expected)
interval = stats.poisson.interval(0.95, 0)
assert_equal(interval, (0, 0))
def test_rvs(self):
vals = stats.poisson.rvs(0.5, size=(2, 50))
assert_(numpy.all(vals >= 0))
assert_(numpy.shape(vals) == (2, 50))
assert_(vals.dtype.char in typecodes['AllInteger'])
val = stats.poisson.rvs(0.5)
assert_(isinstance(val, int))
val = stats.poisson(0.5).rvs(3)
assert_(isinstance(val, numpy.ndarray))
assert_(val.dtype.char in typecodes['AllInteger'])
def test_stats(self):
mu = 16.0
result = stats.poisson.stats(mu, moments='mvsk')
assert_allclose(result, [mu, mu, np.sqrt(1.0/mu), 1.0/mu])
mu = np.array([0.0, 1.0, 2.0])
result = stats.poisson.stats(mu, moments='mvsk')
expected = (mu, mu, [np.inf, 1, 1/np.sqrt(2)], [np.inf, 1, 0.5])
assert_allclose(result, expected)
class TestZipf(object):
def setup_method(self):
np.random.seed(1234)
def test_rvs(self):
vals = stats.zipf.rvs(1.5, size=(2, 50))
assert_(numpy.all(vals >= 1))
assert_(numpy.shape(vals) == (2, 50))
assert_(vals.dtype.char in typecodes['AllInteger'])
val = stats.zipf.rvs(1.5)
assert_(isinstance(val, int))
val = stats.zipf(1.5).rvs(3)
assert_(isinstance(val, numpy.ndarray))
assert_(val.dtype.char in typecodes['AllInteger'])
def test_moments(self):
# n-th moment is finite iff a > n + 1
m, v = stats.zipf.stats(a=2.8)
assert_(np.isfinite(m))
assert_equal(v, np.inf)
s, k = stats.zipf.stats(a=4.8, moments='sk')
assert_(not np.isfinite([s, k]).all())
class TestDLaplace(object):
def setup_method(self):
np.random.seed(1234)
def test_rvs(self):
vals = stats.dlaplace.rvs(1.5, size=(2, 50))
assert_(numpy.shape(vals) == (2, 50))
assert_(vals.dtype.char in typecodes['AllInteger'])
val = stats.dlaplace.rvs(1.5)
assert_(isinstance(val, int))
val = stats.dlaplace(1.5).rvs(3)
assert_(isinstance(val, numpy.ndarray))
assert_(val.dtype.char in typecodes['AllInteger'])
assert_(stats.dlaplace.rvs(0.8) is not None)
def test_stats(self):
# compare the explicit formulas w/ direct summation using pmf
a = 1.
dl = stats.dlaplace(a)
m, v, s, k = dl.stats('mvsk')
N = 37
xx = np.arange(-N, N+1)
pp = dl.pmf(xx)
m2, m4 = np.sum(pp*xx**2), np.sum(pp*xx**4)
assert_equal((m, s), (0, 0))
assert_allclose((v, k), (m2, m4/m2**2 - 3.), atol=1e-14, rtol=1e-8)
def test_stats2(self):
a = np.log(2.)
dl = stats.dlaplace(a)
m, v, s, k = dl.stats('mvsk')
assert_equal((m, s), (0., 0.))
assert_allclose((v, k), (4., 3.25))
class TestInvGamma(object):
def test_invgamma_inf_gh_1866(self):
# invgamma's moments are only finite for a>n
# specific numbers checked w/ boost 1.54
with warnings.catch_warnings():
warnings.simplefilter('error', RuntimeWarning)
mvsk = stats.invgamma.stats(a=19.31, moments='mvsk')
expected = [0.05461496450, 0.0001723162534, 1.020362676,
2.055616582]
assert_allclose(mvsk, expected)
a = [1.1, 3.1, 5.6]
mvsk = stats.invgamma.stats(a=a, moments='mvsk')
expected = ([10., 0.476190476, 0.2173913043], # mmm
[np.inf, 0.2061430632, 0.01312749422], # vvv
[np.nan, 41.95235392, 2.919025532], # sss
[np.nan, np.nan, 24.51923076]) # kkk
for x, y in zip(mvsk, expected):
assert_almost_equal(x, y)
def test_cdf_ppf(self):
# gh-6245
x = np.logspace(-2.6, 0)
y = stats.invgamma.cdf(x, 1)
xx = stats.invgamma.ppf(y, 1)
assert_allclose(x, xx)
def test_sf_isf(self):
# gh-6245
if sys.maxsize > 2**32:
x = np.logspace(2, 100)
else:
# Invgamme roundtrip on 32-bit systems has relative accuracy
# ~1e-15 until x=1e+15, and becomes inf above x=1e+18
x = np.logspace(2, 18)
y = stats.invgamma.sf(x, 1)
xx = stats.invgamma.isf(y, 1)
assert_allclose(x, xx, rtol=1.0)
class TestF(object):
def test_f_moments(self):
# n-th moment of F distributions is only finite for n < dfd / 2
m, v, s, k = stats.f.stats(11, 6.5, moments='mvsk')
assert_(np.isfinite(m))
assert_(np.isfinite(v))
assert_(np.isfinite(s))
assert_(not np.isfinite(k))
def test_moments_warnings(self):
# no warnings should be generated for dfd = 2, 4, 6, 8 (div by zero)
with warnings.catch_warnings():
warnings.simplefilter('error', RuntimeWarning)
stats.f.stats(dfn=[11]*4, dfd=[2, 4, 6, 8], moments='mvsk')
@pytest.mark.xfail(reason='f stats does not properly broadcast')
def test_stats_broadcast(self):
# stats do not fully broadcast just yet
mv = stats.f.stats(dfn=11, dfd=[11, 12])
def test_rvgeneric_std():
# Regression test for #1191
assert_array_almost_equal(stats.t.std([5, 6]), [1.29099445, 1.22474487])
class TestRvDiscrete(object):
def setup_method(self):
np.random.seed(1234)
def test_rvs(self):
states = [-1, 0, 1, 2, 3, 4]
probability = [0.0, 0.3, 0.4, 0.0, 0.3, 0.0]
samples = 1000
r = stats.rv_discrete(name='sample', values=(states, probability))
x = r.rvs(size=samples)
assert_(isinstance(x, numpy.ndarray))
for s, p in zip(states, probability):
assert_(abs(sum(x == s)/float(samples) - p) < 0.05)
x = r.rvs()
assert_(isinstance(x, int))
def test_entropy(self):
# Basic tests of entropy.
pvals = np.array([0.25, 0.45, 0.3])
p = stats.rv_discrete(values=([0, 1, 2], pvals))
expected_h = -sum(xlogy(pvals, pvals))
h = p.entropy()
assert_allclose(h, expected_h)
p = stats.rv_discrete(values=([0, 1, 2], [1.0, 0, 0]))
h = p.entropy()
assert_equal(h, 0.0)
def test_pmf(self):
xk = [1, 2, 4]
pk = [0.5, 0.3, 0.2]
rv = stats.rv_discrete(values=(xk, pk))
x = [[1., 4.],
[3., 2]]
assert_allclose(rv.pmf(x),
[[0.5, 0.2],
[0., 0.3]], atol=1e-14)
def test_cdf(self):
xk = [1, 2, 4]
pk = [0.5, 0.3, 0.2]
rv = stats.rv_discrete(values=(xk, pk))
x_values = [-2, 1., 1.1, 1.5, 2.0, 3.0, 4, 5]
expected = [0, 0.5, 0.5, 0.5, 0.8, 0.8, 1, 1]
assert_allclose(rv.cdf(x_values), expected, atol=1e-14)
# also check scalar arguments
assert_allclose([rv.cdf(xx) for xx in x_values],
expected, atol=1e-14)
def test_ppf(self):
xk = [1, 2, 4]
pk = [0.5, 0.3, 0.2]
rv = stats.rv_discrete(values=(xk, pk))
q_values = [0.1, 0.5, 0.6, 0.8, 0.9, 1.]
expected = [1, 1, 2, 2, 4, 4]
assert_allclose(rv.ppf(q_values), expected, atol=1e-14)
# also check scalar arguments
assert_allclose([rv.ppf(q) for q in q_values],
expected, atol=1e-14)
def test_cdf_ppf_next(self):
# copied and special cased from test_discrete_basic
vals = ([1, 2, 4, 7, 8], [0.1, 0.2, 0.3, 0.3, 0.1])
rv = stats.rv_discrete(values=vals)
assert_array_equal(rv.ppf(rv.cdf(rv.xk[:-1]) + 1e-8),
rv.xk[1:])
def test_expect(self):
xk = [1, 2, 4, 6, 7, 11]
pk = [0.1, 0.2, 0.2, 0.2, 0.2, 0.1]
rv = stats.rv_discrete(values=(xk, pk))
assert_allclose(rv.expect(), np.sum(rv.xk * rv.pk), atol=1e-14)
def test_bad_input(self):
xk = [1, 2, 3]
pk = [0.5, 0.5]
assert_raises(ValueError, stats.rv_discrete, **dict(values=(xk, pk)))
pk = [1, 2, 3]
assert_raises(ValueError, stats.rv_discrete, **dict(values=(xk, pk)))
class TestSkewNorm(object):
def setup_method(self):
np.random.seed(1234)
def test_normal(self):
# When the skewness is 0 the distribution is normal
x = np.linspace(-5, 5, 100)
assert_array_almost_equal(stats.skewnorm.pdf(x, a=0),
stats.norm.pdf(x))
def test_rvs(self):
shape = (3, 4, 5)
x = stats.skewnorm.rvs(a=0.75, size=shape)
assert_equal(shape, x.shape)
x = stats.skewnorm.rvs(a=-3, size=shape)
assert_equal(shape, x.shape)
def test_moments(self):
X = stats.skewnorm.rvs(a=4, size=int(1e6), loc=5, scale=2)
expected = [np.mean(X), np.var(X), stats.skew(X), stats.kurtosis(X)]
computed = stats.skewnorm.stats(a=4, loc=5, scale=2, moments='mvsk')
assert_array_almost_equal(computed, expected, decimal=2)
X = stats.skewnorm.rvs(a=-4, size=int(1e6), loc=5, scale=2)
expected = [np.mean(X), np.var(X), stats.skew(X), stats.kurtosis(X)]
computed = stats.skewnorm.stats(a=-4, loc=5, scale=2, moments='mvsk')
assert_array_almost_equal(computed, expected, decimal=2)
def test_cdf_large_x(self):
# Regression test for gh-7746.
# The x values are large enough that the closest 64 bit floating
# point representation of the exact CDF is 1.0.
p = stats.skewnorm.cdf([10, 20, 30], -1)
assert_allclose(p, np.ones(3), rtol=1e-14)
p = stats.skewnorm.cdf(25, 2.5)
assert_allclose(p, 1.0, rtol=1e-14)
def test_cdf_sf_small_values(self):
# Triples are [x, a, cdf(x, a)]. These values were computed
# using CDF[SkewNormDistribution[0, 1, a], x] in Wolfram Alpha.
cdfvals = [
[-8, 1, 3.870035046664392611e-31],
[-4, 2, 8.1298399188811398e-21],
[-2, 5, 1.55326826787106273e-26],
[-9, -1, 2.257176811907681295e-19],
[-10, -4, 1.523970604832105213e-23],
]
for x, a, cdfval in cdfvals:
p = stats.skewnorm.cdf(x, a)
assert_allclose(p, cdfval, rtol=1e-8)
# For the skew normal distribution, sf(-x, -a) = cdf(x, a).
p = stats.skewnorm.sf(-x, -a)
assert_allclose(p, cdfval, rtol=1e-8)
class TestExpon(object):
def test_zero(self):
assert_equal(stats.expon.pdf(0), 1)
def test_tail(self): # Regression test for ticket 807
assert_equal(stats.expon.cdf(1e-18), 1e-18)
assert_equal(stats.expon.isf(stats.expon.sf(40)), 40)
class TestExponNorm(object):
def test_moments(self):
# Some moment test cases based on non-loc/scaled formula
def get_moms(lam, sig, mu):
# See wikipedia for these formulae
# where it is listed as an exponentially modified gaussian
opK2 = 1.0 + 1 / (lam*sig)**2
exp_skew = 2 / (lam * sig)**3 * opK2**(-1.5)
exp_kurt = 6.0 * (1 + (lam * sig)**2)**(-2)
return [mu + 1/lam, sig*sig + 1.0/(lam*lam), exp_skew, exp_kurt]
mu, sig, lam = 0, 1, 1
K = 1.0 / (lam * sig)
sts = stats.exponnorm.stats(K, loc=mu, scale=sig, moments='mvsk')
assert_almost_equal(sts, get_moms(lam, sig, mu))
mu, sig, lam = -3, 2, 0.1
K = 1.0 / (lam * sig)
sts = stats.exponnorm.stats(K, loc=mu, scale=sig, moments='mvsk')
assert_almost_equal(sts, get_moms(lam, sig, mu))
mu, sig, lam = 0, 3, 1
K = 1.0 / (lam * sig)
sts = stats.exponnorm.stats(K, loc=mu, scale=sig, moments='mvsk')
assert_almost_equal(sts, get_moms(lam, sig, mu))
mu, sig, lam = -5, 11, 3.5
K = 1.0 / (lam * sig)
sts = stats.exponnorm.stats(K, loc=mu, scale=sig, moments='mvsk')
assert_almost_equal(sts, get_moms(lam, sig, mu))
def test_extremes_x(self):
# Test for extreme values against overflows
assert_almost_equal(stats.exponnorm.pdf(-900, 1), 0.0)
assert_almost_equal(stats.exponnorm.pdf(+900, 1), 0.0)
class TestGenExpon(object):
def test_pdf_unity_area(self):
from scipy.integrate import simps
# PDF should integrate to one
p = stats.genexpon.pdf(numpy.arange(0, 10, 0.01), 0.5, 0.5, 2.0)
assert_almost_equal(simps(p, dx=0.01), 1, 1)
def test_cdf_bounds(self):
# CDF should always be positive
cdf = stats.genexpon.cdf(numpy.arange(0, 10, 0.01), 0.5, 0.5, 2.0)
assert_(numpy.all((0 <= cdf) & (cdf <= 1)))
class TestExponpow(object):
def test_tail(self):
assert_almost_equal(stats.exponpow.cdf(1e-10, 2.), 1e-20)
assert_almost_equal(stats.exponpow.isf(stats.exponpow.sf(5, .8), .8),
5)
class TestSkellam(object):
def test_pmf(self):
# comparison to R
k = numpy.arange(-10, 15)
mu1, mu2 = 10, 5
skpmfR = numpy.array(
[4.2254582961926893e-005, 1.1404838449648488e-004,
2.8979625801752660e-004, 6.9177078182101231e-004,
1.5480716105844708e-003, 3.2412274963433889e-003,
6.3373707175123292e-003, 1.1552351566696643e-002,
1.9606152375042644e-002, 3.0947164083410337e-002,
4.5401737566767360e-002, 6.1894328166820688e-002,
7.8424609500170578e-002, 9.2418812533573133e-002,
1.0139793148019728e-001, 1.0371927988298846e-001,
9.9076583077406091e-002, 8.8546660073089561e-002,
7.4187842052486810e-002, 5.8392772862200251e-002,
4.3268692953013159e-002, 3.0248159818374226e-002,
1.9991434305603021e-002, 1.2516877303301180e-002,
7.4389876226229707e-003])
assert_almost_equal(stats.skellam.pmf(k, mu1, mu2), skpmfR, decimal=15)
def test_cdf(self):
# comparison to R, only 5 decimals
k = numpy.arange(-10, 15)
mu1, mu2 = 10, 5
skcdfR = numpy.array(
[6.4061475386192104e-005, 1.7810985988267694e-004,
4.6790611790020336e-004, 1.1596768997212152e-003,
2.7077485103056847e-003, 5.9489760066490718e-003,
1.2286346724161398e-002, 2.3838698290858034e-002,
4.3444850665900668e-002, 7.4392014749310995e-002,
1.1979375231607835e-001, 1.8168808048289900e-001,
2.6011268998306952e-001, 3.5253150251664261e-001,
4.5392943399683988e-001, 5.5764871387982828e-001,
6.5672529695723436e-001, 7.4527195703032389e-001,
8.1945979908281064e-001, 8.7785257194501087e-001,
9.2112126489802404e-001, 9.5136942471639818e-001,
9.7136085902200120e-001, 9.8387773632530240e-001,
9.9131672394792536e-001])
assert_almost_equal(stats.skellam.cdf(k, mu1, mu2), skcdfR, decimal=5)
class TestLognorm(object):
def test_pdf(self):
# Regression test for Ticket #1471: avoid nan with 0/0 situation
# Also make sure there are no warnings at x=0, cf gh-5202
with warnings.catch_warnings():
warnings.simplefilter('error', RuntimeWarning)
pdf = stats.lognorm.pdf([0, 0.5, 1], 1)
assert_array_almost_equal(pdf, [0.0, 0.62749608, 0.39894228])
def test_logcdf(self):
# Regression test for gh-5940: sf et al would underflow too early
x2, mu, sigma = 201.68, 195, 0.149
assert_allclose(stats.lognorm.sf(x2-mu, s=sigma),
stats.norm.sf(np.log(x2-mu)/sigma))
assert_allclose(stats.lognorm.logsf(x2-mu, s=sigma),
stats.norm.logsf(np.log(x2-mu)/sigma))
class TestBeta(object):
def test_logpdf(self):
# Regression test for Ticket #1326: avoid nan with 0*log(0) situation
logpdf = stats.beta.logpdf(0, 1, 0.5)
assert_almost_equal(logpdf, -0.69314718056)
logpdf = stats.beta.logpdf(0, 0.5, 1)
assert_almost_equal(logpdf, np.inf)
def test_logpdf_ticket_1866(self):
alpha, beta = 267, 1472
x = np.array([0.2, 0.5, 0.6])
b = stats.beta(alpha, beta)
assert_allclose(b.logpdf(x).sum(), -1201.699061824062)
assert_allclose(b.pdf(x), np.exp(b.logpdf(x)))
class TestBetaPrime(object):
def test_logpdf(self):
alpha, beta = 267, 1472
x = np.array([0.2, 0.5, 0.6])
b = stats.betaprime(alpha, beta)
assert_(np.isfinite(b.logpdf(x)).all())
assert_allclose(b.pdf(x), np.exp(b.logpdf(x)))
def test_cdf(self):
# regression test for gh-4030: Implementation of
# scipy.stats.betaprime.cdf()
x = stats.betaprime.cdf(0, 0.2, 0.3)
assert_equal(x, 0.0)
alpha, beta = 267, 1472
x = np.array([0.2, 0.5, 0.6])
cdfs = stats.betaprime.cdf(x, alpha, beta)
assert_(np.isfinite(cdfs).all())
# check the new cdf implementation vs generic one:
gen_cdf = stats.rv_continuous._cdf_single
cdfs_g = [gen_cdf(stats.betaprime, val, alpha, beta) for val in x]
assert_allclose(cdfs, cdfs_g, atol=0, rtol=2e-12)
class TestGamma(object):
def test_pdf(self):
# a few test cases to compare with R
pdf = stats.gamma.pdf(90, 394, scale=1./5)
assert_almost_equal(pdf, 0.002312341)
pdf = stats.gamma.pdf(3, 10, scale=1./5)
assert_almost_equal(pdf, 0.1620358)
def test_logpdf(self):
# Regression test for Ticket #1326: cornercase avoid nan with 0*log(0)
# situation
logpdf = stats.gamma.logpdf(0, 1)
assert_almost_equal(logpdf, 0)
class TestChi2(object):
# regression tests after precision improvements, ticket:1041, not verified
def test_precision(self):
assert_almost_equal(stats.chi2.pdf(1000, 1000), 8.919133934753128e-003,
decimal=14)
assert_almost_equal(stats.chi2.pdf(100, 100), 0.028162503162596778,
decimal=14)
class TestGumbelL(object):
# gh-6228
def test_cdf_ppf(self):
x = np.linspace(-100, -4)
y = stats.gumbel_l.cdf(x)
xx = stats.gumbel_l.ppf(y)
assert_allclose(x, xx)
def test_logcdf_logsf(self):
x = np.linspace(-100, -4)
y = stats.gumbel_l.logcdf(x)
z = stats.gumbel_l.logsf(x)
u = np.exp(y)
v = -special.expm1(z)
assert_allclose(u, v)
def test_sf_isf(self):
x = np.linspace(-20, 5)
y = stats.gumbel_l.sf(x)
xx = stats.gumbel_l.isf(y)
assert_allclose(x, xx)
class TestArrayArgument(object): # test for ticket:992
def setup_method(self):
np.random.seed(1234)
def test_noexception(self):
rvs = stats.norm.rvs(loc=(np.arange(5)), scale=np.ones(5),
size=(10, 5))
assert_equal(rvs.shape, (10, 5))
class TestDocstring(object):
def test_docstrings(self):
# See ticket #761
if stats.rayleigh.__doc__ is not None:
assert_("rayleigh" in stats.rayleigh.__doc__.lower())
if stats.bernoulli.__doc__ is not None:
assert_("bernoulli" in stats.bernoulli.__doc__.lower())
def test_no_name_arg(self):
# If name is not given, construction shouldn't fail. See #1508.
stats.rv_continuous()
stats.rv_discrete()
class TestEntropy(object):
def test_entropy_positive(self):
# See ticket #497
pk = [0.5, 0.2, 0.3]
qk = [0.1, 0.25, 0.65]
eself = stats.entropy(pk, pk)
edouble = stats.entropy(pk, qk)
assert_(0.0 == eself)
assert_(edouble >= 0.0)
def test_entropy_base(self):
pk = np.ones(16, float)
S = stats.entropy(pk, base=2.)
assert_(abs(S - 4.) < 1.e-5)
qk = np.ones(16, float)
qk[:8] = 2.
S = stats.entropy(pk, qk)
S2 = stats.entropy(pk, qk, base=2.)
assert_(abs(S/S2 - np.log(2.)) < 1.e-5)
def test_entropy_zero(self):
# Test for PR-479
assert_almost_equal(stats.entropy([0, 1, 2]), 0.63651416829481278,
decimal=12)
def test_entropy_2d(self):
pk = [[0.1, 0.2], [0.6, 0.3], [0.3, 0.5]]
qk = [[0.2, 0.1], [0.3, 0.6], [0.5, 0.3]]
assert_array_almost_equal(stats.entropy(pk, qk),
[0.1933259, 0.18609809])
def test_entropy_2d_zero(self):
pk = [[0.1, 0.2], [0.6, 0.3], [0.3, 0.5]]
qk = [[0.0, 0.1], [0.3, 0.6], [0.5, 0.3]]
assert_array_almost_equal(stats.entropy(pk, qk),
[np.inf, 0.18609809])
pk[0][0] = 0.0
assert_array_almost_equal(stats.entropy(pk, qk),
[0.17403988, 0.18609809])
def TestArgsreduce():
a = array([1, 3, 2, 1, 2, 3, 3])
b, c = argsreduce(a > 1, a, 2)
assert_array_equal(b, [3, 2, 2, 3, 3])
assert_array_equal(c, [2, 2, 2, 2, 2])
b, c = argsreduce(2 > 1, a, 2)
assert_array_equal(b, a[0])
assert_array_equal(c, [2])
b, c = argsreduce(a > 0, a, 2)
assert_array_equal(b, a)
assert_array_equal(c, [2] * numpy.size(a))
class TestFitMethod(object):
skip = ['ncf']
def setup_method(self):
np.random.seed(1234)
@pytest.mark.slow
@pytest.mark.parametrize('dist,args,alpha', cases_test_all_distributions())
def test_fit(self, dist, args, alpha):
if dist in self.skip:
pytest.skip("%s fit known to fail" % dist)
distfunc = getattr(stats, dist)
with np.errstate(all='ignore'), suppress_warnings() as sup:
sup.filter(category=DeprecationWarning, message=".*frechet_")
res = distfunc.rvs(*args, **{'size': 200})
vals = distfunc.fit(res)
vals2 = distfunc.fit(res, optimizer='powell')
# Only check the length of the return
# FIXME: should check the actual results to see if we are 'close'
# to what was created --- but what is 'close' enough
assert_(len(vals) == 2+len(args))
assert_(len(vals2) == 2+len(args))
@pytest.mark.slow
@pytest.mark.parametrize('dist,args,alpha', cases_test_all_distributions())
def test_fix_fit(self, dist, args, alpha):
# Not sure why 'ncf', and 'beta' are failing
# frechet has different len(args) than distfunc.numargs
if dist in self.skip + ['frechet']:
pytest.skip("%s fit known to fail" % dist)
distfunc = getattr(stats, dist)
with np.errstate(all='ignore'), suppress_warnings() as sup:
sup.filter(category=DeprecationWarning, message=".*frechet_")
res = distfunc.rvs(*args, **{'size': 200})
vals = distfunc.fit(res, floc=0)
vals2 = distfunc.fit(res, fscale=1)
assert_(len(vals) == 2+len(args))
assert_(vals[-2] == 0)
assert_(vals2[-1] == 1)
assert_(len(vals2) == 2+len(args))
if len(args) > 0:
vals3 = distfunc.fit(res, f0=args[0])
assert_(len(vals3) == 2+len(args))
assert_(vals3[0] == args[0])
if len(args) > 1:
vals4 = distfunc.fit(res, f1=args[1])
assert_(len(vals4) == 2+len(args))
assert_(vals4[1] == args[1])
if len(args) > 2:
vals5 = distfunc.fit(res, f2=args[2])
assert_(len(vals5) == 2+len(args))
assert_(vals5[2] == args[2])
def test_fix_fit_2args_lognorm(self):
# Regression test for #1551.
np.random.seed(12345)
with np.errstate(all='ignore'):
x = stats.lognorm.rvs(0.25, 0., 20.0, size=20)
expected_shape = np.sqrt(((np.log(x) - np.log(20))**2).mean())
assert_allclose(np.array(stats.lognorm.fit(x, floc=0, fscale=20)),
[expected_shape, 0, 20], atol=1e-8)
def test_fix_fit_norm(self):
x = np.arange(1, 6)
loc, scale = stats.norm.fit(x)
assert_almost_equal(loc, 3)
assert_almost_equal(scale, np.sqrt(2))
loc, scale = stats.norm.fit(x, floc=2)
assert_equal(loc, 2)
assert_equal(scale, np.sqrt(3))
loc, scale = stats.norm.fit(x, fscale=2)
assert_almost_equal(loc, 3)
assert_equal(scale, 2)
def test_fix_fit_gamma(self):
x = np.arange(1, 6)
meanlog = np.log(x).mean()
# A basic test of gamma.fit with floc=0.
floc = 0
a, loc, scale = stats.gamma.fit(x, floc=floc)
s = np.log(x.mean()) - meanlog
assert_almost_equal(np.log(a) - special.digamma(a), s, decimal=5)
assert_equal(loc, floc)
assert_almost_equal(scale, x.mean()/a, decimal=8)
# Regression tests for gh-2514.
# The problem was that if `floc=0` was given, any other fixed
# parameters were ignored.
f0 = 1
floc = 0
a, loc, scale = stats.gamma.fit(x, f0=f0, floc=floc)
assert_equal(a, f0)
assert_equal(loc, floc)
assert_almost_equal(scale, x.mean()/a, decimal=8)
f0 = 2
floc = 0
a, loc, scale = stats.gamma.fit(x, f0=f0, floc=floc)
assert_equal(a, f0)
assert_equal(loc, floc)
assert_almost_equal(scale, x.mean()/a, decimal=8)
# loc and scale fixed.
floc = 0
fscale = 2
a, loc, scale = stats.gamma.fit(x, floc=floc, fscale=fscale)
assert_equal(loc, floc)
assert_equal(scale, fscale)
c = meanlog - np.log(fscale)
assert_almost_equal(special.digamma(a), c)
def test_fix_fit_beta(self):
# Test beta.fit when both floc and fscale are given.
def mlefunc(a, b, x):
# Zeros of this function are critical points of
# the maximum likelihood function.
n = len(x)
s1 = np.log(x).sum()
s2 = np.log(1-x).sum()
psiab = special.psi(a + b)
func = [s1 - n * (-psiab + special.psi(a)),
s2 - n * (-psiab + special.psi(b))]
return func
# Basic test with floc and fscale given.
x = np.array([0.125, 0.25, 0.5])
a, b, loc, scale = stats.beta.fit(x, floc=0, fscale=1)
assert_equal(loc, 0)
assert_equal(scale, 1)
assert_allclose(mlefunc(a, b, x), [0, 0], atol=1e-6)
# Basic test with f0, floc and fscale given.
# This is also a regression test for gh-2514.
x = np.array([0.125, 0.25, 0.5])
a, b, loc, scale = stats.beta.fit(x, f0=2, floc=0, fscale=1)
assert_equal(a, 2)
assert_equal(loc, 0)
assert_equal(scale, 1)
da, db = mlefunc(a, b, x)
assert_allclose(db, 0, atol=1e-5)
# Same floc and fscale values as above, but reverse the data
# and fix b (f1).
x2 = 1 - x
a2, b2, loc2, scale2 = stats.beta.fit(x2, f1=2, floc=0, fscale=1)
assert_equal(b2, 2)
assert_equal(loc2, 0)
assert_equal(scale2, 1)
da, db = mlefunc(a2, b2, x2)
assert_allclose(da, 0, atol=1e-5)
# a2 of this test should equal b from above.
assert_almost_equal(a2, b)
# Check for detection of data out of bounds when floc and fscale
# are given.
assert_raises(ValueError, stats.beta.fit, x, floc=0.5, fscale=1)
y = np.array([0, .5, 1])
assert_raises(ValueError, stats.beta.fit, y, floc=0, fscale=1)
assert_raises(ValueError, stats.beta.fit, y, floc=0, fscale=1, f0=2)
assert_raises(ValueError, stats.beta.fit, y, floc=0, fscale=1, f1=2)
# Check that attempting to fix all the parameters raises a ValueError.
assert_raises(ValueError, stats.beta.fit, y, f0=0, f1=1,
floc=2, fscale=3)
def test_expon_fit(self):
x = np.array([2, 2, 4, 4, 4, 4, 4, 8])
loc, scale = stats.expon.fit(x)
assert_equal(loc, 2) # x.min()
assert_equal(scale, 2) # x.mean() - x.min()
loc, scale = stats.expon.fit(x, fscale=3)
assert_equal(loc, 2) # x.min()
assert_equal(scale, 3) # fscale
loc, scale = stats.expon.fit(x, floc=0)
assert_equal(loc, 0) # floc
assert_equal(scale, 4) # x.mean() - loc
def test_lognorm_fit(self):
x = np.array([1.5, 3, 10, 15, 23, 59])
lnxm1 = np.log(x - 1)
shape, loc, scale = stats.lognorm.fit(x, floc=1)
assert_allclose(shape, lnxm1.std(), rtol=1e-12)
assert_equal(loc, 1)
assert_allclose(scale, np.exp(lnxm1.mean()), rtol=1e-12)
shape, loc, scale = stats.lognorm.fit(x, floc=1, fscale=6)
assert_allclose(shape, np.sqrt(((lnxm1 - np.log(6))**2).mean()),
rtol=1e-12)
assert_equal(loc, 1)
assert_equal(scale, 6)
shape, loc, scale = stats.lognorm.fit(x, floc=1, fix_s=0.75)
assert_equal(shape, 0.75)
assert_equal(loc, 1)
assert_allclose(scale, np.exp(lnxm1.mean()), rtol=1e-12)
def test_uniform_fit(self):
x = np.array([1.0, 1.1, 1.2, 9.0])
loc, scale = stats.uniform.fit(x)
assert_equal(loc, x.min())
assert_equal(scale, x.ptp())
loc, scale = stats.uniform.fit(x, floc=0)
assert_equal(loc, 0)
assert_equal(scale, x.max())
loc, scale = stats.uniform.fit(x, fscale=10)
assert_equal(loc, 0)
assert_equal(scale, 10)
assert_raises(ValueError, stats.uniform.fit, x, floc=2.0)
assert_raises(ValueError, stats.uniform.fit, x, fscale=5.0)
def test_fshapes(self):
# take a beta distribution, with shapes='a, b', and make sure that
# fa is equivalent to f0, and fb is equivalent to f1
a, b = 3., 4.
x = stats.beta.rvs(a, b, size=100, random_state=1234)
res_1 = stats.beta.fit(x, f0=3.)
res_2 = stats.beta.fit(x, fa=3.)
assert_allclose(res_1, res_2, atol=1e-12, rtol=1e-12)
res_2 = stats.beta.fit(x, fix_a=3.)
assert_allclose(res_1, res_2, atol=1e-12, rtol=1e-12)
res_3 = stats.beta.fit(x, f1=4.)
res_4 = stats.beta.fit(x, fb=4.)
assert_allclose(res_3, res_4, atol=1e-12, rtol=1e-12)
res_4 = stats.beta.fit(x, fix_b=4.)
assert_allclose(res_3, res_4, atol=1e-12, rtol=1e-12)
# cannot specify both positional and named args at the same time
assert_raises(ValueError, stats.beta.fit, x, fa=1, f0=2)
# check that attempting to fix all parameters raises a ValueError
assert_raises(ValueError, stats.beta.fit, x, fa=0, f1=1,
floc=2, fscale=3)
# check that specifying floc, fscale and fshapes works for
# beta and gamma which override the generic fit method
res_5 = stats.beta.fit(x, fa=3., floc=0, fscale=1)
aa, bb, ll, ss = res_5
assert_equal([aa, ll, ss], [3., 0, 1])
# gamma distribution
a = 3.
data = stats.gamma.rvs(a, size=100)
aa, ll, ss = stats.gamma.fit(data, fa=a)
assert_equal(aa, a)
def test_extra_params(self):
# unknown parameters should raise rather than be silently ignored
dist = stats.exponnorm
data = dist.rvs(K=2, size=100)
dct = dict(enikibeniki=-101)
assert_raises(TypeError, dist.fit, data, **dct)
class TestFrozen(object):
def setup_method(self):
np.random.seed(1234)
# Test that a frozen distribution gives the same results as the original
# object.
#
# Only tested for the normal distribution (with loc and scale specified)
# and for the gamma distribution (with a shape parameter specified).
def test_norm(self):
dist = stats.norm
frozen = stats.norm(loc=10.0, scale=3.0)
result_f = frozen.pdf(20.0)
result = dist.pdf(20.0, loc=10.0, scale=3.0)
assert_equal(result_f, result)
result_f = frozen.cdf(20.0)
result = dist.cdf(20.0, loc=10.0, scale=3.0)
assert_equal(result_f, result)
result_f = frozen.ppf(0.25)
result = dist.ppf(0.25, loc=10.0, scale=3.0)
assert_equal(result_f, result)
result_f = frozen.isf(0.25)
result = dist.isf(0.25, loc=10.0, scale=3.0)
assert_equal(result_f, result)
result_f = frozen.sf(10.0)
result = dist.sf(10.0, loc=10.0, scale=3.0)
assert_equal(result_f, result)
result_f = frozen.median()
result = dist.median(loc=10.0, scale=3.0)
assert_equal(result_f, result)
result_f = frozen.mean()
result = dist.mean(loc=10.0, scale=3.0)
assert_equal(result_f, result)
result_f = frozen.var()
result = dist.var(loc=10.0, scale=3.0)
assert_equal(result_f, result)
result_f = frozen.std()
result = dist.std(loc=10.0, scale=3.0)
assert_equal(result_f, result)
result_f = frozen.entropy()
result = dist.entropy(loc=10.0, scale=3.0)
assert_equal(result_f, result)
result_f = frozen.moment(2)
result = dist.moment(2, loc=10.0, scale=3.0)
assert_equal(result_f, result)
assert_equal(frozen.a, dist.a)
assert_equal(frozen.b, dist.b)
def test_gamma(self):
a = 2.0
dist = stats.gamma
frozen = stats.gamma(a)
result_f = frozen.pdf(20.0)
result = dist.pdf(20.0, a)
assert_equal(result_f, result)
result_f = frozen.cdf(20.0)
result = dist.cdf(20.0, a)
assert_equal(result_f, result)
result_f = frozen.ppf(0.25)
result = dist.ppf(0.25, a)
assert_equal(result_f, result)
result_f = frozen.isf(0.25)
result = dist.isf(0.25, a)
assert_equal(result_f, result)
result_f = frozen.sf(10.0)
result = dist.sf(10.0, a)
assert_equal(result_f, result)
result_f = frozen.median()
result = dist.median(a)
assert_equal(result_f, result)
result_f = frozen.mean()
result = dist.mean(a)
assert_equal(result_f, result)
result_f = frozen.var()
result = dist.var(a)
assert_equal(result_f, result)
result_f = frozen.std()
result = dist.std(a)
assert_equal(result_f, result)
result_f = frozen.entropy()
result = dist.entropy(a)
assert_equal(result_f, result)
result_f = frozen.moment(2)
result = dist.moment(2, a)
assert_equal(result_f, result)
assert_equal(frozen.a, frozen.dist.a)
assert_equal(frozen.b, frozen.dist.b)
def test_regression_ticket_1293(self):
# Create a frozen distribution.
frozen = stats.lognorm(1)
# Call one of its methods that does not take any keyword arguments.
m1 = frozen.moment(2)
# Now call a method that takes a keyword argument.
frozen.stats(moments='mvsk')
# Call moment(2) again.
# After calling stats(), the following was raising an exception.
# So this test passes if the following does not raise an exception.
m2 = frozen.moment(2)
# The following should also be true, of course. But it is not
# the focus of this test.
assert_equal(m1, m2)
def test_ab(self):
# test that the support of a frozen distribution
# (i) remains frozen even if it changes for the original one
# (ii) is actually correct if the shape parameters are such that
# the values of [a, b] are not the default [0, inf]
# take a genpareto as an example where the support
# depends on the value of the shape parameter:
# for c > 0: a, b = 0, inf
# for c < 0: a, b = 0, -1/c
rv = stats.genpareto(c=-0.1)
a, b = rv.dist.a, rv.dist.b
assert_equal([a, b], [0., 10.])
assert_equal([rv.a, rv.b], [0., 10.])
stats.genpareto.pdf(0, c=0.1) # this changes genpareto.b
assert_equal([rv.dist.a, rv.dist.b], [a, b])
assert_equal([rv.a, rv.b], [a, b])
rv1 = stats.genpareto(c=0.1)
assert_(rv1.dist is not rv.dist)
def test_rv_frozen_in_namespace(self):
# Regression test for gh-3522
assert_(hasattr(stats.distributions, 'rv_frozen'))
def test_random_state(self):
# only check that the random_state attribute exists,
frozen = stats.norm()
assert_(hasattr(frozen, 'random_state'))
# ... that it can be set,
frozen.random_state = 42
assert_equal(frozen.random_state.get_state(),
np.random.RandomState(42).get_state())
# ... and that .rvs method accepts it as an argument
rndm = np.random.RandomState(1234)
frozen.rvs(size=8, random_state=rndm)
def test_pickling(self):
# test that a frozen instance pickles and unpickles
# (this method is a clone of common_tests.check_pickling)
beta = stats.beta(2.3098496451481823, 0.62687954300963677)
poiss = stats.poisson(3.)
sample = stats.rv_discrete(values=([0, 1, 2, 3],
[0.1, 0.2, 0.3, 0.4]))
for distfn in [beta, poiss, sample]:
distfn.random_state = 1234
distfn.rvs(size=8)
s = pickle.dumps(distfn)
r0 = distfn.rvs(size=8)
unpickled = pickle.loads(s)
r1 = unpickled.rvs(size=8)
assert_equal(r0, r1)
# also smoke test some methods
medians = [distfn.ppf(0.5), unpickled.ppf(0.5)]
assert_equal(medians[0], medians[1])
assert_equal(distfn.cdf(medians[0]),
unpickled.cdf(medians[1]))
def test_expect(self):
# smoke test the expect method of the frozen distribution
# only take a gamma w/loc and scale and poisson with loc specified
def func(x):
return x
gm = stats.gamma(a=2, loc=3, scale=4)
gm_val = gm.expect(func, lb=1, ub=2, conditional=True)
gamma_val = stats.gamma.expect(func, args=(2,), loc=3, scale=4,
lb=1, ub=2, conditional=True)
assert_allclose(gm_val, gamma_val)
p = stats.poisson(3, loc=4)
p_val = p.expect(func)
poisson_val = stats.poisson.expect(func, args=(3,), loc=4)
assert_allclose(p_val, poisson_val)
class TestExpect(object):
# Test for expect method.
#
# Uses normal distribution and beta distribution for finite bounds, and
# hypergeom for discrete distribution with finite support
def test_norm(self):
v = stats.norm.expect(lambda x: (x-5)*(x-5), loc=5, scale=2)
assert_almost_equal(v, 4, decimal=14)
m = stats.norm.expect(lambda x: (x), loc=5, scale=2)
assert_almost_equal(m, 5, decimal=14)
lb = stats.norm.ppf(0.05, loc=5, scale=2)
ub = stats.norm.ppf(0.95, loc=5, scale=2)
prob90 = stats.norm.expect(lambda x: 1, loc=5, scale=2, lb=lb, ub=ub)
assert_almost_equal(prob90, 0.9, decimal=14)
prob90c = stats.norm.expect(lambda x: 1, loc=5, scale=2, lb=lb, ub=ub,
conditional=True)
assert_almost_equal(prob90c, 1., decimal=14)
def test_beta(self):
# case with finite support interval
v = stats.beta.expect(lambda x: (x-19/3.)*(x-19/3.), args=(10, 5),
loc=5, scale=2)
assert_almost_equal(v, 1./18., decimal=13)
m = stats.beta.expect(lambda x: x, args=(10, 5), loc=5., scale=2.)
assert_almost_equal(m, 19/3., decimal=13)
ub = stats.beta.ppf(0.95, 10, 10, loc=5, scale=2)
lb = stats.beta.ppf(0.05, 10, 10, loc=5, scale=2)
prob90 = stats.beta.expect(lambda x: 1., args=(10, 10), loc=5.,
scale=2., lb=lb, ub=ub, conditional=False)
assert_almost_equal(prob90, 0.9, decimal=13)
prob90c = stats.beta.expect(lambda x: 1, args=(10, 10), loc=5,
scale=2, lb=lb, ub=ub, conditional=True)
assert_almost_equal(prob90c, 1., decimal=13)
def test_hypergeom(self):
# test case with finite bounds
# without specifying bounds
m_true, v_true = stats.hypergeom.stats(20, 10, 8, loc=5.)
m = stats.hypergeom.expect(lambda x: x, args=(20, 10, 8), loc=5.)
assert_almost_equal(m, m_true, decimal=13)
v = stats.hypergeom.expect(lambda x: (x-9.)**2, args=(20, 10, 8),
loc=5.)
assert_almost_equal(v, v_true, decimal=14)
# with bounds, bounds equal to shifted support
v_bounds = stats.hypergeom.expect(lambda x: (x-9.)**2,
args=(20, 10, 8),
loc=5., lb=5, ub=13)
assert_almost_equal(v_bounds, v_true, decimal=14)
# drop boundary points
prob_true = 1-stats.hypergeom.pmf([5, 13], 20, 10, 8, loc=5).sum()
prob_bounds = stats.hypergeom.expect(lambda x: 1, args=(20, 10, 8),
loc=5., lb=6, ub=12)
assert_almost_equal(prob_bounds, prob_true, decimal=13)
# conditional
prob_bc = stats.hypergeom.expect(lambda x: 1, args=(20, 10, 8), loc=5.,
lb=6, ub=12, conditional=True)
assert_almost_equal(prob_bc, 1, decimal=14)
# check simple integral
prob_b = stats.hypergeom.expect(lambda x: 1, args=(20, 10, 8),
lb=0, ub=8)
assert_almost_equal(prob_b, 1, decimal=13)
def test_poisson(self):
# poisson, use lower bound only
prob_bounds = stats.poisson.expect(lambda x: 1, args=(2,), lb=3,
conditional=False)
prob_b_true = 1-stats.poisson.cdf(2, 2)
assert_almost_equal(prob_bounds, prob_b_true, decimal=14)
prob_lb = stats.poisson.expect(lambda x: 1, args=(2,), lb=2,
conditional=True)
assert_almost_equal(prob_lb, 1, decimal=14)
def test_genhalflogistic(self):
# genhalflogistic, changes upper bound of support in _argcheck
# regression test for gh-2622
halflog = stats.genhalflogistic
# check consistency when calling expect twice with the same input
res1 = halflog.expect(args=(1.5,))
halflog.expect(args=(0.5,))
res2 = halflog.expect(args=(1.5,))
assert_almost_equal(res1, res2, decimal=14)
def test_rice_overflow(self):
# rice.pdf(999, 0.74) was inf since special.i0 silentyly overflows
# check that using i0e fixes it
assert_(np.isfinite(stats.rice.pdf(999, 0.74)))
assert_(np.isfinite(stats.rice.expect(lambda x: 1, args=(0.74,))))
assert_(np.isfinite(stats.rice.expect(lambda x: 2, args=(0.74,))))
assert_(np.isfinite(stats.rice.expect(lambda x: 3, args=(0.74,))))
def test_logser(self):
# test a discrete distribution with infinite support and loc
p, loc = 0.3, 3
res_0 = stats.logser.expect(lambda k: k, args=(p,))
# check against the correct answer (sum of a geom series)
assert_allclose(res_0,
p / (p - 1.) / np.log(1. - p), atol=1e-15)
# now check it with `loc`
res_l = stats.logser.expect(lambda k: k, args=(p,), loc=loc)
assert_allclose(res_l, res_0 + loc, atol=1e-15)
def test_skellam(self):
# Use a discrete distribution w/ bi-infinite support. Compute two first
# moments and compare to known values (cf skellam.stats)
p1, p2 = 18, 22
m1 = stats.skellam.expect(lambda x: x, args=(p1, p2))
m2 = stats.skellam.expect(lambda x: x**2, args=(p1, p2))
assert_allclose(m1, p1 - p2, atol=1e-12)
assert_allclose(m2 - m1**2, p1 + p2, atol=1e-12)
def test_randint(self):
# Use a discrete distribution w/ parameter-dependent support, which
# is larger than the default chunksize
lo, hi = 0, 113
res = stats.randint.expect(lambda x: x, (lo, hi))
assert_allclose(res,
sum(_ for _ in range(lo, hi)) / (hi - lo), atol=1e-15)
def test_zipf(self):
# Test that there is no infinite loop even if the sum diverges
assert_warns(RuntimeWarning, stats.zipf.expect,
lambda x: x**2, (2,))
def test_discrete_kwds(self):
# check that discrete expect accepts keywords to control the summation
n0 = stats.poisson.expect(lambda x: 1, args=(2,))
n1 = stats.poisson.expect(lambda x: 1, args=(2,),
maxcount=1001, chunksize=32, tolerance=1e-8)
assert_almost_equal(n0, n1, decimal=14)
def test_moment(self):
# test the .moment() method: compute a higher moment and compare to
# a known value
def poiss_moment5(mu):
return mu**5 + 10*mu**4 + 25*mu**3 + 15*mu**2 + mu
for mu in [5, 7]:
m5 = stats.poisson.moment(5, mu)
assert_allclose(m5, poiss_moment5(mu), rtol=1e-10)
class TestNct(object):
def test_nc_parameter(self):
# Parameter values c<=0 were not enabled (gh-2402).
# For negative values c and for c=0 results of rv.cdf(0) below were nan
rv = stats.nct(5, 0)
assert_equal(rv.cdf(0), 0.5)
rv = stats.nct(5, -1)
assert_almost_equal(rv.cdf(0), 0.841344746069, decimal=10)
def test_broadcasting(self):
res = stats.nct.pdf(5, np.arange(4, 7)[:, None],
np.linspace(0.1, 1, 4))
expected = array([[0.00321886, 0.00557466, 0.00918418, 0.01442997],
[0.00217142, 0.00395366, 0.00683888, 0.01126276],
[0.00153078, 0.00291093, 0.00525206, 0.00900815]])
assert_allclose(res, expected, rtol=1e-5)
def test_variance_gh_issue_2401(self):
# Computation of the variance of a non-central t-distribution resulted
# in a TypeError: ufunc 'isinf' not supported for the input types,
# and the inputs could not be safely coerced to any supported types
# according to the casting rule 'safe'
rv = stats.nct(4, 0)
assert_equal(rv.var(), 2.0)
def test_nct_inf_moments(self):
# n-th moment of nct only exists for df > n
m, v, s, k = stats.nct.stats(df=1.9, nc=0.3, moments='mvsk')
assert_(np.isfinite(m))
assert_equal([v, s, k], [np.inf, np.nan, np.nan])
m, v, s, k = stats.nct.stats(df=3.1, nc=0.3, moments='mvsk')
assert_(np.isfinite([m, v, s]).all())
assert_equal(k, np.nan)
class TestRice(object):
def test_rice_zero_b(self):
# rice distribution should work with b=0, cf gh-2164
x = [0.2, 1., 5.]
assert_(np.isfinite(stats.rice.pdf(x, b=0.)).all())
assert_(np.isfinite(stats.rice.logpdf(x, b=0.)).all())
assert_(np.isfinite(stats.rice.cdf(x, b=0.)).all())
assert_(np.isfinite(stats.rice.logcdf(x, b=0.)).all())
q = [0.1, 0.1, 0.5, 0.9]
assert_(np.isfinite(stats.rice.ppf(q, b=0.)).all())
mvsk = stats.rice.stats(0, moments='mvsk')
assert_(np.isfinite(mvsk).all())
# furthermore, pdf is continuous as b\to 0
# rice.pdf(x, b\to 0) = x exp(-x^2/2) + O(b^2)
# see e.g. Abramovich & Stegun 9.6.7 & 9.6.10
b = 1e-8
assert_allclose(stats.rice.pdf(x, 0), stats.rice.pdf(x, b),
atol=b, rtol=0)
def test_rice_rvs(self):
rvs = stats.rice.rvs
assert_equal(rvs(b=3.).size, 1)
assert_equal(rvs(b=3., size=(3, 5)).shape, (3, 5))
class TestErlang(object):
def setup_method(self):
np.random.seed(1234)
def test_erlang_runtimewarning(self):
# erlang should generate a RuntimeWarning if a non-integer
# shape parameter is used.
with warnings.catch_warnings():
warnings.simplefilter("error", RuntimeWarning)
# The non-integer shape parameter 1.3 should trigger a
# RuntimeWarning
assert_raises(RuntimeWarning,
stats.erlang.rvs, 1.3, loc=0, scale=1, size=4)
# Calling the fit method with `f0` set to an integer should
# *not* trigger a RuntimeWarning. It should return the same
# values as gamma.fit(...).
data = [0.5, 1.0, 2.0, 4.0]
result_erlang = stats.erlang.fit(data, f0=1)
result_gamma = stats.gamma.fit(data, f0=1)
assert_allclose(result_erlang, result_gamma, rtol=1e-3)
class TestRayleigh(object):
# gh-6227
def test_logpdf(self):
y = stats.rayleigh.logpdf(50)
assert_allclose(y, -1246.0879769945718)
def test_logsf(self):
y = stats.rayleigh.logsf(50)
assert_allclose(y, -1250)
class TestExponWeib(object):
def test_pdf_logpdf(self):
# Regression test for gh-3508.
x = 0.1
a = 1.0
c = 100.0
p = stats.exponweib.pdf(x, a, c)
logp = stats.exponweib.logpdf(x, a, c)
# Expected values were computed with mpmath.
assert_allclose([p, logp],
[1.0000000000000054e-97, -223.35075402042244])
def test_a_is_1(self):
# For issue gh-3508.
# Check that when a=1, the pdf and logpdf methods of exponweib are the
# same as those of weibull_min.
x = np.logspace(-4, -1, 4)
a = 1
c = 100
p = stats.exponweib.pdf(x, a, c)
expected = stats.weibull_min.pdf(x, c)
assert_allclose(p, expected)
logp = stats.exponweib.logpdf(x, a, c)
expected = stats.weibull_min.logpdf(x, c)
assert_allclose(logp, expected)
def test_a_is_1_c_is_1(self):
# When a = 1 and c = 1, the distribution is exponential.
x = np.logspace(-8, 1, 10)
a = 1
c = 1
p = stats.exponweib.pdf(x, a, c)
expected = stats.expon.pdf(x)
assert_allclose(p, expected)
logp = stats.exponweib.logpdf(x, a, c)
expected = stats.expon.logpdf(x)
assert_allclose(logp, expected)
class TestWeibull(object):
def test_logpdf(self):
# gh-6217
y = stats.weibull_min.logpdf(0, 1)
assert_equal(y, 0)
def test_with_maxima_distrib(self):
# Tests for weibull_min and weibull_max.
# The expected values were computed using the symbolic algebra
# program 'maxima' with the package 'distrib', which has
# 'pdf_weibull' and 'cdf_weibull'. The mapping between the
# scipy and maxima functions is as follows:
# -----------------------------------------------------------------
# scipy maxima
# --------------------------------- ------------------------------
# weibull_min.pdf(x, a, scale=b) pdf_weibull(x, a, b)
# weibull_min.logpdf(x, a, scale=b) log(pdf_weibull(x, a, b))
# weibull_min.cdf(x, a, scale=b) cdf_weibull(x, a, b)
# weibull_min.logcdf(x, a, scale=b) log(cdf_weibull(x, a, b))
# weibull_min.sf(x, a, scale=b) 1 - cdf_weibull(x, a, b)
# weibull_min.logsf(x, a, scale=b) log(1 - cdf_weibull(x, a, b))
#
# weibull_max.pdf(x, a, scale=b) pdf_weibull(-x, a, b)
# weibull_max.logpdf(x, a, scale=b) log(pdf_weibull(-x, a, b))
# weibull_max.cdf(x, a, scale=b) 1 - cdf_weibull(-x, a, b)
# weibull_max.logcdf(x, a, scale=b) log(1 - cdf_weibull(-x, a, b))
# weibull_max.sf(x, a, scale=b) cdf_weibull(-x, a, b)
# weibull_max.logsf(x, a, scale=b) log(cdf_weibull(-x, a, b))
# -----------------------------------------------------------------
x = 1.5
a = 2.0
b = 3.0
# weibull_min
p = stats.weibull_min.pdf(x, a, scale=b)
assert_allclose(p, np.exp(-0.25)/3)
lp = stats.weibull_min.logpdf(x, a, scale=b)
assert_allclose(lp, -0.25 - np.log(3))
c = stats.weibull_min.cdf(x, a, scale=b)
assert_allclose(c, -special.expm1(-0.25))
lc = stats.weibull_min.logcdf(x, a, scale=b)
assert_allclose(lc, np.log(-special.expm1(-0.25)))
s = stats.weibull_min.sf(x, a, scale=b)
assert_allclose(s, np.exp(-0.25))
ls = stats.weibull_min.logsf(x, a, scale=b)
assert_allclose(ls, -0.25)
# Also test using a large value x, for which computing the survival
# function using the CDF would result in 0.
s = stats.weibull_min.sf(30, 2, scale=3)
assert_allclose(s, np.exp(-100))
ls = stats.weibull_min.logsf(30, 2, scale=3)
assert_allclose(ls, -100)
# weibull_max
x = -1.5
p = stats.weibull_max.pdf(x, a, scale=b)
assert_allclose(p, np.exp(-0.25)/3)
lp = stats.weibull_max.logpdf(x, a, scale=b)
assert_allclose(lp, -0.25 - np.log(3))
c = stats.weibull_max.cdf(x, a, scale=b)
assert_allclose(c, np.exp(-0.25))
lc = stats.weibull_max.logcdf(x, a, scale=b)
assert_allclose(lc, -0.25)
s = stats.weibull_max.sf(x, a, scale=b)
assert_allclose(s, -special.expm1(-0.25))
ls = stats.weibull_max.logsf(x, a, scale=b)
assert_allclose(ls, np.log(-special.expm1(-0.25)))
# Also test using a value of x close to 0, for which computing the
# survival function using the CDF would result in 0.
s = stats.weibull_max.sf(-1e-9, 2, scale=3)
assert_allclose(s, -special.expm1(-1/9000000000000000000))
ls = stats.weibull_max.logsf(-1e-9, 2, scale=3)
assert_allclose(ls, np.log(-special.expm1(-1/9000000000000000000)))
class TestRdist(object):
@pytest.mark.slow
def test_rdist_cdf_gh1285(self):
# check workaround in rdist._cdf for issue gh-1285.
distfn = stats.rdist
values = [0.001, 0.5, 0.999]
assert_almost_equal(distfn.cdf(distfn.ppf(values, 541.0), 541.0),
values, decimal=5)
class TestTrapz(object):
def test_reduces_to_triang(self):
modes = [0, 0.3, 0.5, 1]
for mode in modes:
x = [0, mode, 1]
assert_almost_equal(stats.trapz.pdf(x, mode, mode),
stats.triang.pdf(x, mode))
assert_almost_equal(stats.trapz.cdf(x, mode, mode),
stats.triang.cdf(x, mode))
def test_reduces_to_uniform(self):
x = np.linspace(0, 1, 10)
assert_almost_equal(stats.trapz.pdf(x, 0, 1), stats.uniform.pdf(x))
assert_almost_equal(stats.trapz.cdf(x, 0, 1), stats.uniform.cdf(x))
def test_cases(self):
# edge cases
assert_almost_equal(stats.trapz.pdf(0, 0, 0), 2)
assert_almost_equal(stats.trapz.pdf(1, 1, 1), 2)
assert_almost_equal(stats.trapz.pdf(0.5, 0, 0.8), 1.11111111111111111)
assert_almost_equal(stats.trapz.pdf(0.5, 0.2, 1.0), 1.11111111111111111)
# straightforward case
assert_almost_equal(stats.trapz.pdf(0.1, 0.2, 0.8), 0.625)
assert_almost_equal(stats.trapz.pdf(0.5, 0.2, 0.8), 1.25)
assert_almost_equal(stats.trapz.pdf(0.9, 0.2, 0.8), 0.625)
assert_almost_equal(stats.trapz.cdf(0.1, 0.2, 0.8), 0.03125)
assert_almost_equal(stats.trapz.cdf(0.2, 0.2, 0.8), 0.125)
assert_almost_equal(stats.trapz.cdf(0.5, 0.2, 0.8), 0.5)
assert_almost_equal(stats.trapz.cdf(0.9, 0.2, 0.8), 0.96875)
assert_almost_equal(stats.trapz.cdf(1.0, 0.2, 0.8), 1.0)
def test_trapz_vect(self):
# test that array-valued shapes and arguments are handled
c = np.array([0.1, 0.2, 0.3])
d = np.array([0.5, 0.6])[:, None]
x = np.array([0.15, 0.25, 0.9])
v = stats.trapz.pdf(x, c, d)
cc, dd, xx = np.broadcast_arrays(c, d, x)
res = np.empty(xx.size, dtype=xx.dtype)
ind = np.arange(xx.size)
for i, x1, c1, d1 in zip(ind, xx.ravel(), cc.ravel(), dd.ravel()):
res[i] = stats.trapz.pdf(x1, c1, d1)
assert_allclose(v, res.reshape(v.shape), atol=1e-15)
class TestTriang(object):
def test_edge_cases(self):
with np.errstate(all='raise'):
assert_equal(stats.triang.pdf(0, 0), 2.)
assert_equal(stats.triang.pdf(0.5, 0), 1.)
assert_equal(stats.triang.pdf(1, 0), 0.)
assert_equal(stats.triang.pdf(0, 1), 0)
assert_equal(stats.triang.pdf(0.5, 1), 1.)
assert_equal(stats.triang.pdf(1, 1), 2)
assert_equal(stats.triang.cdf(0., 0.), 0.)
assert_equal(stats.triang.cdf(0.5, 0.), 0.75)
assert_equal(stats.triang.cdf(1.0, 0.), 1.0)
assert_equal(stats.triang.cdf(0., 1.), 0.)
assert_equal(stats.triang.cdf(0.5, 1.), 0.25)
assert_equal(stats.triang.cdf(1., 1.), 1)
def test_540_567():
# test for nan returned in tickets 540, 567
assert_almost_equal(stats.norm.cdf(-1.7624320982), 0.03899815971089126,
decimal=10, err_msg='test_540_567')
assert_almost_equal(stats.norm.cdf(-1.7624320983), 0.038998159702449846,
decimal=10, err_msg='test_540_567')
assert_almost_equal(stats.norm.cdf(1.38629436112, loc=0.950273420309,
scale=0.204423758009),
0.98353464004309321,
decimal=10, err_msg='test_540_567')
def test_regression_ticket_1316():
# The following was raising an exception, because _construct_default_doc()
# did not handle the default keyword extradoc=None. See ticket #1316.
g = stats._continuous_distns.gamma_gen(name='gamma')
def test_regression_ticket_1326():
# adjust to avoid nan with 0*log(0)
assert_almost_equal(stats.chi2.pdf(0.0, 2), 0.5, 14)
def test_regression_tukey_lambda():
# Make sure that Tukey-Lambda distribution correctly handles
# non-positive lambdas.
x = np.linspace(-5.0, 5.0, 101)
olderr = np.seterr(divide='ignore')
try:
for lam in [0.0, -1.0, -2.0, np.array([[-1.0], [0.0], [-2.0]])]:
p = stats.tukeylambda.pdf(x, lam)
assert_((p != 0.0).all())
assert_(~np.isnan(p).all())
lam = np.array([[-1.0], [0.0], [2.0]])
p = stats.tukeylambda.pdf(x, lam)
finally:
np.seterr(**olderr)
assert_(~np.isnan(p).all())
assert_((p[0] != 0.0).all())
assert_((p[1] != 0.0).all())
assert_((p[2] != 0.0).any())
assert_((p[2] == 0.0).any())
@pytest.mark.skipif(DOCSTRINGS_STRIPPED, reason="docstrings stripped")
def test_regression_ticket_1421():
assert_('pdf(x, mu, loc=0, scale=1)' not in stats.poisson.__doc__)
assert_('pmf(x,' in stats.poisson.__doc__)
def test_nan_arguments_gh_issue_1362():
with np.errstate(invalid='ignore'):
assert_(np.isnan(stats.t.logcdf(1, np.nan)))
assert_(np.isnan(stats.t.cdf(1, np.nan)))
assert_(np.isnan(stats.t.logsf(1, np.nan)))
assert_(np.isnan(stats.t.sf(1, np.nan)))
assert_(np.isnan(stats.t.pdf(1, np.nan)))
assert_(np.isnan(stats.t.logpdf(1, np.nan)))
assert_(np.isnan(stats.t.ppf(1, np.nan)))
assert_(np.isnan(stats.t.isf(1, np.nan)))
assert_(np.isnan(stats.bernoulli.logcdf(np.nan, 0.5)))
assert_(np.isnan(stats.bernoulli.cdf(np.nan, 0.5)))
assert_(np.isnan(stats.bernoulli.logsf(np.nan, 0.5)))
assert_(np.isnan(stats.bernoulli.sf(np.nan, 0.5)))
assert_(np.isnan(stats.bernoulli.pmf(np.nan, 0.5)))
assert_(np.isnan(stats.bernoulli.logpmf(np.nan, 0.5)))
assert_(np.isnan(stats.bernoulli.ppf(np.nan, 0.5)))
assert_(np.isnan(stats.bernoulli.isf(np.nan, 0.5)))
def test_frozen_fit_ticket_1536():
np.random.seed(5678)
true = np.array([0.25, 0., 0.5])
x = stats.lognorm.rvs(true[0], true[1], true[2], size=100)
olderr = np.seterr(divide='ignore')
try:
params = np.array(stats.lognorm.fit(x, floc=0.))
finally:
np.seterr(**olderr)
assert_almost_equal(params, true, decimal=2)
params = np.array(stats.lognorm.fit(x, fscale=0.5, loc=0))
assert_almost_equal(params, true, decimal=2)
params = np.array(stats.lognorm.fit(x, f0=0.25, loc=0))
assert_almost_equal(params, true, decimal=2)
params = np.array(stats.lognorm.fit(x, f0=0.25, floc=0))
assert_almost_equal(params, true, decimal=2)
np.random.seed(5678)
loc = 1
floc = 0.9
x = stats.norm.rvs(loc, 2., size=100)
params = np.array(stats.norm.fit(x, floc=floc))
expected = np.array([floc, np.sqrt(((x-floc)**2).mean())])
assert_almost_equal(params, expected, decimal=4)
def test_regression_ticket_1530():
# Check the starting value works for Cauchy distribution fit.
np.random.seed(654321)
rvs = stats.cauchy.rvs(size=100)
params = stats.cauchy.fit(rvs)
expected = (0.045, 1.142)
assert_almost_equal(params, expected, decimal=1)
def test_gh_pr_4806():
# Check starting values for Cauchy distribution fit.
np.random.seed(1234)
x = np.random.randn(42)
for offset in 10000.0, 1222333444.0:
loc, scale = stats.cauchy.fit(x + offset)
assert_allclose(loc, offset, atol=1.0)
assert_allclose(scale, 0.6, atol=1.0)
def test_tukeylambda_stats_ticket_1545():
# Some test for the variance and kurtosis of the Tukey Lambda distr.
# See test_tukeylamdba_stats.py for more tests.
mv = stats.tukeylambda.stats(0, moments='mvsk')
# Known exact values:
expected = [0, np.pi**2/3, 0, 1.2]
assert_almost_equal(mv, expected, decimal=10)
mv = stats.tukeylambda.stats(3.13, moments='mvsk')
# 'expected' computed with mpmath.
expected = [0, 0.0269220858861465102, 0, -0.898062386219224104]
assert_almost_equal(mv, expected, decimal=10)
mv = stats.tukeylambda.stats(0.14, moments='mvsk')
# 'expected' computed with mpmath.
expected = [0, 2.11029702221450250, 0, -0.02708377353223019456]
assert_almost_equal(mv, expected, decimal=10)
def test_poisson_logpmf_ticket_1436():
assert_(np.isfinite(stats.poisson.logpmf(1500, 200)))
def test_powerlaw_stats():
"""Test the powerlaw stats function.
This unit test is also a regression test for ticket 1548.
The exact values are:
mean:
mu = a / (a + 1)
variance:
sigma**2 = a / ((a + 2) * (a + 1) ** 2)
skewness:
One formula (see http://en.wikipedia.org/wiki/Skewness) is
gamma_1 = (E[X**3] - 3*mu*E[X**2] + 2*mu**3) / sigma**3
A short calculation shows that E[X**k] is a / (a + k), so gamma_1
can be implemented as
n = a/(a+3) - 3*(a/(a+1))*a/(a+2) + 2*(a/(a+1))**3
d = sqrt(a/((a+2)*(a+1)**2)) ** 3
gamma_1 = n/d
Either by simplifying, or by a direct calculation of mu_3 / sigma**3,
one gets the more concise formula:
gamma_1 = -2.0 * ((a - 1) / (a + 3)) * sqrt((a + 2) / a)
kurtosis: (See http://en.wikipedia.org/wiki/Kurtosis)
The excess kurtosis is
gamma_2 = mu_4 / sigma**4 - 3
A bit of calculus and algebra (sympy helps) shows that
mu_4 = 3*a*(3*a**2 - a + 2) / ((a+1)**4 * (a+2) * (a+3) * (a+4))
so
gamma_2 = 3*(3*a**2 - a + 2) * (a+2) / (a*(a+3)*(a+4)) - 3
which can be rearranged to
gamma_2 = 6 * (a**3 - a**2 - 6*a + 2) / (a*(a+3)*(a+4))
"""
cases = [(1.0, (0.5, 1./12, 0.0, -1.2)),
(2.0, (2./3, 2./36, -0.56568542494924734, -0.6))]
for a, exact_mvsk in cases:
mvsk = stats.powerlaw.stats(a, moments="mvsk")
assert_array_almost_equal(mvsk, exact_mvsk)
def test_powerlaw_edge():
# Regression test for gh-3986.
p = stats.powerlaw.logpdf(0, 1)
assert_equal(p, 0.0)
def test_exponpow_edge():
# Regression test for gh-3982.
p = stats.exponpow.logpdf(0, 1)
assert_equal(p, 0.0)
# Check pdf and logpdf at x = 0 for other values of b.
p = stats.exponpow.pdf(0, [0.25, 1.0, 1.5])
assert_equal(p, [np.inf, 1.0, 0.0])
p = stats.exponpow.logpdf(0, [0.25, 1.0, 1.5])
assert_equal(p, [np.inf, 0.0, -np.inf])
def test_gengamma_edge():
# Regression test for gh-3985.
p = stats.gengamma.pdf(0, 1, 1)
assert_equal(p, 1.0)
# Regression tests for gh-4724.
p = stats.gengamma._munp(-2, 200, 1.)
assert_almost_equal(p, 1./199/198)
p = stats.gengamma._munp(-2, 10, 1.)
assert_almost_equal(p, 1./9/8)
def test_ksone_fit_freeze():
# Regression test for ticket #1638.
d = np.array(
[-0.18879233, 0.15734249, 0.18695107, 0.27908787, -0.248649,
-0.2171497, 0.12233512, 0.15126419, 0.03119282, 0.4365294,
0.08930393, -0.23509903, 0.28231224, -0.09974875, -0.25196048,
0.11102028, 0.1427649, 0.10176452, 0.18754054, 0.25826724,
0.05988819, 0.0531668, 0.21906056, 0.32106729, 0.2117662,
0.10886442, 0.09375789, 0.24583286, -0.22968366, -0.07842391,
-0.31195432, -0.21271196, 0.1114243, -0.13293002, 0.01331725,
-0.04330977, -0.09485776, -0.28434547, 0.22245721, -0.18518199,
-0.10943985, -0.35243174, 0.06897665, -0.03553363, -0.0701746,
-0.06037974, 0.37670779, -0.21684405])
try:
olderr = np.seterr(invalid='ignore')
with suppress_warnings() as sup:
sup.filter(IntegrationWarning,
"The maximum number of subdivisions .50. has been achieved.")
sup.filter(RuntimeWarning,
"floating point number truncated to an integer")
stats.ksone.fit(d)
finally:
np.seterr(**olderr)
def test_norm_logcdf():
# Test precision of the logcdf of the normal distribution.
# This precision was enhanced in ticket 1614.
x = -np.asarray(list(range(0, 120, 4)))
# Values from R
expected = [-0.69314718, -10.36010149, -35.01343716, -75.41067300,
-131.69539607, -203.91715537, -292.09872100, -396.25241451,
-516.38564863, -652.50322759, -804.60844201, -972.70364403,
-1156.79057310, -1356.87055173, -1572.94460885, -1805.01356068,
-2053.07806561, -2317.13866238, -2597.19579746, -2893.24984493,
-3205.30112136, -3533.34989701, -3877.39640444, -4237.44084522,
-4613.48339520, -5005.52420869, -5413.56342187, -5837.60115548,
-6277.63751711, -6733.67260303]
assert_allclose(stats.norm().logcdf(x), expected, atol=1e-8)
# also test the complex-valued code path
assert_allclose(stats.norm().logcdf(x + 1e-14j).real, expected, atol=1e-8)
# test the accuracy: d(logcdf)/dx = pdf / cdf \equiv exp(logpdf - logcdf)
deriv = (stats.norm.logcdf(x + 1e-10j)/1e-10).imag
deriv_expected = np.exp(stats.norm.logpdf(x) - stats.norm.logcdf(x))
assert_allclose(deriv, deriv_expected, atol=1e-10)
def test_levy_cdf_ppf():
# Test levy.cdf, including small arguments.
x = np.array([1000, 1.0, 0.5, 0.1, 0.01, 0.001])
# Expected values were calculated separately with mpmath.
# E.g.
# >>> mpmath.mp.dps = 100
# >>> x = mpmath.mp.mpf('0.01')
# >>> cdf = mpmath.erfc(mpmath.sqrt(1/(2*x)))
expected = np.array([0.9747728793699604,
0.3173105078629141,
0.1572992070502851,
0.0015654022580025495,
1.523970604832105e-23,
1.795832784800726e-219])
y = stats.levy.cdf(x)
assert_allclose(y, expected, rtol=1e-10)
# ppf(expected) should get us back to x.
xx = stats.levy.ppf(expected)
assert_allclose(xx, x, rtol=1e-13)
def test_hypergeom_interval_1802():
# these two had endless loops
assert_equal(stats.hypergeom.interval(.95, 187601, 43192, 757),
(152.0, 197.0))
assert_equal(stats.hypergeom.interval(.945, 187601, 43192, 757),
(152.0, 197.0))
# this was working also before
assert_equal(stats.hypergeom.interval(.94, 187601, 43192, 757),
(153.0, 196.0))
# degenerate case .a == .b
assert_equal(stats.hypergeom.ppf(0.02, 100, 100, 8), 8)
assert_equal(stats.hypergeom.ppf(1, 100, 100, 8), 8)
def test_distribution_too_many_args():
np.random.seed(1234)
# Check that a TypeError is raised when too many args are given to a method
# Regression test for ticket 1815.
x = np.linspace(0.1, 0.7, num=5)
assert_raises(TypeError, stats.gamma.pdf, x, 2, 3, loc=1.0)
assert_raises(TypeError, stats.gamma.pdf, x, 2, 3, 4, loc=1.0)
assert_raises(TypeError, stats.gamma.pdf, x, 2, 3, 4, 5)
assert_raises(TypeError, stats.gamma.pdf, x, 2, 3, loc=1.0, scale=0.5)
assert_raises(TypeError, stats.gamma.rvs, 2., 3, loc=1.0, scale=0.5)
assert_raises(TypeError, stats.gamma.cdf, x, 2., 3, loc=1.0, scale=0.5)
assert_raises(TypeError, stats.gamma.ppf, x, 2., 3, loc=1.0, scale=0.5)
assert_raises(TypeError, stats.gamma.stats, 2., 3, loc=1.0, scale=0.5)
assert_raises(TypeError, stats.gamma.entropy, 2., 3, loc=1.0, scale=0.5)
assert_raises(TypeError, stats.gamma.fit, x, 2., 3, loc=1.0, scale=0.5)
# These should not give errors
stats.gamma.pdf(x, 2, 3) # loc=3
stats.gamma.pdf(x, 2, 3, 4) # loc=3, scale=4
stats.gamma.stats(2., 3)
stats.gamma.stats(2., 3, 4)
stats.gamma.stats(2., 3, 4, 'mv')
stats.gamma.rvs(2., 3, 4, 5)
stats.gamma.fit(stats.gamma.rvs(2., size=7), 2.)
# Also for a discrete distribution
stats.geom.pmf(x, 2, loc=3) # no error, loc=3
assert_raises(TypeError, stats.geom.pmf, x, 2, 3, 4)
assert_raises(TypeError, stats.geom.pmf, x, 2, 3, loc=4)
# And for distributions with 0, 2 and 3 args respectively
assert_raises(TypeError, stats.expon.pdf, x, 3, loc=1.0)
assert_raises(TypeError, stats.exponweib.pdf, x, 3, 4, 5, loc=1.0)
assert_raises(TypeError, stats.exponweib.pdf, x, 3, 4, 5, 0.1, 0.1)
assert_raises(TypeError, stats.ncf.pdf, x, 3, 4, 5, 6, loc=1.0)
assert_raises(TypeError, stats.ncf.pdf, x, 3, 4, 5, 6, 1.0, scale=0.5)
stats.ncf.pdf(x, 3, 4, 5, 6, 1.0) # 3 args, plus loc/scale
def test_ncx2_tails_ticket_955():
# Trac #955 -- check that the cdf computed by special functions
# matches the integrated pdf
a = stats.ncx2.cdf(np.arange(20, 25, 0.2), 2, 1.07458615e+02)
b = stats.ncx2._cdfvec(np.arange(20, 25, 0.2), 2, 1.07458615e+02)
assert_allclose(a, b, rtol=1e-3, atol=0)
def test_ncx2_tails_pdf():
# ncx2.pdf does not return nans in extreme tails(example from gh-1577)
# NB: this is to check that nan_to_num is not needed in ncx2.pdf
with suppress_warnings() as sup:
sup.filter(RuntimeWarning, "divide by zero encountered in log")
assert_equal(stats.ncx2.pdf(1, np.arange(340, 350), 2), 0)
logval = stats.ncx2.logpdf(1, np.arange(340, 350), 2)
assert_(np.isneginf(logval).all())
def test_foldnorm_zero():
# Parameter value c=0 was not enabled, see gh-2399.
rv = stats.foldnorm(0, scale=1)
assert_equal(rv.cdf(0), 0) # rv.cdf(0) previously resulted in: nan
def test_stats_shapes_argcheck():
# stats method was failing for vector shapes if some of the values
# were outside of the allowed range, see gh-2678
mv3 = stats.invgamma.stats([0.0, 0.5, 1.0], 1, 0.5) # 0 is not a legal `a`
mv2 = stats.invgamma.stats([0.5, 1.0], 1, 0.5)
mv2_augmented = tuple(np.r_[np.nan, _] for _ in mv2)
assert_equal(mv2_augmented, mv3)
# -1 is not a legal shape parameter
mv3 = stats.lognorm.stats([2, 2.4, -1])
mv2 = stats.lognorm.stats([2, 2.4])
mv2_augmented = tuple(np.r_[_, np.nan] for _ in mv2)
assert_equal(mv2_augmented, mv3)
# FIXME: this is only a quick-and-dirty test of a quick-and-dirty bugfix.
# stats method with multiple shape parameters is not properly vectorized
# anyway, so some distributions may or may not fail.
# Test subclassing distributions w/ explicit shapes
class _distr_gen(stats.rv_continuous):
def _pdf(self, x, a):
return 42
class _distr2_gen(stats.rv_continuous):
def _cdf(self, x, a):
return 42 * a + x
class _distr3_gen(stats.rv_continuous):
def _pdf(self, x, a, b):
return a + b
def _cdf(self, x, a):
# Different # of shape params from _pdf, to be able to check that
# inspection catches the inconsistency."""
return 42 * a + x
class _distr6_gen(stats.rv_continuous):
# Two shape parameters (both _pdf and _cdf defined, consistent shapes.)
def _pdf(self, x, a, b):
return a*x + b
def _cdf(self, x, a, b):
return 42 * a + x
class TestSubclassingExplicitShapes(object):
# Construct a distribution w/ explicit shapes parameter and test it.
def test_correct_shapes(self):
dummy_distr = _distr_gen(name='dummy', shapes='a')
assert_equal(dummy_distr.pdf(1, a=1), 42)
def test_wrong_shapes_1(self):
dummy_distr = _distr_gen(name='dummy', shapes='A')
assert_raises(TypeError, dummy_distr.pdf, 1, **dict(a=1))
def test_wrong_shapes_2(self):
dummy_distr = _distr_gen(name='dummy', shapes='a, b, c')
dct = dict(a=1, b=2, c=3)
assert_raises(TypeError, dummy_distr.pdf, 1, **dct)
def test_shapes_string(self):
# shapes must be a string
dct = dict(name='dummy', shapes=42)
assert_raises(TypeError, _distr_gen, **dct)
def test_shapes_identifiers_1(self):
# shapes must be a comma-separated list of valid python identifiers
dct = dict(name='dummy', shapes='(!)')
assert_raises(SyntaxError, _distr_gen, **dct)
def test_shapes_identifiers_2(self):
dct = dict(name='dummy', shapes='4chan')
assert_raises(SyntaxError, _distr_gen, **dct)
def test_shapes_identifiers_3(self):
dct = dict(name='dummy', shapes='m(fti)')
assert_raises(SyntaxError, _distr_gen, **dct)
def test_shapes_identifiers_nodefaults(self):
dct = dict(name='dummy', shapes='a=2')
assert_raises(SyntaxError, _distr_gen, **dct)
def test_shapes_args(self):
dct = dict(name='dummy', shapes='*args')
assert_raises(SyntaxError, _distr_gen, **dct)
def test_shapes_kwargs(self):
dct = dict(name='dummy', shapes='**kwargs')
assert_raises(SyntaxError, _distr_gen, **dct)
def test_shapes_keywords(self):
# python keywords cannot be used for shape parameters
dct = dict(name='dummy', shapes='a, b, c, lambda')
assert_raises(SyntaxError, _distr_gen, **dct)
def test_shapes_signature(self):
# test explicit shapes which agree w/ the signature of _pdf
class _dist_gen(stats.rv_continuous):
def _pdf(self, x, a):
return stats.norm._pdf(x) * a
dist = _dist_gen(shapes='a')
assert_equal(dist.pdf(0.5, a=2), stats.norm.pdf(0.5)*2)
def test_shapes_signature_inconsistent(self):
# test explicit shapes which do not agree w/ the signature of _pdf
class _dist_gen(stats.rv_continuous):
def _pdf(self, x, a):
return stats.norm._pdf(x) * a
dist = _dist_gen(shapes='a, b')
assert_raises(TypeError, dist.pdf, 0.5, **dict(a=1, b=2))
def test_star_args(self):
# test _pdf with only starargs
# NB: **kwargs of pdf will never reach _pdf
class _dist_gen(stats.rv_continuous):
def _pdf(self, x, *args):
extra_kwarg = args[0]
return stats.norm._pdf(x) * extra_kwarg
dist = _dist_gen(shapes='extra_kwarg')
assert_equal(dist.pdf(0.5, extra_kwarg=33), stats.norm.pdf(0.5)*33)
assert_equal(dist.pdf(0.5, 33), stats.norm.pdf(0.5)*33)
assert_raises(TypeError, dist.pdf, 0.5, **dict(xxx=33))
def test_star_args_2(self):
# test _pdf with named & starargs
# NB: **kwargs of pdf will never reach _pdf
class _dist_gen(stats.rv_continuous):
def _pdf(self, x, offset, *args):
extra_kwarg = args[0]
return stats.norm._pdf(x) * extra_kwarg + offset
dist = _dist_gen(shapes='offset, extra_kwarg')
assert_equal(dist.pdf(0.5, offset=111, extra_kwarg=33),
stats.norm.pdf(0.5)*33 + 111)
assert_equal(dist.pdf(0.5, 111, 33),
stats.norm.pdf(0.5)*33 + 111)
def test_extra_kwarg(self):
# **kwargs to _pdf are ignored.
# this is a limitation of the framework (_pdf(x, *goodargs))
class _distr_gen(stats.rv_continuous):
def _pdf(self, x, *args, **kwargs):
# _pdf should handle *args, **kwargs itself. Here "handling"
# is ignoring *args and looking for ``extra_kwarg`` and using
# that.
extra_kwarg = kwargs.pop('extra_kwarg', 1)
return stats.norm._pdf(x) * extra_kwarg
dist = _distr_gen(shapes='extra_kwarg')
assert_equal(dist.pdf(1, extra_kwarg=3), stats.norm.pdf(1))
def shapes_empty_string(self):
# shapes='' is equivalent to shapes=None
class _dist_gen(stats.rv_continuous):
def _pdf(self, x):
return stats.norm.pdf(x)
dist = _dist_gen(shapes='')
assert_equal(dist.pdf(0.5), stats.norm.pdf(0.5))
class TestSubclassingNoShapes(object):
# Construct a distribution w/o explicit shapes parameter and test it.
def test_only__pdf(self):
dummy_distr = _distr_gen(name='dummy')
assert_equal(dummy_distr.pdf(1, a=1), 42)
def test_only__cdf(self):
# _pdf is determined from _cdf by taking numerical derivative
dummy_distr = _distr2_gen(name='dummy')
assert_almost_equal(dummy_distr.pdf(1, a=1), 1)
@pytest.mark.skipif(DOCSTRINGS_STRIPPED, reason="docstring stripped")
def test_signature_inspection(self):
# check that _pdf signature inspection works correctly, and is used in
# the class docstring
dummy_distr = _distr_gen(name='dummy')
assert_equal(dummy_distr.numargs, 1)
assert_equal(dummy_distr.shapes, 'a')
res = re.findall(r'logpdf\(x, a, loc=0, scale=1\)',
dummy_distr.__doc__)
assert_(len(res) == 1)
@pytest.mark.skipif(DOCSTRINGS_STRIPPED, reason="docstring stripped")
def test_signature_inspection_2args(self):
# same for 2 shape params and both _pdf and _cdf defined
dummy_distr = _distr6_gen(name='dummy')
assert_equal(dummy_distr.numargs, 2)
assert_equal(dummy_distr.shapes, 'a, b')
res = re.findall(r'logpdf\(x, a, b, loc=0, scale=1\)',
dummy_distr.__doc__)
assert_(len(res) == 1)
def test_signature_inspection_2args_incorrect_shapes(self):
# both _pdf and _cdf defined, but shapes are inconsistent: raises
try:
_distr3_gen(name='dummy')
except TypeError:
pass
else:
raise AssertionError('TypeError not raised.')
def test_defaults_raise(self):
# default arguments should raise
class _dist_gen(stats.rv_continuous):
def _pdf(self, x, a=42):
return 42
assert_raises(TypeError, _dist_gen, **dict(name='dummy'))
def test_starargs_raise(self):
# without explicit shapes, *args are not allowed
class _dist_gen(stats.rv_continuous):
def _pdf(self, x, a, *args):
return 42
assert_raises(TypeError, _dist_gen, **dict(name='dummy'))
def test_kwargs_raise(self):
# without explicit shapes, **kwargs are not allowed
class _dist_gen(stats.rv_continuous):
def _pdf(self, x, a, **kwargs):
return 42
assert_raises(TypeError, _dist_gen, **dict(name='dummy'))
@pytest.mark.skipif(DOCSTRINGS_STRIPPED, reason="docstring stripped")
def test_docstrings():
badones = [r',\s*,', r'\(\s*,', r'^\s*:']
for distname in stats.__all__:
dist = getattr(stats, distname)
if isinstance(dist, (stats.rv_discrete, stats.rv_continuous)):
for regex in badones:
assert_(re.search(regex, dist.__doc__) is None)
def test_infinite_input():
assert_almost_equal(stats.skellam.sf(np.inf, 10, 11), 0)
assert_almost_equal(stats.ncx2._cdf(np.inf, 8, 0.1), 1)
def test_lomax_accuracy():
# regression test for gh-4033
p = stats.lomax.ppf(stats.lomax.cdf(1e-100, 1), 1)
assert_allclose(p, 1e-100)
def test_gompertz_accuracy():
# Regression test for gh-4031
p = stats.gompertz.ppf(stats.gompertz.cdf(1e-100, 1), 1)
assert_allclose(p, 1e-100)
def test_truncexpon_accuracy():
# regression test for gh-4035
p = stats.truncexpon.ppf(stats.truncexpon.cdf(1e-100, 1), 1)
assert_allclose(p, 1e-100)
def test_rayleigh_accuracy():
# regression test for gh-4034
p = stats.rayleigh.isf(stats.rayleigh.sf(9, 1), 1)
assert_almost_equal(p, 9.0, decimal=15)
def test_genextreme_give_no_warnings():
"""regression test for gh-6219"""
with warnings.catch_warnings(record=True) as w:
warnings.simplefilter("always")
p = stats.genextreme.cdf(.5, 0)
p = stats.genextreme.pdf(.5, 0)
p = stats.genextreme.ppf(.5, 0)
p = stats.genextreme.logpdf(-np.inf, 0.0)
number_of_warnings_thrown = len(w)
assert_equal(number_of_warnings_thrown, 0)
def test_genextreme_entropy():
# regression test for gh-5181
euler_gamma = 0.5772156649015329
h = stats.genextreme.entropy(-1.0)
assert_allclose(h, 2*euler_gamma + 1, rtol=1e-14)
h = stats.genextreme.entropy(0)
assert_allclose(h, euler_gamma + 1, rtol=1e-14)
h = stats.genextreme.entropy(1.0)
assert_equal(h, 1)
h = stats.genextreme.entropy(-2.0, scale=10)
assert_allclose(h, euler_gamma*3 + np.log(10) + 1, rtol=1e-14)
h = stats.genextreme.entropy(10)
assert_allclose(h, -9*euler_gamma + 1, rtol=1e-14)
h = stats.genextreme.entropy(-10)
assert_allclose(h, 11*euler_gamma + 1, rtol=1e-14)
def test_genextreme_sf_isf():
# Expected values were computed using mpmath:
#
# import mpmath
#
# def mp_genextreme_sf(x, xi, mu=0, sigma=1):
# # Formula from wikipedia, which has a sign convention for xi that
# # is the opposite of scipy's shape parameter.
# if xi != 0:
# t = mpmath.power(1 + ((x - mu)/sigma)*xi, -1/xi)
# else:
# t = mpmath.exp(-(x - mu)/sigma)
# return 1 - mpmath.exp(-t)
#
# >>> mpmath.mp.dps = 1000
# >>> s = mp_genextreme_sf(mpmath.mp.mpf("1e8"), mpmath.mp.mpf("0.125"))
# >>> float(s)
# 1.6777205262585625e-57
# >>> s = mp_genextreme_sf(mpmath.mp.mpf("7.98"), mpmath.mp.mpf("-0.125"))
# >>> float(s)
# 1.52587890625e-21
# >>> s = mp_genextreme_sf(mpmath.mp.mpf("7.98"), mpmath.mp.mpf("0"))
# >>> float(s)
# 0.00034218086528426593
x = 1e8
s = stats.genextreme.sf(x, -0.125)
assert_allclose(s, 1.6777205262585625e-57)
x2 = stats.genextreme.isf(s, -0.125)
assert_allclose(x2, x)
x = 7.98
s = stats.genextreme.sf(x, 0.125)
assert_allclose(s, 1.52587890625e-21)
x2 = stats.genextreme.isf(s, 0.125)
assert_allclose(x2, x)
x = 7.98
s = stats.genextreme.sf(x, 0)
assert_allclose(s, 0.00034218086528426593)
x2 = stats.genextreme.isf(s, 0)
assert_allclose(x2, x)
def test_burr12_ppf_small_arg():
prob = 1e-16
quantile = stats.burr12.ppf(prob, 2, 3)
# The expected quantile was computed using mpmath:
# >>> import mpmath
# >>> prob = mpmath.mpf('1e-16')
# >>> c = mpmath.mpf(2)
# >>> d = mpmath.mpf(3)
# >>> float(((1-q)**(-1/d) - 1)**(1/c))
# 5.7735026918962575e-09
assert_allclose(quantile, 5.7735026918962575e-09)
def test_crystalball_function():
"""
All values are calculated using the independent implementation of the ROOT framework (see https://root.cern.ch/).
Corresponding ROOT code is given in the comments.
"""
X = np.linspace(-5.0, 5.0, 21)[:-1]
# for(float x = -5.0; x < 5.0; x+=0.5)
# std::cout << ROOT::Math::crystalball_pdf(x, 1.0, 2.0, 1.0) << ", ";
calculated = stats.crystalball.pdf(X, beta=1.0, m=2.0)
expected = np.array([0.0202867, 0.0241428, 0.0292128, 0.0360652, 0.045645,
0.059618, 0.0811467, 0.116851, 0.18258, 0.265652,
0.301023, 0.265652, 0.18258, 0.097728, 0.0407391,
0.013226, 0.00334407, 0.000658486, 0.000100982,
1.20606e-05])
assert_allclose(expected, calculated, rtol=0.001)
# for(float x = -5.0; x < 5.0; x+=0.5)
# std::cout << ROOT::Math::crystalball_pdf(x, 2.0, 3.0, 1.0) << ", ";
calculated = stats.crystalball.pdf(X, beta=2.0, m=3.0)
expected = np.array([0.0019648, 0.00279754, 0.00417592, 0.00663121,
0.0114587, 0.0223803, 0.0530497, 0.12726, 0.237752,
0.345928, 0.391987, 0.345928, 0.237752, 0.12726,
0.0530497, 0.0172227, 0.00435458, 0.000857469,
0.000131497, 1.57051e-05])
assert_allclose(expected, calculated, rtol=0.001)
# for(float x = -5.0; x < 5.0; x+=0.5)
# std::cout << ROOT::Math::crystalball_pdf(x, 2.0, 3.0, 2.0, 0.5) << ", ";
calculated = stats.crystalball.pdf(X, beta=2.0, m=3.0, loc=0.5, scale=2.0)
expected = np.array([0.00785921, 0.0111902, 0.0167037, 0.0265249,
0.0423866, 0.0636298, 0.0897324, 0.118876, 0.147944,
0.172964, 0.189964, 0.195994, 0.189964, 0.172964,
0.147944, 0.118876, 0.0897324, 0.0636298, 0.0423866,
0.0265249])
assert_allclose(expected, calculated, rtol=0.001)
# for(float x = -5.0; x < 5.0; x+=0.5)
# std::cout << ROOT::Math::crystalball_cdf(x, 1.0, 2.0, 1.0) << ", ";
calculated = stats.crystalball.cdf(X, beta=1.0, m=2.0)
expected = np.array([0.12172, 0.132785, 0.146064, 0.162293, 0.18258,
0.208663, 0.24344, 0.292128, 0.36516, 0.478254,
0.622723, 0.767192, 0.880286, 0.94959, 0.982834,
0.995314, 0.998981, 0.999824, 0.999976, 0.999997])
assert_allclose(expected, calculated, rtol=0.001)
# for(float x = -5.0; x < 5.0; x+=0.5)
# std::cout << ROOT::Math::crystalball_cdf(x, 2.0, 3.0, 1.0) << ", ";
calculated = stats.crystalball.cdf(X, beta=2.0, m=3.0)
expected = np.array([0.00442081, 0.00559509, 0.00730787, 0.00994682,
0.0143234, 0.0223803, 0.0397873, 0.0830763, 0.173323,
0.320592, 0.508717, 0.696841, 0.844111, 0.934357,
0.977646, 0.993899, 0.998674, 0.999771, 0.999969,
0.999997])
assert_allclose(expected, calculated, rtol=0.001)
# for(float x = -5.0; x < 5.0; x+=0.5)
# std::cout << ROOT::Math::crystalball_cdf(x, 2.0, 3.0, 2.0, 0.5) << ", ";
calculated = stats.crystalball.cdf(X, beta=2.0, m=3.0, loc=0.5, scale=2.0)
expected = np.array([0.0176832, 0.0223803, 0.0292315, 0.0397873, 0.0567945,
0.0830763, 0.121242, 0.173323, 0.24011, 0.320592,
0.411731, 0.508717, 0.605702, 0.696841, 0.777324,
0.844111, 0.896192, 0.934357, 0.960639, 0.977646])
assert_allclose(expected, calculated, rtol=0.001)
def test_crystalball_function_moments():
"""
All values are calculated using the pdf formula and the integrate function of Mathematica
"""
# The Last two (alpha, n) pairs test the special case n == alpha**2
beta = np.array([2.0, 1.0, 3.0, 2.0, 3.0])
m = np.array([3.0, 3.0, 2.0, 4.0, 9.0])
# The distribution should be correctly normalised
expected_0th_moment = np.array([1.0, 1.0, 1.0, 1.0, 1.0])
calculated_0th_moment = stats.crystalball._munp(0, beta, m)
assert_allclose(expected_0th_moment, calculated_0th_moment, rtol=0.001)
# calculated using wolframalpha.com
# e.g. for beta = 2 and m = 3 we calculate the norm like this:
# integrate exp(-x^2/2) from -2 to infinity + integrate (3/2)^3*exp(-2^2/2)*(3/2-2-x)^(-3) from -infinity to -2
norm = np.array([2.5511, 3.01873, 2.51065, 2.53983, 2.507410455])
expected_1th_moment = np.array([-0.21992, -3.03265, np.inf, -0.135335, -0.003174]) / norm
calculated_1th_moment = stats.crystalball._munp(1, beta, m)
assert_allclose(expected_1th_moment, calculated_1th_moment, rtol=0.001)
expected_2th_moment = np.array([np.inf, np.inf, np.inf, 3.2616, 2.519908]) / norm
calculated_2th_moment = stats.crystalball._munp(2, beta, m)
assert_allclose(expected_2th_moment, calculated_2th_moment, rtol=0.001)
expected_3th_moment = np.array([np.inf, np.inf, np.inf, np.inf, -0.0577668]) / norm
calculated_3th_moment = stats.crystalball._munp(3, beta, m)
assert_allclose(expected_3th_moment, calculated_3th_moment, rtol=0.001)
expected_4th_moment = np.array([np.inf, np.inf, np.inf, np.inf, 7.78468]) / norm
calculated_4th_moment = stats.crystalball._munp(4, beta, m)
assert_allclose(expected_4th_moment, calculated_4th_moment, rtol=0.001)
expected_5th_moment = np.array([np.inf, np.inf, np.inf, np.inf, -1.31086]) / norm
calculated_5th_moment = stats.crystalball._munp(5, beta, m)
assert_allclose(expected_5th_moment, calculated_5th_moment, rtol=0.001)
def test_argus_function():
# There is no usable reference implementation.
# (RooFit implementation returns unreasonable results which are not normalized correctly)
# Instead we do some tests if the distribution behaves as expected for different shapes and scales
for i in range(1, 10):
for j in range(1, 10):
assert_equal(stats.argus.pdf(i + 0.001, chi=j, scale=i), 0.0)
assert_(stats.argus.pdf(i - 0.001, chi=j, scale=i) > 0.0)
assert_equal(stats.argus.pdf(-0.001, chi=j, scale=i), 0.0)
assert_(stats.argus.pdf(+0.001, chi=j, scale=i) > 0.0)
for i in range(1, 10):
assert_equal(stats.argus.cdf(1.0, chi=i), 1.0)
assert_equal(stats.argus.cdf(1.0, chi=i), 1.0 - stats.argus.sf(1.0, chi=i))
class TestHistogram(object):
def setup_method(self):
np.random.seed(1234)
# We have 8 bins
# [1,2), [2,3), [3,4), [4,5), [5,6), [6,7), [7,8), [8,9)
# But actually np.histogram will put the last 9 also in the [8,9) bin!
# Therefore there is a slight difference below for the last bin, from
# what you might have expected.
histogram = np.histogram([1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5,
6, 6, 6, 6, 7, 7, 7, 8, 8, 9], bins=8)
self.template = stats.rv_histogram(histogram)
data = stats.norm.rvs(loc=1.0, scale=2.5,size=10000, random_state=123)
norm_histogram = np.histogram(data, bins=50)
self.norm_template = stats.rv_histogram(norm_histogram)
def test_pdf(self):
values = np.array([0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5,
5.0, 5.5, 6.0, 6.5, 7.0, 7.5, 8.0, 8.5, 9.0, 9.5])
pdf_values = np.asarray([0.0/25.0, 0.0/25.0, 1.0/25.0, 1.0/25.0,
2.0/25.0, 2.0/25.0, 3.0/25.0, 3.0/25.0,
4.0/25.0, 4.0/25.0, 5.0/25.0, 5.0/25.0,
4.0/25.0, 4.0/25.0, 3.0/25.0, 3.0/25.0,
3.0/25.0, 3.0/25.0, 0.0/25.0, 0.0/25.0])
assert_allclose(self.template.pdf(values), pdf_values)
# Test explicitly the corner cases:
# As stated above the pdf in the bin [8,9) is greater than
# one would naively expect because np.histogram putted the 9
# into the [8,9) bin.
assert_almost_equal(self.template.pdf(8.0), 3.0/25.0)
assert_almost_equal(self.template.pdf(8.5), 3.0/25.0)
# 9 is outside our defined bins [8,9) hence the pdf is already 0
# for a continuous distribution this is fine, because a single value
# does not have a finite probability!
assert_almost_equal(self.template.pdf(9.0), 0.0/25.0)
assert_almost_equal(self.template.pdf(10.0), 0.0/25.0)
x = np.linspace(-2, 2, 10)
assert_allclose(self.norm_template.pdf(x),
stats.norm.pdf(x, loc=1.0, scale=2.5), rtol=0.1)
def test_cdf_ppf(self):
values = np.array([0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5,
5.0, 5.5, 6.0, 6.5, 7.0, 7.5, 8.0, 8.5, 9.0, 9.5])
cdf_values = np.asarray([0.0/25.0, 0.0/25.0, 0.0/25.0, 0.5/25.0,
1.0/25.0, 2.0/25.0, 3.0/25.0, 4.5/25.0,
6.0/25.0, 8.0/25.0, 10.0/25.0, 12.5/25.0,
15.0/25.0, 17.0/25.0, 19.0/25.0, 20.5/25.0,
22.0/25.0, 23.5/25.0, 25.0/25.0, 25.0/25.0])
assert_allclose(self.template.cdf(values), cdf_values)
# First three and last two values in cdf_value are not unique
assert_allclose(self.template.ppf(cdf_values[2:-1]), values[2:-1])
# Test of cdf and ppf are inverse functions
x = np.linspace(1.0, 9.0, 100)
assert_allclose(self.template.ppf(self.template.cdf(x)), x)
x = np.linspace(0.0, 1.0, 100)
assert_allclose(self.template.cdf(self.template.ppf(x)), x)
x = np.linspace(-2, 2, 10)
assert_allclose(self.norm_template.cdf(x),
stats.norm.cdf(x, loc=1.0, scale=2.5), rtol=0.1)
def test_rvs(self):
N = 10000
sample = self.template.rvs(size=N, random_state=123)
assert_equal(np.sum(sample < 1.0), 0.0)
assert_allclose(np.sum(sample <= 2.0), 1.0/25.0 * N, rtol=0.2)
assert_allclose(np.sum(sample <= 2.5), 2.0/25.0 * N, rtol=0.2)
assert_allclose(np.sum(sample <= 3.0), 3.0/25.0 * N, rtol=0.1)
assert_allclose(np.sum(sample <= 3.5), 4.5/25.0 * N, rtol=0.1)
assert_allclose(np.sum(sample <= 4.0), 6.0/25.0 * N, rtol=0.1)
assert_allclose(np.sum(sample <= 4.5), 8.0/25.0 * N, rtol=0.1)
assert_allclose(np.sum(sample <= 5.0), 10.0/25.0 * N, rtol=0.05)
assert_allclose(np.sum(sample <= 5.5), 12.5/25.0 * N, rtol=0.05)
assert_allclose(np.sum(sample <= 6.0), 15.0/25.0 * N, rtol=0.05)
assert_allclose(np.sum(sample <= 6.5), 17.0/25.0 * N, rtol=0.05)
assert_allclose(np.sum(sample <= 7.0), 19.0/25.0 * N, rtol=0.05)
assert_allclose(np.sum(sample <= 7.5), 20.5/25.0 * N, rtol=0.05)
assert_allclose(np.sum(sample <= 8.0), 22.0/25.0 * N, rtol=0.05)
assert_allclose(np.sum(sample <= 8.5), 23.5/25.0 * N, rtol=0.05)
assert_allclose(np.sum(sample <= 9.0), 25.0/25.0 * N, rtol=0.05)
assert_allclose(np.sum(sample <= 9.0), 25.0/25.0 * N, rtol=0.05)
assert_equal(np.sum(sample > 9.0), 0.0)
def test_munp(self):
for n in range(4):
assert_allclose(self.norm_template._munp(n),
stats.norm._munp(n, 1.0, 2.5), rtol=0.05)
def test_entropy(self):
assert_allclose(self.norm_template.entropy(),
stats.norm.entropy(loc=1.0, scale=2.5), rtol=0.05)
| 129,718 | 36.75291 | 117 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/tests/test_multivariate.py
|
"""
Test functions for multivariate normal distributions.
"""
from __future__ import division, print_function, absolute_import
import pickle
from numpy.testing import (assert_allclose, assert_almost_equal,
assert_array_almost_equal, assert_equal,
assert_array_less, assert_)
from pytest import raises as assert_raises
from .test_continuous_basic import check_distribution_rvs
import numpy
import numpy as np
import scipy.linalg
from scipy.stats._multivariate import _PSD, _lnB
from scipy.stats import multivariate_normal
from scipy.stats import matrix_normal
from scipy.stats import special_ortho_group, ortho_group
from scipy.stats import random_correlation
from scipy.stats import unitary_group
from scipy.stats import dirichlet, beta
from scipy.stats import wishart, multinomial, invwishart, chi2, invgamma
from scipy.stats import norm, uniform
from scipy.stats import ks_2samp, kstest
from scipy.stats import binom
from scipy.integrate import romb
from .common_tests import check_random_state_property
class TestMultivariateNormal(object):
def test_input_shape(self):
mu = np.arange(3)
cov = np.identity(2)
assert_raises(ValueError, multivariate_normal.pdf, (0, 1), mu, cov)
assert_raises(ValueError, multivariate_normal.pdf, (0, 1, 2), mu, cov)
assert_raises(ValueError, multivariate_normal.cdf, (0, 1), mu, cov)
assert_raises(ValueError, multivariate_normal.cdf, (0, 1, 2), mu, cov)
def test_scalar_values(self):
np.random.seed(1234)
# When evaluated on scalar data, the pdf should return a scalar
x, mean, cov = 1.5, 1.7, 2.5
pdf = multivariate_normal.pdf(x, mean, cov)
assert_equal(pdf.ndim, 0)
# When evaluated on a single vector, the pdf should return a scalar
x = np.random.randn(5)
mean = np.random.randn(5)
cov = np.abs(np.random.randn(5)) # Diagonal values for cov. matrix
pdf = multivariate_normal.pdf(x, mean, cov)
assert_equal(pdf.ndim, 0)
# When evaluated on scalar data, the cdf should return a scalar
x, mean, cov = 1.5, 1.7, 2.5
cdf = multivariate_normal.cdf(x, mean, cov)
assert_equal(cdf.ndim, 0)
# When evaluated on a single vector, the cdf should return a scalar
x = np.random.randn(5)
mean = np.random.randn(5)
cov = np.abs(np.random.randn(5)) # Diagonal values for cov. matrix
cdf = multivariate_normal.cdf(x, mean, cov)
assert_equal(cdf.ndim, 0)
def test_logpdf(self):
# Check that the log of the pdf is in fact the logpdf
np.random.seed(1234)
x = np.random.randn(5)
mean = np.random.randn(5)
cov = np.abs(np.random.randn(5))
d1 = multivariate_normal.logpdf(x, mean, cov)
d2 = multivariate_normal.pdf(x, mean, cov)
assert_allclose(d1, np.log(d2))
def test_logpdf_default_values(self):
# Check that the log of the pdf is in fact the logpdf
# with default parameters Mean=None and cov = 1
np.random.seed(1234)
x = np.random.randn(5)
d1 = multivariate_normal.logpdf(x)
d2 = multivariate_normal.pdf(x)
# check whether default values are being used
d3 = multivariate_normal.logpdf(x, None, 1)
d4 = multivariate_normal.pdf(x, None, 1)
assert_allclose(d1, np.log(d2))
assert_allclose(d3, np.log(d4))
def test_logcdf(self):
# Check that the log of the cdf is in fact the logcdf
np.random.seed(1234)
x = np.random.randn(5)
mean = np.random.randn(5)
cov = np.abs(np.random.randn(5))
d1 = multivariate_normal.logcdf(x, mean, cov)
d2 = multivariate_normal.cdf(x, mean, cov)
assert_allclose(d1, np.log(d2))
def test_logcdf_default_values(self):
# Check that the log of the cdf is in fact the logcdf
# with default parameters Mean=None and cov = 1
np.random.seed(1234)
x = np.random.randn(5)
d1 = multivariate_normal.logcdf(x)
d2 = multivariate_normal.cdf(x)
# check whether default values are being used
d3 = multivariate_normal.logcdf(x, None, 1)
d4 = multivariate_normal.cdf(x, None, 1)
assert_allclose(d1, np.log(d2))
assert_allclose(d3, np.log(d4))
def test_rank(self):
# Check that the rank is detected correctly.
np.random.seed(1234)
n = 4
mean = np.random.randn(n)
for expected_rank in range(1, n + 1):
s = np.random.randn(n, expected_rank)
cov = np.dot(s, s.T)
distn = multivariate_normal(mean, cov, allow_singular=True)
assert_equal(distn.cov_info.rank, expected_rank)
def test_degenerate_distributions(self):
def _sample_orthonormal_matrix(n):
M = np.random.randn(n, n)
u, s, v = scipy.linalg.svd(M)
return u
for n in range(1, 5):
x = np.random.randn(n)
for k in range(1, n + 1):
# Sample a small covariance matrix.
s = np.random.randn(k, k)
cov_kk = np.dot(s, s.T)
# Embed the small covariance matrix into a larger low rank matrix.
cov_nn = np.zeros((n, n))
cov_nn[:k, :k] = cov_kk
# Define a rotation of the larger low rank matrix.
u = _sample_orthonormal_matrix(n)
cov_rr = np.dot(u, np.dot(cov_nn, u.T))
y = np.dot(u, x)
# Check some identities.
distn_kk = multivariate_normal(np.zeros(k), cov_kk,
allow_singular=True)
distn_nn = multivariate_normal(np.zeros(n), cov_nn,
allow_singular=True)
distn_rr = multivariate_normal(np.zeros(n), cov_rr,
allow_singular=True)
assert_equal(distn_kk.cov_info.rank, k)
assert_equal(distn_nn.cov_info.rank, k)
assert_equal(distn_rr.cov_info.rank, k)
pdf_kk = distn_kk.pdf(x[:k])
pdf_nn = distn_nn.pdf(x)
pdf_rr = distn_rr.pdf(y)
assert_allclose(pdf_kk, pdf_nn)
assert_allclose(pdf_kk, pdf_rr)
logpdf_kk = distn_kk.logpdf(x[:k])
logpdf_nn = distn_nn.logpdf(x)
logpdf_rr = distn_rr.logpdf(y)
assert_allclose(logpdf_kk, logpdf_nn)
assert_allclose(logpdf_kk, logpdf_rr)
def test_large_pseudo_determinant(self):
# Check that large pseudo-determinants are handled appropriately.
# Construct a singular diagonal covariance matrix
# whose pseudo determinant overflows double precision.
large_total_log = 1000.0
npos = 100
nzero = 2
large_entry = np.exp(large_total_log / npos)
n = npos + nzero
cov = np.zeros((n, n), dtype=float)
np.fill_diagonal(cov, large_entry)
cov[-nzero:, -nzero:] = 0
# Check some determinants.
assert_equal(scipy.linalg.det(cov), 0)
assert_equal(scipy.linalg.det(cov[:npos, :npos]), np.inf)
assert_allclose(np.linalg.slogdet(cov[:npos, :npos]),
(1, large_total_log))
# Check the pseudo-determinant.
psd = _PSD(cov)
assert_allclose(psd.log_pdet, large_total_log)
def test_broadcasting(self):
np.random.seed(1234)
n = 4
# Construct a random covariance matrix.
data = np.random.randn(n, n)
cov = np.dot(data, data.T)
mean = np.random.randn(n)
# Construct an ndarray which can be interpreted as
# a 2x3 array whose elements are random data vectors.
X = np.random.randn(2, 3, n)
# Check that multiple data points can be evaluated at once.
desired_pdf = multivariate_normal.pdf(X, mean, cov)
desired_cdf = multivariate_normal.cdf(X, mean, cov)
for i in range(2):
for j in range(3):
actual = multivariate_normal.pdf(X[i, j], mean, cov)
assert_allclose(actual, desired_pdf[i,j])
# Repeat for cdf
actual = multivariate_normal.cdf(X[i, j], mean, cov)
assert_allclose(actual, desired_cdf[i,j], rtol=1e-3)
def test_normal_1D(self):
# The probability density function for a 1D normal variable should
# agree with the standard normal distribution in scipy.stats.distributions
x = np.linspace(0, 2, 10)
mean, cov = 1.2, 0.9
scale = cov**0.5
d1 = norm.pdf(x, mean, scale)
d2 = multivariate_normal.pdf(x, mean, cov)
assert_allclose(d1, d2)
# The same should hold for the cumulative distribution function
d1 = norm.cdf(x, mean, scale)
d2 = multivariate_normal.cdf(x, mean, cov)
assert_allclose(d1, d2)
def test_marginalization(self):
# Integrating out one of the variables of a 2D Gaussian should
# yield a 1D Gaussian
mean = np.array([2.5, 3.5])
cov = np.array([[.5, 0.2], [0.2, .6]])
n = 2 ** 8 + 1 # Number of samples
delta = 6 / (n - 1) # Grid spacing
v = np.linspace(0, 6, n)
xv, yv = np.meshgrid(v, v)
pos = np.empty((n, n, 2))
pos[:, :, 0] = xv
pos[:, :, 1] = yv
pdf = multivariate_normal.pdf(pos, mean, cov)
# Marginalize over x and y axis
margin_x = romb(pdf, delta, axis=0)
margin_y = romb(pdf, delta, axis=1)
# Compare with standard normal distribution
gauss_x = norm.pdf(v, loc=mean[0], scale=cov[0, 0] ** 0.5)
gauss_y = norm.pdf(v, loc=mean[1], scale=cov[1, 1] ** 0.5)
assert_allclose(margin_x, gauss_x, rtol=1e-2, atol=1e-2)
assert_allclose(margin_y, gauss_y, rtol=1e-2, atol=1e-2)
def test_frozen(self):
# The frozen distribution should agree with the regular one
np.random.seed(1234)
x = np.random.randn(5)
mean = np.random.randn(5)
cov = np.abs(np.random.randn(5))
norm_frozen = multivariate_normal(mean, cov)
assert_allclose(norm_frozen.pdf(x), multivariate_normal.pdf(x, mean, cov))
assert_allclose(norm_frozen.logpdf(x),
multivariate_normal.logpdf(x, mean, cov))
assert_allclose(norm_frozen.cdf(x), multivariate_normal.cdf(x, mean, cov))
assert_allclose(norm_frozen.logcdf(x),
multivariate_normal.logcdf(x, mean, cov))
def test_pseudodet_pinv(self):
# Make sure that pseudo-inverse and pseudo-det agree on cutoff
# Assemble random covariance matrix with large and small eigenvalues
np.random.seed(1234)
n = 7
x = np.random.randn(n, n)
cov = np.dot(x, x.T)
s, u = scipy.linalg.eigh(cov)
s = 0.5 * np.ones(n)
s[0] = 1.0
s[-1] = 1e-7
cov = np.dot(u, np.dot(np.diag(s), u.T))
# Set cond so that the lowest eigenvalue is below the cutoff
cond = 1e-5
psd = _PSD(cov, cond=cond)
psd_pinv = _PSD(psd.pinv, cond=cond)
# Check that the log pseudo-determinant agrees with the sum
# of the logs of all but the smallest eigenvalue
assert_allclose(psd.log_pdet, np.sum(np.log(s[:-1])))
# Check that the pseudo-determinant of the pseudo-inverse
# agrees with 1 / pseudo-determinant
assert_allclose(-psd.log_pdet, psd_pinv.log_pdet)
def test_exception_nonsquare_cov(self):
cov = [[1, 2, 3], [4, 5, 6]]
assert_raises(ValueError, _PSD, cov)
def test_exception_nonfinite_cov(self):
cov_nan = [[1, 0], [0, np.nan]]
assert_raises(ValueError, _PSD, cov_nan)
cov_inf = [[1, 0], [0, np.inf]]
assert_raises(ValueError, _PSD, cov_inf)
def test_exception_non_psd_cov(self):
cov = [[1, 0], [0, -1]]
assert_raises(ValueError, _PSD, cov)
def test_exception_singular_cov(self):
np.random.seed(1234)
x = np.random.randn(5)
mean = np.random.randn(5)
cov = np.ones((5, 5))
e = np.linalg.LinAlgError
assert_raises(e, multivariate_normal, mean, cov)
assert_raises(e, multivariate_normal.pdf, x, mean, cov)
assert_raises(e, multivariate_normal.logpdf, x, mean, cov)
assert_raises(e, multivariate_normal.cdf, x, mean, cov)
assert_raises(e, multivariate_normal.logcdf, x, mean, cov)
def test_R_values(self):
# Compare the multivariate pdf with some values precomputed
# in R version 3.0.1 (2013-05-16) on Mac OS X 10.6.
# The values below were generated by the following R-script:
# > library(mnormt)
# > x <- seq(0, 2, length=5)
# > y <- 3*x - 2
# > z <- x + cos(y)
# > mu <- c(1, 3, 2)
# > Sigma <- matrix(c(1,2,0,2,5,0.5,0,0.5,3), 3, 3)
# > r_pdf <- dmnorm(cbind(x,y,z), mu, Sigma)
r_pdf = np.array([0.0002214706, 0.0013819953, 0.0049138692,
0.0103803050, 0.0140250800])
x = np.linspace(0, 2, 5)
y = 3 * x - 2
z = x + np.cos(y)
r = np.array([x, y, z]).T
mean = np.array([1, 3, 2], 'd')
cov = np.array([[1, 2, 0], [2, 5, .5], [0, .5, 3]], 'd')
pdf = multivariate_normal.pdf(r, mean, cov)
assert_allclose(pdf, r_pdf, atol=1e-10)
# Compare the multivariate cdf with some values precomputed
# in R version 3.3.2 (2016-10-31) on Debian GNU/Linux.
# The values below were generated by the following R-script:
# > library(mnormt)
# > x <- seq(0, 2, length=5)
# > y <- 3*x - 2
# > z <- x + cos(y)
# > mu <- c(1, 3, 2)
# > Sigma <- matrix(c(1,2,0,2,5,0.5,0,0.5,3), 3, 3)
# > r_cdf <- pmnorm(cbind(x,y,z), mu, Sigma)
r_cdf = np.array([0.0017866215, 0.0267142892, 0.0857098761,
0.1063242573, 0.2501068509])
cdf = multivariate_normal.cdf(r, mean, cov)
assert_allclose(cdf, r_cdf, atol=1e-5)
# Also test bivariate cdf with some values precomputed
# in R version 3.3.2 (2016-10-31) on Debian GNU/Linux.
# The values below were generated by the following R-script:
# > library(mnormt)
# > x <- seq(0, 2, length=5)
# > y <- 3*x - 2
# > mu <- c(1, 3)
# > Sigma <- matrix(c(1,2,2,5), 2, 2)
# > r_cdf2 <- pmnorm(cbind(x,y), mu, Sigma)
r_cdf2 = np.array([0.01262147, 0.05838989, 0.18389571,
0.40696599, 0.66470577])
r2 = np.array([x, y]).T
mean2 = np.array([1, 3], 'd')
cov2 = np.array([[1, 2], [2, 5]], 'd')
cdf2 = multivariate_normal.cdf(r2, mean2, cov2)
assert_allclose(cdf2, r_cdf2, atol=1e-5)
def test_multivariate_normal_rvs_zero_covariance(self):
mean = np.zeros(2)
covariance = np.zeros((2, 2))
model = multivariate_normal(mean, covariance, allow_singular=True)
sample = model.rvs()
assert_equal(sample, [0, 0])
def test_rvs_shape(self):
# Check that rvs parses the mean and covariance correctly, and returns
# an array of the right shape
N = 300
d = 4
sample = multivariate_normal.rvs(mean=np.zeros(d), cov=1, size=N)
assert_equal(sample.shape, (N, d))
sample = multivariate_normal.rvs(mean=None,
cov=np.array([[2, .1], [.1, 1]]),
size=N)
assert_equal(sample.shape, (N, 2))
u = multivariate_normal(mean=0, cov=1)
sample = u.rvs(N)
assert_equal(sample.shape, (N, ))
def test_large_sample(self):
# Generate large sample and compare sample mean and sample covariance
# with mean and covariance matrix.
np.random.seed(2846)
n = 3
mean = np.random.randn(n)
M = np.random.randn(n, n)
cov = np.dot(M, M.T)
size = 5000
sample = multivariate_normal.rvs(mean, cov, size)
assert_allclose(numpy.cov(sample.T), cov, rtol=1e-1)
assert_allclose(sample.mean(0), mean, rtol=1e-1)
def test_entropy(self):
np.random.seed(2846)
n = 3
mean = np.random.randn(n)
M = np.random.randn(n, n)
cov = np.dot(M, M.T)
rv = multivariate_normal(mean, cov)
# Check that frozen distribution agrees with entropy function
assert_almost_equal(rv.entropy(), multivariate_normal.entropy(mean, cov))
# Compare entropy with manually computed expression involving
# the sum of the logs of the eigenvalues of the covariance matrix
eigs = np.linalg.eig(cov)[0]
desired = 1 / 2 * (n * (np.log(2 * np.pi) + 1) + np.sum(np.log(eigs)))
assert_almost_equal(desired, rv.entropy())
def test_lnB(self):
alpha = np.array([1, 1, 1])
desired = .5 # e^lnB = 1/2 for [1, 1, 1]
assert_almost_equal(np.exp(_lnB(alpha)), desired)
class TestMatrixNormal(object):
def test_bad_input(self):
# Check that bad inputs raise errors
num_rows = 4
num_cols = 3
M = 0.3 * np.ones((num_rows,num_cols))
U = 0.5 * np.identity(num_rows) + 0.5 * np.ones((num_rows, num_rows))
V = 0.7 * np.identity(num_cols) + 0.3 * np.ones((num_cols, num_cols))
# Incorrect dimensions
assert_raises(ValueError, matrix_normal, np.zeros((5,4,3)))
assert_raises(ValueError, matrix_normal, M, np.zeros(10), V)
assert_raises(ValueError, matrix_normal, M, U, np.zeros(10))
assert_raises(ValueError, matrix_normal, M, U, U)
assert_raises(ValueError, matrix_normal, M, V, V)
assert_raises(ValueError, matrix_normal, M.T, U, V)
# Singular covariance
e = np.linalg.LinAlgError
assert_raises(e, matrix_normal, M, U, np.ones((num_cols, num_cols)))
assert_raises(e, matrix_normal, M, np.ones((num_rows, num_rows)), V)
def test_default_inputs(self):
# Check that default argument handling works
num_rows = 4
num_cols = 3
M = 0.3 * np.ones((num_rows,num_cols))
U = 0.5 * np.identity(num_rows) + 0.5 * np.ones((num_rows, num_rows))
V = 0.7 * np.identity(num_cols) + 0.3 * np.ones((num_cols, num_cols))
Z = np.zeros((num_rows, num_cols))
Zr = np.zeros((num_rows, 1))
Zc = np.zeros((1, num_cols))
Ir = np.identity(num_rows)
Ic = np.identity(num_cols)
I1 = np.identity(1)
assert_equal(matrix_normal.rvs(mean=M, rowcov=U, colcov=V).shape,
(num_rows, num_cols))
assert_equal(matrix_normal.rvs(mean=M).shape,
(num_rows, num_cols))
assert_equal(matrix_normal.rvs(rowcov=U).shape,
(num_rows, 1))
assert_equal(matrix_normal.rvs(colcov=V).shape,
(1, num_cols))
assert_equal(matrix_normal.rvs(mean=M, colcov=V).shape,
(num_rows, num_cols))
assert_equal(matrix_normal.rvs(mean=M, rowcov=U).shape,
(num_rows, num_cols))
assert_equal(matrix_normal.rvs(rowcov=U, colcov=V).shape,
(num_rows, num_cols))
assert_equal(matrix_normal(mean=M).rowcov, Ir)
assert_equal(matrix_normal(mean=M).colcov, Ic)
assert_equal(matrix_normal(rowcov=U).mean, Zr)
assert_equal(matrix_normal(rowcov=U).colcov, I1)
assert_equal(matrix_normal(colcov=V).mean, Zc)
assert_equal(matrix_normal(colcov=V).rowcov, I1)
assert_equal(matrix_normal(mean=M, rowcov=U).colcov, Ic)
assert_equal(matrix_normal(mean=M, colcov=V).rowcov, Ir)
assert_equal(matrix_normal(rowcov=U, colcov=V).mean, Z)
def test_covariance_expansion(self):
# Check that covariance can be specified with scalar or vector
num_rows = 4
num_cols = 3
M = 0.3 * np.ones((num_rows,num_cols))
Uv = 0.2*np.ones(num_rows)
Us = 0.2
Vv = 0.1*np.ones(num_cols)
Vs = 0.1
Ir = np.identity(num_rows)
Ic = np.identity(num_cols)
assert_equal(matrix_normal(mean=M, rowcov=Uv, colcov=Vv).rowcov,
0.2*Ir)
assert_equal(matrix_normal(mean=M, rowcov=Uv, colcov=Vv).colcov,
0.1*Ic)
assert_equal(matrix_normal(mean=M, rowcov=Us, colcov=Vs).rowcov,
0.2*Ir)
assert_equal(matrix_normal(mean=M, rowcov=Us, colcov=Vs).colcov,
0.1*Ic)
def test_frozen_matrix_normal(self):
for i in range(1,5):
for j in range(1,5):
M = 0.3 * np.ones((i,j))
U = 0.5 * np.identity(i) + 0.5 * np.ones((i,i))
V = 0.7 * np.identity(j) + 0.3 * np.ones((j,j))
frozen = matrix_normal(mean=M, rowcov=U, colcov=V)
rvs1 = frozen.rvs(random_state=1234)
rvs2 = matrix_normal.rvs(mean=M, rowcov=U, colcov=V,
random_state=1234)
assert_equal(rvs1, rvs2)
X = frozen.rvs(random_state=1234)
pdf1 = frozen.pdf(X)
pdf2 = matrix_normal.pdf(X, mean=M, rowcov=U, colcov=V)
assert_equal(pdf1, pdf2)
logpdf1 = frozen.logpdf(X)
logpdf2 = matrix_normal.logpdf(X, mean=M, rowcov=U, colcov=V)
assert_equal(logpdf1, logpdf2)
def test_matches_multivariate(self):
# Check that the pdfs match those obtained by vectorising and
# treating as a multivariate normal.
for i in range(1,5):
for j in range(1,5):
M = 0.3 * np.ones((i,j))
U = 0.5 * np.identity(i) + 0.5 * np.ones((i,i))
V = 0.7 * np.identity(j) + 0.3 * np.ones((j,j))
frozen = matrix_normal(mean=M, rowcov=U, colcov=V)
X = frozen.rvs(random_state=1234)
pdf1 = frozen.pdf(X)
logpdf1 = frozen.logpdf(X)
vecX = X.T.flatten()
vecM = M.T.flatten()
cov = np.kron(V,U)
pdf2 = multivariate_normal.pdf(vecX, mean=vecM, cov=cov)
logpdf2 = multivariate_normal.logpdf(vecX, mean=vecM, cov=cov)
assert_allclose(pdf1, pdf2, rtol=1E-10)
assert_allclose(logpdf1, logpdf2, rtol=1E-10)
def test_array_input(self):
# Check array of inputs has the same output as the separate entries.
num_rows = 4
num_cols = 3
M = 0.3 * np.ones((num_rows,num_cols))
U = 0.5 * np.identity(num_rows) + 0.5 * np.ones((num_rows, num_rows))
V = 0.7 * np.identity(num_cols) + 0.3 * np.ones((num_cols, num_cols))
N = 10
frozen = matrix_normal(mean=M, rowcov=U, colcov=V)
X1 = frozen.rvs(size=N, random_state=1234)
X2 = frozen.rvs(size=N, random_state=4321)
X = np.concatenate((X1[np.newaxis,:,:,:],X2[np.newaxis,:,:,:]), axis=0)
assert_equal(X.shape, (2, N, num_rows, num_cols))
array_logpdf = frozen.logpdf(X)
assert_equal(array_logpdf.shape, (2, N))
for i in range(2):
for j in range(N):
separate_logpdf = matrix_normal.logpdf(X[i,j], mean=M,
rowcov=U, colcov=V)
assert_allclose(separate_logpdf, array_logpdf[i,j], 1E-10)
def test_moments(self):
# Check that the sample moments match the parameters
num_rows = 4
num_cols = 3
M = 0.3 * np.ones((num_rows,num_cols))
U = 0.5 * np.identity(num_rows) + 0.5 * np.ones((num_rows, num_rows))
V = 0.7 * np.identity(num_cols) + 0.3 * np.ones((num_cols, num_cols))
N = 1000
frozen = matrix_normal(mean=M, rowcov=U, colcov=V)
X = frozen.rvs(size=N, random_state=1234)
sample_mean = np.mean(X,axis=0)
assert_allclose(sample_mean, M, atol=0.1)
sample_colcov = np.cov(X.reshape(N*num_rows,num_cols).T)
assert_allclose(sample_colcov, V, atol=0.1)
sample_rowcov = np.cov(np.swapaxes(X,1,2).reshape(
N*num_cols,num_rows).T)
assert_allclose(sample_rowcov, U, atol=0.1)
class TestDirichlet(object):
def test_frozen_dirichlet(self):
np.random.seed(2846)
n = np.random.randint(1, 32)
alpha = np.random.uniform(10e-10, 100, n)
d = dirichlet(alpha)
assert_equal(d.var(), dirichlet.var(alpha))
assert_equal(d.mean(), dirichlet.mean(alpha))
assert_equal(d.entropy(), dirichlet.entropy(alpha))
num_tests = 10
for i in range(num_tests):
x = np.random.uniform(10e-10, 100, n)
x /= np.sum(x)
assert_equal(d.pdf(x[:-1]), dirichlet.pdf(x[:-1], alpha))
assert_equal(d.logpdf(x[:-1]), dirichlet.logpdf(x[:-1], alpha))
def test_numpy_rvs_shape_compatibility(self):
np.random.seed(2846)
alpha = np.array([1.0, 2.0, 3.0])
x = np.random.dirichlet(alpha, size=7)
assert_equal(x.shape, (7, 3))
assert_raises(ValueError, dirichlet.pdf, x, alpha)
assert_raises(ValueError, dirichlet.logpdf, x, alpha)
dirichlet.pdf(x.T, alpha)
dirichlet.pdf(x.T[:-1], alpha)
dirichlet.logpdf(x.T, alpha)
dirichlet.logpdf(x.T[:-1], alpha)
def test_alpha_with_zeros(self):
np.random.seed(2846)
alpha = [1.0, 0.0, 3.0]
# don't pass invalid alpha to np.random.dirichlet
x = np.random.dirichlet(np.maximum(1e-9, alpha), size=7).T
assert_raises(ValueError, dirichlet.pdf, x, alpha)
assert_raises(ValueError, dirichlet.logpdf, x, alpha)
def test_alpha_with_negative_entries(self):
np.random.seed(2846)
alpha = [1.0, -2.0, 3.0]
# don't pass invalid alpha to np.random.dirichlet
x = np.random.dirichlet(np.maximum(1e-9, alpha), size=7).T
assert_raises(ValueError, dirichlet.pdf, x, alpha)
assert_raises(ValueError, dirichlet.logpdf, x, alpha)
def test_data_with_zeros(self):
alpha = np.array([1.0, 2.0, 3.0, 4.0])
x = np.array([0.1, 0.0, 0.2, 0.7])
dirichlet.pdf(x, alpha)
dirichlet.logpdf(x, alpha)
alpha = np.array([1.0, 1.0, 1.0, 1.0])
assert_almost_equal(dirichlet.pdf(x, alpha), 6)
assert_almost_equal(dirichlet.logpdf(x, alpha), np.log(6))
def test_data_with_zeros_and_small_alpha(self):
alpha = np.array([1.0, 0.5, 3.0, 4.0])
x = np.array([0.1, 0.0, 0.2, 0.7])
assert_raises(ValueError, dirichlet.pdf, x, alpha)
assert_raises(ValueError, dirichlet.logpdf, x, alpha)
def test_data_with_negative_entries(self):
alpha = np.array([1.0, 2.0, 3.0, 4.0])
x = np.array([0.1, -0.1, 0.3, 0.7])
assert_raises(ValueError, dirichlet.pdf, x, alpha)
assert_raises(ValueError, dirichlet.logpdf, x, alpha)
def test_data_with_too_large_entries(self):
alpha = np.array([1.0, 2.0, 3.0, 4.0])
x = np.array([0.1, 1.1, 0.3, 0.7])
assert_raises(ValueError, dirichlet.pdf, x, alpha)
assert_raises(ValueError, dirichlet.logpdf, x, alpha)
def test_data_too_deep_c(self):
alpha = np.array([1.0, 2.0, 3.0])
x = np.ones((2, 7, 7)) / 14
assert_raises(ValueError, dirichlet.pdf, x, alpha)
assert_raises(ValueError, dirichlet.logpdf, x, alpha)
def test_alpha_too_deep(self):
alpha = np.array([[1.0, 2.0], [3.0, 4.0]])
x = np.ones((2, 2, 7)) / 4
assert_raises(ValueError, dirichlet.pdf, x, alpha)
assert_raises(ValueError, dirichlet.logpdf, x, alpha)
def test_alpha_correct_depth(self):
alpha = np.array([1.0, 2.0, 3.0])
x = np.ones((3, 7)) / 3
dirichlet.pdf(x, alpha)
dirichlet.logpdf(x, alpha)
def test_non_simplex_data(self):
alpha = np.array([1.0, 2.0, 3.0])
x = np.ones((3, 7)) / 2
assert_raises(ValueError, dirichlet.pdf, x, alpha)
assert_raises(ValueError, dirichlet.logpdf, x, alpha)
def test_data_vector_too_short(self):
alpha = np.array([1.0, 2.0, 3.0, 4.0])
x = np.ones((2, 7)) / 2
assert_raises(ValueError, dirichlet.pdf, x, alpha)
assert_raises(ValueError, dirichlet.logpdf, x, alpha)
def test_data_vector_too_long(self):
alpha = np.array([1.0, 2.0, 3.0, 4.0])
x = np.ones((5, 7)) / 5
assert_raises(ValueError, dirichlet.pdf, x, alpha)
assert_raises(ValueError, dirichlet.logpdf, x, alpha)
def test_mean_and_var(self):
alpha = np.array([1., 0.8, 0.2])
d = dirichlet(alpha)
expected_var = [1. / 12., 0.08, 0.03]
expected_mean = [0.5, 0.4, 0.1]
assert_array_almost_equal(d.var(), expected_var)
assert_array_almost_equal(d.mean(), expected_mean)
def test_scalar_values(self):
alpha = np.array([0.2])
d = dirichlet(alpha)
# For alpha of length 1, mean and var should be scalar instead of array
assert_equal(d.mean().ndim, 0)
assert_equal(d.var().ndim, 0)
assert_equal(d.pdf([1.]).ndim, 0)
assert_equal(d.logpdf([1.]).ndim, 0)
def test_K_and_K_minus_1_calls_equal(self):
# Test that calls with K and K-1 entries yield the same results.
np.random.seed(2846)
n = np.random.randint(1, 32)
alpha = np.random.uniform(10e-10, 100, n)
d = dirichlet(alpha)
num_tests = 10
for i in range(num_tests):
x = np.random.uniform(10e-10, 100, n)
x /= np.sum(x)
assert_almost_equal(d.pdf(x[:-1]), d.pdf(x))
def test_multiple_entry_calls(self):
# Test that calls with multiple x vectors as matrix work
np.random.seed(2846)
n = np.random.randint(1, 32)
alpha = np.random.uniform(10e-10, 100, n)
d = dirichlet(alpha)
num_tests = 10
num_multiple = 5
xm = None
for i in range(num_tests):
for m in range(num_multiple):
x = np.random.uniform(10e-10, 100, n)
x /= np.sum(x)
if xm is not None:
xm = np.vstack((xm, x))
else:
xm = x
rm = d.pdf(xm.T)
rs = None
for xs in xm:
r = d.pdf(xs)
if rs is not None:
rs = np.append(rs, r)
else:
rs = r
assert_array_almost_equal(rm, rs)
def test_2D_dirichlet_is_beta(self):
np.random.seed(2846)
alpha = np.random.uniform(10e-10, 100, 2)
d = dirichlet(alpha)
b = beta(alpha[0], alpha[1])
num_tests = 10
for i in range(num_tests):
x = np.random.uniform(10e-10, 100, 2)
x /= np.sum(x)
assert_almost_equal(b.pdf(x), d.pdf([x]))
assert_almost_equal(b.mean(), d.mean()[0])
assert_almost_equal(b.var(), d.var()[0])
def test_multivariate_normal_dimensions_mismatch():
# Regression test for GH #3493. Check that setting up a PDF with a mean of
# length M and a covariance matrix of size (N, N), where M != N, raises a
# ValueError with an informative error message.
mu = np.array([0.0, 0.0])
sigma = np.array([[1.0]])
assert_raises(ValueError, multivariate_normal, mu, sigma)
# A simple check that the right error message was passed along. Checking
# that the entire message is there, word for word, would be somewhat
# fragile, so we just check for the leading part.
try:
multivariate_normal(mu, sigma)
except ValueError as e:
msg = "Dimension mismatch"
assert_equal(str(e)[:len(msg)], msg)
class TestWishart(object):
def test_scale_dimensions(self):
# Test that we can call the Wishart with various scale dimensions
# Test case: dim=1, scale=1
true_scale = np.array(1, ndmin=2)
scales = [
1, # scalar
[1], # iterable
np.array(1), # 0-dim
np.r_[1], # 1-dim
np.array(1, ndmin=2) # 2-dim
]
for scale in scales:
w = wishart(1, scale)
assert_equal(w.scale, true_scale)
assert_equal(w.scale.shape, true_scale.shape)
# Test case: dim=2, scale=[[1,0]
# [0,2]
true_scale = np.array([[1,0],
[0,2]])
scales = [
[1,2], # iterable
np.r_[1,2], # 1-dim
np.array([[1,0], # 2-dim
[0,2]])
]
for scale in scales:
w = wishart(2, scale)
assert_equal(w.scale, true_scale)
assert_equal(w.scale.shape, true_scale.shape)
# We cannot call with a df < dim
assert_raises(ValueError, wishart, 1, np.eye(2))
# We cannot call with a 3-dimension array
scale = np.array(1, ndmin=3)
assert_raises(ValueError, wishart, 1, scale)
def test_quantile_dimensions(self):
# Test that we can call the Wishart rvs with various quantile dimensions
# If dim == 1, consider x.shape = [1,1,1]
X = [
1, # scalar
[1], # iterable
np.array(1), # 0-dim
np.r_[1], # 1-dim
np.array(1, ndmin=2), # 2-dim
np.array([1], ndmin=3) # 3-dim
]
w = wishart(1,1)
density = w.pdf(np.array(1, ndmin=3))
for x in X:
assert_equal(w.pdf(x), density)
# If dim == 1, consider x.shape = [1,1,*]
X = [
[1,2,3], # iterable
np.r_[1,2,3], # 1-dim
np.array([1,2,3], ndmin=3) # 3-dim
]
w = wishart(1,1)
density = w.pdf(np.array([1,2,3], ndmin=3))
for x in X:
assert_equal(w.pdf(x), density)
# If dim == 2, consider x.shape = [2,2,1]
# where x[:,:,*] = np.eye(1)*2
X = [
2, # scalar
[2,2], # iterable
np.array(2), # 0-dim
np.r_[2,2], # 1-dim
np.array([[2,0],
[0,2]]), # 2-dim
np.array([[2,0],
[0,2]])[:,:,np.newaxis] # 3-dim
]
w = wishart(2,np.eye(2))
density = w.pdf(np.array([[2,0],
[0,2]])[:,:,np.newaxis])
for x in X:
assert_equal(w.pdf(x), density)
def test_frozen(self):
# Test that the frozen and non-frozen Wishart gives the same answers
# Construct an arbitrary positive definite scale matrix
dim = 4
scale = np.diag(np.arange(dim)+1)
scale[np.tril_indices(dim, k=-1)] = np.arange(dim * (dim-1) // 2)
scale = np.dot(scale.T, scale)
# Construct a collection of positive definite matrices to test the PDF
X = []
for i in range(5):
x = np.diag(np.arange(dim)+(i+1)**2)
x[np.tril_indices(dim, k=-1)] = np.arange(dim * (dim-1) // 2)
x = np.dot(x.T, x)
X.append(x)
X = np.array(X).T
# Construct a 1D and 2D set of parameters
parameters = [
(10, 1, np.linspace(0.1, 10, 5)), # 1D case
(10, scale, X)
]
for (df, scale, x) in parameters:
w = wishart(df, scale)
assert_equal(w.var(), wishart.var(df, scale))
assert_equal(w.mean(), wishart.mean(df, scale))
assert_equal(w.mode(), wishart.mode(df, scale))
assert_equal(w.entropy(), wishart.entropy(df, scale))
assert_equal(w.pdf(x), wishart.pdf(x, df, scale))
def test_1D_is_chisquared(self):
# The 1-dimensional Wishart with an identity scale matrix is just a
# chi-squared distribution.
# Test variance, mean, entropy, pdf
# Kolgomorov-Smirnov test for rvs
np.random.seed(482974)
sn = 500
dim = 1
scale = np.eye(dim)
df_range = np.arange(1, 10, 2, dtype=float)
X = np.linspace(0.1,10,num=10)
for df in df_range:
w = wishart(df, scale)
c = chi2(df)
# Statistics
assert_allclose(w.var(), c.var())
assert_allclose(w.mean(), c.mean())
assert_allclose(w.entropy(), c.entropy())
# PDF
assert_allclose(w.pdf(X), c.pdf(X))
# rvs
rvs = w.rvs(size=sn)
args = (df,)
alpha = 0.01
check_distribution_rvs('chi2', args, alpha, rvs)
def test_is_scaled_chisquared(self):
# The 2-dimensional Wishart with an arbitrary scale matrix can be
# transformed to a scaled chi-squared distribution.
# For :math:`S \sim W_p(V,n)` and :math:`\lambda \in \mathbb{R}^p` we have
# :math:`\lambda' S \lambda \sim \lambda' V \lambda \times \chi^2(n)`
np.random.seed(482974)
sn = 500
df = 10
dim = 4
# Construct an arbitrary positive definite matrix
scale = np.diag(np.arange(4)+1)
scale[np.tril_indices(4, k=-1)] = np.arange(6)
scale = np.dot(scale.T, scale)
# Use :math:`\lambda = [1, \dots, 1]'`
lamda = np.ones((dim,1))
sigma_lamda = lamda.T.dot(scale).dot(lamda).squeeze()
w = wishart(df, sigma_lamda)
c = chi2(df, scale=sigma_lamda)
# Statistics
assert_allclose(w.var(), c.var())
assert_allclose(w.mean(), c.mean())
assert_allclose(w.entropy(), c.entropy())
# PDF
X = np.linspace(0.1,10,num=10)
assert_allclose(w.pdf(X), c.pdf(X))
# rvs
rvs = w.rvs(size=sn)
args = (df,0,sigma_lamda)
alpha = 0.01
check_distribution_rvs('chi2', args, alpha, rvs)
class TestMultinomial(object):
def test_logpmf(self):
vals1 = multinomial.logpmf((3,4), 7, (0.3, 0.7))
assert_allclose(vals1, -1.483270127243324, rtol=1e-8)
vals2 = multinomial.logpmf([3, 4], 0, [.3, .7])
assert_allclose(vals2, np.NAN, rtol=1e-8)
vals3 = multinomial.logpmf([3, 4], 0, [-2, 3])
assert_allclose(vals3, np.NAN, rtol=1e-8)
def test_reduces_binomial(self):
# test that the multinomial pmf reduces to the binomial pmf in the 2d
# case
val1 = multinomial.logpmf((3, 4), 7, (0.3, 0.7))
val2 = binom.logpmf(3, 7, 0.3)
assert_allclose(val1, val2, rtol=1e-8)
val1 = multinomial.pmf((6, 8), 14, (0.1, 0.9))
val2 = binom.pmf(6, 14, 0.1)
assert_allclose(val1, val2, rtol=1e-8)
def test_R(self):
# test against the values produced by this R code
# (https://stat.ethz.ch/R-manual/R-devel/library/stats/html/Multinom.html)
# X <- t(as.matrix(expand.grid(0:3, 0:3))); X <- X[, colSums(X) <= 3]
# X <- rbind(X, 3:3 - colSums(X)); dimnames(X) <- list(letters[1:3], NULL)
# X
# apply(X, 2, function(x) dmultinom(x, prob = c(1,2,5)))
n, p = 3, [1./8, 2./8, 5./8]
r_vals = {(0, 0, 3): 0.244140625, (1, 0, 2): 0.146484375,
(2, 0, 1): 0.029296875, (3, 0, 0): 0.001953125,
(0, 1, 2): 0.292968750, (1, 1, 1): 0.117187500,
(2, 1, 0): 0.011718750, (0, 2, 1): 0.117187500,
(1, 2, 0): 0.023437500, (0, 3, 0): 0.015625000}
for x in r_vals:
assert_allclose(multinomial.pmf(x, n, p), r_vals[x], atol=1e-14)
def test_rvs_np(self):
# test that .rvs agrees w/numpy
sc_rvs = multinomial.rvs(3, [1/4.]*3, size=7, random_state=123)
rndm = np.random.RandomState(123)
np_rvs = rndm.multinomial(3, [1/4.]*3, size=7)
assert_equal(sc_rvs, np_rvs)
def test_pmf(self):
vals0 = multinomial.pmf((5,), 5, (1,))
assert_allclose(vals0, 1, rtol=1e-8)
vals1 = multinomial.pmf((3,4), 7, (.3, .7))
assert_allclose(vals1, .22689449999999994, rtol=1e-8)
vals2 = multinomial.pmf([[[3,5],[0,8]], [[-1, 9], [1, 1]]], 8,
(.1, .9))
assert_allclose(vals2, [[.03306744, .43046721], [0, 0]], rtol=1e-8)
x = np.empty((0,2), dtype=np.float64)
vals3 = multinomial.pmf(x, 4, (.3, .7))
assert_equal(vals3, np.empty([], dtype=np.float64))
vals4 = multinomial.pmf([1,2], 4, (.3, .7))
assert_allclose(vals4, 0, rtol=1e-8)
vals5 = multinomial.pmf([3, 3, 0], 6, [2/3.0, 1/3.0, 0])
assert_allclose(vals5, 0.219478737997, rtol=1e-8)
def test_pmf_broadcasting(self):
vals0 = multinomial.pmf([1, 2], 3, [[.1, .9], [.2, .8]])
assert_allclose(vals0, [.243, .384], rtol=1e-8)
vals1 = multinomial.pmf([1, 2], [3, 4], [.1, .9])
assert_allclose(vals1, [.243, 0], rtol=1e-8)
vals2 = multinomial.pmf([[[1, 2], [1, 1]]], 3, [.1, .9])
assert_allclose(vals2, [[.243, 0]], rtol=1e-8)
vals3 = multinomial.pmf([1, 2], [[[3], [4]]], [.1, .9])
assert_allclose(vals3, [[[.243], [0]]], rtol=1e-8)
vals4 = multinomial.pmf([[1, 2], [1,1]], [[[[3]]]], [.1, .9])
assert_allclose(vals4, [[[[.243, 0]]]], rtol=1e-8)
def test_cov(self):
cov1 = multinomial.cov(5, (.2, .3, .5))
cov2 = [[5*.2*.8, -5*.2*.3, -5*.2*.5],
[-5*.3*.2, 5*.3*.7, -5*.3*.5],
[-5*.5*.2, -5*.5*.3, 5*.5*.5]]
assert_allclose(cov1, cov2, rtol=1e-8)
def test_cov_broadcasting(self):
cov1 = multinomial.cov(5, [[.1, .9], [.2, .8]])
cov2 = [[[.45, -.45],[-.45, .45]], [[.8, -.8], [-.8, .8]]]
assert_allclose(cov1, cov2, rtol=1e-8)
cov3 = multinomial.cov([4, 5], [.1, .9])
cov4 = [[[.36, -.36], [-.36, .36]], [[.45, -.45], [-.45, .45]]]
assert_allclose(cov3, cov4, rtol=1e-8)
cov5 = multinomial.cov([4, 5], [[.3, .7], [.4, .6]])
cov6 = [[[4*.3*.7, -4*.3*.7], [-4*.3*.7, 4*.3*.7]],
[[5*.4*.6, -5*.4*.6], [-5*.4*.6, 5*.4*.6]]]
assert_allclose(cov5, cov6, rtol=1e-8)
def test_entropy(self):
# this is equivalent to a binomial distribution with n=2, so the
# entropy .77899774929 is easily computed "by hand"
ent0 = multinomial.entropy(2, [.2, .8])
assert_allclose(ent0, binom.entropy(2, .2), rtol=1e-8)
def test_entropy_broadcasting(self):
ent0 = multinomial.entropy([2, 3], [.2, .3])
assert_allclose(ent0, [binom.entropy(2, .2), binom.entropy(3, .2)],
rtol=1e-8)
ent1 = multinomial.entropy([7, 8], [[.3, .7], [.4, .6]])
assert_allclose(ent1, [binom.entropy(7, .3), binom.entropy(8, .4)],
rtol=1e-8)
ent2 = multinomial.entropy([[7], [8]], [[.3, .7], [.4, .6]])
assert_allclose(ent2,
[[binom.entropy(7, .3), binom.entropy(7, .4)],
[binom.entropy(8, .3), binom.entropy(8, .4)]],
rtol=1e-8)
def test_mean(self):
mean1 = multinomial.mean(5, [.2, .8])
assert_allclose(mean1, [5*.2, 5*.8], rtol=1e-8)
def test_mean_broadcasting(self):
mean1 = multinomial.mean([5, 6], [.2, .8])
assert_allclose(mean1, [[5*.2, 5*.8], [6*.2, 6*.8]], rtol=1e-8)
def test_frozen(self):
# The frozen distribution should agree with the regular one
np.random.seed(1234)
n = 12
pvals = (.1, .2, .3, .4)
x = [[0,0,0,12],[0,0,1,11],[0,1,1,10],[1,1,1,9],[1,1,2,8]]
x = np.asarray(x, dtype=np.float64)
mn_frozen = multinomial(n, pvals)
assert_allclose(mn_frozen.pmf(x), multinomial.pmf(x, n, pvals))
assert_allclose(mn_frozen.logpmf(x), multinomial.logpmf(x, n, pvals))
assert_allclose(mn_frozen.entropy(), multinomial.entropy(n, pvals))
class TestInvwishart(object):
def test_frozen(self):
# Test that the frozen and non-frozen inverse Wishart gives the same
# answers
# Construct an arbitrary positive definite scale matrix
dim = 4
scale = np.diag(np.arange(dim)+1)
scale[np.tril_indices(dim, k=-1)] = np.arange(dim*(dim-1)/2)
scale = np.dot(scale.T, scale)
# Construct a collection of positive definite matrices to test the PDF
X = []
for i in range(5):
x = np.diag(np.arange(dim)+(i+1)**2)
x[np.tril_indices(dim, k=-1)] = np.arange(dim*(dim-1)/2)
x = np.dot(x.T, x)
X.append(x)
X = np.array(X).T
# Construct a 1D and 2D set of parameters
parameters = [
(10, 1, np.linspace(0.1, 10, 5)), # 1D case
(10, scale, X)
]
for (df, scale, x) in parameters:
iw = invwishart(df, scale)
assert_equal(iw.var(), invwishart.var(df, scale))
assert_equal(iw.mean(), invwishart.mean(df, scale))
assert_equal(iw.mode(), invwishart.mode(df, scale))
assert_allclose(iw.pdf(x), invwishart.pdf(x, df, scale))
def test_1D_is_invgamma(self):
# The 1-dimensional inverse Wishart with an identity scale matrix is
# just an inverse gamma distribution.
# Test variance, mean, pdf
# Kolgomorov-Smirnov test for rvs
np.random.seed(482974)
sn = 500
dim = 1
scale = np.eye(dim)
df_range = np.arange(5, 20, 2, dtype=float)
X = np.linspace(0.1,10,num=10)
for df in df_range:
iw = invwishart(df, scale)
ig = invgamma(df/2, scale=1./2)
# Statistics
assert_allclose(iw.var(), ig.var())
assert_allclose(iw.mean(), ig.mean())
# PDF
assert_allclose(iw.pdf(X), ig.pdf(X))
# rvs
rvs = iw.rvs(size=sn)
args = (df/2, 0, 1./2)
alpha = 0.01
check_distribution_rvs('invgamma', args, alpha, rvs)
def test_wishart_invwishart_2D_rvs(self):
dim = 3
df = 10
# Construct a simple non-diagonal positive definite matrix
scale = np.eye(dim)
scale[0,1] = 0.5
scale[1,0] = 0.5
# Construct frozen Wishart and inverse Wishart random variables
w = wishart(df, scale)
iw = invwishart(df, scale)
# Get the generated random variables from a known seed
np.random.seed(248042)
w_rvs = wishart.rvs(df, scale)
np.random.seed(248042)
frozen_w_rvs = w.rvs()
np.random.seed(248042)
iw_rvs = invwishart.rvs(df, scale)
np.random.seed(248042)
frozen_iw_rvs = iw.rvs()
# Manually calculate what it should be, based on the Bartlett (1933)
# decomposition of a Wishart into D A A' D', where D is the Cholesky
# factorization of the scale matrix and A is the lower triangular matrix
# with the square root of chi^2 variates on the diagonal and N(0,1)
# variates in the lower triangle.
np.random.seed(248042)
covariances = np.random.normal(size=3)
variances = np.r_[
np.random.chisquare(df),
np.random.chisquare(df-1),
np.random.chisquare(df-2),
]**0.5
# Construct the lower-triangular A matrix
A = np.diag(variances)
A[np.tril_indices(dim, k=-1)] = covariances
# Wishart random variate
D = np.linalg.cholesky(scale)
DA = D.dot(A)
manual_w_rvs = np.dot(DA, DA.T)
# inverse Wishart random variate
# Supposing that the inverse wishart has scale matrix `scale`, then the
# random variate is the inverse of a random variate drawn from a Wishart
# distribution with scale matrix `inv_scale = np.linalg.inv(scale)`
iD = np.linalg.cholesky(np.linalg.inv(scale))
iDA = iD.dot(A)
manual_iw_rvs = np.linalg.inv(np.dot(iDA, iDA.T))
# Test for equality
assert_allclose(w_rvs, manual_w_rvs)
assert_allclose(frozen_w_rvs, manual_w_rvs)
assert_allclose(iw_rvs, manual_iw_rvs)
assert_allclose(frozen_iw_rvs, manual_iw_rvs)
class TestSpecialOrthoGroup(object):
def test_reproducibility(self):
np.random.seed(514)
x = special_ortho_group.rvs(3)
expected = np.array([[-0.99394515, -0.04527879, 0.10011432],
[0.04821555, -0.99846897, 0.02711042],
[0.09873351, 0.03177334, 0.99460653]])
assert_array_almost_equal(x, expected)
random_state = np.random.RandomState(seed=514)
x = special_ortho_group.rvs(3, random_state=random_state)
assert_array_almost_equal(x, expected)
def test_invalid_dim(self):
assert_raises(ValueError, special_ortho_group.rvs, None)
assert_raises(ValueError, special_ortho_group.rvs, (2, 2))
assert_raises(ValueError, special_ortho_group.rvs, 1)
assert_raises(ValueError, special_ortho_group.rvs, 2.5)
def test_frozen_matrix(self):
dim = 7
frozen = special_ortho_group(dim)
rvs1 = frozen.rvs(random_state=1234)
rvs2 = special_ortho_group.rvs(dim, random_state=1234)
assert_equal(rvs1, rvs2)
def test_det_and_ortho(self):
xs = [special_ortho_group.rvs(dim)
for dim in range(2,12)
for i in range(3)]
# Test that determinants are always +1
dets = [np.linalg.det(x) for x in xs]
assert_allclose(dets, [1.]*30, rtol=1e-13)
# Test that these are orthogonal matrices
for x in xs:
assert_array_almost_equal(np.dot(x, x.T),
np.eye(x.shape[0]))
def test_haar(self):
# Test that the distribution is constant under rotation
# Every column should have the same distribution
# Additionally, the distribution should be invariant under another rotation
# Generate samples
dim = 5
samples = 1000 # Not too many, or the test takes too long
ks_prob = .05
np.random.seed(514)
xs = special_ortho_group.rvs(dim, size=samples)
# Dot a few rows (0, 1, 2) with unit vectors (0, 2, 4, 3),
# effectively picking off entries in the matrices of xs.
# These projections should all have the same disribution,
# establishing rotational invariance. We use the two-sided
# KS test to confirm this.
# We could instead test that angles between random vectors
# are uniformly distributed, but the below is sufficient.
# It is not feasible to consider all pairs, so pick a few.
els = ((0,0), (0,2), (1,4), (2,3))
#proj = {(er, ec): [x[er][ec] for x in xs] for er, ec in els}
proj = dict(((er, ec), sorted([x[er][ec] for x in xs])) for er, ec in els)
pairs = [(e0, e1) for e0 in els for e1 in els if e0 > e1]
ks_tests = [ks_2samp(proj[p0], proj[p1])[1] for (p0, p1) in pairs]
assert_array_less([ks_prob]*len(pairs), ks_tests)
class TestOrthoGroup(object):
def test_reproducibility(self):
np.random.seed(515)
x = ortho_group.rvs(3)
x2 = ortho_group.rvs(3, random_state=515)
# Note this matrix has det -1, distinguishing O(N) from SO(N)
assert_almost_equal(np.linalg.det(x), -1)
expected = np.array([[0.94449759, -0.21678569, -0.24683651],
[-0.13147569, -0.93800245, 0.3207266],
[0.30106219, 0.27047251, 0.9144431]])
assert_array_almost_equal(x, expected)
assert_array_almost_equal(x2, expected)
def test_invalid_dim(self):
assert_raises(ValueError, ortho_group.rvs, None)
assert_raises(ValueError, ortho_group.rvs, (2, 2))
assert_raises(ValueError, ortho_group.rvs, 1)
assert_raises(ValueError, ortho_group.rvs, 2.5)
def test_det_and_ortho(self):
xs = [[ortho_group.rvs(dim)
for i in range(10)]
for dim in range(2,12)]
# Test that abs determinants are always +1
dets = np.array([[np.linalg.det(x) for x in xx] for xx in xs])
assert_allclose(np.fabs(dets), np.ones(dets.shape), rtol=1e-13)
# Test that we get both positive and negative determinants
# Check that we have at least one and less than 10 negative dets in a sample of 10. The rest are positive by the previous test.
# Test each dimension separately
assert_array_less([0]*10, [np.where(d < 0)[0].shape[0] for d in dets])
assert_array_less([np.where(d < 0)[0].shape[0] for d in dets], [10]*10)
# Test that these are orthogonal matrices
for xx in xs:
for x in xx:
assert_array_almost_equal(np.dot(x, x.T),
np.eye(x.shape[0]))
def test_haar(self):
# Test that the distribution is constant under rotation
# Every column should have the same distribution
# Additionally, the distribution should be invariant under another rotation
# Generate samples
dim = 5
samples = 1000 # Not too many, or the test takes too long
ks_prob = .05
np.random.seed(518) # Note that the test is sensitive to seed too
xs = ortho_group.rvs(dim, size=samples)
# Dot a few rows (0, 1, 2) with unit vectors (0, 2, 4, 3),
# effectively picking off entries in the matrices of xs.
# These projections should all have the same disribution,
# establishing rotational invariance. We use the two-sided
# KS test to confirm this.
# We could instead test that angles between random vectors
# are uniformly distributed, but the below is sufficient.
# It is not feasible to consider all pairs, so pick a few.
els = ((0,0), (0,2), (1,4), (2,3))
#proj = {(er, ec): [x[er][ec] for x in xs] for er, ec in els}
proj = dict(((er, ec), sorted([x[er][ec] for x in xs])) for er, ec in els)
pairs = [(e0, e1) for e0 in els for e1 in els if e0 > e1]
ks_tests = [ks_2samp(proj[p0], proj[p1])[1] for (p0, p1) in pairs]
assert_array_less([ks_prob]*len(pairs), ks_tests)
def test_pairwise_distances(self):
# Test that the distribution of pairwise distances is close to correct.
np.random.seed(514)
def random_ortho(dim):
u, _s, v = np.linalg.svd(np.random.normal(size=(dim, dim)))
return np.dot(u, v)
for dim in range(2, 6):
def generate_test_statistics(rvs, N=1000, eps=1e-10):
stats = np.array([
np.sum((rvs(dim=dim) - rvs(dim=dim))**2)
for _ in range(N)
])
# Add a bit of noise to account for numeric accuracy.
stats += np.random.uniform(-eps, eps, size=stats.shape)
return stats
expected = generate_test_statistics(random_ortho)
actual = generate_test_statistics(scipy.stats.ortho_group.rvs)
_D, p = scipy.stats.ks_2samp(expected, actual)
assert_array_less(.05, p)
class TestRandomCorrelation(object):
def test_reproducibility(self):
np.random.seed(514)
eigs = (.5, .8, 1.2, 1.5)
x = random_correlation.rvs((.5, .8, 1.2, 1.5))
x2 = random_correlation.rvs((.5, .8, 1.2, 1.5), random_state=514)
expected = np.array([[1., -0.20387311, 0.18366501, -0.04953711],
[-0.20387311, 1., -0.24351129, 0.06703474],
[0.18366501, -0.24351129, 1., 0.38530195],
[-0.04953711, 0.06703474, 0.38530195, 1.]])
assert_array_almost_equal(x, expected)
assert_array_almost_equal(x2, expected)
def test_invalid_eigs(self):
assert_raises(ValueError, random_correlation.rvs, None)
assert_raises(ValueError, random_correlation.rvs, 'test')
assert_raises(ValueError, random_correlation.rvs, 2.5)
assert_raises(ValueError, random_correlation.rvs, [2.5])
assert_raises(ValueError, random_correlation.rvs, [[1,2],[3,4]])
assert_raises(ValueError, random_correlation.rvs, [2.5, -.5])
assert_raises(ValueError, random_correlation.rvs, [1, 2, .1])
def test_definition(self):
# Test the definition of a correlation matrix in several dimensions:
#
# 1. Det is product of eigenvalues (and positive by construction
# in examples)
# 2. 1's on diagonal
# 3. Matrix is symmetric
def norm(i, e):
return i*e/sum(e)
np.random.seed(123)
eigs = [norm(i, np.random.uniform(size=i)) for i in range(2, 6)]
eigs.append([4,0,0,0])
ones = [[1.]*len(e) for e in eigs]
xs = [random_correlation.rvs(e) for e in eigs]
# Test that determinants are products of eigenvalues
# These are positive by construction
# Could also test that the eigenvalues themselves are correct,
# but this seems sufficient.
dets = [np.fabs(np.linalg.det(x)) for x in xs]
dets_known = [np.prod(e) for e in eigs]
assert_allclose(dets, dets_known, rtol=1e-13, atol=1e-13)
# Test for 1's on the diagonal
diags = [np.diag(x) for x in xs]
for a, b in zip(diags, ones):
assert_allclose(a, b, rtol=1e-13)
# Correlation matrices are symmetric
for x in xs:
assert_allclose(x, x.T, rtol=1e-13)
def test_to_corr(self):
# Check some corner cases in to_corr
# ajj == 1
m = np.array([[0.1, 0], [0, 1]], dtype=float)
m = random_correlation._to_corr(m)
assert_allclose(m, np.array([[1, 0], [0, 0.1]]))
# Floating point overflow; fails to compute the correct
# rotation, but should still produce some valid rotation
# rather than infs/nans
with np.errstate(over='ignore'):
g = np.array([[0, 1], [-1, 0]])
m0 = np.array([[1e300, 0], [0, np.nextafter(1, 0)]], dtype=float)
m = random_correlation._to_corr(m0.copy())
assert_allclose(m, g.T.dot(m0).dot(g))
m0 = np.array([[0.9, 1e300], [1e300, 1.1]], dtype=float)
m = random_correlation._to_corr(m0.copy())
assert_allclose(m, g.T.dot(m0).dot(g))
# Zero discriminant; should set the first diag entry to 1
m0 = np.array([[2, 1], [1, 2]], dtype=float)
m = random_correlation._to_corr(m0.copy())
assert_allclose(m[0,0], 1)
# Slightly negative discriminant; should be approx correct still
m0 = np.array([[2 + 1e-7, 1], [1, 2]], dtype=float)
m = random_correlation._to_corr(m0.copy())
assert_allclose(m[0,0], 1)
class TestUnitaryGroup(object):
def test_reproducibility(self):
np.random.seed(514)
x = unitary_group.rvs(3)
x2 = unitary_group.rvs(3, random_state=514)
expected = np.array([[0.308771+0.360312j, 0.044021+0.622082j, 0.160327+0.600173j],
[0.732757+0.297107j, 0.076692-0.4614j, -0.394349+0.022613j],
[-0.148844+0.357037j, -0.284602-0.557949j, 0.607051+0.299257j]])
assert_array_almost_equal(x, expected)
assert_array_almost_equal(x2, expected)
def test_invalid_dim(self):
assert_raises(ValueError, unitary_group.rvs, None)
assert_raises(ValueError, unitary_group.rvs, (2, 2))
assert_raises(ValueError, unitary_group.rvs, 1)
assert_raises(ValueError, unitary_group.rvs, 2.5)
def test_unitarity(self):
xs = [unitary_group.rvs(dim)
for dim in range(2,12)
for i in range(3)]
# Test that these are unitary matrices
for x in xs:
assert_allclose(np.dot(x, x.conj().T), np.eye(x.shape[0]), atol=1e-15)
def test_haar(self):
# Test that the eigenvalues, which lie on the unit circle in
# the complex plane, are uncorrelated.
# Generate samples
dim = 5
samples = 1000 # Not too many, or the test takes too long
np.random.seed(514) # Note that the test is sensitive to seed too
xs = unitary_group.rvs(dim, size=samples)
# The angles "x" of the eigenvalues should be uniformly distributed
# Overall this seems to be a necessary but weak test of the distribution.
eigs = np.vstack(scipy.linalg.eigvals(x) for x in xs)
x = np.arctan2(eigs.imag, eigs.real)
res = kstest(x.ravel(), uniform(-np.pi, 2*np.pi).cdf)
assert_(res.pvalue > 0.05)
def check_pickling(distfn, args):
# check that a distribution instance pickles and unpickles
# pay special attention to the random_state property
# save the random_state (restore later)
rndm = distfn.random_state
distfn.random_state = 1234
distfn.rvs(*args, size=8)
s = pickle.dumps(distfn)
r0 = distfn.rvs(*args, size=8)
unpickled = pickle.loads(s)
r1 = unpickled.rvs(*args, size=8)
assert_equal(r0, r1)
# restore the random_state
distfn.random_state = rndm
def test_random_state_property():
scale = np.eye(3)
scale[0, 1] = 0.5
scale[1, 0] = 0.5
dists = [
[multivariate_normal, ()],
[dirichlet, (np.array([1.]), )],
[wishart, (10, scale)],
[invwishart, (10, scale)],
[multinomial, (5, [0.5, 0.4, 0.1])],
[ortho_group, (2,)],
[special_ortho_group, (2,)]
]
for distfn, args in dists:
check_random_state_property(distfn, args)
check_pickling(distfn, args)
| 62,607 | 37.362745 | 135 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/tests/test_discrete_distns.py
|
from __future__ import division, print_function, absolute_import
from scipy.stats import hypergeom, bernoulli
import numpy as np
from numpy.testing import assert_almost_equal
def test_hypergeom_logpmf():
# symmetries test
# f(k,N,K,n) = f(n-k,N,N-K,n) = f(K-k,N,K,N-n) = f(k,N,n,K)
k = 5
N = 50
K = 10
n = 5
logpmf1 = hypergeom.logpmf(k,N,K,n)
logpmf2 = hypergeom.logpmf(n-k,N,N-K,n)
logpmf3 = hypergeom.logpmf(K-k,N,K,N-n)
logpmf4 = hypergeom.logpmf(k,N,n,K)
assert_almost_equal(logpmf1, logpmf2, decimal=12)
assert_almost_equal(logpmf1, logpmf3, decimal=12)
assert_almost_equal(logpmf1, logpmf4, decimal=12)
# test related distribution
# Bernoulli distribution if n = 1
k = 1
N = 10
K = 7
n = 1
hypergeom_logpmf = hypergeom.logpmf(k,N,K,n)
bernoulli_logpmf = bernoulli.logpmf(k,K/N)
assert_almost_equal(hypergeom_logpmf, bernoulli_logpmf, decimal=12)
| 946 | 28.59375 | 71 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/tests/test_mstats_extras.py
|
from __future__ import division, print_function, absolute_import
import numpy as np
import numpy.ma as ma
import scipy.stats.mstats as ms
from numpy.testing import (assert_equal, assert_almost_equal, assert_,
assert_allclose)
def test_compare_medians_ms():
x = np.arange(7)
y = x + 10
assert_almost_equal(ms.compare_medians_ms(x, y), 0)
y2 = np.linspace(0, 1, num=10)
assert_almost_equal(ms.compare_medians_ms(x, y2), 0.017116406778)
def test_hdmedian():
# 1-D array
x = ma.arange(11)
assert_allclose(ms.hdmedian(x), 5, rtol=1e-14)
x.mask = ma.make_mask(x)
x.mask[:7] = False
assert_allclose(ms.hdmedian(x), 3, rtol=1e-14)
# Check that `var` keyword returns a value. TODO: check whether returned
# value is actually correct.
assert_(ms.hdmedian(x, var=True).size == 2)
# 2-D array
x2 = ma.arange(22).reshape((11, 2))
assert_allclose(ms.hdmedian(x2, axis=0), [10, 11])
x2.mask = ma.make_mask(x2)
x2.mask[:7, :] = False
assert_allclose(ms.hdmedian(x2, axis=0), [6, 7])
def test_rsh():
np.random.seed(132345)
x = np.random.randn(100)
res = ms.rsh(x)
# Just a sanity check that the code runs and output shape is correct.
# TODO: check that implementation is correct.
assert_(res.shape == x.shape)
# Check points keyword
res = ms.rsh(x, points=[0, 1.])
assert_(res.size == 2)
def test_mjci():
# Tests the Marits-Jarrett estimator
data = ma.array([77, 87, 88,114,151,210,219,246,253,262,
296,299,306,376,428,515,666,1310,2611])
assert_almost_equal(ms.mjci(data),[55.76819,45.84028,198.87875],5)
def test_trimmed_mean_ci():
# Tests the confidence intervals of the trimmed mean.
data = ma.array([545,555,558,572,575,576,578,580,
594,605,635,651,653,661,666])
assert_almost_equal(ms.trimmed_mean(data,0.2), 596.2, 1)
assert_equal(np.round(ms.trimmed_mean_ci(data,(0.2,0.2)),1),
[561.8, 630.6])
def test_idealfourths():
# Tests ideal-fourths
test = np.arange(100)
assert_almost_equal(np.asarray(ms.idealfourths(test)),
[24.416667,74.583333],6)
test_2D = test.repeat(3).reshape(-1,3)
assert_almost_equal(ms.idealfourths(test_2D, axis=0),
[[24.416667,24.416667,24.416667],
[74.583333,74.583333,74.583333]],6)
assert_almost_equal(ms.idealfourths(test_2D, axis=1),
test.repeat(2).reshape(-1,2))
test = [0, 0]
_result = ms.idealfourths(test)
assert_(np.isnan(_result).all())
class TestQuantiles(object):
data = [0.706560797,0.727229578,0.990399276,0.927065621,0.158953014,
0.887764025,0.239407086,0.349638551,0.972791145,0.149789972,
0.936947700,0.132359948,0.046041972,0.641675031,0.945530547,
0.224218684,0.771450991,0.820257774,0.336458052,0.589113496,
0.509736129,0.696838829,0.491323573,0.622767425,0.775189248,
0.641461450,0.118455200,0.773029450,0.319280007,0.752229111,
0.047841438,0.466295911,0.583850781,0.840581845,0.550086491,
0.466470062,0.504765074,0.226855960,0.362641207,0.891620942,
0.127898691,0.490094097,0.044882048,0.041441695,0.317976349,
0.504135618,0.567353033,0.434617473,0.636243375,0.231803616,
0.230154113,0.160011327,0.819464108,0.854706985,0.438809221,
0.487427267,0.786907310,0.408367937,0.405534192,0.250444460,
0.995309248,0.144389588,0.739947527,0.953543606,0.680051621,
0.388382017,0.863530727,0.006514031,0.118007779,0.924024803,
0.384236354,0.893687694,0.626534881,0.473051932,0.750134705,
0.241843555,0.432947602,0.689538104,0.136934797,0.150206859,
0.474335206,0.907775349,0.525869295,0.189184225,0.854284286,
0.831089744,0.251637345,0.587038213,0.254475554,0.237781276,
0.827928620,0.480283781,0.594514455,0.213641488,0.024194386,
0.536668589,0.699497811,0.892804071,0.093835427,0.731107772]
def test_hdquantiles(self):
data = self.data
assert_almost_equal(ms.hdquantiles(data,[0., 1.]),
[0.006514031, 0.995309248])
hdq = ms.hdquantiles(data,[0.25, 0.5, 0.75])
assert_almost_equal(hdq, [0.253210762, 0.512847491, 0.762232442,])
hdq = ms.hdquantiles_sd(data,[0.25, 0.5, 0.75])
assert_almost_equal(hdq, [0.03786954, 0.03805389, 0.03800152,], 4)
data = np.array(data).reshape(10,10)
hdq = ms.hdquantiles(data,[0.25,0.5,0.75],axis=0)
assert_almost_equal(hdq[:,0], ms.hdquantiles(data[:,0],[0.25,0.5,0.75]))
assert_almost_equal(hdq[:,-1], ms.hdquantiles(data[:,-1],[0.25,0.5,0.75]))
hdq = ms.hdquantiles(data,[0.25,0.5,0.75],axis=0,var=True)
assert_almost_equal(hdq[...,0],
ms.hdquantiles(data[:,0],[0.25,0.5,0.75],var=True))
assert_almost_equal(hdq[...,-1],
ms.hdquantiles(data[:,-1],[0.25,0.5,0.75], var=True))
def test_hdquantiles_sd(self):
# Only test that code runs, implementation not checked for correctness
res = ms.hdquantiles_sd(self.data)
assert_(res.size == 3)
def test_mquantiles_cimj(self):
# Only test that code runs, implementation not checked for correctness
ci_lower, ci_upper = ms.mquantiles_cimj(self.data)
assert_(ci_lower.size == ci_upper.size == 3)
| 5,464 | 38.890511 | 82 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/tests/__init__.py
| 0 | 0 | 0 |
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cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/tests/test_fit.py
|
from __future__ import division, print_function, absolute_import
import os
import numpy as np
from numpy.testing import assert_allclose
from scipy._lib._numpy_compat import suppress_warnings
import pytest
from scipy import stats
from .test_continuous_basic import distcont
# this is not a proper statistical test for convergence, but only
# verifies that the estimate and true values don't differ by too much
fit_sizes = [1000, 5000] # sample sizes to try
thresh_percent = 0.25 # percent of true parameters for fail cut-off
thresh_min = 0.75 # minimum difference estimate - true to fail test
failing_fits = [
'burr',
'chi2',
'gausshyper',
'genexpon',
'gengamma',
'kappa4',
'ksone',
'mielke',
'ncf',
'ncx2',
'pearson3',
'powerlognorm',
'truncexpon',
'tukeylambda',
'vonmises',
'wrapcauchy',
'levy_stable',
'trapz'
]
# Don't run the fit test on these:
skip_fit = [
'erlang', # Subclass of gamma, generates a warning.
]
def cases_test_cont_fit():
# this tests the closeness of the estimated parameters to the true
# parameters with fit method of continuous distributions
# Note: is slow, some distributions don't converge with sample size <= 10000
for distname, arg in distcont:
if distname not in skip_fit:
yield distname, arg
@pytest.mark.slow
@pytest.mark.parametrize('distname,arg', cases_test_cont_fit())
def test_cont_fit(distname, arg):
if distname in failing_fits:
# Skip failing fits unless overridden
xfail = True
try:
xfail = not int(os.environ['SCIPY_XFAIL'])
except:
pass
if xfail:
msg = "Fitting %s doesn't work reliably yet" % distname
msg += " [Set environment variable SCIPY_XFAIL=1 to run this test nevertheless.]"
pytest.xfail(msg)
distfn = getattr(stats, distname)
truearg = np.hstack([arg, [0.0, 1.0]])
diffthreshold = np.max(np.vstack([truearg*thresh_percent,
np.ones(distfn.numargs+2)*thresh_min]),
0)
for fit_size in fit_sizes:
# Note that if a fit succeeds, the other fit_sizes are skipped
np.random.seed(1234)
with np.errstate(all='ignore'), suppress_warnings() as sup:
sup.filter(category=DeprecationWarning, message=".*frechet_")
rvs = distfn.rvs(size=fit_size, *arg)
est = distfn.fit(rvs) # start with default values
diff = est - truearg
# threshold for location
diffthreshold[-2] = np.max([np.abs(rvs.mean())*thresh_percent,thresh_min])
if np.any(np.isnan(est)):
raise AssertionError('nan returned in fit')
else:
if np.all(np.abs(diff) <= diffthreshold):
break
else:
txt = 'parameter: %s\n' % str(truearg)
txt += 'estimated: %s\n' % str(est)
txt += 'diff : %s\n' % str(diff)
raise AssertionError('fit not very good in %s\n' % distfn.name + txt)
def _check_loc_scale_mle_fit(name, data, desired, atol=None):
d = getattr(stats, name)
actual = d.fit(data)[-2:]
assert_allclose(actual, desired, atol=atol,
err_msg='poor mle fit of (loc, scale) in %s' % name)
def test_non_default_loc_scale_mle_fit():
data = np.array([1.01, 1.78, 1.78, 1.78, 1.88, 1.88, 1.88, 2.00])
_check_loc_scale_mle_fit('uniform', data, [1.01, 0.99], 1e-3)
_check_loc_scale_mle_fit('expon', data, [1.01, 0.73875], 1e-3)
def test_expon_fit():
"""gh-6167"""
data = [0, 0, 0, 0, 2, 2, 2, 2]
phat = stats.expon.fit(data, floc=0)
assert_allclose(phat, [0, 1.0], atol=1e-3)
| 3,803 | 29.677419 | 93 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/tests/test_rank.py
|
from __future__ import division, print_function, absolute_import
import numpy as np
from numpy.testing import assert_equal, assert_array_equal
from scipy.stats import rankdata, tiecorrect
class TestTieCorrect(object):
def test_empty(self):
"""An empty array requires no correction, should return 1.0."""
ranks = np.array([], dtype=np.float64)
c = tiecorrect(ranks)
assert_equal(c, 1.0)
def test_one(self):
"""A single element requires no correction, should return 1.0."""
ranks = np.array([1.0], dtype=np.float64)
c = tiecorrect(ranks)
assert_equal(c, 1.0)
def test_no_correction(self):
"""Arrays with no ties require no correction."""
ranks = np.arange(2.0)
c = tiecorrect(ranks)
assert_equal(c, 1.0)
ranks = np.arange(3.0)
c = tiecorrect(ranks)
assert_equal(c, 1.0)
def test_basic(self):
"""Check a few basic examples of the tie correction factor."""
# One tie of two elements
ranks = np.array([1.0, 2.5, 2.5])
c = tiecorrect(ranks)
T = 2.0
N = ranks.size
expected = 1.0 - (T**3 - T) / (N**3 - N)
assert_equal(c, expected)
# One tie of two elements (same as above, but tie is not at the end)
ranks = np.array([1.5, 1.5, 3.0])
c = tiecorrect(ranks)
T = 2.0
N = ranks.size
expected = 1.0 - (T**3 - T) / (N**3 - N)
assert_equal(c, expected)
# One tie of three elements
ranks = np.array([1.0, 3.0, 3.0, 3.0])
c = tiecorrect(ranks)
T = 3.0
N = ranks.size
expected = 1.0 - (T**3 - T) / (N**3 - N)
assert_equal(c, expected)
# Two ties, lengths 2 and 3.
ranks = np.array([1.5, 1.5, 4.0, 4.0, 4.0])
c = tiecorrect(ranks)
T1 = 2.0
T2 = 3.0
N = ranks.size
expected = 1.0 - ((T1**3 - T1) + (T2**3 - T2)) / (N**3 - N)
assert_equal(c, expected)
def test_overflow(self):
ntie, k = 2000, 5
a = np.repeat(np.arange(k), ntie)
n = a.size # ntie * k
out = tiecorrect(rankdata(a))
assert_equal(out, 1.0 - k * (ntie**3 - ntie) / float(n**3 - n))
class TestRankData(object):
def test_empty(self):
"""stats.rankdata([]) should return an empty array."""
a = np.array([], dtype=int)
r = rankdata(a)
assert_array_equal(r, np.array([], dtype=np.float64))
r = rankdata([])
assert_array_equal(r, np.array([], dtype=np.float64))
def test_one(self):
"""Check stats.rankdata with an array of length 1."""
data = [100]
a = np.array(data, dtype=int)
r = rankdata(a)
assert_array_equal(r, np.array([1.0], dtype=np.float64))
r = rankdata(data)
assert_array_equal(r, np.array([1.0], dtype=np.float64))
def test_basic(self):
"""Basic tests of stats.rankdata."""
data = [100, 10, 50]
expected = np.array([3.0, 1.0, 2.0], dtype=np.float64)
a = np.array(data, dtype=int)
r = rankdata(a)
assert_array_equal(r, expected)
r = rankdata(data)
assert_array_equal(r, expected)
data = [40, 10, 30, 10, 50]
expected = np.array([4.0, 1.5, 3.0, 1.5, 5.0], dtype=np.float64)
a = np.array(data, dtype=int)
r = rankdata(a)
assert_array_equal(r, expected)
r = rankdata(data)
assert_array_equal(r, expected)
data = [20, 20, 20, 10, 10, 10]
expected = np.array([5.0, 5.0, 5.0, 2.0, 2.0, 2.0], dtype=np.float64)
a = np.array(data, dtype=int)
r = rankdata(a)
assert_array_equal(r, expected)
r = rankdata(data)
assert_array_equal(r, expected)
# The docstring states explicitly that the argument is flattened.
a2d = a.reshape(2, 3)
r = rankdata(a2d)
assert_array_equal(r, expected)
def test_rankdata_object_string(self):
min_rank = lambda a: [1 + sum(i < j for i in a) for j in a]
max_rank = lambda a: [sum(i <= j for i in a) for j in a]
ordinal_rank = lambda a: min_rank([(x, i) for i, x in enumerate(a)])
def average_rank(a):
return [(i + j) / 2.0 for i, j in zip(min_rank(a), max_rank(a))]
def dense_rank(a):
b = np.unique(a)
return [1 + sum(i < j for i in b) for j in a]
rankf = dict(min=min_rank, max=max_rank, ordinal=ordinal_rank,
average=average_rank, dense=dense_rank)
def check_ranks(a):
for method in 'min', 'max', 'dense', 'ordinal', 'average':
out = rankdata(a, method=method)
assert_array_equal(out, rankf[method](a))
val = ['foo', 'bar', 'qux', 'xyz', 'abc', 'efg', 'ace', 'qwe', 'qaz']
check_ranks(np.random.choice(val, 200))
check_ranks(np.random.choice(val, 200).astype('object'))
val = np.array([0, 1, 2, 2.718, 3, 3.141], dtype='object')
check_ranks(np.random.choice(val, 200).astype('object'))
def test_large_int(self):
data = np.array([2**60, 2**60+1], dtype=np.uint64)
r = rankdata(data)
assert_array_equal(r, [1.0, 2.0])
data = np.array([2**60, 2**60+1], dtype=np.int64)
r = rankdata(data)
assert_array_equal(r, [1.0, 2.0])
data = np.array([2**60, -2**60+1], dtype=np.int64)
r = rankdata(data)
assert_array_equal(r, [2.0, 1.0])
def test_big_tie(self):
for n in [10000, 100000, 1000000]:
data = np.ones(n, dtype=int)
r = rankdata(data)
expected_rank = 0.5 * (n + 1)
assert_array_equal(r, expected_rank * data,
"test failed with n=%d" % n)
_cases = (
# values, method, expected
([], 'average', []),
([], 'min', []),
([], 'max', []),
([], 'dense', []),
([], 'ordinal', []),
#
([100], 'average', [1.0]),
([100], 'min', [1.0]),
([100], 'max', [1.0]),
([100], 'dense', [1.0]),
([100], 'ordinal', [1.0]),
#
([100, 100, 100], 'average', [2.0, 2.0, 2.0]),
([100, 100, 100], 'min', [1.0, 1.0, 1.0]),
([100, 100, 100], 'max', [3.0, 3.0, 3.0]),
([100, 100, 100], 'dense', [1.0, 1.0, 1.0]),
([100, 100, 100], 'ordinal', [1.0, 2.0, 3.0]),
#
([100, 300, 200], 'average', [1.0, 3.0, 2.0]),
([100, 300, 200], 'min', [1.0, 3.0, 2.0]),
([100, 300, 200], 'max', [1.0, 3.0, 2.0]),
([100, 300, 200], 'dense', [1.0, 3.0, 2.0]),
([100, 300, 200], 'ordinal', [1.0, 3.0, 2.0]),
#
([100, 200, 300, 200], 'average', [1.0, 2.5, 4.0, 2.5]),
([100, 200, 300, 200], 'min', [1.0, 2.0, 4.0, 2.0]),
([100, 200, 300, 200], 'max', [1.0, 3.0, 4.0, 3.0]),
([100, 200, 300, 200], 'dense', [1.0, 2.0, 3.0, 2.0]),
([100, 200, 300, 200], 'ordinal', [1.0, 2.0, 4.0, 3.0]),
#
([100, 200, 300, 200, 100], 'average', [1.5, 3.5, 5.0, 3.5, 1.5]),
([100, 200, 300, 200, 100], 'min', [1.0, 3.0, 5.0, 3.0, 1.0]),
([100, 200, 300, 200, 100], 'max', [2.0, 4.0, 5.0, 4.0, 2.0]),
([100, 200, 300, 200, 100], 'dense', [1.0, 2.0, 3.0, 2.0, 1.0]),
([100, 200, 300, 200, 100], 'ordinal', [1.0, 3.0, 5.0, 4.0, 2.0]),
#
([10] * 30, 'ordinal', np.arange(1.0, 31.0)),
)
def test_cases():
for values, method, expected in _cases:
r = rankdata(values, method=method)
assert_array_equal(r, expected)
| 7,514 | 33.315068 | 77 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/tests/test_stats.py
|
""" Test functions for stats module
WRITTEN BY LOUIS LUANGKESORN <lluang@yahoo.com> FOR THE STATS MODULE
BASED ON WILKINSON'S STATISTICS QUIZ
http://www.stanford.edu/~clint/bench/wilk.txt
Additional tests by a host of SciPy developers.
"""
from __future__ import division, print_function, absolute_import
import os
import sys
import warnings
from collections import namedtuple
from numpy.testing import (assert_, assert_equal,
assert_almost_equal, assert_array_almost_equal,
assert_array_equal, assert_approx_equal,
assert_allclose)
import pytest
from pytest import raises as assert_raises
from scipy._lib._numpy_compat import suppress_warnings
import numpy.ma.testutils as mat
from numpy import array, arange, float32, float64, power
import numpy as np
import scipy.stats as stats
import scipy.stats.mstats as mstats
import scipy.stats.mstats_basic as mstats_basic
from scipy._lib._version import NumpyVersion
from scipy._lib.six import xrange
from .common_tests import check_named_results
""" Numbers in docstrings beginning with 'W' refer to the section numbers
and headings found in the STATISTICS QUIZ of Leland Wilkinson. These are
considered to be essential functionality. True testing and
evaluation of a statistics package requires use of the
NIST Statistical test data. See McCoullough(1999) Assessing The Reliability
of Statistical Software for a test methodology and its
implementation in testing SAS, SPSS, and S-Plus
"""
# Datasets
# These data sets are from the nasty.dat sets used by Wilkinson
# For completeness, I should write the relevant tests and count them as failures
# Somewhat acceptable, since this is still beta software. It would count as a
# good target for 1.0 status
X = array([1,2,3,4,5,6,7,8,9], float)
ZERO = array([0,0,0,0,0,0,0,0,0], float)
BIG = array([99999991,99999992,99999993,99999994,99999995,99999996,99999997,
99999998,99999999], float)
LITTLE = array([0.99999991,0.99999992,0.99999993,0.99999994,0.99999995,0.99999996,
0.99999997,0.99999998,0.99999999], float)
HUGE = array([1e+12,2e+12,3e+12,4e+12,5e+12,6e+12,7e+12,8e+12,9e+12], float)
TINY = array([1e-12,2e-12,3e-12,4e-12,5e-12,6e-12,7e-12,8e-12,9e-12], float)
ROUND = array([0.5,1.5,2.5,3.5,4.5,5.5,6.5,7.5,8.5], float)
class TestTrimmedStats(object):
# TODO: write these tests to handle missing values properly
dprec = np.finfo(np.float64).precision
def test_tmean(self):
y = stats.tmean(X, (2, 8), (True, True))
assert_approx_equal(y, 5.0, significant=self.dprec)
y1 = stats.tmean(X, limits=(2, 8), inclusive=(False, False))
y2 = stats.tmean(X, limits=None)
assert_approx_equal(y1, y2, significant=self.dprec)
def test_tvar(self):
y = stats.tvar(X, limits=(2, 8), inclusive=(True, True))
assert_approx_equal(y, 4.6666666666666661, significant=self.dprec)
y = stats.tvar(X, limits=None)
assert_approx_equal(y, X.var(ddof=1), significant=self.dprec)
def test_tstd(self):
y = stats.tstd(X, (2, 8), (True, True))
assert_approx_equal(y, 2.1602468994692865, significant=self.dprec)
y = stats.tstd(X, limits=None)
assert_approx_equal(y, X.std(ddof=1), significant=self.dprec)
def test_tmin(self):
assert_equal(stats.tmin(4), 4)
x = np.arange(10)
assert_equal(stats.tmin(x), 0)
assert_equal(stats.tmin(x, lowerlimit=0), 0)
assert_equal(stats.tmin(x, lowerlimit=0, inclusive=False), 1)
x = x.reshape((5, 2))
assert_equal(stats.tmin(x, lowerlimit=0, inclusive=False), [2, 1])
assert_equal(stats.tmin(x, axis=1), [0, 2, 4, 6, 8])
assert_equal(stats.tmin(x, axis=None), 0)
x = np.arange(10.)
x[9] = np.nan
with suppress_warnings() as sup:
r = sup.record(RuntimeWarning, "invalid value*")
assert_equal(stats.tmin(x), np.nan)
assert_equal(stats.tmin(x, nan_policy='omit'), 0.)
assert_raises(ValueError, stats.tmin, x, nan_policy='raise')
assert_raises(ValueError, stats.tmin, x, nan_policy='foobar')
msg = "'propagate', 'raise', 'omit'"
with assert_raises(ValueError, message=msg):
stats.tmin(x, nan_policy='foo')
def test_tmax(self):
assert_equal(stats.tmax(4), 4)
x = np.arange(10)
assert_equal(stats.tmax(x), 9)
assert_equal(stats.tmax(x, upperlimit=9), 9)
assert_equal(stats.tmax(x, upperlimit=9, inclusive=False), 8)
x = x.reshape((5, 2))
assert_equal(stats.tmax(x, upperlimit=9, inclusive=False), [8, 7])
assert_equal(stats.tmax(x, axis=1), [1, 3, 5, 7, 9])
assert_equal(stats.tmax(x, axis=None), 9)
x = np.arange(10.)
x[6] = np.nan
with suppress_warnings() as sup:
r = sup.record(RuntimeWarning, "invalid value*")
assert_equal(stats.tmax(x), np.nan)
assert_equal(stats.tmax(x, nan_policy='omit'), 9.)
assert_raises(ValueError, stats.tmax, x, nan_policy='raise')
assert_raises(ValueError, stats.tmax, x, nan_policy='foobar')
def test_tsem(self):
y = stats.tsem(X, limits=(3, 8), inclusive=(False, True))
y_ref = np.array([4, 5, 6, 7, 8])
assert_approx_equal(y, y_ref.std(ddof=1) / np.sqrt(y_ref.size),
significant=self.dprec)
assert_approx_equal(stats.tsem(X, limits=[-1, 10]),
stats.tsem(X, limits=None),
significant=self.dprec)
class TestCorrPearsonr(object):
""" W.II.D. Compute a correlation matrix on all the variables.
All the correlations, except for ZERO and MISS, should be exactly 1.
ZERO and MISS should have undefined or missing correlations with the
other variables. The same should go for SPEARMAN correlations, if
your program has them.
"""
def test_pXX(self):
y = stats.pearsonr(X,X)
r = y[0]
assert_approx_equal(r,1.0)
def test_pXBIG(self):
y = stats.pearsonr(X,BIG)
r = y[0]
assert_approx_equal(r,1.0)
def test_pXLITTLE(self):
y = stats.pearsonr(X,LITTLE)
r = y[0]
assert_approx_equal(r,1.0)
def test_pXHUGE(self):
y = stats.pearsonr(X,HUGE)
r = y[0]
assert_approx_equal(r,1.0)
def test_pXTINY(self):
y = stats.pearsonr(X,TINY)
r = y[0]
assert_approx_equal(r,1.0)
def test_pXROUND(self):
y = stats.pearsonr(X,ROUND)
r = y[0]
assert_approx_equal(r,1.0)
def test_pBIGBIG(self):
y = stats.pearsonr(BIG,BIG)
r = y[0]
assert_approx_equal(r,1.0)
def test_pBIGLITTLE(self):
y = stats.pearsonr(BIG,LITTLE)
r = y[0]
assert_approx_equal(r,1.0)
def test_pBIGHUGE(self):
y = stats.pearsonr(BIG,HUGE)
r = y[0]
assert_approx_equal(r,1.0)
def test_pBIGTINY(self):
y = stats.pearsonr(BIG,TINY)
r = y[0]
assert_approx_equal(r,1.0)
def test_pBIGROUND(self):
y = stats.pearsonr(BIG,ROUND)
r = y[0]
assert_approx_equal(r,1.0)
def test_pLITTLELITTLE(self):
y = stats.pearsonr(LITTLE,LITTLE)
r = y[0]
assert_approx_equal(r,1.0)
def test_pLITTLEHUGE(self):
y = stats.pearsonr(LITTLE,HUGE)
r = y[0]
assert_approx_equal(r,1.0)
def test_pLITTLETINY(self):
y = stats.pearsonr(LITTLE,TINY)
r = y[0]
assert_approx_equal(r,1.0)
def test_pLITTLEROUND(self):
y = stats.pearsonr(LITTLE,ROUND)
r = y[0]
assert_approx_equal(r,1.0)
def test_pHUGEHUGE(self):
y = stats.pearsonr(HUGE,HUGE)
r = y[0]
assert_approx_equal(r,1.0)
def test_pHUGETINY(self):
y = stats.pearsonr(HUGE,TINY)
r = y[0]
assert_approx_equal(r,1.0)
def test_pHUGEROUND(self):
y = stats.pearsonr(HUGE,ROUND)
r = y[0]
assert_approx_equal(r,1.0)
def test_pTINYTINY(self):
y = stats.pearsonr(TINY,TINY)
r = y[0]
assert_approx_equal(r,1.0)
def test_pTINYROUND(self):
y = stats.pearsonr(TINY,ROUND)
r = y[0]
assert_approx_equal(r,1.0)
def test_pROUNDROUND(self):
y = stats.pearsonr(ROUND,ROUND)
r = y[0]
assert_approx_equal(r,1.0)
def test_r_exactly_pos1(self):
a = arange(3.0)
b = a
r, prob = stats.pearsonr(a,b)
assert_equal(r, 1.0)
assert_equal(prob, 0.0)
def test_r_exactly_neg1(self):
a = arange(3.0)
b = -a
r, prob = stats.pearsonr(a,b)
assert_equal(r, -1.0)
assert_equal(prob, 0.0)
def test_basic(self):
# A basic test, with a correlation coefficient
# that is not 1 or -1.
a = array([-1, 0, 1])
b = array([0, 0, 3])
r, prob = stats.pearsonr(a, b)
assert_approx_equal(r, np.sqrt(3)/2)
assert_approx_equal(prob, 1.0/3)
class TestFisherExact(object):
"""Some tests to show that fisher_exact() works correctly.
Note that in SciPy 0.9.0 this was not working well for large numbers due to
inaccuracy of the hypergeom distribution (see #1218). Fixed now.
Also note that R and Scipy have different argument formats for their
hypergeometric distribution functions.
R:
> phyper(18999, 99000, 110000, 39000, lower.tail = FALSE)
[1] 1.701815e-09
"""
def test_basic(self):
fisher_exact = stats.fisher_exact
res = fisher_exact([[14500, 20000], [30000, 40000]])[1]
assert_approx_equal(res, 0.01106, significant=4)
res = fisher_exact([[100, 2], [1000, 5]])[1]
assert_approx_equal(res, 0.1301, significant=4)
res = fisher_exact([[2, 7], [8, 2]])[1]
assert_approx_equal(res, 0.0230141, significant=6)
res = fisher_exact([[5, 1], [10, 10]])[1]
assert_approx_equal(res, 0.1973244, significant=6)
res = fisher_exact([[5, 15], [20, 20]])[1]
assert_approx_equal(res, 0.0958044, significant=6)
res = fisher_exact([[5, 16], [20, 25]])[1]
assert_approx_equal(res, 0.1725862, significant=6)
res = fisher_exact([[10, 5], [10, 1]])[1]
assert_approx_equal(res, 0.1973244, significant=6)
res = fisher_exact([[5, 0], [1, 4]])[1]
assert_approx_equal(res, 0.04761904, significant=6)
res = fisher_exact([[0, 1], [3, 2]])[1]
assert_approx_equal(res, 1.0)
res = fisher_exact([[0, 2], [6, 4]])[1]
assert_approx_equal(res, 0.4545454545)
res = fisher_exact([[2, 7], [8, 2]])
assert_approx_equal(res[1], 0.0230141, significant=6)
assert_approx_equal(res[0], 4.0 / 56)
def test_precise(self):
# results from R
#
# R defines oddsratio differently (see Notes section of fisher_exact
# docstring), so those will not match. We leave them in anyway, in
# case they will be useful later on. We test only the p-value.
tablist = [
([[100, 2], [1000, 5]], (2.505583993422285e-001, 1.300759363430016e-001)),
([[2, 7], [8, 2]], (8.586235135736206e-002, 2.301413756522114e-002)),
([[5, 1], [10, 10]], (4.725646047336584e+000, 1.973244147157190e-001)),
([[5, 15], [20, 20]], (3.394396617440852e-001, 9.580440012477637e-002)),
([[5, 16], [20, 25]], (3.960558326183334e-001, 1.725864953812994e-001)),
([[10, 5], [10, 1]], (2.116112781158483e-001, 1.973244147157190e-001)),
([[10, 5], [10, 0]], (0.000000000000000e+000, 6.126482213438734e-002)),
([[5, 0], [1, 4]], (np.inf, 4.761904761904762e-002)),
([[0, 5], [1, 4]], (0.000000000000000e+000, 1.000000000000000e+000)),
([[5, 1], [0, 4]], (np.inf, 4.761904761904758e-002)),
([[0, 1], [3, 2]], (0.000000000000000e+000, 1.000000000000000e+000))
]
for table, res_r in tablist:
res = stats.fisher_exact(np.asarray(table))
np.testing.assert_almost_equal(res[1], res_r[1], decimal=11,
verbose=True)
@pytest.mark.slow
def test_large_numbers(self):
# Test with some large numbers. Regression test for #1401
pvals = [5.56e-11, 2.666e-11, 1.363e-11] # from R
for pval, num in zip(pvals, [75, 76, 77]):
res = stats.fisher_exact([[17704, 496], [1065, num]])[1]
assert_approx_equal(res, pval, significant=4)
res = stats.fisher_exact([[18000, 80000], [20000, 90000]])[1]
assert_approx_equal(res, 0.2751, significant=4)
def test_raises(self):
# test we raise an error for wrong shape of input.
assert_raises(ValueError, stats.fisher_exact,
np.arange(6).reshape(2, 3))
def test_row_or_col_zero(self):
tables = ([[0, 0], [5, 10]],
[[5, 10], [0, 0]],
[[0, 5], [0, 10]],
[[5, 0], [10, 0]])
for table in tables:
oddsratio, pval = stats.fisher_exact(table)
assert_equal(pval, 1.0)
assert_equal(oddsratio, np.nan)
def test_less_greater(self):
tables = (
# Some tables to compare with R:
[[2, 7], [8, 2]],
[[200, 7], [8, 300]],
[[28, 21], [6, 1957]],
[[190, 800], [200, 900]],
# Some tables with simple exact values
# (includes regression test for ticket #1568):
[[0, 2], [3, 0]],
[[1, 1], [2, 1]],
[[2, 0], [1, 2]],
[[0, 1], [2, 3]],
[[1, 0], [1, 4]],
)
pvals = (
# from R:
[0.018521725952066501, 0.9990149169715733],
[1.0, 2.0056578803889148e-122],
[1.0, 5.7284374608319831e-44],
[0.7416227, 0.2959826],
# Exact:
[0.1, 1.0],
[0.7, 0.9],
[1.0, 0.3],
[2./3, 1.0],
[1.0, 1./3],
)
for table, pval in zip(tables, pvals):
res = []
res.append(stats.fisher_exact(table, alternative="less")[1])
res.append(stats.fisher_exact(table, alternative="greater")[1])
assert_allclose(res, pval, atol=0, rtol=1e-7)
def test_gh3014(self):
# check if issue #3014 has been fixed.
# before, this would have risen a ValueError
odds, pvalue = stats.fisher_exact([[1, 2], [9, 84419233]])
class TestCorrSpearmanr(object):
""" W.II.D. Compute a correlation matrix on all the variables.
All the correlations, except for ZERO and MISS, should be exactly 1.
ZERO and MISS should have undefined or missing correlations with the
other variables. The same should go for SPEARMAN corelations, if
your program has them.
"""
def test_scalar(self):
y = stats.spearmanr(4., 2.)
assert_(np.isnan(y).all())
def test_uneven_lengths(self):
assert_raises(ValueError, stats.spearmanr, [1, 2, 1], [8, 9])
assert_raises(ValueError, stats.spearmanr, [1, 2, 1], 8)
def test_nan_policy(self):
x = np.arange(10.)
x[9] = np.nan
assert_array_equal(stats.spearmanr(x, x), (np.nan, np.nan))
assert_array_equal(stats.spearmanr(x, x, nan_policy='omit'),
(1.0, 0.0))
assert_raises(ValueError, stats.spearmanr, x, x, nan_policy='raise')
assert_raises(ValueError, stats.spearmanr, x, x, nan_policy='foobar')
def test_sXX(self):
y = stats.spearmanr(X,X)
r = y[0]
assert_approx_equal(r,1.0)
def test_sXBIG(self):
y = stats.spearmanr(X,BIG)
r = y[0]
assert_approx_equal(r,1.0)
def test_sXLITTLE(self):
y = stats.spearmanr(X,LITTLE)
r = y[0]
assert_approx_equal(r,1.0)
def test_sXHUGE(self):
y = stats.spearmanr(X,HUGE)
r = y[0]
assert_approx_equal(r,1.0)
def test_sXTINY(self):
y = stats.spearmanr(X,TINY)
r = y[0]
assert_approx_equal(r,1.0)
def test_sXROUND(self):
y = stats.spearmanr(X,ROUND)
r = y[0]
assert_approx_equal(r,1.0)
def test_sBIGBIG(self):
y = stats.spearmanr(BIG,BIG)
r = y[0]
assert_approx_equal(r,1.0)
def test_sBIGLITTLE(self):
y = stats.spearmanr(BIG,LITTLE)
r = y[0]
assert_approx_equal(r,1.0)
def test_sBIGHUGE(self):
y = stats.spearmanr(BIG,HUGE)
r = y[0]
assert_approx_equal(r,1.0)
def test_sBIGTINY(self):
y = stats.spearmanr(BIG,TINY)
r = y[0]
assert_approx_equal(r,1.0)
def test_sBIGROUND(self):
y = stats.spearmanr(BIG,ROUND)
r = y[0]
assert_approx_equal(r,1.0)
def test_sLITTLELITTLE(self):
y = stats.spearmanr(LITTLE,LITTLE)
r = y[0]
assert_approx_equal(r,1.0)
def test_sLITTLEHUGE(self):
y = stats.spearmanr(LITTLE,HUGE)
r = y[0]
assert_approx_equal(r,1.0)
def test_sLITTLETINY(self):
y = stats.spearmanr(LITTLE,TINY)
r = y[0]
assert_approx_equal(r,1.0)
def test_sLITTLEROUND(self):
y = stats.spearmanr(LITTLE,ROUND)
r = y[0]
assert_approx_equal(r,1.0)
def test_sHUGEHUGE(self):
y = stats.spearmanr(HUGE,HUGE)
r = y[0]
assert_approx_equal(r,1.0)
def test_sHUGETINY(self):
y = stats.spearmanr(HUGE,TINY)
r = y[0]
assert_approx_equal(r,1.0)
def test_sHUGEROUND(self):
y = stats.spearmanr(HUGE,ROUND)
r = y[0]
assert_approx_equal(r,1.0)
def test_sTINYTINY(self):
y = stats.spearmanr(TINY,TINY)
r = y[0]
assert_approx_equal(r,1.0)
def test_sTINYROUND(self):
y = stats.spearmanr(TINY,ROUND)
r = y[0]
assert_approx_equal(r,1.0)
def test_sROUNDROUND(self):
y = stats.spearmanr(ROUND,ROUND)
r = y[0]
assert_approx_equal(r,1.0)
def test_spearmanr_result_attributes(self):
res = stats.spearmanr(X, X)
attributes = ('correlation', 'pvalue')
check_named_results(res, attributes)
def test_spearmanr():
# Cross-check with R:
# cor.test(c(1,2,3,4,5),c(5,6,7,8,7),method="spearmanr")
x1 = [1, 2, 3, 4, 5]
x2 = [5, 6, 7, 8, 7]
expected = (0.82078268166812329, 0.088587005313543798)
res = stats.spearmanr(x1, x2)
assert_approx_equal(res[0], expected[0])
assert_approx_equal(res[1], expected[1])
attributes = ('correlation', 'pvalue')
res = stats.spearmanr(x1, x2)
check_named_results(res, attributes)
# with only ties in one or both inputs
with np.errstate(invalid="ignore"):
assert_equal(stats.spearmanr([2,2,2], [2,2,2]), (np.nan, np.nan))
assert_equal(stats.spearmanr([2,0,2], [2,2,2]), (np.nan, np.nan))
assert_equal(stats.spearmanr([2,2,2], [2,0,2]), (np.nan, np.nan))
# empty arrays provided as input
assert_equal(stats.spearmanr([], []), (np.nan, np.nan))
np.random.seed(7546)
x = np.array([np.random.normal(loc=1, scale=1, size=500),
np.random.normal(loc=1, scale=1, size=500)])
corr = [[1.0, 0.3],
[0.3, 1.0]]
x = np.dot(np.linalg.cholesky(corr), x)
expected = (0.28659685838743354, 6.579862219051161e-11)
res = stats.spearmanr(x[0], x[1])
assert_approx_equal(res[0], expected[0])
assert_approx_equal(res[1], expected[1])
assert_approx_equal(stats.spearmanr([1,1,2], [1,1,2])[0], 1.0)
# test nan_policy
x = np.arange(10.)
x[9] = np.nan
assert_array_equal(stats.spearmanr(x, x), (np.nan, np.nan))
assert_allclose(stats.spearmanr(x, x, nan_policy='omit'),
(1.0, 0))
assert_raises(ValueError, stats.spearmanr, x, x, nan_policy='raise')
assert_raises(ValueError, stats.spearmanr, x, x, nan_policy='foobar')
# test unequal length inputs
x = np.arange(10.)
y = np.arange(20.)
assert_raises(ValueError, stats.spearmanr, x, y)
#test paired value
x1 = [1, 2, 3, 4]
x2 = [8, 7, 6, np.nan]
res1 = stats.spearmanr(x1, x2, nan_policy='omit')
res2 = stats.spearmanr(x1[:3], x2[:3], nan_policy='omit')
assert_equal(res1, res2)
# Regression test for GitHub issue #6061 - Overflow on Windows
x = list(range(2000))
y = list(range(2000))
y[0], y[9] = y[9], y[0]
y[10], y[434] = y[434], y[10]
y[435], y[1509] = y[1509], y[435]
# rho = 1 - 6 * (2 * (9^2 + 424^2 + 1074^2))/(2000 * (2000^2 - 1))
# = 1 - (1 / 500)
# = 0.998
x.append(np.nan)
y.append(3.0)
assert_almost_equal(stats.spearmanr(x, y, nan_policy='omit')[0], 0.998)
class TestCorrSpearmanrTies(object):
"""Some tests of tie-handling by the spearmanr function."""
def test_tie1(self):
# Data
x = [1.0, 2.0, 3.0, 4.0]
y = [1.0, 2.0, 2.0, 3.0]
# Ranks of the data, with tie-handling.
xr = [1.0, 2.0, 3.0, 4.0]
yr = [1.0, 2.5, 2.5, 4.0]
# Result of spearmanr should be the same as applying
# pearsonr to the ranks.
sr = stats.spearmanr(x, y)
pr = stats.pearsonr(xr, yr)
assert_almost_equal(sr, pr)
def test_tie2(self):
# Test tie-handling if inputs contain nan's
# Data without nan's
x1 = [1, 2, 2.5, 2]
y1 = [1, 3, 2.5, 4]
# Same data with nan's
x2 = [1, 2, 2.5, 2, np.nan]
y2 = [1, 3, 2.5, 4, np.nan]
# Results for two data sets should be the same if nan's are ignored
sr1 = stats.spearmanr(x1, y1)
sr2 = stats.spearmanr(x2, y2, nan_policy='omit')
assert_almost_equal(sr1, sr2)
# W.II.E. Tabulate X against X, using BIG as a case weight. The values
# should appear on the diagonal and the total should be 899999955.
# If the table cannot hold these values, forget about working with
# census data. You can also tabulate HUGE against TINY. There is no
# reason a tabulation program should not be able to distinguish
# different values regardless of their magnitude.
# I need to figure out how to do this one.
def test_kendalltau():
# with some ties
# Cross-check with R:
# cor.test(c(12,2,1,12,2),c(1,4,7,1,0),method="kendall",exact=FALSE)
x1 = [12, 2, 1, 12, 2]
x2 = [1, 4, 7, 1, 0]
expected = (-0.47140452079103173, 0.28274545993277478)
res = stats.kendalltau(x1, x2)
assert_approx_equal(res[0], expected[0])
assert_approx_equal(res[1], expected[1])
# test for namedtuple attribute results
attributes = ('correlation', 'pvalue')
res = stats.kendalltau(x1, x2)
check_named_results(res, attributes)
# with only ties in one or both inputs
assert_equal(stats.kendalltau([2,2,2], [2,2,2]), (np.nan, np.nan))
assert_equal(stats.kendalltau([2,0,2], [2,2,2]), (np.nan, np.nan))
assert_equal(stats.kendalltau([2,2,2], [2,0,2]), (np.nan, np.nan))
# empty arrays provided as input
assert_equal(stats.kendalltau([], []), (np.nan, np.nan))
# check with larger arrays
np.random.seed(7546)
x = np.array([np.random.normal(loc=1, scale=1, size=500),
np.random.normal(loc=1, scale=1, size=500)])
corr = [[1.0, 0.3],
[0.3, 1.0]]
x = np.dot(np.linalg.cholesky(corr), x)
expected = (0.19291382765531062, 1.1337095377742629e-10)
res = stats.kendalltau(x[0], x[1])
assert_approx_equal(res[0], expected[0])
assert_approx_equal(res[1], expected[1])
# and do we get a tau of 1 for identical inputs?
assert_approx_equal(stats.kendalltau([1,1,2], [1,1,2])[0], 1.0)
# test nan_policy
x = np.arange(10.)
x[9] = np.nan
assert_array_equal(stats.kendalltau(x, x), (np.nan, np.nan))
assert_allclose(stats.kendalltau(x, x, nan_policy='omit'),
(1.0, 0.00017455009626808976), rtol=1e-06)
assert_raises(ValueError, stats.kendalltau, x, x, nan_policy='raise')
assert_raises(ValueError, stats.kendalltau, x, x, nan_policy='foobar')
# test unequal length inputs
x = np.arange(10.)
y = np.arange(20.)
assert_raises(ValueError, stats.kendalltau, x, y)
# test all ties
tau, p_value = stats.kendalltau([], [])
assert_equal(np.nan, tau)
assert_equal(np.nan, p_value)
tau, p_value = stats.kendalltau([0], [0])
assert_equal(np.nan, tau)
assert_equal(np.nan, p_value)
# Regression test for GitHub issue #6061 - Overflow on Windows
x = np.arange(2000, dtype=float)
x = np.ma.masked_greater(x, 1995)
y = np.arange(2000, dtype=float)
y = np.concatenate((y[1000:], y[:1000]))
assert_(np.isfinite(stats.kendalltau(x,y)[1]))
def test_kendalltau_vs_mstats_basic():
np.random.seed(42)
for s in range(2,10):
a = []
# Generate rankings with ties
for i in range(s):
a += [i]*i
b = list(a)
np.random.shuffle(a)
np.random.shuffle(b)
expected = mstats_basic.kendalltau(a, b)
actual = stats.kendalltau(a, b)
assert_approx_equal(actual[0], expected[0])
assert_approx_equal(actual[1], expected[1])
def test_kendalltau_nan_2nd_arg():
# regression test for gh-6134: nans in the second arg were not handled
x = [1., 2., 3., 4.]
y = [np.nan, 2.4, 3.4, 3.4]
r1 = stats.kendalltau(x, y, nan_policy='omit')
r2 = stats.kendalltau(x[1:], y[1:])
assert_allclose(r1.correlation, r2.correlation, atol=1e-15)
def test_weightedtau():
x = [12, 2, 1, 12, 2]
y = [1, 4, 7, 1, 0]
tau, p_value = stats.weightedtau(x, y)
assert_approx_equal(tau, -0.56694968153682723)
assert_equal(np.nan, p_value)
tau, p_value = stats.weightedtau(x, y, additive=False)
assert_approx_equal(tau, -0.62205716951801038)
assert_equal(np.nan, p_value)
# This must be exactly Kendall's tau
tau, p_value = stats.weightedtau(x, y, weigher=lambda x: 1)
assert_approx_equal(tau, -0.47140452079103173)
assert_equal(np.nan, p_value)
# Asymmetric, ranked version
tau, p_value = stats.weightedtau(x, y, rank=None)
assert_approx_equal(tau, -0.4157652301037516)
assert_equal(np.nan, p_value)
tau, p_value = stats.weightedtau(y, x, rank=None)
assert_approx_equal(tau, -0.7181341329699029)
assert_equal(np.nan, p_value)
tau, p_value = stats.weightedtau(x, y, rank=None, additive=False)
assert_approx_equal(tau, -0.40644850966246893)
assert_equal(np.nan, p_value)
tau, p_value = stats.weightedtau(y, x, rank=None, additive=False)
assert_approx_equal(tau, -0.83766582937355172)
assert_equal(np.nan, p_value)
tau, p_value = stats.weightedtau(x, y, rank=False)
assert_approx_equal(tau, -0.51604397940261848)
assert_equal(np.nan, p_value)
# This must be exactly Kendall's tau
tau, p_value = stats.weightedtau(x, y, rank=True, weigher=lambda x: 1)
assert_approx_equal(tau, -0.47140452079103173)
assert_equal(np.nan, p_value)
tau, p_value = stats.weightedtau(y, x, rank=True, weigher=lambda x: 1)
assert_approx_equal(tau, -0.47140452079103173)
assert_equal(np.nan, p_value)
# Test argument conversion
tau, p_value = stats.weightedtau(np.asarray(x, dtype=np.float64), y)
assert_approx_equal(tau, -0.56694968153682723)
tau, p_value = stats.weightedtau(np.asarray(x, dtype=np.int16), y)
assert_approx_equal(tau, -0.56694968153682723)
tau, p_value = stats.weightedtau(np.asarray(x, dtype=np.float64), np.asarray(y, dtype=np.float64))
assert_approx_equal(tau, -0.56694968153682723)
# All ties
tau, p_value = stats.weightedtau([], [])
assert_equal(np.nan, tau)
assert_equal(np.nan, p_value)
tau, p_value = stats.weightedtau([0], [0])
assert_equal(np.nan, tau)
assert_equal(np.nan, p_value)
# Size mismatches
assert_raises(ValueError, stats.weightedtau, [0, 1], [0, 1, 2])
assert_raises(ValueError, stats.weightedtau, [0, 1], [0, 1], [0])
# NaNs
x = [12, 2, 1, 12, 2]
y = [1, 4, 7, 1, np.nan]
tau, p_value = stats.weightedtau(x, y)
assert_approx_equal(tau, -0.56694968153682723)
x = [12, 2, np.nan, 12, 2]
tau, p_value = stats.weightedtau(x, y)
assert_approx_equal(tau, -0.56694968153682723)
def test_weightedtau_vs_quadratic():
# Trivial quadratic implementation, all parameters mandatory
def wkq(x, y, rank, weigher, add):
tot = conc = disc = u = v = 0
for i in range(len(x)):
for j in range(len(x)):
w = weigher(rank[i]) + weigher(rank[j]) if add else weigher(rank[i]) * weigher(rank[j])
tot += w
if x[i] == x[j]:
u += w
if y[i] == y[j]:
v += w
if x[i] < x[j] and y[i] < y[j] or x[i] > x[j] and y[i] > y[j]:
conc += w
elif x[i] < x[j] and y[i] > y[j] or x[i] > x[j] and y[i] < y[j]:
disc += w
return (conc - disc) / np.sqrt(tot - u) / np.sqrt(tot - v)
np.random.seed(42)
for s in range(3,10):
a = []
# Generate rankings with ties
for i in range(s):
a += [i]*i
b = list(a)
np.random.shuffle(a)
np.random.shuffle(b)
# First pass: use element indices as ranks
rank = np.arange(len(a), dtype=np.intp)
for _ in range(2):
for add in [True, False]:
expected = wkq(a, b, rank, lambda x: 1./(x+1), add)
actual = stats.weightedtau(a, b, rank, lambda x: 1./(x+1), add).correlation
assert_approx_equal(expected, actual)
# Second pass: use a random rank
np.random.shuffle(rank)
class TestFindRepeats(object):
def test_basic(self):
a = [1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 5]
res, nums = stats.find_repeats(a)
assert_array_equal(res, [1, 2, 3, 4])
assert_array_equal(nums, [3, 3, 2, 2])
def test_empty_result(self):
# Check that empty arrays are returned when there are no repeats.
for a in [[10, 20, 50, 30, 40], []]:
repeated, counts = stats.find_repeats(a)
assert_array_equal(repeated, [])
assert_array_equal(counts, [])
class TestRegression(object):
def test_linregressBIGX(self):
# W.II.F. Regress BIG on X.
# The constant should be 99999990 and the regression coefficient should be 1.
y = stats.linregress(X,BIG)
intercept = y[1]
r = y[2]
assert_almost_equal(intercept,99999990)
assert_almost_equal(r,1.0)
def test_regressXX(self):
# W.IV.B. Regress X on X.
# The constant should be exactly 0 and the regression coefficient should be 1.
# This is a perfectly valid regression. The program should not complain.
y = stats.linregress(X,X)
intercept = y[1]
r = y[2]
assert_almost_equal(intercept,0.0)
assert_almost_equal(r,1.0)
# W.IV.C. Regress X on BIG and LITTLE (two predictors). The program
# should tell you that this model is "singular" because BIG and
# LITTLE are linear combinations of each other. Cryptic error
# messages are unacceptable here. Singularity is the most
# fundamental regression error.
# Need to figure out how to handle multiple linear regression. Not obvious
def test_regressZEROX(self):
# W.IV.D. Regress ZERO on X.
# The program should inform you that ZERO has no variance or it should
# go ahead and compute the regression and report a correlation and
# total sum of squares of exactly 0.
y = stats.linregress(X,ZERO)
intercept = y[1]
r = y[2]
assert_almost_equal(intercept,0.0)
assert_almost_equal(r,0.0)
def test_regress_simple(self):
# Regress a line with sinusoidal noise.
x = np.linspace(0, 100, 100)
y = 0.2 * np.linspace(0, 100, 100) + 10
y += np.sin(np.linspace(0, 20, 100))
res = stats.linregress(x, y)
assert_almost_equal(res[4], 2.3957814497838803e-3)
def test_regress_simple_onearg_rows(self):
# Regress a line w sinusoidal noise, with a single input of shape (2, N).
x = np.linspace(0, 100, 100)
y = 0.2 * np.linspace(0, 100, 100) + 10
y += np.sin(np.linspace(0, 20, 100))
rows = np.vstack((x, y))
res = stats.linregress(rows)
assert_almost_equal(res[4], 2.3957814497838803e-3)
def test_regress_simple_onearg_cols(self):
x = np.linspace(0, 100, 100)
y = 0.2 * np.linspace(0, 100, 100) + 10
y += np.sin(np.linspace(0, 20, 100))
cols = np.hstack((np.expand_dims(x, 1), np.expand_dims(y, 1)))
res = stats.linregress(cols)
assert_almost_equal(res[4], 2.3957814497838803e-3)
def test_regress_shape_error(self):
# Check that a single input argument to linregress with wrong shape
# results in a ValueError.
assert_raises(ValueError, stats.linregress, np.ones((3, 3)))
def test_linregress(self):
# compared with multivariate ols with pinv
x = np.arange(11)
y = np.arange(5,16)
y[[(1),(-2)]] -= 1
y[[(0),(-1)]] += 1
res = (1.0, 5.0, 0.98229948625750, 7.45259691e-008, 0.063564172616372733)
assert_array_almost_equal(stats.linregress(x,y),res,decimal=14)
def test_regress_simple_negative_cor(self):
# If the slope of the regression is negative the factor R tend to -1 not 1.
# Sometimes rounding errors makes it < -1 leading to stderr being NaN
a, n = 1e-71, 100000
x = np.linspace(a, 2 * a, n)
y = np.linspace(2 * a, a, n)
stats.linregress(x, y)
res = stats.linregress(x, y)
assert_(res[2] >= -1) # propagated numerical errors were not corrected
assert_almost_equal(res[2], -1) # perfect negative correlation case
assert_(not np.isnan(res[4])) # stderr should stay finite
def test_linregress_result_attributes(self):
# Regress a line with sinusoidal noise.
x = np.linspace(0, 100, 100)
y = 0.2 * np.linspace(0, 100, 100) + 10
y += np.sin(np.linspace(0, 20, 100))
res = stats.linregress(x, y)
attributes = ('slope', 'intercept', 'rvalue', 'pvalue', 'stderr')
check_named_results(res, attributes)
def test_regress_two_inputs(self):
# Regress a simple line formed by two points.
x = np.arange(2)
y = np.arange(3, 5)
res = stats.linregress(x, y)
assert_almost_equal(res[3], 0.0) # non-horizontal line
assert_almost_equal(res[4], 0.0) # zero stderr
def test_regress_two_inputs_horizontal_line(self):
# Regress a horizontal line formed by two points.
x = np.arange(2)
y = np.ones(2)
res = stats.linregress(x, y)
assert_almost_equal(res[3], 1.0) # horizontal line
assert_almost_equal(res[4], 0.0) # zero stderr
def test_nist_norris(self):
x = [0.2, 337.4, 118.2, 884.6, 10.1, 226.5, 666.3, 996.3, 448.6, 777.0,
558.2, 0.4, 0.6, 775.5, 666.9, 338.0, 447.5, 11.6, 556.0, 228.1,
995.8, 887.6, 120.2, 0.3, 0.3, 556.8, 339.1, 887.2, 999.0, 779.0,
11.1, 118.3, 229.2, 669.1, 448.9, 0.5]
y = [0.1, 338.8, 118.1, 888.0, 9.2, 228.1, 668.5, 998.5, 449.1, 778.9,
559.2, 0.3, 0.1, 778.1, 668.8, 339.3, 448.9, 10.8, 557.7, 228.3,
998.0, 888.8, 119.6, 0.3, 0.6, 557.6, 339.3, 888.0, 998.5, 778.9,
10.2, 117.6, 228.9, 668.4, 449.2, 0.2]
# Expected values
exp_slope = 1.00211681802045
exp_intercept = -0.262323073774029
exp_rvalue = 0.999993745883712
actual = stats.linregress(x, y)
assert_almost_equal(actual.slope, exp_slope)
assert_almost_equal(actual.intercept, exp_intercept)
assert_almost_equal(actual.rvalue, exp_rvalue, decimal=5)
def test_empty_input(self):
assert_raises(ValueError, stats.linregress, [], [])
def test_nan_input(self):
x = np.arange(10.)
x[9] = np.nan
with np.errstate(invalid="ignore"):
assert_array_equal(stats.linregress(x, x),
(np.nan, np.nan, np.nan, np.nan, np.nan))
def test_theilslopes():
# Basic slope test.
slope, intercept, lower, upper = stats.theilslopes([0,1,1])
assert_almost_equal(slope, 0.5)
assert_almost_equal(intercept, 0.5)
# Test of confidence intervals.
x = [1, 2, 3, 4, 10, 12, 18]
y = [9, 15, 19, 20, 45, 55, 78]
slope, intercept, lower, upper = stats.theilslopes(y, x, 0.07)
assert_almost_equal(slope, 4)
assert_almost_equal(upper, 4.38, decimal=2)
assert_almost_equal(lower, 3.71, decimal=2)
def test_cumfreq():
x = [1, 4, 2, 1, 3, 1]
cumfreqs, lowlim, binsize, extrapoints = stats.cumfreq(x, numbins=4)
assert_array_almost_equal(cumfreqs, np.array([3., 4., 5., 6.]))
cumfreqs, lowlim, binsize, extrapoints = stats.cumfreq(x, numbins=4,
defaultreallimits=(1.5, 5))
assert_(extrapoints == 3)
# test for namedtuple attribute results
attributes = ('cumcount', 'lowerlimit', 'binsize', 'extrapoints')
res = stats.cumfreq(x, numbins=4, defaultreallimits=(1.5, 5))
check_named_results(res, attributes)
def test_relfreq():
a = np.array([1, 4, 2, 1, 3, 1])
relfreqs, lowlim, binsize, extrapoints = stats.relfreq(a, numbins=4)
assert_array_almost_equal(relfreqs,
array([0.5, 0.16666667, 0.16666667, 0.16666667]))
# test for namedtuple attribute results
attributes = ('frequency', 'lowerlimit', 'binsize', 'extrapoints')
res = stats.relfreq(a, numbins=4)
check_named_results(res, attributes)
# check array_like input is accepted
relfreqs2, lowlim, binsize, extrapoints = stats.relfreq([1, 4, 2, 1, 3, 1],
numbins=4)
assert_array_almost_equal(relfreqs, relfreqs2)
class TestGMean(object):
def test_1D_list(self):
a = (1,2,3,4)
actual = stats.gmean(a)
desired = power(1*2*3*4,1./4.)
assert_almost_equal(actual, desired,decimal=14)
desired1 = stats.gmean(a,axis=-1)
assert_almost_equal(actual, desired1, decimal=14)
def test_1D_array(self):
a = array((1,2,3,4), float32)
actual = stats.gmean(a)
desired = power(1*2*3*4,1./4.)
assert_almost_equal(actual, desired, decimal=7)
desired1 = stats.gmean(a,axis=-1)
assert_almost_equal(actual, desired1, decimal=7)
def test_2D_array_default(self):
a = array(((1,2,3,4),
(1,2,3,4),
(1,2,3,4)))
actual = stats.gmean(a)
desired = array((1,2,3,4))
assert_array_almost_equal(actual, desired, decimal=14)
desired1 = stats.gmean(a,axis=0)
assert_array_almost_equal(actual, desired1, decimal=14)
def test_2D_array_dim1(self):
a = array(((1,2,3,4),
(1,2,3,4),
(1,2,3,4)))
actual = stats.gmean(a, axis=1)
v = power(1*2*3*4,1./4.)
desired = array((v,v,v))
assert_array_almost_equal(actual, desired, decimal=14)
def test_large_values(self):
a = array([1e100, 1e200, 1e300])
actual = stats.gmean(a)
assert_approx_equal(actual, 1e200, significant=13)
class TestHMean(object):
def test_1D_list(self):
a = (1,2,3,4)
actual = stats.hmean(a)
desired = 4. / (1./1 + 1./2 + 1./3 + 1./4)
assert_almost_equal(actual, desired, decimal=14)
desired1 = stats.hmean(array(a),axis=-1)
assert_almost_equal(actual, desired1, decimal=14)
def test_1D_array(self):
a = array((1,2,3,4), float64)
actual = stats.hmean(a)
desired = 4. / (1./1 + 1./2 + 1./3 + 1./4)
assert_almost_equal(actual, desired, decimal=14)
desired1 = stats.hmean(a,axis=-1)
assert_almost_equal(actual, desired1, decimal=14)
def test_2D_array_default(self):
a = array(((1,2,3,4),
(1,2,3,4),
(1,2,3,4)))
actual = stats.hmean(a)
desired = array((1.,2.,3.,4.))
assert_array_almost_equal(actual, desired, decimal=14)
actual1 = stats.hmean(a,axis=0)
assert_array_almost_equal(actual1, desired, decimal=14)
def test_2D_array_dim1(self):
a = array(((1,2,3,4),
(1,2,3,4),
(1,2,3,4)))
v = 4. / (1./1 + 1./2 + 1./3 + 1./4)
desired1 = array((v,v,v))
actual1 = stats.hmean(a, axis=1)
assert_array_almost_equal(actual1, desired1, decimal=14)
class TestScoreatpercentile(object):
def setup_method(self):
self.a1 = [3, 4, 5, 10, -3, -5, 6]
self.a2 = [3, -6, -2, 8, 7, 4, 2, 1]
self.a3 = [3., 4, 5, 10, -3, -5, -6, 7.0]
def test_basic(self):
x = arange(8) * 0.5
assert_equal(stats.scoreatpercentile(x, 0), 0.)
assert_equal(stats.scoreatpercentile(x, 100), 3.5)
assert_equal(stats.scoreatpercentile(x, 50), 1.75)
def test_fraction(self):
scoreatperc = stats.scoreatpercentile
# Test defaults
assert_equal(scoreatperc(list(range(10)), 50), 4.5)
assert_equal(scoreatperc(list(range(10)), 50, (2,7)), 4.5)
assert_equal(scoreatperc(list(range(100)), 50, limit=(1, 8)), 4.5)
assert_equal(scoreatperc(np.array([1, 10,100]), 50, (10,100)), 55)
assert_equal(scoreatperc(np.array([1, 10,100]), 50, (1,10)), 5.5)
# explicitly specify interpolation_method 'fraction' (the default)
assert_equal(scoreatperc(list(range(10)), 50, interpolation_method='fraction'),
4.5)
assert_equal(scoreatperc(list(range(10)), 50, limit=(2, 7),
interpolation_method='fraction'),
4.5)
assert_equal(scoreatperc(list(range(100)), 50, limit=(1, 8),
interpolation_method='fraction'),
4.5)
assert_equal(scoreatperc(np.array([1, 10,100]), 50, (10, 100),
interpolation_method='fraction'),
55)
assert_equal(scoreatperc(np.array([1, 10,100]), 50, (1,10),
interpolation_method='fraction'),
5.5)
def test_lower_higher(self):
scoreatperc = stats.scoreatpercentile
# interpolation_method 'lower'/'higher'
assert_equal(scoreatperc(list(range(10)), 50,
interpolation_method='lower'), 4)
assert_equal(scoreatperc(list(range(10)), 50,
interpolation_method='higher'), 5)
assert_equal(scoreatperc(list(range(10)), 50, (2,7),
interpolation_method='lower'), 4)
assert_equal(scoreatperc(list(range(10)), 50, limit=(2,7),
interpolation_method='higher'), 5)
assert_equal(scoreatperc(list(range(100)), 50, (1,8),
interpolation_method='lower'), 4)
assert_equal(scoreatperc(list(range(100)), 50, (1,8),
interpolation_method='higher'), 5)
assert_equal(scoreatperc(np.array([1, 10, 100]), 50, (10, 100),
interpolation_method='lower'), 10)
assert_equal(scoreatperc(np.array([1, 10, 100]), 50, limit=(10, 100),
interpolation_method='higher'), 100)
assert_equal(scoreatperc(np.array([1, 10, 100]), 50, (1, 10),
interpolation_method='lower'), 1)
assert_equal(scoreatperc(np.array([1, 10, 100]), 50, limit=(1, 10),
interpolation_method='higher'), 10)
def test_sequence_per(self):
x = arange(8) * 0.5
expected = np.array([0, 3.5, 1.75])
res = stats.scoreatpercentile(x, [0, 100, 50])
assert_allclose(res, expected)
assert_(isinstance(res, np.ndarray))
# Test with ndarray. Regression test for gh-2861
assert_allclose(stats.scoreatpercentile(x, np.array([0, 100, 50])),
expected)
# Also test combination of 2-D array, axis not None and array-like per
res2 = stats.scoreatpercentile(np.arange(12).reshape((3,4)),
np.array([0, 1, 100, 100]), axis=1)
expected2 = array([[0, 4, 8],
[0.03, 4.03, 8.03],
[3, 7, 11],
[3, 7, 11]])
assert_allclose(res2, expected2)
def test_axis(self):
scoreatperc = stats.scoreatpercentile
x = arange(12).reshape(3, 4)
assert_equal(scoreatperc(x, (25, 50, 100)), [2.75, 5.5, 11.0])
r0 = [[2, 3, 4, 5], [4, 5, 6, 7], [8, 9, 10, 11]]
assert_equal(scoreatperc(x, (25, 50, 100), axis=0), r0)
r1 = [[0.75, 4.75, 8.75], [1.5, 5.5, 9.5], [3, 7, 11]]
assert_equal(scoreatperc(x, (25, 50, 100), axis=1), r1)
x = array([[1, 1, 1],
[1, 1, 1],
[4, 4, 3],
[1, 1, 1],
[1, 1, 1]])
score = stats.scoreatpercentile(x, 50)
assert_equal(score.shape, ())
assert_equal(score, 1.0)
score = stats.scoreatpercentile(x, 50, axis=0)
assert_equal(score.shape, (3,))
assert_equal(score, [1, 1, 1])
def test_exception(self):
assert_raises(ValueError, stats.scoreatpercentile, [1, 2], 56,
interpolation_method='foobar')
assert_raises(ValueError, stats.scoreatpercentile, [1], 101)
assert_raises(ValueError, stats.scoreatpercentile, [1], -1)
def test_empty(self):
assert_equal(stats.scoreatpercentile([], 50), np.nan)
assert_equal(stats.scoreatpercentile(np.array([[], []]), 50), np.nan)
assert_equal(stats.scoreatpercentile([], [50, 99]), [np.nan, np.nan])
class TestItemfreq(object):
a = [5, 7, 1, 2, 1, 5, 7] * 10
b = [1, 2, 5, 7]
def test_numeric_types(self):
# Check itemfreq works for all dtypes (adapted from np.unique tests)
def _check_itemfreq(dt):
a = np.array(self.a, dt)
with suppress_warnings() as sup:
sup.filter(DeprecationWarning)
v = stats.itemfreq(a)
assert_array_equal(v[:, 0], [1, 2, 5, 7])
assert_array_equal(v[:, 1], np.array([20, 10, 20, 20], dtype=dt))
dtypes = [np.int32, np.int64, np.float32, np.float64,
np.complex64, np.complex128]
for dt in dtypes:
_check_itemfreq(dt)
def test_object_arrays(self):
a, b = self.a, self.b
dt = 'O'
aa = np.empty(len(a), dt)
aa[:] = a
bb = np.empty(len(b), dt)
bb[:] = b
with suppress_warnings() as sup:
sup.filter(DeprecationWarning)
v = stats.itemfreq(aa)
assert_array_equal(v[:, 0], bb)
def test_structured_arrays(self):
a, b = self.a, self.b
dt = [('', 'i'), ('', 'i')]
aa = np.array(list(zip(a, a)), dt)
bb = np.array(list(zip(b, b)), dt)
with suppress_warnings() as sup:
sup.filter(DeprecationWarning)
v = stats.itemfreq(aa)
# Arrays don't compare equal because v[:,0] is object array
assert_equal(tuple(v[2, 0]), tuple(bb[2]))
class TestMode(object):
def test_empty(self):
vals, counts = stats.mode([])
assert_equal(vals, np.array([]))
assert_equal(counts, np.array([]))
def test_scalar(self):
vals, counts = stats.mode(4.)
assert_equal(vals, np.array([4.]))
assert_equal(counts, np.array([1]))
def test_basic(self):
data1 = [3, 5, 1, 10, 23, 3, 2, 6, 8, 6, 10, 6]
vals = stats.mode(data1)
assert_equal(vals[0][0], 6)
assert_equal(vals[1][0], 3)
def test_axes(self):
data1 = [10, 10, 30, 40]
data2 = [10, 10, 10, 10]
data3 = [20, 10, 20, 20]
data4 = [30, 30, 30, 30]
data5 = [40, 30, 30, 30]
arr = np.array([data1, data2, data3, data4, data5])
vals = stats.mode(arr, axis=None)
assert_equal(vals[0], np.array([30]))
assert_equal(vals[1], np.array([8]))
vals = stats.mode(arr, axis=0)
assert_equal(vals[0], np.array([[10, 10, 30, 30]]))
assert_equal(vals[1], np.array([[2, 3, 3, 2]]))
vals = stats.mode(arr, axis=1)
assert_equal(vals[0], np.array([[10], [10], [20], [30], [30]]))
assert_equal(vals[1], np.array([[2], [4], [3], [4], [3]]))
def test_strings(self):
data1 = ['rain', 'showers', 'showers']
with suppress_warnings() as sup:
r = sup.record(RuntimeWarning, ".*checked for nan values")
vals = stats.mode(data1)
assert_equal(len(r), 1)
assert_equal(vals[0][0], 'showers')
assert_equal(vals[1][0], 2)
@pytest.mark.xfail(sys.version_info > (3,), reason='numpy github issue 641')
def test_mixed_objects(self):
objects = [10, True, np.nan, 'hello', 10]
arr = np.empty((5,), dtype=object)
arr[:] = objects
with suppress_warnings() as sup:
r = sup.record(RuntimeWarning, ".*checked for nan values")
vals = stats.mode(arr)
assert_equal(len(r), 1)
assert_equal(vals[0][0], 10)
assert_equal(vals[1][0], 2)
def test_objects(self):
# Python objects must be sortable (le + eq) and have ne defined
# for np.unique to work. hash is for set.
class Point(object):
def __init__(self, x):
self.x = x
def __eq__(self, other):
return self.x == other.x
def __ne__(self, other):
return self.x != other.x
def __lt__(self, other):
return self.x < other.x
def __hash__(self):
return hash(self.x)
points = [Point(x) for x in [1, 2, 3, 4, 3, 2, 2, 2]]
arr = np.empty((8,), dtype=object)
arr[:] = points
assert_(len(set(points)) == 4)
assert_equal(np.unique(arr).shape, (4,))
with suppress_warnings() as sup:
r = sup.record(RuntimeWarning, ".*checked for nan values")
vals = stats.mode(arr)
assert_equal(len(r), 1)
assert_equal(vals[0][0], Point(2))
assert_equal(vals[1][0], 4)
def test_mode_result_attributes(self):
data1 = [3, 5, 1, 10, 23, 3, 2, 6, 8, 6, 10, 6]
data2 = []
actual = stats.mode(data1)
attributes = ('mode', 'count')
check_named_results(actual, attributes)
actual2 = stats.mode(data2)
check_named_results(actual2, attributes)
def test_mode_nan(self):
data1 = [3, np.nan, 5, 1, 10, 23, 3, 2, 6, 8, 6, 10, 6]
actual = stats.mode(data1)
assert_equal(actual, (6, 3))
actual = stats.mode(data1, nan_policy='omit')
assert_equal(actual, (6, 3))
assert_raises(ValueError, stats.mode, data1, nan_policy='raise')
assert_raises(ValueError, stats.mode, data1, nan_policy='foobar')
class TestVariability(object):
testcase = [1,2,3,4]
scalar_testcase = 4.
def test_sem(self):
# This is not in R, so used:
# sqrt(var(testcase)*3/4)/sqrt(3)
# y = stats.sem(self.shoes[0])
# assert_approx_equal(y,0.775177399)
with suppress_warnings() as sup, np.errstate(invalid="ignore"):
sup.filter(RuntimeWarning, "Degrees of freedom <= 0 for slice")
y = stats.sem(self.scalar_testcase)
assert_(np.isnan(y))
y = stats.sem(self.testcase)
assert_approx_equal(y, 0.6454972244)
n = len(self.testcase)
assert_allclose(stats.sem(self.testcase, ddof=0) * np.sqrt(n/(n-2)),
stats.sem(self.testcase, ddof=2))
x = np.arange(10.)
x[9] = np.nan
assert_equal(stats.sem(x), np.nan)
assert_equal(stats.sem(x, nan_policy='omit'), 0.9128709291752769)
assert_raises(ValueError, stats.sem, x, nan_policy='raise')
assert_raises(ValueError, stats.sem, x, nan_policy='foobar')
def test_zmap(self):
# not in R, so tested by using:
# (testcase[i] - mean(testcase, axis=0)) / sqrt(var(testcase) * 3/4)
y = stats.zmap(self.testcase,self.testcase)
desired = ([-1.3416407864999, -0.44721359549996, 0.44721359549996, 1.3416407864999])
assert_array_almost_equal(desired,y,decimal=12)
def test_zmap_axis(self):
# Test use of 'axis' keyword in zmap.
x = np.array([[0.0, 0.0, 1.0, 1.0],
[1.0, 1.0, 1.0, 2.0],
[2.0, 0.0, 2.0, 0.0]])
t1 = 1.0/np.sqrt(2.0/3)
t2 = np.sqrt(3.)/3
t3 = np.sqrt(2.)
z0 = stats.zmap(x, x, axis=0)
z1 = stats.zmap(x, x, axis=1)
z0_expected = [[-t1, -t3/2, -t3/2, 0.0],
[0.0, t3, -t3/2, t1],
[t1, -t3/2, t3, -t1]]
z1_expected = [[-1.0, -1.0, 1.0, 1.0],
[-t2, -t2, -t2, np.sqrt(3.)],
[1.0, -1.0, 1.0, -1.0]]
assert_array_almost_equal(z0, z0_expected)
assert_array_almost_equal(z1, z1_expected)
def test_zmap_ddof(self):
# Test use of 'ddof' keyword in zmap.
x = np.array([[0.0, 0.0, 1.0, 1.0],
[0.0, 1.0, 2.0, 3.0]])
z = stats.zmap(x, x, axis=1, ddof=1)
z0_expected = np.array([-0.5, -0.5, 0.5, 0.5])/(1.0/np.sqrt(3))
z1_expected = np.array([-1.5, -0.5, 0.5, 1.5])/(np.sqrt(5./3))
assert_array_almost_equal(z[0], z0_expected)
assert_array_almost_equal(z[1], z1_expected)
def test_zscore(self):
# not in R, so tested by using:
# (testcase[i] - mean(testcase, axis=0)) / sqrt(var(testcase) * 3/4)
y = stats.zscore(self.testcase)
desired = ([-1.3416407864999, -0.44721359549996, 0.44721359549996, 1.3416407864999])
assert_array_almost_equal(desired,y,decimal=12)
def test_zscore_axis(self):
# Test use of 'axis' keyword in zscore.
x = np.array([[0.0, 0.0, 1.0, 1.0],
[1.0, 1.0, 1.0, 2.0],
[2.0, 0.0, 2.0, 0.0]])
t1 = 1.0/np.sqrt(2.0/3)
t2 = np.sqrt(3.)/3
t3 = np.sqrt(2.)
z0 = stats.zscore(x, axis=0)
z1 = stats.zscore(x, axis=1)
z0_expected = [[-t1, -t3/2, -t3/2, 0.0],
[0.0, t3, -t3/2, t1],
[t1, -t3/2, t3, -t1]]
z1_expected = [[-1.0, -1.0, 1.0, 1.0],
[-t2, -t2, -t2, np.sqrt(3.)],
[1.0, -1.0, 1.0, -1.0]]
assert_array_almost_equal(z0, z0_expected)
assert_array_almost_equal(z1, z1_expected)
def test_zscore_ddof(self):
# Test use of 'ddof' keyword in zscore.
x = np.array([[0.0, 0.0, 1.0, 1.0],
[0.0, 1.0, 2.0, 3.0]])
z = stats.zscore(x, axis=1, ddof=1)
z0_expected = np.array([-0.5, -0.5, 0.5, 0.5])/(1.0/np.sqrt(3))
z1_expected = np.array([-1.5, -0.5, 0.5, 1.5])/(np.sqrt(5./3))
assert_array_almost_equal(z[0], z0_expected)
assert_array_almost_equal(z[1], z1_expected)
class _numpy_version_warn_context_mgr(object):
"""
A simple context maneger class to avoid retyping the same code for
different versions of numpy when the only difference is that older
versions raise warnings.
This manager does not apply for cases where the old code returns
different values.
"""
def __init__(self, min_numpy_version, warning_type, num_warnings):
if NumpyVersion(np.__version__) < min_numpy_version:
self.numpy_is_old = True
self.warning_type = warning_type
self.num_warnings = num_warnings
self.delegate = warnings.catch_warnings(record = True)
else:
self.numpy_is_old = False
def __enter__(self):
if self.numpy_is_old:
self.warn_list = self.delegate.__enter__()
warnings.simplefilter("always")
return None
def __exit__(self, exc_type, exc_value, traceback):
if self.numpy_is_old:
self.delegate.__exit__(exc_type, exc_value, traceback)
_check_warnings(self.warn_list, self.warning_type, self.num_warnings)
def _check_warnings(warn_list, expected_type, expected_len):
"""
Checks that all of the warnings from a list returned by
`warnings.catch_all(record=True)` are of the required type and that the list
contains expected number of warnings.
"""
assert_equal(len(warn_list), expected_len, "number of warnings")
for warn_ in warn_list:
assert_(warn_.category is expected_type)
class TestIQR(object):
def test_basic(self):
x = np.arange(8) * 0.5
np.random.shuffle(x)
assert_equal(stats.iqr(x), 1.75)
def test_api(self):
d = np.ones((5, 5))
stats.iqr(d)
stats.iqr(d, None)
stats.iqr(d, 1)
stats.iqr(d, (0, 1))
stats.iqr(d, None, (10, 90))
stats.iqr(d, None, (30, 20), 'raw')
stats.iqr(d, None, (25, 75), 1.5, 'propagate')
if NumpyVersion(np.__version__) >= '1.9.0a':
stats.iqr(d, None, (50, 50), 'normal', 'raise', 'linear')
stats.iqr(d, None, (25, 75), -0.4, 'omit', 'lower', True)
def test_empty(self):
assert_equal(stats.iqr([]), np.nan)
assert_equal(stats.iqr(np.arange(0)), np.nan)
def test_constant(self):
# Constant array always gives 0
x = np.ones((7, 4))
assert_equal(stats.iqr(x), 0.0)
assert_array_equal(stats.iqr(x, axis=0), np.zeros(4))
assert_array_equal(stats.iqr(x, axis=1), np.zeros(7))
# Even for older versions, 'linear' does not raise a warning
with _numpy_version_warn_context_mgr('1.9.0a', RuntimeWarning, 4):
assert_equal(stats.iqr(x, interpolation='linear'), 0.0)
assert_equal(stats.iqr(x, interpolation='midpoint'), 0.0)
assert_equal(stats.iqr(x, interpolation='nearest'), 0.0)
assert_equal(stats.iqr(x, interpolation='lower'), 0.0)
assert_equal(stats.iqr(x, interpolation='higher'), 0.0)
# 0 only along constant dimensions
# This also tests much of `axis`
y = np.ones((4, 5, 6)) * np.arange(6)
assert_array_equal(stats.iqr(y, axis=0), np.zeros((5, 6)))
assert_array_equal(stats.iqr(y, axis=1), np.zeros((4, 6)))
assert_array_equal(stats.iqr(y, axis=2), 2.5 * np.ones((4, 5)))
assert_array_equal(stats.iqr(y, axis=(0, 1)), np.zeros(6))
assert_array_equal(stats.iqr(y, axis=(0, 2)), 3. * np.ones(5))
assert_array_equal(stats.iqr(y, axis=(1, 2)), 3. * np.ones(4))
def test_scalarlike(self):
x = np.arange(1) + 7.0
assert_equal(stats.iqr(x[0]), 0.0)
assert_equal(stats.iqr(x), 0.0)
if NumpyVersion(np.__version__) >= '1.9.0a':
assert_array_equal(stats.iqr(x, keepdims=True), [0.0])
else:
with warnings.catch_warnings(record=True) as w:
warnings.simplefilter("always")
assert_array_equal(stats.iqr(x, keepdims=True), 0.0)
_check_warnings(w, RuntimeWarning, 1)
def test_2D(self):
x = np.arange(15).reshape((3, 5))
assert_equal(stats.iqr(x), 7.0)
assert_array_equal(stats.iqr(x, axis=0), 5. * np.ones(5))
assert_array_equal(stats.iqr(x, axis=1), 2. * np.ones(3))
assert_array_equal(stats.iqr(x, axis=(0, 1)), 7.0)
assert_array_equal(stats.iqr(x, axis=(1, 0)), 7.0)
def test_axis(self):
# The `axis` keyword is also put through its paces in `test_keepdims`.
o = np.random.normal(size=(71, 23))
x = np.dstack([o] * 10) # x.shape = (71, 23, 10)
q = stats.iqr(o)
assert_equal(stats.iqr(x, axis=(0, 1)), q)
x = np.rollaxis(x, -1, 0) # x.shape = (10, 71, 23)
assert_equal(stats.iqr(x, axis=(2, 1)), q)
x = x.swapaxes(0, 1) # x.shape = (71, 10, 23)
assert_equal(stats.iqr(x, axis=(0, 2)), q)
x = x.swapaxes(0, 1) # x.shape = (10, 71, 23)
assert_equal(stats.iqr(x, axis=(0, 1, 2)),
stats.iqr(x, axis=None))
assert_equal(stats.iqr(x, axis=(0,)),
stats.iqr(x, axis=0))
d = np.arange(3 * 5 * 7 * 11)
# Older versions of numpy only shuffle along axis=0.
# Not sure about newer, don't care.
np.random.shuffle(d)
d = d.reshape((3, 5, 7, 11))
assert_equal(stats.iqr(d, axis=(0, 1, 2))[0],
stats.iqr(d[:,:,:, 0].ravel()))
assert_equal(stats.iqr(d, axis=(0, 1, 3))[1],
stats.iqr(d[:,:, 1,:].ravel()))
assert_equal(stats.iqr(d, axis=(3, 1, -4))[2],
stats.iqr(d[:,:, 2,:].ravel()))
assert_equal(stats.iqr(d, axis=(3, 1, 2))[2],
stats.iqr(d[2,:,:,:].ravel()))
assert_equal(stats.iqr(d, axis=(3, 2))[2, 1],
stats.iqr(d[2, 1,:,:].ravel()))
assert_equal(stats.iqr(d, axis=(1, -2))[2, 1],
stats.iqr(d[2, :, :, 1].ravel()))
assert_equal(stats.iqr(d, axis=(1, 3))[2, 2],
stats.iqr(d[2, :, 2,:].ravel()))
if NumpyVersion(np.__version__) >= '1.9.0a':
assert_raises(IndexError, stats.iqr, d, axis=4)
else:
assert_raises(ValueError, stats.iqr, d, axis=4)
assert_raises(ValueError, stats.iqr, d, axis=(0, 0))
def test_rng(self):
x = np.arange(5)
assert_equal(stats.iqr(x), 2)
assert_equal(stats.iqr(x, rng=(25, 87.5)), 2.5)
assert_equal(stats.iqr(x, rng=(12.5, 75)), 2.5)
assert_almost_equal(stats.iqr(x, rng=(10, 50)), 1.6) # 3-1.4
assert_raises(ValueError, stats.iqr, x, rng=(0, 101))
assert_raises(ValueError, stats.iqr, x, rng=(np.nan, 25))
assert_raises(TypeError, stats.iqr, x, rng=(0, 50, 60))
def test_interpolation(self):
x = np.arange(5)
y = np.arange(4)
# Default
assert_equal(stats.iqr(x), 2)
assert_equal(stats.iqr(y), 1.5)
if NumpyVersion(np.__version__) >= '1.9.0a':
# Linear
assert_equal(stats.iqr(x, interpolation='linear'), 2)
assert_equal(stats.iqr(y, interpolation='linear'), 1.5)
# Higher
assert_equal(stats.iqr(x, interpolation='higher'), 2)
assert_equal(stats.iqr(x, rng=(25, 80), interpolation='higher'), 3)
assert_equal(stats.iqr(y, interpolation='higher'), 2)
# Lower (will generally, but not always be the same as higher)
assert_equal(stats.iqr(x, interpolation='lower'), 2)
assert_equal(stats.iqr(x, rng=(25, 80), interpolation='lower'), 2)
assert_equal(stats.iqr(y, interpolation='lower'), 2)
# Nearest
assert_equal(stats.iqr(x, interpolation='nearest'), 2)
assert_equal(stats.iqr(y, interpolation='nearest'), 1)
# Midpoint
if NumpyVersion(np.__version__) >= '1.11.0a':
assert_equal(stats.iqr(x, interpolation='midpoint'), 2)
assert_equal(stats.iqr(x, rng=(25, 80), interpolation='midpoint'), 2.5)
assert_equal(stats.iqr(y, interpolation='midpoint'), 2)
else:
# midpoint did not work correctly before numpy 1.11.0
assert_equal(stats.iqr(x, interpolation='midpoint'), 2)
assert_equal(stats.iqr(x, rng=(25, 80), interpolation='midpoint'), 2)
assert_equal(stats.iqr(y, interpolation='midpoint'), 2)
else:
with warnings.catch_warnings(record=True) as w:
warnings.simplefilter("always")
# Linear
assert_equal(stats.iqr(x, interpolation='linear'), 2)
assert_equal(stats.iqr(y, interpolation='linear'), 1.5)
# Higher
assert_equal(stats.iqr(x, interpolation='higher'), 2)
assert_almost_equal(stats.iqr(x, rng=(25, 80), interpolation='higher'), 2.2)
assert_equal(stats.iqr(y, interpolation='higher'), 1.5)
# Lower
assert_equal(stats.iqr(x, interpolation='lower'), 2)
assert_almost_equal(stats.iqr(x, rng=(25, 80), interpolation='lower'), 2.2)
assert_equal(stats.iqr(y, interpolation='lower'), 1.5)
# Nearest
assert_equal(stats.iqr(x, interpolation='nearest'), 2)
assert_equal(stats.iqr(y, interpolation='nearest'), 1.5)
# Midpoint
assert_equal(stats.iqr(x, interpolation='midpoint'), 2)
assert_almost_equal(stats.iqr(x, rng=(25, 80), interpolation='midpoint'), 2.2)
assert_equal(stats.iqr(y, interpolation='midpoint'), 1.5)
_check_warnings(w, RuntimeWarning, 11)
if NumpyVersion(np.__version__) >= '1.9.0a':
assert_raises(ValueError, stats.iqr, x, interpolation='foobar')
else:
with warnings.catch_warnings(record=True) as w:
warnings.simplefilter("always")
assert_equal(stats.iqr(x, interpolation='foobar'), 2)
_check_warnings(w, RuntimeWarning, 1)
def test_keepdims(self):
numpy_version = NumpyVersion(np.__version__)
# Also tests most of `axis`
x = np.ones((3, 5, 7, 11))
assert_equal(stats.iqr(x, axis=None, keepdims=False).shape, ())
assert_equal(stats.iqr(x, axis=2, keepdims=False).shape, (3, 5, 11))
assert_equal(stats.iqr(x, axis=(0, 1), keepdims=False).shape, (7, 11))
assert_equal(stats.iqr(x, axis=(0, 3), keepdims=False).shape, (5, 7))
assert_equal(stats.iqr(x, axis=(1,), keepdims=False).shape, (3, 7, 11))
assert_equal(stats.iqr(x, (0, 1, 2, 3), keepdims=False).shape, ())
assert_equal(stats.iqr(x, axis=(0, 1, 3), keepdims=False).shape, (7,))
if numpy_version >= '1.9.0a':
assert_equal(stats.iqr(x, axis=None, keepdims=True).shape, (1, 1, 1, 1))
assert_equal(stats.iqr(x, axis=2, keepdims=True).shape, (3, 5, 1, 11))
assert_equal(stats.iqr(x, axis=(0, 1), keepdims=True).shape, (1, 1, 7, 11))
assert_equal(stats.iqr(x, axis=(0, 3), keepdims=True).shape, (1, 5, 7, 1))
assert_equal(stats.iqr(x, axis=(1,), keepdims=True).shape, (3, 1, 7, 11))
assert_equal(stats.iqr(x, (0, 1, 2, 3), keepdims=True).shape, (1, 1, 1, 1))
assert_equal(stats.iqr(x, axis=(0, 1, 3), keepdims=True).shape, (1, 1, 7, 1))
else:
with warnings.catch_warnings(record=True) as w:
warnings.simplefilter("always")
assert_equal(stats.iqr(x, axis=None, keepdims=True).shape, ())
assert_equal(stats.iqr(x, axis=2, keepdims=True).shape, (3, 5, 11))
assert_equal(stats.iqr(x, axis=(0, 1), keepdims=True).shape, (7, 11))
assert_equal(stats.iqr(x, axis=(0, 3), keepdims=True).shape, (5, 7))
assert_equal(stats.iqr(x, axis=(1,), keepdims=True).shape, (3, 7, 11))
assert_equal(stats.iqr(x, (0, 1, 2, 3), keepdims=True).shape, ())
assert_equal(stats.iqr(x, axis=(0, 1, 3), keepdims=True).shape, (7,))
_check_warnings(w, RuntimeWarning, 7)
def test_nanpolicy(self):
numpy_version = NumpyVersion(np.__version__)
x = np.arange(15.0).reshape((3, 5))
# No NaNs
assert_equal(stats.iqr(x, nan_policy='propagate'), 7)
assert_equal(stats.iqr(x, nan_policy='omit'), 7)
assert_equal(stats.iqr(x, nan_policy='raise'), 7)
# Yes NaNs
x[1, 2] = np.nan
with warnings.catch_warnings(record=True) as w:
warnings.simplefilter("always")
if numpy_version < '1.10.0a':
# Fails over to mishmash of omit/propagate, but mostly omit
# The first case showcases the "incorrect" behavior of np.percentile
assert_equal(stats.iqr(x, nan_policy='propagate'), 8)
assert_equal(stats.iqr(x, axis=0, nan_policy='propagate'), [5, 5, np.nan, 5, 5])
if numpy_version < '1.9.0a':
assert_equal(stats.iqr(x, axis=1, nan_policy='propagate'), [2, 3, 2])
else:
# some fixes to percentile nan handling in 1.9
assert_equal(stats.iqr(x, axis=1, nan_policy='propagate'), [2, np.nan, 2])
_check_warnings(w, RuntimeWarning, 3)
else:
assert_equal(stats.iqr(x, nan_policy='propagate'), np.nan)
assert_equal(stats.iqr(x, axis=0, nan_policy='propagate'), [5, 5, np.nan, 5, 5])
assert_equal(stats.iqr(x, axis=1, nan_policy='propagate'), [2, np.nan, 2])
with warnings.catch_warnings(record=True) as w:
warnings.simplefilter("always")
if numpy_version < '1.9.0a':
# Fails over to mishmash of omit/propagate, but mostly omit
assert_equal(stats.iqr(x, nan_policy='omit'), 8)
assert_equal(stats.iqr(x, axis=0, nan_policy='omit'), [5, 5, np.nan, 5, 5])
assert_equal(stats.iqr(x, axis=1, nan_policy='omit'), [2, 3, 2])
_check_warnings(w, RuntimeWarning, 3)
else:
assert_equal(stats.iqr(x, nan_policy='omit'), 7.5)
assert_equal(stats.iqr(x, axis=0, nan_policy='omit'), 5 * np.ones(5))
assert_equal(stats.iqr(x, axis=1, nan_policy='omit'), [2, 2.5, 2])
assert_raises(ValueError, stats.iqr, x, nan_policy='raise')
assert_raises(ValueError, stats.iqr, x, axis=0, nan_policy='raise')
assert_raises(ValueError, stats.iqr, x, axis=1, nan_policy='raise')
# Bad policy
assert_raises(ValueError, stats.iqr, x, nan_policy='barfood')
def test_scale(self):
numpy_version = NumpyVersion(np.__version__)
x = np.arange(15.0).reshape((3, 5))
# No NaNs
assert_equal(stats.iqr(x, scale='raw'), 7)
assert_almost_equal(stats.iqr(x, scale='normal'), 7 / 1.3489795)
assert_equal(stats.iqr(x, scale=2.0), 3.5)
# Yes NaNs
x[1, 2] = np.nan
with warnings.catch_warnings(record=True) as w:
warnings.simplefilter("always")
if numpy_version < '1.10.0a':
# Fails over to mishmash of omit/propagate, but mostly omit
assert_equal(stats.iqr(x, scale='raw', nan_policy='propagate'), 8)
assert_almost_equal(stats.iqr(x, scale='normal',
nan_policy='propagate'),
8 / 1.3489795)
assert_equal(stats.iqr(x, scale=2.0, nan_policy='propagate'), 4)
# axis=1 chosen to show behavior with both nans and without
if numpy_version < '1.9.0a':
assert_equal(stats.iqr(x, axis=1, nan_policy='propagate'), [2, 3, 2])
assert_almost_equal(stats.iqr(x, axis=1, scale='normal',
nan_policy='propagate'),
np.array([2, 3, 2]) / 1.3489795)
assert_equal(stats.iqr(x, axis=1, scale=2.0,
nan_policy='propagate'), [1, 1.5, 1])
else:
# some fixes to percentile nan handling in 1.9
assert_equal(stats.iqr(x, axis=1, nan_policy='propagate'), [2, np.nan, 2])
assert_almost_equal(stats.iqr(x, axis=1, scale='normal',
nan_policy='propagate'),
np.array([2, np.nan, 2]) / 1.3489795)
assert_equal(stats.iqr(x, axis=1, scale=2.0,
nan_policy='propagate'), [1, np.nan, 1])
_check_warnings(w, RuntimeWarning, 6)
else:
assert_equal(stats.iqr(x, scale='raw', nan_policy='propagate'), np.nan)
assert_equal(stats.iqr(x, scale='normal', nan_policy='propagate'), np.nan)
assert_equal(stats.iqr(x, scale=2.0, nan_policy='propagate'), np.nan)
# axis=1 chosen to show behavior with both nans and without
assert_equal(stats.iqr(x, axis=1, scale='raw',
nan_policy='propagate'), [2, np.nan, 2])
assert_almost_equal(stats.iqr(x, axis=1, scale='normal',
nan_policy='propagate'),
np.array([2, np.nan, 2]) / 1.3489795)
assert_equal(stats.iqr(x, axis=1, scale=2.0, nan_policy='propagate'),
[1, np.nan, 1])
_check_warnings(w, RuntimeWarning, 6)
if numpy_version < '1.9.0a':
with warnings.catch_warnings(record=True) as w:
warnings.simplefilter("always")
# Fails over to mishmash of omit/propagate, but mostly omit
assert_equal(stats.iqr(x, scale='raw', nan_policy='omit'), 8)
assert_almost_equal(stats.iqr(x, scale='normal', nan_policy='omit'),
8 / 1.3489795)
assert_equal(stats.iqr(x, scale=2.0, nan_policy='omit'), 4)
_check_warnings(w, RuntimeWarning, 3)
else:
assert_equal(stats.iqr(x, scale='raw', nan_policy='omit'), 7.5)
assert_almost_equal(stats.iqr(x, scale='normal', nan_policy='omit'),
7.5 / 1.3489795)
assert_equal(stats.iqr(x, scale=2.0, nan_policy='omit'), 3.75)
# Bad scale
assert_raises(ValueError, stats.iqr, x, scale='foobar')
class TestMoments(object):
"""
Comparison numbers are found using R v.1.5.1
note that length(testcase) = 4
testmathworks comes from documentation for the
Statistics Toolbox for Matlab and can be found at both
http://www.mathworks.com/access/helpdesk/help/toolbox/stats/kurtosis.shtml
http://www.mathworks.com/access/helpdesk/help/toolbox/stats/skewness.shtml
Note that both test cases came from here.
"""
testcase = [1,2,3,4]
scalar_testcase = 4.
np.random.seed(1234)
testcase_moment_accuracy = np.random.rand(42)
testmathworks = [1.165, 0.6268, 0.0751, 0.3516, -0.6965]
def test_moment(self):
# mean((testcase-mean(testcase))**power,axis=0),axis=0))**power))
y = stats.moment(self.scalar_testcase)
assert_approx_equal(y, 0.0)
y = stats.moment(self.testcase, 0)
assert_approx_equal(y, 1.0)
y = stats.moment(self.testcase, 1)
assert_approx_equal(y, 0.0, 10)
y = stats.moment(self.testcase, 2)
assert_approx_equal(y, 1.25)
y = stats.moment(self.testcase, 3)
assert_approx_equal(y, 0.0)
y = stats.moment(self.testcase, 4)
assert_approx_equal(y, 2.5625)
# check array_like input for moment
y = stats.moment(self.testcase, [1, 2, 3, 4])
assert_allclose(y, [0, 1.25, 0, 2.5625])
# check moment input consists only of integers
y = stats.moment(self.testcase, 0.0)
assert_approx_equal(y, 1.0)
assert_raises(ValueError, stats.moment, self.testcase, 1.2)
y = stats.moment(self.testcase, [1.0, 2, 3, 4.0])
assert_allclose(y, [0, 1.25, 0, 2.5625])
# test empty input
y = stats.moment([])
assert_equal(y, np.nan)
x = np.arange(10.)
x[9] = np.nan
assert_equal(stats.moment(x, 2), np.nan)
assert_almost_equal(stats.moment(x, nan_policy='omit'), 0.0)
assert_raises(ValueError, stats.moment, x, nan_policy='raise')
assert_raises(ValueError, stats.moment, x, nan_policy='foobar')
def test_moment_propagate_nan(self):
# Check that the shape of the result is the same for inputs
# with and without nans, cf gh-5817
a = np.arange(8).reshape(2, -1).astype(float)
a[1, 0] = np.nan
mm = stats.moment(a, 2, axis=1, nan_policy="propagate")
np.testing.assert_allclose(mm, [1.25, np.nan], atol=1e-15)
def test_variation(self):
# variation = samplestd / mean
y = stats.variation(self.scalar_testcase)
assert_approx_equal(y, 0.0)
y = stats.variation(self.testcase)
assert_approx_equal(y, 0.44721359549996, 10)
x = np.arange(10.)
x[9] = np.nan
assert_equal(stats.variation(x), np.nan)
assert_almost_equal(stats.variation(x, nan_policy='omit'),
0.6454972243679028)
assert_raises(ValueError, stats.variation, x, nan_policy='raise')
assert_raises(ValueError, stats.variation, x, nan_policy='foobar')
def test_variation_propagate_nan(self):
# Check that the shape of the result is the same for inputs
# with and without nans, cf gh-5817
a = np.arange(8).reshape(2, -1).astype(float)
a[1, 0] = np.nan
vv = stats.variation(a, axis=1, nan_policy="propagate")
np.testing.assert_allclose(vv, [0.7453559924999299, np.nan], atol=1e-15)
def test_skewness(self):
# Scalar test case
y = stats.skew(self.scalar_testcase)
assert_approx_equal(y, 0.0)
# sum((testmathworks-mean(testmathworks,axis=0))**3,axis=0) /
# ((sqrt(var(testmathworks)*4/5))**3)/5
y = stats.skew(self.testmathworks)
assert_approx_equal(y, -0.29322304336607, 10)
y = stats.skew(self.testmathworks, bias=0)
assert_approx_equal(y, -0.437111105023940, 10)
y = stats.skew(self.testcase)
assert_approx_equal(y, 0.0, 10)
x = np.arange(10.)
x[9] = np.nan
with np.errstate(invalid='ignore'):
assert_equal(stats.skew(x), np.nan)
assert_equal(stats.skew(x, nan_policy='omit'), 0.)
assert_raises(ValueError, stats.skew, x, nan_policy='raise')
assert_raises(ValueError, stats.skew, x, nan_policy='foobar')
def test_skewness_scalar(self):
# `skew` must return a scalar for 1-dim input
assert_equal(stats.skew(arange(10)), 0.0)
def test_skew_propagate_nan(self):
# Check that the shape of the result is the same for inputs
# with and without nans, cf gh-5817
a = np.arange(8).reshape(2, -1).astype(float)
a[1, 0] = np.nan
with np.errstate(invalid='ignore'):
s = stats.skew(a, axis=1, nan_policy="propagate")
np.testing.assert_allclose(s, [0, np.nan], atol=1e-15)
def test_kurtosis(self):
# Scalar test case
y = stats.kurtosis(self.scalar_testcase)
assert_approx_equal(y, -3.0)
# sum((testcase-mean(testcase,axis=0))**4,axis=0)/((sqrt(var(testcase)*3/4))**4)/4
# sum((test2-mean(testmathworks,axis=0))**4,axis=0)/((sqrt(var(testmathworks)*4/5))**4)/5
# Set flags for axis = 0 and
# fisher=0 (Pearson's defn of kurtosis for compatibility with Matlab)
y = stats.kurtosis(self.testmathworks, 0, fisher=0, bias=1)
assert_approx_equal(y, 2.1658856802973, 10)
# Note that MATLAB has confusing docs for the following case
# kurtosis(x,0) gives an unbiased estimate of Pearson's skewness
# kurtosis(x) gives a biased estimate of Fisher's skewness (Pearson-3)
# The MATLAB docs imply that both should give Fisher's
y = stats.kurtosis(self.testmathworks, fisher=0, bias=0)
assert_approx_equal(y, 3.663542721189047, 10)
y = stats.kurtosis(self.testcase, 0, 0)
assert_approx_equal(y, 1.64)
x = np.arange(10.)
x[9] = np.nan
assert_equal(stats.kurtosis(x), np.nan)
assert_almost_equal(stats.kurtosis(x, nan_policy='omit'), -1.230000)
assert_raises(ValueError, stats.kurtosis, x, nan_policy='raise')
assert_raises(ValueError, stats.kurtosis, x, nan_policy='foobar')
def test_kurtosis_array_scalar(self):
assert_equal(type(stats.kurtosis([1,2,3])), float)
def test_kurtosis_propagate_nan(self):
# Check that the shape of the result is the same for inputs
# with and without nans, cf gh-5817
a = np.arange(8).reshape(2, -1).astype(float)
a[1, 0] = np.nan
k = stats.kurtosis(a, axis=1, nan_policy="propagate")
np.testing.assert_allclose(k, [-1.36, np.nan], atol=1e-15)
def test_moment_accuracy(self):
# 'moment' must have a small enough error compared to the slower
# but very accurate numpy.power() implementation.
tc_no_mean = self.testcase_moment_accuracy - \
np.mean(self.testcase_moment_accuracy)
assert_allclose(np.power(tc_no_mean, 42).mean(),
stats.moment(self.testcase_moment_accuracy, 42))
class TestStudentTest(object):
X1 = np.array([-1, 0, 1])
X2 = np.array([0, 1, 2])
T1_0 = 0
P1_0 = 1
T1_1 = -1.732051
P1_1 = 0.2254033
T1_2 = -3.464102
P1_2 = 0.0741799
T2_0 = 1.732051
P2_0 = 0.2254033
def test_onesample(self):
with suppress_warnings() as sup, np.errstate(invalid="ignore"):
sup.filter(RuntimeWarning, "Degrees of freedom <= 0 for slice")
t, p = stats.ttest_1samp(4., 3.)
assert_(np.isnan(t))
assert_(np.isnan(p))
t, p = stats.ttest_1samp(self.X1, 0)
assert_array_almost_equal(t, self.T1_0)
assert_array_almost_equal(p, self.P1_0)
res = stats.ttest_1samp(self.X1, 0)
attributes = ('statistic', 'pvalue')
check_named_results(res, attributes)
t, p = stats.ttest_1samp(self.X2, 0)
assert_array_almost_equal(t, self.T2_0)
assert_array_almost_equal(p, self.P2_0)
t, p = stats.ttest_1samp(self.X1, 1)
assert_array_almost_equal(t, self.T1_1)
assert_array_almost_equal(p, self.P1_1)
t, p = stats.ttest_1samp(self.X1, 2)
assert_array_almost_equal(t, self.T1_2)
assert_array_almost_equal(p, self.P1_2)
# check nan policy
np.random.seed(7654567)
x = stats.norm.rvs(loc=5, scale=10, size=51)
x[50] = np.nan
with np.errstate(invalid="ignore"):
assert_array_equal(stats.ttest_1samp(x, 5.0), (np.nan, np.nan))
assert_array_almost_equal(stats.ttest_1samp(x, 5.0, nan_policy='omit'),
(-1.6412624074367159, 0.107147027334048005))
assert_raises(ValueError, stats.ttest_1samp, x, 5.0, nan_policy='raise')
assert_raises(ValueError, stats.ttest_1samp, x, 5.0,
nan_policy='foobar')
def test_percentileofscore():
pcos = stats.percentileofscore
assert_equal(pcos([1,2,3,4,5,6,7,8,9,10],4), 40.0)
for (kind, result) in [('mean', 35.0),
('strict', 30.0),
('weak', 40.0)]:
assert_equal(pcos(np.arange(10) + 1, 4, kind=kind), result)
# multiple - 2
for (kind, result) in [('rank', 45.0),
('strict', 30.0),
('weak', 50.0),
('mean', 40.0)]:
assert_equal(pcos([1,2,3,4,4,5,6,7,8,9], 4, kind=kind), result)
# multiple - 3
assert_equal(pcos([1,2,3,4,4,4,5,6,7,8], 4), 50.0)
for (kind, result) in [('rank', 50.0),
('mean', 45.0),
('strict', 30.0),
('weak', 60.0)]:
assert_equal(pcos([1,2,3,4,4,4,5,6,7,8], 4, kind=kind), result)
# missing
for kind in ('rank', 'mean', 'strict', 'weak'):
assert_equal(pcos([1,2,3,5,6,7,8,9,10,11], 4, kind=kind), 30)
# larger numbers
for (kind, result) in [('mean', 35.0),
('strict', 30.0),
('weak', 40.0)]:
assert_equal(
pcos([10, 20, 30, 40, 50, 60, 70, 80, 90, 100], 40,
kind=kind), result)
for (kind, result) in [('mean', 45.0),
('strict', 30.0),
('weak', 60.0)]:
assert_equal(
pcos([10, 20, 30, 40, 40, 40, 50, 60, 70, 80],
40, kind=kind), result)
for kind in ('rank', 'mean', 'strict', 'weak'):
assert_equal(
pcos([10, 20, 30, 50, 60, 70, 80, 90, 100, 110],
40, kind=kind), 30.0)
# boundaries
for (kind, result) in [('rank', 10.0),
('mean', 5.0),
('strict', 0.0),
('weak', 10.0)]:
assert_equal(
pcos([10, 20, 30, 50, 60, 70, 80, 90, 100, 110],
10, kind=kind), result)
for (kind, result) in [('rank', 100.0),
('mean', 95.0),
('strict', 90.0),
('weak', 100.0)]:
assert_equal(
pcos([10, 20, 30, 50, 60, 70, 80, 90, 100, 110],
110, kind=kind), result)
# out of bounds
for (kind, score, result) in [('rank', 200, 100.0),
('mean', 200, 100.0),
('mean', 0, 0.0)]:
assert_equal(
pcos([10, 20, 30, 50, 60, 70, 80, 90, 100, 110],
score, kind=kind), result)
assert_raises(ValueError, pcos, [1, 2, 3, 3, 4], 3, kind='unrecognized')
PowerDivCase = namedtuple('Case', ['f_obs', 'f_exp', 'ddof', 'axis',
'chi2', # Pearson's
'log', # G-test (log-likelihood)
'mod_log', # Modified log-likelihood
'cr', # Cressie-Read (lambda=2/3)
])
# The details of the first two elements in power_div_1d_cases are used
# in a test in TestPowerDivergence. Check that code before making
# any changes here.
power_div_1d_cases = [
# Use the default f_exp.
PowerDivCase(f_obs=[4, 8, 12, 8], f_exp=None, ddof=0, axis=None,
chi2=4,
log=2*(4*np.log(4/8) + 12*np.log(12/8)),
mod_log=2*(8*np.log(8/4) + 8*np.log(8/12)),
cr=(4*((4/8)**(2/3) - 1) + 12*((12/8)**(2/3) - 1))/(5/9)),
# Give a non-uniform f_exp.
PowerDivCase(f_obs=[4, 8, 12, 8], f_exp=[2, 16, 12, 2], ddof=0, axis=None,
chi2=24,
log=2*(4*np.log(4/2) + 8*np.log(8/16) + 8*np.log(8/2)),
mod_log=2*(2*np.log(2/4) + 16*np.log(16/8) + 2*np.log(2/8)),
cr=(4*((4/2)**(2/3) - 1) + 8*((8/16)**(2/3) - 1) +
8*((8/2)**(2/3) - 1))/(5/9)),
# f_exp is a scalar.
PowerDivCase(f_obs=[4, 8, 12, 8], f_exp=8, ddof=0, axis=None,
chi2=4,
log=2*(4*np.log(4/8) + 12*np.log(12/8)),
mod_log=2*(8*np.log(8/4) + 8*np.log(8/12)),
cr=(4*((4/8)**(2/3) - 1) + 12*((12/8)**(2/3) - 1))/(5/9)),
# f_exp equal to f_obs.
PowerDivCase(f_obs=[3, 5, 7, 9], f_exp=[3, 5, 7, 9], ddof=0, axis=0,
chi2=0, log=0, mod_log=0, cr=0),
]
power_div_empty_cases = [
# Shape is (0,)--a data set with length 0. The computed
# test statistic should be 0.
PowerDivCase(f_obs=[],
f_exp=None, ddof=0, axis=0,
chi2=0, log=0, mod_log=0, cr=0),
# Shape is (0, 3). This is 3 data sets, but each data set has
# length 0, so the computed test statistic should be [0, 0, 0].
PowerDivCase(f_obs=np.array([[],[],[]]).T,
f_exp=None, ddof=0, axis=0,
chi2=[0, 0, 0],
log=[0, 0, 0],
mod_log=[0, 0, 0],
cr=[0, 0, 0]),
# Shape is (3, 0). This represents an empty collection of
# data sets in which each data set has length 3. The test
# statistic should be an empty array.
PowerDivCase(f_obs=np.array([[],[],[]]),
f_exp=None, ddof=0, axis=0,
chi2=[],
log=[],
mod_log=[],
cr=[]),
]
class TestPowerDivergence(object):
def check_power_divergence(self, f_obs, f_exp, ddof, axis, lambda_,
expected_stat):
f_obs = np.asarray(f_obs)
if axis is None:
num_obs = f_obs.size
else:
b = np.broadcast(f_obs, f_exp)
num_obs = b.shape[axis]
with suppress_warnings() as sup:
sup.filter(RuntimeWarning, "Mean of empty slice")
stat, p = stats.power_divergence(
f_obs=f_obs, f_exp=f_exp, ddof=ddof,
axis=axis, lambda_=lambda_)
assert_allclose(stat, expected_stat)
if lambda_ == 1 or lambda_ == "pearson":
# Also test stats.chisquare.
stat, p = stats.chisquare(f_obs=f_obs, f_exp=f_exp, ddof=ddof,
axis=axis)
assert_allclose(stat, expected_stat)
ddof = np.asarray(ddof)
expected_p = stats.distributions.chi2.sf(expected_stat,
num_obs - 1 - ddof)
assert_allclose(p, expected_p)
def test_basic(self):
for case in power_div_1d_cases:
self.check_power_divergence(
case.f_obs, case.f_exp, case.ddof, case.axis,
None, case.chi2)
self.check_power_divergence(
case.f_obs, case.f_exp, case.ddof, case.axis,
"pearson", case.chi2)
self.check_power_divergence(
case.f_obs, case.f_exp, case.ddof, case.axis,
1, case.chi2)
self.check_power_divergence(
case.f_obs, case.f_exp, case.ddof, case.axis,
"log-likelihood", case.log)
self.check_power_divergence(
case.f_obs, case.f_exp, case.ddof, case.axis,
"mod-log-likelihood", case.mod_log)
self.check_power_divergence(
case.f_obs, case.f_exp, case.ddof, case.axis,
"cressie-read", case.cr)
self.check_power_divergence(
case.f_obs, case.f_exp, case.ddof, case.axis,
2/3, case.cr)
def test_basic_masked(self):
for case in power_div_1d_cases:
mobs = np.ma.array(case.f_obs)
self.check_power_divergence(
mobs, case.f_exp, case.ddof, case.axis,
None, case.chi2)
self.check_power_divergence(
mobs, case.f_exp, case.ddof, case.axis,
"pearson", case.chi2)
self.check_power_divergence(
mobs, case.f_exp, case.ddof, case.axis,
1, case.chi2)
self.check_power_divergence(
mobs, case.f_exp, case.ddof, case.axis,
"log-likelihood", case.log)
self.check_power_divergence(
mobs, case.f_exp, case.ddof, case.axis,
"mod-log-likelihood", case.mod_log)
self.check_power_divergence(
mobs, case.f_exp, case.ddof, case.axis,
"cressie-read", case.cr)
self.check_power_divergence(
mobs, case.f_exp, case.ddof, case.axis,
2/3, case.cr)
def test_axis(self):
case0 = power_div_1d_cases[0]
case1 = power_div_1d_cases[1]
f_obs = np.vstack((case0.f_obs, case1.f_obs))
f_exp = np.vstack((np.ones_like(case0.f_obs)*np.mean(case0.f_obs),
case1.f_exp))
# Check the four computational code paths in power_divergence
# using a 2D array with axis=1.
self.check_power_divergence(
f_obs, f_exp, 0, 1,
"pearson", [case0.chi2, case1.chi2])
self.check_power_divergence(
f_obs, f_exp, 0, 1,
"log-likelihood", [case0.log, case1.log])
self.check_power_divergence(
f_obs, f_exp, 0, 1,
"mod-log-likelihood", [case0.mod_log, case1.mod_log])
self.check_power_divergence(
f_obs, f_exp, 0, 1,
"cressie-read", [case0.cr, case1.cr])
# Reshape case0.f_obs to shape (2,2), and use axis=None.
# The result should be the same.
self.check_power_divergence(
np.array(case0.f_obs).reshape(2, 2), None, 0, None,
"pearson", case0.chi2)
def test_ddof_broadcasting(self):
# Test that ddof broadcasts correctly.
# ddof does not affect the test statistic. It is broadcast
# with the computed test statistic for the computation of
# the p value.
case0 = power_div_1d_cases[0]
case1 = power_div_1d_cases[1]
# Create 4x2 arrays of observed and expected frequencies.
f_obs = np.vstack((case0.f_obs, case1.f_obs)).T
f_exp = np.vstack((np.ones_like(case0.f_obs)*np.mean(case0.f_obs),
case1.f_exp)).T
expected_chi2 = [case0.chi2, case1.chi2]
# ddof has shape (2, 1). This is broadcast with the computed
# statistic, so p will have shape (2,2).
ddof = np.array([[0], [1]])
stat, p = stats.power_divergence(f_obs, f_exp, ddof=ddof)
assert_allclose(stat, expected_chi2)
# Compute the p values separately, passing in scalars for ddof.
stat0, p0 = stats.power_divergence(f_obs, f_exp, ddof=ddof[0,0])
stat1, p1 = stats.power_divergence(f_obs, f_exp, ddof=ddof[1,0])
assert_array_equal(p, np.vstack((p0, p1)))
def test_empty_cases(self):
with warnings.catch_warnings():
for case in power_div_empty_cases:
self.check_power_divergence(
case.f_obs, case.f_exp, case.ddof, case.axis,
"pearson", case.chi2)
self.check_power_divergence(
case.f_obs, case.f_exp, case.ddof, case.axis,
"log-likelihood", case.log)
self.check_power_divergence(
case.f_obs, case.f_exp, case.ddof, case.axis,
"mod-log-likelihood", case.mod_log)
self.check_power_divergence(
case.f_obs, case.f_exp, case.ddof, case.axis,
"cressie-read", case.cr)
def test_power_divergence_result_attributes(self):
f_obs = power_div_1d_cases[0].f_obs
f_exp = power_div_1d_cases[0].f_exp
ddof = power_div_1d_cases[0].ddof
axis = power_div_1d_cases[0].axis
res = stats.power_divergence(f_obs=f_obs, f_exp=f_exp, ddof=ddof,
axis=axis, lambda_="pearson")
attributes = ('statistic', 'pvalue')
check_named_results(res, attributes)
def test_chisquare_masked_arrays():
# Test masked arrays.
obs = np.array([[8, 8, 16, 32, -1], [-1, -1, 3, 4, 5]]).T
mask = np.array([[0, 0, 0, 0, 1], [1, 1, 0, 0, 0]]).T
mobs = np.ma.masked_array(obs, mask)
expected_chisq = np.array([24.0, 0.5])
expected_g = np.array([2*(2*8*np.log(0.5) + 32*np.log(2.0)),
2*(3*np.log(0.75) + 5*np.log(1.25))])
chi2 = stats.distributions.chi2
chisq, p = stats.chisquare(mobs)
mat.assert_array_equal(chisq, expected_chisq)
mat.assert_array_almost_equal(p, chi2.sf(expected_chisq,
mobs.count(axis=0) - 1))
g, p = stats.power_divergence(mobs, lambda_='log-likelihood')
mat.assert_array_almost_equal(g, expected_g, decimal=15)
mat.assert_array_almost_equal(p, chi2.sf(expected_g,
mobs.count(axis=0) - 1))
chisq, p = stats.chisquare(mobs.T, axis=1)
mat.assert_array_equal(chisq, expected_chisq)
mat.assert_array_almost_equal(p, chi2.sf(expected_chisq,
mobs.T.count(axis=1) - 1))
g, p = stats.power_divergence(mobs.T, axis=1, lambda_="log-likelihood")
mat.assert_array_almost_equal(g, expected_g, decimal=15)
mat.assert_array_almost_equal(p, chi2.sf(expected_g,
mobs.count(axis=0) - 1))
obs1 = np.ma.array([3, 5, 6, 99, 10], mask=[0, 0, 0, 1, 0])
exp1 = np.ma.array([2, 4, 8, 10, 99], mask=[0, 0, 0, 0, 1])
chi2, p = stats.chisquare(obs1, f_exp=exp1)
# Because of the mask at index 3 of obs1 and at index 4 of exp1,
# only the first three elements are included in the calculation
# of the statistic.
mat.assert_array_equal(chi2, 1/2 + 1/4 + 4/8)
# When axis=None, the two values should have type np.float64.
chisq, p = stats.chisquare(np.ma.array([1,2,3]), axis=None)
assert_(isinstance(chisq, np.float64))
assert_(isinstance(p, np.float64))
assert_equal(chisq, 1.0)
assert_almost_equal(p, stats.distributions.chi2.sf(1.0, 2))
# Empty arrays:
# A data set with length 0 returns a masked scalar.
with np.errstate(invalid='ignore'):
with suppress_warnings() as sup:
sup.filter(RuntimeWarning, "Mean of empty slice")
chisq, p = stats.chisquare(np.ma.array([]))
assert_(isinstance(chisq, np.ma.MaskedArray))
assert_equal(chisq.shape, ())
assert_(chisq.mask)
empty3 = np.ma.array([[],[],[]])
# empty3 is a collection of 0 data sets (whose lengths would be 3, if
# there were any), so the return value is an array with length 0.
chisq, p = stats.chisquare(empty3)
assert_(isinstance(chisq, np.ma.MaskedArray))
mat.assert_array_equal(chisq, [])
# empty3.T is an array containing 3 data sets, each with length 0,
# so an array of size (3,) is returned, with all values masked.
with np.errstate(invalid='ignore'):
with suppress_warnings() as sup:
sup.filter(RuntimeWarning, "Mean of empty slice")
chisq, p = stats.chisquare(empty3.T)
assert_(isinstance(chisq, np.ma.MaskedArray))
assert_equal(chisq.shape, (3,))
assert_(np.all(chisq.mask))
def test_power_divergence_against_cressie_read_data():
# Test stats.power_divergence against tables 4 and 5 from
# Cressie and Read, "Multimonial Goodness-of-Fit Tests",
# J. R. Statist. Soc. B (1984), Vol 46, No. 3, pp. 440-464.
# This tests the calculation for several values of lambda.
# `table4` holds just the second and third columns from Table 4.
table4 = np.array([
# observed, expected,
15, 15.171,
11, 13.952,
14, 12.831,
17, 11.800,
5, 10.852,
11, 9.9796,
10, 9.1777,
4, 8.4402,
8, 7.7620,
10, 7.1383,
7, 6.5647,
9, 6.0371,
11, 5.5520,
3, 5.1059,
6, 4.6956,
1, 4.3183,
1, 3.9713,
4, 3.6522,
]).reshape(-1, 2)
table5 = np.array([
# lambda, statistic
-10.0, 72.2e3,
-5.0, 28.9e1,
-3.0, 65.6,
-2.0, 40.6,
-1.5, 34.0,
-1.0, 29.5,
-0.5, 26.5,
0.0, 24.6,
0.5, 23.4,
0.67, 23.1,
1.0, 22.7,
1.5, 22.6,
2.0, 22.9,
3.0, 24.8,
5.0, 35.5,
10.0, 21.4e1,
]).reshape(-1, 2)
for lambda_, expected_stat in table5:
stat, p = stats.power_divergence(table4[:,0], table4[:,1],
lambda_=lambda_)
assert_allclose(stat, expected_stat, rtol=5e-3)
def test_friedmanchisquare():
# see ticket:113
# verified with matlab and R
# From Demsar "Statistical Comparisons of Classifiers over Multiple Data Sets"
# 2006, Xf=9.28 (no tie handling, tie corrected Xf >=9.28)
x1 = [array([0.763, 0.599, 0.954, 0.628, 0.882, 0.936, 0.661, 0.583,
0.775, 1.0, 0.94, 0.619, 0.972, 0.957]),
array([0.768, 0.591, 0.971, 0.661, 0.888, 0.931, 0.668, 0.583,
0.838, 1.0, 0.962, 0.666, 0.981, 0.978]),
array([0.771, 0.590, 0.968, 0.654, 0.886, 0.916, 0.609, 0.563,
0.866, 1.0, 0.965, 0.614, 0.9751, 0.946]),
array([0.798, 0.569, 0.967, 0.657, 0.898, 0.931, 0.685, 0.625,
0.875, 1.0, 0.962, 0.669, 0.975, 0.970])]
# From "Bioestadistica para las ciencias de la salud" Xf=18.95 p<0.001:
x2 = [array([4,3,5,3,5,3,2,5,4,4,4,3]),
array([2,2,1,2,3,1,2,3,2,1,1,3]),
array([2,4,3,3,4,3,3,4,4,1,2,1]),
array([3,5,4,3,4,4,3,3,3,4,4,4])]
# From Jerrorl H. Zar, "Biostatistical Analysis"(example 12.6), Xf=10.68, 0.005 < p < 0.01:
# Probability from this example is inexact using Chisquare approximation of Friedman Chisquare.
x3 = [array([7.0,9.9,8.5,5.1,10.3]),
array([5.3,5.7,4.7,3.5,7.7]),
array([4.9,7.6,5.5,2.8,8.4]),
array([8.8,8.9,8.1,3.3,9.1])]
assert_array_almost_equal(stats.friedmanchisquare(x1[0],x1[1],x1[2],x1[3]),
(10.2283464566929, 0.0167215803284414))
assert_array_almost_equal(stats.friedmanchisquare(x2[0],x2[1],x2[2],x2[3]),
(18.9428571428571, 0.000280938375189499))
assert_array_almost_equal(stats.friedmanchisquare(x3[0],x3[1],x3[2],x3[3]),
(10.68, 0.0135882729582176))
assert_raises(ValueError, stats.friedmanchisquare,x3[0],x3[1])
# test for namedtuple attribute results
attributes = ('statistic', 'pvalue')
res = stats.friedmanchisquare(*x1)
check_named_results(res, attributes)
# test using mstats
assert_array_almost_equal(mstats.friedmanchisquare(x1[0], x1[1],
x1[2], x1[3]),
(10.2283464566929, 0.0167215803284414))
# the following fails
# assert_array_almost_equal(mstats.friedmanchisquare(x2[0],x2[1],x2[2],x2[3]),
# (18.9428571428571, 0.000280938375189499))
assert_array_almost_equal(mstats.friedmanchisquare(x3[0], x3[1],
x3[2], x3[3]),
(10.68, 0.0135882729582176))
assert_raises(ValueError, mstats.friedmanchisquare,x3[0],x3[1])
def test_kstest():
# from numpy.testing import assert_almost_equal
# comparing with values from R
x = np.linspace(-1,1,9)
D,p = stats.kstest(x,'norm')
assert_almost_equal(D, 0.15865525393145705, 12)
assert_almost_equal(p, 0.95164069201518386, 1)
x = np.linspace(-15,15,9)
D,p = stats.kstest(x,'norm')
assert_almost_equal(D, 0.44435602715924361, 15)
assert_almost_equal(p, 0.038850140086788665, 8)
# test for namedtuple attribute results
attributes = ('statistic', 'pvalue')
res = stats.kstest(x, 'norm')
check_named_results(res, attributes)
# the following tests rely on deterministicaly replicated rvs
np.random.seed(987654321)
x = stats.norm.rvs(loc=0.2, size=100)
D,p = stats.kstest(x, 'norm', mode='asymp')
assert_almost_equal(D, 0.12464329735846891, 15)
assert_almost_equal(p, 0.089444888711820769, 15)
assert_almost_equal(np.array(stats.kstest(x, 'norm', mode='asymp')),
np.array((0.12464329735846891, 0.089444888711820769)), 15)
assert_almost_equal(np.array(stats.kstest(x,'norm', alternative='less')),
np.array((0.12464329735846891, 0.040989164077641749)), 15)
# this 'greater' test fails with precision of decimal=14
assert_almost_equal(np.array(stats.kstest(x,'norm', alternative='greater')),
np.array((0.0072115233216310994, 0.98531158590396228)), 12)
# missing: no test that uses *args
def test_ks_2samp():
# exact small sample solution
data1 = np.array([1.0,2.0])
data2 = np.array([1.0,2.0,3.0])
assert_almost_equal(np.array(stats.ks_2samp(data1+0.01,data2)),
np.array((0.33333333333333337, 0.99062316386915694)))
assert_almost_equal(np.array(stats.ks_2samp(data1-0.01,data2)),
np.array((0.66666666666666674, 0.42490954988801982)))
# these can also be verified graphically
assert_almost_equal(
np.array(stats.ks_2samp(np.linspace(1,100,100),
np.linspace(1,100,100)+2+0.1)),
np.array((0.030000000000000027, 0.99999999996005062)))
assert_almost_equal(
np.array(stats.ks_2samp(np.linspace(1,100,100),
np.linspace(1,100,100)+2-0.1)),
np.array((0.020000000000000018, 0.99999999999999933)))
# these are just regression tests
assert_almost_equal(
np.array(stats.ks_2samp(np.linspace(1,100,100),
np.linspace(1,100,110)+20.1)),
np.array((0.21090909090909091, 0.015880386730710221)))
assert_almost_equal(
np.array(stats.ks_2samp(np.linspace(1,100,100),
np.linspace(1,100,110)+20-0.1)),
np.array((0.20818181818181825, 0.017981441789762638)))
# test for namedtuple attribute results
attributes = ('statistic', 'pvalue')
res = stats.ks_2samp(data1 - 0.01, data2)
check_named_results(res, attributes)
def test_ttest_rel():
# regression test
tr,pr = 0.81248591389165692, 0.41846234511362157
tpr = ([tr,-tr],[pr,pr])
rvs1 = np.linspace(1,100,100)
rvs2 = np.linspace(1.01,99.989,100)
rvs1_2D = np.array([np.linspace(1,100,100), np.linspace(1.01,99.989,100)])
rvs2_2D = np.array([np.linspace(1.01,99.989,100), np.linspace(1,100,100)])
t,p = stats.ttest_rel(rvs1, rvs2, axis=0)
assert_array_almost_equal([t,p],(tr,pr))
t,p = stats.ttest_rel(rvs1_2D.T, rvs2_2D.T, axis=0)
assert_array_almost_equal([t,p],tpr)
t,p = stats.ttest_rel(rvs1_2D, rvs2_2D, axis=1)
assert_array_almost_equal([t,p],tpr)
# test scalars
with suppress_warnings() as sup, np.errstate(invalid="ignore"):
sup.filter(RuntimeWarning, "Degrees of freedom <= 0 for slice")
t, p = stats.ttest_rel(4., 3.)
assert_(np.isnan(t))
assert_(np.isnan(p))
# test for namedtuple attribute results
attributes = ('statistic', 'pvalue')
res = stats.ttest_rel(rvs1, rvs2, axis=0)
check_named_results(res, attributes)
# test on 3 dimensions
rvs1_3D = np.dstack([rvs1_2D,rvs1_2D,rvs1_2D])
rvs2_3D = np.dstack([rvs2_2D,rvs2_2D,rvs2_2D])
t,p = stats.ttest_rel(rvs1_3D, rvs2_3D, axis=1)
assert_array_almost_equal(np.abs(t), tr)
assert_array_almost_equal(np.abs(p), pr)
assert_equal(t.shape, (2, 3))
t,p = stats.ttest_rel(np.rollaxis(rvs1_3D,2), np.rollaxis(rvs2_3D,2), axis=2)
assert_array_almost_equal(np.abs(t), tr)
assert_array_almost_equal(np.abs(p), pr)
assert_equal(t.shape, (3, 2))
# check nan policy
np.random.seed(12345678)
x = stats.norm.rvs(loc=5, scale=10, size=501)
x[500] = np.nan
y = (stats.norm.rvs(loc=5, scale=10, size=501) +
stats.norm.rvs(scale=0.2, size=501))
y[500] = np.nan
with np.errstate(invalid="ignore"):
assert_array_equal(stats.ttest_rel(x, x), (np.nan, np.nan))
assert_array_almost_equal(stats.ttest_rel(x, y, nan_policy='omit'),
(0.25299925303978066, 0.8003729814201519))
assert_raises(ValueError, stats.ttest_rel, x, y, nan_policy='raise')
assert_raises(ValueError, stats.ttest_rel, x, y, nan_policy='foobar')
# test zero division problem
t, p = stats.ttest_rel([0, 0, 0], [1, 1, 1])
assert_equal((np.abs(t), p), (np.inf, 0))
with np.errstate(invalid="ignore"):
assert_equal(stats.ttest_rel([0, 0, 0], [0, 0, 0]), (np.nan, np.nan))
# check that nan in input array result in nan output
anan = np.array([[1, np.nan], [-1, 1]])
assert_equal(stats.ttest_rel(anan, np.zeros((2, 2))),
([0, np.nan], [1, np.nan]))
# test incorrect input shape raise an error
x = np.arange(24)
assert_raises(ValueError, stats.ttest_rel, x.reshape((8, 3)),
x.reshape((2, 3, 4)))
def test_ttest_rel_nan_2nd_arg():
# regression test for gh-6134: nans in the second arg were not handled
x = [np.nan, 2.0, 3.0, 4.0]
y = [1.0, 2.0, 1.0, 2.0]
r1 = stats.ttest_rel(x, y, nan_policy='omit')
r2 = stats.ttest_rel(y, x, nan_policy='omit')
assert_allclose(r2.statistic, -r1.statistic, atol=1e-15)
assert_allclose(r2.pvalue, r1.pvalue, atol=1e-15)
# NB: arguments are paired when NaNs are dropped
r3 = stats.ttest_rel(y[1:], x[1:])
assert_allclose(r2, r3, atol=1e-15)
# .. and this is consistent with R. R code:
# x = c(NA, 2.0, 3.0, 4.0)
# y = c(1.0, 2.0, 1.0, 2.0)
# t.test(x, y, paired=TRUE)
assert_allclose(r2, (-2, 0.1835), atol=1e-4)
def _desc_stats(x1, x2, axis=0):
def _stats(x, axis=0):
x = np.asarray(x)
mu = np.mean(x, axis=axis)
std = np.std(x, axis=axis, ddof=1)
nobs = x.shape[axis]
return mu, std, nobs
return _stats(x1, axis) + _stats(x2, axis)
def test_ttest_ind():
# regression test
tr = 1.0912746897927283
pr = 0.27647818616351882
tpr = ([tr,-tr],[pr,pr])
rvs2 = np.linspace(1,100,100)
rvs1 = np.linspace(5,105,100)
rvs1_2D = np.array([rvs1, rvs2])
rvs2_2D = np.array([rvs2, rvs1])
t,p = stats.ttest_ind(rvs1, rvs2, axis=0)
assert_array_almost_equal([t,p],(tr,pr))
# test from_stats API
assert_array_almost_equal(stats.ttest_ind_from_stats(*_desc_stats(rvs1,
rvs2)),
[t, p])
t,p = stats.ttest_ind(rvs1_2D.T, rvs2_2D.T, axis=0)
assert_array_almost_equal([t,p],tpr)
args = _desc_stats(rvs1_2D.T, rvs2_2D.T)
assert_array_almost_equal(stats.ttest_ind_from_stats(*args),
[t, p])
t,p = stats.ttest_ind(rvs1_2D, rvs2_2D, axis=1)
assert_array_almost_equal([t,p],tpr)
args = _desc_stats(rvs1_2D, rvs2_2D, axis=1)
assert_array_almost_equal(stats.ttest_ind_from_stats(*args),
[t, p])
# test scalars
with suppress_warnings() as sup, np.errstate(invalid="ignore"):
sup.filter(RuntimeWarning, "Degrees of freedom <= 0 for slice")
t, p = stats.ttest_ind(4., 3.)
assert_(np.isnan(t))
assert_(np.isnan(p))
# test on 3 dimensions
rvs1_3D = np.dstack([rvs1_2D,rvs1_2D,rvs1_2D])
rvs2_3D = np.dstack([rvs2_2D,rvs2_2D,rvs2_2D])
t,p = stats.ttest_ind(rvs1_3D, rvs2_3D, axis=1)
assert_almost_equal(np.abs(t), np.abs(tr))
assert_array_almost_equal(np.abs(p), pr)
assert_equal(t.shape, (2, 3))
t,p = stats.ttest_ind(np.rollaxis(rvs1_3D,2), np.rollaxis(rvs2_3D,2), axis=2)
assert_array_almost_equal(np.abs(t), np.abs(tr))
assert_array_almost_equal(np.abs(p), pr)
assert_equal(t.shape, (3, 2))
# check nan policy
np.random.seed(12345678)
x = stats.norm.rvs(loc=5, scale=10, size=501)
x[500] = np.nan
y = stats.norm.rvs(loc=5, scale=10, size=500)
with np.errstate(invalid="ignore"):
assert_array_equal(stats.ttest_ind(x, y), (np.nan, np.nan))
assert_array_almost_equal(stats.ttest_ind(x, y, nan_policy='omit'),
(0.24779670949091914, 0.80434267337517906))
assert_raises(ValueError, stats.ttest_ind, x, y, nan_policy='raise')
assert_raises(ValueError, stats.ttest_ind, x, y, nan_policy='foobar')
# test zero division problem
t, p = stats.ttest_ind([0, 0, 0], [1, 1, 1])
assert_equal((np.abs(t), p), (np.inf, 0))
with np.errstate(invalid="ignore"):
assert_equal(stats.ttest_ind([0, 0, 0], [0, 0, 0]), (np.nan, np.nan))
# check that nan in input array result in nan output
anan = np.array([[1, np.nan], [-1, 1]])
assert_equal(stats.ttest_ind(anan, np.zeros((2, 2))),
([0, np.nan], [1, np.nan]))
def test_ttest_ind_with_uneq_var():
# check vs. R
a = (1, 2, 3)
b = (1.1, 2.9, 4.2)
pr = 0.53619490753126731
tr = -0.68649512735572582
t, p = stats.ttest_ind(a, b, equal_var=False)
assert_array_almost_equal([t,p], [tr, pr])
# test from desc stats API
assert_array_almost_equal(stats.ttest_ind_from_stats(*_desc_stats(a, b),
equal_var=False),
[t, p])
a = (1, 2, 3, 4)
pr = 0.84354139131608286
tr = -0.2108663315950719
t, p = stats.ttest_ind(a, b, equal_var=False)
assert_array_almost_equal([t,p], [tr, pr])
assert_array_almost_equal(stats.ttest_ind_from_stats(*_desc_stats(a, b),
equal_var=False),
[t, p])
# regression test
tr = 1.0912746897927283
tr_uneq_n = 0.66745638708050492
pr = 0.27647831993021388
pr_uneq_n = 0.50873585065616544
tpr = ([tr,-tr],[pr,pr])
rvs3 = np.linspace(1,100, 25)
rvs2 = np.linspace(1,100,100)
rvs1 = np.linspace(5,105,100)
rvs1_2D = np.array([rvs1, rvs2])
rvs2_2D = np.array([rvs2, rvs1])
t,p = stats.ttest_ind(rvs1, rvs2, axis=0, equal_var=False)
assert_array_almost_equal([t,p],(tr,pr))
assert_array_almost_equal(stats.ttest_ind_from_stats(*_desc_stats(rvs1,
rvs2),
equal_var=False),
(t, p))
t,p = stats.ttest_ind(rvs1, rvs3, axis=0, equal_var=False)
assert_array_almost_equal([t,p], (tr_uneq_n, pr_uneq_n))
assert_array_almost_equal(stats.ttest_ind_from_stats(*_desc_stats(rvs1,
rvs3),
equal_var=False),
(t, p))
t,p = stats.ttest_ind(rvs1_2D.T, rvs2_2D.T, axis=0, equal_var=False)
assert_array_almost_equal([t,p],tpr)
args = _desc_stats(rvs1_2D.T, rvs2_2D.T)
assert_array_almost_equal(stats.ttest_ind_from_stats(*args,
equal_var=False),
(t, p))
t,p = stats.ttest_ind(rvs1_2D, rvs2_2D, axis=1, equal_var=False)
assert_array_almost_equal([t,p],tpr)
args = _desc_stats(rvs1_2D, rvs2_2D, axis=1)
assert_array_almost_equal(stats.ttest_ind_from_stats(*args,
equal_var=False),
(t, p))
# test for namedtuple attribute results
attributes = ('statistic', 'pvalue')
res = stats.ttest_ind(rvs1, rvs2, axis=0, equal_var=False)
check_named_results(res, attributes)
# test on 3 dimensions
rvs1_3D = np.dstack([rvs1_2D,rvs1_2D,rvs1_2D])
rvs2_3D = np.dstack([rvs2_2D,rvs2_2D,rvs2_2D])
t,p = stats.ttest_ind(rvs1_3D, rvs2_3D, axis=1, equal_var=False)
assert_almost_equal(np.abs(t), np.abs(tr))
assert_array_almost_equal(np.abs(p), pr)
assert_equal(t.shape, (2, 3))
args = _desc_stats(rvs1_3D, rvs2_3D, axis=1)
t, p = stats.ttest_ind_from_stats(*args, equal_var=False)
assert_almost_equal(np.abs(t), np.abs(tr))
assert_array_almost_equal(np.abs(p), pr)
assert_equal(t.shape, (2, 3))
t,p = stats.ttest_ind(np.rollaxis(rvs1_3D,2), np.rollaxis(rvs2_3D,2),
axis=2, equal_var=False)
assert_array_almost_equal(np.abs(t), np.abs(tr))
assert_array_almost_equal(np.abs(p), pr)
assert_equal(t.shape, (3, 2))
args = _desc_stats(np.rollaxis(rvs1_3D, 2),
np.rollaxis(rvs2_3D, 2), axis=2)
t, p = stats.ttest_ind_from_stats(*args, equal_var=False)
assert_array_almost_equal(np.abs(t), np.abs(tr))
assert_array_almost_equal(np.abs(p), pr)
assert_equal(t.shape, (3, 2))
# test zero division problem
t, p = stats.ttest_ind([0, 0, 0], [1, 1, 1], equal_var=False)
assert_equal((np.abs(t), p), (np.inf, 0))
with np.errstate(all='ignore'):
assert_equal(stats.ttest_ind([0, 0, 0], [0, 0, 0], equal_var=False),
(np.nan, np.nan))
# check that nan in input array result in nan output
anan = np.array([[1, np.nan], [-1, 1]])
assert_equal(stats.ttest_ind(anan, np.zeros((2, 2)), equal_var=False),
([0, np.nan], [1, np.nan]))
def test_ttest_ind_nan_2nd_arg():
# regression test for gh-6134: nans in the second arg were not handled
x = [np.nan, 2.0, 3.0, 4.0]
y = [1.0, 2.0, 1.0, 2.0]
r1 = stats.ttest_ind(x, y, nan_policy='omit')
r2 = stats.ttest_ind(y, x, nan_policy='omit')
assert_allclose(r2.statistic, -r1.statistic, atol=1e-15)
assert_allclose(r2.pvalue, r1.pvalue, atol=1e-15)
# NB: arguments are not paired when NaNs are dropped
r3 = stats.ttest_ind(y, x[1:])
assert_allclose(r2, r3, atol=1e-15)
# .. and this is consistent with R. R code:
# x = c(NA, 2.0, 3.0, 4.0)
# y = c(1.0, 2.0, 1.0, 2.0)
# t.test(x, y, var.equal=TRUE)
assert_allclose(r2, (-2.5354627641855498, 0.052181400457057901), atol=1e-15)
def test_gh5686():
mean1, mean2 = np.array([1, 2]), np.array([3, 4])
std1, std2 = np.array([5, 3]), np.array([4, 5])
nobs1, nobs2 = np.array([130, 140]), np.array([100, 150])
# This will raise a TypeError unless gh-5686 is fixed.
stats.ttest_ind_from_stats(mean1, std1, nobs1, mean2, std2, nobs2)
def test_ttest_1samp_new():
n1, n2, n3 = (10,15,20)
rvn1 = stats.norm.rvs(loc=5,scale=10,size=(n1,n2,n3))
# check multidimensional array and correct axis handling
# deterministic rvn1 and rvn2 would be better as in test_ttest_rel
t1,p1 = stats.ttest_1samp(rvn1[:,:,:], np.ones((n2,n3)),axis=0)
t2,p2 = stats.ttest_1samp(rvn1[:,:,:], 1,axis=0)
t3,p3 = stats.ttest_1samp(rvn1[:,0,0], 1)
assert_array_almost_equal(t1,t2, decimal=14)
assert_almost_equal(t1[0,0],t3, decimal=14)
assert_equal(t1.shape, (n2,n3))
t1,p1 = stats.ttest_1samp(rvn1[:,:,:], np.ones((n1,n3)),axis=1)
t2,p2 = stats.ttest_1samp(rvn1[:,:,:], 1,axis=1)
t3,p3 = stats.ttest_1samp(rvn1[0,:,0], 1)
assert_array_almost_equal(t1,t2, decimal=14)
assert_almost_equal(t1[0,0],t3, decimal=14)
assert_equal(t1.shape, (n1,n3))
t1,p1 = stats.ttest_1samp(rvn1[:,:,:], np.ones((n1,n2)),axis=2)
t2,p2 = stats.ttest_1samp(rvn1[:,:,:], 1,axis=2)
t3,p3 = stats.ttest_1samp(rvn1[0,0,:], 1)
assert_array_almost_equal(t1,t2, decimal=14)
assert_almost_equal(t1[0,0],t3, decimal=14)
assert_equal(t1.shape, (n1,n2))
# test zero division problem
t, p = stats.ttest_1samp([0, 0, 0], 1)
assert_equal((np.abs(t), p), (np.inf, 0))
with np.errstate(all='ignore'):
assert_equal(stats.ttest_1samp([0, 0, 0], 0), (np.nan, np.nan))
# check that nan in input array result in nan output
anan = np.array([[1, np.nan],[-1, 1]])
assert_equal(stats.ttest_1samp(anan, 0), ([0, np.nan], [1, np.nan]))
class TestDescribe(object):
def test_describe_scalar(self):
with suppress_warnings() as sup, np.errstate(invalid="ignore"):
sup.filter(RuntimeWarning, "Degrees of freedom <= 0 for slice")
n, mm, m, v, sk, kurt = stats.describe(4.)
assert_equal(n, 1)
assert_equal(mm, (4.0, 4.0))
assert_equal(m, 4.0)
assert_(np.isnan(v))
assert_array_almost_equal(sk, 0.0, decimal=13)
assert_array_almost_equal(kurt, -3.0, decimal=13)
def test_describe_numbers(self):
x = np.vstack((np.ones((3,4)), 2 * np.ones((2,4))))
nc, mmc = (5, ([1., 1., 1., 1.], [2., 2., 2., 2.]))
mc = np.array([1.4, 1.4, 1.4, 1.4])
vc = np.array([0.3, 0.3, 0.3, 0.3])
skc = [0.40824829046386357] * 4
kurtc = [-1.833333333333333] * 4
n, mm, m, v, sk, kurt = stats.describe(x)
assert_equal(n, nc)
assert_equal(mm, mmc)
assert_equal(m, mc)
assert_equal(v, vc)
assert_array_almost_equal(sk, skc, decimal=13)
assert_array_almost_equal(kurt, kurtc, decimal=13)
n, mm, m, v, sk, kurt = stats.describe(x.T, axis=1)
assert_equal(n, nc)
assert_equal(mm, mmc)
assert_equal(m, mc)
assert_equal(v, vc)
assert_array_almost_equal(sk, skc, decimal=13)
assert_array_almost_equal(kurt, kurtc, decimal=13)
x = np.arange(10.)
x[9] = np.nan
nc, mmc = (9, (0.0, 8.0))
mc = 4.0
vc = 7.5
skc = 0.0
kurtc = -1.2300000000000002
n, mm, m, v, sk, kurt = stats.describe(x, nan_policy='omit')
assert_equal(n, nc)
assert_equal(mm, mmc)
assert_equal(m, mc)
assert_equal(v, vc)
assert_array_almost_equal(sk, skc)
assert_array_almost_equal(kurt, kurtc, decimal=13)
assert_raises(ValueError, stats.describe, x, nan_policy='raise')
assert_raises(ValueError, stats.describe, x, nan_policy='foobar')
def test_describe_result_attributes(self):
actual = stats.describe(np.arange(5))
attributes = ('nobs', 'minmax', 'mean', 'variance', 'skewness',
'kurtosis')
check_named_results(actual, attributes)
def test_describe_ddof(self):
x = np.vstack((np.ones((3, 4)), 2 * np.ones((2, 4))))
nc, mmc = (5, ([1., 1., 1., 1.], [2., 2., 2., 2.]))
mc = np.array([1.4, 1.4, 1.4, 1.4])
vc = np.array([0.24, 0.24, 0.24, 0.24])
skc = [0.40824829046386357] * 4
kurtc = [-1.833333333333333] * 4
n, mm, m, v, sk, kurt = stats.describe(x, ddof=0)
assert_equal(n, nc)
assert_allclose(mm, mmc, rtol=1e-15)
assert_allclose(m, mc, rtol=1e-15)
assert_allclose(v, vc, rtol=1e-15)
assert_array_almost_equal(sk, skc, decimal=13)
assert_array_almost_equal(kurt, kurtc, decimal=13)
def test_describe_axis_none(self):
x = np.vstack((np.ones((3, 4)), 2 * np.ones((2, 4))))
# expected values
e_nobs, e_minmax = (20, (1.0, 2.0))
e_mean = 1.3999999999999999
e_var = 0.25263157894736848
e_skew = 0.4082482904638634
e_kurt = -1.8333333333333333
# actual values
a = stats.describe(x, axis=None)
assert_equal(a.nobs, e_nobs)
assert_almost_equal(a.minmax, e_minmax)
assert_almost_equal(a.mean, e_mean)
assert_almost_equal(a.variance, e_var)
assert_array_almost_equal(a.skewness, e_skew, decimal=13)
assert_array_almost_equal(a.kurtosis, e_kurt, decimal=13)
def test_describe_empty(self):
assert_raises(ValueError, stats.describe, [])
def test_normalitytests():
assert_raises(ValueError, stats.skewtest, 4.)
assert_raises(ValueError, stats.kurtosistest, 4.)
assert_raises(ValueError, stats.normaltest, 4.)
# numbers verified with R: dagoTest in package fBasics
st_normal, st_skew, st_kurt = (3.92371918, 1.98078826, -0.01403734)
pv_normal, pv_skew, pv_kurt = (0.14059673, 0.04761502, 0.98880019)
x = np.array((-2,-1,0,1,2,3)*4)**2
attributes = ('statistic', 'pvalue')
assert_array_almost_equal(stats.normaltest(x), (st_normal, pv_normal))
check_named_results(stats.normaltest(x), attributes)
assert_array_almost_equal(stats.skewtest(x), (st_skew, pv_skew))
check_named_results(stats.skewtest(x), attributes)
assert_array_almost_equal(stats.kurtosistest(x), (st_kurt, pv_kurt))
check_named_results(stats.kurtosistest(x), attributes)
# Test axis=None (equal to axis=0 for 1-D input)
assert_array_almost_equal(stats.normaltest(x, axis=None),
(st_normal, pv_normal))
assert_array_almost_equal(stats.skewtest(x, axis=None),
(st_skew, pv_skew))
assert_array_almost_equal(stats.kurtosistest(x, axis=None),
(st_kurt, pv_kurt))
x = np.arange(10.)
x[9] = np.nan
with np.errstate(invalid="ignore"):
assert_array_equal(stats.skewtest(x), (np.nan, np.nan))
expected = (1.0184643553962129, 0.30845733195153502)
assert_array_almost_equal(stats.skewtest(x, nan_policy='omit'), expected)
with np.errstate(all='ignore'):
assert_raises(ValueError, stats.skewtest, x, nan_policy='raise')
assert_raises(ValueError, stats.skewtest, x, nan_policy='foobar')
x = np.arange(30.)
x[29] = np.nan
with np.errstate(all='ignore'):
assert_array_equal(stats.kurtosistest(x), (np.nan, np.nan))
expected = (-2.2683547379505273, 0.023307594135872967)
assert_array_almost_equal(stats.kurtosistest(x, nan_policy='omit'),
expected)
assert_raises(ValueError, stats.kurtosistest, x, nan_policy='raise')
assert_raises(ValueError, stats.kurtosistest, x, nan_policy='foobar')
with np.errstate(all='ignore'):
assert_array_equal(stats.normaltest(x), (np.nan, np.nan))
expected = (6.2260409514287449, 0.04446644248650191)
assert_array_almost_equal(stats.normaltest(x, nan_policy='omit'), expected)
assert_raises(ValueError, stats.normaltest, x, nan_policy='raise')
assert_raises(ValueError, stats.normaltest, x, nan_policy='foobar')
class TestRankSums(object):
def test_ranksums_result_attributes(self):
res = stats.ranksums(np.arange(5), np.arange(25))
attributes = ('statistic', 'pvalue')
check_named_results(res, attributes)
class TestJarqueBera(object):
def test_jarque_bera_stats(self):
np.random.seed(987654321)
x = np.random.normal(0, 1, 100000)
y = np.random.chisquare(10000, 100000)
z = np.random.rayleigh(1, 100000)
assert_(stats.jarque_bera(x)[1] > stats.jarque_bera(y)[1])
assert_(stats.jarque_bera(x)[1] > stats.jarque_bera(z)[1])
assert_(stats.jarque_bera(y)[1] > stats.jarque_bera(z)[1])
def test_jarque_bera_array_like(self):
np.random.seed(987654321)
x = np.random.normal(0, 1, 100000)
JB1, p1 = stats.jarque_bera(list(x))
JB2, p2 = stats.jarque_bera(tuple(x))
JB3, p3 = stats.jarque_bera(x.reshape(2, 50000))
assert_(JB1 == JB2 == JB3)
assert_(p1 == p2 == p3)
def test_jarque_bera_size(self):
assert_raises(ValueError, stats.jarque_bera, [])
def test_skewtest_too_few_samples():
# Regression test for ticket #1492.
# skewtest requires at least 8 samples; 7 should raise a ValueError.
x = np.arange(7.0)
assert_raises(ValueError, stats.skewtest, x)
def test_kurtosistest_too_few_samples():
# Regression test for ticket #1425.
# kurtosistest requires at least 5 samples; 4 should raise a ValueError.
x = np.arange(4.0)
assert_raises(ValueError, stats.kurtosistest, x)
class TestMannWhitneyU(object):
X = [19.8958398126694, 19.5452691647182, 19.0577309166425, 21.716543054589,
20.3269502208702, 20.0009273294025, 19.3440043632957, 20.4216806548105,
19.0649894736528, 18.7808043120398, 19.3680942943298, 19.4848044069953,
20.7514611265663, 19.0894948874598, 19.4975522356628, 18.9971170734274,
20.3239606288208, 20.6921298083835, 19.0724259532507, 18.9825187935021,
19.5144462609601, 19.8256857844223, 20.5174677102032, 21.1122407995892,
17.9490854922535, 18.2847521114727, 20.1072217648826, 18.6439891962179,
20.4970638083542, 19.5567594734914]
Y = [19.2790668029091, 16.993808441865, 18.5416338448258, 17.2634018833575,
19.1577183624616, 18.5119655377495, 18.6068455037221, 18.8358343362655,
19.0366413269742, 18.1135025515417, 19.2201873866958, 17.8344909022841,
18.2894380745856, 18.6661374133922, 19.9688601693252, 16.0672254617636,
19.00596360572, 19.201561539032, 19.0487501090183, 19.0847908674356]
significant = 14
def test_mannwhitneyu_one_sided(self):
u1, p1 = stats.mannwhitneyu(self.X, self.Y, alternative='less')
u2, p2 = stats.mannwhitneyu(self.Y, self.X, alternative='greater')
u3, p3 = stats.mannwhitneyu(self.X, self.Y, alternative='greater')
u4, p4 = stats.mannwhitneyu(self.Y, self.X, alternative='less')
assert_equal(p1, p2)
assert_equal(p3, p4)
assert_(p1 != p3)
assert_equal(u1, 498)
assert_equal(u2, 102)
assert_equal(u3, 498)
assert_equal(u4, 102)
assert_approx_equal(p1, 0.999957683256589, significant=self.significant)
assert_approx_equal(p3, 4.5941632666275e-05, significant=self.significant)
def test_mannwhitneyu_two_sided(self):
u1, p1 = stats.mannwhitneyu(self.X, self.Y, alternative='two-sided')
u2, p2 = stats.mannwhitneyu(self.Y, self.X, alternative='two-sided')
assert_equal(p1, p2)
assert_equal(u1, 498)
assert_equal(u2, 102)
assert_approx_equal(p1, 9.188326533255e-05,
significant=self.significant)
def test_mannwhitneyu_default(self):
# The default value for alternative is None
with suppress_warnings() as sup:
sup.filter(DeprecationWarning,
"Calling `mannwhitneyu` without .*`alternative`")
u1, p1 = stats.mannwhitneyu(self.X, self.Y)
u2, p2 = stats.mannwhitneyu(self.Y, self.X)
u3, p3 = stats.mannwhitneyu(self.X, self.Y, alternative=None)
assert_equal(p1, p2)
assert_equal(p1, p3)
assert_equal(u1, 102)
assert_equal(u2, 102)
assert_equal(u3, 102)
assert_approx_equal(p1, 4.5941632666275e-05,
significant=self.significant)
def test_mannwhitneyu_no_correct_one_sided(self):
u1, p1 = stats.mannwhitneyu(self.X, self.Y, False,
alternative='less')
u2, p2 = stats.mannwhitneyu(self.Y, self.X, False,
alternative='greater')
u3, p3 = stats.mannwhitneyu(self.X, self.Y, False,
alternative='greater')
u4, p4 = stats.mannwhitneyu(self.Y, self.X, False,
alternative='less')
assert_equal(p1, p2)
assert_equal(p3, p4)
assert_(p1 != p3)
assert_equal(u1, 498)
assert_equal(u2, 102)
assert_equal(u3, 498)
assert_equal(u4, 102)
assert_approx_equal(p1, 0.999955905990004, significant=self.significant)
assert_approx_equal(p3, 4.40940099958089e-05, significant=self.significant)
def test_mannwhitneyu_no_correct_two_sided(self):
u1, p1 = stats.mannwhitneyu(self.X, self.Y, False,
alternative='two-sided')
u2, p2 = stats.mannwhitneyu(self.Y, self.X, False,
alternative='two-sided')
assert_equal(p1, p2)
assert_equal(u1, 498)
assert_equal(u2, 102)
assert_approx_equal(p1, 8.81880199916178e-05,
significant=self.significant)
def test_mannwhitneyu_no_correct_default(self):
# The default value for alternative is None
with suppress_warnings() as sup:
sup.filter(DeprecationWarning,
"Calling `mannwhitneyu` without .*`alternative`")
u1, p1 = stats.mannwhitneyu(self.X, self.Y, False)
u2, p2 = stats.mannwhitneyu(self.Y, self.X, False)
u3, p3 = stats.mannwhitneyu(self.X, self.Y, False,
alternative=None)
assert_equal(p1, p2)
assert_equal(p1, p3)
assert_equal(u1, 102)
assert_equal(u2, 102)
assert_equal(u3, 102)
assert_approx_equal(p1, 4.40940099958089e-05,
significant=self.significant)
def test_mannwhitneyu_ones(self):
x = np.array([1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 2., 1., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 2., 1., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 2.,
1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
1., 1., 2., 1., 1., 1., 1., 2., 1., 1., 2., 1., 1., 2.,
1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 2., 1.,
1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 2., 1., 1., 1., 1., 1., 1.,
1., 1., 1., 2., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 1., 3., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1.])
y = np.array([1., 1., 1., 1., 1., 1., 1., 2., 1., 2., 1., 1., 1., 1.,
2., 1., 1., 1., 2., 1., 1., 1., 1., 1., 2., 1., 1., 3.,
1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 2., 1., 2., 1.,
1., 1., 1., 1., 1., 2., 1., 1., 1., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 2., 1., 1., 1., 1., 1., 2.,
2., 1., 1., 2., 1., 1., 2., 1., 2., 1., 1., 1., 1., 2.,
2., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
1., 2., 1., 1., 1., 1., 1., 2., 2., 2., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
2., 1., 1., 2., 1., 1., 1., 1., 2., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 2., 1., 1., 1., 2., 1., 1.,
1., 1., 1., 1.])
# p-value verified with matlab and R to 5 significant digits
assert_array_almost_equal(stats.stats.mannwhitneyu(x, y,
alternative='less'),
(16980.5, 2.8214327656317373e-005),
decimal=12)
def test_mannwhitneyu_result_attributes(self):
# test for namedtuple attribute results
attributes = ('statistic', 'pvalue')
res = stats.mannwhitneyu(self.X, self.Y, alternative="less")
check_named_results(res, attributes)
def test_pointbiserial():
# same as mstats test except for the nan
# Test data: http://support.sas.com/ctx/samples/index.jsp?sid=490&tab=output
x = [1,0,1,1,1,1,0,1,0,0,0,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0,0,1,0,
0,0,0,0,1]
y = [14.8,13.8,12.4,10.1,7.1,6.1,5.8,4.6,4.3,3.5,3.3,3.2,3.0,
2.8,2.8,2.5,2.4,2.3,2.1,1.7,1.7,1.5,1.3,1.3,1.2,1.2,1.1,
0.8,0.7,0.6,0.5,0.2,0.2,0.1]
assert_almost_equal(stats.pointbiserialr(x, y)[0], 0.36149, 5)
# test for namedtuple attribute results
attributes = ('correlation', 'pvalue')
res = stats.pointbiserialr(x, y)
check_named_results(res, attributes)
def test_obrientransform():
# A couple tests calculated by hand.
x1 = np.array([0, 2, 4])
t1 = stats.obrientransform(x1)
expected = [7, -2, 7]
assert_allclose(t1[0], expected)
x2 = np.array([0, 3, 6, 9])
t2 = stats.obrientransform(x2)
expected = np.array([30, 0, 0, 30])
assert_allclose(t2[0], expected)
# Test two arguments.
a, b = stats.obrientransform(x1, x2)
assert_equal(a, t1[0])
assert_equal(b, t2[0])
# Test three arguments.
a, b, c = stats.obrientransform(x1, x2, x1)
assert_equal(a, t1[0])
assert_equal(b, t2[0])
assert_equal(c, t1[0])
# This is a regression test to check np.var replacement.
# The author of this test didn't separately verify the numbers.
x1 = np.arange(5)
result = np.array(
[[5.41666667, 1.04166667, -0.41666667, 1.04166667, 5.41666667],
[21.66666667, 4.16666667, -1.66666667, 4.16666667, 21.66666667]])
assert_array_almost_equal(stats.obrientransform(x1, 2*x1), result, decimal=8)
# Example from "O'Brien Test for Homogeneity of Variance"
# by Herve Abdi.
values = range(5, 11)
reps = np.array([5, 11, 9, 3, 2, 2])
data = np.repeat(values, reps)
transformed_values = np.array([3.1828, 0.5591, 0.0344,
1.6086, 5.2817, 11.0538])
expected = np.repeat(transformed_values, reps)
result = stats.obrientransform(data)
assert_array_almost_equal(result[0], expected, decimal=4)
class HarMeanTestCase:
def test_1dlist(self):
# Test a 1d list
a = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100]
b = 34.1417152147
self.do(a, b)
def test_1darray(self):
# Test a 1d array
a = np.array([10, 20, 30, 40, 50, 60, 70, 80, 90, 100])
b = 34.1417152147
self.do(a, b)
def test_1dma(self):
# Test a 1d masked array
a = np.ma.array([10, 20, 30, 40, 50, 60, 70, 80, 90, 100])
b = 34.1417152147
self.do(a, b)
def test_1dmavalue(self):
# Test a 1d masked array with a masked value
a = np.ma.array([10, 20, 30, 40, 50, 60, 70, 80, 90, 100],
mask=[0,0,0,0,0,0,0,0,0,1])
b = 31.8137186141
self.do(a, b)
# Note the next tests use axis=None as default, not axis=0
def test_2dlist(self):
# Test a 2d list
a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
b = 38.6696271841
self.do(a, b)
def test_2darray(self):
# Test a 2d array
a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
b = 38.6696271841
self.do(np.array(a), b)
def test_2dma(self):
# Test a 2d masked array
a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
b = 38.6696271841
self.do(np.ma.array(a), b)
def test_2daxis0(self):
# Test a 2d list with axis=0
a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
b = np.array([22.88135593, 39.13043478, 52.90076336, 65.45454545])
self.do(a, b, axis=0)
def test_2daxis1(self):
# Test a 2d list with axis=1
a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
b = np.array([19.2, 63.03939962, 103.80078637])
self.do(a, b, axis=1)
def test_2dmatrixdaxis0(self):
# Test a 2d list with axis=0
a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
b = np.matrix([[22.88135593, 39.13043478, 52.90076336, 65.45454545]])
self.do(np.matrix(a), b, axis=0)
def test_2dmatrixaxis1(self):
# Test a 2d list with axis=1
a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
b = np.matrix([[19.2, 63.03939962, 103.80078637]]).T
self.do(np.matrix(a), b, axis=1)
class TestHarMean(HarMeanTestCase):
def do(self, a, b, axis=None, dtype=None):
x = stats.hmean(a, axis=axis, dtype=dtype)
assert_almost_equal(b, x)
assert_equal(x.dtype, dtype)
class GeoMeanTestCase:
def test_1dlist(self):
# Test a 1d list
a = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100]
b = 45.2872868812
self.do(a, b)
def test_1darray(self):
# Test a 1d array
a = np.array([10, 20, 30, 40, 50, 60, 70, 80, 90, 100])
b = 45.2872868812
self.do(a, b)
def test_1dma(self):
# Test a 1d masked array
a = np.ma.array([10, 20, 30, 40, 50, 60, 70, 80, 90, 100])
b = 45.2872868812
self.do(a, b)
def test_1dmavalue(self):
# Test a 1d masked array with a masked value
a = np.ma.array([10, 20, 30, 40, 50, 60, 70, 80, 90, 100], mask=[0,0,0,0,0,0,0,0,0,1])
b = 41.4716627439
self.do(a, b)
# Note the next tests use axis=None as default, not axis=0
def test_2dlist(self):
# Test a 2d list
a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
b = 52.8885199
self.do(a, b)
def test_2darray(self):
# Test a 2d array
a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
b = 52.8885199
self.do(np.array(a), b)
def test_2dma(self):
# Test a 2d masked array
a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
b = 52.8885199
self.do(np.ma.array(a), b)
def test_2daxis0(self):
# Test a 2d list with axis=0
a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
b = np.array([35.56893304, 49.32424149, 61.3579244, 72.68482371])
self.do(a, b, axis=0)
def test_2daxis1(self):
# Test a 2d list with axis=1
a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
b = np.array([22.13363839, 64.02171746, 104.40086817])
self.do(a, b, axis=1)
def test_2dmatrixdaxis0(self):
# Test a 2d list with axis=0
a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
b = np.matrix([[35.56893304, 49.32424149, 61.3579244, 72.68482371]])
self.do(np.matrix(a), b, axis=0)
def test_2dmatrixaxis1(self):
# Test a 2d list with axis=1
a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
b = np.matrix([[22.13363839, 64.02171746, 104.40086817]]).T
self.do(np.matrix(a), b, axis=1)
def test_1dlist0(self):
# Test a 1d list with zero element
a = [10, 20, 30, 40, 50, 60, 70, 80, 90, 0]
b = 0.0 # due to exp(-inf)=0
olderr = np.seterr(all='ignore')
try:
self.do(a, b)
finally:
np.seterr(**olderr)
def test_1darray0(self):
# Test a 1d array with zero element
a = np.array([10, 20, 30, 40, 50, 60, 70, 80, 90, 0])
b = 0.0 # due to exp(-inf)=0
olderr = np.seterr(all='ignore')
try:
self.do(a, b)
finally:
np.seterr(**olderr)
def test_1dma0(self):
# Test a 1d masked array with zero element
a = np.ma.array([10, 20, 30, 40, 50, 60, 70, 80, 90, 0])
b = 41.4716627439
olderr = np.seterr(all='ignore')
try:
self.do(a, b)
finally:
np.seterr(**olderr)
def test_1dmainf(self):
# Test a 1d masked array with negative element
a = np.ma.array([10, 20, 30, 40, 50, 60, 70, 80, 90, -1])
b = 41.4716627439
olderr = np.seterr(all='ignore')
try:
self.do(a, b)
finally:
np.seterr(**olderr)
class TestGeoMean(GeoMeanTestCase):
def do(self, a, b, axis=None, dtype=None):
# Note this doesn't test when axis is not specified
x = stats.gmean(a, axis=axis, dtype=dtype)
assert_almost_equal(b, x)
assert_equal(x.dtype, dtype)
def test_binomtest():
# precision tests compared to R for ticket:986
pp = np.concatenate((np.linspace(0.1,0.2,5), np.linspace(0.45,0.65,5),
np.linspace(0.85,0.95,5)))
n = 501
x = 450
results = [0.0, 0.0, 1.0159969301994141e-304,
2.9752418572150531e-275, 7.7668382922535275e-250,
2.3381250925167094e-099, 7.8284591587323951e-081,
9.9155947819961383e-065, 2.8729390725176308e-050,
1.7175066298388421e-037, 0.0021070691951093692,
0.12044570587262322, 0.88154763174802508, 0.027120993063129286,
2.6102587134694721e-006]
for p, res in zip(pp,results):
assert_approx_equal(stats.binom_test(x, n, p), res,
significant=12, err_msg='fail forp=%f' % p)
assert_approx_equal(stats.binom_test(50,100,0.1), 5.8320387857343647e-024,
significant=12, err_msg='fail forp=%f' % p)
def test_binomtest2():
# test added for issue #2384
res2 = [
[1.0, 1.0],
[0.5,1.0,0.5],
[0.25,1.00,1.00,0.25],
[0.125,0.625,1.000,0.625,0.125],
[0.0625,0.3750,1.0000,1.0000,0.3750,0.0625],
[0.03125,0.21875,0.68750,1.00000,0.68750,0.21875,0.03125],
[0.015625,0.125000,0.453125,1.000000,1.000000,0.453125,0.125000,0.015625],
[0.0078125,0.0703125,0.2890625,0.7265625,1.0000000,0.7265625,0.2890625,
0.0703125,0.0078125],
[0.00390625,0.03906250,0.17968750,0.50781250,1.00000000,1.00000000,
0.50781250,0.17968750,0.03906250,0.00390625],
[0.001953125,0.021484375,0.109375000,0.343750000,0.753906250,1.000000000,
0.753906250,0.343750000,0.109375000,0.021484375,0.001953125]
]
for k in range(1, 11):
res1 = [stats.binom_test(v, k, 0.5) for v in range(k + 1)]
assert_almost_equal(res1, res2[k-1], decimal=10)
def test_binomtest3():
# test added for issue #2384
# test when x == n*p and neighbors
res3 = [stats.binom_test(v, v*k, 1./k) for v in range(1, 11)
for k in range(2, 11)]
assert_equal(res3, np.ones(len(res3), int))
#> bt=c()
#> for(i in as.single(1:10)){for(k in as.single(2:10)){bt = c(bt, binom.test(i-1, k*i,(1/k))$p.value); print(c(i+1, k*i,(1/k)))}}
binom_testm1 = np.array([
0.5, 0.5555555555555556, 0.578125, 0.5904000000000003,
0.5981224279835393, 0.603430543396034, 0.607304096221924,
0.610255656871054, 0.612579511000001, 0.625, 0.670781893004115,
0.68853759765625, 0.6980101120000006, 0.703906431368616,
0.70793209416498, 0.7108561134173507, 0.713076544331419,
0.714820192935702, 0.6875, 0.7268709038256367, 0.7418963909149174,
0.74986110468096, 0.7548015520398076, 0.7581671424768577,
0.760607984787832, 0.762459425024199, 0.7639120677676575, 0.7265625,
0.761553963657302, 0.774800934828818, 0.7818005980538996,
0.78613491480358, 0.789084353140195, 0.7912217659828884,
0.79284214559524, 0.794112956558801, 0.75390625, 0.7856929451142176,
0.7976688481430754, 0.8039848974727624, 0.807891868948366,
0.8105487660137676, 0.812473307174702, 0.8139318233591120,
0.815075399104785, 0.7744140625, 0.8037322594985427,
0.814742863657656, 0.8205425178645808, 0.8241275984172285,
0.8265645374416, 0.8283292196088257, 0.829666291102775,
0.8307144686362666, 0.7905273437499996, 0.8178712053954738,
0.828116983756619, 0.833508948940494, 0.8368403871552892,
0.839104213210105, 0.840743186196171, 0.84198481438049,
0.8429580531563676, 0.803619384765625, 0.829338573944648,
0.8389591907548646, 0.84401876783902, 0.84714369697889,
0.8492667010581667, 0.850803474598719, 0.851967542858308,
0.8528799045949524, 0.8145294189453126, 0.838881732845347,
0.847979024541911, 0.852760894015685, 0.8557134656773457,
0.8577190131799202, 0.85917058278431, 0.860270010472127,
0.861131648404582, 0.823802947998047, 0.846984756807511,
0.855635653643743, 0.860180994825685, 0.86298688573253,
0.864892525675245, 0.866271647085603, 0.867316125625004,
0.8681346531755114
])
# > bt=c()
# > for(i in as.single(1:10)){for(k in as.single(2:10)){bt = c(bt, binom.test(i+1, k*i,(1/k))$p.value); print(c(i+1, k*i,(1/k)))}}
binom_testp1 = np.array([
0.5, 0.259259259259259, 0.26171875, 0.26272, 0.2632244513031551,
0.2635138663069203, 0.2636951804161073, 0.2638162407564354,
0.2639010709000002, 0.625, 0.4074074074074074, 0.42156982421875,
0.4295746560000003, 0.43473045988554, 0.4383309503172684,
0.4409884859402103, 0.4430309389962837, 0.444649849401104, 0.6875,
0.4927602499618962, 0.5096031427383425, 0.5189636628480,
0.5249280070771274, 0.5290623300865124, 0.5320974248125793,
0.5344204730474308, 0.536255847400756, 0.7265625, 0.5496019313526808,
0.5669248746708034, 0.576436455045805, 0.5824538812831795,
0.5866053321547824, 0.589642781414643, 0.5919618019300193,
0.593790427805202, 0.75390625, 0.590868349763505, 0.607983393277209,
0.617303847446822, 0.623172512167948, 0.627208862156123,
0.6301556891501057, 0.632401894928977, 0.6341708982290303,
0.7744140625, 0.622562037497196, 0.639236102912278, 0.648263335014579,
0.65392850011132, 0.657816519817211, 0.660650782947676,
0.662808780346311, 0.6645068560246006, 0.7905273437499996,
0.6478843304312477, 0.6640468318879372, 0.6727589686071775,
0.6782129857784873, 0.681950188903695, 0.684671508668418,
0.686741824999918, 0.688369886732168, 0.803619384765625,
0.668716055304315, 0.684360013879534, 0.6927642396829181,
0.6980155964704895, 0.701609591890657, 0.7042244320992127,
0.7062125081341817, 0.707775152962577, 0.8145294189453126,
0.686243374488305, 0.7013873696358975, 0.709501223328243,
0.714563595144314, 0.718024953392931, 0.7205416252126137,
0.722454130389843, 0.723956813292035, 0.823802947998047,
0.701255953767043, 0.715928221686075, 0.723772209289768,
0.7286603031173616, 0.7319999279787631, 0.7344267920995765,
0.736270323773157, 0.737718376096348
])
res4_p1 = [stats.binom_test(v+1, v*k, 1./k) for v in range(1, 11)
for k in range(2, 11)]
res4_m1 = [stats.binom_test(v-1, v*k, 1./k) for v in range(1, 11)
for k in range(2, 11)]
assert_almost_equal(res4_p1, binom_testp1, decimal=13)
assert_almost_equal(res4_m1, binom_testm1, decimal=13)
class TestTrim(object):
# test trim functions
def test_trim1(self):
a = np.arange(11)
assert_equal(np.sort(stats.trim1(a, 0.1)), np.arange(10))
assert_equal(np.sort(stats.trim1(a, 0.2)), np.arange(9))
assert_equal(np.sort(stats.trim1(a, 0.2, tail='left')),
np.arange(2, 11))
assert_equal(np.sort(stats.trim1(a, 3/11., tail='left')),
np.arange(3, 11))
assert_equal(stats.trim1(a, 1.0), [])
assert_equal(stats.trim1(a, 1.0, tail='left'), [])
# empty input
assert_equal(stats.trim1([], 0.1), [])
assert_equal(stats.trim1([], 3/11., tail='left'), [])
assert_equal(stats.trim1([], 4/6.), [])
def test_trimboth(self):
a = np.arange(11)
assert_equal(np.sort(stats.trimboth(a, 3/11.)), np.arange(3, 8))
assert_equal(np.sort(stats.trimboth(a, 0.2)),
np.array([2, 3, 4, 5, 6, 7, 8]))
assert_equal(np.sort(stats.trimboth(np.arange(24).reshape(6, 4), 0.2)),
np.arange(4, 20).reshape(4, 4))
assert_equal(np.sort(stats.trimboth(np.arange(24).reshape(4, 6).T,
2/6.)),
np.array([[2, 8, 14, 20], [3, 9, 15, 21]]))
assert_raises(ValueError, stats.trimboth,
np.arange(24).reshape(4, 6).T, 4/6.)
# empty input
assert_equal(stats.trimboth([], 0.1), [])
assert_equal(stats.trimboth([], 3/11.), [])
assert_equal(stats.trimboth([], 4/6.), [])
def test_trim_mean(self):
# don't use pre-sorted arrays
a = np.array([4, 8, 2, 0, 9, 5, 10, 1, 7, 3, 6])
idx = np.array([3, 5, 0, 1, 2, 4])
a2 = np.arange(24).reshape(6, 4)[idx, :]
a3 = np.arange(24).reshape(6, 4, order='F')[idx, :]
assert_equal(stats.trim_mean(a3, 2/6.),
np.array([2.5, 8.5, 14.5, 20.5]))
assert_equal(stats.trim_mean(a2, 2/6.),
np.array([10., 11., 12., 13.]))
idx4 = np.array([1, 0, 3, 2])
a4 = np.arange(24).reshape(4, 6)[idx4, :]
assert_equal(stats.trim_mean(a4, 2/6.),
np.array([9., 10., 11., 12., 13., 14.]))
# shuffled arange(24) as array_like
a = [7, 11, 12, 21, 16, 6, 22, 1, 5, 0, 18, 10, 17, 9, 19, 15, 23,
20, 2, 14, 4, 13, 8, 3]
assert_equal(stats.trim_mean(a, 2/6.), 11.5)
assert_equal(stats.trim_mean([5,4,3,1,2,0], 2/6.), 2.5)
# check axis argument
np.random.seed(1234)
a = np.random.randint(20, size=(5, 6, 4, 7))
for axis in [0, 1, 2, 3, -1]:
res1 = stats.trim_mean(a, 2/6., axis=axis)
res2 = stats.trim_mean(np.rollaxis(a, axis), 2/6.)
assert_equal(res1, res2)
res1 = stats.trim_mean(a, 2/6., axis=None)
res2 = stats.trim_mean(a.ravel(), 2/6.)
assert_equal(res1, res2)
assert_raises(ValueError, stats.trim_mean, a, 0.6)
# empty input
assert_equal(stats.trim_mean([], 0.0), np.nan)
assert_equal(stats.trim_mean([], 0.6), np.nan)
class TestSigmaClip(object):
def test_sigmaclip1(self):
a = np.concatenate((np.linspace(9.5, 10.5, 31), np.linspace(0, 20, 5)))
fact = 4 # default
c, low, upp = stats.sigmaclip(a)
assert_(c.min() > low)
assert_(c.max() < upp)
assert_equal(low, c.mean() - fact*c.std())
assert_equal(upp, c.mean() + fact*c.std())
assert_equal(c.size, a.size)
def test_sigmaclip2(self):
a = np.concatenate((np.linspace(9.5, 10.5, 31), np.linspace(0, 20, 5)))
fact = 1.5
c, low, upp = stats.sigmaclip(a, fact, fact)
assert_(c.min() > low)
assert_(c.max() < upp)
assert_equal(low, c.mean() - fact*c.std())
assert_equal(upp, c.mean() + fact*c.std())
assert_equal(c.size, 4)
assert_equal(a.size, 36) # check original array unchanged
def test_sigmaclip3(self):
a = np.concatenate((np.linspace(9.5, 10.5, 11),
np.linspace(-100, -50, 3)))
fact = 1.8
c, low, upp = stats.sigmaclip(a, fact, fact)
assert_(c.min() > low)
assert_(c.max() < upp)
assert_equal(low, c.mean() - fact*c.std())
assert_equal(upp, c.mean() + fact*c.std())
assert_equal(c, np.linspace(9.5, 10.5, 11))
def test_sigmaclip_result_attributes(self):
a = np.concatenate((np.linspace(9.5, 10.5, 11),
np.linspace(-100, -50, 3)))
fact = 1.8
res = stats.sigmaclip(a, fact, fact)
attributes = ('clipped', 'lower', 'upper')
check_named_results(res, attributes)
def test_std_zero(self):
# regression test #8632
x = np.ones(10)
assert_equal(stats.sigmaclip(x)[0], x)
class TestFOneWay(object):
def test_trivial(self):
# A trivial test of stats.f_oneway, with F=0.
F, p = stats.f_oneway([0,2], [0,2])
assert_equal(F, 0.0)
def test_basic(self):
# Despite being a floating point calculation, this data should
# result in F being exactly 2.0.
F, p = stats.f_oneway([0,2], [2,4])
assert_equal(F, 2.0)
def test_large_integer_array(self):
a = np.array([655, 788], dtype=np.uint16)
b = np.array([789, 772], dtype=np.uint16)
F, p = stats.f_oneway(a, b)
assert_almost_equal(F, 0.77450216931805538)
def test_result_attributes(self):
a = np.array([655, 788], dtype=np.uint16)
b = np.array([789, 772], dtype=np.uint16)
res = stats.f_oneway(a, b)
attributes = ('statistic', 'pvalue')
check_named_results(res, attributes)
def test_nist(self):
# These are the nist ANOVA files. They can be found at:
# http://www.itl.nist.gov/div898/strd/anova/anova.html
filenames = ['SiRstv.dat', 'SmLs01.dat', 'SmLs02.dat', 'SmLs03.dat',
'AtmWtAg.dat', 'SmLs04.dat', 'SmLs05.dat', 'SmLs06.dat',
'SmLs07.dat', 'SmLs08.dat', 'SmLs09.dat']
for test_case in filenames:
rtol = 1e-7
fname = os.path.abspath(os.path.join(os.path.dirname(__file__),
'data/nist_anova', test_case))
with open(fname, 'r') as f:
content = f.read().split('\n')
certified = [line.split() for line in content[40:48]
if line.strip()]
dataf = np.loadtxt(fname, skiprows=60)
y, x = dataf.T
y = y.astype(int)
caty = np.unique(y)
f = float(certified[0][-1])
xlist = [x[y == i] for i in caty]
res = stats.f_oneway(*xlist)
# With the hard test cases we relax the tolerance a bit.
hard_tc = ('SmLs07.dat', 'SmLs08.dat', 'SmLs09.dat')
if test_case in hard_tc:
rtol = 1e-4
assert_allclose(res[0], f, rtol=rtol,
err_msg='Failing testcase: %s' % test_case)
class TestKruskal(object):
def test_simple(self):
x = [1]
y = [2]
h, p = stats.kruskal(x, y)
assert_equal(h, 1.0)
assert_approx_equal(p, stats.distributions.chi2.sf(h, 1))
h, p = stats.kruskal(np.array(x), np.array(y))
assert_equal(h, 1.0)
assert_approx_equal(p, stats.distributions.chi2.sf(h, 1))
def test_basic(self):
x = [1, 3, 5, 7, 9]
y = [2, 4, 6, 8, 10]
h, p = stats.kruskal(x, y)
assert_approx_equal(h, 3./11, significant=10)
assert_approx_equal(p, stats.distributions.chi2.sf(3./11, 1))
h, p = stats.kruskal(np.array(x), np.array(y))
assert_approx_equal(h, 3./11, significant=10)
assert_approx_equal(p, stats.distributions.chi2.sf(3./11, 1))
def test_simple_tie(self):
x = [1]
y = [1, 2]
h_uncorr = 1.5**2 + 2*2.25**2 - 12
corr = 0.75
expected = h_uncorr / corr # 0.5
h, p = stats.kruskal(x, y)
# Since the expression is simple and the exact answer is 0.5, it
# should be safe to use assert_equal().
assert_equal(h, expected)
def test_another_tie(self):
x = [1, 1, 1, 2]
y = [2, 2, 2, 2]
h_uncorr = (12. / 8. / 9.) * 4 * (3**2 + 6**2) - 3 * 9
corr = 1 - float(3**3 - 3 + 5**3 - 5) / (8**3 - 8)
expected = h_uncorr / corr
h, p = stats.kruskal(x, y)
assert_approx_equal(h, expected)
def test_three_groups(self):
# A test of stats.kruskal with three groups, with ties.
x = [1, 1, 1]
y = [2, 2, 2]
z = [2, 2]
h_uncorr = (12. / 8. / 9.) * (3*2**2 + 3*6**2 + 2*6**2) - 3 * 9 # 5.0
corr = 1 - float(3**3 - 3 + 5**3 - 5) / (8**3 - 8)
expected = h_uncorr / corr # 7.0
h, p = stats.kruskal(x, y, z)
assert_approx_equal(h, expected)
assert_approx_equal(p, stats.distributions.chi2.sf(h, 2))
def test_empty(self):
# A test of stats.kruskal with three groups, with ties.
x = [1, 1, 1]
y = [2, 2, 2]
z = []
assert_equal(stats.kruskal(x, y, z), (np.nan, np.nan))
def test_kruskal_result_attributes(self):
x = [1, 3, 5, 7, 9]
y = [2, 4, 6, 8, 10]
res = stats.kruskal(x, y)
attributes = ('statistic', 'pvalue')
check_named_results(res, attributes)
def test_nan_policy(self):
x = np.arange(10.)
x[9] = np.nan
assert_equal(stats.kruskal(x, x), (np.nan, np.nan))
assert_almost_equal(stats.kruskal(x, x, nan_policy='omit'), (0.0, 1.0))
assert_raises(ValueError, stats.kruskal, x, x, nan_policy='raise')
assert_raises(ValueError, stats.kruskal, x, x, nan_policy='foobar')
class TestCombinePvalues(object):
def test_fisher(self):
# Example taken from http://en.wikipedia.org/wiki/Fisher's_exact_test#Example
xsq, p = stats.combine_pvalues([.01, .2, .3], method='fisher')
assert_approx_equal(p, 0.02156, significant=4)
def test_stouffer(self):
Z, p = stats.combine_pvalues([.01, .2, .3], method='stouffer')
assert_approx_equal(p, 0.01651, significant=4)
def test_stouffer2(self):
Z, p = stats.combine_pvalues([.5, .5, .5], method='stouffer')
assert_approx_equal(p, 0.5, significant=4)
def test_weighted_stouffer(self):
Z, p = stats.combine_pvalues([.01, .2, .3], method='stouffer',
weights=np.ones(3))
assert_approx_equal(p, 0.01651, significant=4)
def test_weighted_stouffer2(self):
Z, p = stats.combine_pvalues([.01, .2, .3], method='stouffer',
weights=np.array((1, 4, 9)))
assert_approx_equal(p, 0.1464, significant=4)
class TestCdfDistanceValidation(object):
"""
Test that _cdf_distance() (via wasserstein_distance()) raises ValueErrors
for bad inputs.
"""
def test_distinct_value_and_weight_lengths(self):
# When the number of weights does not match the number of values,
# a ValueError should be raised.
assert_raises(ValueError, stats.wasserstein_distance,
[1], [2], [4], [3, 1])
assert_raises(ValueError, stats.wasserstein_distance, [1], [2], [1, 0])
def test_zero_weight(self):
# When a distribution is given zero weight, a ValueError should be
# raised.
assert_raises(ValueError, stats.wasserstein_distance,
[0, 1], [2], [0, 0])
assert_raises(ValueError, stats.wasserstein_distance,
[0, 1], [2], [3, 1], [0])
def test_negative_weights(self):
# A ValueError should be raised if there are any negative weights.
assert_raises(ValueError, stats.wasserstein_distance,
[0, 1], [2, 2], [1, 1], [3, -1])
def test_empty_distribution(self):
# A ValueError should be raised when trying to measure the distance
# between something and nothing.
assert_raises(ValueError, stats.wasserstein_distance, [], [2, 2])
assert_raises(ValueError, stats.wasserstein_distance, [1], [])
def test_inf_weight(self):
# An inf weight is not valid.
assert_raises(ValueError, stats.wasserstein_distance,
[1, 2, 1], [1, 1], [1, np.inf, 1], [1, 1])
class TestWassersteinDistance(object):
""" Tests for wasserstein_distance() output values.
"""
def test_simple(self):
# For basic distributions, the value of the Wasserstein distance is
# straightforward.
assert_almost_equal(
stats.wasserstein_distance([0, 1], [0], [1, 1], [1]),
.5)
assert_almost_equal(stats.wasserstein_distance(
[0, 1], [0], [3, 1], [1]),
.25)
assert_almost_equal(stats.wasserstein_distance(
[0, 2], [0], [1, 1], [1]),
1)
assert_almost_equal(stats.wasserstein_distance(
[0, 1, 2], [1, 2, 3]),
1)
def test_same_distribution(self):
# Any distribution moved to itself should have a Wasserstein distance of
# zero.
assert_equal(stats.wasserstein_distance([1, 2, 3], [2, 1, 3]), 0)
assert_equal(
stats.wasserstein_distance([1, 1, 1, 4], [4, 1],
[1, 1, 1, 1], [1, 3]),
0)
def test_shift(self):
# If the whole distribution is shifted by x, then the Wasserstein
# distance should be x.
assert_almost_equal(stats.wasserstein_distance([0], [1]), 1)
assert_almost_equal(stats.wasserstein_distance([-5], [5]), 10)
assert_almost_equal(
stats.wasserstein_distance([1, 2, 3, 4, 5], [11, 12, 13, 14, 15]),
10)
assert_almost_equal(
stats.wasserstein_distance([4.5, 6.7, 2.1], [4.6, 7, 9.2],
[3, 1, 1], [1, 3, 1]),
2.5)
def test_combine_weights(self):
# Assigning a weight w to a value is equivalent to including that value
# w times in the value array with weight of 1.
assert_almost_equal(
stats.wasserstein_distance(
[0, 0, 1, 1, 1, 1, 5], [0, 3, 3, 3, 3, 4, 4],
[1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1]),
stats.wasserstein_distance([5, 0, 1], [0, 4, 3],
[1, 2, 4], [1, 2, 4]))
def test_collapse(self):
# Collapsing a distribution to a point distribution at zero is
# equivalent to taking the average of the absolute values of the values.
u = np.arange(-10, 30, 0.3)
v = np.zeros_like(u)
assert_almost_equal(
stats.wasserstein_distance(u, v),
np.mean(np.abs(u)))
u_weights = np.arange(len(u))
v_weights = u_weights[::-1]
assert_almost_equal(
stats.wasserstein_distance(u, v, u_weights, v_weights),
np.average(np.abs(u), weights=u_weights))
def test_zero_weight(self):
# Values with zero weight have no impact on the Wasserstein distance.
assert_almost_equal(
stats.wasserstein_distance([1, 2, 100000], [1, 1],
[1, 1, 0], [1, 1]),
stats.wasserstein_distance([1, 2], [1, 1], [1, 1], [1, 1]))
def test_inf_values(self):
# Inf values can lead to an inf distance or trigger a RuntimeWarning
# (and return NaN) if the distance is undefined.
assert_equal(
stats.wasserstein_distance([1, 2, np.inf], [1, 1]),
np.inf)
assert_equal(
stats.wasserstein_distance([1, 2, np.inf], [-np.inf, 1]),
np.inf)
assert_equal(
stats.wasserstein_distance([1, -np.inf, np.inf], [1, 1]),
np.inf)
with suppress_warnings() as sup:
r = sup.record(RuntimeWarning, "invalid value*")
assert_equal(
stats.wasserstein_distance([1, 2, np.inf], [np.inf, 1]),
np.nan)
class TestEnergyDistance(object):
""" Tests for energy_distance() output values.
"""
def test_simple(self):
# For basic distributions, the value of the energy distance is
# straightforward.
assert_almost_equal(
stats.energy_distance([0, 1], [0], [1, 1], [1]),
np.sqrt(2) * .5)
assert_almost_equal(stats.energy_distance(
[0, 1], [0], [3, 1], [1]),
np.sqrt(2) * .25)
assert_almost_equal(stats.energy_distance(
[0, 2], [0], [1, 1], [1]),
2 * .5)
assert_almost_equal(
stats.energy_distance([0, 1, 2], [1, 2, 3]),
np.sqrt(2) * (3*(1./3**2))**.5)
def test_same_distribution(self):
# Any distribution moved to itself should have a energy distance of
# zero.
assert_equal(stats.energy_distance([1, 2, 3], [2, 1, 3]), 0)
assert_equal(
stats.energy_distance([1, 1, 1, 4], [4, 1], [1, 1, 1, 1], [1, 3]),
0)
def test_shift(self):
# If a single-point distribution is shifted by x, then the energy
# distance should be sqrt(2) * sqrt(x).
assert_almost_equal(stats.energy_distance([0], [1]), np.sqrt(2))
assert_almost_equal(
stats.energy_distance([-5], [5]),
np.sqrt(2) * 10**.5)
def test_combine_weights(self):
# Assigning a weight w to a value is equivalent to including that value
# w times in the value array with weight of 1.
assert_almost_equal(
stats.energy_distance([0, 0, 1, 1, 1, 1, 5], [0, 3, 3, 3, 3, 4, 4],
[1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1]),
stats.energy_distance([5, 0, 1], [0, 4, 3], [1, 2, 4], [1, 2, 4]))
def test_zero_weight(self):
# Values with zero weight have no impact on the energy distance.
assert_almost_equal(
stats.energy_distance([1, 2, 100000], [1, 1], [1, 1, 0], [1, 1]),
stats.energy_distance([1, 2], [1, 1], [1, 1], [1, 1]))
def test_inf_values(self):
# Inf values can lead to an inf distance or trigger a RuntimeWarning
# (and return NaN) if the distance is undefined.
assert_equal(stats.energy_distance([1, 2, np.inf], [1, 1]), np.inf)
assert_equal(
stats.energy_distance([1, 2, np.inf], [-np.inf, 1]),
np.inf)
assert_equal(
stats.energy_distance([1, -np.inf, np.inf], [1, 1]),
np.inf)
with suppress_warnings() as sup:
r = sup.record(RuntimeWarning, "invalid value*")
assert_equal(
stats.energy_distance([1, 2, np.inf], [np.inf, 1]),
np.nan)
| 170,720 | 39.000234 | 134 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/scipy/stats/tests/test_mstats_basic.py
|
"""
Tests for the stats.mstats module (support for masked arrays)
"""
from __future__ import division, print_function, absolute_import
import warnings
import numpy as np
from numpy import nan
import numpy.ma as ma
from numpy.ma import masked, nomask
import scipy.stats.mstats as mstats
from scipy import stats
from .common_tests import check_named_results
import pytest
from pytest import raises as assert_raises
from numpy.ma.testutils import (assert_equal, assert_almost_equal,
assert_array_almost_equal, assert_array_almost_equal_nulp, assert_,
assert_allclose, assert_array_equal)
from scipy._lib._numpy_compat import suppress_warnings
class TestMquantiles(object):
def test_mquantiles_limit_keyword(self):
# Regression test for Trac ticket #867
data = np.array([[6., 7., 1.],
[47., 15., 2.],
[49., 36., 3.],
[15., 39., 4.],
[42., 40., -999.],
[41., 41., -999.],
[7., -999., -999.],
[39., -999., -999.],
[43., -999., -999.],
[40., -999., -999.],
[36., -999., -999.]])
desired = [[19.2, 14.6, 1.45],
[40.0, 37.5, 2.5],
[42.8, 40.05, 3.55]]
quants = mstats.mquantiles(data, axis=0, limit=(0, 50))
assert_almost_equal(quants, desired)
class TestGMean(object):
def test_1D(self):
a = (1,2,3,4)
actual = mstats.gmean(a)
desired = np.power(1*2*3*4,1./4.)
assert_almost_equal(actual, desired, decimal=14)
desired1 = mstats.gmean(a,axis=-1)
assert_almost_equal(actual, desired1, decimal=14)
assert_(not isinstance(desired1, ma.MaskedArray))
a = ma.array((1,2,3,4),mask=(0,0,0,1))
actual = mstats.gmean(a)
desired = np.power(1*2*3,1./3.)
assert_almost_equal(actual, desired,decimal=14)
desired1 = mstats.gmean(a,axis=-1)
assert_almost_equal(actual, desired1, decimal=14)
@pytest.mark.skipif(not hasattr(np, 'float96'), reason='cannot find float96 so skipping')
def test_1D_float96(self):
a = ma.array((1,2,3,4), mask=(0,0,0,1))
actual_dt = mstats.gmean(a, dtype=np.float96)
desired_dt = np.power(1 * 2 * 3, 1. / 3.).astype(np.float96)
assert_almost_equal(actual_dt, desired_dt, decimal=14)
assert_(actual_dt.dtype == desired_dt.dtype)
def test_2D(self):
a = ma.array(((1, 2, 3, 4), (1, 2, 3, 4), (1, 2, 3, 4)),
mask=((0, 0, 0, 0), (1, 0, 0, 1), (0, 1, 1, 0)))
actual = mstats.gmean(a)
desired = np.array((1,2,3,4))
assert_array_almost_equal(actual, desired, decimal=14)
desired1 = mstats.gmean(a,axis=0)
assert_array_almost_equal(actual, desired1, decimal=14)
actual = mstats.gmean(a, -1)
desired = ma.array((np.power(1*2*3*4,1./4.),
np.power(2*3,1./2.),
np.power(1*4,1./2.)))
assert_array_almost_equal(actual, desired, decimal=14)
class TestHMean(object):
def test_1D(self):
a = (1,2,3,4)
actual = mstats.hmean(a)
desired = 4. / (1./1 + 1./2 + 1./3 + 1./4)
assert_almost_equal(actual, desired, decimal=14)
desired1 = mstats.hmean(ma.array(a),axis=-1)
assert_almost_equal(actual, desired1, decimal=14)
a = ma.array((1,2,3,4),mask=(0,0,0,1))
actual = mstats.hmean(a)
desired = 3. / (1./1 + 1./2 + 1./3)
assert_almost_equal(actual, desired,decimal=14)
desired1 = mstats.hmean(a,axis=-1)
assert_almost_equal(actual, desired1, decimal=14)
@pytest.mark.skipif(not hasattr(np, 'float96'), reason='cannot find float96 so skipping')
def test_1D_float96(self):
a = ma.array((1,2,3,4), mask=(0,0,0,1))
actual_dt = mstats.hmean(a, dtype=np.float96)
desired_dt = np.asarray(3. / (1./1 + 1./2 + 1./3),
dtype=np.float96)
assert_almost_equal(actual_dt, desired_dt, decimal=14)
assert_(actual_dt.dtype == desired_dt.dtype)
def test_2D(self):
a = ma.array(((1,2,3,4),(1,2,3,4),(1,2,3,4)),
mask=((0,0,0,0),(1,0,0,1),(0,1,1,0)))
actual = mstats.hmean(a)
desired = ma.array((1,2,3,4))
assert_array_almost_equal(actual, desired, decimal=14)
actual1 = mstats.hmean(a,axis=-1)
desired = (4./(1/1.+1/2.+1/3.+1/4.),
2./(1/2.+1/3.),
2./(1/1.+1/4.)
)
assert_array_almost_equal(actual1, desired, decimal=14)
class TestRanking(object):
def test_ranking(self):
x = ma.array([0,1,1,1,2,3,4,5,5,6,])
assert_almost_equal(mstats.rankdata(x),
[1,3,3,3,5,6,7,8.5,8.5,10])
x[[3,4]] = masked
assert_almost_equal(mstats.rankdata(x),
[1,2.5,2.5,0,0,4,5,6.5,6.5,8])
assert_almost_equal(mstats.rankdata(x, use_missing=True),
[1,2.5,2.5,4.5,4.5,4,5,6.5,6.5,8])
x = ma.array([0,1,5,1,2,4,3,5,1,6,])
assert_almost_equal(mstats.rankdata(x),
[1,3,8.5,3,5,7,6,8.5,3,10])
x = ma.array([[0,1,1,1,2], [3,4,5,5,6,]])
assert_almost_equal(mstats.rankdata(x),
[[1,3,3,3,5], [6,7,8.5,8.5,10]])
assert_almost_equal(mstats.rankdata(x, axis=1),
[[1,3,3,3,5], [1,2,3.5,3.5,5]])
assert_almost_equal(mstats.rankdata(x,axis=0),
[[1,1,1,1,1], [2,2,2,2,2,]])
class TestCorr(object):
def test_pearsonr(self):
# Tests some computations of Pearson's r
x = ma.arange(10)
with warnings.catch_warnings():
# The tests in this context are edge cases, with perfect
# correlation or anticorrelation, or totally masked data.
# None of these should trigger a RuntimeWarning.
warnings.simplefilter("error", RuntimeWarning)
assert_almost_equal(mstats.pearsonr(x, x)[0], 1.0)
assert_almost_equal(mstats.pearsonr(x, x[::-1])[0], -1.0)
x = ma.array(x, mask=True)
pr = mstats.pearsonr(x, x)
assert_(pr[0] is masked)
assert_(pr[1] is masked)
x1 = ma.array([-1.0, 0.0, 1.0])
y1 = ma.array([0, 0, 3])
r, p = mstats.pearsonr(x1, y1)
assert_almost_equal(r, np.sqrt(3)/2)
assert_almost_equal(p, 1.0/3)
# (x2, y2) have the same unmasked data as (x1, y1).
mask = [False, False, False, True]
x2 = ma.array([-1.0, 0.0, 1.0, 99.0], mask=mask)
y2 = ma.array([0, 0, 3, -1], mask=mask)
r, p = mstats.pearsonr(x2, y2)
assert_almost_equal(r, np.sqrt(3)/2)
assert_almost_equal(p, 1.0/3)
def test_spearmanr(self):
# Tests some computations of Spearman's rho
(x, y) = ([5.05,6.75,3.21,2.66],[1.65,2.64,2.64,6.95])
assert_almost_equal(mstats.spearmanr(x,y)[0], -0.6324555)
(x, y) = ([5.05,6.75,3.21,2.66,np.nan],[1.65,2.64,2.64,6.95,np.nan])
(x, y) = (ma.fix_invalid(x), ma.fix_invalid(y))
assert_almost_equal(mstats.spearmanr(x,y)[0], -0.6324555)
x = [2.0, 47.4, 42.0, 10.8, 60.1, 1.7, 64.0, 63.1,
1.0, 1.4, 7.9, 0.3, 3.9, 0.3, 6.7]
y = [22.6, 8.3, 44.4, 11.9, 24.6, 0.6, 5.7, 41.6,
0.0, 0.6, 6.7, 3.8, 1.0, 1.2, 1.4]
assert_almost_equal(mstats.spearmanr(x,y)[0], 0.6887299)
x = [2.0, 47.4, 42.0, 10.8, 60.1, 1.7, 64.0, 63.1,
1.0, 1.4, 7.9, 0.3, 3.9, 0.3, 6.7, np.nan]
y = [22.6, 8.3, 44.4, 11.9, 24.6, 0.6, 5.7, 41.6,
0.0, 0.6, 6.7, 3.8, 1.0, 1.2, 1.4, np.nan]
(x, y) = (ma.fix_invalid(x), ma.fix_invalid(y))
assert_almost_equal(mstats.spearmanr(x,y)[0], 0.6887299)
# Next test is to make sure calculation uses sufficient precision.
# The denominator's value is ~n^3 and used to be represented as an
# int. 2000**3 > 2**32 so these arrays would cause overflow on
# some machines.
x = list(range(2000))
y = list(range(2000))
y[0], y[9] = y[9], y[0]
y[10], y[434] = y[434], y[10]
y[435], y[1509] = y[1509], y[435]
# rho = 1 - 6 * (2 * (9^2 + 424^2 + 1074^2))/(2000 * (2000^2 - 1))
# = 1 - (1 / 500)
# = 0.998
assert_almost_equal(mstats.spearmanr(x,y)[0], 0.998)
# test for namedtuple attributes
res = mstats.spearmanr(x, y)
attributes = ('correlation', 'pvalue')
check_named_results(res, attributes, ma=True)
def test_kendalltau(self):
# Tests some computations of Kendall's tau
x = ma.fix_invalid([5.05, 6.75, 3.21, 2.66,np.nan])
y = ma.fix_invalid([1.65, 26.5, -5.93, 7.96, np.nan])
z = ma.fix_invalid([1.65, 2.64, 2.64, 6.95, np.nan])
assert_almost_equal(np.asarray(mstats.kendalltau(x,y)),
[+0.3333333,0.4969059])
assert_almost_equal(np.asarray(mstats.kendalltau(x,z)),
[-0.5477226,0.2785987])
#
x = ma.fix_invalid([0, 0, 0, 0,20,20, 0,60, 0,20,
10,10, 0,40, 0,20, 0, 0, 0, 0, 0, np.nan])
y = ma.fix_invalid([0,80,80,80,10,33,60, 0,67,27,
25,80,80,80,80,80,80, 0,10,45, np.nan, 0])
result = mstats.kendalltau(x,y)
assert_almost_equal(np.asarray(result), [-0.1585188, 0.4128009])
# make sure internal variable use correct precision with
# larger arrays
x = np.arange(2000, dtype=float)
x = ma.masked_greater(x, 1995)
y = np.arange(2000, dtype=float)
y = np.concatenate((y[1000:], y[:1000]))
assert_(np.isfinite(mstats.kendalltau(x,y)[1]))
# test for namedtuple attributes
res = mstats.kendalltau(x, y)
attributes = ('correlation', 'pvalue')
check_named_results(res, attributes, ma=True)
def test_kendalltau_seasonal(self):
# Tests the seasonal Kendall tau.
x = [[nan,nan, 4, 2, 16, 26, 5, 1, 5, 1, 2, 3, 1],
[4, 3, 5, 3, 2, 7, 3, 1, 1, 2, 3, 5, 3],
[3, 2, 5, 6, 18, 4, 9, 1, 1,nan, 1, 1,nan],
[nan, 6, 11, 4, 17,nan, 6, 1, 1, 2, 5, 1, 1]]
x = ma.fix_invalid(x).T
output = mstats.kendalltau_seasonal(x)
assert_almost_equal(output['global p-value (indep)'], 0.008, 3)
assert_almost_equal(output['seasonal p-value'].round(2),
[0.18,0.53,0.20,0.04])
def test_pointbiserial(self):
x = [1,0,1,1,1,1,0,1,0,0,0,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0,0,1,0,
0,0,0,0,1,-1]
y = [14.8,13.8,12.4,10.1,7.1,6.1,5.8,4.6,4.3,3.5,3.3,3.2,3.0,
2.8,2.8,2.5,2.4,2.3,2.1,1.7,1.7,1.5,1.3,1.3,1.2,1.2,1.1,
0.8,0.7,0.6,0.5,0.2,0.2,0.1,np.nan]
assert_almost_equal(mstats.pointbiserialr(x, y)[0], 0.36149, 5)
# test for namedtuple attributes
res = mstats.pointbiserialr(x, y)
attributes = ('correlation', 'pvalue')
check_named_results(res, attributes, ma=True)
class TestTrimming(object):
def test_trim(self):
a = ma.arange(10)
assert_equal(mstats.trim(a), [0,1,2,3,4,5,6,7,8,9])
a = ma.arange(10)
assert_equal(mstats.trim(a,(2,8)), [None,None,2,3,4,5,6,7,8,None])
a = ma.arange(10)
assert_equal(mstats.trim(a,limits=(2,8),inclusive=(False,False)),
[None,None,None,3,4,5,6,7,None,None])
a = ma.arange(10)
assert_equal(mstats.trim(a,limits=(0.1,0.2),relative=True),
[None,1,2,3,4,5,6,7,None,None])
a = ma.arange(12)
a[[0,-1]] = a[5] = masked
assert_equal(mstats.trim(a, (2,8)),
[None, None, 2, 3, 4, None, 6, 7, 8, None, None, None])
x = ma.arange(100).reshape(10, 10)
expected = [1]*10 + [0]*70 + [1]*20
trimx = mstats.trim(x, (0.1,0.2), relative=True, axis=None)
assert_equal(trimx._mask.ravel(), expected)
trimx = mstats.trim(x, (0.1,0.2), relative=True, axis=0)
assert_equal(trimx._mask.ravel(), expected)
trimx = mstats.trim(x, (0.1,0.2), relative=True, axis=-1)
assert_equal(trimx._mask.T.ravel(), expected)
# same as above, but with an extra masked row inserted
x = ma.arange(110).reshape(11, 10)
x[1] = masked
expected = [1]*20 + [0]*70 + [1]*20
trimx = mstats.trim(x, (0.1,0.2), relative=True, axis=None)
assert_equal(trimx._mask.ravel(), expected)
trimx = mstats.trim(x, (0.1,0.2), relative=True, axis=0)
assert_equal(trimx._mask.ravel(), expected)
trimx = mstats.trim(x.T, (0.1,0.2), relative=True, axis=-1)
assert_equal(trimx.T._mask.ravel(), expected)
def test_trim_old(self):
x = ma.arange(100)
assert_equal(mstats.trimboth(x).count(), 60)
assert_equal(mstats.trimtail(x,tail='r').count(), 80)
x[50:70] = masked
trimx = mstats.trimboth(x)
assert_equal(trimx.count(), 48)
assert_equal(trimx._mask, [1]*16 + [0]*34 + [1]*20 + [0]*14 + [1]*16)
x._mask = nomask
x.shape = (10,10)
assert_equal(mstats.trimboth(x).count(), 60)
assert_equal(mstats.trimtail(x).count(), 80)
def test_trimmedmean(self):
data = ma.array([77, 87, 88,114,151,210,219,246,253,262,
296,299,306,376,428,515,666,1310,2611])
assert_almost_equal(mstats.trimmed_mean(data,0.1), 343, 0)
assert_almost_equal(mstats.trimmed_mean(data,(0.1,0.1)), 343, 0)
assert_almost_equal(mstats.trimmed_mean(data,(0.2,0.2)), 283, 0)
def test_trimmed_stde(self):
data = ma.array([77, 87, 88,114,151,210,219,246,253,262,
296,299,306,376,428,515,666,1310,2611])
assert_almost_equal(mstats.trimmed_stde(data,(0.2,0.2)), 56.13193, 5)
assert_almost_equal(mstats.trimmed_stde(data,0.2), 56.13193, 5)
def test_winsorization(self):
data = ma.array([77, 87, 88,114,151,210,219,246,253,262,
296,299,306,376,428,515,666,1310,2611])
assert_almost_equal(mstats.winsorize(data,(0.2,0.2)).var(ddof=1),
21551.4, 1)
assert_almost_equal(
mstats.winsorize(data, (0.2,0.2),(False,False)).var(ddof=1),
11887.3, 1)
data[5] = masked
winsorized = mstats.winsorize(data)
assert_equal(winsorized.mask, data.mask)
class TestMoments(object):
# Comparison numbers are found using R v.1.5.1
# note that length(testcase) = 4
# testmathworks comes from documentation for the
# Statistics Toolbox for Matlab and can be found at both
# http://www.mathworks.com/access/helpdesk/help/toolbox/stats/kurtosis.shtml
# http://www.mathworks.com/access/helpdesk/help/toolbox/stats/skewness.shtml
# Note that both test cases came from here.
testcase = [1,2,3,4]
testmathworks = ma.fix_invalid([1.165, 0.6268, 0.0751, 0.3516, -0.6965,
np.nan])
testcase_2d = ma.array(
np.array([[0.05245846, 0.50344235, 0.86589117, 0.36936353, 0.46961149],
[0.11574073, 0.31299969, 0.45925772, 0.72618805, 0.75194407],
[0.67696689, 0.91878127, 0.09769044, 0.04645137, 0.37615733],
[0.05903624, 0.29908861, 0.34088298, 0.66216337, 0.83160998],
[0.64619526, 0.94894632, 0.27855892, 0.0706151, 0.39962917]]),
mask=np.array([[True, False, False, True, False],
[True, True, True, False, True],
[False, False, False, False, False],
[True, True, True, True, True],
[False, False, True, False, False]], dtype=bool))
def test_moment(self):
y = mstats.moment(self.testcase,1)
assert_almost_equal(y,0.0,10)
y = mstats.moment(self.testcase,2)
assert_almost_equal(y,1.25)
y = mstats.moment(self.testcase,3)
assert_almost_equal(y,0.0)
y = mstats.moment(self.testcase,4)
assert_almost_equal(y,2.5625)
def test_variation(self):
y = mstats.variation(self.testcase)
assert_almost_equal(y,0.44721359549996, 10)
def test_skewness(self):
y = mstats.skew(self.testmathworks)
assert_almost_equal(y,-0.29322304336607,10)
y = mstats.skew(self.testmathworks,bias=0)
assert_almost_equal(y,-0.437111105023940,10)
y = mstats.skew(self.testcase)
assert_almost_equal(y,0.0,10)
def test_kurtosis(self):
# Set flags for axis = 0 and fisher=0 (Pearson's definition of kurtosis
# for compatibility with Matlab)
y = mstats.kurtosis(self.testmathworks,0,fisher=0,bias=1)
assert_almost_equal(y, 2.1658856802973,10)
# Note that MATLAB has confusing docs for the following case
# kurtosis(x,0) gives an unbiased estimate of Pearson's skewness
# kurtosis(x) gives a biased estimate of Fisher's skewness (Pearson-3)
# The MATLAB docs imply that both should give Fisher's
y = mstats.kurtosis(self.testmathworks,fisher=0, bias=0)
assert_almost_equal(y, 3.663542721189047,10)
y = mstats.kurtosis(self.testcase,0,0)
assert_almost_equal(y,1.64)
# test that kurtosis works on multidimensional masked arrays
correct_2d = ma.array(np.array([-1.5, -3., -1.47247052385, 0.,
-1.26979517952]),
mask=np.array([False, False, False, True,
False], dtype=bool))
assert_array_almost_equal(mstats.kurtosis(self.testcase_2d, 1),
correct_2d)
for i, row in enumerate(self.testcase_2d):
assert_almost_equal(mstats.kurtosis(row), correct_2d[i])
correct_2d_bias_corrected = ma.array(
np.array([-1.5, -3., -1.88988209538, 0., -0.5234638463918877]),
mask=np.array([False, False, False, True, False], dtype=bool))
assert_array_almost_equal(mstats.kurtosis(self.testcase_2d, 1,
bias=False),
correct_2d_bias_corrected)
for i, row in enumerate(self.testcase_2d):
assert_almost_equal(mstats.kurtosis(row, bias=False),
correct_2d_bias_corrected[i])
# Check consistency between stats and mstats implementations
assert_array_almost_equal_nulp(mstats.kurtosis(self.testcase_2d[2, :]),
stats.kurtosis(self.testcase_2d[2, :]),
nulp=4)
def test_mode(self):
a1 = [0,0,0,1,1,1,2,3,3,3,3,4,5,6,7]
a2 = np.reshape(a1, (3,5))
a3 = np.array([1,2,3,4,5,6])
a4 = np.reshape(a3, (3,2))
ma1 = ma.masked_where(ma.array(a1) > 2, a1)
ma2 = ma.masked_where(a2 > 2, a2)
ma3 = ma.masked_where(a3 < 2, a3)
ma4 = ma.masked_where(ma.array(a4) < 2, a4)
assert_equal(mstats.mode(a1, axis=None), (3,4))
assert_equal(mstats.mode(a1, axis=0), (3,4))
assert_equal(mstats.mode(ma1, axis=None), (0,3))
assert_equal(mstats.mode(a2, axis=None), (3,4))
assert_equal(mstats.mode(ma2, axis=None), (0,3))
assert_equal(mstats.mode(a3, axis=None), (1,1))
assert_equal(mstats.mode(ma3, axis=None), (2,1))
assert_equal(mstats.mode(a2, axis=0), ([[0,0,0,1,1]], [[1,1,1,1,1]]))
assert_equal(mstats.mode(ma2, axis=0), ([[0,0,0,1,1]], [[1,1,1,1,1]]))
assert_equal(mstats.mode(a2, axis=-1), ([[0],[3],[3]], [[3],[3],[1]]))
assert_equal(mstats.mode(ma2, axis=-1), ([[0],[1],[0]], [[3],[1],[0]]))
assert_equal(mstats.mode(ma4, axis=0), ([[3,2]], [[1,1]]))
assert_equal(mstats.mode(ma4, axis=-1), ([[2],[3],[5]], [[1],[1],[1]]))
a1_res = mstats.mode(a1, axis=None)
# test for namedtuple attributes
attributes = ('mode', 'count')
check_named_results(a1_res, attributes, ma=True)
def test_mode_modifies_input(self):
# regression test for gh-6428: mode(..., axis=None) may not modify
# the input array
im = np.zeros((100, 100))
im[:50, :] += 1
im[:, :50] += 1
cp = im.copy()
a = mstats.mode(im, None)
assert_equal(im, cp)
class TestPercentile(object):
def setup_method(self):
self.a1 = [3,4,5,10,-3,-5,6]
self.a2 = [3,-6,-2,8,7,4,2,1]
self.a3 = [3.,4,5,10,-3,-5,-6,7.0]
def test_percentile(self):
x = np.arange(8) * 0.5
assert_equal(mstats.scoreatpercentile(x, 0), 0.)
assert_equal(mstats.scoreatpercentile(x, 100), 3.5)
assert_equal(mstats.scoreatpercentile(x, 50), 1.75)
def test_2D(self):
x = ma.array([[1, 1, 1],
[1, 1, 1],
[4, 4, 3],
[1, 1, 1],
[1, 1, 1]])
assert_equal(mstats.scoreatpercentile(x,50), [1,1,1])
class TestVariability(object):
""" Comparison numbers are found using R v.1.5.1
note that length(testcase) = 4
"""
testcase = ma.fix_invalid([1,2,3,4,np.nan])
def test_sem(self):
# This is not in R, so used: sqrt(var(testcase)*3/4) / sqrt(3)
y = mstats.sem(self.testcase)
assert_almost_equal(y, 0.6454972244)
n = self.testcase.count()
assert_allclose(mstats.sem(self.testcase, ddof=0) * np.sqrt(n/(n-2)),
mstats.sem(self.testcase, ddof=2))
def test_zmap(self):
# This is not in R, so tested by using:
# (testcase[i]-mean(testcase,axis=0)) / sqrt(var(testcase)*3/4)
y = mstats.zmap(self.testcase, self.testcase)
desired_unmaskedvals = ([-1.3416407864999, -0.44721359549996,
0.44721359549996, 1.3416407864999])
assert_array_almost_equal(desired_unmaskedvals,
y.data[y.mask == False], decimal=12)
def test_zscore(self):
# This is not in R, so tested by using:
# (testcase[i]-mean(testcase,axis=0)) / sqrt(var(testcase)*3/4)
y = mstats.zscore(self.testcase)
desired = ma.fix_invalid([-1.3416407864999, -0.44721359549996,
0.44721359549996, 1.3416407864999, np.nan])
assert_almost_equal(desired, y, decimal=12)
class TestMisc(object):
def test_obrientransform(self):
args = [[5]*5+[6]*11+[7]*9+[8]*3+[9]*2+[10]*2,
[6]+[7]*2+[8]*4+[9]*9+[10]*16]
result = [5*[3.1828]+11*[0.5591]+9*[0.0344]+3*[1.6086]+2*[5.2817]+2*[11.0538],
[10.4352]+2*[4.8599]+4*[1.3836]+9*[0.0061]+16*[0.7277]]
assert_almost_equal(np.round(mstats.obrientransform(*args).T,4),
result,4)
def test_kstwosamp(self):
x = [[nan,nan, 4, 2, 16, 26, 5, 1, 5, 1, 2, 3, 1],
[4, 3, 5, 3, 2, 7, 3, 1, 1, 2, 3, 5, 3],
[3, 2, 5, 6, 18, 4, 9, 1, 1,nan, 1, 1,nan],
[nan, 6, 11, 4, 17,nan, 6, 1, 1, 2, 5, 1, 1]]
x = ma.fix_invalid(x).T
(winter,spring,summer,fall) = x.T
assert_almost_equal(np.round(mstats.ks_twosamp(winter,spring),4),
(0.1818,0.9892))
assert_almost_equal(np.round(mstats.ks_twosamp(winter,spring,'g'),4),
(0.1469,0.7734))
assert_almost_equal(np.round(mstats.ks_twosamp(winter,spring,'l'),4),
(0.1818,0.6744))
def test_friedmanchisq(self):
# No missing values
args = ([9.0,9.5,5.0,7.5,9.5,7.5,8.0,7.0,8.5,6.0],
[7.0,6.5,7.0,7.5,5.0,8.0,6.0,6.5,7.0,7.0],
[6.0,8.0,4.0,6.0,7.0,6.5,6.0,4.0,6.5,3.0])
result = mstats.friedmanchisquare(*args)
assert_almost_equal(result[0], 10.4737, 4)
assert_almost_equal(result[1], 0.005317, 6)
# Missing values
x = [[nan,nan, 4, 2, 16, 26, 5, 1, 5, 1, 2, 3, 1],
[4, 3, 5, 3, 2, 7, 3, 1, 1, 2, 3, 5, 3],
[3, 2, 5, 6, 18, 4, 9, 1, 1,nan, 1, 1,nan],
[nan, 6, 11, 4, 17,nan, 6, 1, 1, 2, 5, 1, 1]]
x = ma.fix_invalid(x)
result = mstats.friedmanchisquare(*x)
assert_almost_equal(result[0], 2.0156, 4)
assert_almost_equal(result[1], 0.5692, 4)
# test for namedtuple attributes
attributes = ('statistic', 'pvalue')
check_named_results(result, attributes, ma=True)
def test_regress_simple():
# Regress a line with sinusoidal noise. Test for #1273.
x = np.linspace(0, 100, 100)
y = 0.2 * np.linspace(0, 100, 100) + 10
y += np.sin(np.linspace(0, 20, 100))
slope, intercept, r_value, p_value, sterr = mstats.linregress(x, y)
assert_almost_equal(slope, 0.19644990055858422)
assert_almost_equal(intercept, 10.211269918932341)
# test for namedtuple attributes
res = mstats.linregress(x, y)
attributes = ('slope', 'intercept', 'rvalue', 'pvalue', 'stderr')
check_named_results(res, attributes, ma=True)
def test_theilslopes():
# Test for basic slope and intercept.
slope, intercept, lower, upper = mstats.theilslopes([0,1,1])
assert_almost_equal(slope, 0.5)
assert_almost_equal(intercept, 0.5)
# Test for correct masking.
y = np.ma.array([0,1,100,1], mask=[False, False, True, False])
slope, intercept, lower, upper = mstats.theilslopes(y)
assert_almost_equal(slope, 1./3)
assert_almost_equal(intercept, 2./3)
# Test of confidence intervals from example in Sen (1968).
x = [1, 2, 3, 4, 10, 12, 18]
y = [9, 15, 19, 20, 45, 55, 78]
slope, intercept, lower, upper = mstats.theilslopes(y, x, 0.07)
assert_almost_equal(slope, 4)
assert_almost_equal(upper, 4.38, decimal=2)
assert_almost_equal(lower, 3.71, decimal=2)
def test_plotting_positions():
# Regression test for #1256
pos = mstats.plotting_positions(np.arange(3), 0, 0)
assert_array_almost_equal(pos.data, np.array([0.25, 0.5, 0.75]))
class TestNormalitytests():
def test_vs_nonmasked(self):
x = np.array((-2,-1,0,1,2,3)*4)**2
assert_array_almost_equal(mstats.normaltest(x),
stats.normaltest(x))
assert_array_almost_equal(mstats.skewtest(x),
stats.skewtest(x))
assert_array_almost_equal(mstats.kurtosistest(x),
stats.kurtosistest(x))
funcs = [stats.normaltest, stats.skewtest, stats.kurtosistest]
mfuncs = [mstats.normaltest, mstats.skewtest, mstats.kurtosistest]
x = [1, 2, 3, 4]
for func, mfunc in zip(funcs, mfuncs):
assert_raises(ValueError, func, x)
assert_raises(ValueError, mfunc, x)
def test_axis_None(self):
# Test axis=None (equal to axis=0 for 1-D input)
x = np.array((-2,-1,0,1,2,3)*4)**2
assert_allclose(mstats.normaltest(x, axis=None), mstats.normaltest(x))
assert_allclose(mstats.skewtest(x, axis=None), mstats.skewtest(x))
assert_allclose(mstats.kurtosistest(x, axis=None),
mstats.kurtosistest(x))
def test_maskedarray_input(self):
# Add some masked values, test result doesn't change
x = np.array((-2,-1,0,1,2,3)*4)**2
xm = np.ma.array(np.r_[np.inf, x, 10],
mask=np.r_[True, [False] * x.size, True])
assert_allclose(mstats.normaltest(xm), stats.normaltest(x))
assert_allclose(mstats.skewtest(xm), stats.skewtest(x))
assert_allclose(mstats.kurtosistest(xm), stats.kurtosistest(x))
def test_nd_input(self):
x = np.array((-2,-1,0,1,2,3)*4)**2
x_2d = np.vstack([x] * 2).T
for func in [mstats.normaltest, mstats.skewtest, mstats.kurtosistest]:
res_1d = func(x)
res_2d = func(x_2d)
assert_allclose(res_2d[0], [res_1d[0]] * 2)
assert_allclose(res_2d[1], [res_1d[1]] * 2)
def test_normaltest_result_attributes(self):
x = np.array((-2, -1, 0, 1, 2, 3)*4)**2
res = mstats.normaltest(x)
attributes = ('statistic', 'pvalue')
check_named_results(res, attributes, ma=True)
def test_kurtosistest_result_attributes(self):
x = np.array((-2, -1, 0, 1, 2, 3)*4)**2
res = mstats.kurtosistest(x)
attributes = ('statistic', 'pvalue')
check_named_results(res, attributes, ma=True)
class TestFOneway():
def test_result_attributes(self):
a = np.array([655, 788], dtype=np.uint16)
b = np.array([789, 772], dtype=np.uint16)
res = mstats.f_oneway(a, b)
attributes = ('statistic', 'pvalue')
check_named_results(res, attributes, ma=True)
class TestMannwhitneyu():
def test_result_attributes(self):
x = np.array([1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 2., 1., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 2., 1., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 2.,
1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
1., 1., 2., 1., 1., 1., 1., 2., 1., 1., 2., 1., 1., 2.,
1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 2., 1.,
1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 2., 1., 1., 1., 1., 1., 1.,
1., 1., 1., 2., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 1., 3., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1.])
y = np.array([1., 1., 1., 1., 1., 1., 1., 2., 1., 2., 1., 1., 1., 1.,
2., 1., 1., 1., 2., 1., 1., 1., 1., 1., 2., 1., 1., 3.,
1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 2., 1., 2., 1.,
1., 1., 1., 1., 1., 2., 1., 1., 1., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 2., 1., 1., 1., 1., 1., 2.,
2., 1., 1., 2., 1., 1., 2., 1., 2., 1., 1., 1., 1., 2.,
2., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
1., 2., 1., 1., 1., 1., 1., 2., 2., 2., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
2., 1., 1., 2., 1., 1., 1., 1., 2., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 2., 1., 1., 1., 2., 1., 1.,
1., 1., 1., 1.])
res = mstats.mannwhitneyu(x, y)
attributes = ('statistic', 'pvalue')
check_named_results(res, attributes, ma=True)
class TestKruskal():
def test_result_attributes(self):
x = [1, 3, 5, 7, 9]
y = [2, 4, 6, 8, 10]
res = mstats.kruskal(x, y)
attributes = ('statistic', 'pvalue')
check_named_results(res, attributes, ma=True)
#TODO: for all ttest functions, add tests with masked array inputs
class TestTtest_rel():
def test_vs_nonmasked(self):
np.random.seed(1234567)
outcome = np.random.randn(20, 4) + [0, 0, 1, 2]
# 1-D inputs
res1 = stats.ttest_rel(outcome[:, 0], outcome[:, 1])
res2 = mstats.ttest_rel(outcome[:, 0], outcome[:, 1])
assert_allclose(res1, res2)
# 2-D inputs
res1 = stats.ttest_rel(outcome[:, 0], outcome[:, 1], axis=None)
res2 = mstats.ttest_rel(outcome[:, 0], outcome[:, 1], axis=None)
assert_allclose(res1, res2)
res1 = stats.ttest_rel(outcome[:, :2], outcome[:, 2:], axis=0)
res2 = mstats.ttest_rel(outcome[:, :2], outcome[:, 2:], axis=0)
assert_allclose(res1, res2)
# Check default is axis=0
res3 = mstats.ttest_rel(outcome[:, :2], outcome[:, 2:])
assert_allclose(res2, res3)
def test_fully_masked(self):
np.random.seed(1234567)
outcome = ma.masked_array(np.random.randn(3, 2),
mask=[[1, 1, 1], [0, 0, 0]])
with suppress_warnings() as sup:
sup.filter(RuntimeWarning, "invalid value encountered in absolute")
for pair in [(outcome[:, 0], outcome[:, 1]), ([np.nan, np.nan], [1.0, 2.0])]:
t, p = mstats.ttest_rel(*pair)
assert_array_equal(t, (np.nan, np.nan))
assert_array_equal(p, (np.nan, np.nan))
def test_result_attributes(self):
np.random.seed(1234567)
outcome = np.random.randn(20, 4) + [0, 0, 1, 2]
res = mstats.ttest_rel(outcome[:, 0], outcome[:, 1])
attributes = ('statistic', 'pvalue')
check_named_results(res, attributes, ma=True)
def test_invalid_input_size(self):
assert_raises(ValueError, mstats.ttest_rel,
np.arange(10), np.arange(11))
x = np.arange(24)
assert_raises(ValueError, mstats.ttest_rel,
x.reshape(2, 3, 4), x.reshape(2, 4, 3), axis=1)
assert_raises(ValueError, mstats.ttest_rel,
x.reshape(2, 3, 4), x.reshape(2, 4, 3), axis=2)
def test_empty(self):
res1 = mstats.ttest_rel([], [])
assert_(np.all(np.isnan(res1)))
def test_zero_division(self):
t, p = mstats.ttest_ind([0, 0, 0], [1, 1, 1])
assert_equal((np.abs(t), p), (np.inf, 0))
with suppress_warnings() as sup:
sup.filter(RuntimeWarning, "invalid value encountered in absolute")
t, p = mstats.ttest_ind([0, 0, 0], [0, 0, 0])
assert_array_equal(t, np.array([np.nan, np.nan]))
assert_array_equal(p, np.array([np.nan, np.nan]))
class TestTtest_ind():
def test_vs_nonmasked(self):
np.random.seed(1234567)
outcome = np.random.randn(20, 4) + [0, 0, 1, 2]
# 1-D inputs
res1 = stats.ttest_ind(outcome[:, 0], outcome[:, 1])
res2 = mstats.ttest_ind(outcome[:, 0], outcome[:, 1])
assert_allclose(res1, res2)
# 2-D inputs
res1 = stats.ttest_ind(outcome[:, 0], outcome[:, 1], axis=None)
res2 = mstats.ttest_ind(outcome[:, 0], outcome[:, 1], axis=None)
assert_allclose(res1, res2)
res1 = stats.ttest_ind(outcome[:, :2], outcome[:, 2:], axis=0)
res2 = mstats.ttest_ind(outcome[:, :2], outcome[:, 2:], axis=0)
assert_allclose(res1, res2)
# Check default is axis=0
res3 = mstats.ttest_ind(outcome[:, :2], outcome[:, 2:])
assert_allclose(res2, res3)
# Check equal_var
res4 = stats.ttest_ind(outcome[:, 0], outcome[:, 1], equal_var=True)
res5 = mstats.ttest_ind(outcome[:, 0], outcome[:, 1], equal_var=True)
assert_allclose(res4, res5)
res4 = stats.ttest_ind(outcome[:, 0], outcome[:, 1], equal_var=False)
res5 = mstats.ttest_ind(outcome[:, 0], outcome[:, 1], equal_var=False)
assert_allclose(res4, res5)
def test_fully_masked(self):
np.random.seed(1234567)
outcome = ma.masked_array(np.random.randn(3, 2), mask=[[1, 1, 1], [0, 0, 0]])
with suppress_warnings() as sup:
sup.filter(RuntimeWarning, "invalid value encountered in absolute")
for pair in [(outcome[:, 0], outcome[:, 1]), ([np.nan, np.nan], [1.0, 2.0])]:
t, p = mstats.ttest_ind(*pair)
assert_array_equal(t, (np.nan, np.nan))
assert_array_equal(p, (np.nan, np.nan))
def test_result_attributes(self):
np.random.seed(1234567)
outcome = np.random.randn(20, 4) + [0, 0, 1, 2]
res = mstats.ttest_ind(outcome[:, 0], outcome[:, 1])
attributes = ('statistic', 'pvalue')
check_named_results(res, attributes, ma=True)
def test_empty(self):
res1 = mstats.ttest_ind([], [])
assert_(np.all(np.isnan(res1)))
def test_zero_division(self):
t, p = mstats.ttest_ind([0, 0, 0], [1, 1, 1])
assert_equal((np.abs(t), p), (np.inf, 0))
with suppress_warnings() as sup:
sup.filter(RuntimeWarning, "invalid value encountered in absolute")
t, p = mstats.ttest_ind([0, 0, 0], [0, 0, 0])
assert_array_equal(t, (np.nan, np.nan))
assert_array_equal(p, (np.nan, np.nan))
t, p = mstats.ttest_ind([0, 0, 0], [1, 1, 1], equal_var=False)
assert_equal((np.abs(t), p), (np.inf, 0))
assert_array_equal(mstats.ttest_ind([0, 0, 0], [0, 0, 0],
equal_var=False), (np.nan, np.nan))
class TestTtest_1samp():
def test_vs_nonmasked(self):
np.random.seed(1234567)
outcome = np.random.randn(20, 4) + [0, 0, 1, 2]
# 1-D inputs
res1 = stats.ttest_1samp(outcome[:, 0], 1)
res2 = mstats.ttest_1samp(outcome[:, 0], 1)
assert_allclose(res1, res2)
# 2-D inputs
res1 = stats.ttest_1samp(outcome[:, 0], outcome[:, 1], axis=None)
res2 = mstats.ttest_1samp(outcome[:, 0], outcome[:, 1], axis=None)
assert_allclose(res1, res2)
res1 = stats.ttest_1samp(outcome[:, :2], outcome[:, 2:], axis=0)
res2 = mstats.ttest_1samp(outcome[:, :2], outcome[:, 2:], axis=0)
assert_allclose(res1, res2)
# Check default is axis=0
res3 = mstats.ttest_1samp(outcome[:, :2], outcome[:, 2:])
assert_allclose(res2, res3)
def test_fully_masked(self):
np.random.seed(1234567)
outcome = ma.masked_array(np.random.randn(3), mask=[1, 1, 1])
expected = (np.nan, np.nan)
with suppress_warnings() as sup:
sup.filter(RuntimeWarning, "invalid value encountered in absolute")
for pair in [((np.nan, np.nan), 0.0), (outcome, 0.0)]:
t, p = mstats.ttest_1samp(*pair)
assert_array_equal(p, expected)
assert_array_equal(t, expected)
def test_result_attributes(self):
np.random.seed(1234567)
outcome = np.random.randn(20, 4) + [0, 0, 1, 2]
res = mstats.ttest_1samp(outcome[:, 0], 1)
attributes = ('statistic', 'pvalue')
check_named_results(res, attributes, ma=True)
def test_empty(self):
res1 = mstats.ttest_1samp([], 1)
assert_(np.all(np.isnan(res1)))
def test_zero_division(self):
t, p = mstats.ttest_1samp([0, 0, 0], 1)
assert_equal((np.abs(t), p), (np.inf, 0))
with suppress_warnings() as sup:
sup.filter(RuntimeWarning, "invalid value encountered in absolute")
t, p = mstats.ttest_1samp([0, 0, 0], 0)
assert_(np.isnan(t))
assert_array_equal(p, (np.nan, np.nan))
class TestCompareWithStats(object):
"""
Class to compare mstats results with stats results.
It is in general assumed that scipy.stats is at a more mature stage than
stats.mstats. If a routine in mstats results in similar results like in
scipy.stats, this is considered also as a proper validation of scipy.mstats
routine.
Different sample sizes are used for testing, as some problems between stats
and mstats are dependent on sample size.
Author: Alexander Loew
NOTE that some tests fail. This might be caused by
a) actual differences or bugs between stats and mstats
b) numerical inaccuracies
c) different definitions of routine interfaces
These failures need to be checked. Current workaround is to have disabled these tests,
but issuing reports on scipy-dev
"""
def get_n(self):
""" Returns list of sample sizes to be used for comparison. """
return [1000, 100, 10, 5]
def generate_xy_sample(self, n):
# This routine generates numpy arrays and corresponding masked arrays
# with the same data, but additional masked values
np.random.seed(1234567)
x = np.random.randn(n)
y = x + np.random.randn(n)
xm = np.ones(len(x) + 5) * 1e16
ym = np.ones(len(y) + 5) * 1e16
xm[0:len(x)] = x
ym[0:len(y)] = y
mask = xm > 9e15
xm = np.ma.array(xm, mask=mask)
ym = np.ma.array(ym, mask=mask)
return x, y, xm, ym
def generate_xy_sample2D(self, n, nx):
x = np.ones((n, nx)) * np.nan
y = np.ones((n, nx)) * np.nan
xm = np.ones((n+5, nx)) * np.nan
ym = np.ones((n+5, nx)) * np.nan
for i in range(nx):
x[:,i], y[:,i], dx, dy = self.generate_xy_sample(n)
xm[0:n, :] = x[0:n]
ym[0:n, :] = y[0:n]
xm = np.ma.array(xm, mask=np.isnan(xm))
ym = np.ma.array(ym, mask=np.isnan(ym))
return x, y, xm, ym
def test_linregress(self):
for n in self.get_n():
x, y, xm, ym = self.generate_xy_sample(n)
res1 = stats.linregress(x, y)
res2 = stats.mstats.linregress(xm, ym)
assert_allclose(np.asarray(res1), np.asarray(res2))
def test_pearsonr(self):
for n in self.get_n():
x, y, xm, ym = self.generate_xy_sample(n)
r, p = stats.pearsonr(x, y)
rm, pm = stats.mstats.pearsonr(xm, ym)
assert_almost_equal(r, rm, decimal=14)
assert_almost_equal(p, pm, decimal=14)
def test_spearmanr(self):
for n in self.get_n():
x, y, xm, ym = self.generate_xy_sample(n)
r, p = stats.spearmanr(x, y)
rm, pm = stats.mstats.spearmanr(xm, ym)
assert_almost_equal(r, rm, 14)
assert_almost_equal(p, pm, 14)
def test_gmean(self):
for n in self.get_n():
x, y, xm, ym = self.generate_xy_sample(n)
r = stats.gmean(abs(x))
rm = stats.mstats.gmean(abs(xm))
assert_allclose(r, rm, rtol=1e-13)
r = stats.gmean(abs(y))
rm = stats.mstats.gmean(abs(ym))
assert_allclose(r, rm, rtol=1e-13)
def test_hmean(self):
for n in self.get_n():
x, y, xm, ym = self.generate_xy_sample(n)
r = stats.hmean(abs(x))
rm = stats.mstats.hmean(abs(xm))
assert_almost_equal(r, rm, 10)
r = stats.hmean(abs(y))
rm = stats.mstats.hmean(abs(ym))
assert_almost_equal(r, rm, 10)
def test_skew(self):
for n in self.get_n():
x, y, xm, ym = self.generate_xy_sample(n)
r = stats.skew(x)
rm = stats.mstats.skew(xm)
assert_almost_equal(r, rm, 10)
r = stats.skew(y)
rm = stats.mstats.skew(ym)
assert_almost_equal(r, rm, 10)
def test_moment(self):
for n in self.get_n():
x, y, xm, ym = self.generate_xy_sample(n)
r = stats.moment(x)
rm = stats.mstats.moment(xm)
assert_almost_equal(r, rm, 10)
r = stats.moment(y)
rm = stats.mstats.moment(ym)
assert_almost_equal(r, rm, 10)
def test_zscore(self):
for n in self.get_n():
x, y, xm, ym = self.generate_xy_sample(n)
#reference solution
zx = (x - x.mean()) / x.std()
zy = (y - y.mean()) / y.std()
#validate stats
assert_allclose(stats.zscore(x), zx, rtol=1e-10)
assert_allclose(stats.zscore(y), zy, rtol=1e-10)
#compare stats and mstats
assert_allclose(stats.zscore(x), stats.mstats.zscore(xm[0:len(x)]),
rtol=1e-10)
assert_allclose(stats.zscore(y), stats.mstats.zscore(ym[0:len(y)]),
rtol=1e-10)
def test_kurtosis(self):
for n in self.get_n():
x, y, xm, ym = self.generate_xy_sample(n)
r = stats.kurtosis(x)
rm = stats.mstats.kurtosis(xm)
assert_almost_equal(r, rm, 10)
r = stats.kurtosis(y)
rm = stats.mstats.kurtosis(ym)
assert_almost_equal(r, rm, 10)
def test_sem(self):
# example from stats.sem doc
a = np.arange(20).reshape(5,4)
am = np.ma.array(a)
r = stats.sem(a,ddof=1)
rm = stats.mstats.sem(am, ddof=1)
assert_allclose(r, 2.82842712, atol=1e-5)
assert_allclose(rm, 2.82842712, atol=1e-5)
for n in self.get_n():
x, y, xm, ym = self.generate_xy_sample(n)
assert_almost_equal(stats.mstats.sem(xm, axis=None, ddof=0),
stats.sem(x, axis=None, ddof=0), decimal=13)
assert_almost_equal(stats.mstats.sem(ym, axis=None, ddof=0),
stats.sem(y, axis=None, ddof=0), decimal=13)
assert_almost_equal(stats.mstats.sem(xm, axis=None, ddof=1),
stats.sem(x, axis=None, ddof=1), decimal=13)
assert_almost_equal(stats.mstats.sem(ym, axis=None, ddof=1),
stats.sem(y, axis=None, ddof=1), decimal=13)
def test_describe(self):
for n in self.get_n():
x, y, xm, ym = self.generate_xy_sample(n)
r = stats.describe(x, ddof=1)
rm = stats.mstats.describe(xm, ddof=1)
for ii in range(6):
assert_almost_equal(np.asarray(r[ii]),
np.asarray(rm[ii]),
decimal=12)
def test_describe_result_attributes(self):
actual = mstats.describe(np.arange(5))
attributes = ('nobs', 'minmax', 'mean', 'variance', 'skewness',
'kurtosis')
check_named_results(actual, attributes, ma=True)
def test_rankdata(self):
for n in self.get_n():
x, y, xm, ym = self.generate_xy_sample(n)
r = stats.rankdata(x)
rm = stats.mstats.rankdata(x)
assert_allclose(r, rm)
def test_tmean(self):
for n in self.get_n():
x, y, xm, ym = self.generate_xy_sample(n)
assert_almost_equal(stats.tmean(x),stats.mstats.tmean(xm), 14)
assert_almost_equal(stats.tmean(y),stats.mstats.tmean(ym), 14)
def test_tmax(self):
for n in self.get_n():
x, y, xm, ym = self.generate_xy_sample(n)
assert_almost_equal(stats.tmax(x,2.),
stats.mstats.tmax(xm,2.), 10)
assert_almost_equal(stats.tmax(y,2.),
stats.mstats.tmax(ym,2.), 10)
assert_almost_equal(stats.tmax(x, upperlimit=3.),
stats.mstats.tmax(xm, upperlimit=3.), 10)
assert_almost_equal(stats.tmax(y, upperlimit=3.),
stats.mstats.tmax(ym, upperlimit=3.), 10)
def test_tmin(self):
for n in self.get_n():
x, y, xm, ym = self.generate_xy_sample(n)
assert_equal(stats.tmin(x),stats.mstats.tmin(xm))
assert_equal(stats.tmin(y),stats.mstats.tmin(ym))
assert_almost_equal(stats.tmin(x,lowerlimit=-1.),
stats.mstats.tmin(xm,lowerlimit=-1.), 10)
assert_almost_equal(stats.tmin(y,lowerlimit=-1.),
stats.mstats.tmin(ym,lowerlimit=-1.), 10)
def test_zmap(self):
for n in self.get_n():
x, y, xm, ym = self.generate_xy_sample(n)
z = stats.zmap(x,y)
zm = stats.mstats.zmap(xm,ym)
assert_allclose(z, zm[0:len(z)], atol=1e-10)
def test_variation(self):
for n in self.get_n():
x, y, xm, ym = self.generate_xy_sample(n)
assert_almost_equal(stats.variation(x), stats.mstats.variation(xm),
decimal=12)
assert_almost_equal(stats.variation(y), stats.mstats.variation(ym),
decimal=12)
def test_tvar(self):
for n in self.get_n():
x, y, xm, ym = self.generate_xy_sample(n)
assert_almost_equal(stats.tvar(x), stats.mstats.tvar(xm),
decimal=12)
assert_almost_equal(stats.tvar(y), stats.mstats.tvar(ym),
decimal=12)
def test_trimboth(self):
a = np.arange(20)
b = stats.trimboth(a, 0.1)
bm = stats.mstats.trimboth(a, 0.1)
assert_allclose(np.sort(b), bm.data[~bm.mask])
def test_tsem(self):
for n in self.get_n():
x, y, xm, ym = self.generate_xy_sample(n)
assert_almost_equal(stats.tsem(x),stats.mstats.tsem(xm), decimal=14)
assert_almost_equal(stats.tsem(y),stats.mstats.tsem(ym), decimal=14)
assert_almost_equal(stats.tsem(x,limits=(-2.,2.)),
stats.mstats.tsem(xm,limits=(-2.,2.)),
decimal=14)
def test_skewtest(self):
# this test is for 1D data
for n in self.get_n():
if n > 8:
x, y, xm, ym = self.generate_xy_sample(n)
r = stats.skewtest(x)
rm = stats.mstats.skewtest(xm)
assert_allclose(r[0], rm[0], rtol=1e-15)
# TODO this test is not performed as it is a known issue that
# mstats returns a slightly different p-value what is a bit
# strange is that other tests like test_maskedarray_input don't
# fail!
#~ assert_almost_equal(r[1], rm[1])
def test_skewtest_result_attributes(self):
x = np.array((-2, -1, 0, 1, 2, 3)*4)**2
res = mstats.skewtest(x)
attributes = ('statistic', 'pvalue')
check_named_results(res, attributes, ma=True)
def test_skewtest_2D_notmasked(self):
# a normal ndarray is passed to the masked function
x = np.random.random((20, 2)) * 20.
r = stats.skewtest(x)
rm = stats.mstats.skewtest(x)
assert_allclose(np.asarray(r), np.asarray(rm))
def test_skewtest_2D_WithMask(self):
nx = 2
for n in self.get_n():
if n > 8:
x, y, xm, ym = self.generate_xy_sample2D(n, nx)
r = stats.skewtest(x)
rm = stats.mstats.skewtest(xm)
assert_equal(r[0][0],rm[0][0])
assert_equal(r[0][1],rm[0][1])
def test_normaltest(self):
np.seterr(over='raise')
with suppress_warnings() as sup:
sup.filter(UserWarning, "kurtosistest only valid for n>=20")
for n in self.get_n():
if n > 8:
x, y, xm, ym = self.generate_xy_sample(n)
r = stats.normaltest(x)
rm = stats.mstats.normaltest(xm)
assert_allclose(np.asarray(r), np.asarray(rm))
def test_find_repeats(self):
x = np.asarray([1,1,2,2,3,3,3,4,4,4,4]).astype('float')
tmp = np.asarray([1,1,2,2,3,3,3,4,4,4,4,5,5,5,5]).astype('float')
mask = (tmp == 5.)
xm = np.ma.array(tmp, mask=mask)
x_orig, xm_orig = x.copy(), xm.copy()
r = stats.find_repeats(x)
rm = stats.mstats.find_repeats(xm)
assert_equal(r, rm)
assert_equal(x, x_orig)
assert_equal(xm, xm_orig)
# This crazy behavior is expected by count_tied_groups, but is not
# in the docstring...
_, counts = stats.mstats.find_repeats([])
assert_equal(counts, np.array(0, dtype=np.intp))
def test_kendalltau(self):
for n in self.get_n():
x, y, xm, ym = self.generate_xy_sample(n)
r = stats.kendalltau(x, y)
rm = stats.mstats.kendalltau(xm, ym)
assert_almost_equal(r[0], rm[0], decimal=10)
assert_almost_equal(r[1], rm[1], decimal=7)
def test_obrientransform(self):
for n in self.get_n():
x, y, xm, ym = self.generate_xy_sample(n)
r = stats.obrientransform(x)
rm = stats.mstats.obrientransform(xm)
assert_almost_equal(r.T, rm[0:len(x)])
| 52,906 | 40.398279 | 93 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/function_docs.py
|
"""
Extended docstrings for functions.py
"""
pi = r"""
`\pi`, roughly equal to 3.141592654, represents the area of the unit
circle, the half-period of trigonometric functions, and many other
things in mathematics.
Mpmath can evaluate `\pi` to arbitrary precision::
>>> from mpmath import *
>>> mp.dps = 50; mp.pretty = True
>>> +pi
3.1415926535897932384626433832795028841971693993751
This shows digits 99991-100000 of `\pi`::
>>> mp.dps = 100000
>>> str(pi)[-10:]
'5549362464'
**Possible issues**
:data:`pi` always rounds to the nearest floating-point
number when used. This means that exact mathematical identities
involving `\pi` will generally not be preserved in floating-point
arithmetic. In particular, multiples of :data:`pi` (except for
the trivial case ``0*pi``) are *not* the exact roots of
:func:`~mpmath.sin`, but differ roughly by the current epsilon::
>>> mp.dps = 15
>>> sin(pi)
1.22464679914735e-16
One solution is to use the :func:`~mpmath.sinpi` function instead::
>>> sinpi(1)
0.0
See the documentation of trigonometric functions for additional
details.
"""
degree = r"""
Represents one degree of angle, `1^{\circ} = \pi/180`, or
about 0.01745329. This constant may be evaluated to arbitrary
precision::
>>> from mpmath import *
>>> mp.dps = 50; mp.pretty = True
>>> +degree
0.017453292519943295769236907684886127134428718885417
The :data:`degree` object is convenient for conversion
to radians::
>>> sin(30 * degree)
0.5
>>> asin(0.5) / degree
30.0
"""
e = r"""
The transcendental number `e` = 2.718281828... is the base of the
natural logarithm (:func:`~mpmath.ln`) and of the exponential function
(:func:`~mpmath.exp`).
Mpmath can be evaluate `e` to arbitrary precision::
>>> from mpmath import *
>>> mp.dps = 50; mp.pretty = True
>>> +e
2.7182818284590452353602874713526624977572470937
This shows digits 99991-100000 of `e`::
>>> mp.dps = 100000
>>> str(e)[-10:]
'2100427165'
**Possible issues**
:data:`e` always rounds to the nearest floating-point number
when used, and mathematical identities involving `e` may not
hold in floating-point arithmetic. For example, ``ln(e)``
might not evaluate exactly to 1.
In particular, don't use ``e**x`` to compute the exponential
function. Use ``exp(x)`` instead; this is both faster and more
accurate.
"""
phi = r"""
Represents the golden ratio `\phi = (1+\sqrt 5)/2`,
approximately equal to 1.6180339887. To high precision,
its value is::
>>> from mpmath import *
>>> mp.dps = 50; mp.pretty = True
>>> +phi
1.6180339887498948482045868343656381177203091798058
Formulas for the golden ratio include the following::
>>> (1+sqrt(5))/2
1.6180339887498948482045868343656381177203091798058
>>> findroot(lambda x: x**2-x-1, 1)
1.6180339887498948482045868343656381177203091798058
>>> limit(lambda n: fib(n+1)/fib(n), inf)
1.6180339887498948482045868343656381177203091798058
"""
euler = r"""
Euler's constant or the Euler-Mascheroni constant `\gamma`
= 0.57721566... is a number of central importance to
number theory and special functions. It is defined as the limit
.. math ::
\gamma = \lim_{n\to\infty} H_n - \log n
where `H_n = 1 + \frac{1}{2} + \ldots + \frac{1}{n}` is a harmonic
number (see :func:`~mpmath.harmonic`).
Evaluation of `\gamma` is supported at arbitrary precision::
>>> from mpmath import *
>>> mp.dps = 50; mp.pretty = True
>>> +euler
0.57721566490153286060651209008240243104215933593992
We can also compute `\gamma` directly from the definition,
although this is less efficient::
>>> limit(lambda n: harmonic(n)-log(n), inf)
0.57721566490153286060651209008240243104215933593992
This shows digits 9991-10000 of `\gamma`::
>>> mp.dps = 10000
>>> str(euler)[-10:]
'4679858165'
Integrals, series, and representations for `\gamma` in terms of
special functions include the following (there are many others)::
>>> mp.dps = 25
>>> -quad(lambda x: exp(-x)*log(x), [0,inf])
0.5772156649015328606065121
>>> quad(lambda x,y: (x-1)/(1-x*y)/log(x*y), [0,1], [0,1])
0.5772156649015328606065121
>>> nsum(lambda k: 1/k-log(1+1/k), [1,inf])
0.5772156649015328606065121
>>> nsum(lambda k: (-1)**k*zeta(k)/k, [2,inf])
0.5772156649015328606065121
>>> -diff(gamma, 1)
0.5772156649015328606065121
>>> limit(lambda x: 1/x-gamma(x), 0)
0.5772156649015328606065121
>>> limit(lambda x: zeta(x)-1/(x-1), 1)
0.5772156649015328606065121
>>> (log(2*pi*nprod(lambda n:
... exp(-2+2/n)*(1+2/n)**n, [1,inf]))-3)/2
0.5772156649015328606065121
For generalizations of the identities `\gamma = -\Gamma'(1)`
and `\gamma = \lim_{x\to1} \zeta(x)-1/(x-1)`, see
:func:`~mpmath.psi` and :func:`~mpmath.stieltjes` respectively.
"""
catalan = r"""
Catalan's constant `K` = 0.91596559... is given by the infinite
series
.. math ::
K = \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^2}.
Mpmath can evaluate it to arbitrary precision::
>>> from mpmath import *
>>> mp.dps = 50; mp.pretty = True
>>> +catalan
0.91596559417721901505460351493238411077414937428167
One can also compute `K` directly from the definition, although
this is significantly less efficient::
>>> nsum(lambda k: (-1)**k/(2*k+1)**2, [0, inf])
0.91596559417721901505460351493238411077414937428167
This shows digits 9991-10000 of `K`::
>>> mp.dps = 10000
>>> str(catalan)[-10:]
'9537871503'
Catalan's constant has numerous integral representations::
>>> mp.dps = 50
>>> quad(lambda x: -log(x)/(1+x**2), [0, 1])
0.91596559417721901505460351493238411077414937428167
>>> quad(lambda x: atan(x)/x, [0, 1])
0.91596559417721901505460351493238411077414937428167
>>> quad(lambda x: ellipk(x**2)/2, [0, 1])
0.91596559417721901505460351493238411077414937428167
>>> quad(lambda x,y: 1/(1+(x*y)**2), [0, 1], [0, 1])
0.91596559417721901505460351493238411077414937428167
As well as series representations::
>>> pi*log(sqrt(3)+2)/8 + 3*nsum(lambda n:
... (fac(n)/(2*n+1))**2/fac(2*n), [0, inf])/8
0.91596559417721901505460351493238411077414937428167
>>> 1-nsum(lambda n: n*zeta(2*n+1)/16**n, [1,inf])
0.91596559417721901505460351493238411077414937428167
"""
khinchin = r"""
Khinchin's constant `K` = 2.68542... is a number that
appears in the theory of continued fractions. Mpmath can evaluate
it to arbitrary precision::
>>> from mpmath import *
>>> mp.dps = 50; mp.pretty = True
>>> +khinchin
2.6854520010653064453097148354817956938203822939945
An integral representation is::
>>> I = quad(lambda x: log((1-x**2)/sincpi(x))/x/(1+x), [0, 1])
>>> 2*exp(1/log(2)*I)
2.6854520010653064453097148354817956938203822939945
The computation of ``khinchin`` is based on an efficient
implementation of the following series::
>>> f = lambda n: (zeta(2*n)-1)/n*sum((-1)**(k+1)/mpf(k)
... for k in range(1,2*int(n)))
>>> exp(nsum(f, [1,inf])/log(2))
2.6854520010653064453097148354817956938203822939945
"""
glaisher = r"""
Glaisher's constant `A`, also known as the Glaisher-Kinkelin
constant, is a number approximately equal to 1.282427129 that
sometimes appears in formulas related to gamma and zeta functions.
It is also related to the Barnes G-function (see :func:`~mpmath.barnesg`).
The constant is defined as `A = \exp(1/12-\zeta'(-1))` where
`\zeta'(s)` denotes the derivative of the Riemann zeta function
(see :func:`~mpmath.zeta`).
Mpmath can evaluate Glaisher's constant to arbitrary precision:
>>> from mpmath import *
>>> mp.dps = 50; mp.pretty = True
>>> +glaisher
1.282427129100622636875342568869791727767688927325
We can verify that the value computed by :data:`glaisher` is
correct using mpmath's facilities for numerical
differentiation and arbitrary evaluation of the zeta function:
>>> exp(mpf(1)/12 - diff(zeta, -1))
1.282427129100622636875342568869791727767688927325
Here is an example of an integral that can be evaluated in
terms of Glaisher's constant:
>>> mp.dps = 15
>>> quad(lambda x: log(gamma(x)), [1, 1.5])
-0.0428537406502909
>>> -0.5 - 7*log(2)/24 + log(pi)/4 + 3*log(glaisher)/2
-0.042853740650291
Mpmath computes Glaisher's constant by applying Euler-Maclaurin
summation to a slowly convergent series. The implementation is
reasonably efficient up to about 10,000 digits. See the source
code for additional details.
References:
http://mathworld.wolfram.com/Glaisher-KinkelinConstant.html
"""
apery = r"""
Represents Apery's constant, which is the irrational number
approximately equal to 1.2020569 given by
.. math ::
\zeta(3) = \sum_{k=1}^\infty\frac{1}{k^3}.
The calculation is based on an efficient hypergeometric
series. To 50 decimal places, the value is given by::
>>> from mpmath import *
>>> mp.dps = 50; mp.pretty = True
>>> +apery
1.2020569031595942853997381615114499907649862923405
Other ways to evaluate Apery's constant using mpmath
include::
>>> zeta(3)
1.2020569031595942853997381615114499907649862923405
>>> -psi(2,1)/2
1.2020569031595942853997381615114499907649862923405
>>> 8*nsum(lambda k: 1/(2*k+1)**3, [0,inf])/7
1.2020569031595942853997381615114499907649862923405
>>> f = lambda k: 2/k**3/(exp(2*pi*k)-1)
>>> 7*pi**3/180 - nsum(f, [1,inf])
1.2020569031595942853997381615114499907649862923405
This shows digits 9991-10000 of Apery's constant::
>>> mp.dps = 10000
>>> str(apery)[-10:]
'3189504235'
"""
mertens = r"""
Represents the Mertens or Meissel-Mertens constant, which is the
prime number analog of Euler's constant:
.. math ::
B_1 = \lim_{N\to\infty}
\left(\sum_{p_k \le N} \frac{1}{p_k} - \log \log N \right)
Here `p_k` denotes the `k`-th prime number. Other names for this
constant include the Hadamard-de la Vallee-Poussin constant or
the prime reciprocal constant.
The following gives the Mertens constant to 50 digits::
>>> from mpmath import *
>>> mp.dps = 50; mp.pretty = True
>>> +mertens
0.2614972128476427837554268386086958590515666482612
References:
http://mathworld.wolfram.com/MertensConstant.html
"""
twinprime = r"""
Represents the twin prime constant, which is the factor `C_2`
featuring in the Hardy-Littlewood conjecture for the growth of the
twin prime counting function,
.. math ::
\pi_2(n) \sim 2 C_2 \frac{n}{\log^2 n}.
It is given by the product over primes
.. math ::
C_2 = \prod_{p\ge3} \frac{p(p-2)}{(p-1)^2} \approx 0.66016
Computing `C_2` to 50 digits::
>>> from mpmath import *
>>> mp.dps = 50; mp.pretty = True
>>> +twinprime
0.66016181584686957392781211001455577843262336028473
References:
http://mathworld.wolfram.com/TwinPrimesConstant.html
"""
ln = r"""
Computes the natural logarithm of `x`, `\ln x`.
See :func:`~mpmath.log` for additional documentation."""
sqrt = r"""
``sqrt(x)`` gives the principal square root of `x`, `\sqrt x`.
For positive real numbers, the principal root is simply the
positive square root. For arbitrary complex numbers, the principal
square root is defined to satisfy `\sqrt x = \exp(\log(x)/2)`.
The function thus has a branch cut along the negative half real axis.
For all mpmath numbers ``x``, calling ``sqrt(x)`` is equivalent to
performing ``x**0.5``.
**Examples**
Basic examples and limits::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> sqrt(10)
3.16227766016838
>>> sqrt(100)
10.0
>>> sqrt(-4)
(0.0 + 2.0j)
>>> sqrt(1+1j)
(1.09868411346781 + 0.455089860562227j)
>>> sqrt(inf)
+inf
Square root evaluation is fast at huge precision::
>>> mp.dps = 50000
>>> a = sqrt(3)
>>> str(a)[-10:]
'9329332814'
:func:`mpmath.iv.sqrt` supports interval arguments::
>>> iv.dps = 15; iv.pretty = True
>>> iv.sqrt([16,100])
[4.0, 10.0]
>>> iv.sqrt(2)
[1.4142135623730949234, 1.4142135623730951455]
>>> iv.sqrt(2) ** 2
[1.9999999999999995559, 2.0000000000000004441]
"""
cbrt = r"""
``cbrt(x)`` computes the cube root of `x`, `x^{1/3}`. This
function is faster and more accurate than raising to a floating-point
fraction::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = False
>>> 125**(mpf(1)/3)
mpf('4.9999999999999991')
>>> cbrt(125)
mpf('5.0')
Every nonzero complex number has three cube roots. This function
returns the cube root defined by `\exp(\log(x)/3)` where the
principal branch of the natural logarithm is used. Note that this
does not give a real cube root for negative real numbers::
>>> mp.pretty = True
>>> cbrt(-1)
(0.5 + 0.866025403784439j)
"""
exp = r"""
Computes the exponential function,
.. math ::
\exp(x) = e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!}.
For complex numbers, the exponential function also satisfies
.. math ::
\exp(x+yi) = e^x (\cos y + i \sin y).
**Basic examples**
Some values of the exponential function::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> exp(0)
1.0
>>> exp(1)
2.718281828459045235360287
>>> exp(-1)
0.3678794411714423215955238
>>> exp(inf)
+inf
>>> exp(-inf)
0.0
Arguments can be arbitrarily large::
>>> exp(10000)
8.806818225662921587261496e+4342
>>> exp(-10000)
1.135483865314736098540939e-4343
Evaluation is supported for interval arguments via
:func:`mpmath.iv.exp`::
>>> iv.dps = 25; iv.pretty = True
>>> iv.exp([-inf,0])
[0.0, 1.0]
>>> iv.exp([0,1])
[1.0, 2.71828182845904523536028749558]
The exponential function can be evaluated efficiently to arbitrary
precision::
>>> mp.dps = 10000
>>> exp(pi) #doctest: +ELLIPSIS
23.140692632779269005729...8984304016040616
**Functional properties**
Numerical verification of Euler's identity for the complex
exponential function::
>>> mp.dps = 15
>>> exp(j*pi)+1
(0.0 + 1.22464679914735e-16j)
>>> chop(exp(j*pi)+1)
0.0
This recovers the coefficients (reciprocal factorials) in the
Maclaurin series expansion of exp::
>>> nprint(taylor(exp, 0, 5))
[1.0, 1.0, 0.5, 0.166667, 0.0416667, 0.00833333]
The exponential function is its own derivative and antiderivative::
>>> exp(pi)
23.1406926327793
>>> diff(exp, pi)
23.1406926327793
>>> quad(exp, [-inf, pi])
23.1406926327793
The exponential function can be evaluated using various methods,
including direct summation of the series, limits, and solving
the defining differential equation::
>>> nsum(lambda k: pi**k/fac(k), [0,inf])
23.1406926327793
>>> limit(lambda k: (1+pi/k)**k, inf)
23.1406926327793
>>> odefun(lambda t, x: x, 0, 1)(pi)
23.1406926327793
"""
cosh = r"""
Computes the hyperbolic cosine of `x`,
`\cosh(x) = (e^x + e^{-x})/2`. Values and limits include::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> cosh(0)
1.0
>>> cosh(1)
1.543080634815243778477906
>>> cosh(-inf), cosh(+inf)
(+inf, +inf)
The hyperbolic cosine is an even, convex function with
a global minimum at `x = 0`, having a Maclaurin series
that starts::
>>> nprint(chop(taylor(cosh, 0, 5)))
[1.0, 0.0, 0.5, 0.0, 0.0416667, 0.0]
Generalized to complex numbers, the hyperbolic cosine is
equivalent to a cosine with the argument rotated
in the imaginary direction, or `\cosh x = \cos ix`::
>>> cosh(2+3j)
(-3.724545504915322565473971 + 0.5118225699873846088344638j)
>>> cos(3-2j)
(-3.724545504915322565473971 + 0.5118225699873846088344638j)
"""
sinh = r"""
Computes the hyperbolic sine of `x`,
`\sinh(x) = (e^x - e^{-x})/2`. Values and limits include::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> sinh(0)
0.0
>>> sinh(1)
1.175201193643801456882382
>>> sinh(-inf), sinh(+inf)
(-inf, +inf)
The hyperbolic sine is an odd function, with a Maclaurin
series that starts::
>>> nprint(chop(taylor(sinh, 0, 5)))
[0.0, 1.0, 0.0, 0.166667, 0.0, 0.00833333]
Generalized to complex numbers, the hyperbolic sine is
essentially a sine with a rotation `i` applied to
the argument; more precisely, `\sinh x = -i \sin ix`::
>>> sinh(2+3j)
(-3.590564589985779952012565 + 0.5309210862485198052670401j)
>>> j*sin(3-2j)
(-3.590564589985779952012565 + 0.5309210862485198052670401j)
"""
tanh = r"""
Computes the hyperbolic tangent of `x`,
`\tanh(x) = \sinh(x)/\cosh(x)`. Values and limits include::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> tanh(0)
0.0
>>> tanh(1)
0.7615941559557648881194583
>>> tanh(-inf), tanh(inf)
(-1.0, 1.0)
The hyperbolic tangent is an odd, sigmoidal function, similar
to the inverse tangent and error function. Its Maclaurin
series is::
>>> nprint(chop(taylor(tanh, 0, 5)))
[0.0, 1.0, 0.0, -0.333333, 0.0, 0.133333]
Generalized to complex numbers, the hyperbolic tangent is
essentially a tangent with a rotation `i` applied to
the argument; more precisely, `\tanh x = -i \tan ix`::
>>> tanh(2+3j)
(0.9653858790221331242784803 - 0.009884375038322493720314034j)
>>> j*tan(3-2j)
(0.9653858790221331242784803 - 0.009884375038322493720314034j)
"""
cos = r"""
Computes the cosine of `x`, `\cos(x)`.
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> cos(pi/3)
0.5
>>> cos(100000001)
-0.9802850113244713353133243
>>> cos(2+3j)
(-4.189625690968807230132555 - 9.109227893755336597979197j)
>>> cos(inf)
nan
>>> nprint(chop(taylor(cos, 0, 6)))
[1.0, 0.0, -0.5, 0.0, 0.0416667, 0.0, -0.00138889]
Intervals are supported via :func:`mpmath.iv.cos`::
>>> iv.dps = 25; iv.pretty = True
>>> iv.cos([0,1])
[0.540302305868139717400936602301, 1.0]
>>> iv.cos([0,2])
[-0.41614683654714238699756823214, 1.0]
"""
sin = r"""
Computes the sine of `x`, `\sin(x)`.
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> sin(pi/3)
0.8660254037844386467637232
>>> sin(100000001)
0.1975887055794968911438743
>>> sin(2+3j)
(9.1544991469114295734673 - 4.168906959966564350754813j)
>>> sin(inf)
nan
>>> nprint(chop(taylor(sin, 0, 6)))
[0.0, 1.0, 0.0, -0.166667, 0.0, 0.00833333, 0.0]
Intervals are supported via :func:`mpmath.iv.sin`::
>>> iv.dps = 25; iv.pretty = True
>>> iv.sin([0,1])
[0.0, 0.841470984807896506652502331201]
>>> iv.sin([0,2])
[0.0, 1.0]
"""
tan = r"""
Computes the tangent of `x`, `\tan(x) = \frac{\sin(x)}{\cos(x)}`.
The tangent function is singular at `x = (n+1/2)\pi`, but
``tan(x)`` always returns a finite result since `(n+1/2)\pi`
cannot be represented exactly using floating-point arithmetic.
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> tan(pi/3)
1.732050807568877293527446
>>> tan(100000001)
-0.2015625081449864533091058
>>> tan(2+3j)
(-0.003764025641504248292751221 + 1.003238627353609801446359j)
>>> tan(inf)
nan
>>> nprint(chop(taylor(tan, 0, 6)))
[0.0, 1.0, 0.0, 0.333333, 0.0, 0.133333, 0.0]
Intervals are supported via :func:`mpmath.iv.tan`::
>>> iv.dps = 25; iv.pretty = True
>>> iv.tan([0,1])
[0.0, 1.55740772465490223050697482944]
>>> iv.tan([0,2]) # Interval includes a singularity
[-inf, +inf]
"""
sec = r"""
Computes the secant of `x`, `\mathrm{sec}(x) = \frac{1}{\cos(x)}`.
The secant function is singular at `x = (n+1/2)\pi`, but
``sec(x)`` always returns a finite result since `(n+1/2)\pi`
cannot be represented exactly using floating-point arithmetic.
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> sec(pi/3)
2.0
>>> sec(10000001)
-1.184723164360392819100265
>>> sec(2+3j)
(-0.04167496441114427004834991 + 0.0906111371962375965296612j)
>>> sec(inf)
nan
>>> nprint(chop(taylor(sec, 0, 6)))
[1.0, 0.0, 0.5, 0.0, 0.208333, 0.0, 0.0847222]
Intervals are supported via :func:`mpmath.iv.sec`::
>>> iv.dps = 25; iv.pretty = True
>>> iv.sec([0,1])
[1.0, 1.85081571768092561791175326276]
>>> iv.sec([0,2]) # Interval includes a singularity
[-inf, +inf]
"""
csc = r"""
Computes the cosecant of `x`, `\mathrm{csc}(x) = \frac{1}{\sin(x)}`.
This cosecant function is singular at `x = n \pi`, but with the
exception of the point `x = 0`, ``csc(x)`` returns a finite result
since `n \pi` cannot be represented exactly using floating-point
arithmetic.
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> csc(pi/3)
1.154700538379251529018298
>>> csc(10000001)
-1.864910497503629858938891
>>> csc(2+3j)
(0.09047320975320743980579048 + 0.04120098628857412646300981j)
>>> csc(inf)
nan
Intervals are supported via :func:`mpmath.iv.csc`::
>>> iv.dps = 25; iv.pretty = True
>>> iv.csc([0,1]) # Interval includes a singularity
[1.18839510577812121626159943988, +inf]
>>> iv.csc([0,2])
[1.0, +inf]
"""
cot = r"""
Computes the cotangent of `x`,
`\mathrm{cot}(x) = \frac{1}{\tan(x)} = \frac{\cos(x)}{\sin(x)}`.
This cotangent function is singular at `x = n \pi`, but with the
exception of the point `x = 0`, ``cot(x)`` returns a finite result
since `n \pi` cannot be represented exactly using floating-point
arithmetic.
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> cot(pi/3)
0.5773502691896257645091488
>>> cot(10000001)
1.574131876209625656003562
>>> cot(2+3j)
(-0.003739710376336956660117409 - 0.9967577965693583104609688j)
>>> cot(inf)
nan
Intervals are supported via :func:`mpmath.iv.cot`::
>>> iv.dps = 25; iv.pretty = True
>>> iv.cot([0,1]) # Interval includes a singularity
[0.642092615934330703006419974862, +inf]
>>> iv.cot([1,2])
[-inf, +inf]
"""
acos = r"""
Computes the inverse cosine or arccosine of `x`, `\cos^{-1}(x)`.
Since `-1 \le \cos(x) \le 1` for real `x`, the inverse
cosine is real-valued only for `-1 \le x \le 1`. On this interval,
:func:`~mpmath.acos` is defined to be a monotonically decreasing
function assuming values between `+\pi` and `0`.
Basic values are::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> acos(-1)
3.141592653589793238462643
>>> acos(0)
1.570796326794896619231322
>>> acos(1)
0.0
>>> nprint(chop(taylor(acos, 0, 6)))
[1.5708, -1.0, 0.0, -0.166667, 0.0, -0.075, 0.0]
:func:`~mpmath.acos` is defined so as to be a proper inverse function of
`\cos(\theta)` for `0 \le \theta < \pi`.
We have `\cos(\cos^{-1}(x)) = x` for all `x`, but
`\cos^{-1}(\cos(x)) = x` only for `0 \le \Re[x] < \pi`::
>>> for x in [1, 10, -1, 2+3j, 10+3j]:
... print("%s %s" % (cos(acos(x)), acos(cos(x))))
...
1.0 1.0
(10.0 + 0.0j) 2.566370614359172953850574
-1.0 1.0
(2.0 + 3.0j) (2.0 + 3.0j)
(10.0 + 3.0j) (2.566370614359172953850574 - 3.0j)
The inverse cosine has two branch points: `x = \pm 1`. :func:`~mpmath.acos`
places the branch cuts along the line segments `(-\infty, -1)` and
`(+1, +\infty)`. In general,
.. math ::
\cos^{-1}(x) = \frac{\pi}{2} + i \log\left(ix + \sqrt{1-x^2} \right)
where the principal-branch log and square root are implied.
"""
asin = r"""
Computes the inverse sine or arcsine of `x`, `\sin^{-1}(x)`.
Since `-1 \le \sin(x) \le 1` for real `x`, the inverse
sine is real-valued only for `-1 \le x \le 1`.
On this interval, it is defined to be a monotonically increasing
function assuming values between `-\pi/2` and `\pi/2`.
Basic values are::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> asin(-1)
-1.570796326794896619231322
>>> asin(0)
0.0
>>> asin(1)
1.570796326794896619231322
>>> nprint(chop(taylor(asin, 0, 6)))
[0.0, 1.0, 0.0, 0.166667, 0.0, 0.075, 0.0]
:func:`~mpmath.asin` is defined so as to be a proper inverse function of
`\sin(\theta)` for `-\pi/2 < \theta < \pi/2`.
We have `\sin(\sin^{-1}(x)) = x` for all `x`, but
`\sin^{-1}(\sin(x)) = x` only for `-\pi/2 < \Re[x] < \pi/2`::
>>> for x in [1, 10, -1, 1+3j, -2+3j]:
... print("%s %s" % (chop(sin(asin(x))), asin(sin(x))))
...
1.0 1.0
10.0 -0.5752220392306202846120698
-1.0 -1.0
(1.0 + 3.0j) (1.0 + 3.0j)
(-2.0 + 3.0j) (-1.141592653589793238462643 - 3.0j)
The inverse sine has two branch points: `x = \pm 1`. :func:`~mpmath.asin`
places the branch cuts along the line segments `(-\infty, -1)` and
`(+1, +\infty)`. In general,
.. math ::
\sin^{-1}(x) = -i \log\left(ix + \sqrt{1-x^2} \right)
where the principal-branch log and square root are implied.
"""
atan = r"""
Computes the inverse tangent or arctangent of `x`, `\tan^{-1}(x)`.
This is a real-valued function for all real `x`, with range
`(-\pi/2, \pi/2)`.
Basic values are::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> atan(-inf)
-1.570796326794896619231322
>>> atan(-1)
-0.7853981633974483096156609
>>> atan(0)
0.0
>>> atan(1)
0.7853981633974483096156609
>>> atan(inf)
1.570796326794896619231322
>>> nprint(chop(taylor(atan, 0, 6)))
[0.0, 1.0, 0.0, -0.333333, 0.0, 0.2, 0.0]
The inverse tangent is often used to compute angles. However,
the atan2 function is often better for this as it preserves sign
(see :func:`~mpmath.atan2`).
:func:`~mpmath.atan` is defined so as to be a proper inverse function of
`\tan(\theta)` for `-\pi/2 < \theta < \pi/2`.
We have `\tan(\tan^{-1}(x)) = x` for all `x`, but
`\tan^{-1}(\tan(x)) = x` only for `-\pi/2 < \Re[x] < \pi/2`::
>>> mp.dps = 25
>>> for x in [1, 10, -1, 1+3j, -2+3j]:
... print("%s %s" % (tan(atan(x)), atan(tan(x))))
...
1.0 1.0
10.0 0.5752220392306202846120698
-1.0 -1.0
(1.0 + 3.0j) (1.000000000000000000000001 + 3.0j)
(-2.0 + 3.0j) (1.141592653589793238462644 + 3.0j)
The inverse tangent has two branch points: `x = \pm i`. :func:`~mpmath.atan`
places the branch cuts along the line segments `(-i \infty, -i)` and
`(+i, +i \infty)`. In general,
.. math ::
\tan^{-1}(x) = \frac{i}{2}\left(\log(1-ix)-\log(1+ix)\right)
where the principal-branch log is implied.
"""
acot = r"""Computes the inverse cotangent of `x`,
`\mathrm{cot}^{-1}(x) = \tan^{-1}(1/x)`."""
asec = r"""Computes the inverse secant of `x`,
`\mathrm{sec}^{-1}(x) = \cos^{-1}(1/x)`."""
acsc = r"""Computes the inverse cosecant of `x`,
`\mathrm{csc}^{-1}(x) = \sin^{-1}(1/x)`."""
coth = r"""Computes the hyperbolic cotangent of `x`,
`\mathrm{coth}(x) = \frac{\cosh(x)}{\sinh(x)}`.
"""
sech = r"""Computes the hyperbolic secant of `x`,
`\mathrm{sech}(x) = \frac{1}{\cosh(x)}`.
"""
csch = r"""Computes the hyperbolic cosecant of `x`,
`\mathrm{csch}(x) = \frac{1}{\sinh(x)}`.
"""
acosh = r"""Computes the inverse hyperbolic cosine of `x`,
`\mathrm{cosh}^{-1}(x) = \log(x+\sqrt{x+1}\sqrt{x-1})`.
"""
asinh = r"""Computes the inverse hyperbolic sine of `x`,
`\mathrm{sinh}^{-1}(x) = \log(x+\sqrt{1+x^2})`.
"""
atanh = r"""Computes the inverse hyperbolic tangent of `x`,
`\mathrm{tanh}^{-1}(x) = \frac{1}{2}\left(\log(1+x)-\log(1-x)\right)`.
"""
acoth = r"""Computes the inverse hyperbolic cotangent of `x`,
`\mathrm{coth}^{-1}(x) = \tanh^{-1}(1/x)`."""
asech = r"""Computes the inverse hyperbolic secant of `x`,
`\mathrm{sech}^{-1}(x) = \cosh^{-1}(1/x)`."""
acsch = r"""Computes the inverse hyperbolic cosecant of `x`,
`\mathrm{csch}^{-1}(x) = \sinh^{-1}(1/x)`."""
sinpi = r"""
Computes `\sin(\pi x)`, more accurately than the expression
``sin(pi*x)``::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> sinpi(10**10), sin(pi*(10**10))
(0.0, -2.23936276195592e-6)
>>> sinpi(10**10+0.5), sin(pi*(10**10+0.5))
(1.0, 0.999999999998721)
"""
cospi = r"""
Computes `\cos(\pi x)`, more accurately than the expression
``cos(pi*x)``::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> cospi(10**10), cos(pi*(10**10))
(1.0, 0.999999999997493)
>>> cospi(10**10+0.5), cos(pi*(10**10+0.5))
(0.0, 1.59960492420134e-6)
"""
sinc = r"""
``sinc(x)`` computes the unnormalized sinc function, defined as
.. math ::
\mathrm{sinc}(x) = \begin{cases}
\sin(x)/x, & \mbox{if } x \ne 0 \\
1, & \mbox{if } x = 0.
\end{cases}
See :func:`~mpmath.sincpi` for the normalized sinc function.
Simple values and limits include::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> sinc(0)
1.0
>>> sinc(1)
0.841470984807897
>>> sinc(inf)
0.0
The integral of the sinc function is the sine integral Si::
>>> quad(sinc, [0, 1])
0.946083070367183
>>> si(1)
0.946083070367183
"""
sincpi = r"""
``sincpi(x)`` computes the normalized sinc function, defined as
.. math ::
\mathrm{sinc}_{\pi}(x) = \begin{cases}
\sin(\pi x)/(\pi x), & \mbox{if } x \ne 0 \\
1, & \mbox{if } x = 0.
\end{cases}
Equivalently, we have
`\mathrm{sinc}_{\pi}(x) = \mathrm{sinc}(\pi x)`.
The normalization entails that the function integrates
to unity over the entire real line::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> quadosc(sincpi, [-inf, inf], period=2.0)
1.0
Like, :func:`~mpmath.sinpi`, :func:`~mpmath.sincpi` is evaluated accurately
at its roots::
>>> sincpi(10)
0.0
"""
expj = r"""
Convenience function for computing `e^{ix}`::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> expj(0)
(1.0 + 0.0j)
>>> expj(-1)
(0.5403023058681397174009366 - 0.8414709848078965066525023j)
>>> expj(j)
(0.3678794411714423215955238 + 0.0j)
>>> expj(1+j)
(0.1987661103464129406288032 + 0.3095598756531121984439128j)
"""
expjpi = r"""
Convenience function for computing `e^{i \pi x}`.
Evaluation is accurate near zeros (see also :func:`~mpmath.cospi`,
:func:`~mpmath.sinpi`)::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> expjpi(0)
(1.0 + 0.0j)
>>> expjpi(1)
(-1.0 + 0.0j)
>>> expjpi(0.5)
(0.0 + 1.0j)
>>> expjpi(-1)
(-1.0 + 0.0j)
>>> expjpi(j)
(0.04321391826377224977441774 + 0.0j)
>>> expjpi(1+j)
(-0.04321391826377224977441774 + 0.0j)
"""
floor = r"""
Computes the floor of `x`, `\lfloor x \rfloor`, defined as
the largest integer less than or equal to `x`::
>>> from mpmath import *
>>> mp.pretty = False
>>> floor(3.5)
mpf('3.0')
.. note ::
:func:`~mpmath.floor`, :func:`~mpmath.ceil` and :func:`~mpmath.nint` return a
floating-point number, not a Python ``int``. If `\lfloor x \rfloor` is
too large to be represented exactly at the present working precision,
the result will be rounded, not necessarily in the direction
implied by the mathematical definition of the function.
To avoid rounding, use *prec=0*::
>>> mp.dps = 15
>>> print(int(floor(10**30+1)))
1000000000000000019884624838656
>>> print(int(floor(10**30+1, prec=0)))
1000000000000000000000000000001
The floor function is defined for complex numbers and
acts on the real and imaginary parts separately::
>>> floor(3.25+4.75j)
mpc(real='3.0', imag='4.0')
"""
ceil = r"""
Computes the ceiling of `x`, `\lceil x \rceil`, defined as
the smallest integer greater than or equal to `x`::
>>> from mpmath import *
>>> mp.pretty = False
>>> ceil(3.5)
mpf('4.0')
The ceiling function is defined for complex numbers and
acts on the real and imaginary parts separately::
>>> ceil(3.25+4.75j)
mpc(real='4.0', imag='5.0')
See notes about rounding for :func:`~mpmath.floor`.
"""
nint = r"""
Evaluates the nearest integer function, `\mathrm{nint}(x)`.
This gives the nearest integer to `x`; on a tie, it
gives the nearest even integer::
>>> from mpmath import *
>>> mp.pretty = False
>>> nint(3.2)
mpf('3.0')
>>> nint(3.8)
mpf('4.0')
>>> nint(3.5)
mpf('4.0')
>>> nint(4.5)
mpf('4.0')
The nearest integer function is defined for complex numbers and
acts on the real and imaginary parts separately::
>>> nint(3.25+4.75j)
mpc(real='3.0', imag='5.0')
See notes about rounding for :func:`~mpmath.floor`.
"""
frac = r"""
Gives the fractional part of `x`, defined as
`\mathrm{frac}(x) = x - \lfloor x \rfloor` (see :func:`~mpmath.floor`).
In effect, this computes `x` modulo 1, or `x+n` where
`n \in \mathbb{Z}` is such that `x+n \in [0,1)`::
>>> from mpmath import *
>>> mp.pretty = False
>>> frac(1.25)
mpf('0.25')
>>> frac(3)
mpf('0.0')
>>> frac(-1.25)
mpf('0.75')
For a complex number, the fractional part function applies to
the real and imaginary parts separately::
>>> frac(2.25+3.75j)
mpc(real='0.25', imag='0.75')
Plotted, the fractional part function gives a sawtooth
wave. The Fourier series coefficients have a simple
form::
>>> mp.dps = 15
>>> nprint(fourier(lambda x: frac(x)-0.5, [0,1], 4))
([0.0, 0.0, 0.0, 0.0, 0.0], [0.0, -0.31831, -0.159155, -0.106103, -0.0795775])
>>> nprint([-1/(pi*k) for k in range(1,5)])
[-0.31831, -0.159155, -0.106103, -0.0795775]
.. note::
The fractional part is sometimes defined as a symmetric
function, i.e. returning `-\mathrm{frac}(-x)` if `x < 0`.
This convention is used, for instance, by Mathematica's
``FractionalPart``.
"""
sign = r"""
Returns the sign of `x`, defined as `\mathrm{sign}(x) = x / |x|`
(with the special case `\mathrm{sign}(0) = 0`)::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = False
>>> sign(10)
mpf('1.0')
>>> sign(-10)
mpf('-1.0')
>>> sign(0)
mpf('0.0')
Note that the sign function is also defined for complex numbers,
for which it gives the projection onto the unit circle::
>>> mp.dps = 15; mp.pretty = True
>>> sign(1+j)
(0.707106781186547 + 0.707106781186547j)
"""
arg = r"""
Computes the complex argument (phase) of `x`, defined as the
signed angle between the positive real axis and `x` in the
complex plane::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> arg(3)
0.0
>>> arg(3+3j)
0.785398163397448
>>> arg(3j)
1.5707963267949
>>> arg(-3)
3.14159265358979
>>> arg(-3j)
-1.5707963267949
The angle is defined to satisfy `-\pi < \arg(x) \le \pi` and
with the sign convention that a nonnegative imaginary part
results in a nonnegative argument.
The value returned by :func:`~mpmath.arg` is an ``mpf`` instance.
"""
fabs = r"""
Returns the absolute value of `x`, `|x|`. Unlike :func:`abs`,
:func:`~mpmath.fabs` converts non-mpmath numbers (such as ``int``)
into mpmath numbers::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = False
>>> fabs(3)
mpf('3.0')
>>> fabs(-3)
mpf('3.0')
>>> fabs(3+4j)
mpf('5.0')
"""
re = r"""
Returns the real part of `x`, `\Re(x)`. :func:`~mpmath.re`
converts a non-mpmath number to an mpmath number::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = False
>>> re(3)
mpf('3.0')
>>> re(-1+4j)
mpf('-1.0')
"""
im = r"""
Returns the imaginary part of `x`, `\Im(x)`. :func:`~mpmath.im`
converts a non-mpmath number to an mpmath number::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = False
>>> im(3)
mpf('0.0')
>>> im(-1+4j)
mpf('4.0')
"""
conj = r"""
Returns the complex conjugate of `x`, `\overline{x}`. Unlike
``x.conjugate()``, :func:`~mpmath.im` converts `x` to a mpmath number::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = False
>>> conj(3)
mpf('3.0')
>>> conj(-1+4j)
mpc(real='-1.0', imag='-4.0')
"""
polar = r"""
Returns the polar representation of the complex number `z`
as a pair `(r, \phi)` such that `z = r e^{i \phi}`::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> polar(-2)
(2.0, 3.14159265358979)
>>> polar(3-4j)
(5.0, -0.927295218001612)
"""
rect = r"""
Returns the complex number represented by polar
coordinates `(r, \phi)`::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> chop(rect(2, pi))
-2.0
>>> rect(sqrt(2), -pi/4)
(1.0 - 1.0j)
"""
expm1 = r"""
Computes `e^x - 1`, accurately for small `x`.
Unlike the expression ``exp(x) - 1``, ``expm1(x)`` does not suffer from
potentially catastrophic cancellation::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> exp(1e-10)-1; print(expm1(1e-10))
1.00000008274037e-10
1.00000000005e-10
>>> exp(1e-20)-1; print(expm1(1e-20))
0.0
1.0e-20
>>> 1/(exp(1e-20)-1)
Traceback (most recent call last):
...
ZeroDivisionError
>>> 1/expm1(1e-20)
1.0e+20
Evaluation works for extremely tiny values::
>>> expm1(0)
0.0
>>> expm1('1e-10000000')
1.0e-10000000
"""
powm1 = r"""
Computes `x^y - 1`, accurately when `x^y` is very close to 1.
This avoids potentially catastrophic cancellation::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> power(0.99999995, 1e-10) - 1
0.0
>>> powm1(0.99999995, 1e-10)
-5.00000012791934e-18
Powers exactly equal to 1, and only those powers, yield 0 exactly::
>>> powm1(-j, 4)
(0.0 + 0.0j)
>>> powm1(3, 0)
0.0
>>> powm1(fadd(-1, 1e-100, exact=True), 4)
-4.0e-100
Evaluation works for extremely tiny `y`::
>>> powm1(2, '1e-100000')
6.93147180559945e-100001
>>> powm1(j, '1e-1000')
(-1.23370055013617e-2000 + 1.5707963267949e-1000j)
"""
root = r"""
``root(z, n, k=0)`` computes an `n`-th root of `z`, i.e. returns a number
`r` that (up to possible approximation error) satisfies `r^n = z`.
(``nthroot`` is available as an alias for ``root``.)
Every complex number `z \ne 0` has `n` distinct `n`-th roots, which are
equidistant points on a circle with radius `|z|^{1/n}`, centered around the
origin. A specific root may be selected using the optional index
`k`. The roots are indexed counterclockwise, starting with `k = 0` for the root
closest to the positive real half-axis.
The `k = 0` root is the so-called principal `n`-th root, often denoted by
`\sqrt[n]{z}` or `z^{1/n}`, and also given by `\exp(\log(z) / n)`. If `z` is
a positive real number, the principal root is just the unique positive
`n`-th root of `z`. Under some circumstances, non-principal real roots exist:
for positive real `z`, `n` even, there is a negative root given by `k = n/2`;
for negative real `z`, `n` odd, there is a negative root given by `k = (n-1)/2`.
To obtain all roots with a simple expression, use
``[root(z,n,k) for k in range(n)]``.
An important special case, ``root(1, n, k)`` returns the `k`-th `n`-th root of
unity, `\zeta_k = e^{2 \pi i k / n}`. Alternatively, :func:`~mpmath.unitroots`
provides a slightly more convenient way to obtain the roots of unity,
including the option to compute only the primitive roots of unity.
Both `k` and `n` should be integers; `k` outside of ``range(n)`` will be
reduced modulo `n`. If `n` is negative, `x^{-1/n} = 1/x^{1/n}` (or
the equivalent reciprocal for a non-principal root with `k \ne 0`) is computed.
:func:`~mpmath.root` is implemented to use Newton's method for small
`n`. At high precision, this makes `x^{1/n}` not much more
expensive than the regular exponentiation, `x^n`. For very large
`n`, :func:`~mpmath.nthroot` falls back to use the exponential function.
**Examples**
:func:`~mpmath.nthroot`/:func:`~mpmath.root` is faster and more accurate than raising to a
floating-point fraction::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = False
>>> 16807 ** (mpf(1)/5)
mpf('7.0000000000000009')
>>> root(16807, 5)
mpf('7.0')
>>> nthroot(16807, 5) # Alias
mpf('7.0')
A high-precision root::
>>> mp.dps = 50; mp.pretty = True
>>> nthroot(10, 5)
1.584893192461113485202101373391507013269442133825
>>> nthroot(10, 5) ** 5
10.0
Computing principal and non-principal square and cube roots::
>>> mp.dps = 15
>>> root(10, 2)
3.16227766016838
>>> root(10, 2, 1)
-3.16227766016838
>>> root(-10, 3)
(1.07721734501594 + 1.86579517236206j)
>>> root(-10, 3, 1)
-2.15443469003188
>>> root(-10, 3, 2)
(1.07721734501594 - 1.86579517236206j)
All the 7th roots of a complex number::
>>> for r in [root(3+4j, 7, k) for k in range(7)]:
... print("%s %s" % (r, r**7))
...
(1.24747270589553 + 0.166227124177353j) (3.0 + 4.0j)
(0.647824911301003 + 1.07895435170559j) (3.0 + 4.0j)
(-0.439648254723098 + 1.17920694574172j) (3.0 + 4.0j)
(-1.19605731775069 + 0.391492658196305j) (3.0 + 4.0j)
(-1.05181082538903 - 0.691023585965793j) (3.0 + 4.0j)
(-0.115529328478668 - 1.25318497558335j) (3.0 + 4.0j)
(0.907748109144957 - 0.871672518271819j) (3.0 + 4.0j)
Cube roots of unity::
>>> for k in range(3): print(root(1, 3, k))
...
1.0
(-0.5 + 0.866025403784439j)
(-0.5 - 0.866025403784439j)
Some exact high order roots::
>>> root(75**210, 105)
5625.0
>>> root(1, 128, 96)
(0.0 - 1.0j)
>>> root(4**128, 128, 96)
(0.0 - 4.0j)
"""
unitroots = r"""
``unitroots(n)`` returns `\zeta_0, \zeta_1, \ldots, \zeta_{n-1}`,
all the distinct `n`-th roots of unity, as a list. If the option
*primitive=True* is passed, only the primitive roots are returned.
Every `n`-th root of unity satisfies `(\zeta_k)^n = 1`. There are `n` distinct
roots for each `n` (`\zeta_k` and `\zeta_j` are the same when
`k = j \pmod n`), which form a regular polygon with vertices on the unit
circle. They are ordered counterclockwise with increasing `k`, starting
with `\zeta_0 = 1`.
**Examples**
The roots of unity up to `n = 4`::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> nprint(unitroots(1))
[1.0]
>>> nprint(unitroots(2))
[1.0, -1.0]
>>> nprint(unitroots(3))
[1.0, (-0.5 + 0.866025j), (-0.5 - 0.866025j)]
>>> nprint(unitroots(4))
[1.0, (0.0 + 1.0j), -1.0, (0.0 - 1.0j)]
Roots of unity form a geometric series that sums to 0::
>>> mp.dps = 50
>>> chop(fsum(unitroots(25)))
0.0
Primitive roots up to `n = 4`::
>>> mp.dps = 15
>>> nprint(unitroots(1, primitive=True))
[1.0]
>>> nprint(unitroots(2, primitive=True))
[-1.0]
>>> nprint(unitroots(3, primitive=True))
[(-0.5 + 0.866025j), (-0.5 - 0.866025j)]
>>> nprint(unitroots(4, primitive=True))
[(0.0 + 1.0j), (0.0 - 1.0j)]
There are only four primitive 12th roots::
>>> nprint(unitroots(12, primitive=True))
[(0.866025 + 0.5j), (-0.866025 + 0.5j), (-0.866025 - 0.5j), (0.866025 - 0.5j)]
The `n`-th roots of unity form a group, the cyclic group of order `n`.
Any primitive root `r` is a generator for this group, meaning that
`r^0, r^1, \ldots, r^{n-1}` gives the whole set of unit roots (in
some permuted order)::
>>> for r in unitroots(6): print(r)
...
1.0
(0.5 + 0.866025403784439j)
(-0.5 + 0.866025403784439j)
-1.0
(-0.5 - 0.866025403784439j)
(0.5 - 0.866025403784439j)
>>> r = unitroots(6, primitive=True)[1]
>>> for k in range(6): print(chop(r**k))
...
1.0
(0.5 - 0.866025403784439j)
(-0.5 - 0.866025403784439j)
-1.0
(-0.5 + 0.866025403784438j)
(0.5 + 0.866025403784438j)
The number of primitive roots equals the Euler totient function `\phi(n)`::
>>> [len(unitroots(n, primitive=True)) for n in range(1,20)]
[1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6, 18]
"""
log = r"""
Computes the base-`b` logarithm of `x`, `\log_b(x)`. If `b` is
unspecified, :func:`~mpmath.log` computes the natural (base `e`) logarithm
and is equivalent to :func:`~mpmath.ln`. In general, the base `b` logarithm
is defined in terms of the natural logarithm as
`\log_b(x) = \ln(x)/\ln(b)`.
By convention, we take `\log(0) = -\infty`.
The natural logarithm is real if `x > 0` and complex if `x < 0` or if
`x` is complex. The principal branch of the complex logarithm is
used, meaning that `\Im(\ln(x)) = -\pi < \arg(x) \le \pi`.
**Examples**
Some basic values and limits::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> log(1)
0.0
>>> log(2)
0.693147180559945
>>> log(1000,10)
3.0
>>> log(4, 16)
0.5
>>> log(j)
(0.0 + 1.5707963267949j)
>>> log(-1)
(0.0 + 3.14159265358979j)
>>> log(0)
-inf
>>> log(inf)
+inf
The natural logarithm is the antiderivative of `1/x`::
>>> quad(lambda x: 1/x, [1, 5])
1.6094379124341
>>> log(5)
1.6094379124341
>>> diff(log, 10)
0.1
The Taylor series expansion of the natural logarithm around
`x = 1` has coefficients `(-1)^{n+1}/n`::
>>> nprint(taylor(log, 1, 7))
[0.0, 1.0, -0.5, 0.333333, -0.25, 0.2, -0.166667, 0.142857]
:func:`~mpmath.log` supports arbitrary precision evaluation::
>>> mp.dps = 50
>>> log(pi)
1.1447298858494001741434273513530587116472948129153
>>> log(pi, pi**3)
0.33333333333333333333333333333333333333333333333333
>>> mp.dps = 25
>>> log(3+4j)
(1.609437912434100374600759 + 0.9272952180016122324285125j)
"""
log10 = r"""
Computes the base-10 logarithm of `x`, `\log_{10}(x)`. ``log10(x)``
is equivalent to ``log(x, 10)``.
"""
fmod = r"""
Converts `x` and `y` to mpmath numbers and returns `x \mod y`.
For mpmath numbers, this is equivalent to ``x % y``.
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> fmod(100, pi)
2.61062773871641
You can use :func:`~mpmath.fmod` to compute fractional parts of numbers::
>>> fmod(10.25, 1)
0.25
"""
radians = r"""
Converts the degree angle `x` to radians::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> radians(60)
1.0471975511966
"""
degrees = r"""
Converts the radian angle `x` to a degree angle::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> degrees(pi/3)
60.0
"""
atan2 = r"""
Computes the two-argument arctangent, `\mathrm{atan2}(y, x)`,
giving the signed angle between the positive `x`-axis and the
point `(x, y)` in the 2D plane. This function is defined for
real `x` and `y` only.
The two-argument arctangent essentially computes
`\mathrm{atan}(y/x)`, but accounts for the signs of both
`x` and `y` to give the angle for the correct quadrant. The
following examples illustrate the difference::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> atan2(1,1), atan(1/1.)
(0.785398163397448, 0.785398163397448)
>>> atan2(1,-1), atan(1/-1.)
(2.35619449019234, -0.785398163397448)
>>> atan2(-1,1), atan(-1/1.)
(-0.785398163397448, -0.785398163397448)
>>> atan2(-1,-1), atan(-1/-1.)
(-2.35619449019234, 0.785398163397448)
The angle convention is the same as that used for the complex
argument; see :func:`~mpmath.arg`.
"""
fibonacci = r"""
``fibonacci(n)`` computes the `n`-th Fibonacci number, `F(n)`. The
Fibonacci numbers are defined by the recurrence `F(n) = F(n-1) + F(n-2)`
with the initial values `F(0) = 0`, `F(1) = 1`. :func:`~mpmath.fibonacci`
extends this definition to arbitrary real and complex arguments
using the formula
.. math ::
F(z) = \frac{\phi^z - \cos(\pi z) \phi^{-z}}{\sqrt 5}
where `\phi` is the golden ratio. :func:`~mpmath.fibonacci` also uses this
continuous formula to compute `F(n)` for extremely large `n`, where
calculating the exact integer would be wasteful.
For convenience, :func:`~mpmath.fib` is available as an alias for
:func:`~mpmath.fibonacci`.
**Basic examples**
Some small Fibonacci numbers are::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> for i in range(10):
... print(fibonacci(i))
...
0.0
1.0
1.0
2.0
3.0
5.0
8.0
13.0
21.0
34.0
>>> fibonacci(50)
12586269025.0
The recurrence for `F(n)` extends backwards to negative `n`::
>>> for i in range(10):
... print(fibonacci(-i))
...
0.0
1.0
-1.0
2.0
-3.0
5.0
-8.0
13.0
-21.0
34.0
Large Fibonacci numbers will be computed approximately unless
the precision is set high enough::
>>> fib(200)
2.8057117299251e+41
>>> mp.dps = 45
>>> fib(200)
280571172992510140037611932413038677189525.0
:func:`~mpmath.fibonacci` can compute approximate Fibonacci numbers
of stupendous size::
>>> mp.dps = 15
>>> fibonacci(10**25)
3.49052338550226e+2089876402499787337692720
**Real and complex arguments**
The extended Fibonacci function is an analytic function. The
property `F(z) = F(z-1) + F(z-2)` holds for arbitrary `z`::
>>> mp.dps = 15
>>> fib(pi)
2.1170270579161
>>> fib(pi-1) + fib(pi-2)
2.1170270579161
>>> fib(3+4j)
(-5248.51130728372 - 14195.962288353j)
>>> fib(2+4j) + fib(1+4j)
(-5248.51130728372 - 14195.962288353j)
The Fibonacci function has infinitely many roots on the
negative half-real axis. The first root is at 0, the second is
close to -0.18, and then there are infinitely many roots that
asymptotically approach `-n+1/2`::
>>> findroot(fib, -0.2)
-0.183802359692956
>>> findroot(fib, -2)
-1.57077646820395
>>> findroot(fib, -17)
-16.4999999596115
>>> findroot(fib, -24)
-23.5000000000479
**Mathematical relationships**
For large `n`, `F(n+1)/F(n)` approaches the golden ratio::
>>> mp.dps = 50
>>> fibonacci(101)/fibonacci(100)
1.6180339887498948482045868343656381177203127439638
>>> +phi
1.6180339887498948482045868343656381177203091798058
The sum of reciprocal Fibonacci numbers converges to an irrational
number for which no closed form expression is known::
>>> mp.dps = 15
>>> nsum(lambda n: 1/fib(n), [1, inf])
3.35988566624318
Amazingly, however, the sum of odd-index reciprocal Fibonacci
numbers can be expressed in terms of a Jacobi theta function::
>>> nsum(lambda n: 1/fib(2*n+1), [0, inf])
1.82451515740692
>>> sqrt(5)*jtheta(2,0,(3-sqrt(5))/2)**2/4
1.82451515740692
Some related sums can be done in closed form::
>>> nsum(lambda k: 1/(1+fib(2*k+1)), [0, inf])
1.11803398874989
>>> phi - 0.5
1.11803398874989
>>> f = lambda k:(-1)**(k+1) / sum(fib(n)**2 for n in range(1,int(k+1)))
>>> nsum(f, [1, inf])
0.618033988749895
>>> phi-1
0.618033988749895
**References**
1. http://mathworld.wolfram.com/FibonacciNumber.html
"""
altzeta = r"""
Gives the Dirichlet eta function, `\eta(s)`, also known as the
alternating zeta function. This function is defined in analogy
with the Riemann zeta function as providing the sum of the
alternating series
.. math ::
\eta(s) = \sum_{k=0}^{\infty} \frac{(-1)^k}{k^s}
= 1-\frac{1}{2^s}+\frac{1}{3^s}-\frac{1}{4^s}+\ldots
The eta function, unlike the Riemann zeta function, is an entire
function, having a finite value for all complex `s`. The special case
`\eta(1) = \log(2)` gives the value of the alternating harmonic series.
The alternating zeta function may expressed using the Riemann zeta function
as `\eta(s) = (1 - 2^{1-s}) \zeta(s)`. It can also be expressed
in terms of the Hurwitz zeta function, for example using
:func:`~mpmath.dirichlet` (see documentation for that function).
**Examples**
Some special values are::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> altzeta(1)
0.693147180559945
>>> altzeta(0)
0.5
>>> altzeta(-1)
0.25
>>> altzeta(-2)
0.0
An example of a sum that can be computed more accurately and
efficiently via :func:`~mpmath.altzeta` than via numerical summation::
>>> sum(-(-1)**n / mpf(n)**2.5 for n in range(1, 100))
0.867204951503984
>>> altzeta(2.5)
0.867199889012184
At positive even integers, the Dirichlet eta function
evaluates to a rational multiple of a power of `\pi`::
>>> altzeta(2)
0.822467033424113
>>> pi**2/12
0.822467033424113
Like the Riemann zeta function, `\eta(s)`, approaches 1
as `s` approaches positive infinity, although it does
so from below rather than from above::
>>> altzeta(30)
0.999999999068682
>>> altzeta(inf)
1.0
>>> mp.pretty = False
>>> altzeta(1000, rounding='d')
mpf('0.99999999999999989')
>>> altzeta(1000, rounding='u')
mpf('1.0')
**References**
1. http://mathworld.wolfram.com/DirichletEtaFunction.html
2. http://en.wikipedia.org/wiki/Dirichlet_eta_function
"""
factorial = r"""
Computes the factorial, `x!`. For integers `n \ge 0`, we have
`n! = 1 \cdot 2 \cdots (n-1) \cdot n` and more generally the factorial
is defined for real or complex `x` by `x! = \Gamma(x+1)`.
**Examples**
Basic values and limits::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> for k in range(6):
... print("%s %s" % (k, fac(k)))
...
0 1.0
1 1.0
2 2.0
3 6.0
4 24.0
5 120.0
>>> fac(inf)
+inf
>>> fac(0.5), sqrt(pi)/2
(0.886226925452758, 0.886226925452758)
For large positive `x`, `x!` can be approximated by
Stirling's formula::
>>> x = 10**10
>>> fac(x)
2.32579620567308e+95657055186
>>> sqrt(2*pi*x)*(x/e)**x
2.32579597597705e+95657055186
:func:`~mpmath.fac` supports evaluation for astronomically large values::
>>> fac(10**30)
6.22311232304258e+29565705518096748172348871081098
Reciprocal factorials appear in the Taylor series of the
exponential function (among many other contexts)::
>>> nsum(lambda k: 1/fac(k), [0, inf]), exp(1)
(2.71828182845905, 2.71828182845905)
>>> nsum(lambda k: pi**k/fac(k), [0, inf]), exp(pi)
(23.1406926327793, 23.1406926327793)
"""
gamma = r"""
Computes the gamma function, `\Gamma(x)`. The gamma function is a
shifted version of the ordinary factorial, satisfying
`\Gamma(n) = (n-1)!` for integers `n > 0`. More generally, it
is defined by
.. math ::
\Gamma(x) = \int_0^{\infty} t^{x-1} e^{-t}\, dt
for any real or complex `x` with `\Re(x) > 0` and for `\Re(x) < 0`
by analytic continuation.
**Examples**
Basic values and limits::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> for k in range(1, 6):
... print("%s %s" % (k, gamma(k)))
...
1 1.0
2 1.0
3 2.0
4 6.0
5 24.0
>>> gamma(inf)
+inf
>>> gamma(0)
Traceback (most recent call last):
...
ValueError: gamma function pole
The gamma function of a half-integer is a rational multiple of
`\sqrt{\pi}`::
>>> gamma(0.5), sqrt(pi)
(1.77245385090552, 1.77245385090552)
>>> gamma(1.5), sqrt(pi)/2
(0.886226925452758, 0.886226925452758)
We can check the integral definition::
>>> gamma(3.5)
3.32335097044784
>>> quad(lambda t: t**2.5*exp(-t), [0,inf])
3.32335097044784
:func:`~mpmath.gamma` supports arbitrary-precision evaluation and
complex arguments::
>>> mp.dps = 50
>>> gamma(sqrt(3))
0.91510229697308632046045539308226554038315280564184
>>> mp.dps = 25
>>> gamma(2j)
(0.009902440080927490985955066 - 0.07595200133501806872408048j)
Arguments can also be large. Note that the gamma function grows
very quickly::
>>> mp.dps = 15
>>> gamma(10**20)
1.9328495143101e+1956570551809674817225
"""
psi = r"""
Gives the polygamma function of order `m` of `z`, `\psi^{(m)}(z)`.
Special cases are known as the *digamma function* (`\psi^{(0)}(z)`),
the *trigamma function* (`\psi^{(1)}(z)`), etc. The polygamma
functions are defined as the logarithmic derivatives of the gamma
function:
.. math ::
\psi^{(m)}(z) = \left(\frac{d}{dz}\right)^{m+1} \log \Gamma(z)
In particular, `\psi^{(0)}(z) = \Gamma'(z)/\Gamma(z)`. In the
present implementation of :func:`~mpmath.psi`, the order `m` must be a
nonnegative integer, while the argument `z` may be an arbitrary
complex number (with exception for the polygamma function's poles
at `z = 0, -1, -2, \ldots`).
**Examples**
For various rational arguments, the polygamma function reduces to
a combination of standard mathematical constants::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> psi(0, 1), -euler
(-0.5772156649015328606065121, -0.5772156649015328606065121)
>>> psi(1, '1/4'), pi**2+8*catalan
(17.19732915450711073927132, 17.19732915450711073927132)
>>> psi(2, '1/2'), -14*apery
(-16.82879664423431999559633, -16.82879664423431999559633)
The polygamma functions are derivatives of each other::
>>> diff(lambda x: psi(3, x), pi), psi(4, pi)
(-0.1105749312578862734526952, -0.1105749312578862734526952)
>>> quad(lambda x: psi(4, x), [2, 3]), psi(3,3)-psi(3,2)
(-0.375, -0.375)
The digamma function diverges logarithmically as `z \to \infty`,
while higher orders tend to zero::
>>> psi(0,inf), psi(1,inf), psi(2,inf)
(+inf, 0.0, 0.0)
Evaluation for a complex argument::
>>> psi(2, -1-2j)
(0.03902435405364952654838445 + 0.1574325240413029954685366j)
Evaluation is supported for large orders `m` and/or large
arguments `z`::
>>> psi(3, 10**100)
2.0e-300
>>> psi(250, 10**30+10**20*j)
(-1.293142504363642687204865e-7010 + 3.232856260909107391513108e-7018j)
**Application to infinite series**
Any infinite series where the summand is a rational function of
the index `k` can be evaluated in closed form in terms of polygamma
functions of the roots and poles of the summand::
>>> a = sqrt(2)
>>> b = sqrt(3)
>>> nsum(lambda k: 1/((k+a)**2*(k+b)), [0, inf])
0.4049668927517857061917531
>>> (psi(0,a)-psi(0,b)-a*psi(1,a)+b*psi(1,a))/(a-b)**2
0.4049668927517857061917531
This follows from the series representation (`m > 0`)
.. math ::
\psi^{(m)}(z) = (-1)^{m+1} m! \sum_{k=0}^{\infty}
\frac{1}{(z+k)^{m+1}}.
Since the roots of a polynomial may be complex, it is sometimes
necessary to use the complex polygamma function to evaluate
an entirely real-valued sum::
>>> nsum(lambda k: 1/(k**2-2*k+3), [0, inf])
1.694361433907061256154665
>>> nprint(polyroots([1,-2,3]))
[(1.0 - 1.41421j), (1.0 + 1.41421j)]
>>> r1 = 1-sqrt(2)*j
>>> r2 = r1.conjugate()
>>> (psi(0,-r2)-psi(0,-r1))/(r1-r2)
(1.694361433907061256154665 + 0.0j)
"""
digamma = r"""
Shortcut for ``psi(0,z)``.
"""
harmonic = r"""
If `n` is an integer, ``harmonic(n)`` gives a floating-point
approximation of the `n`-th harmonic number `H(n)`, defined as
.. math ::
H(n) = 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n}
The first few harmonic numbers are::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> for n in range(8):
... print("%s %s" % (n, harmonic(n)))
...
0 0.0
1 1.0
2 1.5
3 1.83333333333333
4 2.08333333333333
5 2.28333333333333
6 2.45
7 2.59285714285714
The infinite harmonic series `1 + 1/2 + 1/3 + \ldots` diverges::
>>> harmonic(inf)
+inf
:func:`~mpmath.harmonic` is evaluated using the digamma function rather
than by summing the harmonic series term by term. It can therefore
be computed quickly for arbitrarily large `n`, and even for
nonintegral arguments::
>>> harmonic(10**100)
230.835724964306
>>> harmonic(0.5)
0.613705638880109
>>> harmonic(3+4j)
(2.24757548223494 + 0.850502209186044j)
:func:`~mpmath.harmonic` supports arbitrary precision evaluation::
>>> mp.dps = 50
>>> harmonic(11)
3.0198773448773448773448773448773448773448773448773
>>> harmonic(pi)
1.8727388590273302654363491032336134987519132374152
The harmonic series diverges, but at a glacial pace. It is possible
to calculate the exact number of terms required before the sum
exceeds a given amount, say 100::
>>> mp.dps = 50
>>> v = 10**findroot(lambda x: harmonic(10**x) - 100, 10)
>>> v
15092688622113788323693563264538101449859496.864101
>>> v = int(ceil(v))
>>> print(v)
15092688622113788323693563264538101449859497
>>> harmonic(v-1)
99.999999999999999999999999999999999999999999942747
>>> harmonic(v)
100.000000000000000000000000000000000000000000009
"""
bernoulli = r"""
Computes the nth Bernoulli number, `B_n`, for any integer `n \ge 0`.
The Bernoulli numbers are rational numbers, but this function
returns a floating-point approximation. To obtain an exact
fraction, use :func:`~mpmath.bernfrac` instead.
**Examples**
Numerical values of the first few Bernoulli numbers::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> for n in range(15):
... print("%s %s" % (n, bernoulli(n)))
...
0 1.0
1 -0.5
2 0.166666666666667
3 0.0
4 -0.0333333333333333
5 0.0
6 0.0238095238095238
7 0.0
8 -0.0333333333333333
9 0.0
10 0.0757575757575758
11 0.0
12 -0.253113553113553
13 0.0
14 1.16666666666667
Bernoulli numbers can be approximated with arbitrary precision::
>>> mp.dps = 50
>>> bernoulli(100)
-2.8382249570693706959264156336481764738284680928013e+78
Arbitrarily large `n` are supported::
>>> mp.dps = 15
>>> bernoulli(10**20 + 2)
3.09136296657021e+1876752564973863312327
The Bernoulli numbers are related to the Riemann zeta function
at integer arguments::
>>> -bernoulli(8) * (2*pi)**8 / (2*fac(8))
1.00407735619794
>>> zeta(8)
1.00407735619794
**Algorithm**
For small `n` (`n < 3000`) :func:`~mpmath.bernoulli` uses a recurrence
formula due to Ramanujan. All results in this range are cached,
so sequential computation of small Bernoulli numbers is
guaranteed to be fast.
For larger `n`, `B_n` is evaluated in terms of the Riemann zeta
function.
"""
stieltjes = r"""
For a nonnegative integer `n`, ``stieltjes(n)`` computes the
`n`-th Stieltjes constant `\gamma_n`, defined as the
`n`-th coefficient in the Laurent series expansion of the
Riemann zeta function around the pole at `s = 1`. That is,
we have:
.. math ::
\zeta(s) = \frac{1}{s-1} \sum_{n=0}^{\infty}
\frac{(-1)^n}{n!} \gamma_n (s-1)^n
More generally, ``stieltjes(n, a)`` gives the corresponding
coefficient `\gamma_n(a)` for the Hurwitz zeta function
`\zeta(s,a)` (with `\gamma_n = \gamma_n(1)`).
**Examples**
The zeroth Stieltjes constant is just Euler's constant `\gamma`::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> stieltjes(0)
0.577215664901533
Some more values are::
>>> stieltjes(1)
-0.0728158454836767
>>> stieltjes(10)
0.000205332814909065
>>> stieltjes(30)
0.00355772885557316
>>> stieltjes(1000)
-1.57095384420474e+486
>>> stieltjes(2000)
2.680424678918e+1109
>>> stieltjes(1, 2.5)
-0.23747539175716
An alternative way to compute `\gamma_1`::
>>> diff(extradps(15)(lambda x: 1/(x-1) - zeta(x)), 1)
-0.0728158454836767
:func:`~mpmath.stieltjes` supports arbitrary precision evaluation::
>>> mp.dps = 50
>>> stieltjes(2)
-0.0096903631928723184845303860352125293590658061013408
**Algorithm**
:func:`~mpmath.stieltjes` numerically evaluates the integral in
the following representation due to Ainsworth, Howell and
Coffey [1], [2]:
.. math ::
\gamma_n(a) = \frac{\log^n a}{2a} - \frac{\log^{n+1}(a)}{n+1} +
\frac{2}{a} \Re \int_0^{\infty}
\frac{(x/a-i)\log^n(a-ix)}{(1+x^2/a^2)(e^{2\pi x}-1)} dx.
For some reference values with `a = 1`, see e.g. [4].
**References**
1. O. R. Ainsworth & L. W. Howell, "An integral representation of
the generalized Euler-Mascheroni constants", NASA Technical
Paper 2456 (1985),
http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19850014994_1985014994.pdf
2. M. W. Coffey, "The Stieltjes constants, their relation to the
`\eta_j` coefficients, and representation of the Hurwitz
zeta function", arXiv:0706.0343v1 http://arxiv.org/abs/0706.0343
3. http://mathworld.wolfram.com/StieltjesConstants.html
4. http://pi.lacim.uqam.ca/piDATA/stieltjesgamma.txt
"""
gammaprod = r"""
Given iterables `a` and `b`, ``gammaprod(a, b)`` computes the
product / quotient of gamma functions:
.. math ::
\frac{\Gamma(a_0) \Gamma(a_1) \cdots \Gamma(a_p)}
{\Gamma(b_0) \Gamma(b_1) \cdots \Gamma(b_q)}
Unlike direct calls to :func:`~mpmath.gamma`, :func:`~mpmath.gammaprod` considers
the entire product as a limit and evaluates this limit properly if
any of the numerator or denominator arguments are nonpositive
integers such that poles of the gamma function are encountered.
That is, :func:`~mpmath.gammaprod` evaluates
.. math ::
\lim_{\epsilon \to 0}
\frac{\Gamma(a_0+\epsilon) \Gamma(a_1+\epsilon) \cdots
\Gamma(a_p+\epsilon)}
{\Gamma(b_0+\epsilon) \Gamma(b_1+\epsilon) \cdots
\Gamma(b_q+\epsilon)}
In particular:
* If there are equally many poles in the numerator and the
denominator, the limit is a rational number times the remaining,
regular part of the product.
* If there are more poles in the numerator, :func:`~mpmath.gammaprod`
returns ``+inf``.
* If there are more poles in the denominator, :func:`~mpmath.gammaprod`
returns 0.
**Examples**
The reciprocal gamma function `1/\Gamma(x)` evaluated at `x = 0`::
>>> from mpmath import *
>>> mp.dps = 15
>>> gammaprod([], [0])
0.0
A limit::
>>> gammaprod([-4], [-3])
-0.25
>>> limit(lambda x: gamma(x-1)/gamma(x), -3, direction=1)
-0.25
>>> limit(lambda x: gamma(x-1)/gamma(x), -3, direction=-1)
-0.25
"""
beta = r"""
Computes the beta function,
`B(x,y) = \Gamma(x) \Gamma(y) / \Gamma(x+y)`.
The beta function is also commonly defined by the integral
representation
.. math ::
B(x,y) = \int_0^1 t^{x-1} (1-t)^{y-1} \, dt
**Examples**
For integer and half-integer arguments where all three gamma
functions are finite, the beta function becomes either rational
number or a rational multiple of `\pi`::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> beta(5, 2)
0.0333333333333333
>>> beta(1.5, 2)
0.266666666666667
>>> 16*beta(2.5, 1.5)
3.14159265358979
Where appropriate, :func:`~mpmath.beta` evaluates limits. A pole
of the beta function is taken to result in ``+inf``::
>>> beta(-0.5, 0.5)
0.0
>>> beta(-3, 3)
-0.333333333333333
>>> beta(-2, 3)
+inf
>>> beta(inf, 1)
0.0
>>> beta(inf, 0)
nan
:func:`~mpmath.beta` supports complex numbers and arbitrary precision
evaluation::
>>> beta(1, 2+j)
(0.4 - 0.2j)
>>> mp.dps = 25
>>> beta(j,0.5)
(1.079424249270925780135675 - 1.410032405664160838288752j)
>>> mp.dps = 50
>>> beta(pi, e)
0.037890298781212201348153837138927165984170287886464
Various integrals can be computed by means of the
beta function::
>>> mp.dps = 15
>>> quad(lambda t: t**2.5*(1-t)**2, [0, 1])
0.0230880230880231
>>> beta(3.5, 3)
0.0230880230880231
>>> quad(lambda t: sin(t)**4 * sqrt(cos(t)), [0, pi/2])
0.319504062596158
>>> beta(2.5, 0.75)/2
0.319504062596158
"""
betainc = r"""
``betainc(a, b, x1=0, x2=1, regularized=False)`` gives the generalized
incomplete beta function,
.. math ::
I_{x_1}^{x_2}(a,b) = \int_{x_1}^{x_2} t^{a-1} (1-t)^{b-1} dt.
When `x_1 = 0, x_2 = 1`, this reduces to the ordinary (complete)
beta function `B(a,b)`; see :func:`~mpmath.beta`.
With the keyword argument ``regularized=True``, :func:`~mpmath.betainc`
computes the regularized incomplete beta function
`I_{x_1}^{x_2}(a,b) / B(a,b)`. This is the cumulative distribution of the
beta distribution with parameters `a`, `b`.
.. note :
Implementations of the incomplete beta function in some other
software uses a different argument order. For example, Mathematica uses the
reversed argument order ``Beta[x1,x2,a,b]``. For the equivalent of SciPy's
three-argument incomplete beta integral (implicitly with `x1 = 0`), use
``betainc(a,b,0,x2,regularized=True)``.
**Examples**
Verifying that :func:`~mpmath.betainc` computes the integral in the
definition::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> x,y,a,b = 3, 4, 0, 6
>>> betainc(x, y, a, b)
-4010.4
>>> quad(lambda t: t**(x-1) * (1-t)**(y-1), [a, b])
-4010.4
The arguments may be arbitrary complex numbers::
>>> betainc(0.75, 1-4j, 0, 2+3j)
(0.2241657956955709603655887 + 0.3619619242700451992411724j)
With regularization::
>>> betainc(1, 2, 0, 0.25, regularized=True)
0.4375
>>> betainc(pi, e, 0, 1, regularized=True) # Complete
1.0
The beta integral satisfies some simple argument transformation
symmetries::
>>> mp.dps = 15
>>> betainc(2,3,4,5), -betainc(2,3,5,4), betainc(3,2,1-5,1-4)
(56.0833333333333, 56.0833333333333, 56.0833333333333)
The beta integral can often be evaluated analytically. For integer and
rational arguments, the incomplete beta function typically reduces to a
simple algebraic-logarithmic expression::
>>> mp.dps = 25
>>> identify(chop(betainc(0, 0, 3, 4)))
'-(log((9/8)))'
>>> identify(betainc(2, 3, 4, 5))
'(673/12)'
>>> identify(betainc(1.5, 1, 1, 2))
'((-12+sqrt(1152))/18)'
"""
binomial = r"""
Computes the binomial coefficient
.. math ::
{n \choose k} = \frac{n!}{k!(n-k)!}.
The binomial coefficient gives the number of ways that `k` items
can be chosen from a set of `n` items. More generally, the binomial
coefficient is a well-defined function of arbitrary real or
complex `n` and `k`, via the gamma function.
**Examples**
Generate Pascal's triangle::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> for n in range(5):
... nprint([binomial(n,k) for k in range(n+1)])
...
[1.0]
[1.0, 1.0]
[1.0, 2.0, 1.0]
[1.0, 3.0, 3.0, 1.0]
[1.0, 4.0, 6.0, 4.0, 1.0]
There is 1 way to select 0 items from the empty set, and 0 ways to
select 1 item from the empty set::
>>> binomial(0, 0)
1.0
>>> binomial(0, 1)
0.0
:func:`~mpmath.binomial` supports large arguments::
>>> binomial(10**20, 10**20-5)
8.33333333333333e+97
>>> binomial(10**20, 10**10)
2.60784095465201e+104342944813
Nonintegral binomial coefficients find use in series
expansions::
>>> nprint(taylor(lambda x: (1+x)**0.25, 0, 4))
[1.0, 0.25, -0.09375, 0.0546875, -0.0375977]
>>> nprint([binomial(0.25, k) for k in range(5)])
[1.0, 0.25, -0.09375, 0.0546875, -0.0375977]
An integral representation::
>>> n, k = 5, 3
>>> f = lambda t: exp(-j*k*t)*(1+exp(j*t))**n
>>> chop(quad(f, [-pi,pi])/(2*pi))
10.0
>>> binomial(n,k)
10.0
"""
rf = r"""
Computes the rising factorial or Pochhammer symbol,
.. math ::
x^{(n)} = x (x+1) \cdots (x+n-1) = \frac{\Gamma(x+n)}{\Gamma(x)}
where the rightmost expression is valid for nonintegral `n`.
**Examples**
For integral `n`, the rising factorial is a polynomial::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> for n in range(5):
... nprint(taylor(lambda x: rf(x,n), 0, n))
...
[1.0]
[0.0, 1.0]
[0.0, 1.0, 1.0]
[0.0, 2.0, 3.0, 1.0]
[0.0, 6.0, 11.0, 6.0, 1.0]
Evaluation is supported for arbitrary arguments::
>>> rf(2+3j, 5.5)
(-7202.03920483347 - 3777.58810701527j)
"""
ff = r"""
Computes the falling factorial,
.. math ::
(x)_n = x (x-1) \cdots (x-n+1) = \frac{\Gamma(x+1)}{\Gamma(x-n+1)}
where the rightmost expression is valid for nonintegral `n`.
**Examples**
For integral `n`, the falling factorial is a polynomial::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> for n in range(5):
... nprint(taylor(lambda x: ff(x,n), 0, n))
...
[1.0]
[0.0, 1.0]
[0.0, -1.0, 1.0]
[0.0, 2.0, -3.0, 1.0]
[0.0, -6.0, 11.0, -6.0, 1.0]
Evaluation is supported for arbitrary arguments::
>>> ff(2+3j, 5.5)
(-720.41085888203 + 316.101124983878j)
"""
fac2 = r"""
Computes the double factorial `x!!`, defined for integers
`x > 0` by
.. math ::
x!! = \begin{cases}
1 \cdot 3 \cdots (x-2) \cdot x & x \;\mathrm{odd} \\
2 \cdot 4 \cdots (x-2) \cdot x & x \;\mathrm{even}
\end{cases}
and more generally by [1]
.. math ::
x!! = 2^{x/2} \left(\frac{\pi}{2}\right)^{(\cos(\pi x)-1)/4}
\Gamma\left(\frac{x}{2}+1\right).
**Examples**
The integer sequence of double factorials begins::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> nprint([fac2(n) for n in range(10)])
[1.0, 1.0, 2.0, 3.0, 8.0, 15.0, 48.0, 105.0, 384.0, 945.0]
For large `x`, double factorials follow a Stirling-like asymptotic
approximation::
>>> x = mpf(10000)
>>> fac2(x)
5.97272691416282e+17830
>>> sqrt(pi)*x**((x+1)/2)*exp(-x/2)
5.97262736954392e+17830
The recurrence formula `x!! = x (x-2)!!` can be reversed to
define the double factorial of negative odd integers (but
not negative even integers)::
>>> fac2(-1), fac2(-3), fac2(-5), fac2(-7)
(1.0, -1.0, 0.333333333333333, -0.0666666666666667)
>>> fac2(-2)
Traceback (most recent call last):
...
ValueError: gamma function pole
With the exception of the poles at negative even integers,
:func:`~mpmath.fac2` supports evaluation for arbitrary complex arguments.
The recurrence formula is valid generally::
>>> fac2(pi+2j)
(-1.3697207890154e-12 + 3.93665300979176e-12j)
>>> (pi+2j)*fac2(pi-2+2j)
(-1.3697207890154e-12 + 3.93665300979176e-12j)
Double factorials should not be confused with nested factorials,
which are immensely larger::
>>> fac(fac(20))
5.13805976125208e+43675043585825292774
>>> fac2(20)
3715891200.0
Double factorials appear, among other things, in series expansions
of Gaussian functions and the error function. Infinite series
include::
>>> nsum(lambda k: 1/fac2(k), [0, inf])
3.05940740534258
>>> sqrt(e)*(1+sqrt(pi/2)*erf(sqrt(2)/2))
3.05940740534258
>>> nsum(lambda k: 2**k/fac2(2*k-1), [1, inf])
4.06015693855741
>>> e * erf(1) * sqrt(pi)
4.06015693855741
A beautiful Ramanujan sum::
>>> nsum(lambda k: (-1)**k*(fac2(2*k-1)/fac2(2*k))**3, [0,inf])
0.90917279454693
>>> (gamma('9/8')/gamma('5/4')/gamma('7/8'))**2
0.90917279454693
**References**
1. http://functions.wolfram.com/GammaBetaErf/Factorial2/27/01/0002/
2. http://mathworld.wolfram.com/DoubleFactorial.html
"""
hyper = r"""
Evaluates the generalized hypergeometric function
.. math ::
\,_pF_q(a_1,\ldots,a_p; b_1,\ldots,b_q; z) =
\sum_{n=0}^\infty \frac{(a_1)_n (a_2)_n \ldots (a_p)_n}
{(b_1)_n(b_2)_n\ldots(b_q)_n} \frac{z^n}{n!}
where `(x)_n` denotes the rising factorial (see :func:`~mpmath.rf`).
The parameters lists ``a_s`` and ``b_s`` may contain integers,
real numbers, complex numbers, as well as exact fractions given in
the form of tuples `(p, q)`. :func:`~mpmath.hyper` is optimized to handle
integers and fractions more efficiently than arbitrary
floating-point parameters (since rational parameters are by
far the most common).
**Examples**
Verifying that :func:`~mpmath.hyper` gives the sum in the definition, by
comparison with :func:`~mpmath.nsum`::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> a,b,c,d = 2,3,4,5
>>> x = 0.25
>>> hyper([a,b],[c,d],x)
1.078903941164934876086237
>>> fn = lambda n: rf(a,n)*rf(b,n)/rf(c,n)/rf(d,n)*x**n/fac(n)
>>> nsum(fn, [0, inf])
1.078903941164934876086237
The parameters can be any combination of integers, fractions,
floats and complex numbers::
>>> a, b, c, d, e = 1, (-1,2), pi, 3+4j, (2,3)
>>> x = 0.2j
>>> hyper([a,b],[c,d,e],x)
(0.9923571616434024810831887 - 0.005753848733883879742993122j)
>>> b, e = -0.5, mpf(2)/3
>>> fn = lambda n: rf(a,n)*rf(b,n)/rf(c,n)/rf(d,n)/rf(e,n)*x**n/fac(n)
>>> nsum(fn, [0, inf])
(0.9923571616434024810831887 - 0.005753848733883879742993122j)
The `\,_0F_0` and `\,_1F_0` series are just elementary functions::
>>> a, z = sqrt(2), +pi
>>> hyper([],[],z)
23.14069263277926900572909
>>> exp(z)
23.14069263277926900572909
>>> hyper([a],[],z)
(-0.09069132879922920160334114 + 0.3283224323946162083579656j)
>>> (1-z)**(-a)
(-0.09069132879922920160334114 + 0.3283224323946162083579656j)
If any `a_k` coefficient is a nonpositive integer, the series terminates
into a finite polynomial::
>>> hyper([1,1,1,-3],[2,5],1)
0.7904761904761904761904762
>>> identify(_)
'(83/105)'
If any `b_k` is a nonpositive integer, the function is undefined (unless the
series terminates before the division by zero occurs)::
>>> hyper([1,1,1,-3],[-2,5],1)
Traceback (most recent call last):
...
ZeroDivisionError: pole in hypergeometric series
>>> hyper([1,1,1,-1],[-2,5],1)
1.1
Except for polynomial cases, the radius of convergence `R` of the hypergeometric
series is either `R = \infty` (if `p \le q`), `R = 1` (if `p = q+1`), or
`R = 0` (if `p > q+1`).
The analytic continuations of the functions with `p = q+1`, i.e. `\,_2F_1`,
`\,_3F_2`, `\,_4F_3`, etc, are all implemented and therefore these functions
can be evaluated for `|z| \ge 1`. The shortcuts :func:`~mpmath.hyp2f1`, :func:`~mpmath.hyp3f2`
are available to handle the most common cases (see their documentation),
but functions of higher degree are also supported via :func:`~mpmath.hyper`::
>>> hyper([1,2,3,4], [5,6,7], 1) # 4F3 at finite-valued branch point
1.141783505526870731311423
>>> hyper([4,5,6,7], [1,2,3], 1) # 4F3 at pole
+inf
>>> hyper([1,2,3,4,5], [6,7,8,9], 10) # 5F4
(1.543998916527972259717257 - 0.5876309929580408028816365j)
>>> hyper([1,2,3,4,5,6], [7,8,9,10,11], 1j) # 6F5
(0.9996565821853579063502466 + 0.0129721075905630604445669j)
Near `z = 1` with noninteger parameters::
>>> hyper(['1/3',1,'3/2',2], ['1/5','11/6','41/8'], 1)
2.219433352235586121250027
>>> hyper(['1/3',1,'3/2',2], ['1/5','11/6','5/4'], 1)
+inf
>>> eps1 = extradps(6)(lambda: 1 - mpf('1e-6'))()
>>> hyper(['1/3',1,'3/2',2], ['1/5','11/6','5/4'], eps1)
2923978034.412973409330956
Please note that, as currently implemented, evaluation of `\,_pF_{p-1}`
with `p \ge 3` may be slow or inaccurate when `|z-1|` is small,
for some parameter values.
Evaluation may be aborted if convergence appears to be too slow.
The optional ``maxterms`` (limiting the number of series terms) and ``maxprec``
(limiting the internal precision) keyword arguments can be used
to control evaluation::
>>> hyper([1,2,3], [4,5,6], 10000)
Traceback (most recent call last):
...
NoConvergence: Hypergeometric series converges too slowly. Try increasing maxterms.
>>> hyper([1,2,3], [4,5,6], 10000, maxterms=10**6)
7.622806053177969474396918e+4310
Additional options include ``force_series`` (which forces direct use of
a hypergeometric series even if another evaluation method might work better)
and ``asymp_tol`` which controls the target tolerance for using
asymptotic series.
When `p > q+1`, ``hyper`` computes the (iterated) Borel sum of the divergent
series. For `\,_2F_0` the Borel sum has an analytic solution and can be
computed efficiently (see :func:`~mpmath.hyp2f0`). For higher degrees, the functions
is evaluated first by attempting to sum it directly as an asymptotic
series (this only works for tiny `|z|`), and then by evaluating the Borel
regularized sum using numerical integration. Except for
special parameter combinations, this can be extremely slow.
>>> hyper([1,1], [], 0.5) # regularization of 2F0
(1.340965419580146562086448 + 0.8503366631752726568782447j)
>>> hyper([1,1,1,1], [1], 0.5) # regularization of 4F1
(1.108287213689475145830699 + 0.5327107430640678181200491j)
With the following magnitude of argument, the asymptotic series for `\,_3F_1`
gives only a few digits. Using Borel summation, ``hyper`` can produce
a value with full accuracy::
>>> mp.dps = 15
>>> hyper([2,0.5,4], [5.25], '0.08', force_series=True)
Traceback (most recent call last):
...
NoConvergence: Hypergeometric series converges too slowly. Try increasing maxterms.
>>> hyper([2,0.5,4], [5.25], '0.08', asymp_tol=1e-4)
1.0725535790737
>>> hyper([2,0.5,4], [5.25], '0.08')
(1.07269542893559 + 5.54668863216891e-5j)
>>> hyper([2,0.5,4], [5.25], '-0.08', asymp_tol=1e-4)
0.946344925484879
>>> hyper([2,0.5,4], [5.25], '-0.08')
0.946312503737771
>>> mp.dps = 25
>>> hyper([2,0.5,4], [5.25], '-0.08')
0.9463125037377662296700858
Note that with the positive `z` value, there is a complex part in the
correct result, which falls below the tolerance of the asymptotic series.
By default, a parameter that appears in both ``a_s`` and ``b_s`` will be removed
unless it is a nonpositive integer. This generally speeds up evaluation
by producing a hypergeometric function of lower order.
This optimization can be disabled by passing ``eliminate=False``.
>>> hyper([1,2,3], [4,5,3], 10000)
1.268943190440206905892212e+4321
>>> hyper([1,2,3], [4,5,3], 10000, eliminate=False)
Traceback (most recent call last):
...
NoConvergence: Hypergeometric series converges too slowly. Try increasing maxterms.
>>> hyper([1,2,3], [4,5,3], 10000, eliminate=False, maxterms=10**6)
1.268943190440206905892212e+4321
If a nonpositive integer `-n` appears in both ``a_s`` and ``b_s``, this parameter
cannot be unambiguously removed since it creates a term 0 / 0.
In this case the hypergeometric series is understood to terminate before
the division by zero occurs. This convention is consistent with Mathematica.
An alternative convention of eliminating the parameters can be toggled
with ``eliminate_all=True``:
>>> hyper([2,-1], [-1], 3)
7.0
>>> hyper([2,-1], [-1], 3, eliminate_all=True)
0.25
>>> hyper([2], [], 3)
0.25
"""
hypercomb = r"""
Computes a weighted combination of hypergeometric functions
.. math ::
\sum_{r=1}^N \left[ \prod_{k=1}^{l_r} {w_{r,k}}^{c_{r,k}}
\frac{\prod_{k=1}^{m_r} \Gamma(\alpha_{r,k})}{\prod_{k=1}^{n_r}
\Gamma(\beta_{r,k})}
\,_{p_r}F_{q_r}(a_{r,1},\ldots,a_{r,p}; b_{r,1},
\ldots, b_{r,q}; z_r)\right].
Typically the parameters are linear combinations of a small set of base
parameters; :func:`~mpmath.hypercomb` permits computing a correct value in
the case that some of the `\alpha`, `\beta`, `b` turn out to be
nonpositive integers, or if division by zero occurs for some `w^c`,
assuming that there are opposing singularities that cancel out.
The limit is computed by evaluating the function with the base
parameters perturbed, at a higher working precision.
The first argument should be a function that takes the perturbable
base parameters ``params`` as input and returns `N` tuples
``(w, c, alpha, beta, a, b, z)``, where the coefficients ``w``, ``c``,
gamma factors ``alpha``, ``beta``, and hypergeometric coefficients
``a``, ``b`` each should be lists of numbers, and ``z`` should be a single
number.
**Examples**
The following evaluates
.. math ::
(a-1) \frac{\Gamma(a-3)}{\Gamma(a-4)} \,_1F_1(a,a-1,z) = e^z(a-4)(a+z-1)
with `a=1, z=3`. There is a zero factor, two gamma function poles, and
the 1F1 function is singular; all singularities cancel out to give a finite
value::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> hypercomb(lambda a: [([a-1],[1],[a-3],[a-4],[a],[a-1],3)], [1])
-180.769832308689
>>> -9*exp(3)
-180.769832308689
"""
hyp0f1 = r"""
Gives the hypergeometric function `\,_0F_1`, sometimes known as the
confluent limit function, defined as
.. math ::
\,_0F_1(a,z) = \sum_{k=0}^{\infty} \frac{1}{(a)_k} \frac{z^k}{k!}.
This function satisfies the differential equation `z f''(z) + a f'(z) = f(z)`,
and is related to the Bessel function of the first kind (see :func:`~mpmath.besselj`).
``hyp0f1(a,z)`` is equivalent to ``hyper([],[a],z)``; see documentation for
:func:`~mpmath.hyper` for more information.
**Examples**
Evaluation for arbitrary arguments::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> hyp0f1(2, 0.25)
1.130318207984970054415392
>>> hyp0f1((1,2), 1234567)
6.27287187546220705604627e+964
>>> hyp0f1(3+4j, 1000000j)
(3.905169561300910030267132e+606 + 3.807708544441684513934213e+606j)
Evaluation is supported for arbitrarily large values of `z`,
using asymptotic expansions::
>>> hyp0f1(1, 10**50)
2.131705322874965310390701e+8685889638065036553022565
>>> hyp0f1(1, -10**50)
1.115945364792025420300208e-13
Verifying the differential equation::
>>> a = 2.5
>>> f = lambda z: hyp0f1(a,z)
>>> for z in [0, 10, 3+4j]:
... chop(z*diff(f,z,2) + a*diff(f,z) - f(z))
...
0.0
0.0
0.0
"""
hyp1f1 = r"""
Gives the confluent hypergeometric function of the first kind,
.. math ::
\,_1F_1(a,b,z) = \sum_{k=0}^{\infty} \frac{(a)_k}{(b)_k} \frac{z^k}{k!},
also known as Kummer's function and sometimes denoted by `M(a,b,z)`. This
function gives one solution to the confluent (Kummer's) differential equation
.. math ::
z f''(z) + (b-z) f'(z) - af(z) = 0.
A second solution is given by the `U` function; see :func:`~mpmath.hyperu`.
Solutions are also given in an alternate form by the Whittaker
functions (:func:`~mpmath.whitm`, :func:`~mpmath.whitw`).
``hyp1f1(a,b,z)`` is equivalent
to ``hyper([a],[b],z)``; see documentation for :func:`~mpmath.hyper` for more
information.
**Examples**
Evaluation for real and complex values of the argument `z`, with
fixed parameters `a = 2, b = -1/3`::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> hyp1f1(2, (-1,3), 3.25)
-2815.956856924817275640248
>>> hyp1f1(2, (-1,3), -3.25)
-1.145036502407444445553107
>>> hyp1f1(2, (-1,3), 1000)
-8.021799872770764149793693e+441
>>> hyp1f1(2, (-1,3), -1000)
0.000003131987633006813594535331
>>> hyp1f1(2, (-1,3), 100+100j)
(-3.189190365227034385898282e+48 - 1.106169926814270418999315e+49j)
Parameters may be complex::
>>> hyp1f1(2+3j, -1+j, 10j)
(261.8977905181045142673351 + 160.8930312845682213562172j)
Arbitrarily large values of `z` are supported::
>>> hyp1f1(3, 4, 10**20)
3.890569218254486878220752e+43429448190325182745
>>> hyp1f1(3, 4, -10**20)
6.0e-60
>>> hyp1f1(3, 4, 10**20*j)
(-1.935753855797342532571597e-20 - 2.291911213325184901239155e-20j)
Verifying the differential equation::
>>> a, b = 1.5, 2
>>> f = lambda z: hyp1f1(a,b,z)
>>> for z in [0, -10, 3, 3+4j]:
... chop(z*diff(f,z,2) + (b-z)*diff(f,z) - a*f(z))
...
0.0
0.0
0.0
0.0
An integral representation::
>>> a, b = 1.5, 3
>>> z = 1.5
>>> hyp1f1(a,b,z)
2.269381460919952778587441
>>> g = lambda t: exp(z*t)*t**(a-1)*(1-t)**(b-a-1)
>>> gammaprod([b],[a,b-a])*quad(g, [0,1])
2.269381460919952778587441
"""
hyp1f2 = r"""
Gives the hypergeometric function `\,_1F_2(a_1,a_2;b_1,b_2; z)`.
The call ``hyp1f2(a1,b1,b2,z)`` is equivalent to
``hyper([a1],[b1,b2],z)``.
Evaluation works for complex and arbitrarily large arguments::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> a, b, c = 1.5, (-1,3), 2.25
>>> hyp1f2(a, b, c, 10**20)
-1.159388148811981535941434e+8685889639
>>> hyp1f2(a, b, c, -10**20)
-12.60262607892655945795907
>>> hyp1f2(a, b, c, 10**20*j)
(4.237220401382240876065501e+6141851464 - 2.950930337531768015892987e+6141851464j)
>>> hyp1f2(2+3j, -2j, 0.5j, 10-20j)
(135881.9905586966432662004 - 86681.95885418079535738828j)
"""
hyp2f2 = r"""
Gives the hypergeometric function `\,_2F_2(a_1,a_2;b_1,b_2; z)`.
The call ``hyp2f2(a1,a2,b1,b2,z)`` is equivalent to
``hyper([a1,a2],[b1,b2],z)``.
Evaluation works for complex and arbitrarily large arguments::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> a, b, c, d = 1.5, (-1,3), 2.25, 4
>>> hyp2f2(a, b, c, d, 10**20)
-5.275758229007902299823821e+43429448190325182663
>>> hyp2f2(a, b, c, d, -10**20)
2561445.079983207701073448
>>> hyp2f2(a, b, c, d, 10**20*j)
(2218276.509664121194836667 - 1280722.539991603850462856j)
>>> hyp2f2(2+3j, -2j, 0.5j, 4j, 10-20j)
(80500.68321405666957342788 - 20346.82752982813540993502j)
"""
hyp2f3 = r"""
Gives the hypergeometric function `\,_2F_3(a_1,a_2;b_1,b_2,b_3; z)`.
The call ``hyp2f3(a1,a2,b1,b2,b3,z)`` is equivalent to
``hyper([a1,a2],[b1,b2,b3],z)``.
Evaluation works for arbitrarily large arguments::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> a1,a2,b1,b2,b3 = 1.5, (-1,3), 2.25, 4, (1,5)
>>> hyp2f3(a1,a2,b1,b2,b3,10**20)
-4.169178177065714963568963e+8685889590
>>> hyp2f3(a1,a2,b1,b2,b3,-10**20)
7064472.587757755088178629
>>> hyp2f3(a1,a2,b1,b2,b3,10**20*j)
(-5.163368465314934589818543e+6141851415 + 1.783578125755972803440364e+6141851416j)
>>> hyp2f3(2+3j, -2j, 0.5j, 4j, -1-j, 10-20j)
(-2280.938956687033150740228 + 13620.97336609573659199632j)
>>> hyp2f3(2+3j, -2j, 0.5j, 4j, -1-j, 10000000-20000000j)
(4.849835186175096516193e+3504 - 3.365981529122220091353633e+3504j)
"""
hyp2f1 = r"""
Gives the Gauss hypergeometric function `\,_2F_1` (often simply referred to as
*the* hypergeometric function), defined for `|z| < 1` as
.. math ::
\,_2F_1(a,b,c,z) = \sum_{k=0}^{\infty}
\frac{(a)_k (b)_k}{(c)_k} \frac{z^k}{k!}.
and for `|z| \ge 1` by analytic continuation, with a branch cut on `(1, \infty)`
when necessary.
Special cases of this function include many of the orthogonal polynomials as
well as the incomplete beta function and other functions. Properties of the
Gauss hypergeometric function are documented comprehensively in many references,
for example Abramowitz & Stegun, section 15.
The implementation supports the analytic continuation as well as evaluation
close to the unit circle where `|z| \approx 1`. The syntax ``hyp2f1(a,b,c,z)``
is equivalent to ``hyper([a,b],[c],z)``.
**Examples**
Evaluation with `z` inside, outside and on the unit circle, for
fixed parameters::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> hyp2f1(2, (1,2), 4, 0.75)
1.303703703703703703703704
>>> hyp2f1(2, (1,2), 4, -1.75)
0.7431290566046919177853916
>>> hyp2f1(2, (1,2), 4, 1.75)
(1.418075801749271137026239 - 1.114976146679907015775102j)
>>> hyp2f1(2, (1,2), 4, 1)
1.6
>>> hyp2f1(2, (1,2), 4, -1)
0.8235498012182875315037882
>>> hyp2f1(2, (1,2), 4, j)
(0.9144026291433065674259078 + 0.2050415770437884900574923j)
>>> hyp2f1(2, (1,2), 4, 2+j)
(0.9274013540258103029011549 + 0.7455257875808100868984496j)
>>> hyp2f1(2, (1,2), 4, 0.25j)
(0.9931169055799728251931672 + 0.06154836525312066938147793j)
Evaluation with complex parameter values::
>>> hyp2f1(1+j, 0.75, 10j, 1+5j)
(0.8834833319713479923389638 + 0.7053886880648105068343509j)
Evaluation with `z = 1`::
>>> hyp2f1(-2.5, 3.5, 1.5, 1)
0.0
>>> hyp2f1(-2.5, 3, 4, 1)
0.06926406926406926406926407
>>> hyp2f1(2, 3, 4, 1)
+inf
Evaluation for huge arguments::
>>> hyp2f1((-1,3), 1.75, 4, '1e100')
(7.883714220959876246415651e+32 + 1.365499358305579597618785e+33j)
>>> hyp2f1((-1,3), 1.75, 4, '1e1000000')
(7.883714220959876246415651e+333332 + 1.365499358305579597618785e+333333j)
>>> hyp2f1((-1,3), 1.75, 4, '1e1000000j')
(1.365499358305579597618785e+333333 - 7.883714220959876246415651e+333332j)
An integral representation::
>>> a,b,c,z = -0.5, 1, 2.5, 0.25
>>> g = lambda t: t**(b-1) * (1-t)**(c-b-1) * (1-t*z)**(-a)
>>> gammaprod([c],[b,c-b]) * quad(g, [0,1])
0.9480458814362824478852618
>>> hyp2f1(a,b,c,z)
0.9480458814362824478852618
Verifying the hypergeometric differential equation::
>>> f = lambda z: hyp2f1(a,b,c,z)
>>> chop(z*(1-z)*diff(f,z,2) + (c-(a+b+1)*z)*diff(f,z) - a*b*f(z))
0.0
"""
hyp3f2 = r"""
Gives the generalized hypergeometric function `\,_3F_2`, defined for `|z| < 1`
as
.. math ::
\,_3F_2(a_1,a_2,a_3,b_1,b_2,z) = \sum_{k=0}^{\infty}
\frac{(a_1)_k (a_2)_k (a_3)_k}{(b_1)_k (b_2)_k} \frac{z^k}{k!}.
and for `|z| \ge 1` by analytic continuation. The analytic structure of this
function is similar to that of `\,_2F_1`, generally with a singularity at
`z = 1` and a branch cut on `(1, \infty)`.
Evaluation is supported inside, on, and outside
the circle of convergence `|z| = 1`::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> hyp3f2(1,2,3,4,5,0.25)
1.083533123380934241548707
>>> hyp3f2(1,2+2j,3,4,5,-10+10j)
(0.1574651066006004632914361 - 0.03194209021885226400892963j)
>>> hyp3f2(1,2,3,4,5,-10)
0.3071141169208772603266489
>>> hyp3f2(1,2,3,4,5,10)
(-0.4857045320523947050581423 - 0.5988311440454888436888028j)
>>> hyp3f2(0.25,1,1,2,1.5,1)
1.157370995096772047567631
>>> (8-pi-2*ln2)/3
1.157370995096772047567631
>>> hyp3f2(1+j,0.5j,2,1,-2j,-1)
(1.74518490615029486475959 + 0.1454701525056682297614029j)
>>> hyp3f2(1+j,0.5j,2,1,-2j,sqrt(j))
(0.9829816481834277511138055 - 0.4059040020276937085081127j)
>>> hyp3f2(-3,2,1,-5,4,1)
1.41
>>> hyp3f2(-3,2,1,-5,4,2)
2.12
Evaluation very close to the unit circle::
>>> hyp3f2(1,2,3,4,5,'1.0001')
(1.564877796743282766872279 - 3.76821518787438186031973e-11j)
>>> hyp3f2(1,2,3,4,5,'1+0.0001j')
(1.564747153061671573212831 + 0.0001305757570366084557648482j)
>>> hyp3f2(1,2,3,4,5,'0.9999')
1.564616644881686134983664
>>> hyp3f2(1,2,3,4,5,'-0.9999')
0.7823896253461678060196207
.. note ::
Evaluation for `|z-1|` small can currently be inaccurate or slow
for some parameter combinations.
For various parameter combinations, `\,_3F_2` admits representation in terms
of hypergeometric functions of lower degree, or in terms of
simpler functions::
>>> for a, b, z in [(1,2,-1), (2,0.5,1)]:
... hyp2f1(a,b,a+b+0.5,z)**2
... hyp3f2(2*a,a+b,2*b,a+b+0.5,2*a+2*b,z)
...
0.4246104461966439006086308
0.4246104461966439006086308
7.111111111111111111111111
7.111111111111111111111111
>>> z = 2+3j
>>> hyp3f2(0.5,1,1.5,2,2,z)
(0.7621440939243342419729144 + 0.4249117735058037649915723j)
>>> 4*(pi-2*ellipe(z))/(pi*z)
(0.7621440939243342419729144 + 0.4249117735058037649915723j)
"""
hyperu = r"""
Gives the Tricomi confluent hypergeometric function `U`, also known as
the Kummer or confluent hypergeometric function of the second kind. This
function gives a second linearly independent solution to the confluent
hypergeometric differential equation (the first is provided by `\,_1F_1` --
see :func:`~mpmath.hyp1f1`).
**Examples**
Evaluation for arbitrary complex arguments::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> hyperu(2,3,4)
0.0625
>>> hyperu(0.25, 5, 1000)
0.1779949416140579573763523
>>> hyperu(0.25, 5, -1000)
(0.1256256609322773150118907 - 0.1256256609322773150118907j)
The `U` function may be singular at `z = 0`::
>>> hyperu(1.5, 2, 0)
+inf
>>> hyperu(1.5, -2, 0)
0.1719434921288400112603671
Verifying the differential equation::
>>> a, b = 1.5, 2
>>> f = lambda z: hyperu(a,b,z)
>>> for z in [-10, 3, 3+4j]:
... chop(z*diff(f,z,2) + (b-z)*diff(f,z) - a*f(z))
...
0.0
0.0
0.0
An integral representation::
>>> a,b,z = 2, 3.5, 4.25
>>> hyperu(a,b,z)
0.06674960718150520648014567
>>> quad(lambda t: exp(-z*t)*t**(a-1)*(1+t)**(b-a-1),[0,inf]) / gamma(a)
0.06674960718150520648014567
[1] http://people.math.sfu.ca/~cbm/aands/page_504.htm
"""
hyp2f0 = r"""
Gives the hypergeometric function `\,_2F_0`, defined formally by the
series
.. math ::
\,_2F_0(a,b;;z) = \sum_{n=0}^{\infty} (a)_n (b)_n \frac{z^n}{n!}.
This series usually does not converge. For small enough `z`, it can be viewed
as an asymptotic series that may be summed directly with an appropriate
truncation. When this is not the case, :func:`~mpmath.hyp2f0` gives a regularized sum,
or equivalently, it uses a representation in terms of the
hypergeometric U function [1]. The series also converges when either `a` or `b`
is a nonpositive integer, as it then terminates into a polynomial
after `-a` or `-b` terms.
**Examples**
Evaluation is supported for arbitrary complex arguments::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> hyp2f0((2,3), 1.25, -100)
0.07095851870980052763312791
>>> hyp2f0((2,3), 1.25, 100)
(-0.03254379032170590665041131 + 0.07269254613282301012735797j)
>>> hyp2f0(-0.75, 1-j, 4j)
(-0.3579987031082732264862155 - 3.052951783922142735255881j)
Even with real arguments, the regularized value of 2F0 is often complex-valued,
but the imaginary part decreases exponentially as `z \to 0`. In the following
example, the first call uses complex evaluation while the second has a small
enough `z` to evaluate using the direct series and thus the returned value
is strictly real (this should be taken to indicate that the imaginary
part is less than ``eps``)::
>>> mp.dps = 15
>>> hyp2f0(1.5, 0.5, 0.05)
(1.04166637647907 + 8.34584913683906e-8j)
>>> hyp2f0(1.5, 0.5, 0.0005)
1.00037535207621
The imaginary part can be retrieved by increasing the working precision::
>>> mp.dps = 80
>>> nprint(hyp2f0(1.5, 0.5, 0.009).imag)
1.23828e-46
In the polynomial case (the series terminating), 2F0 can evaluate exactly::
>>> mp.dps = 15
>>> hyp2f0(-6,-6,2)
291793.0
>>> identify(hyp2f0(-2,1,0.25))
'(5/8)'
The coefficients of the polynomials can be recovered using Taylor expansion::
>>> nprint(taylor(lambda x: hyp2f0(-3,0.5,x), 0, 10))
[1.0, -1.5, 2.25, -1.875, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
>>> nprint(taylor(lambda x: hyp2f0(-4,0.5,x), 0, 10))
[1.0, -2.0, 4.5, -7.5, 6.5625, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
[1] http://people.math.sfu.ca/~cbm/aands/page_504.htm
"""
gammainc = r"""
``gammainc(z, a=0, b=inf)`` computes the (generalized) incomplete
gamma function with integration limits `[a, b]`:
.. math ::
\Gamma(z,a,b) = \int_a^b t^{z-1} e^{-t} \, dt
The generalized incomplete gamma function reduces to the
following special cases when one or both endpoints are fixed:
* `\Gamma(z,0,\infty)` is the standard ("complete")
gamma function, `\Gamma(z)` (available directly
as the mpmath function :func:`~mpmath.gamma`)
* `\Gamma(z,a,\infty)` is the "upper" incomplete gamma
function, `\Gamma(z,a)`
* `\Gamma(z,0,b)` is the "lower" incomplete gamma
function, `\gamma(z,b)`.
Of course, we have
`\Gamma(z,0,x) + \Gamma(z,x,\infty) = \Gamma(z)`
for all `z` and `x`.
Note however that some authors reverse the order of the
arguments when defining the lower and upper incomplete
gamma function, so one should be careful to get the correct
definition.
If also given the keyword argument ``regularized=True``,
:func:`~mpmath.gammainc` computes the "regularized" incomplete gamma
function
.. math ::
P(z,a,b) = \frac{\Gamma(z,a,b)}{\Gamma(z)}.
**Examples**
We can compare with numerical quadrature to verify that
:func:`~mpmath.gammainc` computes the integral in the definition::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> gammainc(2+3j, 4, 10)
(0.00977212668627705160602312 - 0.0770637306312989892451977j)
>>> quad(lambda t: t**(2+3j-1) * exp(-t), [4, 10])
(0.00977212668627705160602312 - 0.0770637306312989892451977j)
Argument symmetries follow directly from the integral definition::
>>> gammainc(3, 4, 5) + gammainc(3, 5, 4)
0.0
>>> gammainc(3,0,2) + gammainc(3,2,4); gammainc(3,0,4)
1.523793388892911312363331
1.523793388892911312363331
>>> findroot(lambda z: gammainc(2,z,3), 1)
3.0
Evaluation for arbitrarily large arguments::
>>> gammainc(10, 100)
4.083660630910611272288592e-26
>>> gammainc(10, 10000000000000000)
5.290402449901174752972486e-4342944819032375
>>> gammainc(3+4j, 1000000+1000000j)
(-1.257913707524362408877881e-434284 + 2.556691003883483531962095e-434284j)
Evaluation of a generalized incomplete gamma function automatically chooses
the representation that gives a more accurate result, depending on which
parameter is larger::
>>> gammainc(10000000, 3) - gammainc(10000000, 2) # Bad
0.0
>>> gammainc(10000000, 2, 3) # Good
1.755146243738946045873491e+4771204
>>> gammainc(2, 0, 100000001) - gammainc(2, 0, 100000000) # Bad
0.0
>>> gammainc(2, 100000000, 100000001) # Good
4.078258353474186729184421e-43429441
The incomplete gamma functions satisfy simple recurrence
relations::
>>> mp.dps = 25
>>> z, a = mpf(3.5), mpf(2)
>>> gammainc(z+1, a); z*gammainc(z,a) + a**z*exp(-a)
10.60130296933533459267329
10.60130296933533459267329
>>> gammainc(z+1,0,a); z*gammainc(z,0,a) - a**z*exp(-a)
1.030425427232114336470932
1.030425427232114336470932
Evaluation at integers and poles::
>>> gammainc(-3, -4, -5)
(-0.2214577048967798566234192 + 0.0j)
>>> gammainc(-3, 0, 5)
+inf
If `z` is an integer, the recurrence reduces the incomplete gamma
function to `P(a) \exp(-a) + Q(b) \exp(-b)` where `P` and
`Q` are polynomials::
>>> gammainc(1, 2); exp(-2)
0.1353352832366126918939995
0.1353352832366126918939995
>>> mp.dps = 50
>>> identify(gammainc(6, 1, 2), ['exp(-1)', 'exp(-2)'])
'(326*exp(-1) + (-872)*exp(-2))'
The incomplete gamma functions reduce to functions such as
the exponential integral Ei and the error function for special
arguments::
>>> mp.dps = 25
>>> gammainc(0, 4); -ei(-4)
0.00377935240984890647887486
0.00377935240984890647887486
>>> gammainc(0.5, 0, 2); sqrt(pi)*erf(sqrt(2))
1.691806732945198336509541
1.691806732945198336509541
"""
erf = r"""
Computes the error function, `\mathrm{erf}(x)`. The error
function is the normalized antiderivative of the Gaussian function
`\exp(-t^2)`. More precisely,
.. math::
\mathrm{erf}(x) = \frac{2}{\sqrt \pi} \int_0^x \exp(-t^2) \,dt
**Basic examples**
Simple values and limits include::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> erf(0)
0.0
>>> erf(1)
0.842700792949715
>>> erf(-1)
-0.842700792949715
>>> erf(inf)
1.0
>>> erf(-inf)
-1.0
For large real `x`, `\mathrm{erf}(x)` approaches 1 very
rapidly::
>>> erf(3)
0.999977909503001
>>> erf(5)
0.999999999998463
The error function is an odd function::
>>> nprint(chop(taylor(erf, 0, 5)))
[0.0, 1.12838, 0.0, -0.376126, 0.0, 0.112838]
:func:`~mpmath.erf` implements arbitrary-precision evaluation and
supports complex numbers::
>>> mp.dps = 50
>>> erf(0.5)
0.52049987781304653768274665389196452873645157575796
>>> mp.dps = 25
>>> erf(1+j)
(1.316151281697947644880271 + 0.1904534692378346862841089j)
Evaluation is supported for large arguments::
>>> mp.dps = 25
>>> erf('1e1000')
1.0
>>> erf('-1e1000')
-1.0
>>> erf('1e-1000')
1.128379167095512573896159e-1000
>>> erf('1e7j')
(0.0 + 8.593897639029319267398803e+43429448190317j)
>>> erf('1e7+1e7j')
(0.9999999858172446172631323 + 3.728805278735270407053139e-8j)
**Related functions**
See also :func:`~mpmath.erfc`, which is more accurate for large `x`,
and :func:`~mpmath.erfi` which gives the antiderivative of
`\exp(t^2)`.
The Fresnel integrals :func:`~mpmath.fresnels` and :func:`~mpmath.fresnelc`
are also related to the error function.
"""
erfc = r"""
Computes the complementary error function,
`\mathrm{erfc}(x) = 1-\mathrm{erf}(x)`.
This function avoids cancellation that occurs when naively
computing the complementary error function as ``1-erf(x)``::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> 1 - erf(10)
0.0
>>> erfc(10)
2.08848758376254e-45
:func:`~mpmath.erfc` works accurately even for ludicrously large
arguments::
>>> erfc(10**10)
4.3504398860243e-43429448190325182776
Complex arguments are supported::
>>> erfc(500+50j)
(1.19739830969552e-107492 + 1.46072418957528e-107491j)
"""
erfi = r"""
Computes the imaginary error function, `\mathrm{erfi}(x)`.
The imaginary error function is defined in analogy with the
error function, but with a positive sign in the integrand:
.. math ::
\mathrm{erfi}(x) = \frac{2}{\sqrt \pi} \int_0^x \exp(t^2) \,dt
Whereas the error function rapidly converges to 1 as `x` grows,
the imaginary error function rapidly diverges to infinity.
The functions are related as
`\mathrm{erfi}(x) = -i\,\mathrm{erf}(ix)` for all complex
numbers `x`.
**Examples**
Basic values and limits::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> erfi(0)
0.0
>>> erfi(1)
1.65042575879754
>>> erfi(-1)
-1.65042575879754
>>> erfi(inf)
+inf
>>> erfi(-inf)
-inf
Note the symmetry between erf and erfi::
>>> erfi(3j)
(0.0 + 0.999977909503001j)
>>> erf(3)
0.999977909503001
>>> erf(1+2j)
(-0.536643565778565 - 5.04914370344703j)
>>> erfi(2+1j)
(-5.04914370344703 - 0.536643565778565j)
Large arguments are supported::
>>> erfi(1000)
1.71130938718796e+434291
>>> erfi(10**10)
7.3167287567024e+43429448190325182754
>>> erfi(-10**10)
-7.3167287567024e+43429448190325182754
>>> erfi(1000-500j)
(2.49895233563961e+325717 + 2.6846779342253e+325717j)
>>> erfi(100000j)
(0.0 + 1.0j)
>>> erfi(-100000j)
(0.0 - 1.0j)
"""
erfinv = r"""
Computes the inverse error function, satisfying
.. math ::
\mathrm{erf}(\mathrm{erfinv}(x)) =
\mathrm{erfinv}(\mathrm{erf}(x)) = x.
This function is defined only for `-1 \le x \le 1`.
**Examples**
Special values include::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> erfinv(0)
0.0
>>> erfinv(1)
+inf
>>> erfinv(-1)
-inf
The domain is limited to the standard interval::
>>> erfinv(2)
Traceback (most recent call last):
...
ValueError: erfinv(x) is defined only for -1 <= x <= 1
It is simple to check that :func:`~mpmath.erfinv` computes inverse values of
:func:`~mpmath.erf` as promised::
>>> erf(erfinv(0.75))
0.75
>>> erf(erfinv(-0.995))
-0.995
:func:`~mpmath.erfinv` supports arbitrary-precision evaluation::
>>> mp.dps = 50
>>> x = erf(2)
>>> x
0.99532226501895273416206925636725292861089179704006
>>> erfinv(x)
2.0
A definite integral involving the inverse error function::
>>> mp.dps = 15
>>> quad(erfinv, [0, 1])
0.564189583547756
>>> 1/sqrt(pi)
0.564189583547756
The inverse error function can be used to generate random numbers
with a Gaussian distribution (although this is a relatively
inefficient algorithm)::
>>> nprint([erfinv(2*rand()-1) for n in range(6)]) # doctest: +SKIP
[-0.586747, 1.10233, -0.376796, 0.926037, -0.708142, -0.732012]
"""
npdf = r"""
``npdf(x, mu=0, sigma=1)`` evaluates the probability density
function of a normal distribution with mean value `\mu`
and variance `\sigma^2`.
Elementary properties of the probability distribution can
be verified using numerical integration::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> quad(npdf, [-inf, inf])
1.0
>>> quad(lambda x: npdf(x, 3), [3, inf])
0.5
>>> quad(lambda x: npdf(x, 3, 2), [3, inf])
0.5
See also :func:`~mpmath.ncdf`, which gives the cumulative
distribution.
"""
ncdf = r"""
``ncdf(x, mu=0, sigma=1)`` evaluates the cumulative distribution
function of a normal distribution with mean value `\mu`
and variance `\sigma^2`.
See also :func:`~mpmath.npdf`, which gives the probability density.
Elementary properties include::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> ncdf(pi, mu=pi)
0.5
>>> ncdf(-inf)
0.0
>>> ncdf(+inf)
1.0
The cumulative distribution is the integral of the density
function having identical mu and sigma::
>>> mp.dps = 15
>>> diff(ncdf, 2)
0.053990966513188
>>> npdf(2)
0.053990966513188
>>> diff(lambda x: ncdf(x, 1, 0.5), 0)
0.107981933026376
>>> npdf(0, 1, 0.5)
0.107981933026376
"""
expint = r"""
:func:`~mpmath.expint(n,z)` gives the generalized exponential integral
or En-function,
.. math ::
\mathrm{E}_n(z) = \int_1^{\infty} \frac{e^{-zt}}{t^n} dt,
where `n` and `z` may both be complex numbers. The case with `n = 1` is
also given by :func:`~mpmath.e1`.
**Examples**
Evaluation at real and complex arguments::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> expint(1, 6.25)
0.0002704758872637179088496194
>>> expint(-3, 2+3j)
(0.00299658467335472929656159 + 0.06100816202125885450319632j)
>>> expint(2+3j, 4-5j)
(0.001803529474663565056945248 - 0.002235061547756185403349091j)
At negative integer values of `n`, `E_n(z)` reduces to a
rational-exponential function::
>>> f = lambda n, z: fac(n)*sum(z**k/fac(k-1) for k in range(1,n+2))/\
... exp(z)/z**(n+2)
>>> n = 3
>>> z = 1/pi
>>> expint(-n,z)
584.2604820613019908668219
>>> f(n,z)
584.2604820613019908668219
>>> n = 5
>>> expint(-n,z)
115366.5762594725451811138
>>> f(n,z)
115366.5762594725451811138
"""
e1 = r"""
Computes the exponential integral `\mathrm{E}_1(z)`, given by
.. math ::
\mathrm{E}_1(z) = \int_z^{\infty} \frac{e^{-t}}{t} dt.
This is equivalent to :func:`~mpmath.expint` with `n = 1`.
**Examples**
Two ways to evaluate this function::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> e1(6.25)
0.0002704758872637179088496194
>>> expint(1,6.25)
0.0002704758872637179088496194
The E1-function is essentially the same as the Ei-function (:func:`~mpmath.ei`)
with negated argument, except for an imaginary branch cut term::
>>> e1(2.5)
0.02491491787026973549562801
>>> -ei(-2.5)
0.02491491787026973549562801
>>> e1(-2.5)
(-7.073765894578600711923552 - 3.141592653589793238462643j)
>>> -ei(2.5)
-7.073765894578600711923552
"""
ei = r"""
Computes the exponential integral or Ei-function, `\mathrm{Ei}(x)`.
The exponential integral is defined as
.. math ::
\mathrm{Ei}(x) = \int_{-\infty\,}^x \frac{e^t}{t} \, dt.
When the integration range includes `t = 0`, the exponential
integral is interpreted as providing the Cauchy principal value.
For real `x`, the Ei-function behaves roughly like
`\mathrm{Ei}(x) \approx \exp(x) + \log(|x|)`.
The Ei-function is related to the more general family of exponential
integral functions denoted by `E_n`, which are available as :func:`~mpmath.expint`.
**Basic examples**
Some basic values and limits are::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> ei(0)
-inf
>>> ei(1)
1.89511781635594
>>> ei(inf)
+inf
>>> ei(-inf)
0.0
For `x < 0`, the defining integral can be evaluated
numerically as a reference::
>>> ei(-4)
-0.00377935240984891
>>> quad(lambda t: exp(t)/t, [-inf, -4])
-0.00377935240984891
:func:`~mpmath.ei` supports complex arguments and arbitrary
precision evaluation::
>>> mp.dps = 50
>>> ei(pi)
10.928374389331410348638445906907535171566338835056
>>> mp.dps = 25
>>> ei(3+4j)
(-4.154091651642689822535359 + 4.294418620024357476985535j)
**Related functions**
The exponential integral is closely related to the logarithmic
integral. See :func:`~mpmath.li` for additional information.
The exponential integral is related to the hyperbolic
and trigonometric integrals (see :func:`~mpmath.chi`, :func:`~mpmath.shi`,
:func:`~mpmath.ci`, :func:`~mpmath.si`) similarly to how the ordinary
exponential function is related to the hyperbolic and
trigonometric functions::
>>> mp.dps = 15
>>> ei(3)
9.93383257062542
>>> chi(3) + shi(3)
9.93383257062542
>>> chop(ci(3j) - j*si(3j) - pi*j/2)
9.93383257062542
Beware that logarithmic corrections, as in the last example
above, are required to obtain the correct branch in general.
For details, see [1].
The exponential integral is also a special case of the
hypergeometric function `\,_2F_2`::
>>> z = 0.6
>>> z*hyper([1,1],[2,2],z) + (ln(z)-ln(1/z))/2 + euler
0.769881289937359
>>> ei(z)
0.769881289937359
**References**
1. Relations between Ei and other functions:
http://functions.wolfram.com/GammaBetaErf/ExpIntegralEi/27/01/
2. Abramowitz & Stegun, section 5:
http://people.math.sfu.ca/~cbm/aands/page_228.htm
3. Asymptotic expansion for Ei:
http://mathworld.wolfram.com/En-Function.html
"""
li = r"""
Computes the logarithmic integral or li-function
`\mathrm{li}(x)`, defined by
.. math ::
\mathrm{li}(x) = \int_0^x \frac{1}{\log t} \, dt
The logarithmic integral has a singularity at `x = 1`.
Alternatively, ``li(x, offset=True)`` computes the offset
logarithmic integral (used in number theory)
.. math ::
\mathrm{Li}(x) = \int_2^x \frac{1}{\log t} \, dt.
These two functions are related via the simple identity
`\mathrm{Li}(x) = \mathrm{li}(x) - \mathrm{li}(2)`.
The logarithmic integral should also not be confused with
the polylogarithm (also denoted by Li), which is implemented
as :func:`~mpmath.polylog`.
**Examples**
Some basic values and limits::
>>> from mpmath import *
>>> mp.dps = 30; mp.pretty = True
>>> li(0)
0.0
>>> li(1)
-inf
>>> li(1)
-inf
>>> li(2)
1.04516378011749278484458888919
>>> findroot(li, 2)
1.45136923488338105028396848589
>>> li(inf)
+inf
>>> li(2, offset=True)
0.0
>>> li(1, offset=True)
-inf
>>> li(0, offset=True)
-1.04516378011749278484458888919
>>> li(10, offset=True)
5.12043572466980515267839286347
The logarithmic integral can be evaluated for arbitrary
complex arguments::
>>> mp.dps = 20
>>> li(3+4j)
(3.1343755504645775265 + 2.6769247817778742392j)
The logarithmic integral is related to the exponential integral::
>>> ei(log(3))
2.1635885946671919729
>>> li(3)
2.1635885946671919729
The logarithmic integral grows like `O(x/\log(x))`::
>>> mp.dps = 15
>>> x = 10**100
>>> x/log(x)
4.34294481903252e+97
>>> li(x)
4.3619719871407e+97
The prime number theorem states that the number of primes less
than `x` is asymptotic to `\mathrm{Li}(x)` (equivalently
`\mathrm{li}(x)`). For example, it is known that there are
exactly 1,925,320,391,606,803,968,923 prime numbers less than
`10^{23}` [1]. The logarithmic integral provides a very
accurate estimate::
>>> li(10**23, offset=True)
1.92532039161405e+21
A definite integral is::
>>> quad(li, [0, 1])
-0.693147180559945
>>> -ln(2)
-0.693147180559945
**References**
1. http://mathworld.wolfram.com/PrimeCountingFunction.html
2. http://mathworld.wolfram.com/LogarithmicIntegral.html
"""
ci = r"""
Computes the cosine integral,
.. math ::
\mathrm{Ci}(x) = -\int_x^{\infty} \frac{\cos t}{t}\,dt
= \gamma + \log x + \int_0^x \frac{\cos t - 1}{t}\,dt
**Examples**
Some values and limits::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> ci(0)
-inf
>>> ci(1)
0.3374039229009681346626462
>>> ci(pi)
0.07366791204642548599010096
>>> ci(inf)
0.0
>>> ci(-inf)
(0.0 + 3.141592653589793238462643j)
>>> ci(2+3j)
(1.408292501520849518759125 - 2.983617742029605093121118j)
The cosine integral behaves roughly like the sinc function
(see :func:`~mpmath.sinc`) for large real `x`::
>>> ci(10**10)
-4.875060251748226537857298e-11
>>> sinc(10**10)
-4.875060250875106915277943e-11
>>> chop(limit(ci, inf))
0.0
It has infinitely many roots on the positive real axis::
>>> findroot(ci, 1)
0.6165054856207162337971104
>>> findroot(ci, 2)
3.384180422551186426397851
Evaluation is supported for `z` anywhere in the complex plane::
>>> ci(10**6*(1+j))
(4.449410587611035724984376e+434287 + 9.75744874290013526417059e+434287j)
We can evaluate the defining integral as a reference::
>>> mp.dps = 15
>>> -quadosc(lambda t: cos(t)/t, [5, inf], omega=1)
-0.190029749656644
>>> ci(5)
-0.190029749656644
Some infinite series can be evaluated using the
cosine integral::
>>> nsum(lambda k: (-1)**k/(fac(2*k)*(2*k)), [1,inf])
-0.239811742000565
>>> ci(1) - euler
-0.239811742000565
"""
si = r"""
Computes the sine integral,
.. math ::
\mathrm{Si}(x) = \int_0^x \frac{\sin t}{t}\,dt.
The sine integral is thus the antiderivative of the sinc
function (see :func:`~mpmath.sinc`).
**Examples**
Some values and limits::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> si(0)
0.0
>>> si(1)
0.9460830703671830149413533
>>> si(-1)
-0.9460830703671830149413533
>>> si(pi)
1.851937051982466170361053
>>> si(inf)
1.570796326794896619231322
>>> si(-inf)
-1.570796326794896619231322
>>> si(2+3j)
(4.547513889562289219853204 + 1.399196580646054789459839j)
The sine integral approaches `\pi/2` for large real `x`::
>>> si(10**10)
1.570796326707584656968511
>>> pi/2
1.570796326794896619231322
Evaluation is supported for `z` anywhere in the complex plane::
>>> si(10**6*(1+j))
(-9.75744874290013526417059e+434287 + 4.449410587611035724984376e+434287j)
We can evaluate the defining integral as a reference::
>>> mp.dps = 15
>>> quad(sinc, [0, 5])
1.54993124494467
>>> si(5)
1.54993124494467
Some infinite series can be evaluated using the
sine integral::
>>> nsum(lambda k: (-1)**k/(fac(2*k+1)*(2*k+1)), [0,inf])
0.946083070367183
>>> si(1)
0.946083070367183
"""
chi = r"""
Computes the hyperbolic cosine integral, defined
in analogy with the cosine integral (see :func:`~mpmath.ci`) as
.. math ::
\mathrm{Chi}(x) = -\int_x^{\infty} \frac{\cosh t}{t}\,dt
= \gamma + \log x + \int_0^x \frac{\cosh t - 1}{t}\,dt
Some values and limits::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> chi(0)
-inf
>>> chi(1)
0.8378669409802082408946786
>>> chi(inf)
+inf
>>> findroot(chi, 0.5)
0.5238225713898644064509583
>>> chi(2+3j)
(-0.1683628683277204662429321 + 2.625115880451325002151688j)
Evaluation is supported for `z` anywhere in the complex plane::
>>> chi(10**6*(1+j))
(4.449410587611035724984376e+434287 - 9.75744874290013526417059e+434287j)
"""
shi = r"""
Computes the hyperbolic sine integral, defined
in analogy with the sine integral (see :func:`~mpmath.si`) as
.. math ::
\mathrm{Shi}(x) = \int_0^x \frac{\sinh t}{t}\,dt.
Some values and limits::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> shi(0)
0.0
>>> shi(1)
1.057250875375728514571842
>>> shi(-1)
-1.057250875375728514571842
>>> shi(inf)
+inf
>>> shi(2+3j)
(-0.1931890762719198291678095 + 2.645432555362369624818525j)
Evaluation is supported for `z` anywhere in the complex plane::
>>> shi(10**6*(1+j))
(4.449410587611035724984376e+434287 - 9.75744874290013526417059e+434287j)
"""
fresnels = r"""
Computes the Fresnel sine integral
.. math ::
S(x) = \int_0^x \sin\left(\frac{\pi t^2}{2}\right) \,dt
Note that some sources define this function
without the normalization factor `\pi/2`.
**Examples**
Some basic values and limits::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> fresnels(0)
0.0
>>> fresnels(inf)
0.5
>>> fresnels(-inf)
-0.5
>>> fresnels(1)
0.4382591473903547660767567
>>> fresnels(1+2j)
(36.72546488399143842838788 + 15.58775110440458732748279j)
Comparing with the definition::
>>> fresnels(3)
0.4963129989673750360976123
>>> quad(lambda t: sin(pi*t**2/2), [0,3])
0.4963129989673750360976123
"""
fresnelc = r"""
Computes the Fresnel cosine integral
.. math ::
C(x) = \int_0^x \cos\left(\frac{\pi t^2}{2}\right) \,dt
Note that some sources define this function
without the normalization factor `\pi/2`.
**Examples**
Some basic values and limits::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> fresnelc(0)
0.0
>>> fresnelc(inf)
0.5
>>> fresnelc(-inf)
-0.5
>>> fresnelc(1)
0.7798934003768228294742064
>>> fresnelc(1+2j)
(16.08787137412548041729489 - 36.22568799288165021578758j)
Comparing with the definition::
>>> fresnelc(3)
0.6057207892976856295561611
>>> quad(lambda t: cos(pi*t**2/2), [0,3])
0.6057207892976856295561611
"""
airyai = r"""
Computes the Airy function `\operatorname{Ai}(z)`, which is
the solution of the Airy differential equation `f''(z) - z f(z) = 0`
with initial conditions
.. math ::
\operatorname{Ai}(0) =
\frac{1}{3^{2/3}\Gamma\left(\frac{2}{3}\right)}
\operatorname{Ai}'(0) =
-\frac{1}{3^{1/3}\Gamma\left(\frac{1}{3}\right)}.
Other common ways of defining the Ai-function include
integrals such as
.. math ::
\operatorname{Ai}(x) = \frac{1}{\pi}
\int_0^{\infty} \cos\left(\frac{1}{3}t^3+xt\right) dt
\qquad x \in \mathbb{R}
\operatorname{Ai}(z) = \frac{\sqrt{3}}{2\pi}
\int_0^{\infty}
\exp\left(-\frac{t^3}{3}-\frac{z^3}{3t^3}\right) dt.
The Ai-function is an entire function with a turning point,
behaving roughly like a slowly decaying sine wave for `z < 0` and
like a rapidly decreasing exponential for `z > 0`.
A second solution of the Airy differential equation
is given by `\operatorname{Bi}(z)` (see :func:`~mpmath.airybi`).
Optionally, with *derivative=alpha*, :func:`airyai` can compute the
`\alpha`-th order fractional derivative with respect to `z`.
For `\alpha = n = 1,2,3,\ldots` this gives the derivative
`\operatorname{Ai}^{(n)}(z)`, and for `\alpha = -n = -1,-2,-3,\ldots`
this gives the `n`-fold iterated integral
.. math ::
f_0(z) = \operatorname{Ai}(z)
f_n(z) = \int_0^z f_{n-1}(t) dt.
The Ai-function has infinitely many zeros, all located along the
negative half of the real axis. They can be computed with
:func:`~mpmath.airyaizero`.
**Plots**
.. literalinclude :: /plots/ai.py
.. image :: /plots/ai.png
.. literalinclude :: /plots/ai_c.py
.. image :: /plots/ai_c.png
**Basic examples**
Limits and values include::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> airyai(0); 1/(power(3,'2/3')*gamma('2/3'))
0.3550280538878172392600632
0.3550280538878172392600632
>>> airyai(1)
0.1352924163128814155241474
>>> airyai(-1)
0.5355608832923521187995166
>>> airyai(inf); airyai(-inf)
0.0
0.0
Evaluation is supported for large magnitudes of the argument::
>>> airyai(-100)
0.1767533932395528780908311
>>> airyai(100)
2.634482152088184489550553e-291
>>> airyai(50+50j)
(-5.31790195707456404099817e-68 - 1.163588003770709748720107e-67j)
>>> airyai(-50+50j)
(1.041242537363167632587245e+158 + 3.347525544923600321838281e+157j)
Huge arguments are also fine::
>>> airyai(10**10)
1.162235978298741779953693e-289529654602171
>>> airyai(-10**10)
0.0001736206448152818510510181
>>> w = airyai(10**10*(1+j))
>>> w.real
5.711508683721355528322567e-186339621747698
>>> w.imag
1.867245506962312577848166e-186339621747697
The first root of the Ai-function is::
>>> findroot(airyai, -2)
-2.338107410459767038489197
>>> airyaizero(1)
-2.338107410459767038489197
**Properties and relations**
Verifying the Airy differential equation::
>>> for z in [-3.4, 0, 2.5, 1+2j]:
... chop(airyai(z,2) - z*airyai(z))
...
0.0
0.0
0.0
0.0
The first few terms of the Taylor series expansion around `z = 0`
(every third term is zero)::
>>> nprint(taylor(airyai, 0, 5))
[0.355028, -0.258819, 0.0, 0.0591713, -0.0215683, 0.0]
The Airy functions satisfy the Wronskian relation
`\operatorname{Ai}(z) \operatorname{Bi}'(z) -
\operatorname{Ai}'(z) \operatorname{Bi}(z) = 1/\pi`::
>>> z = -0.5
>>> airyai(z)*airybi(z,1) - airyai(z,1)*airybi(z)
0.3183098861837906715377675
>>> 1/pi
0.3183098861837906715377675
The Airy functions can be expressed in terms of Bessel
functions of order `\pm 1/3`. For `\Re[z] \le 0`, we have::
>>> z = -3
>>> airyai(z)
-0.3788142936776580743472439
>>> y = 2*power(-z,'3/2')/3
>>> (sqrt(-z) * (besselj('1/3',y) + besselj('-1/3',y)))/3
-0.3788142936776580743472439
**Derivatives and integrals**
Derivatives of the Ai-function (directly and using :func:`~mpmath.diff`)::
>>> airyai(-3,1); diff(airyai,-3)
0.3145837692165988136507873
0.3145837692165988136507873
>>> airyai(-3,2); diff(airyai,-3,2)
1.136442881032974223041732
1.136442881032974223041732
>>> airyai(1000,1); diff(airyai,1000)
-2.943133917910336090459748e-9156
-2.943133917910336090459748e-9156
Several derivatives at `z = 0`::
>>> airyai(0,0); airyai(0,1); airyai(0,2)
0.3550280538878172392600632
-0.2588194037928067984051836
0.0
>>> airyai(0,3); airyai(0,4); airyai(0,5)
0.3550280538878172392600632
-0.5176388075856135968103671
0.0
>>> airyai(0,15); airyai(0,16); airyai(0,17)
1292.30211615165475090663
-3188.655054727379756351861
0.0
The integral of the Ai-function::
>>> airyai(3,-1); quad(airyai, [0,3])
0.3299203760070217725002701
0.3299203760070217725002701
>>> airyai(-10,-1); quad(airyai, [0,-10])
-0.765698403134212917425148
-0.765698403134212917425148
Integrals of high or fractional order::
>>> airyai(-2,0.5); differint(airyai,-2,0.5,0)
(0.0 + 0.2453596101351438273844725j)
(0.0 + 0.2453596101351438273844725j)
>>> airyai(-2,-4); differint(airyai,-2,-4,0)
0.2939176441636809580339365
0.2939176441636809580339365
>>> airyai(0,-1); airyai(0,-2); airyai(0,-3)
0.0
0.0
0.0
Integrals of the Ai-function can be evaluated at limit points::
>>> airyai(-1000000,-1); airyai(-inf,-1)
-0.6666843728311539978751512
-0.6666666666666666666666667
>>> airyai(10,-1); airyai(+inf,-1)
0.3333333332991690159427932
0.3333333333333333333333333
>>> airyai(+inf,-2); airyai(+inf,-3)
+inf
+inf
>>> airyai(-1000000,-2); airyai(-inf,-2)
666666.4078472650651209742
+inf
>>> airyai(-1000000,-3); airyai(-inf,-3)
-333333074513.7520264995733
-inf
**References**
1. [DLMF]_ Chapter 9: Airy and Related Functions
2. [WolframFunctions]_ section: Bessel-Type Functions
"""
airybi = r"""
Computes the Airy function `\operatorname{Bi}(z)`, which is
the solution of the Airy differential equation `f''(z) - z f(z) = 0`
with initial conditions
.. math ::
\operatorname{Bi}(0) =
\frac{1}{3^{1/6}\Gamma\left(\frac{2}{3}\right)}
\operatorname{Bi}'(0) =
\frac{3^{1/6}}{\Gamma\left(\frac{1}{3}\right)}.
Like the Ai-function (see :func:`~mpmath.airyai`), the Bi-function
is oscillatory for `z < 0`, but it grows rather than decreases
for `z > 0`.
Optionally, as for :func:`~mpmath.airyai`, derivatives, integrals
and fractional derivatives can be computed with the *derivative*
parameter.
The Bi-function has infinitely many zeros along the negative
half-axis, as well as complex zeros, which can all be computed
with :func:`~mpmath.airybizero`.
**Plots**
.. literalinclude :: /plots/bi.py
.. image :: /plots/bi.png
.. literalinclude :: /plots/bi_c.py
.. image :: /plots/bi_c.png
**Basic examples**
Limits and values include::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> airybi(0); 1/(power(3,'1/6')*gamma('2/3'))
0.6149266274460007351509224
0.6149266274460007351509224
>>> airybi(1)
1.207423594952871259436379
>>> airybi(-1)
0.10399738949694461188869
>>> airybi(inf); airybi(-inf)
+inf
0.0
Evaluation is supported for large magnitudes of the argument::
>>> airybi(-100)
0.02427388768016013160566747
>>> airybi(100)
6.041223996670201399005265e+288
>>> airybi(50+50j)
(-5.322076267321435669290334e+63 + 1.478450291165243789749427e+65j)
>>> airybi(-50+50j)
(-3.347525544923600321838281e+157 + 1.041242537363167632587245e+158j)
Huge arguments::
>>> airybi(10**10)
1.369385787943539818688433e+289529654602165
>>> airybi(-10**10)
0.001775656141692932747610973
>>> w = airybi(10**10*(1+j))
>>> w.real
-6.559955931096196875845858e+186339621747689
>>> w.imag
-6.822462726981357180929024e+186339621747690
The first real root of the Bi-function is::
>>> findroot(airybi, -1); airybizero(1)
-1.17371322270912792491998
-1.17371322270912792491998
**Properties and relations**
Verifying the Airy differential equation::
>>> for z in [-3.4, 0, 2.5, 1+2j]:
... chop(airybi(z,2) - z*airybi(z))
...
0.0
0.0
0.0
0.0
The first few terms of the Taylor series expansion around `z = 0`
(every third term is zero)::
>>> nprint(taylor(airybi, 0, 5))
[0.614927, 0.448288, 0.0, 0.102488, 0.0373574, 0.0]
The Airy functions can be expressed in terms of Bessel
functions of order `\pm 1/3`. For `\Re[z] \le 0`, we have::
>>> z = -3
>>> airybi(z)
-0.1982896263749265432206449
>>> p = 2*power(-z,'3/2')/3
>>> sqrt(-mpf(z)/3)*(besselj('-1/3',p) - besselj('1/3',p))
-0.1982896263749265432206449
**Derivatives and integrals**
Derivatives of the Bi-function (directly and using :func:`~mpmath.diff`)::
>>> airybi(-3,1); diff(airybi,-3)
-0.675611222685258537668032
-0.675611222685258537668032
>>> airybi(-3,2); diff(airybi,-3,2)
0.5948688791247796296619346
0.5948688791247796296619346
>>> airybi(1000,1); diff(airybi,1000)
1.710055114624614989262335e+9156
1.710055114624614989262335e+9156
Several derivatives at `z = 0`::
>>> airybi(0,0); airybi(0,1); airybi(0,2)
0.6149266274460007351509224
0.4482883573538263579148237
0.0
>>> airybi(0,3); airybi(0,4); airybi(0,5)
0.6149266274460007351509224
0.8965767147076527158296474
0.0
>>> airybi(0,15); airybi(0,16); airybi(0,17)
2238.332923903442675949357
5522.912562599140729510628
0.0
The integral of the Bi-function::
>>> airybi(3,-1); quad(airybi, [0,3])
10.06200303130620056316655
10.06200303130620056316655
>>> airybi(-10,-1); quad(airybi, [0,-10])
-0.01504042480614002045135483
-0.01504042480614002045135483
Integrals of high or fractional order::
>>> airybi(-2,0.5); differint(airybi, -2, 0.5, 0)
(0.0 + 0.5019859055341699223453257j)
(0.0 + 0.5019859055341699223453257j)
>>> airybi(-2,-4); differint(airybi,-2,-4,0)
0.2809314599922447252139092
0.2809314599922447252139092
>>> airybi(0,-1); airybi(0,-2); airybi(0,-3)
0.0
0.0
0.0
Integrals of the Bi-function can be evaluated at limit points::
>>> airybi(-1000000,-1); airybi(-inf,-1)
0.000002191261128063434047966873
0.0
>>> airybi(10,-1); airybi(+inf,-1)
147809803.1074067161675853
+inf
>>> airybi(+inf,-2); airybi(+inf,-3)
+inf
+inf
>>> airybi(-1000000,-2); airybi(-inf,-2)
0.4482883750599908479851085
0.4482883573538263579148237
>>> gamma('2/3')*power(3,'2/3')/(2*pi)
0.4482883573538263579148237
>>> airybi(-100000,-3); airybi(-inf,-3)
-44828.52827206932872493133
-inf
>>> airybi(-100000,-4); airybi(-inf,-4)
2241411040.437759489540248
+inf
"""
airyaizero = r"""
Gives the `k`-th zero of the Airy Ai-function,
i.e. the `k`-th number `a_k` ordered by magnitude for which
`\operatorname{Ai}(a_k) = 0`.
Optionally, with *derivative=1*, the corresponding
zero `a'_k` of the derivative function, i.e.
`\operatorname{Ai}'(a'_k) = 0`, is computed.
**Examples**
Some values of `a_k`::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> airyaizero(1)
-2.338107410459767038489197
>>> airyaizero(2)
-4.087949444130970616636989
>>> airyaizero(3)
-5.520559828095551059129856
>>> airyaizero(1000)
-281.0315196125215528353364
Some values of `a'_k`::
>>> airyaizero(1,1)
-1.018792971647471089017325
>>> airyaizero(2,1)
-3.248197582179836537875424
>>> airyaizero(3,1)
-4.820099211178735639400616
>>> airyaizero(1000,1)
-280.9378080358935070607097
Verification::
>>> chop(airyai(airyaizero(1)))
0.0
>>> chop(airyai(airyaizero(1,1),1))
0.0
"""
airybizero = r"""
With *complex=False*, gives the `k`-th real zero of the Airy Bi-function,
i.e. the `k`-th number `b_k` ordered by magnitude for which
`\operatorname{Bi}(b_k) = 0`.
With *complex=True*, gives the `k`-th complex zero in the upper
half plane `\beta_k`. Also the conjugate `\overline{\beta_k}`
is a zero.
Optionally, with *derivative=1*, the corresponding
zero `b'_k` or `\beta'_k` of the derivative function, i.e.
`\operatorname{Bi}'(b'_k) = 0` or `\operatorname{Bi}'(\beta'_k) = 0`,
is computed.
**Examples**
Some values of `b_k`::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> airybizero(1)
-1.17371322270912792491998
>>> airybizero(2)
-3.271093302836352715680228
>>> airybizero(3)
-4.830737841662015932667709
>>> airybizero(1000)
-280.9378112034152401578834
Some values of `b_k`::
>>> airybizero(1,1)
-2.294439682614123246622459
>>> airybizero(2,1)
-4.073155089071828215552369
>>> airybizero(3,1)
-5.512395729663599496259593
>>> airybizero(1000,1)
-281.0315164471118527161362
Some values of `\beta_k`::
>>> airybizero(1,complex=True)
(0.9775448867316206859469927 + 2.141290706038744575749139j)
>>> airybizero(2,complex=True)
(1.896775013895336346627217 + 3.627291764358919410440499j)
>>> airybizero(3,complex=True)
(2.633157739354946595708019 + 4.855468179979844983174628j)
>>> airybizero(1000,complex=True)
(140.4978560578493018899793 + 243.3907724215792121244867j)
Some values of `\beta'_k`::
>>> airybizero(1,1,complex=True)
(0.2149470745374305676088329 + 1.100600143302797880647194j)
>>> airybizero(2,1,complex=True)
(1.458168309223507392028211 + 2.912249367458445419235083j)
>>> airybizero(3,1,complex=True)
(2.273760763013482299792362 + 4.254528549217097862167015j)
>>> airybizero(1000,1,complex=True)
(140.4509972835270559730423 + 243.3096175398562811896208j)
Verification::
>>> chop(airybi(airybizero(1)))
0.0
>>> chop(airybi(airybizero(1,1),1))
0.0
>>> u = airybizero(1,complex=True)
>>> chop(airybi(u))
0.0
>>> chop(airybi(conj(u)))
0.0
The complex zeros (in the upper and lower half-planes respectively)
asymptotically approach the rays `z = R \exp(\pm i \pi /3)`::
>>> arg(airybizero(1,complex=True))
1.142532510286334022305364
>>> arg(airybizero(1000,complex=True))
1.047271114786212061583917
>>> arg(airybizero(1000000,complex=True))
1.047197624741816183341355
>>> pi/3
1.047197551196597746154214
"""
ellipk = r"""
Evaluates the complete elliptic integral of the first kind,
`K(m)`, defined by
.. math ::
K(m) = \int_0^{\pi/2} \frac{dt}{\sqrt{1-m \sin^2 t}} \, = \,
\frac{\pi}{2} \,_2F_1\left(\frac{1}{2}, \frac{1}{2}, 1, m\right).
Note that the argument is the parameter `m = k^2`,
not the modulus `k` which is sometimes used.
**Plots**
.. literalinclude :: /plots/ellipk.py
.. image :: /plots/ellipk.png
**Examples**
Values and limits include::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> ellipk(0)
1.570796326794896619231322
>>> ellipk(inf)
(0.0 + 0.0j)
>>> ellipk(-inf)
0.0
>>> ellipk(1)
+inf
>>> ellipk(-1)
1.31102877714605990523242
>>> ellipk(2)
(1.31102877714605990523242 - 1.31102877714605990523242j)
Verifying the defining integral and hypergeometric
representation::
>>> ellipk(0.5)
1.85407467730137191843385
>>> quad(lambda t: (1-0.5*sin(t)**2)**-0.5, [0, pi/2])
1.85407467730137191843385
>>> pi/2*hyp2f1(0.5,0.5,1,0.5)
1.85407467730137191843385
Evaluation is supported for arbitrary complex `m`::
>>> ellipk(3+4j)
(0.9111955638049650086562171 + 0.6313342832413452438845091j)
A definite integral::
>>> quad(ellipk, [0, 1])
2.0
"""
agm = r"""
``agm(a, b)`` computes the arithmetic-geometric mean of `a` and
`b`, defined as the limit of the following iteration:
.. math ::
a_0 = a
b_0 = b
a_{n+1} = \frac{a_n+b_n}{2}
b_{n+1} = \sqrt{a_n b_n}
This function can be called with a single argument, computing
`\mathrm{agm}(a,1) = \mathrm{agm}(1,a)`.
**Examples**
It is a well-known theorem that the geometric mean of
two distinct positive numbers is less than the arithmetic
mean. It follows that the arithmetic-geometric mean lies
between the two means::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> a = mpf(3)
>>> b = mpf(4)
>>> sqrt(a*b)
3.46410161513775
>>> agm(a,b)
3.48202767635957
>>> (a+b)/2
3.5
The arithmetic-geometric mean is scale-invariant::
>>> agm(10*e, 10*pi)
29.261085515723
>>> 10*agm(e, pi)
29.261085515723
As an order-of-magnitude estimate, `\mathrm{agm}(1,x) \approx x`
for large `x`::
>>> agm(10**10)
643448704.760133
>>> agm(10**50)
1.34814309345871e+48
For tiny `x`, `\mathrm{agm}(1,x) \approx -\pi/(2 \log(x/4))`::
>>> agm('0.01')
0.262166887202249
>>> -pi/2/log('0.0025')
0.262172347753122
The arithmetic-geometric mean can also be computed for complex
numbers::
>>> agm(3, 2+j)
(2.51055133276184 + 0.547394054060638j)
The AGM iteration converges very quickly (each step doubles
the number of correct digits), so :func:`~mpmath.agm` supports efficient
high-precision evaluation::
>>> mp.dps = 10000
>>> a = agm(1,2)
>>> str(a)[-10:]
'1679581912'
**Mathematical relations**
The arithmetic-geometric mean may be used to evaluate the
following two parametric definite integrals:
.. math ::
I_1 = \int_0^{\infty}
\frac{1}{\sqrt{(x^2+a^2)(x^2+b^2)}} \,dx
I_2 = \int_0^{\pi/2}
\frac{1}{\sqrt{a^2 \cos^2(x) + b^2 \sin^2(x)}} \,dx
We have::
>>> mp.dps = 15
>>> a = 3
>>> b = 4
>>> f1 = lambda x: ((x**2+a**2)*(x**2+b**2))**-0.5
>>> f2 = lambda x: ((a*cos(x))**2 + (b*sin(x))**2)**-0.5
>>> quad(f1, [0, inf])
0.451115405388492
>>> quad(f2, [0, pi/2])
0.451115405388492
>>> pi/(2*agm(a,b))
0.451115405388492
A formula for `\Gamma(1/4)`::
>>> gamma(0.25)
3.62560990822191
>>> sqrt(2*sqrt(2*pi**3)/agm(1,sqrt(2)))
3.62560990822191
**Possible issues**
The branch cut chosen for complex `a` and `b` is somewhat
arbitrary.
"""
gegenbauer = r"""
Evaluates the Gegenbauer polynomial, or ultraspherical polynomial,
.. math ::
C_n^{(a)}(z) = {n+2a-1 \choose n} \,_2F_1\left(-n, n+2a;
a+\frac{1}{2}; \frac{1}{2}(1-z)\right).
When `n` is a nonnegative integer, this formula gives a polynomial
in `z` of degree `n`, but all parameters are permitted to be
complex numbers. With `a = 1/2`, the Gegenbauer polynomial
reduces to a Legendre polynomial.
**Examples**
Evaluation for arbitrary arguments::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> gegenbauer(3, 0.5, -10)
-2485.0
>>> gegenbauer(1000, 10, 100)
3.012757178975667428359374e+2322
>>> gegenbauer(2+3j, -0.75, -1000j)
(-5038991.358609026523401901 + 9414549.285447104177860806j)
Evaluation at negative integer orders::
>>> gegenbauer(-4, 2, 1.75)
-1.0
>>> gegenbauer(-4, 3, 1.75)
0.0
>>> gegenbauer(-4, 2j, 1.75)
0.0
>>> gegenbauer(-7, 0.5, 3)
8989.0
The Gegenbauer polynomials solve the differential equation::
>>> n, a = 4.5, 1+2j
>>> f = lambda z: gegenbauer(n, a, z)
>>> for z in [0, 0.75, -0.5j]:
... chop((1-z**2)*diff(f,z,2) - (2*a+1)*z*diff(f,z) + n*(n+2*a)*f(z))
...
0.0
0.0
0.0
The Gegenbauer polynomials have generating function
`(1-2zt+t^2)^{-a}`::
>>> a, z = 2.5, 1
>>> taylor(lambda t: (1-2*z*t+t**2)**(-a), 0, 3)
[1.0, 5.0, 15.0, 35.0]
>>> [gegenbauer(n,a,z) for n in range(4)]
[1.0, 5.0, 15.0, 35.0]
The Gegenbauer polynomials are orthogonal on `[-1, 1]` with respect
to the weight `(1-z^2)^{a-\frac{1}{2}}`::
>>> a, n, m = 2.5, 4, 5
>>> Cn = lambda z: gegenbauer(n, a, z, zeroprec=1000)
>>> Cm = lambda z: gegenbauer(m, a, z, zeroprec=1000)
>>> chop(quad(lambda z: Cn(z)*Cm(z)*(1-z**2)*(a-0.5), [-1, 1]))
0.0
"""
laguerre = r"""
Gives the generalized (associated) Laguerre polynomial, defined by
.. math ::
L_n^a(z) = \frac{\Gamma(n+b+1)}{\Gamma(b+1) \Gamma(n+1)}
\,_1F_1(-n, a+1, z).
With `a = 0` and `n` a nonnegative integer, this reduces to an ordinary
Laguerre polynomial, the sequence of which begins
`L_0(z) = 1, L_1(z) = 1-z, L_2(z) = z^2-2z+1, \ldots`.
The Laguerre polynomials are orthogonal with respect to the weight
`z^a e^{-z}` on `[0, \infty)`.
**Plots**
.. literalinclude :: /plots/laguerre.py
.. image :: /plots/laguerre.png
**Examples**
Evaluation for arbitrary arguments::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> laguerre(5, 0, 0.25)
0.03726399739583333333333333
>>> laguerre(1+j, 0.5, 2+3j)
(4.474921610704496808379097 - 11.02058050372068958069241j)
>>> laguerre(2, 0, 10000)
49980001.0
>>> laguerre(2.5, 0, 10000)
-9.327764910194842158583189e+4328
The first few Laguerre polynomials, normalized to have integer
coefficients::
>>> for n in range(7):
... chop(taylor(lambda z: fac(n)*laguerre(n, 0, z), 0, n))
...
[1.0]
[1.0, -1.0]
[2.0, -4.0, 1.0]
[6.0, -18.0, 9.0, -1.0]
[24.0, -96.0, 72.0, -16.0, 1.0]
[120.0, -600.0, 600.0, -200.0, 25.0, -1.0]
[720.0, -4320.0, 5400.0, -2400.0, 450.0, -36.0, 1.0]
Verifying orthogonality::
>>> Lm = lambda t: laguerre(m,a,t)
>>> Ln = lambda t: laguerre(n,a,t)
>>> a, n, m = 2.5, 2, 3
>>> chop(quad(lambda t: exp(-t)*t**a*Lm(t)*Ln(t), [0,inf]))
0.0
"""
hermite = r"""
Evaluates the Hermite polynomial `H_n(z)`, which may be defined using
the recurrence
.. math ::
H_0(z) = 1
H_1(z) = 2z
H_{n+1} = 2z H_n(z) - 2n H_{n-1}(z).
The Hermite polynomials are orthogonal on `(-\infty, \infty)` with
respect to the weight `e^{-z^2}`. More generally, allowing arbitrary complex
values of `n`, the Hermite function `H_n(z)` is defined as
.. math ::
H_n(z) = (2z)^n \,_2F_0\left(-\frac{n}{2}, \frac{1-n}{2},
-\frac{1}{z^2}\right)
for `\Re{z} > 0`, or generally
.. math ::
H_n(z) = 2^n \sqrt{\pi} \left(
\frac{1}{\Gamma\left(\frac{1-n}{2}\right)}
\,_1F_1\left(-\frac{n}{2}, \frac{1}{2}, z^2\right) -
\frac{2z}{\Gamma\left(-\frac{n}{2}\right)}
\,_1F_1\left(\frac{1-n}{2}, \frac{3}{2}, z^2\right)
\right).
**Plots**
.. literalinclude :: /plots/hermite.py
.. image :: /plots/hermite.png
**Examples**
Evaluation for arbitrary arguments::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> hermite(0, 10)
1.0
>>> hermite(1, 10); hermite(2, 10)
20.0
398.0
>>> hermite(10000, 2)
4.950440066552087387515653e+19334
>>> hermite(3, -10**8)
-7999999999999998800000000.0
>>> hermite(-3, -10**8)
1.675159751729877682920301e+4342944819032534
>>> hermite(2+3j, -1+2j)
(-0.07652130602993513389421901 - 0.1084662449961914580276007j)
Coefficients of the first few Hermite polynomials are::
>>> for n in range(7):
... chop(taylor(lambda z: hermite(n, z), 0, n))
...
[1.0]
[0.0, 2.0]
[-2.0, 0.0, 4.0]
[0.0, -12.0, 0.0, 8.0]
[12.0, 0.0, -48.0, 0.0, 16.0]
[0.0, 120.0, 0.0, -160.0, 0.0, 32.0]
[-120.0, 0.0, 720.0, 0.0, -480.0, 0.0, 64.0]
Values at `z = 0`::
>>> for n in range(-5, 9):
... hermite(n, 0)
...
0.02769459142039868792653387
0.08333333333333333333333333
0.2215567313631895034122709
0.5
0.8862269254527580136490837
1.0
0.0
-2.0
0.0
12.0
0.0
-120.0
0.0
1680.0
Hermite functions satisfy the differential equation::
>>> n = 4
>>> f = lambda z: hermite(n, z)
>>> z = 1.5
>>> chop(diff(f,z,2) - 2*z*diff(f,z) + 2*n*f(z))
0.0
Verifying orthogonality::
>>> chop(quad(lambda t: hermite(2,t)*hermite(4,t)*exp(-t**2), [-inf,inf]))
0.0
"""
jacobi = r"""
``jacobi(n, a, b, x)`` evaluates the Jacobi polynomial
`P_n^{(a,b)}(x)`. The Jacobi polynomials are a special
case of the hypergeometric function `\,_2F_1` given by:
.. math ::
P_n^{(a,b)}(x) = {n+a \choose n}
\,_2F_1\left(-n,1+a+b+n,a+1,\frac{1-x}{2}\right).
Note that this definition generalizes to nonintegral values
of `n`. When `n` is an integer, the hypergeometric series
terminates after a finite number of terms, giving
a polynomial in `x`.
**Evaluation of Jacobi polynomials**
A special evaluation is `P_n^{(a,b)}(1) = {n+a \choose n}`::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> jacobi(4, 0.5, 0.25, 1)
2.4609375
>>> binomial(4+0.5, 4)
2.4609375
A Jacobi polynomial of degree `n` is equal to its
Taylor polynomial of degree `n`. The explicit
coefficients of Jacobi polynomials can therefore
be recovered easily using :func:`~mpmath.taylor`::
>>> for n in range(5):
... nprint(taylor(lambda x: jacobi(n,1,2,x), 0, n))
...
[1.0]
[-0.5, 2.5]
[-0.75, -1.5, 5.25]
[0.5, -3.5, -3.5, 10.5]
[0.625, 2.5, -11.25, -7.5, 20.625]
For nonintegral `n`, the Jacobi "polynomial" is no longer
a polynomial::
>>> nprint(taylor(lambda x: jacobi(0.5,1,2,x), 0, 4))
[0.309983, 1.84119, -1.26933, 1.26699, -1.34808]
**Orthogonality**
The Jacobi polynomials are orthogonal on the interval
`[-1, 1]` with respect to the weight function
`w(x) = (1-x)^a (1+x)^b`. That is,
`w(x) P_n^{(a,b)}(x) P_m^{(a,b)}(x)` integrates to
zero if `m \ne n` and to a nonzero number if `m = n`.
The orthogonality is easy to verify using numerical
quadrature::
>>> P = jacobi
>>> f = lambda x: (1-x)**a * (1+x)**b * P(m,a,b,x) * P(n,a,b,x)
>>> a = 2
>>> b = 3
>>> m, n = 3, 4
>>> chop(quad(f, [-1, 1]), 1)
0.0
>>> m, n = 4, 4
>>> quad(f, [-1, 1])
1.9047619047619
**Differential equation**
The Jacobi polynomials are solutions of the differential
equation
.. math ::
(1-x^2) y'' + (b-a-(a+b+2)x) y' + n (n+a+b+1) y = 0.
We can verify that :func:`~mpmath.jacobi` approximately satisfies
this equation::
>>> from mpmath import *
>>> mp.dps = 15
>>> a = 2.5
>>> b = 4
>>> n = 3
>>> y = lambda x: jacobi(n,a,b,x)
>>> x = pi
>>> A0 = n*(n+a+b+1)*y(x)
>>> A1 = (b-a-(a+b+2)*x)*diff(y,x)
>>> A2 = (1-x**2)*diff(y,x,2)
>>> nprint(A2 + A1 + A0, 1)
4.0e-12
The difference of order `10^{-12}` is as close to zero as
it could be at 15-digit working precision, since the terms
are large::
>>> A0, A1, A2
(26560.2328981879, -21503.7641037294, -5056.46879445852)
"""
legendre = r"""
``legendre(n, x)`` evaluates the Legendre polynomial `P_n(x)`.
The Legendre polynomials are given by the formula
.. math ::
P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 -1)^n.
Alternatively, they can be computed recursively using
.. math ::
P_0(x) = 1
P_1(x) = x
(n+1) P_{n+1}(x) = (2n+1) x P_n(x) - n P_{n-1}(x).
A third definition is in terms of the hypergeometric function
`\,_2F_1`, whereby they can be generalized to arbitrary `n`:
.. math ::
P_n(x) = \,_2F_1\left(-n, n+1, 1, \frac{1-x}{2}\right)
**Plots**
.. literalinclude :: /plots/legendre.py
.. image :: /plots/legendre.png
**Basic evaluation**
The Legendre polynomials assume fixed values at the points
`x = -1` and `x = 1`::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> nprint([legendre(n, 1) for n in range(6)])
[1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
>>> nprint([legendre(n, -1) for n in range(6)])
[1.0, -1.0, 1.0, -1.0, 1.0, -1.0]
The coefficients of Legendre polynomials can be recovered
using degree-`n` Taylor expansion::
>>> for n in range(5):
... nprint(chop(taylor(lambda x: legendre(n, x), 0, n)))
...
[1.0]
[0.0, 1.0]
[-0.5, 0.0, 1.5]
[0.0, -1.5, 0.0, 2.5]
[0.375, 0.0, -3.75, 0.0, 4.375]
The roots of Legendre polynomials are located symmetrically
on the interval `[-1, 1]`::
>>> for n in range(5):
... nprint(polyroots(taylor(lambda x: legendre(n, x), 0, n)[::-1]))
...
[]
[0.0]
[-0.57735, 0.57735]
[-0.774597, 0.0, 0.774597]
[-0.861136, -0.339981, 0.339981, 0.861136]
An example of an evaluation for arbitrary `n`::
>>> legendre(0.75, 2+4j)
(1.94952805264875 + 2.1071073099422j)
**Orthogonality**
The Legendre polynomials are orthogonal on `[-1, 1]` with respect
to the trivial weight `w(x) = 1`. That is, `P_m(x) P_n(x)`
integrates to zero if `m \ne n` and to `2/(2n+1)` if `m = n`::
>>> m, n = 3, 4
>>> quad(lambda x: legendre(m,x)*legendre(n,x), [-1, 1])
0.0
>>> m, n = 4, 4
>>> quad(lambda x: legendre(m,x)*legendre(n,x), [-1, 1])
0.222222222222222
**Differential equation**
The Legendre polynomials satisfy the differential equation
.. math ::
((1-x^2) y')' + n(n+1) y' = 0.
We can verify this numerically::
>>> n = 3.6
>>> x = 0.73
>>> P = legendre
>>> A = diff(lambda t: (1-t**2)*diff(lambda u: P(n,u), t), x)
>>> B = n*(n+1)*P(n,x)
>>> nprint(A+B,1)
9.0e-16
"""
legenp = r"""
Calculates the (associated) Legendre function of the first kind of
degree *n* and order *m*, `P_n^m(z)`. Taking `m = 0` gives the ordinary
Legendre function of the first kind, `P_n(z)`. The parameters may be
complex numbers.
In terms of the Gauss hypergeometric function, the (associated) Legendre
function is defined as
.. math ::
P_n^m(z) = \frac{1}{\Gamma(1-m)} \frac{(1+z)^{m/2}}{(1-z)^{m/2}}
\,_2F_1\left(-n, n+1, 1-m, \frac{1-z}{2}\right).
With *type=3* instead of *type=2*, the alternative
definition
.. math ::
\hat{P}_n^m(z) = \frac{1}{\Gamma(1-m)} \frac{(z+1)^{m/2}}{(z-1)^{m/2}}
\,_2F_1\left(-n, n+1, 1-m, \frac{1-z}{2}\right).
is used. These functions correspond respectively to ``LegendreP[n,m,2,z]``
and ``LegendreP[n,m,3,z]`` in Mathematica.
The general solution of the (associated) Legendre differential equation
.. math ::
(1-z^2) f''(z) - 2zf'(z) + \left(n(n+1)-\frac{m^2}{1-z^2}\right)f(z) = 0
is given by `C_1 P_n^m(z) + C_2 Q_n^m(z)` for arbitrary constants
`C_1`, `C_2`, where `Q_n^m(z)` is a Legendre function of the
second kind as implemented by :func:`~mpmath.legenq`.
**Examples**
Evaluation for arbitrary parameters and arguments::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> legenp(2, 0, 10); legendre(2, 10)
149.5
149.5
>>> legenp(-2, 0.5, 2.5)
(1.972260393822275434196053 - 1.972260393822275434196053j)
>>> legenp(2+3j, 1-j, -0.5+4j)
(-3.335677248386698208736542 - 5.663270217461022307645625j)
>>> chop(legenp(3, 2, -1.5, type=2))
28.125
>>> chop(legenp(3, 2, -1.5, type=3))
-28.125
Verifying the associated Legendre differential equation::
>>> n, m = 2, -0.5
>>> C1, C2 = 1, -3
>>> f = lambda z: C1*legenp(n,m,z) + C2*legenq(n,m,z)
>>> deq = lambda z: (1-z**2)*diff(f,z,2) - 2*z*diff(f,z) + \
... (n*(n+1)-m**2/(1-z**2))*f(z)
>>> for z in [0, 2, -1.5, 0.5+2j]:
... chop(deq(mpmathify(z)))
...
0.0
0.0
0.0
0.0
"""
legenq = r"""
Calculates the (associated) Legendre function of the second kind of
degree *n* and order *m*, `Q_n^m(z)`. Taking `m = 0` gives the ordinary
Legendre function of the second kind, `Q_n(z)`. The parameters may be
complex numbers.
The Legendre functions of the second kind give a second set of
solutions to the (associated) Legendre differential equation.
(See :func:`~mpmath.legenp`.)
Unlike the Legendre functions of the first kind, they are not
polynomials of `z` for integer `n`, `m` but rational or logarithmic
functions with poles at `z = \pm 1`.
There are various ways to define Legendre functions of
the second kind, giving rise to different complex structure.
A version can be selected using the *type* keyword argument.
The *type=2* and *type=3* functions are given respectively by
.. math ::
Q_n^m(z) = \frac{\pi}{2 \sin(\pi m)}
\left( \cos(\pi m) P_n^m(z) -
\frac{\Gamma(1+m+n)}{\Gamma(1-m+n)} P_n^{-m}(z)\right)
\hat{Q}_n^m(z) = \frac{\pi}{2 \sin(\pi m)} e^{\pi i m}
\left( \hat{P}_n^m(z) -
\frac{\Gamma(1+m+n)}{\Gamma(1-m+n)} \hat{P}_n^{-m}(z)\right)
where `P` and `\hat{P}` are the *type=2* and *type=3* Legendre functions
of the first kind. The formulas above should be understood as limits
when `m` is an integer.
These functions correspond to ``LegendreQ[n,m,2,z]`` (or ``LegendreQ[n,m,z]``)
and ``LegendreQ[n,m,3,z]`` in Mathematica. The *type=3* function
is essentially the same as the function defined in
Abramowitz & Stegun (eq. 8.1.3) but with `(z+1)^{m/2}(z-1)^{m/2}` instead
of `(z^2-1)^{m/2}`, giving slightly different branches.
**Examples**
Evaluation for arbitrary parameters and arguments::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> legenq(2, 0, 0.5)
-0.8186632680417568557122028
>>> legenq(-1.5, -2, 2.5)
(0.6655964618250228714288277 + 0.3937692045497259717762649j)
>>> legenq(2-j, 3+4j, -6+5j)
(-10001.95256487468541686564 - 6011.691337610097577791134j)
Different versions of the function::
>>> legenq(2, 1, 0.5)
0.7298060598018049369381857
>>> legenq(2, 1, 1.5)
(-7.902916572420817192300921 + 0.1998650072605976600724502j)
>>> legenq(2, 1, 0.5, type=3)
(2.040524284763495081918338 - 0.7298060598018049369381857j)
>>> chop(legenq(2, 1, 1.5, type=3))
-0.1998650072605976600724502
"""
chebyt = r"""
``chebyt(n, x)`` evaluates the Chebyshev polynomial of the first
kind `T_n(x)`, defined by the identity
.. math ::
T_n(\cos x) = \cos(n x).
The Chebyshev polynomials of the first kind are a special
case of the Jacobi polynomials, and by extension of the
hypergeometric function `\,_2F_1`. They can thus also be
evaluated for nonintegral `n`.
**Plots**
.. literalinclude :: /plots/chebyt.py
.. image :: /plots/chebyt.png
**Basic evaluation**
The coefficients of the `n`-th polynomial can be recovered
using using degree-`n` Taylor expansion::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> for n in range(5):
... nprint(chop(taylor(lambda x: chebyt(n, x), 0, n)))
...
[1.0]
[0.0, 1.0]
[-1.0, 0.0, 2.0]
[0.0, -3.0, 0.0, 4.0]
[1.0, 0.0, -8.0, 0.0, 8.0]
**Orthogonality**
The Chebyshev polynomials of the first kind are orthogonal
on the interval `[-1, 1]` with respect to the weight
function `w(x) = 1/\sqrt{1-x^2}`::
>>> f = lambda x: chebyt(m,x)*chebyt(n,x)/sqrt(1-x**2)
>>> m, n = 3, 4
>>> nprint(quad(f, [-1, 1]),1)
0.0
>>> m, n = 4, 4
>>> quad(f, [-1, 1])
1.57079632596448
"""
chebyu = r"""
``chebyu(n, x)`` evaluates the Chebyshev polynomial of the second
kind `U_n(x)`, defined by the identity
.. math ::
U_n(\cos x) = \frac{\sin((n+1)x)}{\sin(x)}.
The Chebyshev polynomials of the second kind are a special
case of the Jacobi polynomials, and by extension of the
hypergeometric function `\,_2F_1`. They can thus also be
evaluated for nonintegral `n`.
**Plots**
.. literalinclude :: /plots/chebyu.py
.. image :: /plots/chebyu.png
**Basic evaluation**
The coefficients of the `n`-th polynomial can be recovered
using using degree-`n` Taylor expansion::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> for n in range(5):
... nprint(chop(taylor(lambda x: chebyu(n, x), 0, n)))
...
[1.0]
[0.0, 2.0]
[-1.0, 0.0, 4.0]
[0.0, -4.0, 0.0, 8.0]
[1.0, 0.0, -12.0, 0.0, 16.0]
**Orthogonality**
The Chebyshev polynomials of the second kind are orthogonal
on the interval `[-1, 1]` with respect to the weight
function `w(x) = \sqrt{1-x^2}`::
>>> f = lambda x: chebyu(m,x)*chebyu(n,x)*sqrt(1-x**2)
>>> m, n = 3, 4
>>> quad(f, [-1, 1])
0.0
>>> m, n = 4, 4
>>> quad(f, [-1, 1])
1.5707963267949
"""
besselj = r"""
``besselj(n, x, derivative=0)`` gives the Bessel function of the first kind
`J_n(x)`. Bessel functions of the first kind are defined as
solutions of the differential equation
.. math ::
x^2 y'' + x y' + (x^2 - n^2) y = 0
which appears, among other things, when solving the radial
part of Laplace's equation in cylindrical coordinates. This
equation has two solutions for given `n`, where the
`J_n`-function is the solution that is nonsingular at `x = 0`.
For positive integer `n`, `J_n(x)` behaves roughly like a sine
(odd `n`) or cosine (even `n`) multiplied by a magnitude factor
that decays slowly as `x \to \pm\infty`.
Generally, `J_n` is a special case of the hypergeometric
function `\,_0F_1`:
.. math ::
J_n(x) = \frac{x^n}{2^n \Gamma(n+1)}
\,_0F_1\left(n+1,-\frac{x^2}{4}\right)
With *derivative* = `m \ne 0`, the `m`-th derivative
.. math ::
\frac{d^m}{dx^m} J_n(x)
is computed.
**Plots**
.. literalinclude :: /plots/besselj.py
.. image :: /plots/besselj.png
.. literalinclude :: /plots/besselj_c.py
.. image :: /plots/besselj_c.png
**Examples**
Evaluation is supported for arbitrary arguments, and at
arbitrary precision::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> besselj(2, 1000)
-0.024777229528606
>>> besselj(4, 0.75)
0.000801070086542314
>>> besselj(2, 1000j)
(-2.48071721019185e+432 + 6.41567059811949e-437j)
>>> mp.dps = 25
>>> besselj(0.75j, 3+4j)
(-2.778118364828153309919653 - 1.5863603889018621585533j)
>>> mp.dps = 50
>>> besselj(1, pi)
0.28461534317975275734531059968613140570981118184947
Arguments may be large::
>>> mp.dps = 25
>>> besselj(0, 10000)
-0.007096160353388801477265164
>>> besselj(0, 10**10)
0.000002175591750246891726859055
>>> besselj(2, 10**100)
7.337048736538615712436929e-51
>>> besselj(2, 10**5*j)
(-3.540725411970948860173735e+43426 + 4.4949812409615803110051e-43433j)
The Bessel functions of the first kind satisfy simple
symmetries around `x = 0`::
>>> mp.dps = 15
>>> nprint([besselj(n,0) for n in range(5)])
[1.0, 0.0, 0.0, 0.0, 0.0]
>>> nprint([besselj(n,pi) for n in range(5)])
[-0.304242, 0.284615, 0.485434, 0.333458, 0.151425]
>>> nprint([besselj(n,-pi) for n in range(5)])
[-0.304242, -0.284615, 0.485434, -0.333458, 0.151425]
Roots of Bessel functions are often used::
>>> nprint([findroot(j0, k) for k in [2, 5, 8, 11, 14]])
[2.40483, 5.52008, 8.65373, 11.7915, 14.9309]
>>> nprint([findroot(j1, k) for k in [3, 7, 10, 13, 16]])
[3.83171, 7.01559, 10.1735, 13.3237, 16.4706]
The roots are not periodic, but the distance between successive
roots asymptotically approaches `2 \pi`. Bessel functions of
the first kind have the following normalization::
>>> quadosc(j0, [0, inf], period=2*pi)
1.0
>>> quadosc(j1, [0, inf], period=2*pi)
1.0
For `n = 1/2` or `n = -1/2`, the Bessel function reduces to a
trigonometric function::
>>> x = 10
>>> besselj(0.5, x), sqrt(2/(pi*x))*sin(x)
(-0.13726373575505, -0.13726373575505)
>>> besselj(-0.5, x), sqrt(2/(pi*x))*cos(x)
(-0.211708866331398, -0.211708866331398)
Derivatives of any order can be computed (negative orders
correspond to integration)::
>>> mp.dps = 25
>>> besselj(0, 7.5, 1)
-0.1352484275797055051822405
>>> diff(lambda x: besselj(0,x), 7.5)
-0.1352484275797055051822405
>>> besselj(0, 7.5, 10)
-0.1377811164763244890135677
>>> diff(lambda x: besselj(0,x), 7.5, 10)
-0.1377811164763244890135677
>>> besselj(0,7.5,-1) - besselj(0,3.5,-1)
-0.1241343240399987693521378
>>> quad(j0, [3.5, 7.5])
-0.1241343240399987693521378
Differentiation with a noninteger order gives the fractional derivative
in the sense of the Riemann-Liouville differintegral, as computed by
:func:`~mpmath.differint`::
>>> mp.dps = 15
>>> besselj(1, 3.5, 0.75)
-0.385977722939384
>>> differint(lambda x: besselj(1, x), 3.5, 0.75)
-0.385977722939384
"""
besseli = r"""
``besseli(n, x, derivative=0)`` gives the modified Bessel function of the
first kind,
.. math ::
I_n(x) = i^{-n} J_n(ix).
With *derivative* = `m \ne 0`, the `m`-th derivative
.. math ::
\frac{d^m}{dx^m} I_n(x)
is computed.
**Plots**
.. literalinclude :: /plots/besseli.py
.. image :: /plots/besseli.png
.. literalinclude :: /plots/besseli_c.py
.. image :: /plots/besseli_c.png
**Examples**
Some values of `I_n(x)`::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> besseli(0,0)
1.0
>>> besseli(1,0)
0.0
>>> besseli(0,1)
1.266065877752008335598245
>>> besseli(3.5, 2+3j)
(-0.2904369752642538144289025 - 0.4469098397654815837307006j)
Arguments may be large::
>>> besseli(2, 1000)
2.480717210191852440616782e+432
>>> besseli(2, 10**10)
4.299602851624027900335391e+4342944813
>>> besseli(2, 6000+10000j)
(-2.114650753239580827144204e+2603 + 4.385040221241629041351886e+2602j)
For integers `n`, the following integral representation holds::
>>> mp.dps = 15
>>> n = 3
>>> x = 2.3
>>> quad(lambda t: exp(x*cos(t))*cos(n*t), [0,pi])/pi
0.349223221159309
>>> besseli(n,x)
0.349223221159309
Derivatives and antiderivatives of any order can be computed::
>>> mp.dps = 25
>>> besseli(2, 7.5, 1)
195.8229038931399062565883
>>> diff(lambda x: besseli(2,x), 7.5)
195.8229038931399062565883
>>> besseli(2, 7.5, 10)
153.3296508971734525525176
>>> diff(lambda x: besseli(2,x), 7.5, 10)
153.3296508971734525525176
>>> besseli(2,7.5,-1) - besseli(2,3.5,-1)
202.5043900051930141956876
>>> quad(lambda x: besseli(2,x), [3.5, 7.5])
202.5043900051930141956876
"""
bessely = r"""
``bessely(n, x, derivative=0)`` gives the Bessel function of the second kind,
.. math ::
Y_n(x) = \frac{J_n(x) \cos(\pi n) - J_{-n}(x)}{\sin(\pi n)}.
For `n` an integer, this formula should be understood as a
limit. With *derivative* = `m \ne 0`, the `m`-th derivative
.. math ::
\frac{d^m}{dx^m} Y_n(x)
is computed.
**Plots**
.. literalinclude :: /plots/bessely.py
.. image :: /plots/bessely.png
.. literalinclude :: /plots/bessely_c.py
.. image :: /plots/bessely_c.png
**Examples**
Some values of `Y_n(x)`::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> bessely(0,0), bessely(1,0), bessely(2,0)
(-inf, -inf, -inf)
>>> bessely(1, pi)
0.3588729167767189594679827
>>> bessely(0.5, 3+4j)
(9.242861436961450520325216 - 3.085042824915332562522402j)
Arguments may be large::
>>> bessely(0, 10000)
0.00364780555898660588668872
>>> bessely(2.5, 10**50)
-4.8952500412050989295774e-26
>>> bessely(2.5, -10**50)
(0.0 + 4.8952500412050989295774e-26j)
Derivatives and antiderivatives of any order can be computed::
>>> bessely(2, 3.5, 1)
0.3842618820422660066089231
>>> diff(lambda x: bessely(2, x), 3.5)
0.3842618820422660066089231
>>> bessely(0.5, 3.5, 1)
-0.2066598304156764337900417
>>> diff(lambda x: bessely(0.5, x), 3.5)
-0.2066598304156764337900417
>>> diff(lambda x: bessely(2, x), 0.5, 10)
-208173867409.5547350101511
>>> bessely(2, 0.5, 10)
-208173867409.5547350101511
>>> bessely(2, 100.5, 100)
0.02668487547301372334849043
>>> quad(lambda x: bessely(2,x), [1,3])
-1.377046859093181969213262
>>> bessely(2,3,-1) - bessely(2,1,-1)
-1.377046859093181969213262
"""
besselk = r"""
``besselk(n, x)`` gives the modified Bessel function of the
second kind,
.. math ::
K_n(x) = \frac{\pi}{2} \frac{I_{-n}(x)-I_{n}(x)}{\sin(\pi n)}
For `n` an integer, this formula should be understood as a
limit.
**Plots**
.. literalinclude :: /plots/besselk.py
.. image :: /plots/besselk.png
.. literalinclude :: /plots/besselk_c.py
.. image :: /plots/besselk_c.png
**Examples**
Evaluation is supported for arbitrary complex arguments::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> besselk(0,1)
0.4210244382407083333356274
>>> besselk(0, -1)
(0.4210244382407083333356274 - 3.97746326050642263725661j)
>>> besselk(3.5, 2+3j)
(-0.02090732889633760668464128 + 0.2464022641351420167819697j)
>>> besselk(2+3j, 0.5)
(0.9615816021726349402626083 + 0.1918250181801757416908224j)
Arguments may be large::
>>> besselk(0, 100)
4.656628229175902018939005e-45
>>> besselk(1, 10**6)
4.131967049321725588398296e-434298
>>> besselk(1, 10**6*j)
(0.001140348428252385844876706 - 0.0005200017201681152909000961j)
>>> besselk(4.5, fmul(10**50, j, exact=True))
(1.561034538142413947789221e-26 + 1.243554598118700063281496e-25j)
The point `x = 0` is a singularity (logarithmic if `n = 0`)::
>>> besselk(0,0)
+inf
>>> besselk(1,0)
+inf
>>> for n in range(-4, 5):
... print(besselk(n, '1e-1000'))
...
4.8e+4001
8.0e+3000
2.0e+2000
1.0e+1000
2302.701024509704096466802
1.0e+1000
2.0e+2000
8.0e+3000
4.8e+4001
"""
hankel1 = r"""
``hankel1(n,x)`` computes the Hankel function of the first kind,
which is the complex combination of Bessel functions given by
.. math ::
H_n^{(1)}(x) = J_n(x) + i Y_n(x).
**Plots**
.. literalinclude :: /plots/hankel1.py
.. image :: /plots/hankel1.png
.. literalinclude :: /plots/hankel1_c.py
.. image :: /plots/hankel1_c.png
**Examples**
The Hankel function is generally complex-valued::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> hankel1(2, pi)
(0.4854339326315091097054957 - 0.0999007139290278787734903j)
>>> hankel1(3.5, pi)
(0.2340002029630507922628888 - 0.6419643823412927142424049j)
"""
hankel2 = r"""
``hankel2(n,x)`` computes the Hankel function of the second kind,
which is the complex combination of Bessel functions given by
.. math ::
H_n^{(2)}(x) = J_n(x) - i Y_n(x).
**Plots**
.. literalinclude :: /plots/hankel2.py
.. image :: /plots/hankel2.png
.. literalinclude :: /plots/hankel2_c.py
.. image :: /plots/hankel2_c.png
**Examples**
The Hankel function is generally complex-valued::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> hankel2(2, pi)
(0.4854339326315091097054957 + 0.0999007139290278787734903j)
>>> hankel2(3.5, pi)
(0.2340002029630507922628888 + 0.6419643823412927142424049j)
"""
lambertw = r"""
The Lambert W function `W(z)` is defined as the inverse function
of `w \exp(w)`. In other words, the value of `W(z)` is such that
`z = W(z) \exp(W(z))` for any complex number `z`.
The Lambert W function is a multivalued function with infinitely
many branches `W_k(z)`, indexed by `k \in \mathbb{Z}`. Each branch
gives a different solution `w` of the equation `z = w \exp(w)`.
All branches are supported by :func:`~mpmath.lambertw`:
* ``lambertw(z)`` gives the principal solution (branch 0)
* ``lambertw(z, k)`` gives the solution on branch `k`
The Lambert W function has two partially real branches: the
principal branch (`k = 0`) is real for real `z > -1/e`, and the
`k = -1` branch is real for `-1/e < z < 0`. All branches except
`k = 0` have a logarithmic singularity at `z = 0`.
The definition, implementation and choice of branches
is based on [Corless]_.
**Plots**
.. literalinclude :: /plots/lambertw.py
.. image :: /plots/lambertw.png
.. literalinclude :: /plots/lambertw_c.py
.. image :: /plots/lambertw_c.png
**Basic examples**
The Lambert W function is the inverse of `w \exp(w)`::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> w = lambertw(1)
>>> w
0.5671432904097838729999687
>>> w*exp(w)
1.0
Any branch gives a valid inverse::
>>> w = lambertw(1, k=3)
>>> w
(-2.853581755409037807206819 + 17.11353553941214591260783j)
>>> w = lambertw(1, k=25)
>>> w
(-5.047020464221569709378686 + 155.4763860949415867162066j)
>>> chop(w*exp(w))
1.0
**Applications to equation-solving**
The Lambert W function may be used to solve various kinds of
equations, such as finding the value of the infinite power
tower `z^{z^{z^{\ldots}}}`::
>>> def tower(z, n):
... if n == 0:
... return z
... return z ** tower(z, n-1)
...
>>> tower(mpf(0.5), 100)
0.6411857445049859844862005
>>> -lambertw(-log(0.5))/log(0.5)
0.6411857445049859844862005
**Properties**
The Lambert W function grows roughly like the natural logarithm
for large arguments::
>>> lambertw(1000); log(1000)
5.249602852401596227126056
6.907755278982137052053974
>>> lambertw(10**100); log(10**100)
224.8431064451185015393731
230.2585092994045684017991
The principal branch of the Lambert W function has a rational
Taylor series expansion around `z = 0`::
>>> nprint(taylor(lambertw, 0, 6), 10)
[0.0, 1.0, -1.0, 1.5, -2.666666667, 5.208333333, -10.8]
Some special values and limits are::
>>> lambertw(0)
0.0
>>> lambertw(1)
0.5671432904097838729999687
>>> lambertw(e)
1.0
>>> lambertw(inf)
+inf
>>> lambertw(0, k=-1)
-inf
>>> lambertw(0, k=3)
-inf
>>> lambertw(inf, k=2)
(+inf + 12.56637061435917295385057j)
>>> lambertw(inf, k=3)
(+inf + 18.84955592153875943077586j)
>>> lambertw(-inf, k=3)
(+inf + 21.9911485751285526692385j)
The `k = 0` and `k = -1` branches join at `z = -1/e` where
`W(z) = -1` for both branches. Since `-1/e` can only be represented
approximately with binary floating-point numbers, evaluating the
Lambert W function at this point only gives `-1` approximately::
>>> lambertw(-1/e, 0)
-0.9999999999998371330228251
>>> lambertw(-1/e, -1)
-1.000000000000162866977175
If `-1/e` happens to round in the negative direction, there might be
a small imaginary part::
>>> mp.dps = 15
>>> lambertw(-1/e)
(-1.0 + 8.22007971483662e-9j)
>>> lambertw(-1/e+eps)
-0.999999966242188
**References**
1. [Corless]_
"""
barnesg = r"""
Evaluates the Barnes G-function, which generalizes the
superfactorial (:func:`~mpmath.superfac`) and by extension also the
hyperfactorial (:func:`~mpmath.hyperfac`) to the complex numbers
in an analogous way to how the gamma function generalizes
the ordinary factorial.
The Barnes G-function may be defined in terms of a Weierstrass
product:
.. math ::
G(z+1) = (2\pi)^{z/2} e^{-[z(z+1)+\gamma z^2]/2}
\prod_{n=1}^\infty
\left[\left(1+\frac{z}{n}\right)^ne^{-z+z^2/(2n)}\right]
For positive integers `n`, we have have relation to superfactorials
`G(n) = \mathrm{sf}(n-2) = 0! \cdot 1! \cdots (n-2)!`.
**Examples**
Some elementary values and limits of the Barnes G-function::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> barnesg(1), barnesg(2), barnesg(3)
(1.0, 1.0, 1.0)
>>> barnesg(4)
2.0
>>> barnesg(5)
12.0
>>> barnesg(6)
288.0
>>> barnesg(7)
34560.0
>>> barnesg(8)
24883200.0
>>> barnesg(inf)
+inf
>>> barnesg(0), barnesg(-1), barnesg(-2)
(0.0, 0.0, 0.0)
Closed-form values are known for some rational arguments::
>>> barnesg('1/2')
0.603244281209446
>>> sqrt(exp(0.25+log(2)/12)/sqrt(pi)/glaisher**3)
0.603244281209446
>>> barnesg('1/4')
0.29375596533861
>>> nthroot(exp('3/8')/exp(catalan/pi)/
... gamma(0.25)**3/sqrt(glaisher)**9, 4)
0.29375596533861
The Barnes G-function satisfies the functional equation
`G(z+1) = \Gamma(z) G(z)`::
>>> z = pi
>>> barnesg(z+1)
2.39292119327948
>>> gamma(z)*barnesg(z)
2.39292119327948
The asymptotic growth rate of the Barnes G-function is related to
the Glaisher-Kinkelin constant::
>>> limit(lambda n: barnesg(n+1)/(n**(n**2/2-mpf(1)/12)*
... (2*pi)**(n/2)*exp(-3*n**2/4)), inf)
0.847536694177301
>>> exp('1/12')/glaisher
0.847536694177301
The Barnes G-function can be differentiated in closed form::
>>> z = 3
>>> diff(barnesg, z)
0.264507203401607
>>> barnesg(z)*((z-1)*psi(0,z)-z+(log(2*pi)+1)/2)
0.264507203401607
Evaluation is supported for arbitrary arguments and at arbitrary
precision::
>>> barnesg(6.5)
2548.7457695685
>>> barnesg(-pi)
0.00535976768353037
>>> barnesg(3+4j)
(-0.000676375932234244 - 4.42236140124728e-5j)
>>> mp.dps = 50
>>> barnesg(1/sqrt(2))
0.81305501090451340843586085064413533788206204124732
>>> q = barnesg(10j)
>>> q.real
0.000000000021852360840356557241543036724799812371995850552234
>>> q.imag
-0.00000000000070035335320062304849020654215545839053210041457588
>>> mp.dps = 15
>>> barnesg(100)
3.10361006263698e+6626
>>> barnesg(-101)
0.0
>>> barnesg(-10.5)
5.94463017605008e+25
>>> barnesg(-10000.5)
-6.14322868174828e+167480422
>>> barnesg(1000j)
(5.21133054865546e-1173597 + 4.27461836811016e-1173597j)
>>> barnesg(-1000+1000j)
(2.43114569750291e+1026623 + 2.24851410674842e+1026623j)
**References**
1. Whittaker & Watson, *A Course of Modern Analysis*,
Cambridge University Press, 4th edition (1927), p.264
2. http://en.wikipedia.org/wiki/Barnes_G-function
3. http://mathworld.wolfram.com/BarnesG-Function.html
"""
superfac = r"""
Computes the superfactorial, defined as the product of
consecutive factorials
.. math ::
\mathrm{sf}(n) = \prod_{k=1}^n k!
For general complex `z`, `\mathrm{sf}(z)` is defined
in terms of the Barnes G-function (see :func:`~mpmath.barnesg`).
**Examples**
The first few superfactorials are (OEIS A000178)::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> for n in range(10):
... print("%s %s" % (n, superfac(n)))
...
0 1.0
1 1.0
2 2.0
3 12.0
4 288.0
5 34560.0
6 24883200.0
7 125411328000.0
8 5.05658474496e+15
9 1.83493347225108e+21
Superfactorials grow very rapidly::
>>> superfac(1000)
3.24570818422368e+1177245
>>> superfac(10**10)
2.61398543581249e+467427913956904067453
Evaluation is supported for arbitrary arguments::
>>> mp.dps = 25
>>> superfac(pi)
17.20051550121297985285333
>>> superfac(2+3j)
(-0.005915485633199789627466468 + 0.008156449464604044948738263j)
>>> diff(superfac, 1)
0.2645072034016070205673056
**References**
1. http://oeis.org/A000178
"""
hyperfac = r"""
Computes the hyperfactorial, defined for integers as the product
.. math ::
H(n) = \prod_{k=1}^n k^k.
The hyperfactorial satisfies the recurrence formula `H(z) = z^z H(z-1)`.
It can be defined more generally in terms of the Barnes G-function (see
:func:`~mpmath.barnesg`) and the gamma function by the formula
.. math ::
H(z) = \frac{\Gamma(z+1)^z}{G(z)}.
The extension to complex numbers can also be done via
the integral representation
.. math ::
H(z) = (2\pi)^{-z/2} \exp \left[
{z+1 \choose 2} + \int_0^z \log(t!)\,dt
\right].
**Examples**
The rapidly-growing sequence of hyperfactorials begins
(OEIS A002109)::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> for n in range(10):
... print("%s %s" % (n, hyperfac(n)))
...
0 1.0
1 1.0
2 4.0
3 108.0
4 27648.0
5 86400000.0
6 4031078400000.0
7 3.3197663987712e+18
8 5.56964379417266e+25
9 2.15779412229419e+34
Some even larger hyperfactorials are::
>>> hyperfac(1000)
5.46458120882585e+1392926
>>> hyperfac(10**10)
4.60408207642219e+489142638002418704309
The hyperfactorial can be evaluated for arbitrary arguments::
>>> hyperfac(0.5)
0.880449235173423
>>> diff(hyperfac, 1)
0.581061466795327
>>> hyperfac(pi)
205.211134637462
>>> hyperfac(-10+1j)
(3.01144471378225e+46 - 2.45285242480185e+46j)
The recurrence property of the hyperfactorial holds
generally::
>>> z = 3-4*j
>>> hyperfac(z)
(-4.49795891462086e-7 - 6.33262283196162e-7j)
>>> z**z * hyperfac(z-1)
(-4.49795891462086e-7 - 6.33262283196162e-7j)
>>> z = mpf(-0.6)
>>> chop(z**z * hyperfac(z-1))
1.28170142849352
>>> hyperfac(z)
1.28170142849352
The hyperfactorial may also be computed using the integral
definition::
>>> z = 2.5
>>> hyperfac(z)
15.9842119922237
>>> (2*pi)**(-z/2)*exp(binomial(z+1,2) +
... quad(lambda t: loggamma(t+1), [0, z]))
15.9842119922237
:func:`~mpmath.hyperfac` supports arbitrary-precision evaluation::
>>> mp.dps = 50
>>> hyperfac(10)
215779412229418562091680268288000000000000000.0
>>> hyperfac(1/sqrt(2))
0.89404818005227001975423476035729076375705084390942
**References**
1. http://oeis.org/A002109
2. http://mathworld.wolfram.com/Hyperfactorial.html
"""
rgamma = r"""
Computes the reciprocal of the gamma function, `1/\Gamma(z)`. This
function evaluates to zero at the poles
of the gamma function, `z = 0, -1, -2, \ldots`.
**Examples**
Basic examples::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> rgamma(1)
1.0
>>> rgamma(4)
0.1666666666666666666666667
>>> rgamma(0); rgamma(-1)
0.0
0.0
>>> rgamma(1000)
2.485168143266784862783596e-2565
>>> rgamma(inf)
0.0
A definite integral that can be evaluated in terms of elementary
integrals::
>>> quad(rgamma, [0,inf])
2.807770242028519365221501
>>> e + quad(lambda t: exp(-t)/(pi**2+log(t)**2), [0,inf])
2.807770242028519365221501
"""
loggamma = r"""
Computes the principal branch of the log-gamma function,
`\ln \Gamma(z)`. Unlike `\ln(\Gamma(z))`, which has infinitely many
complex branch cuts, the principal log-gamma function only has a single
branch cut along the negative half-axis. The principal branch
continuously matches the asymptotic Stirling expansion
.. math ::
\ln \Gamma(z) \sim \frac{\ln(2 \pi)}{2} +
\left(z-\frac{1}{2}\right) \ln(z) - z + O(z^{-1}).
The real parts of both functions agree, but their imaginary
parts generally differ by `2 n \pi` for some `n \in \mathbb{Z}`.
They coincide for `z \in \mathbb{R}, z > 0`.
Computationally, it is advantageous to use :func:`~mpmath.loggamma`
instead of :func:`~mpmath.gamma` for extremely large arguments.
**Examples**
Comparing with `\ln(\Gamma(z))`::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> loggamma('13.2'); log(gamma('13.2'))
20.49400419456603678498394
20.49400419456603678498394
>>> loggamma(3+4j)
(-1.756626784603784110530604 + 4.742664438034657928194889j)
>>> log(gamma(3+4j))
(-1.756626784603784110530604 - 1.540520869144928548730397j)
>>> log(gamma(3+4j)) + 2*pi*j
(-1.756626784603784110530604 + 4.742664438034657928194889j)
Note the imaginary parts for negative arguments::
>>> loggamma(-0.5); loggamma(-1.5); loggamma(-2.5)
(1.265512123484645396488946 - 3.141592653589793238462643j)
(0.8600470153764810145109327 - 6.283185307179586476925287j)
(-0.05624371649767405067259453 - 9.42477796076937971538793j)
Some special values::
>>> loggamma(1); loggamma(2)
0.0
0.0
>>> loggamma(3); +ln2
0.6931471805599453094172321
0.6931471805599453094172321
>>> loggamma(3.5); log(15*sqrt(pi)/8)
1.200973602347074224816022
1.200973602347074224816022
>>> loggamma(inf)
+inf
Huge arguments are permitted::
>>> loggamma('1e30')
6.807755278982137052053974e+31
>>> loggamma('1e300')
6.897755278982137052053974e+302
>>> loggamma('1e3000')
6.906755278982137052053974e+3003
>>> loggamma('1e100000000000000000000')
2.302585092994045684007991e+100000000000000000020
>>> loggamma('1e30j')
(-1.570796326794896619231322e+30 + 6.807755278982137052053974e+31j)
>>> loggamma('1e300j')
(-1.570796326794896619231322e+300 + 6.897755278982137052053974e+302j)
>>> loggamma('1e3000j')
(-1.570796326794896619231322e+3000 + 6.906755278982137052053974e+3003j)
The log-gamma function can be integrated analytically
on any interval of unit length::
>>> z = 0
>>> quad(loggamma, [z,z+1]); log(2*pi)/2
0.9189385332046727417803297
0.9189385332046727417803297
>>> z = 3+4j
>>> quad(loggamma, [z,z+1]); (log(z)-1)*z + log(2*pi)/2
(-0.9619286014994750641314421 + 5.219637303741238195688575j)
(-0.9619286014994750641314421 + 5.219637303741238195688575j)
The derivatives of the log-gamma function are given by the
polygamma function (:func:`~mpmath.psi`)::
>>> diff(loggamma, -4+3j); psi(0, -4+3j)
(1.688493531222971393607153 + 2.554898911356806978892748j)
(1.688493531222971393607153 + 2.554898911356806978892748j)
>>> diff(loggamma, -4+3j, 2); psi(1, -4+3j)
(-0.1539414829219882371561038 - 0.1020485197430267719746479j)
(-0.1539414829219882371561038 - 0.1020485197430267719746479j)
The log-gamma function satisfies an additive form of the
recurrence relation for the ordinary gamma function::
>>> z = 2+3j
>>> loggamma(z); loggamma(z+1) - log(z)
(-2.092851753092733349564189 + 2.302396543466867626153708j)
(-2.092851753092733349564189 + 2.302396543466867626153708j)
"""
siegeltheta = r"""
Computes the Riemann-Siegel theta function,
.. math ::
\theta(t) = \frac{
\log\Gamma\left(\frac{1+2it}{4}\right) -
\log\Gamma\left(\frac{1-2it}{4}\right)
}{2i} - \frac{\log \pi}{2} t.
The Riemann-Siegel theta function is important in
providing the phase factor for the Z-function
(see :func:`~mpmath.siegelz`). Evaluation is supported for real and
complex arguments::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> siegeltheta(0)
0.0
>>> siegeltheta(inf)
+inf
>>> siegeltheta(-inf)
-inf
>>> siegeltheta(1)
-1.767547952812290388302216
>>> siegeltheta(10+0.25j)
(-3.068638039426838572528867 + 0.05804937947429712998395177j)
Arbitrary derivatives may be computed with derivative = k
>>> siegeltheta(1234, derivative=2)
0.0004051864079114053109473741
>>> diff(siegeltheta, 1234, n=2)
0.0004051864079114053109473741
The Riemann-Siegel theta function has odd symmetry around `t = 0`,
two local extreme points and three real roots including 0 (located
symmetrically)::
>>> nprint(chop(taylor(siegeltheta, 0, 5)))
[0.0, -2.68609, 0.0, 2.69433, 0.0, -6.40218]
>>> findroot(diffun(siegeltheta), 7)
6.28983598883690277966509
>>> findroot(siegeltheta, 20)
17.84559954041086081682634
For large `t`, there is a famous asymptotic formula
for `\theta(t)`, to first order given by::
>>> t = mpf(10**6)
>>> siegeltheta(t)
5488816.353078403444882823
>>> -t*log(2*pi/t)/2-t/2
5488816.745777464310273645
"""
grampoint = r"""
Gives the `n`-th Gram point `g_n`, defined as the solution
to the equation `\theta(g_n) = \pi n` where `\theta(t)`
is the Riemann-Siegel theta function (:func:`~mpmath.siegeltheta`).
The first few Gram points are::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> grampoint(0)
17.84559954041086081682634
>>> grampoint(1)
23.17028270124630927899664
>>> grampoint(2)
27.67018221781633796093849
>>> grampoint(3)
31.71797995476405317955149
Checking the definition::
>>> siegeltheta(grampoint(3))
9.42477796076937971538793
>>> 3*pi
9.42477796076937971538793
A large Gram point::
>>> grampoint(10**10)
3293531632.728335454561153
Gram points are useful when studying the Z-function
(:func:`~mpmath.siegelz`). See the documentation of that function
for additional examples.
:func:`~mpmath.grampoint` can solve the defining equation for
nonintegral `n`. There is a fixed point where `g(x) = x`::
>>> findroot(lambda x: grampoint(x) - x, 10000)
9146.698193171459265866198
**References**
1. http://mathworld.wolfram.com/GramPoint.html
"""
siegelz = r"""
Computes the Z-function, also known as the Riemann-Siegel Z function,
.. math ::
Z(t) = e^{i \theta(t)} \zeta(1/2+it)
where `\zeta(s)` is the Riemann zeta function (:func:`~mpmath.zeta`)
and where `\theta(t)` denotes the Riemann-Siegel theta function
(see :func:`~mpmath.siegeltheta`).
Evaluation is supported for real and complex arguments::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> siegelz(1)
-0.7363054628673177346778998
>>> siegelz(3+4j)
(-0.1852895764366314976003936 - 0.2773099198055652246992479j)
The first four derivatives are supported, using the
optional *derivative* keyword argument::
>>> siegelz(1234567, derivative=3)
56.89689348495089294249178
>>> diff(siegelz, 1234567, n=3)
56.89689348495089294249178
The Z-function has a Maclaurin expansion::
>>> nprint(chop(taylor(siegelz, 0, 4)))
[-1.46035, 0.0, 2.73588, 0.0, -8.39357]
The Z-function `Z(t)` is equal to `\pm |\zeta(s)|` on the
critical line `s = 1/2+it` (i.e. for real arguments `t`
to `Z`). Its zeros coincide with those of the Riemann zeta
function::
>>> findroot(siegelz, 14)
14.13472514173469379045725
>>> findroot(siegelz, 20)
21.02203963877155499262848
>>> findroot(zeta, 0.5+14j)
(0.5 + 14.13472514173469379045725j)
>>> findroot(zeta, 0.5+20j)
(0.5 + 21.02203963877155499262848j)
Since the Z-function is real-valued on the critical line
(and unlike `|\zeta(s)|` analytic), it is useful for
investigating the zeros of the Riemann zeta function.
For example, one can use a root-finding algorithm based
on sign changes::
>>> findroot(siegelz, [100, 200], solver='bisect')
176.4414342977104188888926
To locate roots, Gram points `g_n` which can be computed
by :func:`~mpmath.grampoint` are useful. If `(-1)^n Z(g_n)` is
positive for two consecutive `n`, then `Z(t)` must have
a zero between those points::
>>> g10 = grampoint(10)
>>> g11 = grampoint(11)
>>> (-1)**10 * siegelz(g10) > 0
True
>>> (-1)**11 * siegelz(g11) > 0
True
>>> findroot(siegelz, [g10, g11], solver='bisect')
56.44624769706339480436776
>>> g10, g11
(54.67523744685325626632663, 57.54516517954725443703014)
"""
riemannr = r"""
Evaluates the Riemann R function, a smooth approximation of the
prime counting function `\pi(x)` (see :func:`~mpmath.primepi`). The Riemann
R function gives a fast numerical approximation useful e.g. to
roughly estimate the number of primes in a given interval.
The Riemann R function is computed using the rapidly convergent Gram
series,
.. math ::
R(x) = 1 + \sum_{k=1}^{\infty}
\frac{\log^k x}{k k! \zeta(k+1)}.
From the Gram series, one sees that the Riemann R function is a
well-defined analytic function (except for a branch cut along
the negative real half-axis); it can be evaluated for arbitrary
real or complex arguments.
The Riemann R function gives a very accurate approximation
of the prime counting function. For example, it is wrong by at
most 2 for `x < 1000`, and for `x = 10^9` differs from the exact
value of `\pi(x)` by 79, or less than two parts in a million.
It is about 10 times more accurate than the logarithmic integral
estimate (see :func:`~mpmath.li`), which however is even faster to evaluate.
It is orders of magnitude more accurate than the extremely
fast `x/\log x` estimate.
**Examples**
For small arguments, the Riemann R function almost exactly
gives the prime counting function if rounded to the nearest
integer::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> primepi(50), riemannr(50)
(15, 14.9757023241462)
>>> max(abs(primepi(n)-int(round(riemannr(n)))) for n in range(100))
1
>>> max(abs(primepi(n)-int(round(riemannr(n)))) for n in range(300))
2
The Riemann R function can be evaluated for arguments far too large
for exact determination of `\pi(x)` to be computationally
feasible with any presently known algorithm::
>>> riemannr(10**30)
1.46923988977204e+28
>>> riemannr(10**100)
4.3619719871407e+97
>>> riemannr(10**1000)
4.3448325764012e+996
A comparison of the Riemann R function and logarithmic integral estimates
for `\pi(x)` using exact values of `\pi(10^n)` up to `n = 9`.
The fractional error is shown in parentheses::
>>> exact = [4,25,168,1229,9592,78498,664579,5761455,50847534]
>>> for n, p in enumerate(exact):
... n += 1
... r, l = riemannr(10**n), li(10**n)
... rerr, lerr = nstr((r-p)/p,3), nstr((l-p)/p,3)
... print("%i %i %s(%s) %s(%s)" % (n, p, r, rerr, l, lerr))
...
1 4 4.56458314100509(0.141) 6.1655995047873(0.541)
2 25 25.6616332669242(0.0265) 30.1261415840796(0.205)
3 168 168.359446281167(0.00214) 177.609657990152(0.0572)
4 1229 1226.93121834343(-0.00168) 1246.13721589939(0.0139)
5 9592 9587.43173884197(-0.000476) 9629.8090010508(0.00394)
6 78498 78527.3994291277(0.000375) 78627.5491594622(0.00165)
7 664579 664667.447564748(0.000133) 664918.405048569(0.000511)
8 5761455 5761551.86732017(1.68e-5) 5762209.37544803(0.000131)
9 50847534 50847455.4277214(-1.55e-6) 50849234.9570018(3.35e-5)
The derivative of the Riemann R function gives the approximate
probability for a number of magnitude `x` to be prime::
>>> diff(riemannr, 1000)
0.141903028110784
>>> mpf(primepi(1050) - primepi(950)) / 100
0.15
Evaluation is supported for arbitrary arguments and at arbitrary
precision::
>>> mp.dps = 30
>>> riemannr(7.5)
3.72934743264966261918857135136
>>> riemannr(-4+2j)
(-0.551002208155486427591793957644 + 2.16966398138119450043195899746j)
"""
primepi = r"""
Evaluates the prime counting function, `\pi(x)`, which gives
the number of primes less than or equal to `x`. The argument
`x` may be fractional.
The prime counting function is very expensive to evaluate
precisely for large `x`, and the present implementation is
not optimized in any way. For numerical approximation of the
prime counting function, it is better to use :func:`~mpmath.primepi2`
or :func:`~mpmath.riemannr`.
Some values of the prime counting function::
>>> from mpmath import *
>>> [primepi(k) for k in range(20)]
[0, 0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8]
>>> primepi(3.5)
2
>>> primepi(100000)
9592
"""
primepi2 = r"""
Returns an interval (as an ``mpi`` instance) providing bounds
for the value of the prime counting function `\pi(x)`. For small
`x`, :func:`~mpmath.primepi2` returns an exact interval based on
the output of :func:`~mpmath.primepi`. For `x > 2656`, a loose interval
based on Schoenfeld's inequality
.. math ::
|\pi(x) - \mathrm{li}(x)| < \frac{\sqrt x \log x}{8 \pi}
is returned. This estimate is rigorous assuming the truth of
the Riemann hypothesis, and can be computed very quickly.
**Examples**
Exact values of the prime counting function for small `x`::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> iv.dps = 15; iv.pretty = True
>>> primepi2(10)
[4.0, 4.0]
>>> primepi2(100)
[25.0, 25.0]
>>> primepi2(1000)
[168.0, 168.0]
Loose intervals are generated for moderately large `x`:
>>> primepi2(10000), primepi(10000)
([1209.0, 1283.0], 1229)
>>> primepi2(50000), primepi(50000)
([5070.0, 5263.0], 5133)
As `x` increases, the absolute error gets worse while the relative
error improves. The exact value of `\pi(10^{23})` is
1925320391606803968923, and :func:`~mpmath.primepi2` gives 9 significant
digits::
>>> p = primepi2(10**23)
>>> p
[1.9253203909477020467e+21, 1.925320392280406229e+21]
>>> mpf(p.delta) / mpf(p.a)
6.9219865355293e-10
A more precise, nonrigorous estimate for `\pi(x)` can be
obtained using the Riemann R function (:func:`~mpmath.riemannr`).
For large enough `x`, the value returned by :func:`~mpmath.primepi2`
essentially amounts to a small perturbation of the value returned by
:func:`~mpmath.riemannr`::
>>> primepi2(10**100)
[4.3619719871407024816e+97, 4.3619719871407032404e+97]
>>> riemannr(10**100)
4.3619719871407e+97
"""
primezeta = r"""
Computes the prime zeta function, which is defined
in analogy with the Riemann zeta function (:func:`~mpmath.zeta`)
as
.. math ::
P(s) = \sum_p \frac{1}{p^s}
where the sum is taken over all prime numbers `p`. Although
this sum only converges for `\mathrm{Re}(s) > 1`, the
function is defined by analytic continuation in the
half-plane `\mathrm{Re}(s) > 0`.
**Examples**
Arbitrary-precision evaluation for real and complex arguments is
supported::
>>> from mpmath import *
>>> mp.dps = 30; mp.pretty = True
>>> primezeta(2)
0.452247420041065498506543364832
>>> primezeta(pi)
0.15483752698840284272036497397
>>> mp.dps = 50
>>> primezeta(3)
0.17476263929944353642311331466570670097541212192615
>>> mp.dps = 20
>>> primezeta(3+4j)
(-0.12085382601645763295 - 0.013370403397787023602j)
The prime zeta function has a logarithmic pole at `s = 1`,
with residue equal to the difference of the Mertens and
Euler constants::
>>> primezeta(1)
+inf
>>> extradps(25)(lambda x: primezeta(1+x)+log(x))(+eps)
-0.31571845205389007685
>>> mertens-euler
-0.31571845205389007685
The analytic continuation to `0 < \mathrm{Re}(s) \le 1`
is implemented. In this strip the function exhibits
very complex behavior; on the unit interval, it has poles at
`1/n` for every squarefree integer `n`::
>>> primezeta(0.5) # Pole at s = 1/2
(-inf + 3.1415926535897932385j)
>>> primezeta(0.25)
(-1.0416106801757269036 + 0.52359877559829887308j)
>>> primezeta(0.5+10j)
(0.54892423556409790529 + 0.45626803423487934264j)
Although evaluation works in principle for any `\mathrm{Re}(s) > 0`,
it should be noted that the evaluation time increases exponentially
as `s` approaches the imaginary axis.
For large `\mathrm{Re}(s)`, `P(s)` is asymptotic to `2^{-s}`::
>>> primezeta(inf)
0.0
>>> primezeta(10), mpf(2)**-10
(0.00099360357443698021786, 0.0009765625)
>>> primezeta(1000)
9.3326361850321887899e-302
>>> primezeta(1000+1000j)
(-3.8565440833654995949e-302 - 8.4985390447553234305e-302j)
**References**
Carl-Erik Froberg, "On the prime zeta function",
BIT 8 (1968), pp. 187-202.
"""
bernpoly = r"""
Evaluates the Bernoulli polynomial `B_n(z)`.
The first few Bernoulli polynomials are::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> for n in range(6):
... nprint(chop(taylor(lambda x: bernpoly(n,x), 0, n)))
...
[1.0]
[-0.5, 1.0]
[0.166667, -1.0, 1.0]
[0.0, 0.5, -1.5, 1.0]
[-0.0333333, 0.0, 1.0, -2.0, 1.0]
[0.0, -0.166667, 0.0, 1.66667, -2.5, 1.0]
At `z = 0`, the Bernoulli polynomial evaluates to a
Bernoulli number (see :func:`~mpmath.bernoulli`)::
>>> bernpoly(12, 0), bernoulli(12)
(-0.253113553113553, -0.253113553113553)
>>> bernpoly(13, 0), bernoulli(13)
(0.0, 0.0)
Evaluation is accurate for large `n` and small `z`::
>>> mp.dps = 25
>>> bernpoly(100, 0.5)
2.838224957069370695926416e+78
>>> bernpoly(1000, 10.5)
5.318704469415522036482914e+1769
"""
polylog = r"""
Computes the polylogarithm, defined by the sum
.. math ::
\mathrm{Li}_s(z) = \sum_{k=1}^{\infty} \frac{z^k}{k^s}.
This series is convergent only for `|z| < 1`, so elsewhere
the analytic continuation is implied.
The polylogarithm should not be confused with the logarithmic
integral (also denoted by Li or li), which is implemented
as :func:`~mpmath.li`.
**Examples**
The polylogarithm satisfies a huge number of functional identities.
A sample of polylogarithm evaluations is shown below::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> polylog(1,0.5), log(2)
(0.693147180559945, 0.693147180559945)
>>> polylog(2,0.5), (pi**2-6*log(2)**2)/12
(0.582240526465012, 0.582240526465012)
>>> polylog(2,-phi), -log(phi)**2-pi**2/10
(-1.21852526068613, -1.21852526068613)
>>> polylog(3,0.5), 7*zeta(3)/8-pi**2*log(2)/12+log(2)**3/6
(0.53721319360804, 0.53721319360804)
:func:`~mpmath.polylog` can evaluate the analytic continuation of the
polylogarithm when `s` is an integer::
>>> polylog(2, 10)
(0.536301287357863 - 7.23378441241546j)
>>> polylog(2, -10)
-4.1982778868581
>>> polylog(2, 10j)
(-3.05968879432873 + 3.71678149306807j)
>>> polylog(-2, 10)
-0.150891632373114
>>> polylog(-2, -10)
0.067618332081142
>>> polylog(-2, 10j)
(0.0384353698579347 + 0.0912451798066779j)
Some more examples, with arguments on the unit circle (note that
the series definition cannot be used for computation here)::
>>> polylog(2,j)
(-0.205616758356028 + 0.915965594177219j)
>>> j*catalan-pi**2/48
(-0.205616758356028 + 0.915965594177219j)
>>> polylog(3,exp(2*pi*j/3))
(-0.534247512515375 + 0.765587078525922j)
>>> -4*zeta(3)/9 + 2*j*pi**3/81
(-0.534247512515375 + 0.765587078525921j)
Polylogarithms of different order are related by integration
and differentiation::
>>> s, z = 3, 0.5
>>> polylog(s+1, z)
0.517479061673899
>>> quad(lambda t: polylog(s,t)/t, [0, z])
0.517479061673899
>>> z*diff(lambda t: polylog(s+2,t), z)
0.517479061673899
Taylor series expansions around `z = 0` are::
>>> for n in range(-3, 4):
... nprint(taylor(lambda x: polylog(n,x), 0, 5))
...
[0.0, 1.0, 8.0, 27.0, 64.0, 125.0]
[0.0, 1.0, 4.0, 9.0, 16.0, 25.0]
[0.0, 1.0, 2.0, 3.0, 4.0, 5.0]
[0.0, 1.0, 1.0, 1.0, 1.0, 1.0]
[0.0, 1.0, 0.5, 0.333333, 0.25, 0.2]
[0.0, 1.0, 0.25, 0.111111, 0.0625, 0.04]
[0.0, 1.0, 0.125, 0.037037, 0.015625, 0.008]
The series defining the polylogarithm is simultaneously
a Taylor series and an L-series. For certain values of `z`, the
polylogarithm reduces to a pure zeta function::
>>> polylog(pi, 1), zeta(pi)
(1.17624173838258, 1.17624173838258)
>>> polylog(pi, -1), -altzeta(pi)
(-0.909670702980385, -0.909670702980385)
Evaluation for arbitrary, nonintegral `s` is supported
for `z` within the unit circle:
>>> polylog(3+4j, 0.25)
(0.24258605789446 - 0.00222938275488344j)
>>> nsum(lambda k: 0.25**k / k**(3+4j), [1,inf])
(0.24258605789446 - 0.00222938275488344j)
It is also currently supported outside of the unit circle for `z`
not too large in magnitude::
>>> polylog(1+j, 20+40j)
(-7.1421172179728 - 3.92726697721369j)
>>> polylog(1+j, 200+400j)
Traceback (most recent call last):
...
NotImplementedError: polylog for arbitrary s and z
**References**
1. Richard Crandall, "Note on fast polylogarithm computation"
http://people.reed.edu/~crandall/papers/Polylog.pdf
2. http://en.wikipedia.org/wiki/Polylogarithm
3. http://mathworld.wolfram.com/Polylogarithm.html
"""
bell = r"""
For `n` a nonnegative integer, ``bell(n,x)`` evaluates the Bell
polynomial `B_n(x)`, the first few of which are
.. math ::
B_0(x) = 1
B_1(x) = x
B_2(x) = x^2+x
B_3(x) = x^3+3x^2+x
If `x = 1` or :func:`~mpmath.bell` is called with only one argument, it
gives the `n`-th Bell number `B_n`, which is the number of
partitions of a set with `n` elements. By setting the precision to
at least `\log_{10} B_n` digits, :func:`~mpmath.bell` provides fast
calculation of exact Bell numbers.
In general, :func:`~mpmath.bell` computes
.. math ::
B_n(x) = e^{-x} \left(\mathrm{sinc}(\pi n) + E_n(x)\right)
where `E_n(x)` is the generalized exponential function implemented
by :func:`~mpmath.polyexp`. This is an extension of Dobinski's formula [1],
where the modification is the sinc term ensuring that `B_n(x)` is
continuous in `n`; :func:`~mpmath.bell` can thus be evaluated,
differentiated, etc for arbitrary complex arguments.
**Examples**
Simple evaluations::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> bell(0, 2.5)
1.0
>>> bell(1, 2.5)
2.5
>>> bell(2, 2.5)
8.75
Evaluation for arbitrary complex arguments::
>>> bell(5.75+1j, 2-3j)
(-10767.71345136587098445143 - 15449.55065599872579097221j)
The first few Bell polynomials::
>>> for k in range(7):
... nprint(taylor(lambda x: bell(k,x), 0, k))
...
[1.0]
[0.0, 1.0]
[0.0, 1.0, 1.0]
[0.0, 1.0, 3.0, 1.0]
[0.0, 1.0, 7.0, 6.0, 1.0]
[0.0, 1.0, 15.0, 25.0, 10.0, 1.0]
[0.0, 1.0, 31.0, 90.0, 65.0, 15.0, 1.0]
The first few Bell numbers and complementary Bell numbers::
>>> [int(bell(k)) for k in range(10)]
[1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147]
>>> [int(bell(k,-1)) for k in range(10)]
[1, -1, 0, 1, 1, -2, -9, -9, 50, 267]
Large Bell numbers::
>>> mp.dps = 50
>>> bell(50)
185724268771078270438257767181908917499221852770.0
>>> bell(50,-1)
-29113173035759403920216141265491160286912.0
Some even larger values::
>>> mp.dps = 25
>>> bell(1000,-1)
-1.237132026969293954162816e+1869
>>> bell(1000)
2.989901335682408421480422e+1927
>>> bell(1000,2)
6.591553486811969380442171e+1987
>>> bell(1000,100.5)
9.101014101401543575679639e+2529
A determinant identity satisfied by Bell numbers::
>>> mp.dps = 15
>>> N = 8
>>> det([[bell(k+j) for j in range(N)] for k in range(N)])
125411328000.0
>>> superfac(N-1)
125411328000.0
**References**
1. http://mathworld.wolfram.com/DobinskisFormula.html
"""
polyexp = r"""
Evaluates the polyexponential function, defined for arbitrary
complex `s`, `z` by the series
.. math ::
E_s(z) = \sum_{k=1}^{\infty} \frac{k^s}{k!} z^k.
`E_s(z)` is constructed from the exponential function analogously
to how the polylogarithm is constructed from the ordinary
logarithm; as a function of `s` (with `z` fixed), `E_s` is an L-series
It is an entire function of both `s` and `z`.
The polyexponential function provides a generalization of the
Bell polynomials `B_n(x)` (see :func:`~mpmath.bell`) to noninteger orders `n`.
In terms of the Bell polynomials,
.. math ::
E_s(z) = e^z B_s(z) - \mathrm{sinc}(\pi s).
Note that `B_n(x)` and `e^{-x} E_n(x)` are identical if `n`
is a nonzero integer, but not otherwise. In particular, they differ
at `n = 0`.
**Examples**
Evaluating a series::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> nsum(lambda k: sqrt(k)/fac(k), [1,inf])
2.101755547733791780315904
>>> polyexp(0.5,1)
2.101755547733791780315904
Evaluation for arbitrary arguments::
>>> polyexp(-3-4j, 2.5+2j)
(2.351660261190434618268706 + 1.202966666673054671364215j)
Evaluation is accurate for tiny function values::
>>> polyexp(4, -100)
3.499471750566824369520223e-36
If `n` is a nonpositive integer, `E_n` reduces to a special
instance of the hypergeometric function `\,_pF_q`::
>>> n = 3
>>> x = pi
>>> polyexp(-n,x)
4.042192318847986561771779
>>> x*hyper([1]*(n+1), [2]*(n+1), x)
4.042192318847986561771779
"""
cyclotomic = r"""
Evaluates the cyclotomic polynomial `\Phi_n(x)`, defined by
.. math ::
\Phi_n(x) = \prod_{\zeta} (x - \zeta)
where `\zeta` ranges over all primitive `n`-th roots of unity
(see :func:`~mpmath.unitroots`). An equivalent representation, used
for computation, is
.. math ::
\Phi_n(x) = \prod_{d\mid n}(x^d-1)^{\mu(n/d)} = \Phi_n(x)
where `\mu(m)` denotes the Moebius function. The cyclotomic
polynomials are integer polynomials, the first of which can be
written explicitly as
.. math ::
\Phi_0(x) = 1
\Phi_1(x) = x - 1
\Phi_2(x) = x + 1
\Phi_3(x) = x^3 + x^2 + 1
\Phi_4(x) = x^2 + 1
\Phi_5(x) = x^4 + x^3 + x^2 + x + 1
\Phi_6(x) = x^2 - x + 1
**Examples**
The coefficients of low-order cyclotomic polynomials can be recovered
using Taylor expansion::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> for n in range(9):
... p = chop(taylor(lambda x: cyclotomic(n,x), 0, 10))
... print("%s %s" % (n, nstr(p[:10+1-p[::-1].index(1)])))
...
0 [1.0]
1 [-1.0, 1.0]
2 [1.0, 1.0]
3 [1.0, 1.0, 1.0]
4 [1.0, 0.0, 1.0]
5 [1.0, 1.0, 1.0, 1.0, 1.0]
6 [1.0, -1.0, 1.0]
7 [1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
8 [1.0, 0.0, 0.0, 0.0, 1.0]
The definition as a product over primitive roots may be checked
by computing the product explicitly (for a real argument, this
method will generally introduce numerical noise in the imaginary
part)::
>>> mp.dps = 25
>>> z = 3+4j
>>> cyclotomic(10, z)
(-419.0 - 360.0j)
>>> fprod(z-r for r in unitroots(10, primitive=True))
(-419.0 - 360.0j)
>>> z = 3
>>> cyclotomic(10, z)
61.0
>>> fprod(z-r for r in unitroots(10, primitive=True))
(61.0 - 3.146045605088568607055454e-25j)
Up to permutation, the roots of a given cyclotomic polynomial
can be checked to agree with the list of primitive roots::
>>> p = taylor(lambda x: cyclotomic(6,x), 0, 6)[:3]
>>> for r in polyroots(p[::-1]):
... print(r)
...
(0.5 - 0.8660254037844386467637232j)
(0.5 + 0.8660254037844386467637232j)
>>>
>>> for r in unitroots(6, primitive=True):
... print(r)
...
(0.5 + 0.8660254037844386467637232j)
(0.5 - 0.8660254037844386467637232j)
"""
meijerg = r"""
Evaluates the Meijer G-function, defined as
.. math ::
G^{m,n}_{p,q} \left( \left. \begin{matrix}
a_1, \dots, a_n ; a_{n+1} \dots a_p \\
b_1, \dots, b_m ; b_{m+1} \dots b_q
\end{matrix}\; \right| \; z ; r \right) =
\frac{1}{2 \pi i} \int_L
\frac{\prod_{j=1}^m \Gamma(b_j+s) \prod_{j=1}^n\Gamma(1-a_j-s)}
{\prod_{j=n+1}^{p}\Gamma(a_j+s) \prod_{j=m+1}^q \Gamma(1-b_j-s)}
z^{-s/r} ds
for an appropriate choice of the contour `L` (see references).
There are `p` elements `a_j`.
The argument *a_s* should be a pair of lists, the first containing the
`n` elements `a_1, \ldots, a_n` and the second containing
the `p-n` elements `a_{n+1}, \ldots a_p`.
There are `q` elements `b_j`.
The argument *b_s* should be a pair of lists, the first containing the
`m` elements `b_1, \ldots, b_m` and the second containing
the `q-m` elements `b_{m+1}, \ldots b_q`.
The implicit tuple `(m, n, p, q)` constitutes the order or degree of the
Meijer G-function, and is determined by the lengths of the coefficient
vectors. Confusingly, the indices in this tuple appear in a different order
from the coefficients, but this notation is standard. The many examples
given below should hopefully clear up any potential confusion.
**Algorithm**
The Meijer G-function is evaluated as a combination of hypergeometric series.
There are two versions of the function, which can be selected with
the optional *series* argument.
*series=1* uses a sum of `m` `\,_pF_{q-1}` functions of `z`
*series=2* uses a sum of `n` `\,_qF_{p-1}` functions of `1/z`
The default series is chosen based on the degree and `|z|` in order
to be consistent with Mathematica's. This definition of the Meijer G-function
has a discontinuity at `|z| = 1` for some orders, which can
be avoided by explicitly specifying a series.
Keyword arguments are forwarded to :func:`~mpmath.hypercomb`.
**Examples**
Many standard functions are special cases of the Meijer G-function
(possibly rescaled and/or with branch cut corrections). We define
some test parameters::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> a = mpf(0.75)
>>> b = mpf(1.5)
>>> z = mpf(2.25)
The exponential function:
`e^z = G^{1,0}_{0,1} \left( \left. \begin{matrix} - \\ 0 \end{matrix} \;
\right| \; -z \right)`
>>> meijerg([[],[]], [[0],[]], -z)
9.487735836358525720550369
>>> exp(z)
9.487735836358525720550369
The natural logarithm:
`\log(1+z) = G^{1,2}_{2,2} \left( \left. \begin{matrix} 1, 1 \\ 1, 0
\end{matrix} \; \right| \; -z \right)`
>>> meijerg([[1,1],[]], [[1],[0]], z)
1.178654996341646117219023
>>> log(1+z)
1.178654996341646117219023
A rational function:
`\frac{z}{z+1} = G^{1,2}_{2,2} \left( \left. \begin{matrix} 1, 1 \\ 1, 1
\end{matrix} \; \right| \; z \right)`
>>> meijerg([[1,1],[]], [[1],[1]], z)
0.6923076923076923076923077
>>> z/(z+1)
0.6923076923076923076923077
The sine and cosine functions:
`\frac{1}{\sqrt \pi} \sin(2 \sqrt z) = G^{1,0}_{0,2} \left( \left. \begin{matrix}
- \\ \frac{1}{2}, 0 \end{matrix} \; \right| \; z \right)`
`\frac{1}{\sqrt \pi} \cos(2 \sqrt z) = G^{1,0}_{0,2} \left( \left. \begin{matrix}
- \\ 0, \frac{1}{2} \end{matrix} \; \right| \; z \right)`
>>> meijerg([[],[]], [[0.5],[0]], (z/2)**2)
0.4389807929218676682296453
>>> sin(z)/sqrt(pi)
0.4389807929218676682296453
>>> meijerg([[],[]], [[0],[0.5]], (z/2)**2)
-0.3544090145996275423331762
>>> cos(z)/sqrt(pi)
-0.3544090145996275423331762
Bessel functions:
`J_a(2 \sqrt z) = G^{1,0}_{0,2} \left( \left.
\begin{matrix} - \\ \frac{a}{2}, -\frac{a}{2}
\end{matrix} \; \right| \; z \right)`
`Y_a(2 \sqrt z) = G^{2,0}_{1,3} \left( \left.
\begin{matrix} \frac{-a-1}{2} \\ \frac{a}{2}, -\frac{a}{2}, \frac{-a-1}{2}
\end{matrix} \; \right| \; z \right)`
`(-z)^{a/2} z^{-a/2} I_a(2 \sqrt z) = G^{1,0}_{0,2} \left( \left.
\begin{matrix} - \\ \frac{a}{2}, -\frac{a}{2}
\end{matrix} \; \right| \; -z \right)`
`2 K_a(2 \sqrt z) = G^{2,0}_{0,2} \left( \left.
\begin{matrix} - \\ \frac{a}{2}, -\frac{a}{2}
\end{matrix} \; \right| \; z \right)`
As the example with the Bessel *I* function shows, a branch
factor is required for some arguments when inverting the square root.
>>> meijerg([[],[]], [[a/2],[-a/2]], (z/2)**2)
0.5059425789597154858527264
>>> besselj(a,z)
0.5059425789597154858527264
>>> meijerg([[],[(-a-1)/2]], [[a/2,-a/2],[(-a-1)/2]], (z/2)**2)
0.1853868950066556941442559
>>> bessely(a, z)
0.1853868950066556941442559
>>> meijerg([[],[]], [[a/2],[-a/2]], -(z/2)**2)
(0.8685913322427653875717476 + 2.096964974460199200551738j)
>>> (-z)**(a/2) / z**(a/2) * besseli(a, z)
(0.8685913322427653875717476 + 2.096964974460199200551738j)
>>> 0.5*meijerg([[],[]], [[a/2,-a/2],[]], (z/2)**2)
0.09334163695597828403796071
>>> besselk(a,z)
0.09334163695597828403796071
Error functions:
`\sqrt{\pi} z^{2(a-1)} \mathrm{erfc}(z) = G^{2,0}_{1,2} \left( \left.
\begin{matrix} a \\ a-1, a-\frac{1}{2}
\end{matrix} \; \right| \; z, \frac{1}{2} \right)`
>>> meijerg([[],[a]], [[a-1,a-0.5],[]], z, 0.5)
0.00172839843123091957468712
>>> sqrt(pi) * z**(2*a-2) * erfc(z)
0.00172839843123091957468712
A Meijer G-function of higher degree, (1,1,2,3):
>>> meijerg([[a],[b]], [[a],[b,a-1]], z)
1.55984467443050210115617
>>> sin((b-a)*pi)/pi*(exp(z)-1)*z**(a-1)
1.55984467443050210115617
A Meijer G-function of still higher degree, (4,1,2,4), that can
be expanded as a messy combination of exponential integrals:
>>> meijerg([[a],[2*b-a]], [[b,a,b-0.5,-1-a+2*b],[]], z)
0.3323667133658557271898061
>>> chop(4**(a-b+1)*sqrt(pi)*gamma(2*b-2*a)*z**a*\
... expint(2*b-2*a, -2*sqrt(-z))*expint(2*b-2*a, 2*sqrt(-z)))
0.3323667133658557271898061
In the following case, different series give different values::
>>> chop(meijerg([[1],[0.25]],[[3],[0.5]],-2))
-0.06417628097442437076207337
>>> meijerg([[1],[0.25]],[[3],[0.5]],-2,series=1)
0.1428699426155117511873047
>>> chop(meijerg([[1],[0.25]],[[3],[0.5]],-2,series=2))
-0.06417628097442437076207337
**References**
1. http://en.wikipedia.org/wiki/Meijer_G-function
2. http://mathworld.wolfram.com/MeijerG-Function.html
3. http://functions.wolfram.com/HypergeometricFunctions/MeijerG/
4. http://functions.wolfram.com/HypergeometricFunctions/MeijerG1/
"""
clsin = r"""
Computes the Clausen sine function, defined formally by the series
.. math ::
\mathrm{Cl}_s(z) = \sum_{k=1}^{\infty} \frac{\sin(kz)}{k^s}.
The special case `\mathrm{Cl}_2(z)` (i.e. ``clsin(2,z)``) is the classical
"Clausen function". More generally, the Clausen function is defined for
complex `s` and `z`, even when the series does not converge. The
Clausen function is related to the polylogarithm (:func:`~mpmath.polylog`) as
.. math ::
\mathrm{Cl}_s(z) = \frac{1}{2i}\left(\mathrm{Li}_s\left(e^{iz}\right) -
\mathrm{Li}_s\left(e^{-iz}\right)\right)
= \mathrm{Im}\left[\mathrm{Li}_s(e^{iz})\right] \quad (s, z \in \mathbb{R}),
and this representation can be taken to provide the analytic continuation of the
series. The complementary function :func:`~mpmath.clcos` gives the corresponding
cosine sum.
**Examples**
Evaluation for arbitrarily chosen `s` and `z`::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> s, z = 3, 4
>>> clsin(s, z); nsum(lambda k: sin(z*k)/k**s, [1,inf])
-0.6533010136329338746275795
-0.6533010136329338746275795
Using `z + \pi` instead of `z` gives an alternating series::
>>> clsin(s, z+pi)
0.8860032351260589402871624
>>> nsum(lambda k: (-1)**k*sin(z*k)/k**s, [1,inf])
0.8860032351260589402871624
With `s = 1`, the sum can be expressed in closed form
using elementary functions::
>>> z = 1 + sqrt(3)
>>> clsin(1, z)
0.2047709230104579724675985
>>> chop((log(1-exp(-j*z)) - log(1-exp(j*z)))/(2*j))
0.2047709230104579724675985
>>> nsum(lambda k: sin(k*z)/k, [1,inf])
0.2047709230104579724675985
The classical Clausen function `\mathrm{Cl}_2(\theta)` gives the
value of the integral `\int_0^{\theta} -\ln(2\sin(x/2)) dx` for
`0 < \theta < 2 \pi`::
>>> cl2 = lambda t: clsin(2, t)
>>> cl2(3.5)
-0.2465045302347694216534255
>>> -quad(lambda x: ln(2*sin(0.5*x)), [0, 3.5])
-0.2465045302347694216534255
This function is symmetric about `\theta = \pi` with zeros and extreme
points::
>>> cl2(0); cl2(pi/3); chop(cl2(pi)); cl2(5*pi/3); chop(cl2(2*pi))
0.0
1.014941606409653625021203
0.0
-1.014941606409653625021203
0.0
Catalan's constant is a special value::
>>> cl2(pi/2)
0.9159655941772190150546035
>>> +catalan
0.9159655941772190150546035
The Clausen sine function can be expressed in closed form when
`s` is an odd integer (becoming zero when `s` < 0)::
>>> z = 1 + sqrt(2)
>>> clsin(1, z); (pi-z)/2
0.3636895456083490948304773
0.3636895456083490948304773
>>> clsin(3, z); pi**2/6*z - pi*z**2/4 + z**3/12
0.5661751584451144991707161
0.5661751584451144991707161
>>> clsin(-1, z)
0.0
>>> clsin(-3, z)
0.0
It can also be expressed in closed form for even integer `s \le 0`,
providing a finite sum for series such as
`\sin(z) + \sin(2z) + \sin(3z) + \ldots`::
>>> z = 1 + sqrt(2)
>>> clsin(0, z)
0.1903105029507513881275865
>>> cot(z/2)/2
0.1903105029507513881275865
>>> clsin(-2, z)
-0.1089406163841548817581392
>>> -cot(z/2)*csc(z/2)**2/4
-0.1089406163841548817581392
Call with ``pi=True`` to multiply `z` by `\pi` exactly::
>>> clsin(3, 3*pi)
-8.892316224968072424732898e-26
>>> clsin(3, 3, pi=True)
0.0
Evaluation for complex `s`, `z` in a nonconvergent case::
>>> s, z = -1-j, 1+2j
>>> clsin(s, z)
(-0.593079480117379002516034 + 0.9038644233367868273362446j)
>>> extraprec(20)(nsum)(lambda k: sin(k*z)/k**s, [1,inf])
(-0.593079480117379002516034 + 0.9038644233367868273362446j)
"""
clcos = r"""
Computes the Clausen cosine function, defined formally by the series
.. math ::
\mathrm{\widetilde{Cl}}_s(z) = \sum_{k=1}^{\infty} \frac{\cos(kz)}{k^s}.
This function is complementary to the Clausen sine function
:func:`~mpmath.clsin`. In terms of the polylogarithm,
.. math ::
\mathrm{\widetilde{Cl}}_s(z) =
\frac{1}{2}\left(\mathrm{Li}_s\left(e^{iz}\right) +
\mathrm{Li}_s\left(e^{-iz}\right)\right)
= \mathrm{Re}\left[\mathrm{Li}_s(e^{iz})\right] \quad (s, z \in \mathbb{R}).
**Examples**
Evaluation for arbitrarily chosen `s` and `z`::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> s, z = 3, 4
>>> clcos(s, z); nsum(lambda k: cos(z*k)/k**s, [1,inf])
-0.6518926267198991308332759
-0.6518926267198991308332759
Using `z + \pi` instead of `z` gives an alternating series::
>>> s, z = 3, 0.5
>>> clcos(s, z+pi)
-0.8155530586502260817855618
>>> nsum(lambda k: (-1)**k*cos(z*k)/k**s, [1,inf])
-0.8155530586502260817855618
With `s = 1`, the sum can be expressed in closed form
using elementary functions::
>>> z = 1 + sqrt(3)
>>> clcos(1, z)
-0.6720334373369714849797918
>>> chop(-0.5*(log(1-exp(j*z))+log(1-exp(-j*z))))
-0.6720334373369714849797918
>>> -log(abs(2*sin(0.5*z))) # Equivalent to above when z is real
-0.6720334373369714849797918
>>> nsum(lambda k: cos(k*z)/k, [1,inf])
-0.6720334373369714849797918
It can also be expressed in closed form when `s` is an even integer.
For example,
>>> clcos(2,z)
-0.7805359025135583118863007
>>> pi**2/6 - pi*z/2 + z**2/4
-0.7805359025135583118863007
The case `s = 0` gives the renormalized sum of
`\cos(z) + \cos(2z) + \cos(3z) + \ldots` (which happens to be the same for
any value of `z`)::
>>> clcos(0, z)
-0.5
>>> nsum(lambda k: cos(k*z), [1,inf])
-0.5
Also the sums
.. math ::
\cos(z) + 2\cos(2z) + 3\cos(3z) + \ldots
and
.. math ::
\cos(z) + 2^n \cos(2z) + 3^n \cos(3z) + \ldots
for higher integer powers `n = -s` can be done in closed form. They are zero
when `n` is positive and even (`s` negative and even)::
>>> clcos(-1, z); 1/(2*cos(z)-2)
-0.2607829375240542480694126
-0.2607829375240542480694126
>>> clcos(-3, z); (2+cos(z))*csc(z/2)**4/8
0.1472635054979944390848006
0.1472635054979944390848006
>>> clcos(-2, z); clcos(-4, z); clcos(-6, z)
0.0
0.0
0.0
With `z = \pi`, the series reduces to that of the Riemann zeta function
(more generally, if `z = p \pi/q`, it is a finite sum over Hurwitz zeta
function values)::
>>> clcos(2.5, 0); zeta(2.5)
1.34148725725091717975677
1.34148725725091717975677
>>> clcos(2.5, pi); -altzeta(2.5)
-0.8671998890121841381913472
-0.8671998890121841381913472
Call with ``pi=True`` to multiply `z` by `\pi` exactly::
>>> clcos(-3, 2*pi)
2.997921055881167659267063e+102
>>> clcos(-3, 2, pi=True)
0.008333333333333333333333333
Evaluation for complex `s`, `z` in a nonconvergent case::
>>> s, z = -1-j, 1+2j
>>> clcos(s, z)
(0.9407430121562251476136807 + 0.715826296033590204557054j)
>>> extraprec(20)(nsum)(lambda k: cos(k*z)/k**s, [1,inf])
(0.9407430121562251476136807 + 0.715826296033590204557054j)
"""
whitm = r"""
Evaluates the Whittaker function `M(k,m,z)`, which gives a solution
to the Whittaker differential equation
.. math ::
\frac{d^2f}{dz^2} + \left(-\frac{1}{4}+\frac{k}{z}+
\frac{(\frac{1}{4}-m^2)}{z^2}\right) f = 0.
A second solution is given by :func:`~mpmath.whitw`.
The Whittaker functions are defined in Abramowitz & Stegun, section 13.1.
They are alternate forms of the confluent hypergeometric functions
`\,_1F_1` and `U`:
.. math ::
M(k,m,z) = e^{-\frac{1}{2}z} z^{\frac{1}{2}+m}
\,_1F_1(\tfrac{1}{2}+m-k, 1+2m, z)
W(k,m,z) = e^{-\frac{1}{2}z} z^{\frac{1}{2}+m}
U(\tfrac{1}{2}+m-k, 1+2m, z).
**Examples**
Evaluation for arbitrary real and complex arguments is supported::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> whitm(1, 1, 1)
0.7302596799460411820509668
>>> whitm(1, 1, -1)
(0.0 - 1.417977827655098025684246j)
>>> whitm(j, j/2, 2+3j)
(3.245477713363581112736478 - 0.822879187542699127327782j)
>>> whitm(2, 3, 100000)
4.303985255686378497193063e+21707
Evaluation at zero::
>>> whitm(1,-1,0); whitm(1,-0.5,0); whitm(1,0,0)
+inf
nan
0.0
We can verify that :func:`~mpmath.whitm` numerically satisfies the
differential equation for arbitrarily chosen values::
>>> k = mpf(0.25)
>>> m = mpf(1.5)
>>> f = lambda z: whitm(k,m,z)
>>> for z in [-1, 2.5, 3, 1+2j]:
... chop(diff(f,z,2) + (-0.25 + k/z + (0.25-m**2)/z**2)*f(z))
...
0.0
0.0
0.0
0.0
An integral involving both :func:`~mpmath.whitm` and :func:`~mpmath.whitw`,
verifying evaluation along the real axis::
>>> quad(lambda x: exp(-x)*whitm(3,2,x)*whitw(1,-2,x), [0,inf])
3.438869842576800225207341
>>> 128/(21*sqrt(pi))
3.438869842576800225207341
"""
whitw = r"""
Evaluates the Whittaker function `W(k,m,z)`, which gives a second
solution to the Whittaker differential equation. (See :func:`~mpmath.whitm`.)
**Examples**
Evaluation for arbitrary real and complex arguments is supported::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> whitw(1, 1, 1)
1.19532063107581155661012
>>> whitw(1, 1, -1)
(-0.9424875979222187313924639 - 0.2607738054097702293308689j)
>>> whitw(j, j/2, 2+3j)
(0.1782899315111033879430369 - 0.01609578360403649340169406j)
>>> whitw(2, 3, 100000)
1.887705114889527446891274e-21705
>>> whitw(-1, -1, 100)
1.905250692824046162462058e-24
Evaluation at zero::
>>> for m in [-1, -0.5, 0, 0.5, 1]:
... whitw(1, m, 0)
...
+inf
nan
0.0
nan
+inf
We can verify that :func:`~mpmath.whitw` numerically satisfies the
differential equation for arbitrarily chosen values::
>>> k = mpf(0.25)
>>> m = mpf(1.5)
>>> f = lambda z: whitw(k,m,z)
>>> for z in [-1, 2.5, 3, 1+2j]:
... chop(diff(f,z,2) + (-0.25 + k/z + (0.25-m**2)/z**2)*f(z))
...
0.0
0.0
0.0
0.0
"""
ber = r"""
Computes the Kelvin function ber, which for real arguments gives the real part
of the Bessel J function of a rotated argument
.. math ::
J_n\left(x e^{3\pi i/4}\right) = \mathrm{ber}_n(x) + i \mathrm{bei}_n(x).
The imaginary part is given by :func:`~mpmath.bei`.
**Plots**
.. literalinclude :: /plots/ber.py
.. image :: /plots/ber.png
**Examples**
Verifying the defining relation::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> n, x = 2, 3.5
>>> ber(n,x)
1.442338852571888752631129
>>> bei(n,x)
-0.948359035324558320217678
>>> besselj(n, x*root(1,8,3))
(1.442338852571888752631129 - 0.948359035324558320217678j)
The ber and bei functions are also defined by analytic continuation
for complex arguments::
>>> ber(1+j, 2+3j)
(4.675445984756614424069563 - 15.84901771719130765656316j)
>>> bei(1+j, 2+3j)
(15.83886679193707699364398 + 4.684053288183046528703611j)
"""
bei = r"""
Computes the Kelvin function bei, which for real arguments gives the
imaginary part of the Bessel J function of a rotated argument.
See :func:`~mpmath.ber`.
"""
ker = r"""
Computes the Kelvin function ker, which for real arguments gives the real part
of the (rescaled) Bessel K function of a rotated argument
.. math ::
e^{-\pi i/2} K_n\left(x e^{3\pi i/4}\right) = \mathrm{ker}_n(x) + i \mathrm{kei}_n(x).
The imaginary part is given by :func:`~mpmath.kei`.
**Plots**
.. literalinclude :: /plots/ker.py
.. image :: /plots/ker.png
**Examples**
Verifying the defining relation::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> n, x = 2, 4.5
>>> ker(n,x)
0.02542895201906369640249801
>>> kei(n,x)
-0.02074960467222823237055351
>>> exp(-n*pi*j/2) * besselk(n, x*root(1,8,1))
(0.02542895201906369640249801 - 0.02074960467222823237055351j)
The ker and kei functions are also defined by analytic continuation
for complex arguments::
>>> ker(1+j, 3+4j)
(1.586084268115490421090533 - 2.939717517906339193598719j)
>>> kei(1+j, 3+4j)
(-2.940403256319453402690132 - 1.585621643835618941044855j)
"""
kei = r"""
Computes the Kelvin function kei, which for real arguments gives the
imaginary part of the (rescaled) Bessel K function of a rotated argument.
See :func:`~mpmath.ker`.
"""
struveh = r"""
Gives the Struve function
.. math ::
\,\mathbf{H}_n(z) =
\sum_{k=0}^\infty \frac{(-1)^k}{\Gamma(k+\frac{3}{2})
\Gamma(k+n+\frac{3}{2})} {\left({\frac{z}{2}}\right)}^{2k+n+1}
which is a solution to the Struve differential equation
.. math ::
z^2 f''(z) + z f'(z) + (z^2-n^2) f(z) = \frac{2 z^{n+1}}{\pi (2n-1)!!}.
**Examples**
Evaluation for arbitrary real and complex arguments::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> struveh(0, 3.5)
0.3608207733778295024977797
>>> struveh(-1, 10)
-0.255212719726956768034732
>>> struveh(1, -100.5)
0.5819566816797362287502246
>>> struveh(2.5, 10000000000000)
3153915652525200060.308937
>>> struveh(2.5, -10000000000000)
(0.0 - 3153915652525200060.308937j)
>>> struveh(1+j, 1000000+4000000j)
(-3.066421087689197632388731e+1737173 - 1.596619701076529803290973e+1737173j)
A Struve function of half-integer order is elementary; for example:
>>> z = 3
>>> struveh(0.5, 3)
0.9167076867564138178671595
>>> sqrt(2/(pi*z))*(1-cos(z))
0.9167076867564138178671595
Numerically verifying the differential equation::
>>> z = mpf(4.5)
>>> n = 3
>>> f = lambda z: struveh(n,z)
>>> lhs = z**2*diff(f,z,2) + z*diff(f,z) + (z**2-n**2)*f(z)
>>> rhs = 2*z**(n+1)/fac2(2*n-1)/pi
>>> lhs
17.40359302709875496632744
>>> rhs
17.40359302709875496632744
"""
struvel = r"""
Gives the modified Struve function
.. math ::
\,\mathbf{L}_n(z) = -i e^{-n\pi i/2} \mathbf{H}_n(i z)
which solves to the modified Struve differential equation
.. math ::
z^2 f''(z) + z f'(z) - (z^2+n^2) f(z) = \frac{2 z^{n+1}}{\pi (2n-1)!!}.
**Examples**
Evaluation for arbitrary real and complex arguments::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> struvel(0, 3.5)
7.180846515103737996249972
>>> struvel(-1, 10)
2670.994904980850550721511
>>> struvel(1, -100.5)
1.757089288053346261497686e+42
>>> struvel(2.5, 10000000000000)
4.160893281017115450519948e+4342944819025
>>> struvel(2.5, -10000000000000)
(0.0 - 4.160893281017115450519948e+4342944819025j)
>>> struvel(1+j, 700j)
(-0.1721150049480079451246076 + 0.1240770953126831093464055j)
>>> struvel(1+j, 1000000+4000000j)
(-2.973341637511505389128708e+434290 - 5.164633059729968297147448e+434290j)
Numerically verifying the differential equation::
>>> z = mpf(3.5)
>>> n = 3
>>> f = lambda z: struvel(n,z)
>>> lhs = z**2*diff(f,z,2) + z*diff(f,z) - (z**2+n**2)*f(z)
>>> rhs = 2*z**(n+1)/fac2(2*n-1)/pi
>>> lhs
6.368850306060678353018165
>>> rhs
6.368850306060678353018165
"""
appellf1 = r"""
Gives the Appell F1 hypergeometric function of two variables,
.. math ::
F_1(a,b_1,b_2,c,x,y) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty}
\frac{(a)_{m+n} (b_1)_m (b_2)_n}{(c)_{m+n}}
\frac{x^m y^n}{m! n!}.
This series is only generally convergent when `|x| < 1` and `|y| < 1`,
although :func:`~mpmath.appellf1` can evaluate an analytic continuation
with respecto to either variable, and sometimes both.
**Examples**
Evaluation is supported for real and complex parameters::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> appellf1(1,0,0.5,1,0.5,0.25)
1.154700538379251529018298
>>> appellf1(1,1+j,0.5,1,0.5,0.5j)
(1.138403860350148085179415 + 1.510544741058517621110615j)
For some integer parameters, the F1 series reduces to a polynomial::
>>> appellf1(2,-4,-3,1,2,5)
-816.0
>>> appellf1(-5,1,2,1,4,5)
-20528.0
The analytic continuation with respect to either `x` or `y`,
and sometimes with respect to both, can be evaluated::
>>> appellf1(2,3,4,5,100,0.5)
(0.0006231042714165329279738662 + 0.0000005769149277148425774499857j)
>>> appellf1('1.1', '0.3', '0.2+2j', '0.4', '0.2', 1.5+3j)
(-0.1782604566893954897128702 + 0.002472407104546216117161499j)
>>> appellf1(1,2,3,4,10,12)
-0.07122993830066776374929313
For certain arguments, F1 reduces to an ordinary hypergeometric function::
>>> appellf1(1,2,3,5,0.5,0.25)
1.547902270302684019335555
>>> 4*hyp2f1(1,2,5,'1/3')/3
1.547902270302684019335555
>>> appellf1(1,2,3,4,0,1.5)
(-1.717202506168937502740238 - 2.792526803190927323077905j)
>>> hyp2f1(1,3,4,1.5)
(-1.717202506168937502740238 - 2.792526803190927323077905j)
The F1 function satisfies a system of partial differential equations::
>>> a,b1,b2,c,x,y = map(mpf, [1,0.5,0.25,1.125,0.25,-0.25])
>>> F = lambda x,y: appellf1(a,b1,b2,c,x,y)
>>> chop(x*(1-x)*diff(F,(x,y),(2,0)) +
... y*(1-x)*diff(F,(x,y),(1,1)) +
... (c-(a+b1+1)*x)*diff(F,(x,y),(1,0)) -
... b1*y*diff(F,(x,y),(0,1)) -
... a*b1*F(x,y))
0.0
>>>
>>> chop(y*(1-y)*diff(F,(x,y),(0,2)) +
... x*(1-y)*diff(F,(x,y),(1,1)) +
... (c-(a+b2+1)*y)*diff(F,(x,y),(0,1)) -
... b2*x*diff(F,(x,y),(1,0)) -
... a*b2*F(x,y))
0.0
The Appell F1 function allows for closed-form evaluation of various
integrals, such as any integral of the form
`\int x^r (x+a)^p (x+b)^q dx`::
>>> def integral(a,b,p,q,r,x1,x2):
... a,b,p,q,r,x1,x2 = map(mpmathify, [a,b,p,q,r,x1,x2])
... f = lambda x: x**r * (x+a)**p * (x+b)**q
... def F(x):
... v = x**(r+1)/(r+1) * (a+x)**p * (b+x)**q
... v *= (1+x/a)**(-p)
... v *= (1+x/b)**(-q)
... v *= appellf1(r+1,-p,-q,2+r,-x/a,-x/b)
... return v
... print("Num. quad: %s" % quad(f, [x1,x2]))
... print("Appell F1: %s" % (F(x2)-F(x1)))
...
>>> integral('1/5','4/3','-2','3','1/2',0,1)
Num. quad: 9.073335358785776206576981
Appell F1: 9.073335358785776206576981
>>> integral('3/2','4/3','-2','3','1/2',0,1)
Num. quad: 1.092829171999626454344678
Appell F1: 1.092829171999626454344678
>>> integral('3/2','4/3','-2','3','1/2',12,25)
Num. quad: 1106.323225040235116498927
Appell F1: 1106.323225040235116498927
Also incomplete elliptic integrals fall into this category [1]::
>>> def E(z, m):
... if (pi/2).ae(z):
... return ellipe(m)
... return 2*round(re(z)/pi)*ellipe(m) + mpf(-1)**round(re(z)/pi)*\
... sin(z)*appellf1(0.5,0.5,-0.5,1.5,sin(z)**2,m*sin(z)**2)
...
>>> z, m = 1, 0.5
>>> E(z,m); quad(lambda t: sqrt(1-m*sin(t)**2), [0,pi/4,3*pi/4,z])
0.9273298836244400669659042
0.9273298836244400669659042
>>> z, m = 3, 2
>>> E(z,m); quad(lambda t: sqrt(1-m*sin(t)**2), [0,pi/4,3*pi/4,z])
(1.057495752337234229715836 + 1.198140234735592207439922j)
(1.057495752337234229715836 + 1.198140234735592207439922j)
**References**
1. [WolframFunctions]_ http://functions.wolfram.com/EllipticIntegrals/EllipticE2/26/01/
2. [SrivastavaKarlsson]_
3. [CabralRosetti]_
4. [Vidunas]_
5. [Slater]_
"""
angerj = r"""
Gives the Anger function
.. math ::
\mathbf{J}_{\nu}(z) = \frac{1}{\pi}
\int_0^{\pi} \cos(\nu t - z \sin t) dt
which is an entire function of both the parameter `\nu` and
the argument `z`. It solves the inhomogeneous Bessel differential
equation
.. math ::
f''(z) + \frac{1}{z}f'(z) + \left(1-\frac{\nu^2}{z^2}\right) f(z)
= \frac{(z-\nu)}{\pi z^2} \sin(\pi \nu).
**Examples**
Evaluation for real and complex parameter and argument::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> angerj(2,3)
0.4860912605858910769078311
>>> angerj(-3+4j, 2+5j)
(-5033.358320403384472395612 + 585.8011892476145118551756j)
>>> angerj(3.25, 1e6j)
(4.630743639715893346570743e+434290 - 1.117960409887505906848456e+434291j)
>>> angerj(-1.5, 1e6)
0.0002795719747073879393087011
The Anger function coincides with the Bessel J-function when `\nu`
is an integer::
>>> angerj(1,3); besselj(1,3)
0.3390589585259364589255146
0.3390589585259364589255146
>>> angerj(1.5,3); besselj(1.5,3)
0.4088969848691080859328847
0.4777182150870917715515015
Verifying the differential equation::
>>> v,z = mpf(2.25), 0.75
>>> f = lambda z: angerj(v,z)
>>> diff(f,z,2) + diff(f,z)/z + (1-(v/z)**2)*f(z)
-0.6002108774380707130367995
>>> (z-v)/(pi*z**2) * sinpi(v)
-0.6002108774380707130367995
Verifying the integral representation::
>>> angerj(v,z)
0.1145380759919333180900501
>>> quad(lambda t: cos(v*t-z*sin(t))/pi, [0,pi])
0.1145380759919333180900501
**References**
1. [DLMF]_ section 11.10: Anger-Weber Functions
"""
webere = r"""
Gives the Weber function
.. math ::
\mathbf{E}_{\nu}(z) = \frac{1}{\pi}
\int_0^{\pi} \sin(\nu t - z \sin t) dt
which is an entire function of both the parameter `\nu` and
the argument `z`. It solves the inhomogeneous Bessel differential
equation
.. math ::
f''(z) + \frac{1}{z}f'(z) + \left(1-\frac{\nu^2}{z^2}\right) f(z)
= -\frac{1}{\pi z^2} (z+\nu+(z-\nu)\cos(\pi \nu)).
**Examples**
Evaluation for real and complex parameter and argument::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> webere(2,3)
-0.1057668973099018425662646
>>> webere(-3+4j, 2+5j)
(-585.8081418209852019290498 - 5033.314488899926921597203j)
>>> webere(3.25, 1e6j)
(-1.117960409887505906848456e+434291 - 4.630743639715893346570743e+434290j)
>>> webere(3.25, 1e6)
-0.00002812518265894315604914453
Up to addition of a rational function of `z`, the Weber function coincides
with the Struve H-function when `\nu` is an integer::
>>> webere(1,3); 2/pi-struveh(1,3)
-0.3834897968188690177372881
-0.3834897968188690177372881
>>> webere(5,3); 26/(35*pi)-struveh(5,3)
0.2009680659308154011878075
0.2009680659308154011878075
Verifying the differential equation::
>>> v,z = mpf(2.25), 0.75
>>> f = lambda z: webere(v,z)
>>> diff(f,z,2) + diff(f,z)/z + (1-(v/z)**2)*f(z)
-1.097441848875479535164627
>>> -(z+v+(z-v)*cospi(v))/(pi*z**2)
-1.097441848875479535164627
Verifying the integral representation::
>>> webere(v,z)
0.1486507351534283744485421
>>> quad(lambda t: sin(v*t-z*sin(t))/pi, [0,pi])
0.1486507351534283744485421
**References**
1. [DLMF]_ section 11.10: Anger-Weber Functions
"""
lommels1 = r"""
Gives the Lommel function `s_{\mu,\nu}` or `s^{(1)}_{\mu,\nu}`
.. math ::
s_{\mu,\nu}(z) = \frac{z^{\mu+1}}{(\mu-\nu+1)(\mu+\nu+1)}
\,_1F_2\left(1; \frac{\mu-\nu+3}{2}, \frac{\mu+\nu+3}{2};
-\frac{z^2}{4} \right)
which solves the inhomogeneous Bessel equation
.. math ::
z^2 f''(z) + z f'(z) + (z^2-\nu^2) f(z) = z^{\mu+1}.
A second solution is given by :func:`~mpmath.lommels2`.
**Plots**
.. literalinclude :: /plots/lommels1.py
.. image :: /plots/lommels1.png
**Examples**
An integral representation::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> u,v,z = 0.25, 0.125, mpf(0.75)
>>> lommels1(u,v,z)
0.4276243877565150372999126
>>> (bessely(v,z)*quad(lambda t: t**u*besselj(v,t), [0,z]) - \
... besselj(v,z)*quad(lambda t: t**u*bessely(v,t), [0,z]))*(pi/2)
0.4276243877565150372999126
A special value::
>>> lommels1(v,v,z)
0.5461221367746048054932553
>>> gamma(v+0.5)*sqrt(pi)*power(2,v-1)*struveh(v,z)
0.5461221367746048054932553
Verifying the differential equation::
>>> f = lambda z: lommels1(u,v,z)
>>> z**2*diff(f,z,2) + z*diff(f,z) + (z**2-v**2)*f(z)
0.6979536443265746992059141
>>> z**(u+1)
0.6979536443265746992059141
**References**
1. [GradshteynRyzhik]_
2. [Weisstein]_ http://mathworld.wolfram.com/LommelFunction.html
"""
lommels2 = r"""
Gives the second Lommel function `S_{\mu,\nu}` or `s^{(2)}_{\mu,\nu}`
.. math ::
S_{\mu,\nu}(z) = s_{\mu,\nu}(z) + 2^{\mu-1}
\Gamma\left(\tfrac{1}{2}(\mu-\nu+1)\right)
\Gamma\left(\tfrac{1}{2}(\mu+\nu+1)\right) \times
\left[\sin(\tfrac{1}{2}(\mu-\nu)\pi) J_{\nu}(z) -
\cos(\tfrac{1}{2}(\mu-\nu)\pi) Y_{\nu}(z)
\right]
which solves the same differential equation as
:func:`~mpmath.lommels1`.
**Plots**
.. literalinclude :: /plots/lommels2.py
.. image :: /plots/lommels2.png
**Examples**
For large `|z|`, `S_{\mu,\nu} \sim z^{\mu-1}`::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> lommels2(10,2,30000)
1.968299831601008419949804e+40
>>> power(30000,9)
1.9683e+40
A special value::
>>> u,v,z = 0.5, 0.125, mpf(0.75)
>>> lommels2(v,v,z)
0.9589683199624672099969765
>>> (struveh(v,z)-bessely(v,z))*power(2,v-1)*sqrt(pi)*gamma(v+0.5)
0.9589683199624672099969765
Verifying the differential equation::
>>> f = lambda z: lommels2(u,v,z)
>>> z**2*diff(f,z,2) + z*diff(f,z) + (z**2-v**2)*f(z)
0.6495190528383289850727924
>>> z**(u+1)
0.6495190528383289850727924
**References**
1. [GradshteynRyzhik]_
2. [Weisstein]_ http://mathworld.wolfram.com/LommelFunction.html
"""
appellf2 = r"""
Gives the Appell F2 hypergeometric function of two variables
.. math ::
F_2(a,b_1,b_2,c_1,c_2,x,y) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty}
\frac{(a)_{m+n} (b_1)_m (b_2)_n}{(c_1)_m (c_2)_n}
\frac{x^m y^n}{m! n!}.
The series is generally absolutely convergent for `|x| + |y| < 1`.
**Examples**
Evaluation for real and complex arguments::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> appellf2(1,2,3,4,5,0.25,0.125)
1.257417193533135344785602
>>> appellf2(1,-3,-4,2,3,2,3)
-42.8
>>> appellf2(0.5,0.25,-0.25,2,3,0.25j,0.25)
(0.9880539519421899867041719 + 0.01497616165031102661476978j)
>>> chop(appellf2(1,1+j,1-j,3j,-3j,0.25,0.25))
1.201311219287411337955192
>>> appellf2(1,1,1,4,6,0.125,16)
(-0.09455532250274744282125152 - 0.7647282253046207836769297j)
A transformation formula::
>>> a,b1,b2,c1,c2,x,y = map(mpf, [1,2,0.5,0.25,1.625,-0.125,0.125])
>>> appellf2(a,b1,b2,c1,c2,x,y)
0.2299211717841180783309688
>>> (1-x)**(-a)*appellf2(a,c1-b1,b2,c1,c2,x/(x-1),y/(1-x))
0.2299211717841180783309688
A system of partial differential equations satisfied by F2::
>>> a,b1,b2,c1,c2,x,y = map(mpf, [1,0.5,0.25,1.125,1.5,0.0625,-0.0625])
>>> F = lambda x,y: appellf2(a,b1,b2,c1,c2,x,y)
>>> chop(x*(1-x)*diff(F,(x,y),(2,0)) -
... x*y*diff(F,(x,y),(1,1)) +
... (c1-(a+b1+1)*x)*diff(F,(x,y),(1,0)) -
... b1*y*diff(F,(x,y),(0,1)) -
... a*b1*F(x,y))
0.0
>>> chop(y*(1-y)*diff(F,(x,y),(0,2)) -
... x*y*diff(F,(x,y),(1,1)) +
... (c2-(a+b2+1)*y)*diff(F,(x,y),(0,1)) -
... b2*x*diff(F,(x,y),(1,0)) -
... a*b2*F(x,y))
0.0
**References**
See references for :func:`~mpmath.appellf1`.
"""
appellf3 = r"""
Gives the Appell F3 hypergeometric function of two variables
.. math ::
F_3(a_1,a_2,b_1,b_2,c,x,y) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty}
\frac{(a_1)_m (a_2)_n (b_1)_m (b_2)_n}{(c)_{m+n}}
\frac{x^m y^n}{m! n!}.
The series is generally absolutely convergent for `|x| < 1, |y| < 1`.
**Examples**
Evaluation for various parameters and variables::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> appellf3(1,2,3,4,5,0.5,0.25)
2.221557778107438938158705
>>> appellf3(1,2,3,4,5,6,0); hyp2f1(1,3,5,6)
(-0.5189554589089861284537389 - 0.1454441043328607980769742j)
(-0.5189554589089861284537389 - 0.1454441043328607980769742j)
>>> appellf3(1,-2,-3,1,1,4,6)
-17.4
>>> appellf3(1,2,-3,1,1,4,6)
(17.7876136773677356641825 + 19.54768762233649126154534j)
>>> appellf3(1,2,-3,1,1,6,4)
(85.02054175067929402953645 + 148.4402528821177305173599j)
>>> chop(appellf3(1+j,2,1-j,2,3,0.25,0.25))
1.719992169545200286696007
Many transformations and evaluations for special combinations
of the parameters are possible, e.g.:
>>> a,b,c,x,y = map(mpf, [0.5,0.25,0.125,0.125,-0.125])
>>> appellf3(a,c-a,b,c-b,c,x,y)
1.093432340896087107444363
>>> (1-y)**(a+b-c)*hyp2f1(a,b,c,x+y-x*y)
1.093432340896087107444363
>>> x**2*appellf3(1,1,1,1,3,x,-x)
0.01568646277445385390945083
>>> polylog(2,x**2)
0.01568646277445385390945083
>>> a1,a2,b1,b2,c,x = map(mpf, [0.5,0.25,0.125,0.5,4.25,0.125])
>>> appellf3(a1,a2,b1,b2,c,x,1)
1.03947361709111140096947
>>> gammaprod([c,c-a2-b2],[c-a2,c-b2])*hyp3f2(a1,b1,c-a2-b2,c-a2,c-b2,x)
1.03947361709111140096947
The Appell F3 function satisfies a pair of partial
differential equations::
>>> a1,a2,b1,b2,c,x,y = map(mpf, [0.5,0.25,0.125,0.5,0.625,0.0625,-0.0625])
>>> F = lambda x,y: appellf3(a1,a2,b1,b2,c,x,y)
>>> chop(x*(1-x)*diff(F,(x,y),(2,0)) +
... y*diff(F,(x,y),(1,1)) +
... (c-(a1+b1+1)*x)*diff(F,(x,y),(1,0)) -
... a1*b1*F(x,y))
0.0
>>> chop(y*(1-y)*diff(F,(x,y),(0,2)) +
... x*diff(F,(x,y),(1,1)) +
... (c-(a2+b2+1)*y)*diff(F,(x,y),(0,1)) -
... a2*b2*F(x,y))
0.0
**References**
See references for :func:`~mpmath.appellf1`.
"""
appellf4 = r"""
Gives the Appell F4 hypergeometric function of two variables
.. math ::
F_4(a,b,c_1,c_2,x,y) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty}
\frac{(a)_{m+n} (b)_{m+n}}{(c_1)_m (c_2)_n}
\frac{x^m y^n}{m! n!}.
The series is generally absolutely convergent for
`\sqrt{|x|} + \sqrt{|y|} < 1`.
**Examples**
Evaluation for various parameters and arguments::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> appellf4(1,1,2,2,0.25,0.125)
1.286182069079718313546608
>>> appellf4(-2,-3,4,5,4,5)
34.8
>>> appellf4(5,4,2,3,0.25j,-0.125j)
(-0.2585967215437846642163352 + 2.436102233553582711818743j)
Reduction to `\,_2F_1` in a special case::
>>> a,b,c,x,y = map(mpf, [0.5,0.25,0.125,0.125,-0.125])
>>> appellf4(a,b,c,a+b-c+1,x*(1-y),y*(1-x))
1.129143488466850868248364
>>> hyp2f1(a,b,c,x)*hyp2f1(a,b,a+b-c+1,y)
1.129143488466850868248364
A system of partial differential equations satisfied by F4::
>>> a,b,c1,c2,x,y = map(mpf, [1,0.5,0.25,1.125,0.0625,-0.0625])
>>> F = lambda x,y: appellf4(a,b,c1,c2,x,y)
>>> chop(x*(1-x)*diff(F,(x,y),(2,0)) -
... y**2*diff(F,(x,y),(0,2)) -
... 2*x*y*diff(F,(x,y),(1,1)) +
... (c1-(a+b+1)*x)*diff(F,(x,y),(1,0)) -
... ((a+b+1)*y)*diff(F,(x,y),(0,1)) -
... a*b*F(x,y))
0.0
>>> chop(y*(1-y)*diff(F,(x,y),(0,2)) -
... x**2*diff(F,(x,y),(2,0)) -
... 2*x*y*diff(F,(x,y),(1,1)) +
... (c2-(a+b+1)*y)*diff(F,(x,y),(0,1)) -
... ((a+b+1)*x)*diff(F,(x,y),(1,0)) -
... a*b*F(x,y))
0.0
**References**
See references for :func:`~mpmath.appellf1`.
"""
zeta = r"""
Computes the Riemann zeta function
.. math ::
\zeta(s) = 1+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+\ldots
or, with `a \ne 1`, the more general Hurwitz zeta function
.. math ::
\zeta(s,a) = \sum_{k=0}^\infty \frac{1}{(a+k)^s}.
Optionally, ``zeta(s, a, n)`` computes the `n`-th derivative with
respect to `s`,
.. math ::
\zeta^{(n)}(s,a) = (-1)^n \sum_{k=0}^\infty \frac{\log^n(a+k)}{(a+k)^s}.
Although these series only converge for `\Re(s) > 1`, the Riemann and Hurwitz
zeta functions are defined through analytic continuation for arbitrary
complex `s \ne 1` (`s = 1` is a pole).
The implementation uses three algorithms: the Borwein algorithm for
the Riemann zeta function when `s` is close to the real line;
the Riemann-Siegel formula for the Riemann zeta function when `s` is
large imaginary, and Euler-Maclaurin summation in all other cases.
The reflection formula for `\Re(s) < 0` is implemented in some cases.
The algorithm can be chosen with ``method = 'borwein'``,
``method='riemann-siegel'`` or ``method = 'euler-maclaurin'``.
The parameter `a` is usually a rational number `a = p/q`, and may be specified
as such by passing an integer tuple `(p, q)`. Evaluation is supported for
arbitrary complex `a`, but may be slow and/or inaccurate when `\Re(s) < 0` for
nonrational `a` or when computing derivatives.
**Examples**
Some values of the Riemann zeta function::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> zeta(2); pi**2 / 6
1.644934066848226436472415
1.644934066848226436472415
>>> zeta(0)
-0.5
>>> zeta(-1)
-0.08333333333333333333333333
>>> zeta(-2)
0.0
For large positive `s`, `\zeta(s)` rapidly approaches 1::
>>> zeta(50)
1.000000000000000888178421
>>> zeta(100)
1.0
>>> zeta(inf)
1.0
>>> 1-sum((zeta(k)-1)/k for k in range(2,85)); +euler
0.5772156649015328606065121
0.5772156649015328606065121
>>> nsum(lambda k: zeta(k)-1, [2, inf])
1.0
Evaluation is supported for complex `s` and `a`:
>>> zeta(-3+4j)
(-0.03373057338827757067584698 + 0.2774499251557093745297677j)
>>> zeta(2+3j, -1+j)
(389.6841230140842816370741 + 295.2674610150305334025962j)
The Riemann zeta function has so-called nontrivial zeros on
the critical line `s = 1/2 + it`::
>>> findroot(zeta, 0.5+14j); zetazero(1)
(0.5 + 14.13472514173469379045725j)
(0.5 + 14.13472514173469379045725j)
>>> findroot(zeta, 0.5+21j); zetazero(2)
(0.5 + 21.02203963877155499262848j)
(0.5 + 21.02203963877155499262848j)
>>> findroot(zeta, 0.5+25j); zetazero(3)
(0.5 + 25.01085758014568876321379j)
(0.5 + 25.01085758014568876321379j)
>>> chop(zeta(zetazero(10)))
0.0
Evaluation on and near the critical line is supported for large
heights `t` by means of the Riemann-Siegel formula (currently
for `a = 1`, `n \le 4`)::
>>> zeta(0.5+100000j)
(1.073032014857753132114076 + 5.780848544363503984261041j)
>>> zeta(0.75+1000000j)
(0.9535316058375145020351559 + 0.9525945894834273060175651j)
>>> zeta(0.5+10000000j)
(11.45804061057709254500227 - 8.643437226836021723818215j)
>>> zeta(0.5+100000000j, derivative=1)
(51.12433106710194942681869 + 43.87221167872304520599418j)
>>> zeta(0.5+100000000j, derivative=2)
(-444.2760822795430400549229 - 896.3789978119185981665403j)
>>> zeta(0.5+100000000j, derivative=3)
(3230.72682687670422215339 + 14374.36950073615897616781j)
>>> zeta(0.5+100000000j, derivative=4)
(-11967.35573095046402130602 - 218945.7817789262839266148j)
>>> zeta(1+10000000j) # off the line
(2.859846483332530337008882 + 0.491808047480981808903986j)
>>> zeta(1+10000000j, derivative=1)
(-4.333835494679647915673205 - 0.08405337962602933636096103j)
>>> zeta(1+10000000j, derivative=4)
(453.2764822702057701894278 - 581.963625832768189140995j)
For investigation of the zeta function zeros, the Riemann-Siegel
Z-function is often more convenient than working with the Riemann
zeta function directly (see :func:`~mpmath.siegelz`).
Some values of the Hurwitz zeta function::
>>> zeta(2, 3); -5./4 + pi**2/6
0.3949340668482264364724152
0.3949340668482264364724152
>>> zeta(2, (3,4)); pi**2 - 8*catalan
2.541879647671606498397663
2.541879647671606498397663
For positive integer values of `s`, the Hurwitz zeta function is
equivalent to a polygamma function (except for a normalizing factor)::
>>> zeta(4, (1,5)); psi(3, '1/5')/6
625.5408324774542966919938
625.5408324774542966919938
Evaluation of derivatives::
>>> zeta(0, 3+4j, 1); loggamma(3+4j) - ln(2*pi)/2
(-2.675565317808456852310934 + 4.742664438034657928194889j)
(-2.675565317808456852310934 + 4.742664438034657928194889j)
>>> zeta(2, 1, 20)
2432902008176640000.000242
>>> zeta(3+4j, 5.5+2j, 4)
(-0.140075548947797130681075 - 0.3109263360275413251313634j)
>>> zeta(0.5+100000j, 1, 4)
(-10407.16081931495861539236 + 13777.78669862804508537384j)
>>> zeta(-100+0.5j, (1,3), derivative=4)
(4.007180821099823942702249e+79 + 4.916117957092593868321778e+78j)
Generating a Taylor series at `s = 2` using derivatives::
>>> for k in range(11): print("%s * (s-2)^%i" % (zeta(2,1,k)/fac(k), k))
...
1.644934066848226436472415 * (s-2)^0
-0.9375482543158437537025741 * (s-2)^1
0.9946401171494505117104293 * (s-2)^2
-1.000024300473840810940657 * (s-2)^3
1.000061933072352565457512 * (s-2)^4
-1.000006869443931806408941 * (s-2)^5
1.000000173233769531820592 * (s-2)^6
-0.9999999569989868493432399 * (s-2)^7
0.9999999937218844508684206 * (s-2)^8
-0.9999999996355013916608284 * (s-2)^9
1.000000000004610645020747 * (s-2)^10
Evaluation at zero and for negative integer `s`::
>>> zeta(0, 10)
-9.5
>>> zeta(-2, (2,3)); mpf(1)/81
0.01234567901234567901234568
0.01234567901234567901234568
>>> zeta(-3+4j, (5,4))
(0.2899236037682695182085988 + 0.06561206166091757973112783j)
>>> zeta(-3.25, 1/pi)
-0.0005117269627574430494396877
>>> zeta(-3.5, pi, 1)
11.156360390440003294709
>>> zeta(-100.5, (8,3))
-4.68162300487989766727122e+77
>>> zeta(-10.5, (-8,3))
(-0.01521913704446246609237979 + 29907.72510874248161608216j)
>>> zeta(-1000.5, (-8,3))
(1.031911949062334538202567e+1770 + 1.519555750556794218804724e+426j)
>>> zeta(-1+j, 3+4j)
(-16.32988355630802510888631 - 22.17706465801374033261383j)
>>> zeta(-1+j, 3+4j, 2)
(32.48985276392056641594055 - 51.11604466157397267043655j)
>>> diff(lambda s: zeta(s, 3+4j), -1+j, 2)
(32.48985276392056641594055 - 51.11604466157397267043655j)
**References**
1. http://mathworld.wolfram.com/RiemannZetaFunction.html
2. http://mathworld.wolfram.com/HurwitzZetaFunction.html
3. http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P155.pdf
"""
dirichlet = r"""
Evaluates the Dirichlet L-function
.. math ::
L(s,\chi) = \sum_{k=1}^\infty \frac{\chi(k)}{k^s}.
where `\chi` is a periodic sequence of length `q` which should be supplied
in the form of a list `[\chi(0), \chi(1), \ldots, \chi(q-1)]`.
Strictly, `\chi` should be a Dirichlet character, but any periodic
sequence will work.
For example, ``dirichlet(s, [1])`` gives the ordinary
Riemann zeta function and ``dirichlet(s, [-1,1])`` gives
the alternating zeta function (Dirichlet eta function).
Also the derivative with respect to `s` (currently only a first
derivative) can be evaluated.
**Examples**
The ordinary Riemann zeta function::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> dirichlet(3, [1]); zeta(3)
1.202056903159594285399738
1.202056903159594285399738
>>> dirichlet(1, [1])
+inf
The alternating zeta function::
>>> dirichlet(1, [-1,1]); ln(2)
0.6931471805599453094172321
0.6931471805599453094172321
The following defines the Dirichlet beta function
`\beta(s) = \sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)^s}` and verifies
several values of this function::
>>> B = lambda s, d=0: dirichlet(s, [0, 1, 0, -1], d)
>>> B(0); 1./2
0.5
0.5
>>> B(1); pi/4
0.7853981633974483096156609
0.7853981633974483096156609
>>> B(2); +catalan
0.9159655941772190150546035
0.9159655941772190150546035
>>> B(2,1); diff(B, 2)
0.08158073611659279510291217
0.08158073611659279510291217
>>> B(-1,1); 2*catalan/pi
0.5831218080616375602767689
0.5831218080616375602767689
>>> B(0,1); log(gamma(0.25)**2/(2*pi*sqrt(2)))
0.3915943927068367764719453
0.3915943927068367764719454
>>> B(1,1); 0.25*pi*(euler+2*ln2+3*ln(pi)-4*ln(gamma(0.25)))
0.1929013167969124293631898
0.1929013167969124293631898
A custom L-series of period 3::
>>> dirichlet(2, [2,0,1])
0.7059715047839078092146831
>>> 2*nsum(lambda k: (3*k)**-2, [1,inf]) + \
... nsum(lambda k: (3*k+2)**-2, [0,inf])
0.7059715047839078092146831
"""
coulombf = r"""
Calculates the regular Coulomb wave function
.. math ::
F_l(\eta,z) = C_l(\eta) z^{l+1} e^{-iz} \,_1F_1(l+1-i\eta, 2l+2, 2iz)
where the normalization constant `C_l(\eta)` is as calculated by
:func:`~mpmath.coulombc`. This function solves the differential equation
.. math ::
f''(z) + \left(1-\frac{2\eta}{z}-\frac{l(l+1)}{z^2}\right) f(z) = 0.
A second linearly independent solution is given by the irregular
Coulomb wave function `G_l(\eta,z)` (see :func:`~mpmath.coulombg`)
and thus the general solution is
`f(z) = C_1 F_l(\eta,z) + C_2 G_l(\eta,z)` for arbitrary
constants `C_1`, `C_2`.
Physically, the Coulomb wave functions give the radial solution
to the Schrodinger equation for a point particle in a `1/z` potential; `z` is
then the radius and `l`, `\eta` are quantum numbers.
The Coulomb wave functions with real parameters are defined
in Abramowitz & Stegun, section 14. However, all parameters are permitted
to be complex in this implementation (see references).
**Plots**
.. literalinclude :: /plots/coulombf.py
.. image :: /plots/coulombf.png
.. literalinclude :: /plots/coulombf_c.py
.. image :: /plots/coulombf_c.png
**Examples**
Evaluation is supported for arbitrary magnitudes of `z`::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> coulombf(2, 1.5, 3.5)
0.4080998961088761187426445
>>> coulombf(-2, 1.5, 3.5)
0.7103040849492536747533465
>>> coulombf(2, 1.5, '1e-10')
4.143324917492256448770769e-33
>>> coulombf(2, 1.5, 1000)
0.4482623140325567050716179
>>> coulombf(2, 1.5, 10**10)
-0.066804196437694360046619
Verifying the differential equation::
>>> l, eta, z = 2, 3, mpf(2.75)
>>> A, B = 1, 2
>>> f = lambda z: A*coulombf(l,eta,z) + B*coulombg(l,eta,z)
>>> chop(diff(f,z,2) + (1-2*eta/z - l*(l+1)/z**2)*f(z))
0.0
A Wronskian relation satisfied by the Coulomb wave functions::
>>> l = 2
>>> eta = 1.5
>>> F = lambda z: coulombf(l,eta,z)
>>> G = lambda z: coulombg(l,eta,z)
>>> for z in [3.5, -1, 2+3j]:
... chop(diff(F,z)*G(z) - F(z)*diff(G,z))
...
1.0
1.0
1.0
Another Wronskian relation::
>>> F = coulombf
>>> G = coulombg
>>> for z in [3.5, -1, 2+3j]:
... chop(F(l-1,eta,z)*G(l,eta,z)-F(l,eta,z)*G(l-1,eta,z) - l/sqrt(l**2+eta**2))
...
0.0
0.0
0.0
An integral identity connecting the regular and irregular wave functions::
>>> l, eta, z = 4+j, 2-j, 5+2j
>>> coulombf(l,eta,z) + j*coulombg(l,eta,z)
(0.7997977752284033239714479 + 0.9294486669502295512503127j)
>>> g = lambda t: exp(-t)*t**(l-j*eta)*(t+2*j*z)**(l+j*eta)
>>> j*exp(-j*z)*z**(-l)/fac(2*l+1)/coulombc(l,eta)*quad(g, [0,inf])
(0.7997977752284033239714479 + 0.9294486669502295512503127j)
Some test case with complex parameters, taken from Michel [2]::
>>> mp.dps = 15
>>> coulombf(1+0.1j, 50+50j, 100.156)
(-1.02107292320897e+15 - 2.83675545731519e+15j)
>>> coulombg(1+0.1j, 50+50j, 100.156)
(2.83675545731519e+15 - 1.02107292320897e+15j)
>>> coulombf(1e-5j, 10+1e-5j, 0.1+1e-6j)
(4.30566371247811e-14 - 9.03347835361657e-19j)
>>> coulombg(1e-5j, 10+1e-5j, 0.1+1e-6j)
(778709182061.134 + 18418936.2660553j)
The following reproduces a table in Abramowitz & Stegun, at twice
the precision::
>>> mp.dps = 10
>>> eta = 2; z = 5
>>> for l in [5, 4, 3, 2, 1, 0]:
... print("%s %s %s" % (l, coulombf(l,eta,z),
... diff(lambda z: coulombf(l,eta,z), z)))
...
5 0.09079533488 0.1042553261
4 0.2148205331 0.2029591779
3 0.4313159311 0.320534053
2 0.7212774133 0.3952408216
1 0.9935056752 0.3708676452
0 1.143337392 0.2937960375
**References**
1. I.J. Thompson & A.R. Barnett, "Coulomb and Bessel Functions of Complex
Arguments and Order", J. Comp. Phys., vol 64, no. 2, June 1986.
2. N. Michel, "Precise Coulomb wave functions for a wide range of
complex `l`, `\eta` and `z`", http://arxiv.org/abs/physics/0702051v1
"""
coulombg = r"""
Calculates the irregular Coulomb wave function
.. math ::
G_l(\eta,z) = \frac{F_l(\eta,z) \cos(\chi) - F_{-l-1}(\eta,z)}{\sin(\chi)}
where `\chi = \sigma_l - \sigma_{-l-1} - (l+1/2) \pi`
and `\sigma_l(\eta) = (\ln \Gamma(1+l+i\eta)-\ln \Gamma(1+l-i\eta))/(2i)`.
See :func:`~mpmath.coulombf` for additional information.
**Plots**
.. literalinclude :: /plots/coulombg.py
.. image :: /plots/coulombg.png
.. literalinclude :: /plots/coulombg_c.py
.. image :: /plots/coulombg_c.png
**Examples**
Evaluation is supported for arbitrary magnitudes of `z`::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> coulombg(-2, 1.5, 3.5)
1.380011900612186346255524
>>> coulombg(2, 1.5, 3.5)
1.919153700722748795245926
>>> coulombg(-2, 1.5, '1e-10')
201126715824.7329115106793
>>> coulombg(-2, 1.5, 1000)
0.1802071520691149410425512
>>> coulombg(-2, 1.5, 10**10)
0.652103020061678070929794
The following reproduces a table in Abramowitz & Stegun,
at twice the precision::
>>> mp.dps = 10
>>> eta = 2; z = 5
>>> for l in [1, 2, 3, 4, 5]:
... print("%s %s %s" % (l, coulombg(l,eta,z),
... -diff(lambda z: coulombg(l,eta,z), z)))
...
1 1.08148276 0.6028279961
2 1.496877075 0.5661803178
3 2.048694714 0.7959909551
4 3.09408669 1.731802374
5 5.629840456 4.549343289
Evaluation close to the singularity at `z = 0`::
>>> mp.dps = 15
>>> coulombg(0,10,1)
3088184933.67358
>>> coulombg(0,10,'1e-10')
5554866000719.8
>>> coulombg(0,10,'1e-100')
5554866221524.1
Evaluation with a half-integer value for `l`::
>>> coulombg(1.5, 1, 10)
0.852320038297334
"""
coulombc = r"""
Gives the normalizing Gamow constant for Coulomb wave functions,
.. math ::
C_l(\eta) = 2^l \exp\left(-\pi \eta/2 + [\ln \Gamma(1+l+i\eta) +
\ln \Gamma(1+l-i\eta)]/2 - \ln \Gamma(2l+2)\right),
where the log gamma function with continuous imaginary part
away from the negative half axis (see :func:`~mpmath.loggamma`) is implied.
This function is used internally for the calculation of
Coulomb wave functions, and automatically cached to make multiple
evaluations with fixed `l`, `\eta` fast.
"""
ellipfun = r"""
Computes any of the Jacobi elliptic functions, defined
in terms of Jacobi theta functions as
.. math ::
\mathrm{sn}(u,m) = \frac{\vartheta_3(0,q)}{\vartheta_2(0,q)}
\frac{\vartheta_1(t,q)}{\vartheta_4(t,q)}
\mathrm{cn}(u,m) = \frac{\vartheta_4(0,q)}{\vartheta_2(0,q)}
\frac{\vartheta_2(t,q)}{\vartheta_4(t,q)}
\mathrm{dn}(u,m) = \frac{\vartheta_4(0,q)}{\vartheta_3(0,q)}
\frac{\vartheta_3(t,q)}{\vartheta_4(t,q)},
or more generally computes a ratio of two such functions. Here
`t = u/\vartheta_3(0,q)^2`, and `q = q(m)` denotes the nome (see
:func:`~mpmath.nome`). Optionally, you can specify the nome directly
instead of `m` by passing ``q=<value>``, or you can directly
specify the elliptic parameter `k` with ``k=<value>``.
The first argument should be a two-character string specifying the
function using any combination of ``'s'``, ``'c'``, ``'d'``, ``'n'``. These
letters respectively denote the basic functions
`\mathrm{sn}(u,m)`, `\mathrm{cn}(u,m)`, `\mathrm{dn}(u,m)`, and `1`.
The identifier specifies the ratio of two such functions.
For example, ``'ns'`` identifies the function
.. math ::
\mathrm{ns}(u,m) = \frac{1}{\mathrm{sn}(u,m)}
and ``'cd'`` identifies the function
.. math ::
\mathrm{cd}(u,m) = \frac{\mathrm{cn}(u,m)}{\mathrm{dn}(u,m)}.
If called with only the first argument, a function object
evaluating the chosen function for given arguments is returned.
**Examples**
Basic evaluation::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> ellipfun('cd', 3.5, 0.5)
-0.9891101840595543931308394
>>> ellipfun('cd', 3.5, q=0.25)
0.07111979240214668158441418
The sn-function is doubly periodic in the complex plane with periods
`4 K(m)` and `2 i K(1-m)` (see :func:`~mpmath.ellipk`)::
>>> sn = ellipfun('sn')
>>> sn(2, 0.25)
0.9628981775982774425751399
>>> sn(2+4*ellipk(0.25), 0.25)
0.9628981775982774425751399
>>> chop(sn(2+2*j*ellipk(1-0.25), 0.25))
0.9628981775982774425751399
The cn-function is doubly periodic with periods `4 K(m)` and `4 i K(1-m)`::
>>> cn = ellipfun('cn')
>>> cn(2, 0.25)
-0.2698649654510865792581416
>>> cn(2+4*ellipk(0.25), 0.25)
-0.2698649654510865792581416
>>> chop(cn(2+4*j*ellipk(1-0.25), 0.25))
-0.2698649654510865792581416
The dn-function is doubly periodic with periods `2 K(m)` and `4 i K(1-m)`::
>>> dn = ellipfun('dn')
>>> dn(2, 0.25)
0.8764740583123262286931578
>>> dn(2+2*ellipk(0.25), 0.25)
0.8764740583123262286931578
>>> chop(dn(2+4*j*ellipk(1-0.25), 0.25))
0.8764740583123262286931578
"""
jtheta = r"""
Computes the Jacobi theta function `\vartheta_n(z, q)`, where
`n = 1, 2, 3, 4`, defined by the infinite series:
.. math ::
\vartheta_1(z,q) = 2 q^{1/4} \sum_{n=0}^{\infty}
(-1)^n q^{n^2+n\,} \sin((2n+1)z)
\vartheta_2(z,q) = 2 q^{1/4} \sum_{n=0}^{\infty}
q^{n^{2\,} + n} \cos((2n+1)z)
\vartheta_3(z,q) = 1 + 2 \sum_{n=1}^{\infty}
q^{n^2\,} \cos(2 n z)
\vartheta_4(z,q) = 1 + 2 \sum_{n=1}^{\infty}
(-q)^{n^2\,} \cos(2 n z)
The theta functions are functions of two variables:
* `z` is the *argument*, an arbitrary real or complex number
* `q` is the *nome*, which must be a real or complex number
in the unit disk (i.e. `|q| < 1`). For `|q| \ll 1`, the
series converge very quickly, so the Jacobi theta functions
can efficiently be evaluated to high precision.
The compact notations `\vartheta_n(q) = \vartheta_n(0,q)`
and `\vartheta_n = \vartheta_n(0,q)` are also frequently
encountered. Finally, Jacobi theta functions are frequently
considered as functions of the half-period ratio `\tau`
and then usually denoted by `\vartheta_n(z|\tau)`.
Optionally, ``jtheta(n, z, q, derivative=d)`` with `d > 0` computes
a `d`-th derivative with respect to `z`.
**Examples and basic properties**
Considered as functions of `z`, the Jacobi theta functions may be
viewed as generalizations of the ordinary trigonometric functions
cos and sin. They are periodic functions::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> jtheta(1, 0.25, '0.2')
0.2945120798627300045053104
>>> jtheta(1, 0.25 + 2*pi, '0.2')
0.2945120798627300045053104
Indeed, the series defining the theta functions are essentially
trigonometric Fourier series. The coefficients can be retrieved
using :func:`~mpmath.fourier`::
>>> mp.dps = 10
>>> nprint(fourier(lambda x: jtheta(2, x, 0.5), [-pi, pi], 4))
([0.0, 1.68179, 0.0, 0.420448, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0])
The Jacobi theta functions are also so-called quasiperiodic
functions of `z` and `\tau`, meaning that for fixed `\tau`,
`\vartheta_n(z, q)` and `\vartheta_n(z+\pi \tau, q)` are the same
except for an exponential factor::
>>> mp.dps = 25
>>> tau = 3*j/10
>>> q = exp(pi*j*tau)
>>> z = 10
>>> jtheta(4, z+tau*pi, q)
(-0.682420280786034687520568 + 1.526683999721399103332021j)
>>> -exp(-2*j*z)/q * jtheta(4, z, q)
(-0.682420280786034687520568 + 1.526683999721399103332021j)
The Jacobi theta functions satisfy a huge number of other
functional equations, such as the following identity (valid for
any `q`)::
>>> q = mpf(3)/10
>>> jtheta(3,0,q)**4
6.823744089352763305137427
>>> jtheta(2,0,q)**4 + jtheta(4,0,q)**4
6.823744089352763305137427
Extensive listings of identities satisfied by the Jacobi theta
functions can be found in standard reference works.
The Jacobi theta functions are related to the gamma function
for special arguments::
>>> jtheta(3, 0, exp(-pi))
1.086434811213308014575316
>>> pi**(1/4.) / gamma(3/4.)
1.086434811213308014575316
:func:`~mpmath.jtheta` supports arbitrary precision evaluation and complex
arguments::
>>> mp.dps = 50
>>> jtheta(4, sqrt(2), 0.5)
2.0549510717571539127004115835148878097035750653737
>>> mp.dps = 25
>>> jtheta(4, 1+2j, (1+j)/5)
(7.180331760146805926356634 - 1.634292858119162417301683j)
Evaluation of derivatives::
>>> mp.dps = 25
>>> jtheta(1, 7, 0.25, 1); diff(lambda z: jtheta(1, z, 0.25), 7)
1.209857192844475388637236
1.209857192844475388637236
>>> jtheta(1, 7, 0.25, 2); diff(lambda z: jtheta(1, z, 0.25), 7, 2)
-0.2598718791650217206533052
-0.2598718791650217206533052
>>> jtheta(2, 7, 0.25, 1); diff(lambda z: jtheta(2, z, 0.25), 7)
-1.150231437070259644461474
-1.150231437070259644461474
>>> jtheta(2, 7, 0.25, 2); diff(lambda z: jtheta(2, z, 0.25), 7, 2)
-0.6226636990043777445898114
-0.6226636990043777445898114
>>> jtheta(3, 7, 0.25, 1); diff(lambda z: jtheta(3, z, 0.25), 7)
-0.9990312046096634316587882
-0.9990312046096634316587882
>>> jtheta(3, 7, 0.25, 2); diff(lambda z: jtheta(3, z, 0.25), 7, 2)
-0.1530388693066334936151174
-0.1530388693066334936151174
>>> jtheta(4, 7, 0.25, 1); diff(lambda z: jtheta(4, z, 0.25), 7)
0.9820995967262793943571139
0.9820995967262793943571139
>>> jtheta(4, 7, 0.25, 2); diff(lambda z: jtheta(4, z, 0.25), 7, 2)
0.3936902850291437081667755
0.3936902850291437081667755
**Possible issues**
For `|q| \ge 1` or `\Im(\tau) \le 0`, :func:`~mpmath.jtheta` raises
``ValueError``. This exception is also raised for `|q|` extremely
close to 1 (or equivalently `\tau` very close to 0), since the
series would converge too slowly::
>>> jtheta(1, 10, 0.99999999 * exp(0.5*j))
Traceback (most recent call last):
...
ValueError: abs(q) > THETA_Q_LIM = 1.000000
"""
eulernum = r"""
Gives the `n`-th Euler number, defined as the `n`-th derivative of
`\mathrm{sech}(t) = 1/\cosh(t)` evaluated at `t = 0`. Equivalently, the
Euler numbers give the coefficients of the Taylor series
.. math ::
\mathrm{sech}(t) = \sum_{n=0}^{\infty} \frac{E_n}{n!} t^n.
The Euler numbers are closely related to Bernoulli numbers
and Bernoulli polynomials. They can also be evaluated in terms of
Euler polynomials (see :func:`~mpmath.eulerpoly`) as `E_n = 2^n E_n(1/2)`.
**Examples**
Computing the first few Euler numbers and verifying that they
agree with the Taylor series::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> [eulernum(n) for n in range(11)]
[1.0, 0.0, -1.0, 0.0, 5.0, 0.0, -61.0, 0.0, 1385.0, 0.0, -50521.0]
>>> chop(diffs(sech, 0, 10))
[1.0, 0.0, -1.0, 0.0, 5.0, 0.0, -61.0, 0.0, 1385.0, 0.0, -50521.0]
Euler numbers grow very rapidly. :func:`~mpmath.eulernum` efficiently
computes numerical approximations for large indices::
>>> eulernum(50)
-6.053285248188621896314384e+54
>>> eulernum(1000)
3.887561841253070615257336e+2371
>>> eulernum(10**20)
4.346791453661149089338186e+1936958564106659551331
Comparing with an asymptotic formula for the Euler numbers::
>>> n = 10**5
>>> (-1)**(n//2) * 8 * sqrt(n/(2*pi)) * (2*n/(pi*e))**n
3.69919063017432362805663e+436961
>>> eulernum(n)
3.699193712834466537941283e+436961
Pass ``exact=True`` to obtain exact values of Euler numbers as integers::
>>> print(eulernum(50, exact=True))
-6053285248188621896314383785111649088103498225146815121
>>> print(eulernum(200, exact=True) % 10**10)
1925859625
>>> eulernum(1001, exact=True)
0
"""
eulerpoly = r"""
Evaluates the Euler polynomial `E_n(z)`, defined by the generating function
representation
.. math ::
\frac{2e^{zt}}{e^t+1} = \sum_{n=0}^\infty E_n(z) \frac{t^n}{n!}.
The Euler polynomials may also be represented in terms of
Bernoulli polynomials (see :func:`~mpmath.bernpoly`) using various formulas, for
example
.. math ::
E_n(z) = \frac{2}{n+1} \left(
B_n(z)-2^{n+1}B_n\left(\frac{z}{2}\right)
\right).
Special values include the Euler numbers `E_n = 2^n E_n(1/2)` (see
:func:`~mpmath.eulernum`).
**Examples**
Computing the coefficients of the first few Euler polynomials::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> for n in range(6):
... chop(taylor(lambda z: eulerpoly(n,z), 0, n))
...
[1.0]
[-0.5, 1.0]
[0.0, -1.0, 1.0]
[0.25, 0.0, -1.5, 1.0]
[0.0, 1.0, 0.0, -2.0, 1.0]
[-0.5, 0.0, 2.5, 0.0, -2.5, 1.0]
Evaluation for arbitrary `z`::
>>> eulerpoly(2,3)
6.0
>>> eulerpoly(5,4)
423.5
>>> eulerpoly(35, 11111111112)
3.994957561486776072734601e+351
>>> eulerpoly(4, 10+20j)
(-47990.0 - 235980.0j)
>>> eulerpoly(2, '-3.5e-5')
0.000035001225
>>> eulerpoly(3, 0.5)
0.0
>>> eulerpoly(55, -10**80)
-1.0e+4400
>>> eulerpoly(5, -inf)
-inf
>>> eulerpoly(6, -inf)
+inf
Computing Euler numbers::
>>> 2**26 * eulerpoly(26,0.5)
-4087072509293123892361.0
>>> eulernum(26)
-4087072509293123892361.0
Evaluation is accurate for large `n` and small `z`::
>>> eulerpoly(100, 0.5)
2.29047999988194114177943e+108
>>> eulerpoly(1000, 10.5)
3.628120031122876847764566e+2070
>>> eulerpoly(10000, 10.5)
1.149364285543783412210773e+30688
"""
spherharm = r"""
Evaluates the spherical harmonic `Y_l^m(\theta,\phi)`,
.. math ::
Y_l^m(\theta,\phi) = \sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}}
P_l^m(\cos \theta) e^{i m \phi}
where `P_l^m` is an associated Legendre function (see :func:`~mpmath.legenp`).
Here `\theta \in [0, \pi]` denotes the polar coordinate (ranging
from the north pole to the south pole) and `\phi \in [0, 2 \pi]` denotes the
azimuthal coordinate on a sphere. Care should be used since many different
conventions for spherical coordinate variables are used.
Usually spherical harmonics are considered for `l \in \mathbb{N}`,
`m \in \mathbb{Z}`, `|m| \le l`. More generally, `l,m,\theta,\phi`
are permitted to be complex numbers.
.. note ::
:func:`~mpmath.spherharm` returns a complex number, even if the value is
purely real.
**Plots**
.. literalinclude :: /plots/spherharm40.py
`Y_{4,0}`:
.. image :: /plots/spherharm40.png
`Y_{4,1}`:
.. image :: /plots/spherharm41.png
`Y_{4,2}`:
.. image :: /plots/spherharm42.png
`Y_{4,3}`:
.. image :: /plots/spherharm43.png
`Y_{4,4}`:
.. image :: /plots/spherharm44.png
**Examples**
Some low-order spherical harmonics with reference values::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> theta = pi/4
>>> phi = pi/3
>>> spherharm(0,0,theta,phi); 0.5*sqrt(1/pi)*expj(0)
(0.2820947917738781434740397 + 0.0j)
(0.2820947917738781434740397 + 0.0j)
>>> spherharm(1,-1,theta,phi); 0.5*sqrt(3/(2*pi))*expj(-phi)*sin(theta)
(0.1221506279757299803965962 - 0.2115710938304086076055298j)
(0.1221506279757299803965962 - 0.2115710938304086076055298j)
>>> spherharm(1,0,theta,phi); 0.5*sqrt(3/pi)*cos(theta)*expj(0)
(0.3454941494713354792652446 + 0.0j)
(0.3454941494713354792652446 + 0.0j)
>>> spherharm(1,1,theta,phi); -0.5*sqrt(3/(2*pi))*expj(phi)*sin(theta)
(-0.1221506279757299803965962 - 0.2115710938304086076055298j)
(-0.1221506279757299803965962 - 0.2115710938304086076055298j)
With the normalization convention used, the spherical harmonics are orthonormal
on the unit sphere::
>>> sphere = [0,pi], [0,2*pi]
>>> dS = lambda t,p: fp.sin(t) # differential element
>>> Y1 = lambda t,p: fp.spherharm(l1,m1,t,p)
>>> Y2 = lambda t,p: fp.conj(fp.spherharm(l2,m2,t,p))
>>> l1 = l2 = 3; m1 = m2 = 2
>>> print(fp.quad(lambda t,p: Y1(t,p)*Y2(t,p)*dS(t,p), *sphere))
(1+0j)
>>> m2 = 1 # m1 != m2
>>> print(fp.chop(fp.quad(lambda t,p: Y1(t,p)*Y2(t,p)*dS(t,p), *sphere)))
0.0
Evaluation is accurate for large orders::
>>> spherharm(1000,750,0.5,0.25)
(3.776445785304252879026585e-102 - 5.82441278771834794493484e-102j)
Evaluation works with complex parameter values::
>>> spherharm(1+j, 2j, 2+3j, -0.5j)
(64.44922331113759992154992 + 1981.693919841408089681743j)
"""
scorergi = r"""
Evaluates the Scorer function
.. math ::
\operatorname{Gi}(z) =
\operatorname{Ai}(z) \int_0^z \operatorname{Bi}(t) dt +
\operatorname{Bi}(z) \int_z^{\infty} \operatorname{Ai}(t) dt
which gives a particular solution to the inhomogeneous Airy
differential equation `f''(z) - z f(z) = 1/\pi`. Another
particular solution is given by the Scorer Hi-function
(:func:`~mpmath.scorerhi`). The two functions are related as
`\operatorname{Gi}(z) + \operatorname{Hi}(z) = \operatorname{Bi}(z)`.
**Plots**
.. literalinclude :: /plots/gi.py
.. image :: /plots/gi.png
.. literalinclude :: /plots/gi_c.py
.. image :: /plots/gi_c.png
**Examples**
Some values and limits::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> scorergi(0); 1/(power(3,'7/6')*gamma('2/3'))
0.2049755424820002450503075
0.2049755424820002450503075
>>> diff(scorergi, 0); 1/(power(3,'5/6')*gamma('1/3'))
0.1494294524512754526382746
0.1494294524512754526382746
>>> scorergi(+inf); scorergi(-inf)
0.0
0.0
>>> scorergi(1)
0.2352184398104379375986902
>>> scorergi(-1)
-0.1166722172960152826494198
Evaluation for large arguments::
>>> scorergi(10)
0.03189600510067958798062034
>>> scorergi(100)
0.003183105228162961476590531
>>> scorergi(1000000)
0.0000003183098861837906721743873
>>> 1/(pi*1000000)
0.0000003183098861837906715377675
>>> scorergi(-1000)
-0.08358288400262780392338014
>>> scorergi(-100000)
0.02886866118619660226809581
>>> scorergi(50+10j)
(0.0061214102799778578790984 - 0.001224335676457532180747917j)
>>> scorergi(-50-10j)
(5.236047850352252236372551e+29 - 3.08254224233701381482228e+29j)
>>> scorergi(100000j)
(-8.806659285336231052679025e+6474077 + 8.684731303500835514850962e+6474077j)
Verifying the connection between Gi and Hi::
>>> z = 0.25
>>> scorergi(z) + scorerhi(z)
0.7287469039362150078694543
>>> airybi(z)
0.7287469039362150078694543
Verifying the differential equation::
>>> for z in [-3.4, 0, 2.5, 1+2j]:
... chop(diff(scorergi,z,2) - z*scorergi(z))
...
-0.3183098861837906715377675
-0.3183098861837906715377675
-0.3183098861837906715377675
-0.3183098861837906715377675
Verifying the integral representation::
>>> z = 0.5
>>> scorergi(z)
0.2447210432765581976910539
>>> Ai,Bi = airyai,airybi
>>> Bi(z)*(Ai(inf,-1)-Ai(z,-1)) + Ai(z)*(Bi(z,-1)-Bi(0,-1))
0.2447210432765581976910539
**References**
1. [DLMF]_ section 9.12: Scorer Functions
"""
scorerhi = r"""
Evaluates the second Scorer function
.. math ::
\operatorname{Hi}(z) =
\operatorname{Bi}(z) \int_{-\infty}^z \operatorname{Ai}(t) dt -
\operatorname{Ai}(z) \int_{-\infty}^z \operatorname{Bi}(t) dt
which gives a particular solution to the inhomogeneous Airy
differential equation `f''(z) - z f(z) = 1/\pi`. See also
:func:`~mpmath.scorergi`.
**Plots**
.. literalinclude :: /plots/hi.py
.. image :: /plots/hi.png
.. literalinclude :: /plots/hi_c.py
.. image :: /plots/hi_c.png
**Examples**
Some values and limits::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> scorerhi(0); 2/(power(3,'7/6')*gamma('2/3'))
0.4099510849640004901006149
0.4099510849640004901006149
>>> diff(scorerhi,0); 2/(power(3,'5/6')*gamma('1/3'))
0.2988589049025509052765491
0.2988589049025509052765491
>>> scorerhi(+inf); scorerhi(-inf)
+inf
0.0
>>> scorerhi(1)
0.9722051551424333218376886
>>> scorerhi(-1)
0.2206696067929598945381098
Evaluation for large arguments::
>>> scorerhi(10)
455641153.5163291358991077
>>> scorerhi(100)
6.041223996670201399005265e+288
>>> scorerhi(1000000)
7.138269638197858094311122e+289529652
>>> scorerhi(-10)
0.0317685352825022727415011
>>> scorerhi(-100)
0.003183092495767499864680483
>>> scorerhi(100j)
(-6.366197716545672122983857e-9 + 0.003183098861710582761688475j)
>>> scorerhi(50+50j)
(-5.322076267321435669290334e+63 + 1.478450291165243789749427e+65j)
>>> scorerhi(-1000-1000j)
(0.0001591549432510502796565538 - 0.000159154943091895334973109j)
Verifying the differential equation::
>>> for z in [-3.4, 0, 2, 1+2j]:
... chop(diff(scorerhi,z,2) - z*scorerhi(z))
...
0.3183098861837906715377675
0.3183098861837906715377675
0.3183098861837906715377675
0.3183098861837906715377675
Verifying the integral representation::
>>> z = 0.5
>>> scorerhi(z)
0.6095559998265972956089949
>>> Ai,Bi = airyai,airybi
>>> Bi(z)*(Ai(z,-1)-Ai(-inf,-1)) - Ai(z)*(Bi(z,-1)-Bi(-inf,-1))
0.6095559998265972956089949
"""
stirling1 = r"""
Gives the Stirling number of the first kind `s(n,k)`, defined by
.. math ::
x(x-1)(x-2)\cdots(x-n+1) = \sum_{k=0}^n s(n,k) x^k.
The value is computed using an integer recurrence. The implementation
is not optimized for approximating large values quickly.
**Examples**
Comparing with the generating function::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> taylor(lambda x: ff(x, 5), 0, 5)
[0.0, 24.0, -50.0, 35.0, -10.0, 1.0]
>>> [stirling1(5, k) for k in range(6)]
[0.0, 24.0, -50.0, 35.0, -10.0, 1.0]
Recurrence relation::
>>> n, k = 5, 3
>>> stirling1(n+1,k) + n*stirling1(n,k) - stirling1(n,k-1)
0.0
The matrices of Stirling numbers of first and second kind are inverses
of each other::
>>> A = matrix(5, 5); B = matrix(5, 5)
>>> for n in range(5):
... for k in range(5):
... A[n,k] = stirling1(n,k)
... B[n,k] = stirling2(n,k)
...
>>> A * B
[1.0 0.0 0.0 0.0 0.0]
[0.0 1.0 0.0 0.0 0.0]
[0.0 0.0 1.0 0.0 0.0]
[0.0 0.0 0.0 1.0 0.0]
[0.0 0.0 0.0 0.0 1.0]
Pass ``exact=True`` to obtain exact values of Stirling numbers as integers::
>>> stirling1(42, 5)
-2.864498971768501633736628e+50
>>> print stirling1(42, 5, exact=True)
-286449897176850163373662803014001546235808317440000
"""
stirling2 = r"""
Gives the Stirling number of the second kind `S(n,k)`, defined by
.. math ::
x^n = \sum_{k=0}^n S(n,k) x(x-1)(x-2)\cdots(x-k+1)
The value is computed using integer arithmetic to evaluate a power sum.
The implementation is not optimized for approximating large values quickly.
**Examples**
Comparing with the generating function::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> taylor(lambda x: sum(stirling2(5,k) * ff(x,k) for k in range(6)), 0, 5)
[0.0, 0.0, 0.0, 0.0, 0.0, 1.0]
Recurrence relation::
>>> n, k = 5, 3
>>> stirling2(n+1,k) - k*stirling2(n,k) - stirling2(n,k-1)
0.0
Pass ``exact=True`` to obtain exact values of Stirling numbers as integers::
>>> stirling2(52, 10)
2.641822121003543906807485e+45
>>> print stirling2(52, 10, exact=True)
2641822121003543906807485307053638921722527655
"""
| 279,815 | 26.892344 | 94 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/visualization.py
|
"""
Plotting (requires matplotlib)
"""
from colorsys import hsv_to_rgb, hls_to_rgb
from .libmp import NoConvergence
from .libmp.backend import xrange
class VisualizationMethods(object):
plot_ignore = (ValueError, ArithmeticError, ZeroDivisionError, NoConvergence)
def plot(ctx, f, xlim=[-5,5], ylim=None, points=200, file=None, dpi=None,
singularities=[], axes=None):
r"""
Shows a simple 2D plot of a function `f(x)` or list of functions
`[f_0(x), f_1(x), \ldots, f_n(x)]` over a given interval
specified by *xlim*. Some examples::
plot(lambda x: exp(x)*li(x), [1, 4])
plot([cos, sin], [-4, 4])
plot([fresnels, fresnelc], [-4, 4])
plot([sqrt, cbrt], [-4, 4])
plot(lambda t: zeta(0.5+t*j), [-20, 20])
plot([floor, ceil, abs, sign], [-5, 5])
Points where the function raises a numerical exception or
returns an infinite value are removed from the graph.
Singularities can also be excluded explicitly
as follows (useful for removing erroneous vertical lines)::
plot(cot, ylim=[-5, 5]) # bad
plot(cot, ylim=[-5, 5], singularities=[-pi, 0, pi]) # good
For parts where the function assumes complex values, the
real part is plotted with dashes and the imaginary part
is plotted with dots.
.. note :: This function requires matplotlib (pylab).
"""
if file:
axes = None
fig = None
if not axes:
import pylab
fig = pylab.figure()
axes = fig.add_subplot(111)
if not isinstance(f, (tuple, list)):
f = [f]
a, b = xlim
colors = ['b', 'r', 'g', 'm', 'k']
for n, func in enumerate(f):
x = ctx.arange(a, b, (b-a)/float(points))
segments = []
segment = []
in_complex = False
for i in xrange(len(x)):
try:
if i != 0:
for sing in singularities:
if x[i-1] <= sing and x[i] >= sing:
raise ValueError
v = func(x[i])
if ctx.isnan(v) or abs(v) > 1e300:
raise ValueError
if hasattr(v, "imag") and v.imag:
re = float(v.real)
im = float(v.imag)
if not in_complex:
in_complex = True
segments.append(segment)
segment = []
segment.append((float(x[i]), re, im))
else:
if in_complex:
in_complex = False
segments.append(segment)
segment = []
if hasattr(v, "real"):
v = v.real
segment.append((float(x[i]), v))
except ctx.plot_ignore:
if segment:
segments.append(segment)
segment = []
if segment:
segments.append(segment)
for segment in segments:
x = [s[0] for s in segment]
y = [s[1] for s in segment]
if not x:
continue
c = colors[n % len(colors)]
if len(segment[0]) == 3:
z = [s[2] for s in segment]
axes.plot(x, y, '--'+c, linewidth=3)
axes.plot(x, z, ':'+c, linewidth=3)
else:
axes.plot(x, y, c, linewidth=3)
axes.set_xlim([float(_) for _ in xlim])
if ylim:
axes.set_ylim([float(_) for _ in ylim])
axes.set_xlabel('x')
axes.set_ylabel('f(x)')
axes.grid(True)
if fig:
if file:
pylab.savefig(file, dpi=dpi)
else:
pylab.show()
def default_color_function(ctx, z):
if ctx.isinf(z):
return (1.0, 1.0, 1.0)
if ctx.isnan(z):
return (0.5, 0.5, 0.5)
pi = 3.1415926535898
a = (float(ctx.arg(z)) + ctx.pi) / (2*ctx.pi)
a = (a + 0.5) % 1.0
b = 1.0 - float(1/(1.0+abs(z)**0.3))
return hls_to_rgb(a, b, 0.8)
blue_orange_colors = [
(-1.0, (0.0, 0.0, 0.0)),
(-0.95, (0.1, 0.2, 0.5)), # dark blue
(-0.5, (0.0, 0.5, 1.0)), # blueish
(-0.05, (0.4, 0.8, 0.8)), # cyanish
( 0.0, (1.0, 1.0, 1.0)),
( 0.05, (1.0, 0.9, 0.3)), # yellowish
( 0.5, (0.9, 0.5, 0.0)), # orangeish
( 0.95, (0.7, 0.1, 0.0)), # redish
( 1.0, (0.0, 0.0, 0.0)),
( 2.0, (0.0, 0.0, 0.0)),
]
def phase_color_function(ctx, z):
if ctx.isinf(z):
return (1.0, 1.0, 1.0)
if ctx.isnan(z):
return (0.5, 0.5, 0.5)
pi = 3.1415926535898
w = float(ctx.arg(z)) / pi
w = max(min(w, 1.0), -1.0)
for i in range(1,len(blue_orange_colors)):
if blue_orange_colors[i][0] > w:
a, (ra, ga, ba) = blue_orange_colors[i-1]
b, (rb, gb, bb) = blue_orange_colors[i]
s = (w-a) / (b-a)
return ra+(rb-ra)*s, ga+(gb-ga)*s, ba+(bb-ba)*s
def cplot(ctx, f, re=[-5,5], im=[-5,5], points=2000, color=None,
verbose=False, file=None, dpi=None, axes=None):
"""
Plots the given complex-valued function *f* over a rectangular part
of the complex plane specified by the pairs of intervals *re* and *im*.
For example::
cplot(lambda z: z, [-2, 2], [-10, 10])
cplot(exp)
cplot(zeta, [0, 1], [0, 50])
By default, the complex argument (phase) is shown as color (hue) and
the magnitude is show as brightness. You can also supply a
custom color function (*color*). This function should take a
complex number as input and return an RGB 3-tuple containing
floats in the range 0.0-1.0.
Alternatively, you can select a builtin color function by passing
a string as *color*:
* "default" - default color scheme
* "phase" - a color scheme that only renders the phase of the function,
with white for positive reals, black for negative reals, gold in the
upper half plane, and blue in the lower half plane.
To obtain a sharp image, the number of points may need to be
increased to 100,000 or thereabout. Since evaluating the
function that many times is likely to be slow, the 'verbose'
option is useful to display progress.
.. note :: This function requires matplotlib (pylab).
"""
if color is None or color == "default":
color = ctx.default_color_function
if color == "phase":
color = ctx.phase_color_function
import pylab
if file:
axes = None
fig = None
if not axes:
fig = pylab.figure()
axes = fig.add_subplot(111)
rea, reb = re
ima, imb = im
dre = reb - rea
dim = imb - ima
M = int(ctx.sqrt(points*dre/dim)+1)
N = int(ctx.sqrt(points*dim/dre)+1)
x = pylab.linspace(rea, reb, M)
y = pylab.linspace(ima, imb, N)
# Note: we have to be careful to get the right rotation.
# Test with these plots:
# cplot(lambda z: z if z.real < 0 else 0)
# cplot(lambda z: z if z.imag < 0 else 0)
w = pylab.zeros((N, M, 3))
for n in xrange(N):
for m in xrange(M):
z = ctx.mpc(x[m], y[n])
try:
v = color(f(z))
except ctx.plot_ignore:
v = (0.5, 0.5, 0.5)
w[n,m] = v
if verbose:
print(str(n) + ' of ' + str(N))
rea, reb, ima, imb = [float(_) for _ in [rea, reb, ima, imb]]
axes.imshow(w, extent=(rea, reb, ima, imb), origin='lower')
axes.set_xlabel('Re(z)')
axes.set_ylabel('Im(z)')
if fig:
if file:
pylab.savefig(file, dpi=dpi)
else:
pylab.show()
def splot(ctx, f, u=[-5,5], v=[-5,5], points=100, keep_aspect=True, \
wireframe=False, file=None, dpi=None, axes=None):
"""
Plots the surface defined by `f`.
If `f` returns a single component, then this plots the surface
defined by `z = f(x,y)` over the rectangular domain with
`x = u` and `y = v`.
If `f` returns three components, then this plots the parametric
surface `x, y, z = f(u,v)` over the pairs of intervals `u` and `v`.
For example, to plot a simple function::
>>> from mpmath import *
>>> f = lambda x, y: sin(x+y)*cos(y)
>>> splot(f, [-pi,pi], [-pi,pi]) # doctest: +SKIP
Plotting a donut::
>>> r, R = 1, 2.5
>>> f = lambda u, v: [r*cos(u), (R+r*sin(u))*cos(v), (R+r*sin(u))*sin(v)]
>>> splot(f, [0, 2*pi], [0, 2*pi]) # doctest: +SKIP
.. note :: This function requires matplotlib (pylab) 0.98.5.3 or higher.
"""
import pylab
import mpl_toolkits.mplot3d as mplot3d
if file:
axes = None
fig = None
if not axes:
fig = pylab.figure()
axes = mplot3d.axes3d.Axes3D(fig)
ua, ub = u
va, vb = v
du = ub - ua
dv = vb - va
if not isinstance(points, (list, tuple)):
points = [points, points]
M, N = points
u = pylab.linspace(ua, ub, M)
v = pylab.linspace(va, vb, N)
x, y, z = [pylab.zeros((M, N)) for i in xrange(3)]
xab, yab, zab = [[0, 0] for i in xrange(3)]
for n in xrange(N):
for m in xrange(M):
fdata = f(ctx.convert(u[m]), ctx.convert(v[n]))
try:
x[m,n], y[m,n], z[m,n] = fdata
except TypeError:
x[m,n], y[m,n], z[m,n] = u[m], v[n], fdata
for c, cab in [(x[m,n], xab), (y[m,n], yab), (z[m,n], zab)]:
if c < cab[0]:
cab[0] = c
if c > cab[1]:
cab[1] = c
if wireframe:
axes.plot_wireframe(x, y, z, rstride=4, cstride=4)
else:
axes.plot_surface(x, y, z, rstride=4, cstride=4)
axes.set_xlabel('x')
axes.set_ylabel('y')
axes.set_zlabel('z')
if keep_aspect:
dx, dy, dz = [cab[1] - cab[0] for cab in [xab, yab, zab]]
maxd = max(dx, dy, dz)
if dx < maxd:
delta = maxd - dx
axes.set_xlim3d(xab[0] - delta / 2.0, xab[1] + delta / 2.0)
if dy < maxd:
delta = maxd - dy
axes.set_ylim3d(yab[0] - delta / 2.0, yab[1] + delta / 2.0)
if dz < maxd:
delta = maxd - dz
axes.set_zlim3d(zab[0] - delta / 2.0, zab[1] + delta / 2.0)
if fig:
if file:
pylab.savefig(file, dpi=dpi)
else:
pylab.show()
VisualizationMethods.plot = plot
VisualizationMethods.default_color_function = default_color_function
VisualizationMethods.phase_color_function = phase_color_function
VisualizationMethods.cplot = cplot
VisualizationMethods.splot = splot
| 10,627 | 32.847134 | 81 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/usertools.py
|
def monitor(f, input='print', output='print'):
"""
Returns a wrapped copy of *f* that monitors evaluation by calling
*input* with every input (*args*, *kwargs*) passed to *f* and
*output* with every value returned from *f*. The default action
(specify using the special string value ``'print'``) is to print
inputs and outputs to stdout, along with the total evaluation
count::
>>> from mpmath import *
>>> mp.dps = 5; mp.pretty = False
>>> diff(monitor(exp), 1) # diff will eval f(x-h) and f(x+h)
in 0 (mpf('0.99999999906867742538452148'),) {}
out 0 mpf('2.7182818259274480055282064')
in 1 (mpf('1.0000000009313225746154785'),) {}
out 1 mpf('2.7182818309906424675501024')
mpf('2.7182808')
To disable either the input or the output handler, you may
pass *None* as argument.
Custom input and output handlers may be used e.g. to store
results for later analysis::
>>> mp.dps = 15
>>> input = []
>>> output = []
>>> findroot(monitor(sin, input.append, output.append), 3.0)
mpf('3.1415926535897932')
>>> len(input) # Count number of evaluations
9
>>> print(input[3]); print(output[3])
((mpf('3.1415076583334066'),), {})
8.49952562843408e-5
>>> print(input[4]); print(output[4])
((mpf('3.1415928201669122'),), {})
-1.66577118985331e-7
"""
if not input:
input = lambda v: None
elif input == 'print':
incount = [0]
def input(value):
args, kwargs = value
print("in %s %r %r" % (incount[0], args, kwargs))
incount[0] += 1
if not output:
output = lambda v: None
elif output == 'print':
outcount = [0]
def output(value):
print("out %s %r" % (outcount[0], value))
outcount[0] += 1
def f_monitored(*args, **kwargs):
input((args, kwargs))
v = f(*args, **kwargs)
output(v)
return v
return f_monitored
def timing(f, *args, **kwargs):
"""
Returns time elapsed for evaluating ``f()``. Optionally arguments
may be passed to time the execution of ``f(*args, **kwargs)``.
If the first call is very quick, ``f`` is called
repeatedly and the best time is returned.
"""
once = kwargs.get('once')
if 'once' in kwargs:
del kwargs['once']
if args or kwargs:
if len(args) == 1 and not kwargs:
arg = args[0]
g = lambda: f(arg)
else:
g = lambda: f(*args, **kwargs)
else:
g = f
from timeit import default_timer as clock
t1=clock(); v=g(); t2=clock(); t=t2-t1
if t > 0.05 or once:
return t
for i in range(3):
t1=clock();
# Evaluate multiple times because the timer function
# has a significant overhead
g();g();g();g();g();g();g();g();g();g()
t2=clock()
t=min(t,(t2-t1)/10)
return t
| 3,029 | 31.234043 | 70 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/conftest.py
|
import os
# This makes py.test put mpath directory into the sys.path, so that we can
# import mpmath" from tests nicely
rootdir = os.path.abspath(os.getcwd())
| 160 | 25.833333 | 74 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/ctx_base.py
|
from operator import gt, lt
from .libmp.backend import xrange
from .functions.functions import SpecialFunctions
from .functions.rszeta import RSCache
from .calculus.quadrature import QuadratureMethods
from .calculus.inverselaplace import LaplaceTransformInversionMethods
from .calculus.calculus import CalculusMethods
from .calculus.optimization import OptimizationMethods
from .calculus.odes import ODEMethods
from .matrices.matrices import MatrixMethods
from .matrices.calculus import MatrixCalculusMethods
from .matrices.linalg import LinearAlgebraMethods
from .matrices.eigen import Eigen
from .identification import IdentificationMethods
from .visualization import VisualizationMethods
from . import libmp
class Context(object):
pass
class StandardBaseContext(Context,
SpecialFunctions,
RSCache,
QuadratureMethods,
LaplaceTransformInversionMethods,
CalculusMethods,
MatrixMethods,
MatrixCalculusMethods,
LinearAlgebraMethods,
Eigen,
IdentificationMethods,
OptimizationMethods,
ODEMethods,
VisualizationMethods):
NoConvergence = libmp.NoConvergence
ComplexResult = libmp.ComplexResult
def __init__(ctx):
ctx._aliases = {}
# Call those that need preinitialization (e.g. for wrappers)
SpecialFunctions.__init__(ctx)
RSCache.__init__(ctx)
QuadratureMethods.__init__(ctx)
LaplaceTransformInversionMethods.__init__(ctx)
CalculusMethods.__init__(ctx)
MatrixMethods.__init__(ctx)
def _init_aliases(ctx):
for alias, value in ctx._aliases.items():
try:
setattr(ctx, alias, getattr(ctx, value))
except AttributeError:
pass
_fixed_precision = False
# XXX
verbose = False
def warn(ctx, msg):
print("Warning:", msg)
def bad_domain(ctx, msg):
raise ValueError(msg)
def _re(ctx, x):
if hasattr(x, "real"):
return x.real
return x
def _im(ctx, x):
if hasattr(x, "imag"):
return x.imag
return ctx.zero
def _as_points(ctx, x):
return x
def fneg(ctx, x, **kwargs):
return -ctx.convert(x)
def fadd(ctx, x, y, **kwargs):
return ctx.convert(x)+ctx.convert(y)
def fsub(ctx, x, y, **kwargs):
return ctx.convert(x)-ctx.convert(y)
def fmul(ctx, x, y, **kwargs):
return ctx.convert(x)*ctx.convert(y)
def fdiv(ctx, x, y, **kwargs):
return ctx.convert(x)/ctx.convert(y)
def fsum(ctx, args, absolute=False, squared=False):
if absolute:
if squared:
return sum((abs(x)**2 for x in args), ctx.zero)
return sum((abs(x) for x in args), ctx.zero)
if squared:
return sum((x**2 for x in args), ctx.zero)
return sum(args, ctx.zero)
def fdot(ctx, xs, ys=None, conjugate=False):
if ys is not None:
xs = zip(xs, ys)
if conjugate:
cf = ctx.conj
return sum((x*cf(y) for (x,y) in xs), ctx.zero)
else:
return sum((x*y for (x,y) in xs), ctx.zero)
def fprod(ctx, args):
prod = ctx.one
for arg in args:
prod *= arg
return prod
def nprint(ctx, x, n=6, **kwargs):
"""
Equivalent to ``print(nstr(x, n))``.
"""
print(ctx.nstr(x, n, **kwargs))
def chop(ctx, x, tol=None):
"""
Chops off small real or imaginary parts, or converts
numbers close to zero to exact zeros. The input can be a
single number or an iterable::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = False
>>> chop(5+1e-10j, tol=1e-9)
mpf('5.0')
>>> nprint(chop([1.0, 1e-20, 3+1e-18j, -4, 2]))
[1.0, 0.0, 3.0, -4.0, 2.0]
The tolerance defaults to ``100*eps``.
"""
if tol is None:
tol = 100*ctx.eps
try:
x = ctx.convert(x)
absx = abs(x)
if abs(x) < tol:
return ctx.zero
if ctx._is_complex_type(x):
#part_tol = min(tol, absx*tol)
part_tol = max(tol, absx*tol)
if abs(x.imag) < part_tol:
return x.real
if abs(x.real) < part_tol:
return ctx.mpc(0, x.imag)
except TypeError:
if isinstance(x, ctx.matrix):
return x.apply(lambda a: ctx.chop(a, tol))
if hasattr(x, "__iter__"):
return [ctx.chop(a, tol) for a in x]
return x
def almosteq(ctx, s, t, rel_eps=None, abs_eps=None):
r"""
Determine whether the difference between `s` and `t` is smaller
than a given epsilon, either relatively or absolutely.
Both a maximum relative difference and a maximum difference
('epsilons') may be specified. The absolute difference is
defined as `|s-t|` and the relative difference is defined
as `|s-t|/\max(|s|, |t|)`.
If only one epsilon is given, both are set to the same value.
If none is given, both epsilons are set to `2^{-p+m}` where
`p` is the current working precision and `m` is a small
integer. The default setting typically allows :func:`~mpmath.almosteq`
to be used to check for mathematical equality
in the presence of small rounding errors.
**Examples**
>>> from mpmath import *
>>> mp.dps = 15
>>> almosteq(3.141592653589793, 3.141592653589790)
True
>>> almosteq(3.141592653589793, 3.141592653589700)
False
>>> almosteq(3.141592653589793, 3.141592653589700, 1e-10)
True
>>> almosteq(1e-20, 2e-20)
True
>>> almosteq(1e-20, 2e-20, rel_eps=0, abs_eps=0)
False
"""
t = ctx.convert(t)
if abs_eps is None and rel_eps is None:
rel_eps = abs_eps = ctx.ldexp(1, -ctx.prec+4)
if abs_eps is None:
abs_eps = rel_eps
elif rel_eps is None:
rel_eps = abs_eps
diff = abs(s-t)
if diff <= abs_eps:
return True
abss = abs(s)
abst = abs(t)
if abss < abst:
err = diff/abst
else:
err = diff/abss
return err <= rel_eps
def arange(ctx, *args):
r"""
This is a generalized version of Python's :func:`~mpmath.range` function
that accepts fractional endpoints and step sizes and
returns a list of ``mpf`` instances. Like :func:`~mpmath.range`,
:func:`~mpmath.arange` can be called with 1, 2 or 3 arguments:
``arange(b)``
`[0, 1, 2, \ldots, x]`
``arange(a, b)``
`[a, a+1, a+2, \ldots, x]`
``arange(a, b, h)``
`[a, a+h, a+h, \ldots, x]`
where `b-1 \le x < b` (in the third case, `b-h \le x < b`).
Like Python's :func:`~mpmath.range`, the endpoint is not included. To
produce ranges where the endpoint is included, :func:`~mpmath.linspace`
is more convenient.
**Examples**
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = False
>>> arange(4)
[mpf('0.0'), mpf('1.0'), mpf('2.0'), mpf('3.0')]
>>> arange(1, 2, 0.25)
[mpf('1.0'), mpf('1.25'), mpf('1.5'), mpf('1.75')]
>>> arange(1, -1, -0.75)
[mpf('1.0'), mpf('0.25'), mpf('-0.5')]
"""
if not len(args) <= 3:
raise TypeError('arange expected at most 3 arguments, got %i'
% len(args))
if not len(args) >= 1:
raise TypeError('arange expected at least 1 argument, got %i'
% len(args))
# set default
a = 0
dt = 1
# interpret arguments
if len(args) == 1:
b = args[0]
elif len(args) >= 2:
a = args[0]
b = args[1]
if len(args) == 3:
dt = args[2]
a, b, dt = ctx.mpf(a), ctx.mpf(b), ctx.mpf(dt)
assert a + dt != a, 'dt is too small and would cause an infinite loop'
# adapt code for sign of dt
if a > b:
if dt > 0:
return []
op = gt
else:
if dt < 0:
return []
op = lt
# create list
result = []
i = 0
t = a
while 1:
t = a + dt*i
i += 1
if op(t, b):
result.append(t)
else:
break
return result
def linspace(ctx, *args, **kwargs):
"""
``linspace(a, b, n)`` returns a list of `n` evenly spaced
samples from `a` to `b`. The syntax ``linspace(mpi(a,b), n)``
is also valid.
This function is often more convenient than :func:`~mpmath.arange`
for partitioning an interval into subintervals, since
the endpoint is included::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = False
>>> linspace(1, 4, 4)
[mpf('1.0'), mpf('2.0'), mpf('3.0'), mpf('4.0')]
You may also provide the keyword argument ``endpoint=False``::
>>> linspace(1, 4, 4, endpoint=False)
[mpf('1.0'), mpf('1.75'), mpf('2.5'), mpf('3.25')]
"""
if len(args) == 3:
a = ctx.mpf(args[0])
b = ctx.mpf(args[1])
n = int(args[2])
elif len(args) == 2:
assert hasattr(args[0], '_mpi_')
a = args[0].a
b = args[0].b
n = int(args[1])
else:
raise TypeError('linspace expected 2 or 3 arguments, got %i' \
% len(args))
if n < 1:
raise ValueError('n must be greater than 0')
if not 'endpoint' in kwargs or kwargs['endpoint']:
if n == 1:
return [ctx.mpf(a)]
step = (b - a) / ctx.mpf(n - 1)
y = [i*step + a for i in xrange(n)]
y[-1] = b
else:
step = (b - a) / ctx.mpf(n)
y = [i*step + a for i in xrange(n)]
return y
def cos_sin(ctx, z, **kwargs):
return ctx.cos(z, **kwargs), ctx.sin(z, **kwargs)
def cospi_sinpi(ctx, z, **kwargs):
return ctx.cospi(z, **kwargs), ctx.sinpi(z, **kwargs)
def _default_hyper_maxprec(ctx, p):
return int(1000 * p**0.25 + 4*p)
_gcd = staticmethod(libmp.gcd)
list_primes = staticmethod(libmp.list_primes)
isprime = staticmethod(libmp.isprime)
bernfrac = staticmethod(libmp.bernfrac)
moebius = staticmethod(libmp.moebius)
_ifac = staticmethod(libmp.ifac)
_eulernum = staticmethod(libmp.eulernum)
_stirling1 = staticmethod(libmp.stirling1)
_stirling2 = staticmethod(libmp.stirling2)
def sum_accurately(ctx, terms, check_step=1):
prec = ctx.prec
try:
extraprec = 10
while 1:
ctx.prec = prec + extraprec + 5
max_mag = ctx.ninf
s = ctx.zero
k = 0
for term in terms():
s += term
if (not k % check_step) and term:
term_mag = ctx.mag(term)
max_mag = max(max_mag, term_mag)
sum_mag = ctx.mag(s)
if sum_mag - term_mag > ctx.prec:
break
k += 1
cancellation = max_mag - sum_mag
if cancellation != cancellation:
break
if cancellation < extraprec or ctx._fixed_precision:
break
extraprec += min(ctx.prec, cancellation)
return s
finally:
ctx.prec = prec
def mul_accurately(ctx, factors, check_step=1):
prec = ctx.prec
try:
extraprec = 10
while 1:
ctx.prec = prec + extraprec + 5
max_mag = ctx.ninf
one = ctx.one
s = one
k = 0
for factor in factors():
s *= factor
term = factor - one
if (not k % check_step):
term_mag = ctx.mag(term)
max_mag = max(max_mag, term_mag)
sum_mag = ctx.mag(s-one)
#if sum_mag - term_mag > ctx.prec:
# break
if -term_mag > ctx.prec:
break
k += 1
cancellation = max_mag - sum_mag
if cancellation != cancellation:
break
if cancellation < extraprec or ctx._fixed_precision:
break
extraprec += min(ctx.prec, cancellation)
return s
finally:
ctx.prec = prec
def power(ctx, x, y):
r"""Converts `x` and `y` to mpmath numbers and evaluates
`x^y = \exp(y \log(x))`::
>>> from mpmath import *
>>> mp.dps = 30; mp.pretty = True
>>> power(2, 0.5)
1.41421356237309504880168872421
This shows the leading few digits of a large Mersenne prime
(performing the exact calculation ``2**43112609-1`` and
displaying the result in Python would be very slow)::
>>> power(2, 43112609)-1
3.16470269330255923143453723949e+12978188
"""
return ctx.convert(x) ** ctx.convert(y)
def _zeta_int(ctx, n):
return ctx.zeta(n)
def maxcalls(ctx, f, N):
"""
Return a wrapped copy of *f* that raises ``NoConvergence`` when *f*
has been called more than *N* times::
>>> from mpmath import *
>>> mp.dps = 15
>>> f = maxcalls(sin, 10)
>>> print(sum(f(n) for n in range(10)))
1.95520948210738
>>> f(10) # doctest: +IGNORE_EXCEPTION_DETAIL
Traceback (most recent call last):
...
NoConvergence: maxcalls: function evaluated 10 times
"""
counter = [0]
def f_maxcalls_wrapped(*args, **kwargs):
counter[0] += 1
if counter[0] > N:
raise ctx.NoConvergence("maxcalls: function evaluated %i times" % N)
return f(*args, **kwargs)
return f_maxcalls_wrapped
def memoize(ctx, f):
"""
Return a wrapped copy of *f* that caches computed values, i.e.
a memoized copy of *f*. Values are only reused if the cached precision
is equal to or higher than the working precision::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> f = memoize(maxcalls(sin, 1))
>>> f(2)
0.909297426825682
>>> f(2)
0.909297426825682
>>> mp.dps = 25
>>> f(2) # doctest: +IGNORE_EXCEPTION_DETAIL
Traceback (most recent call last):
...
NoConvergence: maxcalls: function evaluated 1 times
"""
f_cache = {}
def f_cached(*args, **kwargs):
if kwargs:
key = args, tuple(kwargs.items())
else:
key = args
prec = ctx.prec
if key in f_cache:
cprec, cvalue = f_cache[key]
if cprec >= prec:
return +cvalue
value = f(*args, **kwargs)
f_cache[key] = (prec, value)
return value
f_cached.__name__ = f.__name__
f_cached.__doc__ = f.__doc__
return f_cached
| 15,985 | 31.294949 | 84 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/math2.py
|
"""
This module complements the math and cmath builtin modules by providing
fast machine precision versions of some additional functions (gamma, ...)
and wrapping math/cmath functions so that they can be called with either
real or complex arguments.
"""
import operator
import math
import cmath
# Irrational (?) constants
pi = 3.1415926535897932385
e = 2.7182818284590452354
sqrt2 = 1.4142135623730950488
sqrt5 = 2.2360679774997896964
phi = 1.6180339887498948482
ln2 = 0.69314718055994530942
ln10 = 2.302585092994045684
euler = 0.57721566490153286061
catalan = 0.91596559417721901505
khinchin = 2.6854520010653064453
apery = 1.2020569031595942854
logpi = 1.1447298858494001741
def _mathfun_real(f_real, f_complex):
def f(x, **kwargs):
if type(x) is float:
return f_real(x)
if type(x) is complex:
return f_complex(x)
try:
x = float(x)
return f_real(x)
except (TypeError, ValueError):
x = complex(x)
return f_complex(x)
f.__name__ = f_real.__name__
return f
def _mathfun(f_real, f_complex):
def f(x, **kwargs):
if type(x) is complex:
return f_complex(x)
try:
return f_real(float(x))
except (TypeError, ValueError):
return f_complex(complex(x))
f.__name__ = f_real.__name__
return f
def _mathfun_n(f_real, f_complex):
def f(*args, **kwargs):
try:
return f_real(*(float(x) for x in args))
except (TypeError, ValueError):
return f_complex(*(complex(x) for x in args))
f.__name__ = f_real.__name__
return f
# Workaround for non-raising log and sqrt in Python 2.5 and 2.4
# on Unix system
try:
math.log(-2.0)
def math_log(x):
if x <= 0.0:
raise ValueError("math domain error")
return math.log(x)
def math_sqrt(x):
if x < 0.0:
raise ValueError("math domain error")
return math.sqrt(x)
except (ValueError, TypeError):
math_log = math.log
math_sqrt = math.sqrt
pow = _mathfun_n(operator.pow, lambda x, y: complex(x)**y)
log = _mathfun_n(math_log, cmath.log)
sqrt = _mathfun(math_sqrt, cmath.sqrt)
exp = _mathfun_real(math.exp, cmath.exp)
cos = _mathfun_real(math.cos, cmath.cos)
sin = _mathfun_real(math.sin, cmath.sin)
tan = _mathfun_real(math.tan, cmath.tan)
acos = _mathfun(math.acos, cmath.acos)
asin = _mathfun(math.asin, cmath.asin)
atan = _mathfun_real(math.atan, cmath.atan)
cosh = _mathfun_real(math.cosh, cmath.cosh)
sinh = _mathfun_real(math.sinh, cmath.sinh)
tanh = _mathfun_real(math.tanh, cmath.tanh)
floor = _mathfun_real(math.floor,
lambda z: complex(math.floor(z.real), math.floor(z.imag)))
ceil = _mathfun_real(math.ceil,
lambda z: complex(math.ceil(z.real), math.ceil(z.imag)))
cos_sin = _mathfun_real(lambda x: (math.cos(x), math.sin(x)),
lambda z: (cmath.cos(z), cmath.sin(z)))
cbrt = _mathfun(lambda x: x**(1./3), lambda z: z**(1./3))
def nthroot(x, n):
r = 1./n
try:
return float(x) ** r
except (ValueError, TypeError):
return complex(x) ** r
def _sinpi_real(x):
if x < 0:
return -_sinpi_real(-x)
n, r = divmod(x, 0.5)
r *= pi
n %= 4
if n == 0: return math.sin(r)
if n == 1: return math.cos(r)
if n == 2: return -math.sin(r)
if n == 3: return -math.cos(r)
def _cospi_real(x):
if x < 0:
x = -x
n, r = divmod(x, 0.5)
r *= pi
n %= 4
if n == 0: return math.cos(r)
if n == 1: return -math.sin(r)
if n == 2: return -math.cos(r)
if n == 3: return math.sin(r)
def _sinpi_complex(z):
if z.real < 0:
return -_sinpi_complex(-z)
n, r = divmod(z.real, 0.5)
z = pi*complex(r, z.imag)
n %= 4
if n == 0: return cmath.sin(z)
if n == 1: return cmath.cos(z)
if n == 2: return -cmath.sin(z)
if n == 3: return -cmath.cos(z)
def _cospi_complex(z):
if z.real < 0:
z = -z
n, r = divmod(z.real, 0.5)
z = pi*complex(r, z.imag)
n %= 4
if n == 0: return cmath.cos(z)
if n == 1: return -cmath.sin(z)
if n == 2: return -cmath.cos(z)
if n == 3: return cmath.sin(z)
cospi = _mathfun_real(_cospi_real, _cospi_complex)
sinpi = _mathfun_real(_sinpi_real, _sinpi_complex)
def tanpi(x):
try:
return sinpi(x) / cospi(x)
except OverflowError:
if complex(x).imag > 10:
return 1j
if complex(x).imag < 10:
return -1j
raise
def cotpi(x):
try:
return cospi(x) / sinpi(x)
except OverflowError:
if complex(x).imag > 10:
return -1j
if complex(x).imag < 10:
return 1j
raise
INF = 1e300*1e300
NINF = -INF
NAN = INF-INF
EPS = 2.2204460492503131e-16
_exact_gamma = (INF, 1.0, 1.0, 2.0, 6.0, 24.0, 120.0, 720.0, 5040.0, 40320.0,
362880.0, 3628800.0, 39916800.0, 479001600.0, 6227020800.0, 87178291200.0,
1307674368000.0, 20922789888000.0, 355687428096000.0, 6402373705728000.0,
121645100408832000.0, 2432902008176640000.0)
_max_exact_gamma = len(_exact_gamma)-1
# Lanczos coefficients used by the GNU Scientific Library
_lanczos_g = 7
_lanczos_p = (0.99999999999980993, 676.5203681218851, -1259.1392167224028,
771.32342877765313, -176.61502916214059, 12.507343278686905,
-0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7)
def _gamma_real(x):
_intx = int(x)
if _intx == x:
if _intx <= 0:
#return (-1)**_intx * INF
raise ZeroDivisionError("gamma function pole")
if _intx <= _max_exact_gamma:
return _exact_gamma[_intx]
if x < 0.5:
# TODO: sinpi
return pi / (_sinpi_real(x)*_gamma_real(1-x))
else:
x -= 1.0
r = _lanczos_p[0]
for i in range(1, _lanczos_g+2):
r += _lanczos_p[i]/(x+i)
t = x + _lanczos_g + 0.5
return 2.506628274631000502417 * t**(x+0.5) * math.exp(-t) * r
def _gamma_complex(x):
if not x.imag:
return complex(_gamma_real(x.real))
if x.real < 0.5:
# TODO: sinpi
return pi / (_sinpi_complex(x)*_gamma_complex(1-x))
else:
x -= 1.0
r = _lanczos_p[0]
for i in range(1, _lanczos_g+2):
r += _lanczos_p[i]/(x+i)
t = x + _lanczos_g + 0.5
return 2.506628274631000502417 * t**(x+0.5) * cmath.exp(-t) * r
gamma = _mathfun_real(_gamma_real, _gamma_complex)
def rgamma(x):
try:
return 1./gamma(x)
except ZeroDivisionError:
return x*0.0
def factorial(x):
return gamma(x+1.0)
def arg(x):
if type(x) is float:
return math.atan2(0.0,x)
return math.atan2(x.imag,x.real)
# XXX: broken for negatives
def loggamma(x):
if type(x) not in (float, complex):
try:
x = float(x)
except (ValueError, TypeError):
x = complex(x)
try:
xreal = x.real
ximag = x.imag
except AttributeError: # py2.5
xreal = x
ximag = 0.0
# Reflection formula
# http://functions.wolfram.com/GammaBetaErf/LogGamma/16/01/01/0003/
if xreal < 0.0:
if abs(x) < 0.5:
v = log(gamma(x))
if ximag == 0:
v = v.conjugate()
return v
z = 1-x
try:
re = z.real
im = z.imag
except AttributeError: # py2.5
re = z
im = 0.0
refloor = floor(re)
if im == 0.0:
imsign = 0
elif im < 0.0:
imsign = -1
else:
imsign = 1
return (-pi*1j)*abs(refloor)*(1-abs(imsign)) + logpi - \
log(sinpi(z-refloor)) - loggamma(z) + 1j*pi*refloor*imsign
if x == 1.0 or x == 2.0:
return x*0
p = 0.
while abs(x) < 11:
p -= log(x)
x += 1.0
s = 0.918938533204672742 + (x-0.5)*log(x) - x
r = 1./x
r2 = r*r
s += 0.083333333333333333333*r; r *= r2
s += -0.0027777777777777777778*r; r *= r2
s += 0.00079365079365079365079*r; r *= r2
s += -0.0005952380952380952381*r; r *= r2
s += 0.00084175084175084175084*r; r *= r2
s += -0.0019175269175269175269*r; r *= r2
s += 0.0064102564102564102564*r; r *= r2
s += -0.02955065359477124183*r
return s + p
_psi_coeff = [
0.083333333333333333333,
-0.0083333333333333333333,
0.003968253968253968254,
-0.0041666666666666666667,
0.0075757575757575757576,
-0.021092796092796092796,
0.083333333333333333333,
-0.44325980392156862745,
3.0539543302701197438,
-26.456212121212121212]
def _digamma_real(x):
_intx = int(x)
if _intx == x:
if _intx <= 0:
raise ZeroDivisionError("polygamma pole")
if x < 0.5:
x = 1.0-x
s = pi*cotpi(x)
else:
s = 0.0
while x < 10.0:
s -= 1.0/x
x += 1.0
x2 = x**-2
t = x2
for c in _psi_coeff:
s -= c*t
if t < 1e-20:
break
t *= x2
return s + math_log(x) - 0.5/x
def _digamma_complex(x):
if not x.imag:
return complex(_digamma_real(x.real))
if x.real < 0.5:
x = 1.0-x
s = pi*cotpi(x)
else:
s = 0.0
while abs(x) < 10.0:
s -= 1.0/x
x += 1.0
x2 = x**-2
t = x2
for c in _psi_coeff:
s -= c*t
if abs(t) < 1e-20:
break
t *= x2
return s + cmath.log(x) - 0.5/x
digamma = _mathfun_real(_digamma_real, _digamma_complex)
# TODO: could implement complex erf and erfc here. Need
# to find an accurate method (avoiding cancellation)
# for approx. 1 < abs(x) < 9.
_erfc_coeff_P = [
1.0000000161203922312,
2.1275306946297962644,
2.2280433377390253297,
1.4695509105618423961,
0.66275911699770787537,
0.20924776504163751585,
0.045459713768411264339,
0.0063065951710717791934,
0.00044560259661560421715][::-1]
_erfc_coeff_Q = [
1.0000000000000000000,
3.2559100272784894318,
4.9019435608903239131,
4.4971472894498014205,
2.7845640601891186528,
1.2146026030046904138,
0.37647108453729465912,
0.080970149639040548613,
0.011178148899483545902,
0.00078981003831980423513][::-1]
def _polyval(coeffs, x):
p = coeffs[0]
for c in coeffs[1:]:
p = c + x*p
return p
def _erf_taylor(x):
# Taylor series assuming 0 <= x <= 1
x2 = x*x
s = t = x
n = 1
while abs(t) > 1e-17:
t *= x2/n
s -= t/(n+n+1)
n += 1
t *= x2/n
s += t/(n+n+1)
n += 1
return 1.1283791670955125739*s
def _erfc_mid(x):
# Rational approximation assuming 0 <= x <= 9
return exp(-x*x)*_polyval(_erfc_coeff_P,x)/_polyval(_erfc_coeff_Q,x)
def _erfc_asymp(x):
# Asymptotic expansion assuming x >= 9
x2 = x*x
v = exp(-x2)/x*0.56418958354775628695
r = t = 0.5 / x2
s = 1.0
for n in range(1,22,4):
s -= t
t *= r * (n+2)
s += t
t *= r * (n+4)
if abs(t) < 1e-17:
break
return s * v
def erf(x):
"""
erf of a real number.
"""
x = float(x)
if x != x:
return x
if x < 0.0:
return -erf(-x)
if x >= 1.0:
if x >= 6.0:
return 1.0
return 1.0 - _erfc_mid(x)
return _erf_taylor(x)
def erfc(x):
"""
erfc of a real number.
"""
x = float(x)
if x != x:
return x
if x < 0.0:
if x < -6.0:
return 2.0
return 2.0-erfc(-x)
if x > 9.0:
return _erfc_asymp(x)
if x >= 1.0:
return _erfc_mid(x)
return 1.0 - _erf_taylor(x)
gauss42 = [\
(0.99839961899006235, 0.0041059986046490839),
(-0.99839961899006235, 0.0041059986046490839),
(0.9915772883408609, 0.009536220301748501),
(-0.9915772883408609,0.009536220301748501),
(0.97934250806374812, 0.014922443697357493),
(-0.97934250806374812, 0.014922443697357493),
(0.96175936533820439,0.020227869569052644),
(-0.96175936533820439, 0.020227869569052644),
(0.93892355735498811, 0.025422959526113047),
(-0.93892355735498811,0.025422959526113047),
(0.91095972490412735, 0.030479240699603467),
(-0.91095972490412735, 0.030479240699603467),
(0.87802056981217269,0.03536907109759211),
(-0.87802056981217269, 0.03536907109759211),
(0.8402859832618168, 0.040065735180692258),
(-0.8402859832618168,0.040065735180692258),
(0.7979620532554873, 0.044543577771965874),
(-0.7979620532554873, 0.044543577771965874),
(0.75127993568948048,0.048778140792803244),
(-0.75127993568948048, 0.048778140792803244),
(0.70049459055617114, 0.052746295699174064),
(-0.70049459055617114,0.052746295699174064),
(0.64588338886924779, 0.056426369358018376),
(-0.64588338886924779, 0.056426369358018376),
(0.58774459748510932, 0.059798262227586649),
(-0.58774459748510932, 0.059798262227586649),
(0.5263957499311922, 0.062843558045002565),
(-0.5263957499311922, 0.062843558045002565),
(0.46217191207042191, 0.065545624364908975),
(-0.46217191207042191, 0.065545624364908975),
(0.39542385204297503, 0.067889703376521934),
(-0.39542385204297503, 0.067889703376521934),
(0.32651612446541151, 0.069862992492594159),
(-0.32651612446541151, 0.069862992492594159),
(0.25582507934287907, 0.071454714265170971),
(-0.25582507934287907, 0.071454714265170971),
(0.18373680656485453, 0.072656175243804091),
(-0.18373680656485453, 0.072656175243804091),
(0.11064502720851986, 0.073460813453467527),
(-0.11064502720851986, 0.073460813453467527),
(0.036948943165351772, 0.073864234232172879),
(-0.036948943165351772, 0.073864234232172879)]
EI_ASYMP_CONVERGENCE_RADIUS = 40.0
def ei_asymp(z, _e1=False):
r = 1./z
s = t = 1.0
k = 1
while 1:
t *= k*r
s += t
if abs(t) < 1e-16:
break
k += 1
v = s*exp(z)/z
if _e1:
if type(z) is complex:
zreal = z.real
zimag = z.imag
else:
zreal = z
zimag = 0.0
if zimag == 0.0 and zreal > 0.0:
v += pi*1j
else:
if type(z) is complex:
if z.imag > 0:
v += pi*1j
if z.imag < 0:
v -= pi*1j
return v
def ei_taylor(z, _e1=False):
s = t = z
k = 2
while 1:
t = t*z/k
term = t/k
if abs(term) < 1e-17:
break
s += term
k += 1
s += euler
if _e1:
s += log(-z)
else:
if type(z) is float or z.imag == 0.0:
s += math_log(abs(z))
else:
s += cmath.log(z)
return s
def ei(z, _e1=False):
typez = type(z)
if typez not in (float, complex):
try:
z = float(z)
typez = float
except (TypeError, ValueError):
z = complex(z)
typez = complex
if not z:
return -INF
absz = abs(z)
if absz > EI_ASYMP_CONVERGENCE_RADIUS:
return ei_asymp(z, _e1)
elif absz <= 2.0 or (typez is float and z > 0.0):
return ei_taylor(z, _e1)
# Integrate, starting from whichever is smaller of a Taylor
# series value or an asymptotic series value
if typez is complex and z.real > 0.0:
zref = z / absz
ref = ei_taylor(zref, _e1)
else:
zref = EI_ASYMP_CONVERGENCE_RADIUS * z / absz
ref = ei_asymp(zref, _e1)
C = (zref-z)*0.5
D = (zref+z)*0.5
s = 0.0
if type(z) is complex:
_exp = cmath.exp
else:
_exp = math.exp
for x,w in gauss42:
t = C*x+D
s += w*_exp(t)/t
ref -= C*s
return ref
def e1(z):
# hack to get consistent signs if the imaginary part if 0
# and signed
typez = type(z)
if type(z) not in (float, complex):
try:
z = float(z)
typez = float
except (TypeError, ValueError):
z = complex(z)
typez = complex
if typez is complex and not z.imag:
z = complex(z.real, 0.0)
# end hack
return -ei(-z, _e1=True)
_zeta_int = [\
-0.5,
0.0,
1.6449340668482264365,1.2020569031595942854,1.0823232337111381915,
1.0369277551433699263,1.0173430619844491397,1.0083492773819228268,
1.0040773561979443394,1.0020083928260822144,1.0009945751278180853,
1.0004941886041194646,1.0002460865533080483,1.0001227133475784891,
1.0000612481350587048,1.0000305882363070205,1.0000152822594086519,
1.0000076371976378998,1.0000038172932649998,1.0000019082127165539,
1.0000009539620338728,1.0000004769329867878,1.0000002384505027277,
1.0000001192199259653,1.0000000596081890513,1.0000000298035035147,
1.0000000149015548284]
_zeta_P = [-3.50000000087575873, -0.701274355654678147,
-0.0672313458590012612, -0.00398731457954257841,
-0.000160948723019303141, -4.67633010038383371e-6,
-1.02078104417700585e-7, -1.68030037095896287e-9,
-1.85231868742346722e-11][::-1]
_zeta_Q = [1.00000000000000000, -0.936552848762465319,
-0.0588835413263763741, -0.00441498861482948666,
-0.000143416758067432622, -5.10691659585090782e-6,
-9.58813053268913799e-8, -1.72963791443181972e-9,
-1.83527919681474132e-11][::-1]
_zeta_1 = [3.03768838606128127e-10, -1.21924525236601262e-8,
2.01201845887608893e-7, -1.53917240683468381e-6,
-5.09890411005967954e-7, 0.000122464707271619326,
-0.000905721539353130232, -0.00239315326074843037,
0.084239750013159168, 0.418938517907442414, 0.500000001921884009]
_zeta_0 = [-3.46092485016748794e-10, -6.42610089468292485e-9,
1.76409071536679773e-7, -1.47141263991560698e-6, -6.38880222546167613e-7,
0.000122641099800668209, -0.000905894913516772796, -0.00239303348507992713,
0.0842396947501199816, 0.418938533204660256, 0.500000000000000052]
def zeta(s):
"""
Riemann zeta function, real argument
"""
if not isinstance(s, (float, int)):
try:
s = float(s)
except (ValueError, TypeError):
try:
s = complex(s)
if not s.imag:
return complex(zeta(s.real))
except (ValueError, TypeError):
pass
raise NotImplementedError
if s == 1:
raise ValueError("zeta(1) pole")
if s >= 27:
return 1.0 + 2.0**(-s) + 3.0**(-s)
n = int(s)
if n == s:
if n >= 0:
return _zeta_int[n]
if not (n % 2):
return 0.0
if s <= 0.0:
return 2.**s*pi**(s-1)*_sinpi_real(0.5*s)*_gamma_real(1-s)*zeta(1-s)
if s <= 2.0:
if s <= 1.0:
return _polyval(_zeta_0,s)/(s-1)
return _polyval(_zeta_1,s)/(s-1)
z = _polyval(_zeta_P,s) / _polyval(_zeta_Q,s)
return 1.0 + 2.0**(-s) + 3.0**(-s) + 4.0**(-s)*z
| 18,561 | 26.580981 | 77 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/identification.py
|
"""
Implements the PSLQ algorithm for integer relation detection,
and derivative algorithms for constant recognition.
"""
from .libmp.backend import xrange
from .libmp import int_types, sqrt_fixed
# round to nearest integer (can be done more elegantly...)
def round_fixed(x, prec):
return ((x + (1<<(prec-1))) >> prec) << prec
class IdentificationMethods(object):
pass
def pslq(ctx, x, tol=None, maxcoeff=1000, maxsteps=100, verbose=False):
r"""
Given a vector of real numbers `x = [x_0, x_1, ..., x_n]`, ``pslq(x)``
uses the PSLQ algorithm to find a list of integers
`[c_0, c_1, ..., c_n]` such that
.. math ::
|c_1 x_1 + c_2 x_2 + ... + c_n x_n| < \mathrm{tol}
and such that `\max |c_k| < \mathrm{maxcoeff}`. If no such vector
exists, :func:`~mpmath.pslq` returns ``None``. The tolerance defaults to
3/4 of the working precision.
**Examples**
Find rational approximations for `\pi`::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> pslq([-1, pi], tol=0.01)
[22, 7]
>>> pslq([-1, pi], tol=0.001)
[355, 113]
>>> mpf(22)/7; mpf(355)/113; +pi
3.14285714285714
3.14159292035398
3.14159265358979
Pi is not a rational number with denominator less than 1000::
>>> pslq([-1, pi])
>>>
To within the standard precision, it can however be approximated
by at least one rational number with denominator less than `10^{12}`::
>>> p, q = pslq([-1, pi], maxcoeff=10**12)
>>> print(p); print(q)
238410049439
75888275702
>>> mpf(p)/q
3.14159265358979
The PSLQ algorithm can be applied to long vectors. For example,
we can investigate the rational (in)dependence of integer square
roots::
>>> mp.dps = 30
>>> pslq([sqrt(n) for n in range(2, 5+1)])
>>>
>>> pslq([sqrt(n) for n in range(2, 6+1)])
>>>
>>> pslq([sqrt(n) for n in range(2, 8+1)])
[2, 0, 0, 0, 0, 0, -1]
**Machin formulas**
A famous formula for `\pi` is Machin's,
.. math ::
\frac{\pi}{4} = 4 \operatorname{acot} 5 - \operatorname{acot} 239
There are actually infinitely many formulas of this type. Two
others are
.. math ::
\frac{\pi}{4} = \operatorname{acot} 1
\frac{\pi}{4} = 12 \operatorname{acot} 49 + 32 \operatorname{acot} 57
+ 5 \operatorname{acot} 239 + 12 \operatorname{acot} 110443
We can easily verify the formulas using the PSLQ algorithm::
>>> mp.dps = 30
>>> pslq([pi/4, acot(1)])
[1, -1]
>>> pslq([pi/4, acot(5), acot(239)])
[1, -4, 1]
>>> pslq([pi/4, acot(49), acot(57), acot(239), acot(110443)])
[1, -12, -32, 5, -12]
We could try to generate a custom Machin-like formula by running
the PSLQ algorithm with a few inverse cotangent values, for example
acot(2), acot(3) ... acot(10). Unfortunately, there is a linear
dependence among these values, resulting in only that dependence
being detected, with a zero coefficient for `\pi`::
>>> pslq([pi] + [acot(n) for n in range(2,11)])
[0, 1, -1, 0, 0, 0, -1, 0, 0, 0]
We get better luck by removing linearly dependent terms::
>>> pslq([pi] + [acot(n) for n in range(2,11) if n not in (3, 5)])
[1, -8, 0, 0, 4, 0, 0, 0]
In other words, we found the following formula::
>>> 8*acot(2) - 4*acot(7)
3.14159265358979323846264338328
>>> +pi
3.14159265358979323846264338328
**Algorithm**
This is a fairly direct translation to Python of the pseudocode given by
David Bailey, "The PSLQ Integer Relation Algorithm":
http://www.cecm.sfu.ca/organics/papers/bailey/paper/html/node3.html
The present implementation uses fixed-point instead of floating-point
arithmetic, since this is significantly (about 7x) faster.
"""
n = len(x)
if n < 2:
raise ValueError("n cannot be less than 2")
# At too low precision, the algorithm becomes meaningless
prec = ctx.prec
if prec < 53:
raise ValueError("prec cannot be less than 53")
if verbose and prec // max(2,n) < 5:
print("Warning: precision for PSLQ may be too low")
target = int(prec * 0.75)
if tol is None:
tol = ctx.mpf(2)**(-target)
else:
tol = ctx.convert(tol)
extra = 60
prec += extra
if verbose:
print("PSLQ using prec %i and tol %s" % (prec, ctx.nstr(tol)))
tol = ctx.to_fixed(tol, prec)
assert tol
# Convert to fixed-point numbers. The dummy None is added so we can
# use 1-based indexing. (This just allows us to be consistent with
# Bailey's indexing. The algorithm is 100 lines long, so debugging
# a single wrong index can be painful.)
x = [None] + [ctx.to_fixed(ctx.mpf(xk), prec) for xk in x]
# Sanity check on magnitudes
minx = min(abs(xx) for xx in x[1:])
if not minx:
raise ValueError("PSLQ requires a vector of nonzero numbers")
if minx < tol//100:
if verbose:
print("STOPPING: (one number is too small)")
return None
g = sqrt_fixed((4<<prec)//3, prec)
A = {}
B = {}
H = {}
# Initialization
# step 1
for i in xrange(1, n+1):
for j in xrange(1, n+1):
A[i,j] = B[i,j] = (i==j) << prec
H[i,j] = 0
# step 2
s = [None] + [0] * n
for k in xrange(1, n+1):
t = 0
for j in xrange(k, n+1):
t += (x[j]**2 >> prec)
s[k] = sqrt_fixed(t, prec)
t = s[1]
y = x[:]
for k in xrange(1, n+1):
y[k] = (x[k] << prec) // t
s[k] = (s[k] << prec) // t
# step 3
for i in xrange(1, n+1):
for j in xrange(i+1, n):
H[i,j] = 0
if i <= n-1:
if s[i]:
H[i,i] = (s[i+1] << prec) // s[i]
else:
H[i,i] = 0
for j in range(1, i):
sjj1 = s[j]*s[j+1]
if sjj1:
H[i,j] = ((-y[i]*y[j])<<prec)//sjj1
else:
H[i,j] = 0
# step 4
for i in xrange(2, n+1):
for j in xrange(i-1, 0, -1):
#t = floor(H[i,j]/H[j,j] + 0.5)
if H[j,j]:
t = round_fixed((H[i,j] << prec)//H[j,j], prec)
else:
#t = 0
continue
y[j] = y[j] + (t*y[i] >> prec)
for k in xrange(1, j+1):
H[i,k] = H[i,k] - (t*H[j,k] >> prec)
for k in xrange(1, n+1):
A[i,k] = A[i,k] - (t*A[j,k] >> prec)
B[k,j] = B[k,j] + (t*B[k,i] >> prec)
# Main algorithm
for REP in range(maxsteps):
# Step 1
m = -1
szmax = -1
for i in range(1, n):
h = H[i,i]
sz = (g**i * abs(h)) >> (prec*(i-1))
if sz > szmax:
m = i
szmax = sz
# Step 2
y[m], y[m+1] = y[m+1], y[m]
tmp = {}
for i in xrange(1,n+1): H[m,i], H[m+1,i] = H[m+1,i], H[m,i]
for i in xrange(1,n+1): A[m,i], A[m+1,i] = A[m+1,i], A[m,i]
for i in xrange(1,n+1): B[i,m], B[i,m+1] = B[i,m+1], B[i,m]
# Step 3
if m <= n - 2:
t0 = sqrt_fixed((H[m,m]**2 + H[m,m+1]**2)>>prec, prec)
# A zero element probably indicates that the precision has
# been exhausted. XXX: this could be spurious, due to
# using fixed-point arithmetic
if not t0:
break
t1 = (H[m,m] << prec) // t0
t2 = (H[m,m+1] << prec) // t0
for i in xrange(m, n+1):
t3 = H[i,m]
t4 = H[i,m+1]
H[i,m] = (t1*t3+t2*t4) >> prec
H[i,m+1] = (-t2*t3+t1*t4) >> prec
# Step 4
for i in xrange(m+1, n+1):
for j in xrange(min(i-1, m+1), 0, -1):
try:
t = round_fixed((H[i,j] << prec)//H[j,j], prec)
# Precision probably exhausted
except ZeroDivisionError:
break
y[j] = y[j] + ((t*y[i]) >> prec)
for k in xrange(1, j+1):
H[i,k] = H[i,k] - (t*H[j,k] >> prec)
for k in xrange(1, n+1):
A[i,k] = A[i,k] - (t*A[j,k] >> prec)
B[k,j] = B[k,j] + (t*B[k,i] >> prec)
# Until a relation is found, the error typically decreases
# slowly (e.g. a factor 1-10) with each step TODO: we could
# compare err from two successive iterations. If there is a
# large drop (several orders of magnitude), that indicates a
# "high quality" relation was detected. Reporting this to
# the user somehow might be useful.
best_err = maxcoeff<<prec
for i in xrange(1, n+1):
err = abs(y[i])
# Maybe we are done?
if err < tol:
# We are done if the coefficients are acceptable
vec = [int(round_fixed(B[j,i], prec) >> prec) for j in \
range(1,n+1)]
if max(abs(v) for v in vec) < maxcoeff:
if verbose:
print("FOUND relation at iter %i/%i, error: %s" % \
(REP, maxsteps, ctx.nstr(err / ctx.mpf(2)**prec, 1)))
return vec
best_err = min(err, best_err)
# Calculate a lower bound for the norm. We could do this
# more exactly (using the Euclidean norm) but there is probably
# no practical benefit.
recnorm = max(abs(h) for h in H.values())
if recnorm:
norm = ((1 << (2*prec)) // recnorm) >> prec
norm //= 100
else:
norm = ctx.inf
if verbose:
print("%i/%i: Error: %8s Norm: %s" % \
(REP, maxsteps, ctx.nstr(best_err / ctx.mpf(2)**prec, 1), norm))
if norm >= maxcoeff:
break
if verbose:
print("CANCELLING after step %i/%i." % (REP, maxsteps))
print("Could not find an integer relation. Norm bound: %s" % norm)
return None
def findpoly(ctx, x, n=1, **kwargs):
r"""
``findpoly(x, n)`` returns the coefficients of an integer
polynomial `P` of degree at most `n` such that `P(x) \approx 0`.
If no polynomial having `x` as a root can be found,
:func:`~mpmath.findpoly` returns ``None``.
:func:`~mpmath.findpoly` works by successively calling :func:`~mpmath.pslq` with
the vectors `[1, x]`, `[1, x, x^2]`, `[1, x, x^2, x^3]`, ...,
`[1, x, x^2, .., x^n]` as input. Keyword arguments given to
:func:`~mpmath.findpoly` are forwarded verbatim to :func:`~mpmath.pslq`. In
particular, you can specify a tolerance for `P(x)` with ``tol``
and a maximum permitted coefficient size with ``maxcoeff``.
For large values of `n`, it is recommended to run :func:`~mpmath.findpoly`
at high precision; preferably 50 digits or more.
**Examples**
By default (degree `n = 1`), :func:`~mpmath.findpoly` simply finds a linear
polynomial with a rational root::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> findpoly(0.7)
[-10, 7]
The generated coefficient list is valid input to ``polyval`` and
``polyroots``::
>>> nprint(polyval(findpoly(phi, 2), phi), 1)
-2.0e-16
>>> for r in polyroots(findpoly(phi, 2)):
... print(r)
...
-0.618033988749895
1.61803398874989
Numbers of the form `m + n \sqrt p` for integers `(m, n, p)` are
solutions to quadratic equations. As we find here, `1+\sqrt 2`
is a root of the polynomial `x^2 - 2x - 1`::
>>> findpoly(1+sqrt(2), 2)
[1, -2, -1]
>>> findroot(lambda x: x**2 - 2*x - 1, 1)
2.4142135623731
Despite only containing square roots, the following number results
in a polynomial of degree 4::
>>> findpoly(sqrt(2)+sqrt(3), 4)
[1, 0, -10, 0, 1]
In fact, `x^4 - 10x^2 + 1` is the *minimal polynomial* of
`r = \sqrt 2 + \sqrt 3`, meaning that a rational polynomial of
lower degree having `r` as a root does not exist. Given sufficient
precision, :func:`~mpmath.findpoly` will usually find the correct
minimal polynomial of a given algebraic number.
**Non-algebraic numbers**
If :func:`~mpmath.findpoly` fails to find a polynomial with given
coefficient size and tolerance constraints, that means no such
polynomial exists.
We can verify that `\pi` is not an algebraic number of degree 3 with
coefficients less than 1000::
>>> mp.dps = 15
>>> findpoly(pi, 3)
>>>
It is always possible to find an algebraic approximation of a number
using one (or several) of the following methods:
1. Increasing the permitted degree
2. Allowing larger coefficients
3. Reducing the tolerance
One example of each method is shown below::
>>> mp.dps = 15
>>> findpoly(pi, 4)
[95, -545, 863, -183, -298]
>>> findpoly(pi, 3, maxcoeff=10000)
[836, -1734, -2658, -457]
>>> findpoly(pi, 3, tol=1e-7)
[-4, 22, -29, -2]
It is unknown whether Euler's constant is transcendental (or even
irrational). We can use :func:`~mpmath.findpoly` to check that if is
an algebraic number, its minimal polynomial must have degree
at least 7 and a coefficient of magnitude at least 1000000::
>>> mp.dps = 200
>>> findpoly(euler, 6, maxcoeff=10**6, tol=1e-100, maxsteps=1000)
>>>
Note that the high precision and strict tolerance is necessary
for such high-degree runs, since otherwise unwanted low-accuracy
approximations will be detected. It may also be necessary to set
maxsteps high to prevent a premature exit (before the coefficient
bound has been reached). Running with ``verbose=True`` to get an
idea what is happening can be useful.
"""
x = ctx.mpf(x)
if n < 1:
raise ValueError("n cannot be less than 1")
if x == 0:
return [1, 0]
xs = [ctx.mpf(1)]
for i in range(1,n+1):
xs.append(x**i)
a = ctx.pslq(xs, **kwargs)
if a is not None:
return a[::-1]
def fracgcd(p, q):
x, y = p, q
while y:
x, y = y, x % y
if x != 1:
p //= x
q //= x
if q == 1:
return p
return p, q
def pslqstring(r, constants):
q = r[0]
r = r[1:]
s = []
for i in range(len(r)):
p = r[i]
if p:
z = fracgcd(-p,q)
cs = constants[i][1]
if cs == '1':
cs = ''
else:
cs = '*' + cs
if isinstance(z, int_types):
if z > 0: term = str(z) + cs
else: term = ("(%s)" % z) + cs
else:
term = ("(%s/%s)" % z) + cs
s.append(term)
s = ' + '.join(s)
if '+' in s or '*' in s:
s = '(' + s + ')'
return s or '0'
def prodstring(r, constants):
q = r[0]
r = r[1:]
num = []
den = []
for i in range(len(r)):
p = r[i]
if p:
z = fracgcd(-p,q)
cs = constants[i][1]
if isinstance(z, int_types):
if abs(z) == 1: t = cs
else: t = '%s**%s' % (cs, abs(z))
([num,den][z<0]).append(t)
else:
t = '%s**(%s/%s)' % (cs, abs(z[0]), z[1])
([num,den][z[0]<0]).append(t)
num = '*'.join(num)
den = '*'.join(den)
if num and den: return "(%s)/(%s)" % (num, den)
if num: return num
if den: return "1/(%s)" % den
def quadraticstring(ctx,t,a,b,c):
if c < 0:
a,b,c = -a,-b,-c
u1 = (-b+ctx.sqrt(b**2-4*a*c))/(2*c)
u2 = (-b-ctx.sqrt(b**2-4*a*c))/(2*c)
if abs(u1-t) < abs(u2-t):
if b: s = '((%s+sqrt(%s))/%s)' % (-b,b**2-4*a*c,2*c)
else: s = '(sqrt(%s)/%s)' % (-4*a*c,2*c)
else:
if b: s = '((%s-sqrt(%s))/%s)' % (-b,b**2-4*a*c,2*c)
else: s = '(-sqrt(%s)/%s)' % (-4*a*c,2*c)
return s
# Transformation y = f(x,c), with inverse function x = f(y,c)
# The third entry indicates whether the transformation is
# redundant when c = 1
transforms = [
(lambda ctx,x,c: x*c, '$y/$c', 0),
(lambda ctx,x,c: x/c, '$c*$y', 1),
(lambda ctx,x,c: c/x, '$c/$y', 0),
(lambda ctx,x,c: (x*c)**2, 'sqrt($y)/$c', 0),
(lambda ctx,x,c: (x/c)**2, '$c*sqrt($y)', 1),
(lambda ctx,x,c: (c/x)**2, '$c/sqrt($y)', 0),
(lambda ctx,x,c: c*x**2, 'sqrt($y)/sqrt($c)', 1),
(lambda ctx,x,c: x**2/c, 'sqrt($c)*sqrt($y)', 1),
(lambda ctx,x,c: c/x**2, 'sqrt($c)/sqrt($y)', 1),
(lambda ctx,x,c: ctx.sqrt(x*c), '$y**2/$c', 0),
(lambda ctx,x,c: ctx.sqrt(x/c), '$c*$y**2', 1),
(lambda ctx,x,c: ctx.sqrt(c/x), '$c/$y**2', 0),
(lambda ctx,x,c: c*ctx.sqrt(x), '$y**2/$c**2', 1),
(lambda ctx,x,c: ctx.sqrt(x)/c, '$c**2*$y**2', 1),
(lambda ctx,x,c: c/ctx.sqrt(x), '$c**2/$y**2', 1),
(lambda ctx,x,c: ctx.exp(x*c), 'log($y)/$c', 0),
(lambda ctx,x,c: ctx.exp(x/c), '$c*log($y)', 1),
(lambda ctx,x,c: ctx.exp(c/x), '$c/log($y)', 0),
(lambda ctx,x,c: c*ctx.exp(x), 'log($y/$c)', 1),
(lambda ctx,x,c: ctx.exp(x)/c, 'log($c*$y)', 1),
(lambda ctx,x,c: c/ctx.exp(x), 'log($c/$y)', 0),
(lambda ctx,x,c: ctx.ln(x*c), 'exp($y)/$c', 0),
(lambda ctx,x,c: ctx.ln(x/c), '$c*exp($y)', 1),
(lambda ctx,x,c: ctx.ln(c/x), '$c/exp($y)', 0),
(lambda ctx,x,c: c*ctx.ln(x), 'exp($y/$c)', 1),
(lambda ctx,x,c: ctx.ln(x)/c, 'exp($c*$y)', 1),
(lambda ctx,x,c: c/ctx.ln(x), 'exp($c/$y)', 0),
]
def identify(ctx, x, constants=[], tol=None, maxcoeff=1000, full=False,
verbose=False):
"""
Given a real number `x`, ``identify(x)`` attempts to find an exact
formula for `x`. This formula is returned as a string. If no match
is found, ``None`` is returned. With ``full=True``, a list of
matching formulas is returned.
As a simple example, :func:`~mpmath.identify` will find an algebraic
formula for the golden ratio::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> identify(phi)
'((1+sqrt(5))/2)'
:func:`~mpmath.identify` can identify simple algebraic numbers and simple
combinations of given base constants, as well as certain basic
transformations thereof. More specifically, :func:`~mpmath.identify`
looks for the following:
1. Fractions
2. Quadratic algebraic numbers
3. Rational linear combinations of the base constants
4. Any of the above after first transforming `x` into `f(x)` where
`f(x)` is `1/x`, `\sqrt x`, `x^2`, `\log x` or `\exp x`, either
directly or with `x` or `f(x)` multiplied or divided by one of
the base constants
5. Products of fractional powers of the base constants and
small integers
Base constants can be given as a list of strings representing mpmath
expressions (:func:`~mpmath.identify` will ``eval`` the strings to numerical
values and use the original strings for the output), or as a dict of
formula:value pairs.
In order not to produce spurious results, :func:`~mpmath.identify` should
be used with high precision; preferably 50 digits or more.
**Examples**
Simple identifications can be performed safely at standard
precision. Here the default recognition of rational, algebraic,
and exp/log of algebraic numbers is demonstrated::
>>> mp.dps = 15
>>> identify(0.22222222222222222)
'(2/9)'
>>> identify(1.9662210973805663)
'sqrt(((24+sqrt(48))/8))'
>>> identify(4.1132503787829275)
'exp((sqrt(8)/2))'
>>> identify(0.881373587019543)
'log(((2+sqrt(8))/2))'
By default, :func:`~mpmath.identify` does not recognize `\pi`. At standard
precision it finds a not too useful approximation. At slightly
increased precision, this approximation is no longer accurate
enough and :func:`~mpmath.identify` more correctly returns ``None``::
>>> identify(pi)
'(2**(176/117)*3**(20/117)*5**(35/39))/(7**(92/117))'
>>> mp.dps = 30
>>> identify(pi)
>>>
Numbers such as `\pi`, and simple combinations of user-defined
constants, can be identified if they are provided explicitly::
>>> identify(3*pi-2*e, ['pi', 'e'])
'(3*pi + (-2)*e)'
Here is an example using a dict of constants. Note that the
constants need not be "atomic"; :func:`~mpmath.identify` can just
as well express the given number in terms of expressions
given by formulas::
>>> identify(pi+e, {'a':pi+2, 'b':2*e})
'((-2) + 1*a + (1/2)*b)'
Next, we attempt some identifications with a set of base constants.
It is necessary to increase the precision a bit.
>>> mp.dps = 50
>>> base = ['sqrt(2)','pi','log(2)']
>>> identify(0.25, base)
'(1/4)'
>>> identify(3*pi + 2*sqrt(2) + 5*log(2)/7, base)
'(2*sqrt(2) + 3*pi + (5/7)*log(2))'
>>> identify(exp(pi+2), base)
'exp((2 + 1*pi))'
>>> identify(1/(3+sqrt(2)), base)
'((3/7) + (-1/7)*sqrt(2))'
>>> identify(sqrt(2)/(3*pi+4), base)
'sqrt(2)/(4 + 3*pi)'
>>> identify(5**(mpf(1)/3)*pi*log(2)**2, base)
'5**(1/3)*pi*log(2)**2'
An example of an erroneous solution being found when too low
precision is used::
>>> mp.dps = 15
>>> identify(1/(3*pi-4*e+sqrt(8)), ['pi', 'e', 'sqrt(2)'])
'((11/25) + (-158/75)*pi + (76/75)*e + (44/15)*sqrt(2))'
>>> mp.dps = 50
>>> identify(1/(3*pi-4*e+sqrt(8)), ['pi', 'e', 'sqrt(2)'])
'1/(3*pi + (-4)*e + 2*sqrt(2))'
**Finding approximate solutions**
The tolerance ``tol`` defaults to 3/4 of the working precision.
Lowering the tolerance is useful for finding approximate matches.
We can for example try to generate approximations for pi::
>>> mp.dps = 15
>>> identify(pi, tol=1e-2)
'(22/7)'
>>> identify(pi, tol=1e-3)
'(355/113)'
>>> identify(pi, tol=1e-10)
'(5**(339/269))/(2**(64/269)*3**(13/269)*7**(92/269))'
With ``full=True``, and by supplying a few base constants,
``identify`` can generate almost endless lists of approximations
for any number (the output below has been truncated to show only
the first few)::
>>> for p in identify(pi, ['e', 'catalan'], tol=1e-5, full=True):
... print(p)
... # doctest: +ELLIPSIS
e/log((6 + (-4/3)*e))
(3**3*5*e*catalan**2)/(2*7**2)
sqrt(((-13) + 1*e + 22*catalan))
log(((-6) + 24*e + 4*catalan)/e)
exp(catalan*((-1/5) + (8/15)*e))
catalan*(6 + (-6)*e + 15*catalan)
sqrt((5 + 26*e + (-3)*catalan))/e
e*sqrt(((-27) + 2*e + 25*catalan))
log(((-1) + (-11)*e + 59*catalan))
((3/20) + (21/20)*e + (3/20)*catalan)
...
The numerical values are roughly as close to `\pi` as permitted by the
specified tolerance:
>>> e/log(6-4*e/3)
3.14157719846001
>>> 135*e*catalan**2/98
3.14166950419369
>>> sqrt(e-13+22*catalan)
3.14158000062992
>>> log(24*e-6+4*catalan)-1
3.14158791577159
**Symbolic processing**
The output formula can be evaluated as a Python expression.
Note however that if fractions (like '2/3') are present in
the formula, Python's :func:`~mpmath.eval()` may erroneously perform
integer division. Note also that the output is not necessarily
in the algebraically simplest form::
>>> identify(sqrt(2))
'(sqrt(8)/2)'
As a solution to both problems, consider using SymPy's
:func:`~mpmath.sympify` to convert the formula into a symbolic expression.
SymPy can be used to pretty-print or further simplify the formula
symbolically::
>>> from sympy import sympify # doctest: +SKIP
>>> sympify(identify(sqrt(2))) # doctest: +SKIP
2**(1/2)
Sometimes :func:`~mpmath.identify` can simplify an expression further than
a symbolic algorithm::
>>> from sympy import simplify # doctest: +SKIP
>>> x = sympify('-1/(-3/2+(1/2)*5**(1/2))*(3/2-1/2*5**(1/2))**(1/2)') # doctest: +SKIP
>>> x # doctest: +SKIP
(3/2 - 5**(1/2)/2)**(-1/2)
>>> x = simplify(x) # doctest: +SKIP
>>> x # doctest: +SKIP
2/(6 - 2*5**(1/2))**(1/2)
>>> mp.dps = 30 # doctest: +SKIP
>>> x = sympify(identify(x.evalf(30))) # doctest: +SKIP
>>> x # doctest: +SKIP
1/2 + 5**(1/2)/2
(In fact, this functionality is available directly in SymPy as the
function :func:`~mpmath.nsimplify`, which is essentially a wrapper for
:func:`~mpmath.identify`.)
**Miscellaneous issues and limitations**
The input `x` must be a real number. All base constants must be
positive real numbers and must not be rationals or rational linear
combinations of each other.
The worst-case computation time grows quickly with the number of
base constants. Already with 3 or 4 base constants,
:func:`~mpmath.identify` may require several seconds to finish. To search
for relations among a large number of constants, you should
consider using :func:`~mpmath.pslq` directly.
The extended transformations are applied to x, not the constants
separately. As a result, ``identify`` will for example be able to
recognize ``exp(2*pi+3)`` with ``pi`` given as a base constant, but
not ``2*exp(pi)+3``. It will be able to recognize the latter if
``exp(pi)`` is given explicitly as a base constant.
"""
solutions = []
def addsolution(s):
if verbose: print("Found: ", s)
solutions.append(s)
x = ctx.mpf(x)
# Further along, x will be assumed positive
if x == 0:
if full: return ['0']
else: return '0'
if x < 0:
sol = ctx.identify(-x, constants, tol, maxcoeff, full, verbose)
if sol is None:
return sol
if full:
return ["-(%s)"%s for s in sol]
else:
return "-(%s)" % sol
if tol:
tol = ctx.mpf(tol)
else:
tol = ctx.eps**0.7
M = maxcoeff
if constants:
if isinstance(constants, dict):
constants = [(ctx.mpf(v), name) for (name, v) in sorted(constants.items())]
else:
namespace = dict((name, getattr(ctx,name)) for name in dir(ctx))
constants = [(eval(p, namespace), p) for p in constants]
else:
constants = []
# We always want to find at least rational terms
if 1 not in [value for (name, value) in constants]:
constants = [(ctx.mpf(1), '1')] + constants
# PSLQ with simple algebraic and functional transformations
for ft, ftn, red in transforms:
for c, cn in constants:
if red and cn == '1':
continue
t = ft(ctx,x,c)
# Prevent exponential transforms from wreaking havoc
if abs(t) > M**2 or abs(t) < tol:
continue
# Linear combination of base constants
r = ctx.pslq([t] + [a[0] for a in constants], tol, M)
s = None
if r is not None and max(abs(uw) for uw in r) <= M and r[0]:
s = pslqstring(r, constants)
# Quadratic algebraic numbers
else:
q = ctx.pslq([ctx.one, t, t**2], tol, M)
if q is not None and len(q) == 3 and q[2]:
aa, bb, cc = q
if max(abs(aa),abs(bb),abs(cc)) <= M:
s = quadraticstring(ctx,t,aa,bb,cc)
if s:
if cn == '1' and ('/$c' in ftn):
s = ftn.replace('$y', s).replace('/$c', '')
else:
s = ftn.replace('$y', s).replace('$c', cn)
addsolution(s)
if not full: return solutions[0]
if verbose:
print(".")
# Check for a direct multiplicative formula
if x != 1:
# Allow fractional powers of fractions
ilogs = [2,3,5,7]
# Watch out for existing fractional powers of fractions
logs = []
for a, s in constants:
if not sum(bool(ctx.findpoly(ctx.ln(a)/ctx.ln(i),1)) for i in ilogs):
logs.append((ctx.ln(a), s))
logs = [(ctx.ln(i),str(i)) for i in ilogs] + logs
r = ctx.pslq([ctx.ln(x)] + [a[0] for a in logs], tol, M)
if r is not None and max(abs(uw) for uw in r) <= M and r[0]:
addsolution(prodstring(r, logs))
if not full: return solutions[0]
if full:
return sorted(solutions, key=len)
else:
return None
IdentificationMethods.pslq = pslq
IdentificationMethods.findpoly = findpoly
IdentificationMethods.identify = identify
if __name__ == '__main__':
import doctest
doctest.testmod()
| 29,269 | 33.598109 | 94 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/rational.py
|
import operator
import sys
from .libmp import int_types, mpf_hash, bitcount, from_man_exp, HASH_MODULUS
new = object.__new__
def create_reduced(p, q, _cache={}):
key = p, q
if key in _cache:
return _cache[key]
x, y = p, q
while y:
x, y = y, x % y
if x != 1:
p //= x
q //= x
v = new(mpq)
v._mpq_ = p, q
# Speedup integers, half-integers and other small fractions
if q <= 4 and abs(key[0]) < 100:
_cache[key] = v
return v
class mpq(object):
"""
Exact rational type, currently only intended for internal use.
"""
__slots__ = ["_mpq_"]
def __new__(cls, p, q=1):
if type(p) is tuple:
p, q = p
elif hasattr(p, '_mpq_'):
p, q = p._mpq_
return create_reduced(p, q)
def __repr__(s):
return "mpq(%s,%s)" % s._mpq_
def __str__(s):
return "(%s/%s)" % s._mpq_
def __int__(s):
a, b = s._mpq_
return a // b
def __nonzero__(s):
return bool(s._mpq_[0])
__bool__ = __nonzero__
def __hash__(s):
a, b = s._mpq_
if sys.version >= "3.2":
inverse = pow(b, HASH_MODULUS-2, HASH_MODULUS)
if not inverse:
h = sys.hash_info.inf
else:
h = (abs(a) * inverse) % HASH_MODULUS
if a < 0: h = -h
if h == -1: h = -2
return h
else:
if b == 1:
return hash(a)
# Power of two: mpf compatible hash
if not (b & (b-1)):
return mpf_hash(from_man_exp(a, 1-bitcount(b)))
return hash((a,b))
def __eq__(s, t):
ttype = type(t)
if ttype is mpq:
return s._mpq_ == t._mpq_
if ttype in int_types:
a, b = s._mpq_
if b != 1:
return False
return a == t
return NotImplemented
def __ne__(s, t):
ttype = type(t)
if ttype is mpq:
return s._mpq_ != t._mpq_
if ttype in int_types:
a, b = s._mpq_
if b != 1:
return True
return a != t
return NotImplemented
def _cmp(s, t, op):
ttype = type(t)
if ttype in int_types:
a, b = s._mpq_
return op(a, t*b)
if ttype is mpq:
a, b = s._mpq_
c, d = t._mpq_
return op(a*d, b*c)
return NotImplementedError
def __lt__(s, t): return s._cmp(t, operator.lt)
def __le__(s, t): return s._cmp(t, operator.le)
def __gt__(s, t): return s._cmp(t, operator.gt)
def __ge__(s, t): return s._cmp(t, operator.ge)
def __abs__(s):
a, b = s._mpq_
if a >= 0:
return s
v = new(mpq)
v._mpq_ = -a, b
return v
def __neg__(s):
a, b = s._mpq_
v = new(mpq)
v._mpq_ = -a, b
return v
def __pos__(s):
return s
def __add__(s, t):
ttype = type(t)
if ttype is mpq:
a, b = s._mpq_
c, d = t._mpq_
return create_reduced(a*d+b*c, b*d)
if ttype in int_types:
a, b = s._mpq_
v = new(mpq)
v._mpq_ = a+b*t, b
return v
return NotImplemented
__radd__ = __add__
def __sub__(s, t):
ttype = type(t)
if ttype is mpq:
a, b = s._mpq_
c, d = t._mpq_
return create_reduced(a*d-b*c, b*d)
if ttype in int_types:
a, b = s._mpq_
v = new(mpq)
v._mpq_ = a-b*t, b
return v
return NotImplemented
def __rsub__(s, t):
ttype = type(t)
if ttype is mpq:
a, b = s._mpq_
c, d = t._mpq_
return create_reduced(b*c-a*d, b*d)
if ttype in int_types:
a, b = s._mpq_
v = new(mpq)
v._mpq_ = b*t-a, b
return v
return NotImplemented
def __mul__(s, t):
ttype = type(t)
if ttype is mpq:
a, b = s._mpq_
c, d = t._mpq_
return create_reduced(a*c, b*d)
if ttype in int_types:
a, b = s._mpq_
return create_reduced(a*t, b)
return NotImplemented
__rmul__ = __mul__
def __div__(s, t):
ttype = type(t)
if ttype is mpq:
a, b = s._mpq_
c, d = t._mpq_
return create_reduced(a*d, b*c)
if ttype in int_types:
a, b = s._mpq_
return create_reduced(a, b*t)
return NotImplemented
def __rdiv__(s, t):
ttype = type(t)
if ttype is mpq:
a, b = s._mpq_
c, d = t._mpq_
return create_reduced(b*c, a*d)
if ttype in int_types:
a, b = s._mpq_
return create_reduced(b*t, a)
return NotImplemented
def __pow__(s, t):
ttype = type(t)
if ttype in int_types:
a, b = s._mpq_
if t:
if t < 0:
a, b, t = b, a, -t
v = new(mpq)
v._mpq_ = a**t, b**t
return v
raise ZeroDivisionError
return NotImplemented
mpq_1 = mpq((1,1))
mpq_0 = mpq((0,1))
mpq_1_2 = mpq((1,2))
mpq_3_2 = mpq((3,2))
mpq_1_4 = mpq((1,4))
mpq_1_16 = mpq((1,16))
mpq_3_16 = mpq((3,16))
mpq_5_2 = mpq((5,2))
mpq_3_4 = mpq((3,4))
mpq_7_4 = mpq((7,4))
mpq_5_4 = mpq((5,4))
# Register with "numbers" ABC
# We do not subclass, hence we do not use the @abstractmethod checks. While
# this is less invasive it may turn out that we do not actually support
# parts of the expected interfaces. See
# http://docs.python.org/2/library/numbers.html for list of abstract
# methods.
try:
import numbers
numbers.Rational.register(mpq)
except ImportError:
pass
| 5,970 | 23.775934 | 79 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/ctx_fp.py
|
from .ctx_base import StandardBaseContext
import math
import cmath
from . import math2
from . import function_docs
from .libmp import mpf_bernoulli, to_float, int_types
from . import libmp
class FPContext(StandardBaseContext):
"""
Context for fast low-precision arithmetic (53-bit precision, giving at most
about 15-digit accuracy), using Python's builtin float and complex.
"""
def __init__(ctx):
StandardBaseContext.__init__(ctx)
# Override SpecialFunctions implementation
ctx.loggamma = math2.loggamma
ctx._bernoulli_cache = {}
ctx.pretty = False
ctx._init_aliases()
_mpq = lambda cls, x: float(x[0])/x[1]
NoConvergence = libmp.NoConvergence
def _get_prec(ctx): return 53
def _set_prec(ctx, p): return
def _get_dps(ctx): return 15
def _set_dps(ctx, p): return
_fixed_precision = True
prec = property(_get_prec, _set_prec)
dps = property(_get_dps, _set_dps)
zero = 0.0
one = 1.0
eps = math2.EPS
inf = math2.INF
ninf = math2.NINF
nan = math2.NAN
j = 1j
# Called by SpecialFunctions.__init__()
@classmethod
def _wrap_specfun(cls, name, f, wrap):
if wrap:
def f_wrapped(ctx, *args, **kwargs):
convert = ctx.convert
args = [convert(a) for a in args]
return f(ctx, *args, **kwargs)
else:
f_wrapped = f
f_wrapped.__doc__ = function_docs.__dict__.get(name, f.__doc__)
setattr(cls, name, f_wrapped)
def bernoulli(ctx, n):
cache = ctx._bernoulli_cache
if n in cache:
return cache[n]
cache[n] = to_float(mpf_bernoulli(n, 53, 'n'), strict=True)
return cache[n]
pi = math2.pi
e = math2.e
euler = math2.euler
sqrt2 = 1.4142135623730950488
sqrt5 = 2.2360679774997896964
phi = 1.6180339887498948482
ln2 = 0.69314718055994530942
ln10 = 2.302585092994045684
euler = 0.57721566490153286061
catalan = 0.91596559417721901505
khinchin = 2.6854520010653064453
apery = 1.2020569031595942854
glaisher = 1.2824271291006226369
absmin = absmax = abs
def is_special(ctx, x):
return x - x != 0.0
def isnan(ctx, x):
return x != x
def isinf(ctx, x):
return abs(x) == math2.INF
def isnormal(ctx, x):
if x:
return x - x == 0.0
return False
def isnpint(ctx, x):
if type(x) is complex:
if x.imag:
return False
x = x.real
return x <= 0.0 and round(x) == x
mpf = float
mpc = complex
def convert(ctx, x):
try:
return float(x)
except:
return complex(x)
power = staticmethod(math2.pow)
sqrt = staticmethod(math2.sqrt)
exp = staticmethod(math2.exp)
ln = log = staticmethod(math2.log)
cos = staticmethod(math2.cos)
sin = staticmethod(math2.sin)
tan = staticmethod(math2.tan)
cos_sin = staticmethod(math2.cos_sin)
acos = staticmethod(math2.acos)
asin = staticmethod(math2.asin)
atan = staticmethod(math2.atan)
cosh = staticmethod(math2.cosh)
sinh = staticmethod(math2.sinh)
tanh = staticmethod(math2.tanh)
gamma = staticmethod(math2.gamma)
rgamma = staticmethod(math2.rgamma)
fac = factorial = staticmethod(math2.factorial)
floor = staticmethod(math2.floor)
ceil = staticmethod(math2.ceil)
cospi = staticmethod(math2.cospi)
sinpi = staticmethod(math2.sinpi)
cbrt = staticmethod(math2.cbrt)
_nthroot = staticmethod(math2.nthroot)
_ei = staticmethod(math2.ei)
_e1 = staticmethod(math2.e1)
_zeta = _zeta_int = staticmethod(math2.zeta)
# XXX: math2
def arg(ctx, z):
z = complex(z)
return math.atan2(z.imag, z.real)
def expj(ctx, x):
return ctx.exp(ctx.j*x)
def expjpi(ctx, x):
return ctx.exp(ctx.j*ctx.pi*x)
ldexp = math.ldexp
frexp = math.frexp
def mag(ctx, z):
if z:
return ctx.frexp(abs(z))[1]
return ctx.ninf
def isint(ctx, z):
if hasattr(z, "imag"): # float/int don't have .real/.imag in py2.5
if z.imag:
return False
z = z.real
try:
return z == int(z)
except:
return False
def nint_distance(ctx, z):
if hasattr(z, "imag"): # float/int don't have .real/.imag in py2.5
n = round(z.real)
else:
n = round(z)
if n == z:
return n, ctx.ninf
return n, ctx.mag(abs(z-n))
def _convert_param(ctx, z):
if type(z) is tuple:
p, q = z
return ctx.mpf(p) / q, 'R'
if hasattr(z, "imag"): # float/int don't have .real/.imag in py2.5
intz = int(z.real)
else:
intz = int(z)
if z == intz:
return intz, 'Z'
return z, 'R'
def _is_real_type(ctx, z):
return isinstance(z, float) or isinstance(z, int_types)
def _is_complex_type(ctx, z):
return isinstance(z, complex)
def hypsum(ctx, p, q, types, coeffs, z, maxterms=6000, **kwargs):
coeffs = list(coeffs)
num = range(p)
den = range(p,p+q)
tol = ctx.eps
s = t = 1.0
k = 0
while 1:
for i in num: t *= (coeffs[i]+k)
for i in den: t /= (coeffs[i]+k)
k += 1; t /= k; t *= z; s += t
if abs(t) < tol:
return s
if k > maxterms:
raise ctx.NoConvergence
def atan2(ctx, x, y):
return math.atan2(x, y)
def psi(ctx, m, z):
m = int(m)
if m == 0:
return ctx.digamma(z)
return (-1)**(m+1) * ctx.fac(m) * ctx.zeta(m+1, z)
digamma = staticmethod(math2.digamma)
def harmonic(ctx, x):
x = ctx.convert(x)
if x == 0 or x == 1:
return x
return ctx.digamma(x+1) + ctx.euler
nstr = str
def to_fixed(ctx, x, prec):
return int(math.ldexp(x, prec))
def rand(ctx):
import random
return random.random()
_erf = staticmethod(math2.erf)
_erfc = staticmethod(math2.erfc)
def sum_accurately(ctx, terms, check_step=1):
s = ctx.zero
k = 0
for term in terms():
s += term
if (not k % check_step) and term:
if abs(term) <= 1e-18*abs(s):
break
k += 1
return s
| 6,572 | 24.877953 | 79 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/ctx_mp.py
|
"""
This module defines the mpf, mpc classes, and standard functions for
operating with them.
"""
__docformat__ = 'plaintext'
import re
from .ctx_base import StandardBaseContext
from .libmp.backend import basestring, BACKEND
from . import libmp
from .libmp import (MPZ, MPZ_ZERO, MPZ_ONE, int_types, repr_dps,
round_floor, round_ceiling, dps_to_prec, round_nearest, prec_to_dps,
ComplexResult, to_pickable, from_pickable, normalize,
from_int, from_float, from_str, to_int, to_float, to_str,
from_rational, from_man_exp,
fone, fzero, finf, fninf, fnan,
mpf_abs, mpf_pos, mpf_neg, mpf_add, mpf_sub, mpf_mul, mpf_mul_int,
mpf_div, mpf_rdiv_int, mpf_pow_int, mpf_mod,
mpf_eq, mpf_cmp, mpf_lt, mpf_gt, mpf_le, mpf_ge,
mpf_hash, mpf_rand,
mpf_sum,
bitcount, to_fixed,
mpc_to_str,
mpc_to_complex, mpc_hash, mpc_pos, mpc_is_nonzero, mpc_neg, mpc_conjugate,
mpc_abs, mpc_add, mpc_add_mpf, mpc_sub, mpc_sub_mpf, mpc_mul, mpc_mul_mpf,
mpc_mul_int, mpc_div, mpc_div_mpf, mpc_pow, mpc_pow_mpf, mpc_pow_int,
mpc_mpf_div,
mpf_pow,
mpf_pi, mpf_degree, mpf_e, mpf_phi, mpf_ln2, mpf_ln10,
mpf_euler, mpf_catalan, mpf_apery, mpf_khinchin,
mpf_glaisher, mpf_twinprime, mpf_mertens,
int_types)
from . import function_docs
from . import rational
new = object.__new__
get_complex = re.compile(r'^\(?(?P<re>[\+\-]?\d*\.?\d*(e[\+\-]?\d+)?)??'
r'(?P<im>[\+\-]?\d*\.?\d*(e[\+\-]?\d+)?j)?\)?$')
if BACKEND == 'sage':
from sage.libs.mpmath.ext_main import Context as BaseMPContext
# pickle hack
import sage.libs.mpmath.ext_main as _mpf_module
else:
from .ctx_mp_python import PythonMPContext as BaseMPContext
from . import ctx_mp_python as _mpf_module
from .ctx_mp_python import _mpf, _mpc, mpnumeric
class MPContext(BaseMPContext, StandardBaseContext):
"""
Context for multiprecision arithmetic with a global precision.
"""
def __init__(ctx):
BaseMPContext.__init__(ctx)
ctx.trap_complex = False
ctx.pretty = False
ctx.types = [ctx.mpf, ctx.mpc, ctx.constant]
ctx._mpq = rational.mpq
ctx.default()
StandardBaseContext.__init__(ctx)
ctx.mpq = rational.mpq
ctx.init_builtins()
ctx.hyp_summators = {}
ctx._init_aliases()
# XXX: automate
try:
ctx.bernoulli.im_func.func_doc = function_docs.bernoulli
ctx.primepi.im_func.func_doc = function_docs.primepi
ctx.psi.im_func.func_doc = function_docs.psi
ctx.atan2.im_func.func_doc = function_docs.atan2
except AttributeError:
# python 3
ctx.bernoulli.__func__.func_doc = function_docs.bernoulli
ctx.primepi.__func__.func_doc = function_docs.primepi
ctx.psi.__func__.func_doc = function_docs.psi
ctx.atan2.__func__.func_doc = function_docs.atan2
ctx.digamma.func_doc = function_docs.digamma
ctx.cospi.func_doc = function_docs.cospi
ctx.sinpi.func_doc = function_docs.sinpi
def init_builtins(ctx):
mpf = ctx.mpf
mpc = ctx.mpc
# Exact constants
ctx.one = ctx.make_mpf(fone)
ctx.zero = ctx.make_mpf(fzero)
ctx.j = ctx.make_mpc((fzero,fone))
ctx.inf = ctx.make_mpf(finf)
ctx.ninf = ctx.make_mpf(fninf)
ctx.nan = ctx.make_mpf(fnan)
eps = ctx.constant(lambda prec, rnd: (0, MPZ_ONE, 1-prec, 1),
"epsilon of working precision", "eps")
ctx.eps = eps
# Approximate constants
ctx.pi = ctx.constant(mpf_pi, "pi", "pi")
ctx.ln2 = ctx.constant(mpf_ln2, "ln(2)", "ln2")
ctx.ln10 = ctx.constant(mpf_ln10, "ln(10)", "ln10")
ctx.phi = ctx.constant(mpf_phi, "Golden ratio phi", "phi")
ctx.e = ctx.constant(mpf_e, "e = exp(1)", "e")
ctx.euler = ctx.constant(mpf_euler, "Euler's constant", "euler")
ctx.catalan = ctx.constant(mpf_catalan, "Catalan's constant", "catalan")
ctx.khinchin = ctx.constant(mpf_khinchin, "Khinchin's constant", "khinchin")
ctx.glaisher = ctx.constant(mpf_glaisher, "Glaisher's constant", "glaisher")
ctx.apery = ctx.constant(mpf_apery, "Apery's constant", "apery")
ctx.degree = ctx.constant(mpf_degree, "1 deg = pi / 180", "degree")
ctx.twinprime = ctx.constant(mpf_twinprime, "Twin prime constant", "twinprime")
ctx.mertens = ctx.constant(mpf_mertens, "Mertens' constant", "mertens")
# Standard functions
ctx.sqrt = ctx._wrap_libmp_function(libmp.mpf_sqrt, libmp.mpc_sqrt)
ctx.cbrt = ctx._wrap_libmp_function(libmp.mpf_cbrt, libmp.mpc_cbrt)
ctx.ln = ctx._wrap_libmp_function(libmp.mpf_log, libmp.mpc_log)
ctx.atan = ctx._wrap_libmp_function(libmp.mpf_atan, libmp.mpc_atan)
ctx.exp = ctx._wrap_libmp_function(libmp.mpf_exp, libmp.mpc_exp)
ctx.expj = ctx._wrap_libmp_function(libmp.mpf_expj, libmp.mpc_expj)
ctx.expjpi = ctx._wrap_libmp_function(libmp.mpf_expjpi, libmp.mpc_expjpi)
ctx.sin = ctx._wrap_libmp_function(libmp.mpf_sin, libmp.mpc_sin)
ctx.cos = ctx._wrap_libmp_function(libmp.mpf_cos, libmp.mpc_cos)
ctx.tan = ctx._wrap_libmp_function(libmp.mpf_tan, libmp.mpc_tan)
ctx.sinh = ctx._wrap_libmp_function(libmp.mpf_sinh, libmp.mpc_sinh)
ctx.cosh = ctx._wrap_libmp_function(libmp.mpf_cosh, libmp.mpc_cosh)
ctx.tanh = ctx._wrap_libmp_function(libmp.mpf_tanh, libmp.mpc_tanh)
ctx.asin = ctx._wrap_libmp_function(libmp.mpf_asin, libmp.mpc_asin)
ctx.acos = ctx._wrap_libmp_function(libmp.mpf_acos, libmp.mpc_acos)
ctx.atan = ctx._wrap_libmp_function(libmp.mpf_atan, libmp.mpc_atan)
ctx.asinh = ctx._wrap_libmp_function(libmp.mpf_asinh, libmp.mpc_asinh)
ctx.acosh = ctx._wrap_libmp_function(libmp.mpf_acosh, libmp.mpc_acosh)
ctx.atanh = ctx._wrap_libmp_function(libmp.mpf_atanh, libmp.mpc_atanh)
ctx.sinpi = ctx._wrap_libmp_function(libmp.mpf_sin_pi, libmp.mpc_sin_pi)
ctx.cospi = ctx._wrap_libmp_function(libmp.mpf_cos_pi, libmp.mpc_cos_pi)
ctx.floor = ctx._wrap_libmp_function(libmp.mpf_floor, libmp.mpc_floor)
ctx.ceil = ctx._wrap_libmp_function(libmp.mpf_ceil, libmp.mpc_ceil)
ctx.nint = ctx._wrap_libmp_function(libmp.mpf_nint, libmp.mpc_nint)
ctx.frac = ctx._wrap_libmp_function(libmp.mpf_frac, libmp.mpc_frac)
ctx.fib = ctx.fibonacci = ctx._wrap_libmp_function(libmp.mpf_fibonacci, libmp.mpc_fibonacci)
ctx.gamma = ctx._wrap_libmp_function(libmp.mpf_gamma, libmp.mpc_gamma)
ctx.rgamma = ctx._wrap_libmp_function(libmp.mpf_rgamma, libmp.mpc_rgamma)
ctx.loggamma = ctx._wrap_libmp_function(libmp.mpf_loggamma, libmp.mpc_loggamma)
ctx.fac = ctx.factorial = ctx._wrap_libmp_function(libmp.mpf_factorial, libmp.mpc_factorial)
ctx.gamma_old = ctx._wrap_libmp_function(libmp.mpf_gamma_old, libmp.mpc_gamma_old)
ctx.fac_old = ctx.factorial_old = ctx._wrap_libmp_function(libmp.mpf_factorial_old, libmp.mpc_factorial_old)
ctx.digamma = ctx._wrap_libmp_function(libmp.mpf_psi0, libmp.mpc_psi0)
ctx.harmonic = ctx._wrap_libmp_function(libmp.mpf_harmonic, libmp.mpc_harmonic)
ctx.ei = ctx._wrap_libmp_function(libmp.mpf_ei, libmp.mpc_ei)
ctx.e1 = ctx._wrap_libmp_function(libmp.mpf_e1, libmp.mpc_e1)
ctx._ci = ctx._wrap_libmp_function(libmp.mpf_ci, libmp.mpc_ci)
ctx._si = ctx._wrap_libmp_function(libmp.mpf_si, libmp.mpc_si)
ctx.ellipk = ctx._wrap_libmp_function(libmp.mpf_ellipk, libmp.mpc_ellipk)
ctx._ellipe = ctx._wrap_libmp_function(libmp.mpf_ellipe, libmp.mpc_ellipe)
ctx.agm1 = ctx._wrap_libmp_function(libmp.mpf_agm1, libmp.mpc_agm1)
ctx._erf = ctx._wrap_libmp_function(libmp.mpf_erf, None)
ctx._erfc = ctx._wrap_libmp_function(libmp.mpf_erfc, None)
ctx._zeta = ctx._wrap_libmp_function(libmp.mpf_zeta, libmp.mpc_zeta)
ctx._altzeta = ctx._wrap_libmp_function(libmp.mpf_altzeta, libmp.mpc_altzeta)
# Faster versions
ctx.sqrt = getattr(ctx, "_sage_sqrt", ctx.sqrt)
ctx.exp = getattr(ctx, "_sage_exp", ctx.exp)
ctx.ln = getattr(ctx, "_sage_ln", ctx.ln)
ctx.cos = getattr(ctx, "_sage_cos", ctx.cos)
ctx.sin = getattr(ctx, "_sage_sin", ctx.sin)
def to_fixed(ctx, x, prec):
return x.to_fixed(prec)
def hypot(ctx, x, y):
r"""
Computes the Euclidean norm of the vector `(x, y)`, equal
to `\sqrt{x^2 + y^2}`. Both `x` and `y` must be real."""
x = ctx.convert(x)
y = ctx.convert(y)
return ctx.make_mpf(libmp.mpf_hypot(x._mpf_, y._mpf_, *ctx._prec_rounding))
def _gamma_upper_int(ctx, n, z):
n = int(ctx._re(n))
if n == 0:
return ctx.e1(z)
if not hasattr(z, '_mpf_'):
raise NotImplementedError
prec, rounding = ctx._prec_rounding
real, imag = libmp.mpf_expint(n, z._mpf_, prec, rounding, gamma=True)
if imag is None:
return ctx.make_mpf(real)
else:
return ctx.make_mpc((real, imag))
def _expint_int(ctx, n, z):
n = int(n)
if n == 1:
return ctx.e1(z)
if not hasattr(z, '_mpf_'):
raise NotImplementedError
prec, rounding = ctx._prec_rounding
real, imag = libmp.mpf_expint(n, z._mpf_, prec, rounding)
if imag is None:
return ctx.make_mpf(real)
else:
return ctx.make_mpc((real, imag))
def _nthroot(ctx, x, n):
if hasattr(x, '_mpf_'):
try:
return ctx.make_mpf(libmp.mpf_nthroot(x._mpf_, n, *ctx._prec_rounding))
except ComplexResult:
if ctx.trap_complex:
raise
x = (x._mpf_, libmp.fzero)
else:
x = x._mpc_
return ctx.make_mpc(libmp.mpc_nthroot(x, n, *ctx._prec_rounding))
def _besselj(ctx, n, z):
prec, rounding = ctx._prec_rounding
if hasattr(z, '_mpf_'):
return ctx.make_mpf(libmp.mpf_besseljn(n, z._mpf_, prec, rounding))
elif hasattr(z, '_mpc_'):
return ctx.make_mpc(libmp.mpc_besseljn(n, z._mpc_, prec, rounding))
def _agm(ctx, a, b=1):
prec, rounding = ctx._prec_rounding
if hasattr(a, '_mpf_') and hasattr(b, '_mpf_'):
try:
v = libmp.mpf_agm(a._mpf_, b._mpf_, prec, rounding)
return ctx.make_mpf(v)
except ComplexResult:
pass
if hasattr(a, '_mpf_'): a = (a._mpf_, libmp.fzero)
else: a = a._mpc_
if hasattr(b, '_mpf_'): b = (b._mpf_, libmp.fzero)
else: b = b._mpc_
return ctx.make_mpc(libmp.mpc_agm(a, b, prec, rounding))
def bernoulli(ctx, n):
return ctx.make_mpf(libmp.mpf_bernoulli(int(n), *ctx._prec_rounding))
def _zeta_int(ctx, n):
return ctx.make_mpf(libmp.mpf_zeta_int(int(n), *ctx._prec_rounding))
def atan2(ctx, y, x):
x = ctx.convert(x)
y = ctx.convert(y)
return ctx.make_mpf(libmp.mpf_atan2(y._mpf_, x._mpf_, *ctx._prec_rounding))
def psi(ctx, m, z):
z = ctx.convert(z)
m = int(m)
if ctx._is_real_type(z):
return ctx.make_mpf(libmp.mpf_psi(m, z._mpf_, *ctx._prec_rounding))
else:
return ctx.make_mpc(libmp.mpc_psi(m, z._mpc_, *ctx._prec_rounding))
def cos_sin(ctx, x, **kwargs):
if type(x) not in ctx.types:
x = ctx.convert(x)
prec, rounding = ctx._parse_prec(kwargs)
if hasattr(x, '_mpf_'):
c, s = libmp.mpf_cos_sin(x._mpf_, prec, rounding)
return ctx.make_mpf(c), ctx.make_mpf(s)
elif hasattr(x, '_mpc_'):
c, s = libmp.mpc_cos_sin(x._mpc_, prec, rounding)
return ctx.make_mpc(c), ctx.make_mpc(s)
else:
return ctx.cos(x, **kwargs), ctx.sin(x, **kwargs)
def cospi_sinpi(ctx, x, **kwargs):
if type(x) not in ctx.types:
x = ctx.convert(x)
prec, rounding = ctx._parse_prec(kwargs)
if hasattr(x, '_mpf_'):
c, s = libmp.mpf_cos_sin_pi(x._mpf_, prec, rounding)
return ctx.make_mpf(c), ctx.make_mpf(s)
elif hasattr(x, '_mpc_'):
c, s = libmp.mpc_cos_sin_pi(x._mpc_, prec, rounding)
return ctx.make_mpc(c), ctx.make_mpc(s)
else:
return ctx.cos(x, **kwargs), ctx.sin(x, **kwargs)
def clone(ctx):
"""
Create a copy of the context, with the same working precision.
"""
a = ctx.__class__()
a.prec = ctx.prec
return a
# Several helper methods
# TODO: add more of these, make consistent, write docstrings, ...
def _is_real_type(ctx, x):
if hasattr(x, '_mpc_') or type(x) is complex:
return False
return True
def _is_complex_type(ctx, x):
if hasattr(x, '_mpc_') or type(x) is complex:
return True
return False
def isnan(ctx, x):
"""
Return *True* if *x* is a NaN (not-a-number), or for a complex
number, whether either the real or complex part is NaN;
otherwise return *False*::
>>> from mpmath import *
>>> isnan(3.14)
False
>>> isnan(nan)
True
>>> isnan(mpc(3.14,2.72))
False
>>> isnan(mpc(3.14,nan))
True
"""
if hasattr(x, "_mpf_"):
return x._mpf_ == fnan
if hasattr(x, "_mpc_"):
return fnan in x._mpc_
if isinstance(x, int_types) or isinstance(x, rational.mpq):
return False
x = ctx.convert(x)
if hasattr(x, '_mpf_') or hasattr(x, '_mpc_'):
return ctx.isnan(x)
raise TypeError("isnan() needs a number as input")
def isfinite(ctx, x):
"""
Return *True* if *x* is a finite number, i.e. neither
an infinity or a NaN.
>>> from mpmath import *
>>> isfinite(inf)
False
>>> isfinite(-inf)
False
>>> isfinite(3)
True
>>> isfinite(nan)
False
>>> isfinite(3+4j)
True
>>> isfinite(mpc(3,inf))
False
>>> isfinite(mpc(nan,3))
False
"""
if ctx.isinf(x) or ctx.isnan(x):
return False
return True
def isnpint(ctx, x):
"""
Determine if *x* is a nonpositive integer.
"""
if not x:
return True
if hasattr(x, '_mpf_'):
sign, man, exp, bc = x._mpf_
return sign and exp >= 0
if hasattr(x, '_mpc_'):
return not x.imag and ctx.isnpint(x.real)
if type(x) in int_types:
return x <= 0
if isinstance(x, ctx.mpq):
p, q = x._mpq_
if not p:
return True
return q == 1 and p <= 0
return ctx.isnpint(ctx.convert(x))
def __str__(ctx):
lines = ["Mpmath settings:",
(" mp.prec = %s" % ctx.prec).ljust(30) + "[default: 53]",
(" mp.dps = %s" % ctx.dps).ljust(30) + "[default: 15]",
(" mp.trap_complex = %s" % ctx.trap_complex).ljust(30) + "[default: False]",
]
return "\n".join(lines)
@property
def _repr_digits(ctx):
return repr_dps(ctx._prec)
@property
def _str_digits(ctx):
return ctx._dps
def extraprec(ctx, n, normalize_output=False):
"""
The block
with extraprec(n):
<code>
increases the precision n bits, executes <code>, and then
restores the precision.
extraprec(n)(f) returns a decorated version of the function f
that increases the working precision by n bits before execution,
and restores the parent precision afterwards. With
normalize_output=True, it rounds the return value to the parent
precision.
"""
return PrecisionManager(ctx, lambda p: p + n, None, normalize_output)
def extradps(ctx, n, normalize_output=False):
"""
This function is analogous to extraprec (see documentation)
but changes the decimal precision instead of the number of bits.
"""
return PrecisionManager(ctx, None, lambda d: d + n, normalize_output)
def workprec(ctx, n, normalize_output=False):
"""
The block
with workprec(n):
<code>
sets the precision to n bits, executes <code>, and then restores
the precision.
workprec(n)(f) returns a decorated version of the function f
that sets the precision to n bits before execution,
and restores the precision afterwards. With normalize_output=True,
it rounds the return value to the parent precision.
"""
return PrecisionManager(ctx, lambda p: n, None, normalize_output)
def workdps(ctx, n, normalize_output=False):
"""
This function is analogous to workprec (see documentation)
but changes the decimal precision instead of the number of bits.
"""
return PrecisionManager(ctx, None, lambda d: n, normalize_output)
def autoprec(ctx, f, maxprec=None, catch=(), verbose=False):
"""
Return a wrapped copy of *f* that repeatedly evaluates *f*
with increasing precision until the result converges to the
full precision used at the point of the call.
This heuristically protects against rounding errors, at the cost of
roughly a 2x slowdown compared to manually setting the optimal
precision. This method can, however, easily be fooled if the results
from *f* depend "discontinuously" on the precision, for instance
if catastrophic cancellation can occur. Therefore, :func:`~mpmath.autoprec`
should be used judiciously.
**Examples**
Many functions are sensitive to perturbations of the input arguments.
If the arguments are decimal numbers, they may have to be converted
to binary at a much higher precision. If the amount of required
extra precision is unknown, :func:`~mpmath.autoprec` is convenient::
>>> from mpmath import *
>>> mp.dps = 15
>>> mp.pretty = True
>>> besselj(5, 125 * 10**28) # Exact input
-8.03284785591801e-17
>>> besselj(5, '1.25e30') # Bad
7.12954868316652e-16
>>> autoprec(besselj)(5, '1.25e30') # Good
-8.03284785591801e-17
The following fails to converge because `\sin(\pi) = 0` whereas all
finite-precision approximations of `\pi` give nonzero values::
>>> autoprec(sin)(pi) # doctest: +IGNORE_EXCEPTION_DETAIL
Traceback (most recent call last):
...
NoConvergence: autoprec: prec increased to 2910 without convergence
As the following example shows, :func:`~mpmath.autoprec` can protect against
cancellation, but is fooled by too severe cancellation::
>>> x = 1e-10
>>> exp(x)-1; expm1(x); autoprec(lambda t: exp(t)-1)(x)
1.00000008274037e-10
1.00000000005e-10
1.00000000005e-10
>>> x = 1e-50
>>> exp(x)-1; expm1(x); autoprec(lambda t: exp(t)-1)(x)
0.0
1.0e-50
0.0
With *catch*, an exception or list of exceptions to intercept
may be specified. The raised exception is interpreted
as signaling insufficient precision. This permits, for example,
evaluating a function where a too low precision results in a
division by zero::
>>> f = lambda x: 1/(exp(x)-1)
>>> f(1e-30)
Traceback (most recent call last):
...
ZeroDivisionError
>>> autoprec(f, catch=ZeroDivisionError)(1e-30)
1.0e+30
"""
def f_autoprec_wrapped(*args, **kwargs):
prec = ctx.prec
if maxprec is None:
maxprec2 = ctx._default_hyper_maxprec(prec)
else:
maxprec2 = maxprec
try:
ctx.prec = prec + 10
try:
v1 = f(*args, **kwargs)
except catch:
v1 = ctx.nan
prec2 = prec + 20
while 1:
ctx.prec = prec2
try:
v2 = f(*args, **kwargs)
except catch:
v2 = ctx.nan
if v1 == v2:
break
err = ctx.mag(v2-v1) - ctx.mag(v2)
if err < (-prec):
break
if verbose:
print("autoprec: target=%s, prec=%s, accuracy=%s" \
% (prec, prec2, -err))
v1 = v2
if prec2 >= maxprec2:
raise ctx.NoConvergence(\
"autoprec: prec increased to %i without convergence"\
% prec2)
prec2 += int(prec2*2)
prec2 = min(prec2, maxprec2)
finally:
ctx.prec = prec
return +v2
return f_autoprec_wrapped
def nstr(ctx, x, n=6, **kwargs):
"""
Convert an ``mpf`` or ``mpc`` to a decimal string literal with *n*
significant digits. The small default value for *n* is chosen to
make this function useful for printing collections of numbers
(lists, matrices, etc).
If *x* is a list or tuple, :func:`~mpmath.nstr` is applied recursively
to each element. For unrecognized classes, :func:`~mpmath.nstr`
simply returns ``str(x)``.
The companion function :func:`~mpmath.nprint` prints the result
instead of returning it.
The keyword arguments *strip_zeros*, *min_fixed*, *max_fixed*
and *show_zero_exponent* are forwarded to :func:`~mpmath.libmp.to_str`.
The number will be printed in fixed-point format if the position
of the leading digit is strictly between min_fixed
(default = min(-dps/3,-5)) and max_fixed (default = dps).
To force fixed-point format always, set min_fixed = -inf,
max_fixed = +inf. To force floating-point format, set
min_fixed >= max_fixed.
>>> from mpmath import *
>>> nstr([+pi, ldexp(1,-500)])
'[3.14159, 3.05494e-151]'
>>> nprint([+pi, ldexp(1,-500)])
[3.14159, 3.05494e-151]
>>> nstr(mpf("5e-10"), 5)
'5.0e-10'
>>> nstr(mpf("5e-10"), 5, strip_zeros=False)
'5.0000e-10'
>>> nstr(mpf("5e-10"), 5, strip_zeros=False, min_fixed=-11)
'0.00000000050000'
>>> nstr(mpf(0), 5, show_zero_exponent=True)
'0.0e+0'
"""
if isinstance(x, list):
return "[%s]" % (", ".join(ctx.nstr(c, n, **kwargs) for c in x))
if isinstance(x, tuple):
return "(%s)" % (", ".join(ctx.nstr(c, n, **kwargs) for c in x))
if hasattr(x, '_mpf_'):
return to_str(x._mpf_, n, **kwargs)
if hasattr(x, '_mpc_'):
return "(" + mpc_to_str(x._mpc_, n, **kwargs) + ")"
if isinstance(x, basestring):
return repr(x)
if isinstance(x, ctx.matrix):
return x.__nstr__(n, **kwargs)
return str(x)
def _convert_fallback(ctx, x, strings):
if strings and isinstance(x, basestring):
if 'j' in x.lower():
x = x.lower().replace(' ', '')
match = get_complex.match(x)
re = match.group('re')
if not re:
re = 0
im = match.group('im').rstrip('j')
return ctx.mpc(ctx.convert(re), ctx.convert(im))
if hasattr(x, "_mpi_"):
a, b = x._mpi_
if a == b:
return ctx.make_mpf(a)
else:
raise ValueError("can only create mpf from zero-width interval")
raise TypeError("cannot create mpf from " + repr(x))
def mpmathify(ctx, *args, **kwargs):
return ctx.convert(*args, **kwargs)
def _parse_prec(ctx, kwargs):
if kwargs:
if kwargs.get('exact'):
return 0, 'f'
prec, rounding = ctx._prec_rounding
if 'rounding' in kwargs:
rounding = kwargs['rounding']
if 'prec' in kwargs:
prec = kwargs['prec']
if prec == ctx.inf:
return 0, 'f'
else:
prec = int(prec)
elif 'dps' in kwargs:
dps = kwargs['dps']
if dps == ctx.inf:
return 0, 'f'
prec = dps_to_prec(dps)
return prec, rounding
return ctx._prec_rounding
_exact_overflow_msg = "the exact result does not fit in memory"
_hypsum_msg = """hypsum() failed to converge to the requested %i bits of accuracy
using a working precision of %i bits. Try with a higher maxprec,
maxterms, or set zeroprec."""
def hypsum(ctx, p, q, flags, coeffs, z, accurate_small=True, **kwargs):
if hasattr(z, "_mpf_"):
key = p, q, flags, 'R'
v = z._mpf_
elif hasattr(z, "_mpc_"):
key = p, q, flags, 'C'
v = z._mpc_
if key not in ctx.hyp_summators:
ctx.hyp_summators[key] = libmp.make_hyp_summator(key)[1]
summator = ctx.hyp_summators[key]
prec = ctx.prec
maxprec = kwargs.get('maxprec', ctx._default_hyper_maxprec(prec))
extraprec = 50
epsshift = 25
# Jumps in magnitude occur when parameters are close to negative
# integers. We must ensure that these terms are included in
# the sum and added accurately
magnitude_check = {}
max_total_jump = 0
for i, c in enumerate(coeffs):
if flags[i] == 'Z':
if i >= p and c <= 0:
ok = False
for ii, cc in enumerate(coeffs[:p]):
# Note: c <= cc or c < cc, depending on convention
if flags[ii] == 'Z' and cc <= 0 and c <= cc:
ok = True
if not ok:
raise ZeroDivisionError("pole in hypergeometric series")
continue
n, d = ctx.nint_distance(c)
n = -int(n)
d = -d
if i >= p and n >= 0 and d > 4:
if n in magnitude_check:
magnitude_check[n] += d
else:
magnitude_check[n] = d
extraprec = max(extraprec, d - prec + 60)
max_total_jump += abs(d)
while 1:
if extraprec > maxprec:
raise ValueError(ctx._hypsum_msg % (prec, prec+extraprec))
wp = prec + extraprec
if magnitude_check:
mag_dict = dict((n,None) for n in magnitude_check)
else:
mag_dict = {}
zv, have_complex, magnitude = summator(coeffs, v, prec, wp, \
epsshift, mag_dict, **kwargs)
cancel = -magnitude
jumps_resolved = True
if extraprec < max_total_jump:
for n in mag_dict.values():
if (n is None) or (n < prec):
jumps_resolved = False
break
accurate = (cancel < extraprec-25-5 or not accurate_small)
if jumps_resolved:
if accurate:
break
# zero?
zeroprec = kwargs.get('zeroprec')
if zeroprec is not None:
if cancel > zeroprec:
if have_complex:
return ctx.mpc(0)
else:
return ctx.zero
# Some near-singularities were not included, so increase
# precision and repeat until they are
extraprec *= 2
# Possible workaround for bad roundoff in fixed-point arithmetic
epsshift += 5
extraprec += 5
if type(zv) is tuple:
if have_complex:
return ctx.make_mpc(zv)
else:
return ctx.make_mpf(zv)
else:
return zv
def ldexp(ctx, x, n):
r"""
Computes `x 2^n` efficiently. No rounding is performed.
The argument `x` must be a real floating-point number (or
possible to convert into one) and `n` must be a Python ``int``.
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = False
>>> ldexp(1, 10)
mpf('1024.0')
>>> ldexp(1, -3)
mpf('0.125')
"""
x = ctx.convert(x)
return ctx.make_mpf(libmp.mpf_shift(x._mpf_, n))
def frexp(ctx, x):
r"""
Given a real number `x`, returns `(y, n)` with `y \in [0.5, 1)`,
`n` a Python integer, and such that `x = y 2^n`. No rounding is
performed.
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = False
>>> frexp(7.5)
(mpf('0.9375'), 3)
"""
x = ctx.convert(x)
y, n = libmp.mpf_frexp(x._mpf_)
return ctx.make_mpf(y), n
def fneg(ctx, x, **kwargs):
"""
Negates the number *x*, giving a floating-point result, optionally
using a custom precision and rounding mode.
See the documentation of :func:`~mpmath.fadd` for a detailed description
of how to specify precision and rounding.
**Examples**
An mpmath number is returned::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = False
>>> fneg(2.5)
mpf('-2.5')
>>> fneg(-5+2j)
mpc(real='5.0', imag='-2.0')
Precise control over rounding is possible::
>>> x = fadd(2, 1e-100, exact=True)
>>> fneg(x)
mpf('-2.0')
>>> fneg(x, rounding='f')
mpf('-2.0000000000000004')
Negating with and without roundoff::
>>> n = 200000000000000000000001
>>> print(int(-mpf(n)))
-200000000000000016777216
>>> print(int(fneg(n)))
-200000000000000016777216
>>> print(int(fneg(n, prec=log(n,2)+1)))
-200000000000000000000001
>>> print(int(fneg(n, dps=log(n,10)+1)))
-200000000000000000000001
>>> print(int(fneg(n, prec=inf)))
-200000000000000000000001
>>> print(int(fneg(n, dps=inf)))
-200000000000000000000001
>>> print(int(fneg(n, exact=True)))
-200000000000000000000001
"""
prec, rounding = ctx._parse_prec(kwargs)
x = ctx.convert(x)
if hasattr(x, '_mpf_'):
return ctx.make_mpf(mpf_neg(x._mpf_, prec, rounding))
if hasattr(x, '_mpc_'):
return ctx.make_mpc(mpc_neg(x._mpc_, prec, rounding))
raise ValueError("Arguments need to be mpf or mpc compatible numbers")
def fadd(ctx, x, y, **kwargs):
"""
Adds the numbers *x* and *y*, giving a floating-point result,
optionally using a custom precision and rounding mode.
The default precision is the working precision of the context.
You can specify a custom precision in bits by passing the *prec* keyword
argument, or by providing an equivalent decimal precision with the *dps*
keyword argument. If the precision is set to ``+inf``, or if the flag
*exact=True* is passed, an exact addition with no rounding is performed.
When the precision is finite, the optional *rounding* keyword argument
specifies the direction of rounding. Valid options are ``'n'`` for
nearest (default), ``'f'`` for floor, ``'c'`` for ceiling, ``'d'``
for down, ``'u'`` for up.
**Examples**
Using :func:`~mpmath.fadd` with precision and rounding control::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = False
>>> fadd(2, 1e-20)
mpf('2.0')
>>> fadd(2, 1e-20, rounding='u')
mpf('2.0000000000000004')
>>> nprint(fadd(2, 1e-20, prec=100), 25)
2.00000000000000000001
>>> nprint(fadd(2, 1e-20, dps=15), 25)
2.0
>>> nprint(fadd(2, 1e-20, dps=25), 25)
2.00000000000000000001
>>> nprint(fadd(2, 1e-20, exact=True), 25)
2.00000000000000000001
Exact addition avoids cancellation errors, enforcing familiar laws
of numbers such as `x+y-x = y`, which don't hold in floating-point
arithmetic with finite precision::
>>> x, y = mpf(2), mpf('1e-1000')
>>> print(x + y - x)
0.0
>>> print(fadd(x, y, prec=inf) - x)
1.0e-1000
>>> print(fadd(x, y, exact=True) - x)
1.0e-1000
Exact addition can be inefficient and may be impossible to perform
with large magnitude differences::
>>> fadd(1, '1e-100000000000000000000', prec=inf)
Traceback (most recent call last):
...
OverflowError: the exact result does not fit in memory
"""
prec, rounding = ctx._parse_prec(kwargs)
x = ctx.convert(x)
y = ctx.convert(y)
try:
if hasattr(x, '_mpf_'):
if hasattr(y, '_mpf_'):
return ctx.make_mpf(mpf_add(x._mpf_, y._mpf_, prec, rounding))
if hasattr(y, '_mpc_'):
return ctx.make_mpc(mpc_add_mpf(y._mpc_, x._mpf_, prec, rounding))
if hasattr(x, '_mpc_'):
if hasattr(y, '_mpf_'):
return ctx.make_mpc(mpc_add_mpf(x._mpc_, y._mpf_, prec, rounding))
if hasattr(y, '_mpc_'):
return ctx.make_mpc(mpc_add(x._mpc_, y._mpc_, prec, rounding))
except (ValueError, OverflowError):
raise OverflowError(ctx._exact_overflow_msg)
raise ValueError("Arguments need to be mpf or mpc compatible numbers")
def fsub(ctx, x, y, **kwargs):
"""
Subtracts the numbers *x* and *y*, giving a floating-point result,
optionally using a custom precision and rounding mode.
See the documentation of :func:`~mpmath.fadd` for a detailed description
of how to specify precision and rounding.
**Examples**
Using :func:`~mpmath.fsub` with precision and rounding control::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = False
>>> fsub(2, 1e-20)
mpf('2.0')
>>> fsub(2, 1e-20, rounding='d')
mpf('1.9999999999999998')
>>> nprint(fsub(2, 1e-20, prec=100), 25)
1.99999999999999999999
>>> nprint(fsub(2, 1e-20, dps=15), 25)
2.0
>>> nprint(fsub(2, 1e-20, dps=25), 25)
1.99999999999999999999
>>> nprint(fsub(2, 1e-20, exact=True), 25)
1.99999999999999999999
Exact subtraction avoids cancellation errors, enforcing familiar laws
of numbers such as `x-y+y = x`, which don't hold in floating-point
arithmetic with finite precision::
>>> x, y = mpf(2), mpf('1e1000')
>>> print(x - y + y)
0.0
>>> print(fsub(x, y, prec=inf) + y)
2.0
>>> print(fsub(x, y, exact=True) + y)
2.0
Exact addition can be inefficient and may be impossible to perform
with large magnitude differences::
>>> fsub(1, '1e-100000000000000000000', prec=inf)
Traceback (most recent call last):
...
OverflowError: the exact result does not fit in memory
"""
prec, rounding = ctx._parse_prec(kwargs)
x = ctx.convert(x)
y = ctx.convert(y)
try:
if hasattr(x, '_mpf_'):
if hasattr(y, '_mpf_'):
return ctx.make_mpf(mpf_sub(x._mpf_, y._mpf_, prec, rounding))
if hasattr(y, '_mpc_'):
return ctx.make_mpc(mpc_sub((x._mpf_, fzero), y._mpc_, prec, rounding))
if hasattr(x, '_mpc_'):
if hasattr(y, '_mpf_'):
return ctx.make_mpc(mpc_sub_mpf(x._mpc_, y._mpf_, prec, rounding))
if hasattr(y, '_mpc_'):
return ctx.make_mpc(mpc_sub(x._mpc_, y._mpc_, prec, rounding))
except (ValueError, OverflowError):
raise OverflowError(ctx._exact_overflow_msg)
raise ValueError("Arguments need to be mpf or mpc compatible numbers")
def fmul(ctx, x, y, **kwargs):
"""
Multiplies the numbers *x* and *y*, giving a floating-point result,
optionally using a custom precision and rounding mode.
See the documentation of :func:`~mpmath.fadd` for a detailed description
of how to specify precision and rounding.
**Examples**
The result is an mpmath number::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = False
>>> fmul(2, 5.0)
mpf('10.0')
>>> fmul(0.5j, 0.5)
mpc(real='0.0', imag='0.25')
Avoiding roundoff::
>>> x, y = 10**10+1, 10**15+1
>>> print(x*y)
10000000001000010000000001
>>> print(mpf(x) * mpf(y))
1.0000000001e+25
>>> print(int(mpf(x) * mpf(y)))
10000000001000011026399232
>>> print(int(fmul(x, y)))
10000000001000011026399232
>>> print(int(fmul(x, y, dps=25)))
10000000001000010000000001
>>> print(int(fmul(x, y, exact=True)))
10000000001000010000000001
Exact multiplication with complex numbers can be inefficient and may
be impossible to perform with large magnitude differences between
real and imaginary parts::
>>> x = 1+2j
>>> y = mpc(2, '1e-100000000000000000000')
>>> fmul(x, y)
mpc(real='2.0', imag='4.0')
>>> fmul(x, y, rounding='u')
mpc(real='2.0', imag='4.0000000000000009')
>>> fmul(x, y, exact=True)
Traceback (most recent call last):
...
OverflowError: the exact result does not fit in memory
"""
prec, rounding = ctx._parse_prec(kwargs)
x = ctx.convert(x)
y = ctx.convert(y)
try:
if hasattr(x, '_mpf_'):
if hasattr(y, '_mpf_'):
return ctx.make_mpf(mpf_mul(x._mpf_, y._mpf_, prec, rounding))
if hasattr(y, '_mpc_'):
return ctx.make_mpc(mpc_mul_mpf(y._mpc_, x._mpf_, prec, rounding))
if hasattr(x, '_mpc_'):
if hasattr(y, '_mpf_'):
return ctx.make_mpc(mpc_mul_mpf(x._mpc_, y._mpf_, prec, rounding))
if hasattr(y, '_mpc_'):
return ctx.make_mpc(mpc_mul(x._mpc_, y._mpc_, prec, rounding))
except (ValueError, OverflowError):
raise OverflowError(ctx._exact_overflow_msg)
raise ValueError("Arguments need to be mpf or mpc compatible numbers")
def fdiv(ctx, x, y, **kwargs):
"""
Divides the numbers *x* and *y*, giving a floating-point result,
optionally using a custom precision and rounding mode.
See the documentation of :func:`~mpmath.fadd` for a detailed description
of how to specify precision and rounding.
**Examples**
The result is an mpmath number::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = False
>>> fdiv(3, 2)
mpf('1.5')
>>> fdiv(2, 3)
mpf('0.66666666666666663')
>>> fdiv(2+4j, 0.5)
mpc(real='4.0', imag='8.0')
The rounding direction and precision can be controlled::
>>> fdiv(2, 3, dps=3) # Should be accurate to at least 3 digits
mpf('0.6666259765625')
>>> fdiv(2, 3, rounding='d')
mpf('0.66666666666666663')
>>> fdiv(2, 3, prec=60)
mpf('0.66666666666666667')
>>> fdiv(2, 3, rounding='u')
mpf('0.66666666666666674')
Checking the error of a division by performing it at higher precision::
>>> fdiv(2, 3) - fdiv(2, 3, prec=100)
mpf('-3.7007434154172148e-17')
Unlike :func:`~mpmath.fadd`, :func:`~mpmath.fmul`, etc., exact division is not
allowed since the quotient of two floating-point numbers generally
does not have an exact floating-point representation. (In the
future this might be changed to allow the case where the division
is actually exact.)
>>> fdiv(2, 3, exact=True)
Traceback (most recent call last):
...
ValueError: division is not an exact operation
"""
prec, rounding = ctx._parse_prec(kwargs)
if not prec:
raise ValueError("division is not an exact operation")
x = ctx.convert(x)
y = ctx.convert(y)
if hasattr(x, '_mpf_'):
if hasattr(y, '_mpf_'):
return ctx.make_mpf(mpf_div(x._mpf_, y._mpf_, prec, rounding))
if hasattr(y, '_mpc_'):
return ctx.make_mpc(mpc_div((x._mpf_, fzero), y._mpc_, prec, rounding))
if hasattr(x, '_mpc_'):
if hasattr(y, '_mpf_'):
return ctx.make_mpc(mpc_div_mpf(x._mpc_, y._mpf_, prec, rounding))
if hasattr(y, '_mpc_'):
return ctx.make_mpc(mpc_div(x._mpc_, y._mpc_, prec, rounding))
raise ValueError("Arguments need to be mpf or mpc compatible numbers")
def nint_distance(ctx, x):
r"""
Return `(n,d)` where `n` is the nearest integer to `x` and `d` is
an estimate of `\log_2(|x-n|)`. If `d < 0`, `-d` gives the precision
(measured in bits) lost to cancellation when computing `x-n`.
>>> from mpmath import *
>>> n, d = nint_distance(5)
>>> print(n); print(d)
5
-inf
>>> n, d = nint_distance(mpf(5))
>>> print(n); print(d)
5
-inf
>>> n, d = nint_distance(mpf(5.00000001))
>>> print(n); print(d)
5
-26
>>> n, d = nint_distance(mpf(4.99999999))
>>> print(n); print(d)
5
-26
>>> n, d = nint_distance(mpc(5,10))
>>> print(n); print(d)
5
4
>>> n, d = nint_distance(mpc(5,0.000001))
>>> print(n); print(d)
5
-19
"""
typx = type(x)
if typx in int_types:
return int(x), ctx.ninf
elif typx is rational.mpq:
p, q = x._mpq_
n, r = divmod(p, q)
if 2*r >= q:
n += 1
elif not r:
return n, ctx.ninf
# log(p/q-n) = log((p-nq)/q) = log(p-nq) - log(q)
d = bitcount(abs(p-n*q)) - bitcount(q)
return n, d
if hasattr(x, "_mpf_"):
re = x._mpf_
im_dist = ctx.ninf
elif hasattr(x, "_mpc_"):
re, im = x._mpc_
isign, iman, iexp, ibc = im
if iman:
im_dist = iexp + ibc
elif im == fzero:
im_dist = ctx.ninf
else:
raise ValueError("requires a finite number")
else:
x = ctx.convert(x)
if hasattr(x, "_mpf_") or hasattr(x, "_mpc_"):
return ctx.nint_distance(x)
else:
raise TypeError("requires an mpf/mpc")
sign, man, exp, bc = re
mag = exp+bc
# |x| < 0.5
if mag < 0:
n = 0
re_dist = mag
elif man:
# exact integer
if exp >= 0:
n = man << exp
re_dist = ctx.ninf
# exact half-integer
elif exp == -1:
n = (man>>1)+1
re_dist = 0
else:
d = (-exp-1)
t = man >> d
if t & 1:
t += 1
man = (t<<d) - man
else:
man -= (t<<d)
n = t>>1 # int(t)>>1
re_dist = exp+bitcount(man)
if sign:
n = -n
elif re == fzero:
re_dist = ctx.ninf
n = 0
else:
raise ValueError("requires a finite number")
return n, max(re_dist, im_dist)
def fprod(ctx, factors):
r"""
Calculates a product containing a finite number of factors (for
infinite products, see :func:`~mpmath.nprod`). The factors will be
converted to mpmath numbers.
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = False
>>> fprod([1, 2, 0.5, 7])
mpf('7.0')
"""
orig = ctx.prec
try:
v = ctx.one
for p in factors:
v *= p
finally:
ctx.prec = orig
return +v
def rand(ctx):
"""
Returns an ``mpf`` with value chosen randomly from `[0, 1)`.
The number of randomly generated bits in the mantissa is equal
to the working precision.
"""
return ctx.make_mpf(mpf_rand(ctx._prec))
def fraction(ctx, p, q):
"""
Given Python integers `(p, q)`, returns a lazy ``mpf`` representing
the fraction `p/q`. The value is updated with the precision.
>>> from mpmath import *
>>> mp.dps = 15
>>> a = fraction(1,100)
>>> b = mpf(1)/100
>>> print(a); print(b)
0.01
0.01
>>> mp.dps = 30
>>> print(a); print(b) # a will be accurate
0.01
0.0100000000000000002081668171172
>>> mp.dps = 15
"""
return ctx.constant(lambda prec, rnd: from_rational(p, q, prec, rnd),
'%s/%s' % (p, q))
def absmin(ctx, x):
return abs(ctx.convert(x))
def absmax(ctx, x):
return abs(ctx.convert(x))
def _as_points(ctx, x):
# XXX: remove this?
if hasattr(x, '_mpi_'):
a, b = x._mpi_
return [ctx.make_mpf(a), ctx.make_mpf(b)]
return x
'''
def _zetasum(ctx, s, a, b):
"""
Computes sum of k^(-s) for k = a, a+1, ..., b with a, b both small
integers.
"""
a = int(a)
b = int(b)
s = ctx.convert(s)
prec, rounding = ctx._prec_rounding
if hasattr(s, '_mpf_'):
v = ctx.make_mpf(libmp.mpf_zetasum(s._mpf_, a, b, prec))
elif hasattr(s, '_mpc_'):
v = ctx.make_mpc(libmp.mpc_zetasum(s._mpc_, a, b, prec))
return v
'''
def _zetasum_fast(ctx, s, a, n, derivatives=[0], reflect=False):
if not (ctx.isint(a) and hasattr(s, "_mpc_")):
raise NotImplementedError
a = int(a)
prec = ctx._prec
xs, ys = libmp.mpc_zetasum(s._mpc_, a, n, derivatives, reflect, prec)
xs = [ctx.make_mpc(x) for x in xs]
ys = [ctx.make_mpc(y) for y in ys]
return xs, ys
class PrecisionManager:
def __init__(self, ctx, precfun, dpsfun, normalize_output=False):
self.ctx = ctx
self.precfun = precfun
self.dpsfun = dpsfun
self.normalize_output = normalize_output
def __call__(self, f):
def g(*args, **kwargs):
orig = self.ctx.prec
try:
if self.precfun:
self.ctx.prec = self.precfun(self.ctx.prec)
else:
self.ctx.dps = self.dpsfun(self.ctx.dps)
if self.normalize_output:
v = f(*args, **kwargs)
if type(v) is tuple:
return tuple([+a for a in v])
return +v
else:
return f(*args, **kwargs)
finally:
self.ctx.prec = orig
g.__name__ = f.__name__
g.__doc__ = f.__doc__
return g
def __enter__(self):
self.origp = self.ctx.prec
if self.precfun:
self.ctx.prec = self.precfun(self.ctx.prec)
else:
self.ctx.dps = self.dpsfun(self.ctx.dps)
def __exit__(self, exc_type, exc_val, exc_tb):
self.ctx.prec = self.origp
return False
if __name__ == '__main__':
import doctest
doctest.testmod()
| 49,671 | 36.041014 | 116 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/ctx_mp_python.py
|
#from ctx_base import StandardBaseContext
from .libmp.backend import basestring, exec_
from .libmp import (MPZ, MPZ_ZERO, MPZ_ONE, int_types, repr_dps,
round_floor, round_ceiling, dps_to_prec, round_nearest, prec_to_dps,
ComplexResult, to_pickable, from_pickable, normalize,
from_int, from_float, from_str, to_int, to_float, to_str,
from_rational, from_man_exp,
fone, fzero, finf, fninf, fnan,
mpf_abs, mpf_pos, mpf_neg, mpf_add, mpf_sub, mpf_mul, mpf_mul_int,
mpf_div, mpf_rdiv_int, mpf_pow_int, mpf_mod,
mpf_eq, mpf_cmp, mpf_lt, mpf_gt, mpf_le, mpf_ge,
mpf_hash, mpf_rand,
mpf_sum,
bitcount, to_fixed,
mpc_to_str,
mpc_to_complex, mpc_hash, mpc_pos, mpc_is_nonzero, mpc_neg, mpc_conjugate,
mpc_abs, mpc_add, mpc_add_mpf, mpc_sub, mpc_sub_mpf, mpc_mul, mpc_mul_mpf,
mpc_mul_int, mpc_div, mpc_div_mpf, mpc_pow, mpc_pow_mpf, mpc_pow_int,
mpc_mpf_div,
mpf_pow,
mpf_pi, mpf_degree, mpf_e, mpf_phi, mpf_ln2, mpf_ln10,
mpf_euler, mpf_catalan, mpf_apery, mpf_khinchin,
mpf_glaisher, mpf_twinprime, mpf_mertens,
int_types)
from . import rational
from . import function_docs
new = object.__new__
class mpnumeric(object):
"""Base class for mpf and mpc."""
__slots__ = []
def __new__(cls, val):
raise NotImplementedError
class _mpf(mpnumeric):
"""
An mpf instance holds a real-valued floating-point number. mpf:s
work analogously to Python floats, but support arbitrary-precision
arithmetic.
"""
__slots__ = ['_mpf_']
def __new__(cls, val=fzero, **kwargs):
"""A new mpf can be created from a Python float, an int, a
or a decimal string representing a number in floating-point
format."""
prec, rounding = cls.context._prec_rounding
if kwargs:
prec = kwargs.get('prec', prec)
if 'dps' in kwargs:
prec = dps_to_prec(kwargs['dps'])
rounding = kwargs.get('rounding', rounding)
if type(val) is cls:
sign, man, exp, bc = val._mpf_
if (not man) and exp:
return val
v = new(cls)
v._mpf_ = normalize(sign, man, exp, bc, prec, rounding)
return v
elif type(val) is tuple:
if len(val) == 2:
v = new(cls)
v._mpf_ = from_man_exp(val[0], val[1], prec, rounding)
return v
if len(val) == 4:
sign, man, exp, bc = val
v = new(cls)
v._mpf_ = normalize(sign, MPZ(man), exp, bc, prec, rounding)
return v
raise ValueError
else:
v = new(cls)
v._mpf_ = mpf_pos(cls.mpf_convert_arg(val, prec, rounding), prec, rounding)
return v
@classmethod
def mpf_convert_arg(cls, x, prec, rounding):
if isinstance(x, int_types): return from_int(x)
if isinstance(x, float): return from_float(x)
if isinstance(x, basestring): return from_str(x, prec, rounding)
if isinstance(x, cls.context.constant): return x.func(prec, rounding)
if hasattr(x, '_mpf_'): return x._mpf_
if hasattr(x, '_mpmath_'):
t = cls.context.convert(x._mpmath_(prec, rounding))
if hasattr(t, '_mpf_'):
return t._mpf_
if hasattr(x, '_mpi_'):
a, b = x._mpi_
if a == b:
return a
raise ValueError("can only create mpf from zero-width interval")
raise TypeError("cannot create mpf from " + repr(x))
@classmethod
def mpf_convert_rhs(cls, x):
if isinstance(x, int_types): return from_int(x)
if isinstance(x, float): return from_float(x)
if isinstance(x, complex_types): return cls.context.mpc(x)
if isinstance(x, rational.mpq):
p, q = x._mpq_
return from_rational(p, q, cls.context.prec)
if hasattr(x, '_mpf_'): return x._mpf_
if hasattr(x, '_mpmath_'):
t = cls.context.convert(x._mpmath_(*cls.context._prec_rounding))
if hasattr(t, '_mpf_'):
return t._mpf_
return t
return NotImplemented
@classmethod
def mpf_convert_lhs(cls, x):
x = cls.mpf_convert_rhs(x)
if type(x) is tuple:
return cls.context.make_mpf(x)
return x
man_exp = property(lambda self: self._mpf_[1:3])
man = property(lambda self: self._mpf_[1])
exp = property(lambda self: self._mpf_[2])
bc = property(lambda self: self._mpf_[3])
real = property(lambda self: self)
imag = property(lambda self: self.context.zero)
conjugate = lambda self: self
def __getstate__(self): return to_pickable(self._mpf_)
def __setstate__(self, val): self._mpf_ = from_pickable(val)
def __repr__(s):
if s.context.pretty:
return str(s)
return "mpf('%s')" % to_str(s._mpf_, s.context._repr_digits)
def __str__(s): return to_str(s._mpf_, s.context._str_digits)
def __hash__(s): return mpf_hash(s._mpf_)
def __int__(s): return int(to_int(s._mpf_))
def __long__(s): return long(to_int(s._mpf_))
def __float__(s): return to_float(s._mpf_, rnd=s.context._prec_rounding[1])
def __complex__(s): return complex(float(s))
def __nonzero__(s): return s._mpf_ != fzero
__bool__ = __nonzero__
def __abs__(s):
cls, new, (prec, rounding) = s._ctxdata
v = new(cls)
v._mpf_ = mpf_abs(s._mpf_, prec, rounding)
return v
def __pos__(s):
cls, new, (prec, rounding) = s._ctxdata
v = new(cls)
v._mpf_ = mpf_pos(s._mpf_, prec, rounding)
return v
def __neg__(s):
cls, new, (prec, rounding) = s._ctxdata
v = new(cls)
v._mpf_ = mpf_neg(s._mpf_, prec, rounding)
return v
def _cmp(s, t, func):
if hasattr(t, '_mpf_'):
t = t._mpf_
else:
t = s.mpf_convert_rhs(t)
if t is NotImplemented:
return t
return func(s._mpf_, t)
def __cmp__(s, t): return s._cmp(t, mpf_cmp)
def __lt__(s, t): return s._cmp(t, mpf_lt)
def __gt__(s, t): return s._cmp(t, mpf_gt)
def __le__(s, t): return s._cmp(t, mpf_le)
def __ge__(s, t): return s._cmp(t, mpf_ge)
def __ne__(s, t):
v = s.__eq__(t)
if v is NotImplemented:
return v
return not v
def __rsub__(s, t):
cls, new, (prec, rounding) = s._ctxdata
if type(t) in int_types:
v = new(cls)
v._mpf_ = mpf_sub(from_int(t), s._mpf_, prec, rounding)
return v
t = s.mpf_convert_lhs(t)
if t is NotImplemented:
return t
return t - s
def __rdiv__(s, t):
cls, new, (prec, rounding) = s._ctxdata
if isinstance(t, int_types):
v = new(cls)
v._mpf_ = mpf_rdiv_int(t, s._mpf_, prec, rounding)
return v
t = s.mpf_convert_lhs(t)
if t is NotImplemented:
return t
return t / s
def __rpow__(s, t):
t = s.mpf_convert_lhs(t)
if t is NotImplemented:
return t
return t ** s
def __rmod__(s, t):
t = s.mpf_convert_lhs(t)
if t is NotImplemented:
return t
return t % s
def sqrt(s):
return s.context.sqrt(s)
def ae(s, t, rel_eps=None, abs_eps=None):
return s.context.almosteq(s, t, rel_eps, abs_eps)
def to_fixed(self, prec):
return to_fixed(self._mpf_, prec)
def __round__(self, *args):
return round(float(self), *args)
mpf_binary_op = """
def %NAME%(self, other):
mpf, new, (prec, rounding) = self._ctxdata
sval = self._mpf_
if hasattr(other, '_mpf_'):
tval = other._mpf_
%WITH_MPF%
ttype = type(other)
if ttype in int_types:
%WITH_INT%
elif ttype is float:
tval = from_float(other)
%WITH_MPF%
elif hasattr(other, '_mpc_'):
tval = other._mpc_
mpc = type(other)
%WITH_MPC%
elif ttype is complex:
tval = from_float(other.real), from_float(other.imag)
mpc = self.context.mpc
%WITH_MPC%
if isinstance(other, mpnumeric):
return NotImplemented
try:
other = mpf.context.convert(other, strings=False)
except TypeError:
return NotImplemented
return self.%NAME%(other)
"""
return_mpf = "; obj = new(mpf); obj._mpf_ = val; return obj"
return_mpc = "; obj = new(mpc); obj._mpc_ = val; return obj"
mpf_pow_same = """
try:
val = mpf_pow(sval, tval, prec, rounding) %s
except ComplexResult:
if mpf.context.trap_complex:
raise
mpc = mpf.context.mpc
val = mpc_pow((sval, fzero), (tval, fzero), prec, rounding) %s
""" % (return_mpf, return_mpc)
def binary_op(name, with_mpf='', with_int='', with_mpc=''):
code = mpf_binary_op
code = code.replace("%WITH_INT%", with_int)
code = code.replace("%WITH_MPC%", with_mpc)
code = code.replace("%WITH_MPF%", with_mpf)
code = code.replace("%NAME%", name)
np = {}
exec_(code, globals(), np)
return np[name]
_mpf.__eq__ = binary_op('__eq__',
'return mpf_eq(sval, tval)',
'return mpf_eq(sval, from_int(other))',
'return (tval[1] == fzero) and mpf_eq(tval[0], sval)')
_mpf.__add__ = binary_op('__add__',
'val = mpf_add(sval, tval, prec, rounding)' + return_mpf,
'val = mpf_add(sval, from_int(other), prec, rounding)' + return_mpf,
'val = mpc_add_mpf(tval, sval, prec, rounding)' + return_mpc)
_mpf.__sub__ = binary_op('__sub__',
'val = mpf_sub(sval, tval, prec, rounding)' + return_mpf,
'val = mpf_sub(sval, from_int(other), prec, rounding)' + return_mpf,
'val = mpc_sub((sval, fzero), tval, prec, rounding)' + return_mpc)
_mpf.__mul__ = binary_op('__mul__',
'val = mpf_mul(sval, tval, prec, rounding)' + return_mpf,
'val = mpf_mul_int(sval, other, prec, rounding)' + return_mpf,
'val = mpc_mul_mpf(tval, sval, prec, rounding)' + return_mpc)
_mpf.__div__ = binary_op('__div__',
'val = mpf_div(sval, tval, prec, rounding)' + return_mpf,
'val = mpf_div(sval, from_int(other), prec, rounding)' + return_mpf,
'val = mpc_mpf_div(sval, tval, prec, rounding)' + return_mpc)
_mpf.__mod__ = binary_op('__mod__',
'val = mpf_mod(sval, tval, prec, rounding)' + return_mpf,
'val = mpf_mod(sval, from_int(other), prec, rounding)' + return_mpf,
'raise NotImplementedError("complex modulo")')
_mpf.__pow__ = binary_op('__pow__',
mpf_pow_same,
'val = mpf_pow_int(sval, other, prec, rounding)' + return_mpf,
'val = mpc_pow((sval, fzero), tval, prec, rounding)' + return_mpc)
_mpf.__radd__ = _mpf.__add__
_mpf.__rmul__ = _mpf.__mul__
_mpf.__truediv__ = _mpf.__div__
_mpf.__rtruediv__ = _mpf.__rdiv__
class _constant(_mpf):
"""Represents a mathematical constant with dynamic precision.
When printed or used in an arithmetic operation, a constant
is converted to a regular mpf at the working precision. A
regular mpf can also be obtained using the operation +x."""
def __new__(cls, func, name, docname=''):
a = object.__new__(cls)
a.name = name
a.func = func
a.__doc__ = getattr(function_docs, docname, '')
return a
def __call__(self, prec=None, dps=None, rounding=None):
prec2, rounding2 = self.context._prec_rounding
if not prec: prec = prec2
if not rounding: rounding = rounding2
if dps: prec = dps_to_prec(dps)
return self.context.make_mpf(self.func(prec, rounding))
@property
def _mpf_(self):
prec, rounding = self.context._prec_rounding
return self.func(prec, rounding)
def __repr__(self):
return "<%s: %s~>" % (self.name, self.context.nstr(self(dps=15)))
class _mpc(mpnumeric):
"""
An mpc represents a complex number using a pair of mpf:s (one
for the real part and another for the imaginary part.) The mpc
class behaves fairly similarly to Python's complex type.
"""
__slots__ = ['_mpc_']
def __new__(cls, real=0, imag=0):
s = object.__new__(cls)
if isinstance(real, complex_types):
real, imag = real.real, real.imag
elif hasattr(real, '_mpc_'):
s._mpc_ = real._mpc_
return s
real = cls.context.mpf(real)
imag = cls.context.mpf(imag)
s._mpc_ = (real._mpf_, imag._mpf_)
return s
real = property(lambda self: self.context.make_mpf(self._mpc_[0]))
imag = property(lambda self: self.context.make_mpf(self._mpc_[1]))
def __getstate__(self):
return to_pickable(self._mpc_[0]), to_pickable(self._mpc_[1])
def __setstate__(self, val):
self._mpc_ = from_pickable(val[0]), from_pickable(val[1])
def __repr__(s):
if s.context.pretty:
return str(s)
r = repr(s.real)[4:-1]
i = repr(s.imag)[4:-1]
return "%s(real=%s, imag=%s)" % (type(s).__name__, r, i)
def __str__(s):
return "(%s)" % mpc_to_str(s._mpc_, s.context._str_digits)
def __complex__(s):
return mpc_to_complex(s._mpc_, rnd=s.context._prec_rounding[1])
def __pos__(s):
cls, new, (prec, rounding) = s._ctxdata
v = new(cls)
v._mpc_ = mpc_pos(s._mpc_, prec, rounding)
return v
def __abs__(s):
prec, rounding = s.context._prec_rounding
v = new(s.context.mpf)
v._mpf_ = mpc_abs(s._mpc_, prec, rounding)
return v
def __neg__(s):
cls, new, (prec, rounding) = s._ctxdata
v = new(cls)
v._mpc_ = mpc_neg(s._mpc_, prec, rounding)
return v
def conjugate(s):
cls, new, (prec, rounding) = s._ctxdata
v = new(cls)
v._mpc_ = mpc_conjugate(s._mpc_, prec, rounding)
return v
def __nonzero__(s):
return mpc_is_nonzero(s._mpc_)
__bool__ = __nonzero__
def __hash__(s):
return mpc_hash(s._mpc_)
@classmethod
def mpc_convert_lhs(cls, x):
try:
y = cls.context.convert(x)
return y
except TypeError:
return NotImplemented
def __eq__(s, t):
if not hasattr(t, '_mpc_'):
if isinstance(t, str):
return False
t = s.mpc_convert_lhs(t)
if t is NotImplemented:
return t
return s.real == t.real and s.imag == t.imag
def __ne__(s, t):
b = s.__eq__(t)
if b is NotImplemented:
return b
return not b
def _compare(*args):
raise TypeError("no ordering relation is defined for complex numbers")
__gt__ = _compare
__le__ = _compare
__gt__ = _compare
__ge__ = _compare
def __add__(s, t):
cls, new, (prec, rounding) = s._ctxdata
if not hasattr(t, '_mpc_'):
t = s.mpc_convert_lhs(t)
if t is NotImplemented:
return t
if hasattr(t, '_mpf_'):
v = new(cls)
v._mpc_ = mpc_add_mpf(s._mpc_, t._mpf_, prec, rounding)
return v
v = new(cls)
v._mpc_ = mpc_add(s._mpc_, t._mpc_, prec, rounding)
return v
def __sub__(s, t):
cls, new, (prec, rounding) = s._ctxdata
if not hasattr(t, '_mpc_'):
t = s.mpc_convert_lhs(t)
if t is NotImplemented:
return t
if hasattr(t, '_mpf_'):
v = new(cls)
v._mpc_ = mpc_sub_mpf(s._mpc_, t._mpf_, prec, rounding)
return v
v = new(cls)
v._mpc_ = mpc_sub(s._mpc_, t._mpc_, prec, rounding)
return v
def __mul__(s, t):
cls, new, (prec, rounding) = s._ctxdata
if not hasattr(t, '_mpc_'):
if isinstance(t, int_types):
v = new(cls)
v._mpc_ = mpc_mul_int(s._mpc_, t, prec, rounding)
return v
t = s.mpc_convert_lhs(t)
if t is NotImplemented:
return t
if hasattr(t, '_mpf_'):
v = new(cls)
v._mpc_ = mpc_mul_mpf(s._mpc_, t._mpf_, prec, rounding)
return v
t = s.mpc_convert_lhs(t)
v = new(cls)
v._mpc_ = mpc_mul(s._mpc_, t._mpc_, prec, rounding)
return v
def __div__(s, t):
cls, new, (prec, rounding) = s._ctxdata
if not hasattr(t, '_mpc_'):
t = s.mpc_convert_lhs(t)
if t is NotImplemented:
return t
if hasattr(t, '_mpf_'):
v = new(cls)
v._mpc_ = mpc_div_mpf(s._mpc_, t._mpf_, prec, rounding)
return v
v = new(cls)
v._mpc_ = mpc_div(s._mpc_, t._mpc_, prec, rounding)
return v
def __pow__(s, t):
cls, new, (prec, rounding) = s._ctxdata
if isinstance(t, int_types):
v = new(cls)
v._mpc_ = mpc_pow_int(s._mpc_, t, prec, rounding)
return v
t = s.mpc_convert_lhs(t)
if t is NotImplemented:
return t
v = new(cls)
if hasattr(t, '_mpf_'):
v._mpc_ = mpc_pow_mpf(s._mpc_, t._mpf_, prec, rounding)
else:
v._mpc_ = mpc_pow(s._mpc_, t._mpc_, prec, rounding)
return v
__radd__ = __add__
def __rsub__(s, t):
t = s.mpc_convert_lhs(t)
if t is NotImplemented:
return t
return t - s
def __rmul__(s, t):
cls, new, (prec, rounding) = s._ctxdata
if isinstance(t, int_types):
v = new(cls)
v._mpc_ = mpc_mul_int(s._mpc_, t, prec, rounding)
return v
t = s.mpc_convert_lhs(t)
if t is NotImplemented:
return t
return t * s
def __rdiv__(s, t):
t = s.mpc_convert_lhs(t)
if t is NotImplemented:
return t
return t / s
def __rpow__(s, t):
t = s.mpc_convert_lhs(t)
if t is NotImplemented:
return t
return t ** s
__truediv__ = __div__
__rtruediv__ = __rdiv__
def ae(s, t, rel_eps=None, abs_eps=None):
return s.context.almosteq(s, t, rel_eps, abs_eps)
complex_types = (complex, _mpc)
class PythonMPContext(object):
def __init__(ctx):
ctx._prec_rounding = [53, round_nearest]
ctx.mpf = type('mpf', (_mpf,), {})
ctx.mpc = type('mpc', (_mpc,), {})
ctx.mpf._ctxdata = [ctx.mpf, new, ctx._prec_rounding]
ctx.mpc._ctxdata = [ctx.mpc, new, ctx._prec_rounding]
ctx.mpf.context = ctx
ctx.mpc.context = ctx
ctx.constant = type('constant', (_constant,), {})
ctx.constant._ctxdata = [ctx.mpf, new, ctx._prec_rounding]
ctx.constant.context = ctx
def make_mpf(ctx, v):
a = new(ctx.mpf)
a._mpf_ = v
return a
def make_mpc(ctx, v):
a = new(ctx.mpc)
a._mpc_ = v
return a
def default(ctx):
ctx._prec = ctx._prec_rounding[0] = 53
ctx._dps = 15
ctx.trap_complex = False
def _set_prec(ctx, n):
ctx._prec = ctx._prec_rounding[0] = max(1, int(n))
ctx._dps = prec_to_dps(n)
def _set_dps(ctx, n):
ctx._prec = ctx._prec_rounding[0] = dps_to_prec(n)
ctx._dps = max(1, int(n))
prec = property(lambda ctx: ctx._prec, _set_prec)
dps = property(lambda ctx: ctx._dps, _set_dps)
def convert(ctx, x, strings=True):
"""
Converts *x* to an ``mpf`` or ``mpc``. If *x* is of type ``mpf``,
``mpc``, ``int``, ``float``, ``complex``, the conversion
will be performed losslessly.
If *x* is a string, the result will be rounded to the present
working precision. Strings representing fractions or complex
numbers are permitted.
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = False
>>> mpmathify(3.5)
mpf('3.5')
>>> mpmathify('2.1')
mpf('2.1000000000000001')
>>> mpmathify('3/4')
mpf('0.75')
>>> mpmathify('2+3j')
mpc(real='2.0', imag='3.0')
"""
if type(x) in ctx.types: return x
if isinstance(x, int_types): return ctx.make_mpf(from_int(x))
if isinstance(x, float): return ctx.make_mpf(from_float(x))
if isinstance(x, complex):
return ctx.make_mpc((from_float(x.real), from_float(x.imag)))
prec, rounding = ctx._prec_rounding
if isinstance(x, rational.mpq):
p, q = x._mpq_
return ctx.make_mpf(from_rational(p, q, prec))
if strings and isinstance(x, basestring):
try:
_mpf_ = from_str(x, prec, rounding)
return ctx.make_mpf(_mpf_)
except ValueError:
pass
if hasattr(x, '_mpf_'): return ctx.make_mpf(x._mpf_)
if hasattr(x, '_mpc_'): return ctx.make_mpc(x._mpc_)
if hasattr(x, '_mpmath_'):
return ctx.convert(x._mpmath_(prec, rounding))
return ctx._convert_fallback(x, strings)
def isnan(ctx, x):
"""
Return *True* if *x* is a NaN (not-a-number), or for a complex
number, whether either the real or complex part is NaN;
otherwise return *False*::
>>> from mpmath import *
>>> isnan(3.14)
False
>>> isnan(nan)
True
>>> isnan(mpc(3.14,2.72))
False
>>> isnan(mpc(3.14,nan))
True
"""
if hasattr(x, "_mpf_"):
return x._mpf_ == fnan
if hasattr(x, "_mpc_"):
return fnan in x._mpc_
if isinstance(x, int_types) or isinstance(x, rational.mpq):
return False
x = ctx.convert(x)
if hasattr(x, '_mpf_') or hasattr(x, '_mpc_'):
return ctx.isnan(x)
raise TypeError("isnan() needs a number as input")
def isinf(ctx, x):
"""
Return *True* if the absolute value of *x* is infinite;
otherwise return *False*::
>>> from mpmath import *
>>> isinf(inf)
True
>>> isinf(-inf)
True
>>> isinf(3)
False
>>> isinf(3+4j)
False
>>> isinf(mpc(3,inf))
True
>>> isinf(mpc(inf,3))
True
"""
if hasattr(x, "_mpf_"):
return x._mpf_ in (finf, fninf)
if hasattr(x, "_mpc_"):
re, im = x._mpc_
return re in (finf, fninf) or im in (finf, fninf)
if isinstance(x, int_types) or isinstance(x, rational.mpq):
return False
x = ctx.convert(x)
if hasattr(x, '_mpf_') or hasattr(x, '_mpc_'):
return ctx.isinf(x)
raise TypeError("isinf() needs a number as input")
def isnormal(ctx, x):
"""
Determine whether *x* is "normal" in the sense of floating-point
representation; that is, return *False* if *x* is zero, an
infinity or NaN; otherwise return *True*. By extension, a
complex number *x* is considered "normal" if its magnitude is
normal::
>>> from mpmath import *
>>> isnormal(3)
True
>>> isnormal(0)
False
>>> isnormal(inf); isnormal(-inf); isnormal(nan)
False
False
False
>>> isnormal(0+0j)
False
>>> isnormal(0+3j)
True
>>> isnormal(mpc(2,nan))
False
"""
if hasattr(x, "_mpf_"):
return bool(x._mpf_[1])
if hasattr(x, "_mpc_"):
re, im = x._mpc_
re_normal = bool(re[1])
im_normal = bool(im[1])
if re == fzero: return im_normal
if im == fzero: return re_normal
return re_normal and im_normal
if isinstance(x, int_types) or isinstance(x, rational.mpq):
return bool(x)
x = ctx.convert(x)
if hasattr(x, '_mpf_') or hasattr(x, '_mpc_'):
return ctx.isnormal(x)
raise TypeError("isnormal() needs a number as input")
def isint(ctx, x, gaussian=False):
"""
Return *True* if *x* is integer-valued; otherwise return
*False*::
>>> from mpmath import *
>>> isint(3)
True
>>> isint(mpf(3))
True
>>> isint(3.2)
False
>>> isint(inf)
False
Optionally, Gaussian integers can be checked for::
>>> isint(3+0j)
True
>>> isint(3+2j)
False
>>> isint(3+2j, gaussian=True)
True
"""
if isinstance(x, int_types):
return True
if hasattr(x, "_mpf_"):
sign, man, exp, bc = xval = x._mpf_
return bool((man and exp >= 0) or xval == fzero)
if hasattr(x, "_mpc_"):
re, im = x._mpc_
rsign, rman, rexp, rbc = re
isign, iman, iexp, ibc = im
re_isint = (rman and rexp >= 0) or re == fzero
if gaussian:
im_isint = (iman and iexp >= 0) or im == fzero
return re_isint and im_isint
return re_isint and im == fzero
if isinstance(x, rational.mpq):
p, q = x._mpq_
return p % q == 0
x = ctx.convert(x)
if hasattr(x, '_mpf_') or hasattr(x, '_mpc_'):
return ctx.isint(x, gaussian)
raise TypeError("isint() needs a number as input")
def fsum(ctx, terms, absolute=False, squared=False):
"""
Calculates a sum containing a finite number of terms (for infinite
series, see :func:`~mpmath.nsum`). The terms will be converted to
mpmath numbers. For len(terms) > 2, this function is generally
faster and produces more accurate results than the builtin
Python function :func:`sum`.
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = False
>>> fsum([1, 2, 0.5, 7])
mpf('10.5')
With squared=True each term is squared, and with absolute=True
the absolute value of each term is used.
"""
prec, rnd = ctx._prec_rounding
real = []
imag = []
other = 0
for term in terms:
reval = imval = 0
if hasattr(term, "_mpf_"):
reval = term._mpf_
elif hasattr(term, "_mpc_"):
reval, imval = term._mpc_
else:
term = ctx.convert(term)
if hasattr(term, "_mpf_"):
reval = term._mpf_
elif hasattr(term, "_mpc_"):
reval, imval = term._mpc_
else:
if absolute: term = ctx.absmax(term)
if squared: term = term**2
other += term
continue
if imval:
if squared:
if absolute:
real.append(mpf_mul(reval,reval))
real.append(mpf_mul(imval,imval))
else:
reval, imval = mpc_pow_int((reval,imval),2,prec+10)
real.append(reval)
imag.append(imval)
elif absolute:
real.append(mpc_abs((reval,imval), prec))
else:
real.append(reval)
imag.append(imval)
else:
if squared:
reval = mpf_mul(reval, reval)
elif absolute:
reval = mpf_abs(reval)
real.append(reval)
s = mpf_sum(real, prec, rnd, absolute)
if imag:
s = ctx.make_mpc((s, mpf_sum(imag, prec, rnd)))
else:
s = ctx.make_mpf(s)
if other is 0:
return s
else:
return s + other
def fdot(ctx, A, B=None, conjugate=False):
r"""
Computes the dot product of the iterables `A` and `B`,
.. math ::
\sum_{k=0} A_k B_k.
Alternatively, :func:`~mpmath.fdot` accepts a single iterable of pairs.
In other words, ``fdot(A,B)`` and ``fdot(zip(A,B))`` are equivalent.
The elements are automatically converted to mpmath numbers.
With ``conjugate=True``, the elements in the second vector
will be conjugated:
.. math ::
\sum_{k=0} A_k \overline{B_k}
**Examples**
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = False
>>> A = [2, 1.5, 3]
>>> B = [1, -1, 2]
>>> fdot(A, B)
mpf('6.5')
>>> list(zip(A, B))
[(2, 1), (1.5, -1), (3, 2)]
>>> fdot(_)
mpf('6.5')
>>> A = [2, 1.5, 3j]
>>> B = [1+j, 3, -1-j]
>>> fdot(A, B)
mpc(real='9.5', imag='-1.0')
>>> fdot(A, B, conjugate=True)
mpc(real='3.5', imag='-5.0')
"""
if B:
A = zip(A, B)
prec, rnd = ctx._prec_rounding
real = []
imag = []
other = 0
hasattr_ = hasattr
types = (ctx.mpf, ctx.mpc)
for a, b in A:
if type(a) not in types: a = ctx.convert(a)
if type(b) not in types: b = ctx.convert(b)
a_real = hasattr_(a, "_mpf_")
b_real = hasattr_(b, "_mpf_")
if a_real and b_real:
real.append(mpf_mul(a._mpf_, b._mpf_))
continue
a_complex = hasattr_(a, "_mpc_")
b_complex = hasattr_(b, "_mpc_")
if a_real and b_complex:
aval = a._mpf_
bre, bim = b._mpc_
if conjugate:
bim = mpf_neg(bim)
real.append(mpf_mul(aval, bre))
imag.append(mpf_mul(aval, bim))
elif b_real and a_complex:
are, aim = a._mpc_
bval = b._mpf_
real.append(mpf_mul(are, bval))
imag.append(mpf_mul(aim, bval))
elif a_complex and b_complex:
#re, im = mpc_mul(a._mpc_, b._mpc_, prec+20)
are, aim = a._mpc_
bre, bim = b._mpc_
if conjugate:
bim = mpf_neg(bim)
real.append(mpf_mul(are, bre))
real.append(mpf_neg(mpf_mul(aim, bim)))
imag.append(mpf_mul(are, bim))
imag.append(mpf_mul(aim, bre))
else:
if conjugate:
other += a*ctx.conj(b)
else:
other += a*b
s = mpf_sum(real, prec, rnd)
if imag:
s = ctx.make_mpc((s, mpf_sum(imag, prec, rnd)))
else:
s = ctx.make_mpf(s)
if other is 0:
return s
else:
return s + other
def _wrap_libmp_function(ctx, mpf_f, mpc_f=None, mpi_f=None, doc="<no doc>"):
"""
Given a low-level mpf_ function, and optionally similar functions
for mpc_ and mpi_, defines the function as a context method.
It is assumed that the return type is the same as that of
the input; the exception is that propagation from mpf to mpc is possible
by raising ComplexResult.
"""
def f(x, **kwargs):
if type(x) not in ctx.types:
x = ctx.convert(x)
prec, rounding = ctx._prec_rounding
if kwargs:
prec = kwargs.get('prec', prec)
if 'dps' in kwargs:
prec = dps_to_prec(kwargs['dps'])
rounding = kwargs.get('rounding', rounding)
if hasattr(x, '_mpf_'):
try:
return ctx.make_mpf(mpf_f(x._mpf_, prec, rounding))
except ComplexResult:
# Handle propagation to complex
if ctx.trap_complex:
raise
return ctx.make_mpc(mpc_f((x._mpf_, fzero), prec, rounding))
elif hasattr(x, '_mpc_'):
return ctx.make_mpc(mpc_f(x._mpc_, prec, rounding))
raise NotImplementedError("%s of a %s" % (name, type(x)))
name = mpf_f.__name__[4:]
f.__doc__ = function_docs.__dict__.get(name, "Computes the %s of x" % doc)
return f
# Called by SpecialFunctions.__init__()
@classmethod
def _wrap_specfun(cls, name, f, wrap):
if wrap:
def f_wrapped(ctx, *args, **kwargs):
convert = ctx.convert
args = [convert(a) for a in args]
prec = ctx.prec
try:
ctx.prec += 10
retval = f(ctx, *args, **kwargs)
finally:
ctx.prec = prec
return +retval
else:
f_wrapped = f
f_wrapped.__doc__ = function_docs.__dict__.get(name, f.__doc__)
setattr(cls, name, f_wrapped)
def _convert_param(ctx, x):
if hasattr(x, "_mpc_"):
v, im = x._mpc_
if im != fzero:
return x, 'C'
elif hasattr(x, "_mpf_"):
v = x._mpf_
else:
if type(x) in int_types:
return int(x), 'Z'
p = None
if isinstance(x, tuple):
p, q = x
elif hasattr(x, '_mpq_'):
p, q = x._mpq_
elif isinstance(x, basestring) and '/' in x:
p, q = x.split('/')
p = int(p)
q = int(q)
if p is not None:
if not p % q:
return p // q, 'Z'
return ctx.mpq(p,q), 'Q'
x = ctx.convert(x)
if hasattr(x, "_mpc_"):
v, im = x._mpc_
if im != fzero:
return x, 'C'
elif hasattr(x, "_mpf_"):
v = x._mpf_
else:
return x, 'U'
sign, man, exp, bc = v
if man:
if exp >= -4:
if sign:
man = -man
if exp >= 0:
return int(man) << exp, 'Z'
if exp >= -4:
p, q = int(man), (1<<(-exp))
return ctx.mpq(p,q), 'Q'
x = ctx.make_mpf(v)
return x, 'R'
elif not exp:
return 0, 'Z'
else:
return x, 'U'
def _mpf_mag(ctx, x):
sign, man, exp, bc = x
if man:
return exp+bc
if x == fzero:
return ctx.ninf
if x == finf or x == fninf:
return ctx.inf
return ctx.nan
def mag(ctx, x):
"""
Quick logarithmic magnitude estimate of a number. Returns an
integer or infinity `m` such that `|x| <= 2^m`. It is not
guaranteed that `m` is an optimal bound, but it will never
be too large by more than 2 (and probably not more than 1).
**Examples**
>>> from mpmath import *
>>> mp.pretty = True
>>> mag(10), mag(10.0), mag(mpf(10)), int(ceil(log(10,2)))
(4, 4, 4, 4)
>>> mag(10j), mag(10+10j)
(4, 5)
>>> mag(0.01), int(ceil(log(0.01,2)))
(-6, -6)
>>> mag(0), mag(inf), mag(-inf), mag(nan)
(-inf, +inf, +inf, nan)
"""
if hasattr(x, "_mpf_"):
return ctx._mpf_mag(x._mpf_)
elif hasattr(x, "_mpc_"):
r, i = x._mpc_
if r == fzero:
return ctx._mpf_mag(i)
if i == fzero:
return ctx._mpf_mag(r)
return 1+max(ctx._mpf_mag(r), ctx._mpf_mag(i))
elif isinstance(x, int_types):
if x:
return bitcount(abs(x))
return ctx.ninf
elif isinstance(x, rational.mpq):
p, q = x._mpq_
if p:
return 1 + bitcount(abs(p)) - bitcount(q)
return ctx.ninf
else:
x = ctx.convert(x)
if hasattr(x, "_mpf_") or hasattr(x, "_mpc_"):
return ctx.mag(x)
else:
raise TypeError("requires an mpf/mpc")
# Register with "numbers" ABC
# We do not subclass, hence we do not use the @abstractmethod checks. While
# this is less invasive it may turn out that we do not actually support
# parts of the expected interfaces. See
# http://docs.python.org/2/library/numbers.html for list of abstract
# methods.
try:
import numbers
numbers.Complex.register(_mpc)
numbers.Real.register(_mpf)
except ImportError:
pass
| 37,210 | 31.555556 | 87 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/ctx_iv.py
|
import operator
from . import libmp
from .libmp.backend import basestring
from .libmp import (
int_types, MPZ_ONE,
prec_to_dps, dps_to_prec, repr_dps,
round_floor, round_ceiling,
fzero, finf, fninf, fnan,
mpf_le, mpf_neg,
from_int, from_float, from_str, from_rational,
mpi_mid, mpi_delta, mpi_str,
mpi_abs, mpi_pos, mpi_neg, mpi_add, mpi_sub,
mpi_mul, mpi_div, mpi_pow_int, mpi_pow,
mpi_from_str,
mpci_pos, mpci_neg, mpci_add, mpci_sub, mpci_mul, mpci_div, mpci_pow,
mpci_abs, mpci_pow, mpci_exp, mpci_log,
ComplexResult,
mpf_hash, mpc_hash)
mpi_zero = (fzero, fzero)
from .ctx_base import StandardBaseContext
new = object.__new__
def convert_mpf_(x, prec, rounding):
if hasattr(x, "_mpf_"): return x._mpf_
if isinstance(x, int_types): return from_int(x, prec, rounding)
if isinstance(x, float): return from_float(x, prec, rounding)
if isinstance(x, basestring): return from_str(x, prec, rounding)
class ivmpf(object):
"""
Interval arithmetic class. Precision is controlled by iv.prec.
"""
def __new__(cls, x=0):
return cls.ctx.convert(x)
def __int__(self):
a, b = self._mpi_
if a == b:
return int(libmp.to_int(a))
raise ValueError
def __hash__(self):
a, b = self._mpi_
if a == b:
return mpf_hash(a)
else:
return hash(self._mpi_)
@property
def real(self): return self
@property
def imag(self): return self.ctx.zero
def conjugate(self): return self
@property
def a(self):
a, b = self._mpi_
return self.ctx.make_mpf((a, a))
@property
def b(self):
a, b = self._mpi_
return self.ctx.make_mpf((b, b))
@property
def mid(self):
ctx = self.ctx
v = mpi_mid(self._mpi_, ctx.prec)
return ctx.make_mpf((v, v))
@property
def delta(self):
ctx = self.ctx
v = mpi_delta(self._mpi_, ctx.prec)
return ctx.make_mpf((v,v))
@property
def _mpci_(self):
return self._mpi_, mpi_zero
def _compare(*args):
raise TypeError("no ordering relation is defined for intervals")
__gt__ = _compare
__le__ = _compare
__gt__ = _compare
__ge__ = _compare
def __contains__(self, t):
t = self.ctx.mpf(t)
return (self.a <= t.a) and (t.b <= self.b)
def __str__(self):
return mpi_str(self._mpi_, self.ctx.prec)
def __repr__(self):
if self.ctx.pretty:
return str(self)
a, b = self._mpi_
n = repr_dps(self.ctx.prec)
a = libmp.to_str(a, n)
b = libmp.to_str(b, n)
return "mpi(%r, %r)" % (a, b)
def _compare(s, t, cmpfun):
if not hasattr(t, "_mpi_"):
try:
t = s.ctx.convert(t)
except:
return NotImplemented
return cmpfun(s._mpi_, t._mpi_)
def __eq__(s, t): return s._compare(t, libmp.mpi_eq)
def __ne__(s, t): return s._compare(t, libmp.mpi_ne)
def __lt__(s, t): return s._compare(t, libmp.mpi_lt)
def __le__(s, t): return s._compare(t, libmp.mpi_le)
def __gt__(s, t): return s._compare(t, libmp.mpi_gt)
def __ge__(s, t): return s._compare(t, libmp.mpi_ge)
def __abs__(self):
return self.ctx.make_mpf(mpi_abs(self._mpi_, self.ctx.prec))
def __pos__(self):
return self.ctx.make_mpf(mpi_pos(self._mpi_, self.ctx.prec))
def __neg__(self):
return self.ctx.make_mpf(mpi_neg(self._mpi_, self.ctx.prec))
def ae(s, t, rel_eps=None, abs_eps=None):
return s.ctx.almosteq(s, t, rel_eps, abs_eps)
class ivmpc(object):
def __new__(cls, re=0, im=0):
re = cls.ctx.convert(re)
im = cls.ctx.convert(im)
y = new(cls)
y._mpci_ = re._mpi_, im._mpi_
return y
def __hash__(self):
(a, b), (c,d) = self._mpci_
if a == b and c == d:
return mpc_hash((a, c))
else:
return hash(self._mpci_)
def __repr__(s):
if s.ctx.pretty:
return str(s)
return "iv.mpc(%s, %s)" % (repr(s.real), repr(s.imag))
def __str__(s):
return "(%s + %s*j)" % (str(s.real), str(s.imag))
@property
def a(self):
(a, b), (c,d) = self._mpci_
return self.ctx.make_mpf((a, a))
@property
def b(self):
(a, b), (c,d) = self._mpci_
return self.ctx.make_mpf((b, b))
@property
def c(self):
(a, b), (c,d) = self._mpci_
return self.ctx.make_mpf((c, c))
@property
def d(self):
(a, b), (c,d) = self._mpci_
return self.ctx.make_mpf((d, d))
@property
def real(s):
return s.ctx.make_mpf(s._mpci_[0])
@property
def imag(s):
return s.ctx.make_mpf(s._mpci_[1])
def conjugate(s):
a, b = s._mpci_
return s.ctx.make_mpc((a, mpf_neg(b)))
def overlap(s, t):
t = s.ctx.convert(t)
real_overlap = (s.a <= t.a <= s.b) or (s.a <= t.b <= s.b) or (t.a <= s.a <= t.b) or (t.a <= s.b <= t.b)
imag_overlap = (s.c <= t.c <= s.d) or (s.c <= t.d <= s.d) or (t.c <= s.c <= t.d) or (t.c <= s.d <= t.d)
return real_overlap and imag_overlap
def __contains__(s, t):
t = s.ctx.convert(t)
return t.real in s.real and t.imag in s.imag
def _compare(s, t, ne=False):
if not isinstance(t, s.ctx._types):
try:
t = s.ctx.convert(t)
except:
return NotImplemented
if hasattr(t, '_mpi_'):
tval = t._mpi_, mpi_zero
elif hasattr(t, '_mpci_'):
tval = t._mpci_
if ne:
return s._mpci_ != tval
return s._mpci_ == tval
def __eq__(s, t): return s._compare(t)
def __ne__(s, t): return s._compare(t, True)
def __lt__(s, t): raise TypeError("complex intervals cannot be ordered")
__le__ = __gt__ = __ge__ = __lt__
def __neg__(s): return s.ctx.make_mpc(mpci_neg(s._mpci_, s.ctx.prec))
def __pos__(s): return s.ctx.make_mpc(mpci_pos(s._mpci_, s.ctx.prec))
def __abs__(s): return s.ctx.make_mpf(mpci_abs(s._mpci_, s.ctx.prec))
def ae(s, t, rel_eps=None, abs_eps=None):
return s.ctx.almosteq(s, t, rel_eps, abs_eps)
def _binary_op(f_real, f_complex):
def g_complex(ctx, sval, tval):
return ctx.make_mpc(f_complex(sval, tval, ctx.prec))
def g_real(ctx, sval, tval):
try:
return ctx.make_mpf(f_real(sval, tval, ctx.prec))
except ComplexResult:
sval = (sval, mpi_zero)
tval = (tval, mpi_zero)
return g_complex(ctx, sval, tval)
def lop_real(s, t):
ctx = s.ctx
if not isinstance(t, ctx._types): t = ctx.convert(t)
if hasattr(t, "_mpi_"): return g_real(ctx, s._mpi_, t._mpi_)
if hasattr(t, "_mpci_"): return g_complex(ctx, (s._mpi_, mpi_zero), t._mpci_)
return NotImplemented
def rop_real(s, t):
ctx = s.ctx
if not isinstance(t, ctx._types): t = ctx.convert(t)
if hasattr(t, "_mpi_"): return g_real(ctx, t._mpi_, s._mpi_)
if hasattr(t, "_mpci_"): return g_complex(ctx, t._mpci_, (s._mpi_, mpi_zero))
return NotImplemented
def lop_complex(s, t):
ctx = s.ctx
if not isinstance(t, s.ctx._types):
try:
t = s.ctx.convert(t)
except (ValueError, TypeError):
return NotImplemented
return g_complex(ctx, s._mpci_, t._mpci_)
def rop_complex(s, t):
ctx = s.ctx
if not isinstance(t, s.ctx._types):
t = s.ctx.convert(t)
return g_complex(ctx, t._mpci_, s._mpci_)
return lop_real, rop_real, lop_complex, rop_complex
ivmpf.__add__, ivmpf.__radd__, ivmpc.__add__, ivmpc.__radd__ = _binary_op(mpi_add, mpci_add)
ivmpf.__sub__, ivmpf.__rsub__, ivmpc.__sub__, ivmpc.__rsub__ = _binary_op(mpi_sub, mpci_sub)
ivmpf.__mul__, ivmpf.__rmul__, ivmpc.__mul__, ivmpc.__rmul__ = _binary_op(mpi_mul, mpci_mul)
ivmpf.__div__, ivmpf.__rdiv__, ivmpc.__div__, ivmpc.__rdiv__ = _binary_op(mpi_div, mpci_div)
ivmpf.__pow__, ivmpf.__rpow__, ivmpc.__pow__, ivmpc.__rpow__ = _binary_op(mpi_pow, mpci_pow)
ivmpf.__truediv__ = ivmpf.__div__; ivmpf.__rtruediv__ = ivmpf.__rdiv__
ivmpc.__truediv__ = ivmpc.__div__; ivmpc.__rtruediv__ = ivmpc.__rdiv__
class ivmpf_constant(ivmpf):
def __new__(cls, f):
self = new(cls)
self._f = f
return self
def _get_mpi_(self):
prec = self.ctx._prec[0]
a = self._f(prec, round_floor)
b = self._f(prec, round_ceiling)
return a, b
_mpi_ = property(_get_mpi_)
class MPIntervalContext(StandardBaseContext):
def __init__(ctx):
ctx.mpf = type('ivmpf', (ivmpf,), {})
ctx.mpc = type('ivmpc', (ivmpc,), {})
ctx._types = (ctx.mpf, ctx.mpc)
ctx._constant = type('ivmpf_constant', (ivmpf_constant,), {})
ctx._prec = [53]
ctx._set_prec(53)
ctx._constant._ctxdata = ctx.mpf._ctxdata = ctx.mpc._ctxdata = [ctx.mpf, new, ctx._prec]
ctx._constant.ctx = ctx.mpf.ctx = ctx.mpc.ctx = ctx
ctx.pretty = False
StandardBaseContext.__init__(ctx)
ctx._init_builtins()
def _mpi(ctx, a, b=None):
if b is None:
return ctx.mpf(a)
return ctx.mpf((a,b))
def _init_builtins(ctx):
ctx.one = ctx.mpf(1)
ctx.zero = ctx.mpf(0)
ctx.inf = ctx.mpf('inf')
ctx.ninf = -ctx.inf
ctx.nan = ctx.mpf('nan')
ctx.j = ctx.mpc(0,1)
ctx.exp = ctx._wrap_mpi_function(libmp.mpi_exp, libmp.mpci_exp)
ctx.sqrt = ctx._wrap_mpi_function(libmp.mpi_sqrt)
ctx.ln = ctx._wrap_mpi_function(libmp.mpi_log, libmp.mpci_log)
ctx.cos = ctx._wrap_mpi_function(libmp.mpi_cos, libmp.mpci_cos)
ctx.sin = ctx._wrap_mpi_function(libmp.mpi_sin, libmp.mpci_sin)
ctx.tan = ctx._wrap_mpi_function(libmp.mpi_tan)
ctx.gamma = ctx._wrap_mpi_function(libmp.mpi_gamma, libmp.mpci_gamma)
ctx.loggamma = ctx._wrap_mpi_function(libmp.mpi_loggamma, libmp.mpci_loggamma)
ctx.rgamma = ctx._wrap_mpi_function(libmp.mpi_rgamma, libmp.mpci_rgamma)
ctx.factorial = ctx._wrap_mpi_function(libmp.mpi_factorial, libmp.mpci_factorial)
ctx.fac = ctx.factorial
ctx.eps = ctx._constant(lambda prec, rnd: (0, MPZ_ONE, 1-prec, 1))
ctx.pi = ctx._constant(libmp.mpf_pi)
ctx.e = ctx._constant(libmp.mpf_e)
ctx.ln2 = ctx._constant(libmp.mpf_ln2)
ctx.ln10 = ctx._constant(libmp.mpf_ln10)
ctx.phi = ctx._constant(libmp.mpf_phi)
ctx.euler = ctx._constant(libmp.mpf_euler)
ctx.catalan = ctx._constant(libmp.mpf_catalan)
ctx.glaisher = ctx._constant(libmp.mpf_glaisher)
ctx.khinchin = ctx._constant(libmp.mpf_khinchin)
ctx.twinprime = ctx._constant(libmp.mpf_twinprime)
def _wrap_mpi_function(ctx, f_real, f_complex=None):
def g(x, **kwargs):
if kwargs:
prec = kwargs.get('prec', ctx._prec[0])
else:
prec = ctx._prec[0]
x = ctx.convert(x)
if hasattr(x, "_mpi_"):
return ctx.make_mpf(f_real(x._mpi_, prec))
if hasattr(x, "_mpci_"):
return ctx.make_mpc(f_complex(x._mpci_, prec))
raise ValueError
return g
@classmethod
def _wrap_specfun(cls, name, f, wrap):
if wrap:
def f_wrapped(ctx, *args, **kwargs):
convert = ctx.convert
args = [convert(a) for a in args]
prec = ctx.prec
try:
ctx.prec += 10
retval = f(ctx, *args, **kwargs)
finally:
ctx.prec = prec
return +retval
else:
f_wrapped = f
setattr(cls, name, f_wrapped)
def _set_prec(ctx, n):
ctx._prec[0] = max(1, int(n))
ctx._dps = prec_to_dps(n)
def _set_dps(ctx, n):
ctx._prec[0] = dps_to_prec(n)
ctx._dps = max(1, int(n))
prec = property(lambda ctx: ctx._prec[0], _set_prec)
dps = property(lambda ctx: ctx._dps, _set_dps)
def make_mpf(ctx, v):
a = new(ctx.mpf)
a._mpi_ = v
return a
def make_mpc(ctx, v):
a = new(ctx.mpc)
a._mpci_ = v
return a
def _mpq(ctx, pq):
p, q = pq
a = libmp.from_rational(p, q, ctx.prec, round_floor)
b = libmp.from_rational(p, q, ctx.prec, round_ceiling)
return ctx.make_mpf((a, b))
def convert(ctx, x):
if isinstance(x, (ctx.mpf, ctx.mpc)):
return x
if isinstance(x, ctx._constant):
return +x
if isinstance(x, complex) or hasattr(x, "_mpc_"):
re = ctx.convert(x.real)
im = ctx.convert(x.imag)
return ctx.mpc(re,im)
if isinstance(x, basestring):
v = mpi_from_str(x, ctx.prec)
return ctx.make_mpf(v)
if hasattr(x, "_mpi_"):
a, b = x._mpi_
else:
try:
a, b = x
except (TypeError, ValueError):
a = b = x
if hasattr(a, "_mpi_"):
a = a._mpi_[0]
else:
a = convert_mpf_(a, ctx.prec, round_floor)
if hasattr(b, "_mpi_"):
b = b._mpi_[1]
else:
b = convert_mpf_(b, ctx.prec, round_ceiling)
if a == fnan or b == fnan:
a = fninf
b = finf
assert mpf_le(a, b), "endpoints must be properly ordered"
return ctx.make_mpf((a, b))
def nstr(ctx, x, n=5, **kwargs):
x = ctx.convert(x)
if hasattr(x, "_mpi_"):
return libmp.mpi_to_str(x._mpi_, n, **kwargs)
if hasattr(x, "_mpci_"):
re = libmp.mpi_to_str(x._mpci_[0], n, **kwargs)
im = libmp.mpi_to_str(x._mpci_[1], n, **kwargs)
return "(%s + %s*j)" % (re, im)
def mag(ctx, x):
x = ctx.convert(x)
if isinstance(x, ctx.mpc):
return max(ctx.mag(x.real), ctx.mag(x.imag)) + 1
a, b = libmp.mpi_abs(x._mpi_)
sign, man, exp, bc = b
if man:
return exp+bc
if b == fzero:
return ctx.ninf
if b == fnan:
return ctx.nan
return ctx.inf
def isnan(ctx, x):
return False
def isinf(ctx, x):
return x == ctx.inf
def isint(ctx, x):
x = ctx.convert(x)
a, b = x._mpi_
if a == b:
sign, man, exp, bc = a
if man:
return exp >= 0
return a == fzero
return None
def ldexp(ctx, x, n):
a, b = ctx.convert(x)._mpi_
a = libmp.mpf_shift(a, n)
b = libmp.mpf_shift(b, n)
return ctx.make_mpf((a,b))
def absmin(ctx, x):
return abs(ctx.convert(x)).a
def absmax(ctx, x):
return abs(ctx.convert(x)).b
def atan2(ctx, y, x):
y = ctx.convert(y)._mpi_
x = ctx.convert(x)._mpi_
return ctx.make_mpf(libmp.mpi_atan2(y,x,ctx.prec))
def _convert_param(ctx, x):
if isinstance(x, libmp.int_types):
return x, 'Z'
if isinstance(x, tuple):
p, q = x
return (ctx.mpf(p) / ctx.mpf(q), 'R')
x = ctx.convert(x)
if isinstance(x, ctx.mpf):
return x, 'R'
if isinstance(x, ctx.mpc):
return x, 'C'
raise ValueError
def _is_real_type(ctx, z):
return isinstance(z, ctx.mpf) or isinstance(z, int_types)
def _is_complex_type(ctx, z):
return isinstance(z, ctx.mpc)
def hypsum(ctx, p, q, types, coeffs, z, maxterms=6000, **kwargs):
coeffs = list(coeffs)
num = range(p)
den = range(p,p+q)
#tol = ctx.eps
s = t = ctx.one
k = 0
while 1:
for i in num: t *= (coeffs[i]+k)
for i in den: t /= (coeffs[i]+k)
k += 1; t /= k; t *= z; s += t
if t == 0:
return s
#if abs(t) < tol:
# return s
if k > maxterms:
raise ctx.NoConvergence
# Register with "numbers" ABC
# We do not subclass, hence we do not use the @abstractmethod checks. While
# this is less invasive it may turn out that we do not actually support
# parts of the expected interfaces. See
# http://docs.python.org/2/library/numbers.html for list of abstract
# methods.
try:
import numbers
numbers.Complex.register(ivmpc)
numbers.Real.register(ivmpf)
except ImportError:
pass
| 16,798 | 30.166976 | 111 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/__init__.py
|
__version__ = '1.0.0'
from .usertools import monitor, timing
from .ctx_fp import FPContext
from .ctx_mp import MPContext
from .ctx_iv import MPIntervalContext
fp = FPContext()
mp = MPContext()
iv = MPIntervalContext()
fp._mp = mp
mp._mp = mp
iv._mp = mp
mp._fp = fp
fp._fp = fp
mp._iv = iv
fp._iv = iv
iv._iv = iv
# XXX: extremely bad pickle hack
from . import ctx_mp as _ctx_mp
_ctx_mp._mpf_module.mpf = mp.mpf
_ctx_mp._mpf_module.mpc = mp.mpc
make_mpf = mp.make_mpf
make_mpc = mp.make_mpc
extraprec = mp.extraprec
extradps = mp.extradps
workprec = mp.workprec
workdps = mp.workdps
autoprec = mp.autoprec
maxcalls = mp.maxcalls
memoize = mp.memoize
mag = mp.mag
bernfrac = mp.bernfrac
qfrom = mp.qfrom
mfrom = mp.mfrom
kfrom = mp.kfrom
taufrom = mp.taufrom
qbarfrom = mp.qbarfrom
ellipfun = mp.ellipfun
jtheta = mp.jtheta
kleinj = mp.kleinj
qp = mp.qp
qhyper = mp.qhyper
qgamma = mp.qgamma
qfac = mp.qfac
nint_distance = mp.nint_distance
plot = mp.plot
cplot = mp.cplot
splot = mp.splot
odefun = mp.odefun
jacobian = mp.jacobian
findroot = mp.findroot
multiplicity = mp.multiplicity
isinf = mp.isinf
isnan = mp.isnan
isnormal = mp.isnormal
isint = mp.isint
isfinite = mp.isfinite
almosteq = mp.almosteq
nan = mp.nan
rand = mp.rand
absmin = mp.absmin
absmax = mp.absmax
fraction = mp.fraction
linspace = mp.linspace
arange = mp.arange
mpmathify = convert = mp.convert
mpc = mp.mpc
mpi = iv._mpi
nstr = mp.nstr
nprint = mp.nprint
chop = mp.chop
fneg = mp.fneg
fadd = mp.fadd
fsub = mp.fsub
fmul = mp.fmul
fdiv = mp.fdiv
fprod = mp.fprod
quad = mp.quad
quadgl = mp.quadgl
quadts = mp.quadts
quadosc = mp.quadosc
invertlaplace = mp.invertlaplace
invlaptalbot = mp.invlaptalbot
invlapstehfest = mp.invlapstehfest
invlapdehoog = mp.invlapdehoog
pslq = mp.pslq
identify = mp.identify
findpoly = mp.findpoly
richardson = mp.richardson
shanks = mp.shanks
levin = mp.levin
cohen_alt = mp.cohen_alt
nsum = mp.nsum
nprod = mp.nprod
difference = mp.difference
diff = mp.diff
diffs = mp.diffs
diffs_prod = mp.diffs_prod
diffs_exp = mp.diffs_exp
diffun = mp.diffun
differint = mp.differint
taylor = mp.taylor
pade = mp.pade
polyval = mp.polyval
polyroots = mp.polyroots
fourier = mp.fourier
fourierval = mp.fourierval
sumem = mp.sumem
sumap = mp.sumap
chebyfit = mp.chebyfit
limit = mp.limit
matrix = mp.matrix
eye = mp.eye
diag = mp.diag
zeros = mp.zeros
ones = mp.ones
hilbert = mp.hilbert
randmatrix = mp.randmatrix
swap_row = mp.swap_row
extend = mp.extend
norm = mp.norm
mnorm = mp.mnorm
lu_solve = mp.lu_solve
lu = mp.lu
qr = mp.qr
unitvector = mp.unitvector
inverse = mp.inverse
residual = mp.residual
qr_solve = mp.qr_solve
cholesky = mp.cholesky
cholesky_solve = mp.cholesky_solve
det = mp.det
cond = mp.cond
hessenberg = mp.hessenberg
schur = mp.schur
eig = mp.eig
eig_sort = mp.eig_sort
eigsy = mp.eigsy
eighe = mp.eighe
eigh = mp.eigh
svd_r = mp.svd_r
svd_c = mp.svd_c
svd = mp.svd
gauss_quadrature = mp.gauss_quadrature
expm = mp.expm
sqrtm = mp.sqrtm
powm = mp.powm
logm = mp.logm
sinm = mp.sinm
cosm = mp.cosm
mpf = mp.mpf
j = mp.j
exp = mp.exp
expj = mp.expj
expjpi = mp.expjpi
ln = mp.ln
im = mp.im
re = mp.re
inf = mp.inf
ninf = mp.ninf
sign = mp.sign
eps = mp.eps
pi = mp.pi
ln2 = mp.ln2
ln10 = mp.ln10
phi = mp.phi
e = mp.e
euler = mp.euler
catalan = mp.catalan
khinchin = mp.khinchin
glaisher = mp.glaisher
apery = mp.apery
degree = mp.degree
twinprime = mp.twinprime
mertens = mp.mertens
ldexp = mp.ldexp
frexp = mp.frexp
fsum = mp.fsum
fdot = mp.fdot
sqrt = mp.sqrt
cbrt = mp.cbrt
exp = mp.exp
ln = mp.ln
log = mp.log
log10 = mp.log10
power = mp.power
cos = mp.cos
sin = mp.sin
tan = mp.tan
cosh = mp.cosh
sinh = mp.sinh
tanh = mp.tanh
acos = mp.acos
asin = mp.asin
atan = mp.atan
asinh = mp.asinh
acosh = mp.acosh
atanh = mp.atanh
sec = mp.sec
csc = mp.csc
cot = mp.cot
sech = mp.sech
csch = mp.csch
coth = mp.coth
asec = mp.asec
acsc = mp.acsc
acot = mp.acot
asech = mp.asech
acsch = mp.acsch
acoth = mp.acoth
cospi = mp.cospi
sinpi = mp.sinpi
sinc = mp.sinc
sincpi = mp.sincpi
cos_sin = mp.cos_sin
cospi_sinpi = mp.cospi_sinpi
fabs = mp.fabs
re = mp.re
im = mp.im
conj = mp.conj
floor = mp.floor
ceil = mp.ceil
nint = mp.nint
frac = mp.frac
root = mp.root
nthroot = mp.nthroot
hypot = mp.hypot
fmod = mp.fmod
ldexp = mp.ldexp
frexp = mp.frexp
sign = mp.sign
arg = mp.arg
phase = mp.phase
polar = mp.polar
rect = mp.rect
degrees = mp.degrees
radians = mp.radians
atan2 = mp.atan2
fib = mp.fib
fibonacci = mp.fibonacci
lambertw = mp.lambertw
zeta = mp.zeta
altzeta = mp.altzeta
gamma = mp.gamma
rgamma = mp.rgamma
factorial = mp.factorial
fac = mp.fac
fac2 = mp.fac2
beta = mp.beta
betainc = mp.betainc
psi = mp.psi
#psi0 = mp.psi0
#psi1 = mp.psi1
#psi2 = mp.psi2
#psi3 = mp.psi3
polygamma = mp.polygamma
digamma = mp.digamma
#trigamma = mp.trigamma
#tetragamma = mp.tetragamma
#pentagamma = mp.pentagamma
harmonic = mp.harmonic
bernoulli = mp.bernoulli
bernfrac = mp.bernfrac
stieltjes = mp.stieltjes
hurwitz = mp.hurwitz
dirichlet = mp.dirichlet
bernpoly = mp.bernpoly
eulerpoly = mp.eulerpoly
eulernum = mp.eulernum
polylog = mp.polylog
clsin = mp.clsin
clcos = mp.clcos
gammainc = mp.gammainc
gammaprod = mp.gammaprod
binomial = mp.binomial
rf = mp.rf
ff = mp.ff
hyper = mp.hyper
hyp0f1 = mp.hyp0f1
hyp1f1 = mp.hyp1f1
hyp1f2 = mp.hyp1f2
hyp2f1 = mp.hyp2f1
hyp2f2 = mp.hyp2f2
hyp2f0 = mp.hyp2f0
hyp2f3 = mp.hyp2f3
hyp3f2 = mp.hyp3f2
hyperu = mp.hyperu
hypercomb = mp.hypercomb
meijerg = mp.meijerg
appellf1 = mp.appellf1
appellf2 = mp.appellf2
appellf3 = mp.appellf3
appellf4 = mp.appellf4
hyper2d = mp.hyper2d
bihyper = mp.bihyper
erf = mp.erf
erfc = mp.erfc
erfi = mp.erfi
erfinv = mp.erfinv
npdf = mp.npdf
ncdf = mp.ncdf
expint = mp.expint
e1 = mp.e1
ei = mp.ei
li = mp.li
ci = mp.ci
si = mp.si
chi = mp.chi
shi = mp.shi
fresnels = mp.fresnels
fresnelc = mp.fresnelc
airyai = mp.airyai
airybi = mp.airybi
airyaizero = mp.airyaizero
airybizero = mp.airybizero
scorergi = mp.scorergi
scorerhi = mp.scorerhi
ellipk = mp.ellipk
ellipe = mp.ellipe
ellipf = mp.ellipf
ellippi = mp.ellippi
elliprc = mp.elliprc
elliprj = mp.elliprj
elliprf = mp.elliprf
elliprd = mp.elliprd
elliprg = mp.elliprg
agm = mp.agm
jacobi = mp.jacobi
chebyt = mp.chebyt
chebyu = mp.chebyu
legendre = mp.legendre
legenp = mp.legenp
legenq = mp.legenq
hermite = mp.hermite
pcfd = mp.pcfd
pcfu = mp.pcfu
pcfv = mp.pcfv
pcfw = mp.pcfw
gegenbauer = mp.gegenbauer
laguerre = mp.laguerre
spherharm = mp.spherharm
besselj = mp.besselj
j0 = mp.j0
j1 = mp.j1
besseli = mp.besseli
bessely = mp.bessely
besselk = mp.besselk
besseljzero = mp.besseljzero
besselyzero = mp.besselyzero
hankel1 = mp.hankel1
hankel2 = mp.hankel2
struveh = mp.struveh
struvel = mp.struvel
angerj = mp.angerj
webere = mp.webere
lommels1 = mp.lommels1
lommels2 = mp.lommels2
whitm = mp.whitm
whitw = mp.whitw
ber = mp.ber
bei = mp.bei
ker = mp.ker
kei = mp.kei
coulombc = mp.coulombc
coulombf = mp.coulombf
coulombg = mp.coulombg
barnesg = mp.barnesg
superfac = mp.superfac
hyperfac = mp.hyperfac
loggamma = mp.loggamma
siegeltheta = mp.siegeltheta
siegelz = mp.siegelz
grampoint = mp.grampoint
zetazero = mp.zetazero
riemannr = mp.riemannr
primepi = mp.primepi
primepi2 = mp.primepi2
primezeta = mp.primezeta
bell = mp.bell
polyexp = mp.polyexp
expm1 = mp.expm1
powm1 = mp.powm1
unitroots = mp.unitroots
cyclotomic = mp.cyclotomic
mangoldt = mp.mangoldt
secondzeta = mp.secondzeta
nzeros = mp.nzeros
backlunds = mp.backlunds
lerchphi = mp.lerchphi
stirling1 = mp.stirling1
stirling2 = mp.stirling2
# be careful when changing this name, don't use test*!
def runtests():
"""
Run all mpmath tests and print output.
"""
import os.path
from inspect import getsourcefile
from .tests import runtests as tests
testdir = os.path.dirname(os.path.abspath(getsourcefile(tests)))
importdir = os.path.abspath(testdir + '/../..')
tests.testit(importdir, testdir)
def doctests(filter=[]):
try:
import psyco; psyco.full()
except ImportError:
pass
import sys
from timeit import default_timer as clock
for i, arg in enumerate(sys.argv):
if '__init__.py' in arg:
filter = [sn for sn in sys.argv[i+1:] if not sn.startswith("-")]
break
import doctest
globs = globals().copy()
for obj in globs: #sorted(globs.keys()):
if filter:
if not sum([pat in obj for pat in filter]):
continue
sys.stdout.write(str(obj) + " ")
sys.stdout.flush()
t1 = clock()
doctest.run_docstring_examples(globs[obj], {}, verbose=("-v" in sys.argv))
t2 = clock()
print(round(t2-t1, 3))
if __name__ == '__main__':
doctests()
| 8,664 | 17.634409 | 82 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/functions/factorials.py
|
from ..libmp.backend import xrange
from .functions import defun, defun_wrapped
@defun
def gammaprod(ctx, a, b, _infsign=False):
a = [ctx.convert(x) for x in a]
b = [ctx.convert(x) for x in b]
poles_num = []
poles_den = []
regular_num = []
regular_den = []
for x in a: [regular_num, poles_num][ctx.isnpint(x)].append(x)
for x in b: [regular_den, poles_den][ctx.isnpint(x)].append(x)
# One more pole in numerator or denominator gives 0 or inf
if len(poles_num) < len(poles_den): return ctx.zero
if len(poles_num) > len(poles_den):
# Get correct sign of infinity for x+h, h -> 0 from above
# XXX: hack, this should be done properly
if _infsign:
a = [x and x*(1+ctx.eps) or x+ctx.eps for x in poles_num]
b = [x and x*(1+ctx.eps) or x+ctx.eps for x in poles_den]
return ctx.sign(ctx.gammaprod(a+regular_num,b+regular_den)) * ctx.inf
else:
return ctx.inf
# All poles cancel
# lim G(i)/G(j) = (-1)**(i+j) * gamma(1-j) / gamma(1-i)
p = ctx.one
orig = ctx.prec
try:
ctx.prec = orig + 15
while poles_num:
i = poles_num.pop()
j = poles_den.pop()
p *= (-1)**(i+j) * ctx.gamma(1-j) / ctx.gamma(1-i)
for x in regular_num: p *= ctx.gamma(x)
for x in regular_den: p /= ctx.gamma(x)
finally:
ctx.prec = orig
return +p
@defun
def beta(ctx, x, y):
x = ctx.convert(x)
y = ctx.convert(y)
if ctx.isinf(y):
x, y = y, x
if ctx.isinf(x):
if x == ctx.inf and not ctx._im(y):
if y == ctx.ninf:
return ctx.nan
if y > 0:
return ctx.zero
if ctx.isint(y):
return ctx.nan
if y < 0:
return ctx.sign(ctx.gamma(y)) * ctx.inf
return ctx.nan
return ctx.gammaprod([x, y], [x+y])
@defun
def binomial(ctx, n, k):
return ctx.gammaprod([n+1], [k+1, n-k+1])
@defun
def rf(ctx, x, n):
return ctx.gammaprod([x+n], [x])
@defun
def ff(ctx, x, n):
return ctx.gammaprod([x+1], [x-n+1])
@defun_wrapped
def fac2(ctx, x):
if ctx.isinf(x):
if x == ctx.inf:
return x
return ctx.nan
return 2**(x/2)*(ctx.pi/2)**((ctx.cospi(x)-1)/4)*ctx.gamma(x/2+1)
@defun_wrapped
def barnesg(ctx, z):
if ctx.isinf(z):
if z == ctx.inf:
return z
return ctx.nan
if ctx.isnan(z):
return z
if (not ctx._im(z)) and ctx._re(z) <= 0 and ctx.isint(ctx._re(z)):
return z*0
# Account for size (would not be needed if computing log(G))
if abs(z) > 5:
ctx.dps += 2*ctx.log(abs(z),2)
# Reflection formula
if ctx.re(z) < -ctx.dps:
w = 1-z
pi2 = 2*ctx.pi
u = ctx.expjpi(2*w)
v = ctx.j*ctx.pi/12 - ctx.j*ctx.pi*w**2/2 + w*ctx.ln(1-u) - \
ctx.j*ctx.polylog(2, u)/pi2
v = ctx.barnesg(2-z)*ctx.exp(v)/pi2**w
if ctx._is_real_type(z):
v = ctx._re(v)
return v
# Estimate terms for asymptotic expansion
# TODO: fixme, obviously
N = ctx.dps // 2 + 5
G = 1
while abs(z) < N or ctx.re(z) < 1:
G /= ctx.gamma(z)
z += 1
z -= 1
s = ctx.mpf(1)/12
s -= ctx.log(ctx.glaisher)
s += z*ctx.log(2*ctx.pi)/2
s += (z**2/2-ctx.mpf(1)/12)*ctx.log(z)
s -= 3*z**2/4
z2k = z2 = z**2
for k in xrange(1, N+1):
t = ctx.bernoulli(2*k+2) / (4*k*(k+1)*z2k)
if abs(t) < ctx.eps:
#print k, N # check how many terms were needed
break
z2k *= z2
s += t
#if k == N:
# print "warning: series for barnesg failed to converge", ctx.dps
return G*ctx.exp(s)
@defun
def superfac(ctx, z):
return ctx.barnesg(z+2)
@defun_wrapped
def hyperfac(ctx, z):
# XXX: estimate needed extra bits accurately
if z == ctx.inf:
return z
if abs(z) > 5:
extra = 4*int(ctx.log(abs(z),2))
else:
extra = 0
ctx.prec += extra
if not ctx._im(z) and ctx._re(z) < 0 and ctx.isint(ctx._re(z)):
n = int(ctx.re(z))
h = ctx.hyperfac(-n-1)
if ((n+1)//2) & 1:
h = -h
if ctx._is_complex_type(z):
return h + 0j
return h
zp1 = z+1
# Wrong branch cut
#v = ctx.gamma(zp1)**z
#ctx.prec -= extra
#return v / ctx.barnesg(zp1)
v = ctx.exp(z*ctx.loggamma(zp1))
ctx.prec -= extra
return v / ctx.barnesg(zp1)
@defun_wrapped
def loggamma_old(ctx, z):
a = ctx._re(z)
b = ctx._im(z)
if not b and a > 0:
return ctx.ln(ctx.gamma_old(z))
u = ctx.arg(z)
w = ctx.ln(ctx.gamma_old(z))
if b:
gi = -b - u/2 + a*u + b*ctx.ln(abs(z))
n = ctx.floor((gi-ctx._im(w))/(2*ctx.pi)+0.5) * (2*ctx.pi)
return w + n*ctx.j
elif a < 0:
n = int(ctx.floor(a))
w += (n-(n%2))*ctx.pi*ctx.j
return w
'''
@defun
def psi0(ctx, z):
"""Shortcut for psi(0,z) (the digamma function)"""
return ctx.psi(0, z)
@defun
def psi1(ctx, z):
"""Shortcut for psi(1,z) (the trigamma function)"""
return ctx.psi(1, z)
@defun
def psi2(ctx, z):
"""Shortcut for psi(2,z) (the tetragamma function)"""
return ctx.psi(2, z)
@defun
def psi3(ctx, z):
"""Shortcut for psi(3,z) (the pentagamma function)"""
return ctx.psi(3, z)
'''
| 5,404 | 26.29798 | 81 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/functions/hypergeometric.py
|
from ..libmp.backend import xrange
from .functions import defun, defun_wrapped
def _check_need_perturb(ctx, terms, prec, discard_known_zeros):
perturb = recompute = False
extraprec = 0
discard = []
for term_index, term in enumerate(terms):
w_s, c_s, alpha_s, beta_s, a_s, b_s, z = term
have_singular_nongamma_weight = False
# Avoid division by zero in leading factors (TODO:
# also check for near division by zero?)
for k, w in enumerate(w_s):
if not w:
if ctx.re(c_s[k]) <= 0 and c_s[k]:
perturb = recompute = True
have_singular_nongamma_weight = True
pole_count = [0, 0, 0]
# Check for gamma and series poles and near-poles
for data_index, data in enumerate([alpha_s, beta_s, b_s]):
for i, x in enumerate(data):
n, d = ctx.nint_distance(x)
# Poles
if n > 0:
continue
if d == ctx.ninf:
# OK if we have a polynomial
# ------------------------------
ok = False
if data_index == 2:
for u in a_s:
if ctx.isnpint(u) and u >= int(n):
ok = True
break
if ok:
continue
pole_count[data_index] += 1
# ------------------------------
#perturb = recompute = True
#return perturb, recompute, extraprec
elif d < -4:
extraprec += -d
recompute = True
if discard_known_zeros and pole_count[1] > pole_count[0] + pole_count[2] \
and not have_singular_nongamma_weight:
discard.append(term_index)
elif sum(pole_count):
perturb = recompute = True
return perturb, recompute, extraprec, discard
_hypercomb_msg = """
hypercomb() failed to converge to the requested %i bits of accuracy
using a working precision of %i bits. The function value may be zero or
infinite; try passing zeroprec=N or infprec=M to bound finite values between
2^(-N) and 2^M. Otherwise try a higher maxprec or maxterms.
"""
@defun
def hypercomb(ctx, function, params=[], discard_known_zeros=True, **kwargs):
orig = ctx.prec
sumvalue = ctx.zero
dist = ctx.nint_distance
ninf = ctx.ninf
orig_params = params[:]
verbose = kwargs.get('verbose', False)
maxprec = kwargs.get('maxprec', ctx._default_hyper_maxprec(orig))
kwargs['maxprec'] = maxprec # For calls to hypsum
zeroprec = kwargs.get('zeroprec')
infprec = kwargs.get('infprec')
perturbed_reference_value = None
hextra = 0
try:
while 1:
ctx.prec += 10
if ctx.prec > maxprec:
raise ValueError(_hypercomb_msg % (orig, ctx.prec))
orig2 = ctx.prec
params = orig_params[:]
terms = function(*params)
if verbose:
print()
print("ENTERING hypercomb main loop")
print("prec =", ctx.prec)
print("hextra", hextra)
perturb, recompute, extraprec, discard = \
_check_need_perturb(ctx, terms, orig, discard_known_zeros)
ctx.prec += extraprec
if perturb:
if "hmag" in kwargs:
hmag = kwargs["hmag"]
elif ctx._fixed_precision:
hmag = int(ctx.prec*0.3)
else:
hmag = orig + 10 + hextra
h = ctx.ldexp(ctx.one, -hmag)
ctx.prec = orig2 + 10 + hmag + 10
for k in range(len(params)):
params[k] += h
# Heuristically ensure that the perturbations
# are "independent" so that two perturbations
# don't accidentally cancel each other out
# in a subtraction.
h += h/(k+1)
if recompute:
terms = function(*params)
if discard_known_zeros:
terms = [term for (i, term) in enumerate(terms) if i not in discard]
if not terms:
return ctx.zero
evaluated_terms = []
for term_index, term_data in enumerate(terms):
w_s, c_s, alpha_s, beta_s, a_s, b_s, z = term_data
if verbose:
print()
print(" Evaluating term %i/%i : %iF%i" % \
(term_index+1, len(terms), len(a_s), len(b_s)))
print(" powers", ctx.nstr(w_s), ctx.nstr(c_s))
print(" gamma", ctx.nstr(alpha_s), ctx.nstr(beta_s))
print(" hyper", ctx.nstr(a_s), ctx.nstr(b_s))
print(" z", ctx.nstr(z))
#v = ctx.hyper(a_s, b_s, z, **kwargs)
#for a in alpha_s: v *= ctx.gamma(a)
#for b in beta_s: v *= ctx.rgamma(b)
#for w, c in zip(w_s, c_s): v *= ctx.power(w, c)
v = ctx.fprod([ctx.hyper(a_s, b_s, z, **kwargs)] + \
[ctx.gamma(a) for a in alpha_s] + \
[ctx.rgamma(b) for b in beta_s] + \
[ctx.power(w,c) for (w,c) in zip(w_s,c_s)])
if verbose:
print(" Value:", v)
evaluated_terms.append(v)
if len(terms) == 1 and (not perturb):
sumvalue = evaluated_terms[0]
break
if ctx._fixed_precision:
sumvalue = ctx.fsum(evaluated_terms)
break
sumvalue = ctx.fsum(evaluated_terms)
term_magnitudes = [ctx.mag(x) for x in evaluated_terms]
max_magnitude = max(term_magnitudes)
sum_magnitude = ctx.mag(sumvalue)
cancellation = max_magnitude - sum_magnitude
if verbose:
print()
print(" Cancellation:", cancellation, "bits")
print(" Increased precision:", ctx.prec - orig, "bits")
precision_ok = cancellation < ctx.prec - orig
if zeroprec is None:
zero_ok = False
else:
zero_ok = max_magnitude - ctx.prec < -zeroprec
if infprec is None:
inf_ok = False
else:
inf_ok = max_magnitude > infprec
if precision_ok and (not perturb) or ctx.isnan(cancellation):
break
elif precision_ok:
if perturbed_reference_value is None:
hextra += 20
perturbed_reference_value = sumvalue
continue
elif ctx.mag(sumvalue - perturbed_reference_value) <= \
ctx.mag(sumvalue) - orig:
break
elif zero_ok:
sumvalue = ctx.zero
break
elif inf_ok:
sumvalue = ctx.inf
break
elif 'hmag' in kwargs:
break
else:
hextra *= 2
perturbed_reference_value = sumvalue
# Increase precision
else:
increment = min(max(cancellation, orig//2), max(extraprec,orig))
ctx.prec += increment
if verbose:
print(" Must start over with increased precision")
continue
finally:
ctx.prec = orig
return +sumvalue
@defun
def hyper(ctx, a_s, b_s, z, **kwargs):
"""
Hypergeometric function, general case.
"""
z = ctx.convert(z)
p = len(a_s)
q = len(b_s)
a_s = [ctx._convert_param(a) for a in a_s]
b_s = [ctx._convert_param(b) for b in b_s]
# Reduce degree by eliminating common parameters
if kwargs.get('eliminate', True):
elim_nonpositive = kwargs.get('eliminate_all', False)
i = 0
while i < q and a_s:
b = b_s[i]
if b in a_s and (elim_nonpositive or not ctx.isnpint(b[0])):
a_s.remove(b)
b_s.remove(b)
p -= 1
q -= 1
else:
i += 1
# Handle special cases
if p == 0:
if q == 1: return ctx._hyp0f1(b_s, z, **kwargs)
elif q == 0: return ctx.exp(z)
elif p == 1:
if q == 1: return ctx._hyp1f1(a_s, b_s, z, **kwargs)
elif q == 2: return ctx._hyp1f2(a_s, b_s, z, **kwargs)
elif q == 0: return ctx._hyp1f0(a_s[0][0], z)
elif p == 2:
if q == 1: return ctx._hyp2f1(a_s, b_s, z, **kwargs)
elif q == 2: return ctx._hyp2f2(a_s, b_s, z, **kwargs)
elif q == 3: return ctx._hyp2f3(a_s, b_s, z, **kwargs)
elif q == 0: return ctx._hyp2f0(a_s, b_s, z, **kwargs)
elif p == q+1:
return ctx._hypq1fq(p, q, a_s, b_s, z, **kwargs)
elif p > q+1 and not kwargs.get('force_series'):
return ctx._hyp_borel(p, q, a_s, b_s, z, **kwargs)
coeffs, types = zip(*(a_s+b_s))
return ctx.hypsum(p, q, types, coeffs, z, **kwargs)
@defun
def hyp0f1(ctx,b,z,**kwargs):
return ctx.hyper([],[b],z,**kwargs)
@defun
def hyp1f1(ctx,a,b,z,**kwargs):
return ctx.hyper([a],[b],z,**kwargs)
@defun
def hyp1f2(ctx,a1,b1,b2,z,**kwargs):
return ctx.hyper([a1],[b1,b2],z,**kwargs)
@defun
def hyp2f1(ctx,a,b,c,z,**kwargs):
return ctx.hyper([a,b],[c],z,**kwargs)
@defun
def hyp2f2(ctx,a1,a2,b1,b2,z,**kwargs):
return ctx.hyper([a1,a2],[b1,b2],z,**kwargs)
@defun
def hyp2f3(ctx,a1,a2,b1,b2,b3,z,**kwargs):
return ctx.hyper([a1,a2],[b1,b2,b3],z,**kwargs)
@defun
def hyp2f0(ctx,a,b,z,**kwargs):
return ctx.hyper([a,b],[],z,**kwargs)
@defun
def hyp3f2(ctx,a1,a2,a3,b1,b2,z,**kwargs):
return ctx.hyper([a1,a2,a3],[b1,b2],z,**kwargs)
@defun_wrapped
def _hyp1f0(ctx, a, z):
return (1-z) ** (-a)
@defun
def _hyp0f1(ctx, b_s, z, **kwargs):
(b, btype), = b_s
if z:
magz = ctx.mag(z)
else:
magz = 0
if magz >= 8 and not kwargs.get('force_series'):
try:
# http://functions.wolfram.com/HypergeometricFunctions/
# Hypergeometric0F1/06/02/03/0004/
# TODO: handle the all-real case more efficiently!
# TODO: figure out how much precision is needed (exponential growth)
orig = ctx.prec
try:
ctx.prec += 12 + magz//2
def h():
w = ctx.sqrt(-z)
jw = ctx.j*w
u = 1/(4*jw)
c = ctx.mpq_1_2 - b
E = ctx.exp(2*jw)
T1 = ([-jw,E], [c,-1], [], [], [b-ctx.mpq_1_2, ctx.mpq_3_2-b], [], -u)
T2 = ([jw,E], [c,1], [], [], [b-ctx.mpq_1_2, ctx.mpq_3_2-b], [], u)
return T1, T2
v = ctx.hypercomb(h, [], force_series=True)
v = ctx.gamma(b)/(2*ctx.sqrt(ctx.pi))*v
finally:
ctx.prec = orig
if ctx._is_real_type(b) and ctx._is_real_type(z):
v = ctx._re(v)
return +v
except ctx.NoConvergence:
pass
return ctx.hypsum(0, 1, (btype,), [b], z, **kwargs)
@defun
def _hyp1f1(ctx, a_s, b_s, z, **kwargs):
(a, atype), = a_s
(b, btype), = b_s
if not z:
return ctx.one+z
magz = ctx.mag(z)
if magz >= 7 and not (ctx.isint(a) and ctx.re(a) <= 0):
if ctx.isinf(z):
if ctx.sign(a) == ctx.sign(b) == ctx.sign(z) == 1:
return ctx.inf
return ctx.nan * z
try:
try:
ctx.prec += magz
sector = ctx._im(z) < 0
def h(a,b):
if sector:
E = ctx.expjpi(ctx.fneg(a, exact=True))
else:
E = ctx.expjpi(a)
rz = 1/z
T1 = ([E,z], [1,-a], [b], [b-a], [a, 1+a-b], [], -rz)
T2 = ([ctx.exp(z),z], [1,a-b], [b], [a], [b-a, 1-a], [], rz)
return T1, T2
v = ctx.hypercomb(h, [a,b], force_series=True)
if ctx._is_real_type(a) and ctx._is_real_type(b) and ctx._is_real_type(z):
v = ctx._re(v)
return +v
except ctx.NoConvergence:
pass
finally:
ctx.prec -= magz
v = ctx.hypsum(1, 1, (atype, btype), [a, b], z, **kwargs)
return v
def _hyp2f1_gosper(ctx,a,b,c,z,**kwargs):
# Use Gosper's recurrence
# See http://www.math.utexas.edu/pipermail/maxima/2006/000126.html
_a,_b,_c,_z = a, b, c, z
orig = ctx.prec
maxprec = kwargs.get('maxprec', 100*orig)
extra = 10
while 1:
ctx.prec = orig + extra
#a = ctx.convert(_a)
#b = ctx.convert(_b)
#c = ctx.convert(_c)
z = ctx.convert(_z)
d = ctx.mpf(0)
e = ctx.mpf(1)
f = ctx.mpf(0)
k = 0
# Common subexpression elimination, unfortunately making
# things a bit unreadable. The formula is quite messy to begin
# with, though...
abz = a*b*z
ch = c * ctx.mpq_1_2
c1h = (c+1) * ctx.mpq_1_2
nz = 1-z
g = z/nz
abg = a*b*g
cba = c-b-a
z2 = z-2
tol = -ctx.prec - 10
nstr = ctx.nstr
nprint = ctx.nprint
mag = ctx.mag
maxmag = ctx.ninf
while 1:
kch = k+ch
kakbz = (k+a)*(k+b)*z / (4*(k+1)*kch*(k+c1h))
d1 = kakbz*(e-(k+cba)*d*g)
e1 = kakbz*(d*abg+(k+c)*e)
ft = d*(k*(cba*z+k*z2-c)-abz)/(2*kch*nz)
f1 = f + e - ft
maxmag = max(maxmag, mag(f1))
if mag(f1-f) < tol:
break
d, e, f = d1, e1, f1
k += 1
cancellation = maxmag - mag(f1)
if cancellation < extra:
break
else:
extra += cancellation
if extra > maxprec:
raise ctx.NoConvergence
return f1
@defun
def _hyp2f1(ctx, a_s, b_s, z, **kwargs):
(a, atype), (b, btype) = a_s
(c, ctype), = b_s
if z == 1:
# TODO: the following logic can be simplified
convergent = ctx.re(c-a-b) > 0
finite = (ctx.isint(a) and a <= 0) or (ctx.isint(b) and b <= 0)
zerodiv = ctx.isint(c) and c <= 0 and not \
((ctx.isint(a) and c <= a <= 0) or (ctx.isint(b) and c <= b <= 0))
#print "bz", a, b, c, z, convergent, finite, zerodiv
# Gauss's theorem gives the value if convergent
if (convergent or finite) and not zerodiv:
return ctx.gammaprod([c, c-a-b], [c-a, c-b], _infsign=True)
# Otherwise, there is a pole and we take the
# sign to be that when approaching from below
# XXX: this evaluation is not necessarily correct in all cases
return ctx.hyp2f1(a,b,c,1-ctx.eps*2) * ctx.inf
# Equal to 1 (first term), unless there is a subsequent
# division by zero
if not z:
# Division by zero but power of z is higher than
# first order so cancels
if c or a == 0 or b == 0:
return 1+z
# Indeterminate
return ctx.nan
# Hit zero denominator unless numerator goes to 0 first
if ctx.isint(c) and c <= 0:
if (ctx.isint(a) and c <= a <= 0) or \
(ctx.isint(b) and c <= b <= 0):
pass
else:
# Pole in series
return ctx.inf
absz = abs(z)
# Fast case: standard series converges rapidly,
# possibly in finitely many terms
if absz <= 0.8 or (ctx.isint(a) and a <= 0 and a >= -1000) or \
(ctx.isint(b) and b <= 0 and b >= -1000):
return ctx.hypsum(2, 1, (atype, btype, ctype), [a, b, c], z, **kwargs)
orig = ctx.prec
try:
ctx.prec += 10
# Use 1/z transformation
if absz >= 1.3:
def h(a,b):
t = ctx.mpq_1-c; ab = a-b; rz = 1/z
T1 = ([-z],[-a], [c,-ab],[b,c-a], [a,t+a],[ctx.mpq_1+ab], rz)
T2 = ([-z],[-b], [c,ab],[a,c-b], [b,t+b],[ctx.mpq_1-ab], rz)
return T1, T2
v = ctx.hypercomb(h, [a,b], **kwargs)
# Use 1-z transformation
elif abs(1-z) <= 0.75:
def h(a,b):
t = c-a-b; ca = c-a; cb = c-b; rz = 1-z
T1 = [], [], [c,t], [ca,cb], [a,b], [1-t], rz
T2 = [rz], [t], [c,a+b-c], [a,b], [ca,cb], [1+t], rz
return T1, T2
v = ctx.hypercomb(h, [a,b], **kwargs)
# Use z/(z-1) transformation
elif abs(z/(z-1)) <= 0.75:
v = ctx.hyp2f1(a, c-b, c, z/(z-1)) / (1-z)**a
# Remaining part of unit circle
else:
v = _hyp2f1_gosper(ctx,a,b,c,z,**kwargs)
finally:
ctx.prec = orig
return +v
@defun
def _hypq1fq(ctx, p, q, a_s, b_s, z, **kwargs):
r"""
Evaluates 3F2, 4F3, 5F4, ...
"""
a_s, a_types = zip(*a_s)
b_s, b_types = zip(*b_s)
a_s = list(a_s)
b_s = list(b_s)
absz = abs(z)
ispoly = False
for a in a_s:
if ctx.isint(a) and a <= 0:
ispoly = True
break
# Direct summation
if absz < 1 or ispoly:
try:
return ctx.hypsum(p, q, a_types+b_types, a_s+b_s, z, **kwargs)
except ctx.NoConvergence:
if absz > 1.1 or ispoly:
raise
# Use expansion at |z-1| -> 0.
# Reference: Wolfgang Buhring, "Generalized Hypergeometric Functions at
# Unit Argument", Proc. Amer. Math. Soc., Vol. 114, No. 1 (Jan. 1992),
# pp.145-153
# The current implementation has several problems:
# 1. We only implement it for 3F2. The expansion coefficients are
# given by extremely messy nested sums in the higher degree cases
# (see reference). Is efficient sequential generation of the coefficients
# possible in the > 3F2 case?
# 2. Although the series converges, it may do so slowly, so we need
# convergence acceleration. The acceleration implemented by
# nsum does not always help, so results returned are sometimes
# inaccurate! Can we do better?
# 3. We should check conditions for convergence, and possibly
# do a better job of cancelling out gamma poles if possible.
if z == 1:
# XXX: should also check for division by zero in the
# denominator of the series (cf. hyp2f1)
S = ctx.re(sum(b_s)-sum(a_s))
if S <= 0:
#return ctx.hyper(a_s, b_s, 1-ctx.eps*2, **kwargs) * ctx.inf
return ctx.hyper(a_s, b_s, 0.9, **kwargs) * ctx.inf
if (p,q) == (3,2) and abs(z-1) < 0.05: # and kwargs.get('sum1')
#print "Using alternate summation (experimental)"
a1,a2,a3 = a_s
b1,b2 = b_s
u = b1+b2-a3
initial = ctx.gammaprod([b2-a3,b1-a3,a1,a2],[b2-a3,b1-a3,1,u])
def term(k, _cache={0:initial}):
u = b1+b2-a3+k
if k in _cache:
t = _cache[k]
else:
t = _cache[k-1]
t *= (b1+k-a3-1)*(b2+k-a3-1)
t /= k*(u-1)
_cache[k] = t
return t * ctx.hyp2f1(a1,a2,u,z)
try:
S = ctx.nsum(term, [0,ctx.inf], verbose=kwargs.get('verbose'),
strict=kwargs.get('strict', True))
return S * ctx.gammaprod([b1,b2],[a1,a2,a3])
except ctx.NoConvergence:
pass
# Try to use convergence acceleration on and close to the unit circle.
# Problem: the convergence acceleration degenerates as |z-1| -> 0,
# except for special cases. Everywhere else, the Shanks transformation
# is very efficient.
if absz < 1.1 and ctx._re(z) <= 1:
def term(kk, _cache={0:ctx.one}):
k = int(kk)
if k != kk:
t = z ** ctx.mpf(kk) / ctx.fac(kk)
for a in a_s: t *= ctx.rf(a,kk)
for b in b_s: t /= ctx.rf(b,kk)
return t
if k in _cache:
return _cache[k]
t = term(k-1)
m = k-1
for j in xrange(p): t *= (a_s[j]+m)
for j in xrange(q): t /= (b_s[j]+m)
t *= z
t /= k
_cache[k] = t
return t
sum_method = kwargs.get('sum_method', 'r+s+e')
try:
return ctx.nsum(term, [0,ctx.inf], verbose=kwargs.get('verbose'),
strict=kwargs.get('strict', True),
method=sum_method.replace('e',''))
except ctx.NoConvergence:
if 'e' not in sum_method:
raise
pass
if kwargs.get('verbose'):
print("Attempting Euler-Maclaurin summation")
"""
Somewhat slower version (one diffs_exp for each factor).
However, this would be faster with fast direct derivatives
of the gamma function.
def power_diffs(k0):
r = 0
l = ctx.log(z)
while 1:
yield z**ctx.mpf(k0) * l**r
r += 1
def loggamma_diffs(x, reciprocal=False):
sign = (-1) ** reciprocal
yield sign * ctx.loggamma(x)
i = 0
while 1:
yield sign * ctx.psi(i,x)
i += 1
def hyper_diffs(k0):
b2 = b_s + [1]
A = [ctx.diffs_exp(loggamma_diffs(a+k0)) for a in a_s]
B = [ctx.diffs_exp(loggamma_diffs(b+k0,True)) for b in b2]
Z = [power_diffs(k0)]
C = ctx.gammaprod([b for b in b2], [a for a in a_s])
for d in ctx.diffs_prod(A + B + Z):
v = C * d
yield v
"""
def log_diffs(k0):
b2 = b_s + [1]
yield sum(ctx.loggamma(a+k0) for a in a_s) - \
sum(ctx.loggamma(b+k0) for b in b2) + k0*ctx.log(z)
i = 0
while 1:
v = sum(ctx.psi(i,a+k0) for a in a_s) - \
sum(ctx.psi(i,b+k0) for b in b2)
if i == 0:
v += ctx.log(z)
yield v
i += 1
def hyper_diffs(k0):
C = ctx.gammaprod([b for b in b_s], [a for a in a_s])
for d in ctx.diffs_exp(log_diffs(k0)):
v = C * d
yield v
tol = ctx.eps / 1024
prec = ctx.prec
try:
trunc = 50 * ctx.dps
ctx.prec += 20
for i in xrange(5):
head = ctx.fsum(term(k) for k in xrange(trunc))
tail, err = ctx.sumem(term, [trunc, ctx.inf], tol=tol,
adiffs=hyper_diffs(trunc),
verbose=kwargs.get('verbose'),
error=True,
_fast_abort=True)
if err < tol:
v = head + tail
break
trunc *= 2
# Need to increase precision because calculation of
# derivatives may be inaccurate
ctx.prec += ctx.prec//2
if i == 4:
raise ctx.NoConvergence(\
"Euler-Maclaurin summation did not converge")
finally:
ctx.prec = prec
return +v
# Use 1/z transformation
# http://functions.wolfram.com/HypergeometricFunctions/
# HypergeometricPFQ/06/01/05/02/0004/
def h(*args):
a_s = list(args[:p])
b_s = list(args[p:])
Ts = []
recz = ctx.one/z
negz = ctx.fneg(z, exact=True)
for k in range(q+1):
ak = a_s[k]
C = [negz]
Cp = [-ak]
Gn = b_s + [ak] + [a_s[j]-ak for j in range(q+1) if j != k]
Gd = a_s + [b_s[j]-ak for j in range(q)]
Fn = [ak] + [ak-b_s[j]+1 for j in range(q)]
Fd = [1-a_s[j]+ak for j in range(q+1) if j != k]
Ts.append((C, Cp, Gn, Gd, Fn, Fd, recz))
return Ts
return ctx.hypercomb(h, a_s+b_s, **kwargs)
@defun
def _hyp_borel(ctx, p, q, a_s, b_s, z, **kwargs):
if a_s:
a_s, a_types = zip(*a_s)
a_s = list(a_s)
else:
a_s, a_types = [], ()
if b_s:
b_s, b_types = zip(*b_s)
b_s = list(b_s)
else:
b_s, b_types = [], ()
kwargs['maxterms'] = kwargs.get('maxterms', ctx.prec)
try:
return ctx.hypsum(p, q, a_types+b_types, a_s+b_s, z, **kwargs)
except ctx.NoConvergence:
pass
prec = ctx.prec
try:
tol = kwargs.get('asymp_tol', ctx.eps/4)
ctx.prec += 10
# hypsum is has a conservative tolerance. So we try again:
def term(k, cache={0:ctx.one}):
if k in cache:
return cache[k]
t = term(k-1)
for a in a_s: t *= (a+(k-1))
for b in b_s: t /= (b+(k-1))
t *= z
t /= k
cache[k] = t
return t
s = ctx.one
for k in xrange(1, ctx.prec):
t = term(k)
s += t
if abs(t) <= tol:
return s
finally:
ctx.prec = prec
if p <= q+3:
contour = kwargs.get('contour')
if not contour:
if ctx.arg(z) < 0.25:
u = z / max(1, abs(z))
if ctx.arg(z) >= 0:
contour = [0, 2j, (2j+2)/u, 2/u, ctx.inf]
else:
contour = [0, -2j, (-2j+2)/u, 2/u, ctx.inf]
#contour = [0, 2j/z, 2/z, ctx.inf]
#contour = [0, 2j, 2/z, ctx.inf]
#contour = [0, 2j, ctx.inf]
else:
contour = [0, ctx.inf]
quad_kwargs = kwargs.get('quad_kwargs', {})
def g(t):
return ctx.exp(-t)*ctx.hyper(a_s, b_s+[1], t*z)
I, err = ctx.quad(g, contour, error=True, **quad_kwargs)
if err <= abs(I)*ctx.eps*8:
return I
raise ctx.NoConvergence
@defun
def _hyp2f2(ctx, a_s, b_s, z, **kwargs):
(a1, a1type), (a2, a2type) = a_s
(b1, b1type), (b2, b2type) = b_s
absz = abs(z)
magz = ctx.mag(z)
orig = ctx.prec
# Asymptotic expansion is ~ exp(z)
asymp_extraprec = magz
# Asymptotic series is in terms of 3F1
can_use_asymptotic = (not kwargs.get('force_series')) and \
(ctx.mag(absz) > 3)
# TODO: much of the following could be shared with 2F3 instead of
# copypasted
if can_use_asymptotic:
#print "using asymp"
try:
try:
ctx.prec += asymp_extraprec
# http://functions.wolfram.com/HypergeometricFunctions/
# Hypergeometric2F2/06/02/02/0002/
def h(a1,a2,b1,b2):
X = a1+a2-b1-b2
A2 = a1+a2
B2 = b1+b2
c = {}
c[0] = ctx.one
c[1] = (A2-1)*X+b1*b2-a1*a2
s1 = 0
k = 0
tprev = 0
while 1:
if k not in c:
uu1 = 1-B2+2*a1+a1**2+2*a2+a2**2-A2*B2+a1*a2+b1*b2+(2*B2-3*(A2+1))*k+2*k**2
uu2 = (k-A2+b1-1)*(k-A2+b2-1)*(k-X-2)
c[k] = ctx.one/k * (uu1*c[k-1]-uu2*c[k-2])
t1 = c[k] * z**(-k)
if abs(t1) < 0.1*ctx.eps:
#print "Convergence :)"
break
# Quit if the series doesn't converge quickly enough
if k > 5 and abs(tprev) / abs(t1) < 1.5:
#print "No convergence :("
raise ctx.NoConvergence
s1 += t1
tprev = t1
k += 1
S = ctx.exp(z)*s1
T1 = [z,S], [X,1], [b1,b2],[a1,a2],[],[],0
T2 = [-z],[-a1],[b1,b2,a2-a1],[a2,b1-a1,b2-a1],[a1,a1-b1+1,a1-b2+1],[a1-a2+1],-1/z
T3 = [-z],[-a2],[b1,b2,a1-a2],[a1,b1-a2,b2-a2],[a2,a2-b1+1,a2-b2+1],[-a1+a2+1],-1/z
return T1, T2, T3
v = ctx.hypercomb(h, [a1,a2,b1,b2], force_series=True, maxterms=4*ctx.prec)
if sum(ctx._is_real_type(u) for u in [a1,a2,b1,b2,z]) == 5:
v = ctx.re(v)
return v
except ctx.NoConvergence:
pass
finally:
ctx.prec = orig
return ctx.hypsum(2, 2, (a1type, a2type, b1type, b2type), [a1, a2, b1, b2], z, **kwargs)
@defun
def _hyp1f2(ctx, a_s, b_s, z, **kwargs):
(a1, a1type), = a_s
(b1, b1type), (b2, b2type) = b_s
absz = abs(z)
magz = ctx.mag(z)
orig = ctx.prec
# Asymptotic expansion is ~ exp(sqrt(z))
asymp_extraprec = z and magz//2
# Asymptotic series is in terms of 3F0
can_use_asymptotic = (not kwargs.get('force_series')) and \
(ctx.mag(absz) > 19) and \
(ctx.sqrt(absz) > 1.5*orig) # and \
# ctx._hyp_check_convergence([a1, a1-b1+1, a1-b2+1], [],
# 1/absz, orig+40+asymp_extraprec)
# TODO: much of the following could be shared with 2F3 instead of
# copypasted
if can_use_asymptotic:
#print "using asymp"
try:
try:
ctx.prec += asymp_extraprec
# http://functions.wolfram.com/HypergeometricFunctions/
# Hypergeometric1F2/06/02/03/
def h(a1,b1,b2):
X = ctx.mpq_1_2*(a1-b1-b2+ctx.mpq_1_2)
c = {}
c[0] = ctx.one
c[1] = 2*(ctx.mpq_1_4*(3*a1+b1+b2-2)*(a1-b1-b2)+b1*b2-ctx.mpq_3_16)
c[2] = 2*(b1*b2+ctx.mpq_1_4*(a1-b1-b2)*(3*a1+b1+b2-2)-ctx.mpq_3_16)**2+\
ctx.mpq_1_16*(-16*(2*a1-3)*b1*b2 + \
4*(a1-b1-b2)*(-8*a1**2+11*a1+b1+b2-2)-3)
s1 = 0
s2 = 0
k = 0
tprev = 0
while 1:
if k not in c:
uu1 = (3*k**2+(-6*a1+2*b1+2*b2-4)*k + 3*a1**2 - \
(b1-b2)**2 - 2*a1*(b1+b2-2) + ctx.mpq_1_4)
uu2 = (k-a1+b1-b2-ctx.mpq_1_2)*(k-a1-b1+b2-ctx.mpq_1_2)*\
(k-a1+b1+b2-ctx.mpq_5_2)
c[k] = ctx.one/(2*k)*(uu1*c[k-1]-uu2*c[k-2])
w = c[k] * (-z)**(-0.5*k)
t1 = (-ctx.j)**k * ctx.mpf(2)**(-k) * w
t2 = ctx.j**k * ctx.mpf(2)**(-k) * w
if abs(t1) < 0.1*ctx.eps:
#print "Convergence :)"
break
# Quit if the series doesn't converge quickly enough
if k > 5 and abs(tprev) / abs(t1) < 1.5:
#print "No convergence :("
raise ctx.NoConvergence
s1 += t1
s2 += t2
tprev = t1
k += 1
S = ctx.expj(ctx.pi*X+2*ctx.sqrt(-z))*s1 + \
ctx.expj(-(ctx.pi*X+2*ctx.sqrt(-z)))*s2
T1 = [0.5*S, ctx.pi, -z], [1, -0.5, X], [b1, b2], [a1],\
[], [], 0
T2 = [-z], [-a1], [b1,b2],[b1-a1,b2-a1], \
[a1,a1-b1+1,a1-b2+1], [], 1/z
return T1, T2
v = ctx.hypercomb(h, [a1,b1,b2], force_series=True, maxterms=4*ctx.prec)
if sum(ctx._is_real_type(u) for u in [a1,b1,b2,z]) == 4:
v = ctx.re(v)
return v
except ctx.NoConvergence:
pass
finally:
ctx.prec = orig
#print "not using asymp"
return ctx.hypsum(1, 2, (a1type, b1type, b2type), [a1, b1, b2], z, **kwargs)
@defun
def _hyp2f3(ctx, a_s, b_s, z, **kwargs):
(a1, a1type), (a2, a2type) = a_s
(b1, b1type), (b2, b2type), (b3, b3type) = b_s
absz = abs(z)
magz = ctx.mag(z)
# Asymptotic expansion is ~ exp(sqrt(z))
asymp_extraprec = z and magz//2
orig = ctx.prec
# Asymptotic series is in terms of 4F1
# The square root below empirically provides a plausible criterion
# for the leading series to converge
can_use_asymptotic = (not kwargs.get('force_series')) and \
(ctx.mag(absz) > 19) and (ctx.sqrt(absz) > 1.5*orig)
if can_use_asymptotic:
#print "using asymp"
try:
try:
ctx.prec += asymp_extraprec
# http://functions.wolfram.com/HypergeometricFunctions/
# Hypergeometric2F3/06/02/03/01/0002/
def h(a1,a2,b1,b2,b3):
X = ctx.mpq_1_2*(a1+a2-b1-b2-b3+ctx.mpq_1_2)
A2 = a1+a2
B3 = b1+b2+b3
A = a1*a2
B = b1*b2+b3*b2+b1*b3
R = b1*b2*b3
c = {}
c[0] = ctx.one
c[1] = 2*(B - A + ctx.mpq_1_4*(3*A2+B3-2)*(A2-B3) - ctx.mpq_3_16)
c[2] = ctx.mpq_1_2*c[1]**2 + ctx.mpq_1_16*(-16*(2*A2-3)*(B-A) + 32*R +\
4*(-8*A2**2 + 11*A2 + 8*A + B3 - 2)*(A2-B3)-3)
s1 = 0
s2 = 0
k = 0
tprev = 0
while 1:
if k not in c:
uu1 = (k-2*X-3)*(k-2*X-2*b1-1)*(k-2*X-2*b2-1)*\
(k-2*X-2*b3-1)
uu2 = (4*(k-1)**3 - 6*(4*X+B3)*(k-1)**2 + \
2*(24*X**2+12*B3*X+4*B+B3-1)*(k-1) - 32*X**3 - \
24*B3*X**2 - 4*B - 8*R - 4*(4*B+B3-1)*X + 2*B3-1)
uu3 = (5*(k-1)**2+2*(-10*X+A2-3*B3+3)*(k-1)+2*c[1])
c[k] = ctx.one/(2*k)*(uu1*c[k-3]-uu2*c[k-2]+uu3*c[k-1])
w = c[k] * ctx.power(-z, -0.5*k)
t1 = (-ctx.j)**k * ctx.mpf(2)**(-k) * w
t2 = ctx.j**k * ctx.mpf(2)**(-k) * w
if abs(t1) < 0.1*ctx.eps:
break
# Quit if the series doesn't converge quickly enough
if k > 5 and abs(tprev) / abs(t1) < 1.5:
raise ctx.NoConvergence
s1 += t1
s2 += t2
tprev = t1
k += 1
S = ctx.expj(ctx.pi*X+2*ctx.sqrt(-z))*s1 + \
ctx.expj(-(ctx.pi*X+2*ctx.sqrt(-z)))*s2
T1 = [0.5*S, ctx.pi, -z], [1, -0.5, X], [b1, b2, b3], [a1, a2],\
[], [], 0
T2 = [-z], [-a1], [b1,b2,b3,a2-a1],[a2,b1-a1,b2-a1,b3-a1], \
[a1,a1-b1+1,a1-b2+1,a1-b3+1], [a1-a2+1], 1/z
T3 = [-z], [-a2], [b1,b2,b3,a1-a2],[a1,b1-a2,b2-a2,b3-a2], \
[a2,a2-b1+1,a2-b2+1,a2-b3+1],[-a1+a2+1], 1/z
return T1, T2, T3
v = ctx.hypercomb(h, [a1,a2,b1,b2,b3], force_series=True, maxterms=4*ctx.prec)
if sum(ctx._is_real_type(u) for u in [a1,a2,b1,b2,b3,z]) == 6:
v = ctx.re(v)
return v
except ctx.NoConvergence:
pass
finally:
ctx.prec = orig
return ctx.hypsum(2, 3, (a1type, a2type, b1type, b2type, b3type), [a1, a2, b1, b2, b3], z, **kwargs)
@defun
def _hyp2f0(ctx, a_s, b_s, z, **kwargs):
(a, atype), (b, btype) = a_s
# We want to try aggressively to use the asymptotic expansion,
# and fall back only when absolutely necessary
try:
kwargsb = kwargs.copy()
kwargsb['maxterms'] = kwargsb.get('maxterms', ctx.prec)
return ctx.hypsum(2, 0, (atype,btype), [a,b], z, **kwargsb)
except ctx.NoConvergence:
if kwargs.get('force_series'):
raise
pass
def h(a, b):
w = ctx.sinpi(b)
rz = -1/z
T1 = ([ctx.pi,w,rz],[1,-1,a],[],[a-b+1,b],[a],[b],rz)
T2 = ([-ctx.pi,w,rz],[1,-1,1+a-b],[],[a,2-b],[a-b+1],[2-b],rz)
return T1, T2
return ctx.hypercomb(h, [a, 1+a-b], **kwargs)
@defun
def meijerg(ctx, a_s, b_s, z, r=1, series=None, **kwargs):
an, ap = a_s
bm, bq = b_s
n = len(an)
p = n + len(ap)
m = len(bm)
q = m + len(bq)
a = an+ap
b = bm+bq
a = [ctx.convert(_) for _ in a]
b = [ctx.convert(_) for _ in b]
z = ctx.convert(z)
if series is None:
if p < q: series = 1
if p > q: series = 2
if p == q:
if m+n == p and abs(z) > 1:
series = 2
else:
series = 1
if kwargs.get('verbose'):
print("Meijer G m,n,p,q,series =", m,n,p,q,series)
if series == 1:
def h(*args):
a = args[:p]
b = args[p:]
terms = []
for k in range(m):
bases = [z]
expts = [b[k]/r]
gn = [b[j]-b[k] for j in range(m) if j != k]
gn += [1-a[j]+b[k] for j in range(n)]
gd = [a[j]-b[k] for j in range(n,p)]
gd += [1-b[j]+b[k] for j in range(m,q)]
hn = [1-a[j]+b[k] for j in range(p)]
hd = [1-b[j]+b[k] for j in range(q) if j != k]
hz = (-ctx.one)**(p-m-n) * z**(ctx.one/r)
terms.append((bases, expts, gn, gd, hn, hd, hz))
return terms
else:
def h(*args):
a = args[:p]
b = args[p:]
terms = []
for k in range(n):
bases = [z]
if r == 1:
expts = [a[k]-1]
else:
expts = [(a[k]-1)/ctx.convert(r)]
gn = [a[k]-a[j] for j in range(n) if j != k]
gn += [1-a[k]+b[j] for j in range(m)]
gd = [a[k]-b[j] for j in range(m,q)]
gd += [1-a[k]+a[j] for j in range(n,p)]
hn = [1-a[k]+b[j] for j in range(q)]
hd = [1+a[j]-a[k] for j in range(p) if j != k]
hz = (-ctx.one)**(q-m-n) / z**(ctx.one/r)
terms.append((bases, expts, gn, gd, hn, hd, hz))
return terms
return ctx.hypercomb(h, a+b, **kwargs)
@defun_wrapped
def appellf1(ctx,a,b1,b2,c,x,y,**kwargs):
# Assume x smaller
# We will use x for the outer loop
if abs(x) > abs(y):
x, y = y, x
b1, b2 = b2, b1
def ok(x):
return abs(x) < 0.99
# Finite cases
if ctx.isnpint(a):
pass
elif ctx.isnpint(b1):
pass
elif ctx.isnpint(b2):
x, y, b1, b2 = y, x, b2, b1
else:
#print x, y
# Note: ok if |y| > 1, because
# 2F1 implements analytic continuation
if not ok(x):
u1 = (x-y)/(x-1)
if not ok(u1):
raise ValueError("Analytic continuation not implemented")
#print "Using analytic continuation"
return (1-x)**(-b1)*(1-y)**(c-a-b2)*\
ctx.appellf1(c-a,b1,c-b1-b2,c,u1,y,**kwargs)
return ctx.hyper2d({'m+n':[a],'m':[b1],'n':[b2]}, {'m+n':[c]}, x,y, **kwargs)
@defun
def appellf2(ctx,a,b1,b2,c1,c2,x,y,**kwargs):
# TODO: continuation
return ctx.hyper2d({'m+n':[a],'m':[b1],'n':[b2]},
{'m':[c1],'n':[c2]}, x,y, **kwargs)
@defun
def appellf3(ctx,a1,a2,b1,b2,c,x,y,**kwargs):
outer_polynomial = ctx.isnpint(a1) or ctx.isnpint(b1)
inner_polynomial = ctx.isnpint(a2) or ctx.isnpint(b2)
if not outer_polynomial:
if inner_polynomial or abs(x) > abs(y):
x, y = y, x
a1,a2,b1,b2 = a2,a1,b2,b1
return ctx.hyper2d({'m':[a1,b1],'n':[a2,b2]}, {'m+n':[c]},x,y,**kwargs)
@defun
def appellf4(ctx,a,b,c1,c2,x,y,**kwargs):
# TODO: continuation
return ctx.hyper2d({'m+n':[a,b]}, {'m':[c1],'n':[c2]},x,y,**kwargs)
@defun
def hyper2d(ctx, a, b, x, y, **kwargs):
r"""
Sums the generalized 2D hypergeometric series
.. math ::
\sum_{m=0}^{\infty} \sum_{n=0}^{\infty}
\frac{P((a),m,n)}{Q((b),m,n)}
\frac{x^m y^n} {m! n!}
where `(a) = (a_1,\ldots,a_r)`, `(b) = (b_1,\ldots,b_s)` and where
`P` and `Q` are products of rising factorials such as `(a_j)_n` or
`(a_j)_{m+n}`. `P` and `Q` are specified in the form of dicts, with
the `m` and `n` dependence as keys and parameter lists as values.
The supported rising factorials are given in the following table
(note that only a few are supported in `Q`):
+------------+-------------------+--------+
| Key | Rising factorial | `Q` |
+============+===================+========+
| ``'m'`` | `(a_j)_m` | Yes |
+------------+-------------------+--------+
| ``'n'`` | `(a_j)_n` | Yes |
+------------+-------------------+--------+
| ``'m+n'`` | `(a_j)_{m+n}` | Yes |
+------------+-------------------+--------+
| ``'m-n'`` | `(a_j)_{m-n}` | No |
+------------+-------------------+--------+
| ``'n-m'`` | `(a_j)_{n-m}` | No |
+------------+-------------------+--------+
| ``'2m+n'`` | `(a_j)_{2m+n}` | No |
+------------+-------------------+--------+
| ``'2m-n'`` | `(a_j)_{2m-n}` | No |
+------------+-------------------+--------+
| ``'2n-m'`` | `(a_j)_{2n-m}` | No |
+------------+-------------------+--------+
For example, the Appell F1 and F4 functions
.. math ::
F_1 = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty}
\frac{(a)_{m+n} (b)_m (c)_n}{(d)_{m+n}}
\frac{x^m y^n}{m! n!}
F_4 = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty}
\frac{(a)_{m+n} (b)_{m+n}}{(c)_m (d)_{n}}
\frac{x^m y^n}{m! n!}
can be represented respectively as
``hyper2d({'m+n':[a], 'm':[b], 'n':[c]}, {'m+n':[d]}, x, y)``
``hyper2d({'m+n':[a,b]}, {'m':[c], 'n':[d]}, x, y)``
More generally, :func:`~mpmath.hyper2d` can evaluate any of the 34 distinct
convergent second-order (generalized Gaussian) hypergeometric
series enumerated by Horn, as well as the Kampe de Feriet
function.
The series is computed by rewriting it so that the inner
series (i.e. the series containing `n` and `y`) has the form of an
ordinary generalized hypergeometric series and thereby can be
evaluated efficiently using :func:`~mpmath.hyper`. If possible,
manually swapping `x` and `y` and the corresponding parameters
can sometimes give better results.
**Examples**
Two separable cases: a product of two geometric series, and a
product of two Gaussian hypergeometric functions::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> x, y = mpf(0.25), mpf(0.5)
>>> hyper2d({'m':1,'n':1}, {}, x,y)
2.666666666666666666666667
>>> 1/(1-x)/(1-y)
2.666666666666666666666667
>>> hyper2d({'m':[1,2],'n':[3,4]}, {'m':[5],'n':[6]}, x,y)
4.164358531238938319669856
>>> hyp2f1(1,2,5,x)*hyp2f1(3,4,6,y)
4.164358531238938319669856
Some more series that can be done in closed form::
>>> hyper2d({'m':1,'n':1},{'m+n':1},x,y)
2.013417124712514809623881
>>> (exp(x)*x-exp(y)*y)/(x-y)
2.013417124712514809623881
Six of the 34 Horn functions, G1-G3 and H1-H3::
>>> from mpmath import *
>>> mp.dps = 10; mp.pretty = True
>>> x, y = 0.0625, 0.125
>>> a1,a2,b1,b2,c1,c2,d = 1.1,-1.2,-1.3,-1.4,1.5,-1.6,1.7
>>> hyper2d({'m+n':a1,'n-m':b1,'m-n':b2},{},x,y) # G1
1.139090746
>>> nsum(lambda m,n: rf(a1,m+n)*rf(b1,n-m)*rf(b2,m-n)*\
... x**m*y**n/fac(m)/fac(n), [0,inf], [0,inf])
1.139090746
>>> hyper2d({'m':a1,'n':a2,'n-m':b1,'m-n':b2},{},x,y) # G2
0.9503682696
>>> nsum(lambda m,n: rf(a1,m)*rf(a2,n)*rf(b1,n-m)*rf(b2,m-n)*\
... x**m*y**n/fac(m)/fac(n), [0,inf], [0,inf])
0.9503682696
>>> hyper2d({'2n-m':a1,'2m-n':a2},{},x,y) # G3
1.029372029
>>> nsum(lambda m,n: rf(a1,2*n-m)*rf(a2,2*m-n)*\
... x**m*y**n/fac(m)/fac(n), [0,inf], [0,inf])
1.029372029
>>> hyper2d({'m-n':a1,'m+n':b1,'n':c1},{'m':d},x,y) # H1
-1.605331256
>>> nsum(lambda m,n: rf(a1,m-n)*rf(b1,m+n)*rf(c1,n)/rf(d,m)*\
... x**m*y**n/fac(m)/fac(n), [0,inf], [0,inf])
-1.605331256
>>> hyper2d({'m-n':a1,'m':b1,'n':[c1,c2]},{'m':d},x,y) # H2
-2.35405404
>>> nsum(lambda m,n: rf(a1,m-n)*rf(b1,m)*rf(c1,n)*rf(c2,n)/rf(d,m)*\
... x**m*y**n/fac(m)/fac(n), [0,inf], [0,inf])
-2.35405404
>>> hyper2d({'2m+n':a1,'n':b1},{'m+n':c1},x,y) # H3
0.974479074
>>> nsum(lambda m,n: rf(a1,2*m+n)*rf(b1,n)/rf(c1,m+n)*\
... x**m*y**n/fac(m)/fac(n), [0,inf], [0,inf])
0.974479074
**References**
1. [SrivastavaKarlsson]_
2. [Weisstein]_ http://mathworld.wolfram.com/HornFunction.html
3. [Weisstein]_ http://mathworld.wolfram.com/AppellHypergeometricFunction.html
"""
x = ctx.convert(x)
y = ctx.convert(y)
def parse(dct, key):
args = dct.pop(key, [])
try:
args = list(args)
except TypeError:
args = [args]
return [ctx.convert(arg) for arg in args]
a_s = dict(a)
b_s = dict(b)
a_m = parse(a, 'm')
a_n = parse(a, 'n')
a_m_add_n = parse(a, 'm+n')
a_m_sub_n = parse(a, 'm-n')
a_n_sub_m = parse(a, 'n-m')
a_2m_add_n = parse(a, '2m+n')
a_2m_sub_n = parse(a, '2m-n')
a_2n_sub_m = parse(a, '2n-m')
b_m = parse(b, 'm')
b_n = parse(b, 'n')
b_m_add_n = parse(b, 'm+n')
if a: raise ValueError("unsupported key: %r" % a.keys()[0])
if b: raise ValueError("unsupported key: %r" % b.keys()[0])
s = 0
outer = ctx.one
m = ctx.mpf(0)
ok_count = 0
prec = ctx.prec
maxterms = kwargs.get('maxterms', 20*prec)
try:
ctx.prec += 10
tol = +ctx.eps
while 1:
inner_sign = 1
outer_sign = 1
inner_a = list(a_n)
inner_b = list(b_n)
outer_a = [a+m for a in a_m]
outer_b = [b+m for b in b_m]
# (a)_{m+n} = (a)_m (a+m)_n
for a in a_m_add_n:
a = a+m
inner_a.append(a)
outer_a.append(a)
# (b)_{m+n} = (b)_m (b+m)_n
for b in b_m_add_n:
b = b+m
inner_b.append(b)
outer_b.append(b)
# (a)_{n-m} = (a-m)_n / (a-m)_m
for a in a_n_sub_m:
inner_a.append(a-m)
outer_b.append(a-m-1)
# (a)_{m-n} = (-1)^(m+n) (1-a-m)_m / (1-a-m)_n
for a in a_m_sub_n:
inner_sign *= (-1)
outer_sign *= (-1)**(m)
inner_b.append(1-a-m)
outer_a.append(-a-m)
# (a)_{2m+n} = (a)_{2m} (a+2m)_n
for a in a_2m_add_n:
inner_a.append(a+2*m)
outer_a.append((a+2*m)*(1+a+2*m))
# (a)_{2m-n} = (-1)^(2m+n) (1-a-2m)_{2m} / (1-a-2m)_n
for a in a_2m_sub_n:
inner_sign *= (-1)
inner_b.append(1-a-2*m)
outer_a.append((a+2*m)*(1+a+2*m))
# (a)_{2n-m} = 4^n ((a-m)/2)_n ((a-m+1)/2)_n / (a-m)_m
for a in a_2n_sub_m:
inner_sign *= 4
inner_a.append(0.5*(a-m))
inner_a.append(0.5*(a-m+1))
outer_b.append(a-m-1)
inner = ctx.hyper(inner_a, inner_b, inner_sign*y,
zeroprec=ctx.prec, **kwargs)
term = outer * inner * outer_sign
if abs(term) < tol:
ok_count += 1
else:
ok_count = 0
if ok_count >= 3 or not outer:
break
s += term
for a in outer_a: outer *= a
for b in outer_b: outer /= b
m += 1
outer = outer * x / m
if m > maxterms:
raise ctx.NoConvergence("maxterms exceeded in hyper2d")
finally:
ctx.prec = prec
return +s
"""
@defun
def kampe_de_feriet(ctx,a,b,c,d,e,f,x,y,**kwargs):
return ctx.hyper2d({'m+n':a,'m':b,'n':c},
{'m+n':d,'m':e,'n':f}, x,y, **kwargs)
"""
@defun
def bihyper(ctx, a_s, b_s, z, **kwargs):
r"""
Evaluates the bilateral hypergeometric series
.. math ::
\,_AH_B(a_1, \ldots, a_k; b_1, \ldots, b_B; z) =
\sum_{n=-\infty}^{\infty}
\frac{(a_1)_n \ldots (a_A)_n}
{(b_1)_n \ldots (b_B)_n} \, z^n
where, for direct convergence, `A = B` and `|z| = 1`, although a
regularized sum exists more generally by considering the
bilateral series as a sum of two ordinary hypergeometric
functions. In order for the series to make sense, none of the
parameters may be integers.
**Examples**
The value of `\,_2H_2` at `z = 1` is given by Dougall's formula::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> a,b,c,d = 0.5, 1.5, 2.25, 3.25
>>> bihyper([a,b],[c,d],1)
-14.49118026212345786148847
>>> gammaprod([c,d,1-a,1-b,c+d-a-b-1],[c-a,d-a,c-b,d-b])
-14.49118026212345786148847
The regularized function `\,_1H_0` can be expressed as the
sum of one `\,_2F_0` function and one `\,_1F_1` function::
>>> a = mpf(0.25)
>>> z = mpf(0.75)
>>> bihyper([a], [], z)
(0.2454393389657273841385582 + 0.2454393389657273841385582j)
>>> hyper([a,1],[],z) + (hyper([1],[1-a],-1/z)-1)
(0.2454393389657273841385582 + 0.2454393389657273841385582j)
>>> hyper([a,1],[],z) + hyper([1],[2-a],-1/z)/z/(a-1)
(0.2454393389657273841385582 + 0.2454393389657273841385582j)
**References**
1. [Slater]_ (chapter 6: "Bilateral Series", pp. 180-189)
2. [Wikipedia]_ http://en.wikipedia.org/wiki/Bilateral_hypergeometric_series
"""
z = ctx.convert(z)
c_s = a_s + b_s
p = len(a_s)
q = len(b_s)
if (p, q) == (0,0) or (p, q) == (1,1):
return ctx.zero * z
neg = (p-q) % 2
def h(*c_s):
a_s = list(c_s[:p])
b_s = list(c_s[p:])
aa_s = [2-b for b in b_s]
bb_s = [2-a for a in a_s]
rp = [(-1)**neg * z] + [1-b for b in b_s] + [1-a for a in a_s]
rc = [-1] + [1]*len(b_s) + [-1]*len(a_s)
T1 = [], [], [], [], a_s + [1], b_s, z
T2 = rp, rc, [], [], aa_s + [1], bb_s, (-1)**neg / z
return T1, T2
return ctx.hypercomb(h, c_s, **kwargs)
| 51,570 | 35.471711 | 104 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/functions/qfunctions.py
|
from .functions import defun, defun_wrapped
@defun
def qp(ctx, a, q=None, n=None, **kwargs):
r"""
Evaluates the q-Pochhammer symbol (or q-rising factorial)
.. math ::
(a; q)_n = \prod_{k=0}^{n-1} (1-a q^k)
where `n = \infty` is permitted if `|q| < 1`. Called with two arguments,
``qp(a,q)`` computes `(a;q)_{\infty}`; with a single argument, ``qp(q)``
computes `(q;q)_{\infty}`. The special case
.. math ::
\phi(q) = (q; q)_{\infty} = \prod_{k=1}^{\infty} (1-q^k) =
\sum_{k=-\infty}^{\infty} (-1)^k q^{(3k^2-k)/2}
is also known as the Euler function, or (up to a factor `q^{-1/24}`)
the Dedekind eta function.
**Examples**
If `n` is a positive integer, the function amounts to a finite product::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> qp(2,3,5)
-725305.0
>>> fprod(1-2*3**k for k in range(5))
-725305.0
>>> qp(2,3,0)
1.0
Complex arguments are allowed::
>>> qp(2-1j, 0.75j)
(0.4628842231660149089976379 + 4.481821753552703090628793j)
The regular Pochhammer symbol `(a)_n` is obtained in the
following limit as `q \to 1`::
>>> a, n = 4, 7
>>> limit(lambda q: qp(q**a,q,n) / (1-q)**n, 1)
604800.0
>>> rf(a,n)
604800.0
The Taylor series of the reciprocal Euler function gives
the partition function `P(n)`, i.e. the number of ways of writing
`n` as a sum of positive integers::
>>> taylor(lambda q: 1/qp(q), 0, 10)
[1.0, 1.0, 2.0, 3.0, 5.0, 7.0, 11.0, 15.0, 22.0, 30.0, 42.0]
Special values include::
>>> qp(0)
1.0
>>> findroot(diffun(qp), -0.4) # location of maximum
-0.4112484791779547734440257
>>> qp(_)
1.228348867038575112586878
The q-Pochhammer symbol is related to the Jacobi theta functions.
For example, the following identity holds::
>>> q = mpf(0.5) # arbitrary
>>> qp(q)
0.2887880950866024212788997
>>> root(3,-2)*root(q,-24)*jtheta(2,pi/6,root(q,6))
0.2887880950866024212788997
"""
a = ctx.convert(a)
if n is None:
n = ctx.inf
else:
n = ctx.convert(n)
if n < 0:
raise ValueError("n cannot be negative")
if q is None:
q = a
else:
q = ctx.convert(q)
if n == 0:
return ctx.one + 0*(a+q)
infinite = (n == ctx.inf)
same = (a == q)
if infinite:
if abs(q) >= 1:
if same and (q == -1 or q == 1):
return ctx.zero * q
raise ValueError("q-function only defined for |q| < 1")
elif q == 0:
return ctx.one - a
maxterms = kwargs.get('maxterms', 50*ctx.prec)
if infinite and same:
# Euler's pentagonal theorem
def terms():
t = 1
yield t
k = 1
x1 = q
x2 = q**2
while 1:
yield (-1)**k * x1
yield (-1)**k * x2
x1 *= q**(3*k+1)
x2 *= q**(3*k+2)
k += 1
if k > maxterms:
raise ctx.NoConvergence
return ctx.sum_accurately(terms)
# return ctx.nprod(lambda k: 1-a*q**k, [0,n-1])
def factors():
k = 0
r = ctx.one
while 1:
yield 1 - a*r
r *= q
k += 1
if k >= n:
raise StopIteration
if k > maxterms:
raise ctx.NoConvergence
return ctx.mul_accurately(factors)
@defun_wrapped
def qgamma(ctx, z, q, **kwargs):
r"""
Evaluates the q-gamma function
.. math ::
\Gamma_q(z) = \frac{(q; q)_{\infty}}{(q^z; q)_{\infty}} (1-q)^{1-z}.
**Examples**
Evaluation for real and complex arguments::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> qgamma(4,0.75)
4.046875
>>> qgamma(6,6)
121226245.0
>>> qgamma(3+4j, 0.5j)
(0.1663082382255199834630088 + 0.01952474576025952984418217j)
The q-gamma function satisfies a functional equation similar
to that of the ordinary gamma function::
>>> q = mpf(0.25)
>>> z = mpf(2.5)
>>> qgamma(z+1,q)
1.428277424823760954685912
>>> (1-q**z)/(1-q)*qgamma(z,q)
1.428277424823760954685912
"""
if abs(q) > 1:
return ctx.qgamma(z,1/q)*q**((z-2)*(z-1)*0.5)
return ctx.qp(q, q, None, **kwargs) / \
ctx.qp(q**z, q, None, **kwargs) * (1-q)**(1-z)
@defun_wrapped
def qfac(ctx, z, q, **kwargs):
r"""
Evaluates the q-factorial,
.. math ::
[n]_q! = (1+q)(1+q+q^2)\cdots(1+q+\cdots+q^{n-1})
or more generally
.. math ::
[z]_q! = \frac{(q;q)_z}{(1-q)^z}.
**Examples**
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> qfac(0,0)
1.0
>>> qfac(4,3)
2080.0
>>> qfac(5,6)
121226245.0
>>> qfac(1+1j, 2+1j)
(0.4370556551322672478613695 + 0.2609739839216039203708921j)
"""
if ctx.isint(z) and ctx._re(z) > 0:
n = int(ctx._re(z))
return ctx.qp(q, q, n, **kwargs) / (1-q)**n
return ctx.qgamma(z+1, q, **kwargs)
@defun
def qhyper(ctx, a_s, b_s, q, z, **kwargs):
r"""
Evaluates the basic hypergeometric series or hypergeometric q-series
.. math ::
\,_r\phi_s \left[\begin{matrix}
a_1 & a_2 & \ldots & a_r \\
b_1 & b_2 & \ldots & b_s
\end{matrix} ; q,z \right] =
\sum_{n=0}^\infty
\frac{(a_1;q)_n, \ldots, (a_r;q)_n}
{(b_1;q)_n, \ldots, (b_s;q)_n}
\left((-1)^n q^{n\choose 2}\right)^{1+s-r}
\frac{z^n}{(q;q)_n}
where `(a;q)_n` denotes the q-Pochhammer symbol (see :func:`~mpmath.qp`).
**Examples**
Evaluation works for real and complex arguments::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> qhyper([0.5], [2.25], 0.25, 4)
-0.1975849091263356009534385
>>> qhyper([0.5], [2.25], 0.25-0.25j, 4)
(2.806330244925716649839237 + 3.568997623337943121769938j)
>>> qhyper([1+j], [2,3+0.5j], 0.25, 3+4j)
(9.112885171773400017270226 - 1.272756997166375050700388j)
Comparing with a summation of the defining series, using
:func:`~mpmath.nsum`::
>>> b, q, z = 3, 0.25, 0.5
>>> qhyper([], [b], q, z)
0.6221136748254495583228324
>>> nsum(lambda n: z**n / qp(q,q,n)/qp(b,q,n) * q**(n*(n-1)), [0,inf])
0.6221136748254495583228324
"""
#a_s = [ctx._convert_param(a)[0] for a in a_s]
#b_s = [ctx._convert_param(b)[0] for b in b_s]
#q = ctx._convert_param(q)[0]
a_s = [ctx.convert(a) for a in a_s]
b_s = [ctx.convert(b) for b in b_s]
q = ctx.convert(q)
z = ctx.convert(z)
r = len(a_s)
s = len(b_s)
d = 1+s-r
maxterms = kwargs.get('maxterms', 50*ctx.prec)
def terms():
t = ctx.one
yield t
qk = 1
k = 0
x = 1
while 1:
for a in a_s:
p = 1 - a*qk
t *= p
for b in b_s:
p = 1 - b*qk
if not p:
raise ValueError
t /= p
t *= z
x *= (-1)**d * qk ** d
qk *= q
t /= (1 - qk)
k += 1
yield t * x
if k > maxterms:
raise ctx.NoConvergence
return ctx.sum_accurately(terms)
| 7,646 | 26.213523 | 78 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/functions/functions.py
|
from ..libmp.backend import xrange
class SpecialFunctions(object):
"""
This class implements special functions using high-level code.
Elementary and some other functions (e.g. gamma function, basecase
hypergeometric series) are assumed to be predefined by the context as
"builtins" or "low-level" functions.
"""
defined_functions = {}
# The series for the Jacobi theta functions converge for |q| < 1;
# in the current implementation they throw a ValueError for
# abs(q) > THETA_Q_LIM
THETA_Q_LIM = 1 - 10**-7
def __init__(self):
cls = self.__class__
for name in cls.defined_functions:
f, wrap = cls.defined_functions[name]
cls._wrap_specfun(name, f, wrap)
self.mpq_1 = self._mpq((1,1))
self.mpq_0 = self._mpq((0,1))
self.mpq_1_2 = self._mpq((1,2))
self.mpq_3_2 = self._mpq((3,2))
self.mpq_1_4 = self._mpq((1,4))
self.mpq_1_16 = self._mpq((1,16))
self.mpq_3_16 = self._mpq((3,16))
self.mpq_5_2 = self._mpq((5,2))
self.mpq_3_4 = self._mpq((3,4))
self.mpq_7_4 = self._mpq((7,4))
self.mpq_5_4 = self._mpq((5,4))
self.mpq_1_3 = self._mpq((1,3))
self.mpq_2_3 = self._mpq((2,3))
self.mpq_4_3 = self._mpq((4,3))
self.mpq_1_6 = self._mpq((1,6))
self.mpq_5_6 = self._mpq((5,6))
self.mpq_5_3 = self._mpq((5,3))
self._misc_const_cache = {}
self._aliases.update({
'phase' : 'arg',
'conjugate' : 'conj',
'nthroot' : 'root',
'polygamma' : 'psi',
'hurwitz' : 'zeta',
#'digamma' : 'psi0',
#'trigamma' : 'psi1',
#'tetragamma' : 'psi2',
#'pentagamma' : 'psi3',
'fibonacci' : 'fib',
'factorial' : 'fac',
})
self.zetazero_memoized = self.memoize(self.zetazero)
# Default -- do nothing
@classmethod
def _wrap_specfun(cls, name, f, wrap):
setattr(cls, name, f)
# Optional fast versions of common functions in common cases.
# If not overridden, default (generic hypergeometric series)
# implementations will be used
def _besselj(ctx, n, z): raise NotImplementedError
def _erf(ctx, z): raise NotImplementedError
def _erfc(ctx, z): raise NotImplementedError
def _gamma_upper_int(ctx, z, a): raise NotImplementedError
def _expint_int(ctx, n, z): raise NotImplementedError
def _zeta(ctx, s): raise NotImplementedError
def _zetasum_fast(ctx, s, a, n, derivatives, reflect): raise NotImplementedError
def _ei(ctx, z): raise NotImplementedError
def _e1(ctx, z): raise NotImplementedError
def _ci(ctx, z): raise NotImplementedError
def _si(ctx, z): raise NotImplementedError
def _altzeta(ctx, s): raise NotImplementedError
def defun_wrapped(f):
SpecialFunctions.defined_functions[f.__name__] = f, True
def defun(f):
SpecialFunctions.defined_functions[f.__name__] = f, False
def defun_static(f):
setattr(SpecialFunctions, f.__name__, f)
@defun_wrapped
def cot(ctx, z): return ctx.one / ctx.tan(z)
@defun_wrapped
def sec(ctx, z): return ctx.one / ctx.cos(z)
@defun_wrapped
def csc(ctx, z): return ctx.one / ctx.sin(z)
@defun_wrapped
def coth(ctx, z): return ctx.one / ctx.tanh(z)
@defun_wrapped
def sech(ctx, z): return ctx.one / ctx.cosh(z)
@defun_wrapped
def csch(ctx, z): return ctx.one / ctx.sinh(z)
@defun_wrapped
def acot(ctx, z):
if not z:
return ctx.pi * 0.5
else:
return ctx.atan(ctx.one / z)
@defun_wrapped
def asec(ctx, z): return ctx.acos(ctx.one / z)
@defun_wrapped
def acsc(ctx, z): return ctx.asin(ctx.one / z)
@defun_wrapped
def acoth(ctx, z):
if not z:
return ctx.pi * 0.5j
else:
return ctx.atanh(ctx.one / z)
@defun_wrapped
def asech(ctx, z): return ctx.acosh(ctx.one / z)
@defun_wrapped
def acsch(ctx, z): return ctx.asinh(ctx.one / z)
@defun
def sign(ctx, x):
x = ctx.convert(x)
if not x or ctx.isnan(x):
return x
if ctx._is_real_type(x):
if x > 0:
return ctx.one
else:
return -ctx.one
return x / abs(x)
@defun
def agm(ctx, a, b=1):
if b == 1:
return ctx.agm1(a)
a = ctx.convert(a)
b = ctx.convert(b)
return ctx._agm(a, b)
@defun_wrapped
def sinc(ctx, x):
if ctx.isinf(x):
return 1/x
if not x:
return x+1
return ctx.sin(x)/x
@defun_wrapped
def sincpi(ctx, x):
if ctx.isinf(x):
return 1/x
if not x:
return x+1
return ctx.sinpi(x)/(ctx.pi*x)
# TODO: tests; improve implementation
@defun_wrapped
def expm1(ctx, x):
if not x:
return ctx.zero
# exp(x) - 1 ~ x
if ctx.mag(x) < -ctx.prec:
return x + 0.5*x**2
# TODO: accurately eval the smaller of the real/imag parts
return ctx.sum_accurately(lambda: iter([ctx.exp(x),-1]),1)
@defun_wrapped
def powm1(ctx, x, y):
mag = ctx.mag
one = ctx.one
w = x**y - one
M = mag(w)
# Only moderate cancellation
if M > -8:
return w
# Check for the only possible exact cases
if not w:
if (not y) or (x in (1, -1, 1j, -1j) and ctx.isint(y)):
return w
x1 = x - one
magy = mag(y)
lnx = ctx.ln(x)
# Small y: x^y - 1 ~ log(x)*y + O(log(x)^2 * y^2)
if magy + mag(lnx) < -ctx.prec:
return lnx*y + (lnx*y)**2/2
# TODO: accurately eval the smaller of the real/imag part
return ctx.sum_accurately(lambda: iter([x**y, -1]), 1)
@defun
def _rootof1(ctx, k, n):
k = int(k)
n = int(n)
k %= n
if not k:
return ctx.one
elif 2*k == n:
return -ctx.one
elif 4*k == n:
return ctx.j
elif 4*k == 3*n:
return -ctx.j
return ctx.expjpi(2*ctx.mpf(k)/n)
@defun
def root(ctx, x, n, k=0):
n = int(n)
x = ctx.convert(x)
if k:
# Special case: there is an exact real root
if (n & 1 and 2*k == n-1) and (not ctx.im(x)) and (ctx.re(x) < 0):
return -ctx.root(-x, n)
# Multiply by root of unity
prec = ctx.prec
try:
ctx.prec += 10
v = ctx.root(x, n, 0) * ctx._rootof1(k, n)
finally:
ctx.prec = prec
return +v
return ctx._nthroot(x, n)
@defun
def unitroots(ctx, n, primitive=False):
gcd = ctx._gcd
prec = ctx.prec
try:
ctx.prec += 10
if primitive:
v = [ctx._rootof1(k,n) for k in range(n) if gcd(k,n) == 1]
else:
# TODO: this can be done *much* faster
v = [ctx._rootof1(k,n) for k in range(n)]
finally:
ctx.prec = prec
return [+x for x in v]
@defun
def arg(ctx, x):
x = ctx.convert(x)
re = ctx._re(x)
im = ctx._im(x)
return ctx.atan2(im, re)
@defun
def fabs(ctx, x):
return abs(ctx.convert(x))
@defun
def re(ctx, x):
x = ctx.convert(x)
if hasattr(x, "real"): # py2.5 doesn't have .real/.imag for all numbers
return x.real
return x
@defun
def im(ctx, x):
x = ctx.convert(x)
if hasattr(x, "imag"): # py2.5 doesn't have .real/.imag for all numbers
return x.imag
return ctx.zero
@defun
def conj(ctx, x):
x = ctx.convert(x)
try:
return x.conjugate()
except AttributeError:
return x
@defun
def polar(ctx, z):
return (ctx.fabs(z), ctx.arg(z))
@defun_wrapped
def rect(ctx, r, phi):
return r * ctx.mpc(*ctx.cos_sin(phi))
@defun
def log(ctx, x, b=None):
if b is None:
return ctx.ln(x)
wp = ctx.prec + 20
return ctx.ln(x, prec=wp) / ctx.ln(b, prec=wp)
@defun
def log10(ctx, x):
return ctx.log(x, 10)
@defun
def fmod(ctx, x, y):
return ctx.convert(x) % ctx.convert(y)
@defun
def degrees(ctx, x):
return x / ctx.degree
@defun
def radians(ctx, x):
return x * ctx.degree
def _lambertw_special(ctx, z, k):
# W(0,0) = 0; all other branches are singular
if not z:
if not k:
return z
return ctx.ninf + z
if z == ctx.inf:
if k == 0:
return z
else:
return z + 2*k*ctx.pi*ctx.j
if z == ctx.ninf:
return (-z) + (2*k+1)*ctx.pi*ctx.j
# Some kind of nan or complex inf/nan?
return ctx.ln(z)
import math
import cmath
def _lambertw_approx_hybrid(z, k):
imag_sign = 0
if hasattr(z, "imag"):
x = float(z.real)
y = z.imag
if y:
imag_sign = (-1) ** (y < 0)
y = float(y)
else:
x = float(z)
y = 0.0
imag_sign = 0
# hack to work regardless of whether Python supports -0.0
if not y:
y = 0.0
z = complex(x,y)
if k == 0:
if -4.0 < y < 4.0 and -1.0 < x < 2.5:
if imag_sign:
# Taylor series in upper/lower half-plane
if y > 1.00: return (0.876+0.645j) + (0.118-0.174j)*(z-(0.75+2.5j))
if y > 0.25: return (0.505+0.204j) + (0.375-0.132j)*(z-(0.75+0.5j))
if y < -1.00: return (0.876-0.645j) + (0.118+0.174j)*(z-(0.75-2.5j))
if y < -0.25: return (0.505-0.204j) + (0.375+0.132j)*(z-(0.75-0.5j))
# Taylor series near -1
if x < -0.5:
if imag_sign >= 0:
return (-0.318+1.34j) + (-0.697-0.593j)*(z+1)
else:
return (-0.318-1.34j) + (-0.697+0.593j)*(z+1)
# return real type
r = -0.367879441171442
if (not imag_sign) and x > r:
z = x
# Singularity near -1/e
if x < -0.2:
return -1 + 2.33164398159712*(z-r)**0.5 - 1.81218788563936*(z-r)
# Taylor series near 0
if x < 0.5: return z
# Simple linear approximation
return 0.2 + 0.3*z
if (not imag_sign) and x > 0.0:
L1 = math.log(x); L2 = math.log(L1)
else:
L1 = cmath.log(z); L2 = cmath.log(L1)
elif k == -1:
# return real type
r = -0.367879441171442
if (not imag_sign) and r < x < 0.0:
z = x
if (imag_sign >= 0) and y < 0.1 and -0.6 < x < -0.2:
return -1 - 2.33164398159712*(z-r)**0.5 - 1.81218788563936*(z-r)
if (not imag_sign) and -0.2 <= x < 0.0:
L1 = math.log(-x)
return L1 - math.log(-L1)
else:
if imag_sign == -1 and (not y) and x < 0.0:
L1 = cmath.log(z) - 3.1415926535897932j
else:
L1 = cmath.log(z) - 6.2831853071795865j
L2 = cmath.log(L1)
return L1 - L2 + L2/L1 + L2*(L2-2)/(2*L1**2)
def _lambertw_series(ctx, z, k, tol):
"""
Return rough approximation for W_k(z) from an asymptotic series,
sufficiently accurate for the Halley iteration to converge to
the correct value.
"""
magz = ctx.mag(z)
if (-10 < magz < 900) and (-1000 < k < 1000):
# Near the branch point at -1/e
if magz < 1 and abs(z+0.36787944117144) < 0.05:
if k == 0 or (k == -1 and ctx._im(z) >= 0) or \
(k == 1 and ctx._im(z) < 0):
delta = ctx.sum_accurately(lambda: [z, ctx.exp(-1)])
cancellation = -ctx.mag(delta)
ctx.prec += cancellation
# Use series given in Corless et al.
p = ctx.sqrt(2*(ctx.e*z+1))
ctx.prec -= cancellation
u = {0:ctx.mpf(-1), 1:ctx.mpf(1)}
a = {0:ctx.mpf(2), 1:ctx.mpf(-1)}
if k != 0:
p = -p
s = ctx.zero
# The series converges, so we could use it directly, but unless
# *extremely* close, it is better to just use the first few
# terms to get a good approximation for the iteration
for l in xrange(max(2,cancellation)):
if l not in u:
a[l] = ctx.fsum(u[j]*u[l+1-j] for j in xrange(2,l))
u[l] = (l-1)*(u[l-2]/2+a[l-2]/4)/(l+1)-a[l]/2-u[l-1]/(l+1)
term = u[l] * p**l
s += term
if ctx.mag(term) < -tol:
return s, True
l += 1
ctx.prec += cancellation//2
return s, False
if k == 0 or k == -1:
return _lambertw_approx_hybrid(z, k), False
if k == 0:
if magz < -1:
return z*(1-z), False
L1 = ctx.ln(z)
L2 = ctx.ln(L1)
elif k == -1 and (not ctx._im(z)) and (-0.36787944117144 < ctx._re(z) < 0):
L1 = ctx.ln(-z)
return L1 - ctx.ln(-L1), False
else:
# This holds both as z -> 0 and z -> inf.
# Relative error is O(1/log(z)).
L1 = ctx.ln(z) + 2j*ctx.pi*k
L2 = ctx.ln(L1)
return L1 - L2 + L2/L1 + L2*(L2-2)/(2*L1**2), False
@defun
def lambertw(ctx, z, k=0):
z = ctx.convert(z)
k = int(k)
if not ctx.isnormal(z):
return _lambertw_special(ctx, z, k)
prec = ctx.prec
ctx.prec += 20 + ctx.mag(k or 1)
wp = ctx.prec
tol = wp - 5
w, done = _lambertw_series(ctx, z, k, tol)
if not done:
# Use Halley iteration to solve w*exp(w) = z
two = ctx.mpf(2)
for i in xrange(100):
ew = ctx.exp(w)
wew = w*ew
wewz = wew-z
wn = w - wewz/(wew+ew-(w+two)*wewz/(two*w+two))
if ctx.mag(wn-w) <= ctx.mag(wn) - tol:
w = wn
break
else:
w = wn
if i == 100:
ctx.warn("Lambert W iteration failed to converge for z = %s" % z)
ctx.prec = prec
return +w
@defun_wrapped
def bell(ctx, n, x=1):
x = ctx.convert(x)
if not n:
if ctx.isnan(x):
return x
return type(x)(1)
if ctx.isinf(x) or ctx.isinf(n) or ctx.isnan(x) or ctx.isnan(n):
return x**n
if n == 1: return x
if n == 2: return x*(x+1)
if x == 0: return ctx.sincpi(n)
return _polyexp(ctx, n, x, True) / ctx.exp(x)
def _polyexp(ctx, n, x, extra=False):
def _terms():
if extra:
yield ctx.sincpi(n)
t = x
k = 1
while 1:
yield k**n * t
k += 1
t = t*x/k
return ctx.sum_accurately(_terms, check_step=4)
@defun_wrapped
def polyexp(ctx, s, z):
if ctx.isinf(z) or ctx.isinf(s) or ctx.isnan(z) or ctx.isnan(s):
return z**s
if z == 0: return z*s
if s == 0: return ctx.expm1(z)
if s == 1: return ctx.exp(z)*z
if s == 2: return ctx.exp(z)*z*(z+1)
return _polyexp(ctx, s, z)
@defun_wrapped
def cyclotomic(ctx, n, z):
n = int(n)
if n < 0:
raise ValueError("n cannot be negative")
p = ctx.one
if n == 0:
return p
if n == 1:
return z - p
if n == 2:
return z + p
# Use divisor product representation. Unfortunately, this sometimes
# includes singularities for roots of unity, which we have to cancel out.
# Matching zeros/poles pairwise, we have (1-z^a)/(1-z^b) ~ a/b + O(z-1).
a_prod = 1
b_prod = 1
num_zeros = 0
num_poles = 0
for d in range(1,n+1):
if not n % d:
w = ctx.moebius(n//d)
# Use powm1 because it is important that we get 0 only
# if it really is exactly 0
b = -ctx.powm1(z, d)
if b:
p *= b**w
else:
if w == 1:
a_prod *= d
num_zeros += 1
elif w == -1:
b_prod *= d
num_poles += 1
#print n, num_zeros, num_poles
if num_zeros:
if num_zeros > num_poles:
p *= 0
else:
p *= a_prod
p /= b_prod
return p
@defun
def mangoldt(ctx, n):
r"""
Evaluates the von Mangoldt function `\Lambda(n) = \log p`
if `n = p^k` a power of a prime, and `\Lambda(n) = 0` otherwise.
**Examples**
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> [mangoldt(n) for n in range(-2,3)]
[0.0, 0.0, 0.0, 0.0, 0.6931471805599453094172321]
>>> mangoldt(6)
0.0
>>> mangoldt(7)
1.945910149055313305105353
>>> mangoldt(8)
0.6931471805599453094172321
>>> fsum(mangoldt(n) for n in range(101))
94.04531122935739224600493
>>> fsum(mangoldt(n) for n in range(10001))
10013.39669326311478372032
"""
n = int(n)
if n < 2:
return ctx.zero
if n % 2 == 0:
# Must be a power of two
if n & (n-1) == 0:
return +ctx.ln2
else:
return ctx.zero
# TODO: the following could be generalized into a perfect
# power testing function
# ---
# Look for a small factor
for p in (3,5,7,11,13,17,19,23,29,31):
if not n % p:
q, r = n // p, 0
while q > 1:
q, r = divmod(q, p)
if r:
return ctx.zero
return ctx.ln(p)
if ctx.isprime(n):
return ctx.ln(n)
# Obviously, we could use arbitrary-precision arithmetic for this...
if n > 10**30:
raise NotImplementedError
k = 2
while 1:
p = int(n**(1./k) + 0.5)
if p < 2:
return ctx.zero
if p ** k == n:
if ctx.isprime(p):
return ctx.ln(p)
k += 1
@defun
def stirling1(ctx, n, k, exact=False):
v = ctx._stirling1(int(n), int(k))
if exact:
return int(v)
else:
return ctx.mpf(v)
@defun
def stirling2(ctx, n, k, exact=False):
v = ctx._stirling2(int(n), int(k))
if exact:
return int(v)
else:
return ctx.mpf(v)
| 17,877 | 27.154331 | 84 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/functions/elliptic.py
|
r"""
Elliptic functions historically comprise the elliptic integrals
and their inverses, and originate from the problem of computing the
arc length of an ellipse. From a more modern point of view,
an elliptic function is defined as a doubly periodic function, i.e.
a function which satisfies
.. math ::
f(z + 2 \omega_1) = f(z + 2 \omega_2) = f(z)
for some half-periods `\omega_1, \omega_2` with
`\mathrm{Im}[\omega_1 / \omega_2] > 0`. The canonical elliptic
functions are the Jacobi elliptic functions. More broadly, this section
includes quasi-doubly periodic functions (such as the Jacobi theta
functions) and other functions useful in the study of elliptic functions.
Many different conventions for the arguments of
elliptic functions are in use. It is even standard to use
different parameterizations for different functions in the same
text or software (and mpmath is no exception).
The usual parameters are the elliptic nome `q`, which usually
must satisfy `|q| < 1`; the elliptic parameter `m` (an arbitrary
complex number); the elliptic modulus `k` (an arbitrary complex
number); and the half-period ratio `\tau`, which usually must
satisfy `\mathrm{Im}[\tau] > 0`.
These quantities can be expressed in terms of each other
using the following relations:
.. math ::
m = k^2
.. math ::
\tau = i \frac{K(1-m)}{K(m)}
.. math ::
q = e^{i \pi \tau}
.. math ::
k = \frac{\vartheta_2^4(q)}{\vartheta_3^4(q)}
In addition, an alternative definition is used for the nome in
number theory, which we here denote by q-bar:
.. math ::
\bar{q} = q^2 = e^{2 i \pi \tau}
For convenience, mpmath provides functions to convert
between the various parameters (:func:`~mpmath.qfrom`, :func:`~mpmath.mfrom`,
:func:`~mpmath.kfrom`, :func:`~mpmath.taufrom`, :func:`~mpmath.qbarfrom`).
**References**
1. [AbramowitzStegun]_
2. [WhittakerWatson]_
"""
from .functions import defun, defun_wrapped
def nome(ctx, m):
m = ctx.convert(m)
if not m:
return m
if m == ctx.one:
return m
if ctx.isnan(m):
return m
if ctx.isinf(m):
if m == ctx.ninf:
return type(m)(-1)
else:
return ctx.mpc(-1)
a = ctx.ellipk(ctx.one-m)
b = ctx.ellipk(m)
v = ctx.exp(-ctx.pi*a/b)
if not ctx._im(m) and ctx._re(m) < 1:
if ctx._is_real_type(m):
return v.real
else:
return v.real + 0j
elif m == 2:
v = ctx.mpc(0, v.imag)
return v
@defun_wrapped
def qfrom(ctx, q=None, m=None, k=None, tau=None, qbar=None):
r"""
Returns the elliptic nome `q`, given any of `q, m, k, \tau, \bar{q}`::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> qfrom(q=0.25)
0.25
>>> qfrom(m=mfrom(q=0.25))
0.25
>>> qfrom(k=kfrom(q=0.25))
0.25
>>> qfrom(tau=taufrom(q=0.25))
(0.25 + 0.0j)
>>> qfrom(qbar=qbarfrom(q=0.25))
0.25
"""
if q is not None:
return ctx.convert(q)
if m is not None:
return nome(ctx, m)
if k is not None:
return nome(ctx, ctx.convert(k)**2)
if tau is not None:
return ctx.expjpi(tau)
if qbar is not None:
return ctx.sqrt(qbar)
@defun_wrapped
def qbarfrom(ctx, q=None, m=None, k=None, tau=None, qbar=None):
r"""
Returns the number-theoretic nome `\bar q`, given any of
`q, m, k, \tau, \bar{q}`::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> qbarfrom(qbar=0.25)
0.25
>>> qbarfrom(q=qfrom(qbar=0.25))
0.25
>>> qbarfrom(m=extraprec(20)(mfrom)(qbar=0.25)) # ill-conditioned
0.25
>>> qbarfrom(k=extraprec(20)(kfrom)(qbar=0.25)) # ill-conditioned
0.25
>>> qbarfrom(tau=taufrom(qbar=0.25))
(0.25 + 0.0j)
"""
if qbar is not None:
return ctx.convert(qbar)
if q is not None:
return ctx.convert(q) ** 2
if m is not None:
return nome(ctx, m) ** 2
if k is not None:
return nome(ctx, ctx.convert(k)**2) ** 2
if tau is not None:
return ctx.expjpi(2*tau)
@defun_wrapped
def taufrom(ctx, q=None, m=None, k=None, tau=None, qbar=None):
r"""
Returns the elliptic half-period ratio `\tau`, given any of
`q, m, k, \tau, \bar{q}`::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> taufrom(tau=0.5j)
(0.0 + 0.5j)
>>> taufrom(q=qfrom(tau=0.5j))
(0.0 + 0.5j)
>>> taufrom(m=mfrom(tau=0.5j))
(0.0 + 0.5j)
>>> taufrom(k=kfrom(tau=0.5j))
(0.0 + 0.5j)
>>> taufrom(qbar=qbarfrom(tau=0.5j))
(0.0 + 0.5j)
"""
if tau is not None:
return ctx.convert(tau)
if m is not None:
m = ctx.convert(m)
return ctx.j*ctx.ellipk(1-m)/ctx.ellipk(m)
if k is not None:
k = ctx.convert(k)
return ctx.j*ctx.ellipk(1-k**2)/ctx.ellipk(k**2)
if q is not None:
return ctx.log(q) / (ctx.pi*ctx.j)
if qbar is not None:
qbar = ctx.convert(qbar)
return ctx.log(qbar) / (2*ctx.pi*ctx.j)
@defun_wrapped
def kfrom(ctx, q=None, m=None, k=None, tau=None, qbar=None):
r"""
Returns the elliptic modulus `k`, given any of
`q, m, k, \tau, \bar{q}`::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> kfrom(k=0.25)
0.25
>>> kfrom(m=mfrom(k=0.25))
0.25
>>> kfrom(q=qfrom(k=0.25))
0.25
>>> kfrom(tau=taufrom(k=0.25))
(0.25 + 0.0j)
>>> kfrom(qbar=qbarfrom(k=0.25))
0.25
As `q \to 1` and `q \to -1`, `k` rapidly approaches
`1` and `i \infty` respectively::
>>> kfrom(q=0.75)
0.9999999999999899166471767
>>> kfrom(q=-0.75)
(0.0 + 7041781.096692038332790615j)
>>> kfrom(q=1)
1
>>> kfrom(q=-1)
(0.0 + +infj)
"""
if k is not None:
return ctx.convert(k)
if m is not None:
return ctx.sqrt(m)
if tau is not None:
q = ctx.expjpi(tau)
if qbar is not None:
q = ctx.sqrt(qbar)
if q == 1:
return q
if q == -1:
return ctx.mpc(0,'inf')
return (ctx.jtheta(2,0,q)/ctx.jtheta(3,0,q))**2
@defun_wrapped
def mfrom(ctx, q=None, m=None, k=None, tau=None, qbar=None):
r"""
Returns the elliptic parameter `m`, given any of
`q, m, k, \tau, \bar{q}`::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> mfrom(m=0.25)
0.25
>>> mfrom(q=qfrom(m=0.25))
0.25
>>> mfrom(k=kfrom(m=0.25))
0.25
>>> mfrom(tau=taufrom(m=0.25))
(0.25 + 0.0j)
>>> mfrom(qbar=qbarfrom(m=0.25))
0.25
As `q \to 1` and `q \to -1`, `m` rapidly approaches
`1` and `-\infty` respectively::
>>> mfrom(q=0.75)
0.9999999999999798332943533
>>> mfrom(q=-0.75)
-49586681013729.32611558353
>>> mfrom(q=1)
1.0
>>> mfrom(q=-1)
-inf
The inverse nome as a function of `q` has an integer
Taylor series expansion::
>>> taylor(lambda q: mfrom(q), 0, 7)
[0.0, 16.0, -128.0, 704.0, -3072.0, 11488.0, -38400.0, 117632.0]
"""
if m is not None:
return m
if k is not None:
return k**2
if tau is not None:
q = ctx.expjpi(tau)
if qbar is not None:
q = ctx.sqrt(qbar)
if q == 1:
return ctx.convert(q)
if q == -1:
return q*ctx.inf
v = (ctx.jtheta(2,0,q)/ctx.jtheta(3,0,q))**4
if ctx._is_real_type(q) and q < 0:
v = v.real
return v
jacobi_spec = {
'sn' : ([3],[2],[1],[4], 'sin', 'tanh'),
'cn' : ([4],[2],[2],[4], 'cos', 'sech'),
'dn' : ([4],[3],[3],[4], '1', 'sech'),
'ns' : ([2],[3],[4],[1], 'csc', 'coth'),
'nc' : ([2],[4],[4],[2], 'sec', 'cosh'),
'nd' : ([3],[4],[4],[3], '1', 'cosh'),
'sc' : ([3],[4],[1],[2], 'tan', 'sinh'),
'sd' : ([3,3],[2,4],[1],[3], 'sin', 'sinh'),
'cd' : ([3],[2],[2],[3], 'cos', '1'),
'cs' : ([4],[3],[2],[1], 'cot', 'csch'),
'dc' : ([2],[3],[3],[2], 'sec', '1'),
'ds' : ([2,4],[3,3],[3],[1], 'csc', 'csch'),
'cc' : None,
'ss' : None,
'nn' : None,
'dd' : None
}
@defun
def ellipfun(ctx, kind, u=None, m=None, q=None, k=None, tau=None):
try:
S = jacobi_spec[kind]
except KeyError:
raise ValueError("First argument must be a two-character string "
"containing 's', 'c', 'd' or 'n', e.g.: 'sn'")
if u is None:
def f(*args, **kwargs):
return ctx.ellipfun(kind, *args, **kwargs)
f.__name__ = kind
return f
prec = ctx.prec
try:
ctx.prec += 10
u = ctx.convert(u)
q = ctx.qfrom(m=m, q=q, k=k, tau=tau)
if S is None:
v = ctx.one + 0*q*u
elif q == ctx.zero:
if S[4] == '1': v = ctx.one
else: v = getattr(ctx, S[4])(u)
v += 0*q*u
elif q == ctx.one:
if S[5] == '1': v = ctx.one
else: v = getattr(ctx, S[5])(u)
v += 0*q*u
else:
t = u / ctx.jtheta(3, 0, q)**2
v = ctx.one
for a in S[0]: v *= ctx.jtheta(a, 0, q)
for b in S[1]: v /= ctx.jtheta(b, 0, q)
for c in S[2]: v *= ctx.jtheta(c, t, q)
for d in S[3]: v /= ctx.jtheta(d, t, q)
finally:
ctx.prec = prec
return +v
@defun_wrapped
def kleinj(ctx, tau=None, **kwargs):
r"""
Evaluates the Klein j-invariant, which is a modular function defined for
`\tau` in the upper half-plane as
.. math ::
J(\tau) = \frac{g_2^3(\tau)}{g_2^3(\tau) - 27 g_3^2(\tau)}
where `g_2` and `g_3` are the modular invariants of the Weierstrass
elliptic function,
.. math ::
g_2(\tau) = 60 \sum_{(m,n) \in \mathbb{Z}^2 \setminus (0,0)} (m \tau+n)^{-4}
g_3(\tau) = 140 \sum_{(m,n) \in \mathbb{Z}^2 \setminus (0,0)} (m \tau+n)^{-6}.
An alternative, common notation is that of the j-function
`j(\tau) = 1728 J(\tau)`.
**Plots**
.. literalinclude :: /plots/kleinj.py
.. image :: /plots/kleinj.png
.. literalinclude :: /plots/kleinj2.py
.. image :: /plots/kleinj2.png
**Examples**
Verifying the functional equation `J(\tau) = J(\tau+1) = J(-\tau^{-1})`::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> tau = 0.625+0.75*j
>>> tau = 0.625+0.75*j
>>> kleinj(tau)
(-0.1507492166511182267125242 + 0.07595948379084571927228948j)
>>> kleinj(tau+1)
(-0.1507492166511182267125242 + 0.07595948379084571927228948j)
>>> kleinj(-1/tau)
(-0.1507492166511182267125242 + 0.07595948379084571927228946j)
The j-function has a famous Laurent series expansion in terms of the nome
`\bar{q}`, `j(\tau) = \bar{q}^{-1} + 744 + 196884\bar{q} + \ldots`::
>>> mp.dps = 15
>>> taylor(lambda q: 1728*q*kleinj(qbar=q), 0, 5, singular=True)
[1.0, 744.0, 196884.0, 21493760.0, 864299970.0, 20245856256.0]
The j-function admits exact evaluation at special algebraic points
related to the Heegner numbers 1, 2, 3, 7, 11, 19, 43, 67, 163::
>>> @extraprec(10)
... def h(n):
... v = (1+sqrt(n)*j)
... if n > 2:
... v *= 0.5
... return v
...
>>> mp.dps = 25
>>> for n in [1,2,3,7,11,19,43,67,163]:
... n, chop(1728*kleinj(h(n)))
...
(1, 1728.0)
(2, 8000.0)
(3, 0.0)
(7, -3375.0)
(11, -32768.0)
(19, -884736.0)
(43, -884736000.0)
(67, -147197952000.0)
(163, -262537412640768000.0)
Also at other special points, the j-function assumes explicit
algebraic values, e.g.::
>>> chop(1728*kleinj(j*sqrt(5)))
1264538.909475140509320227
>>> identify(cbrt(_)) # note: not simplified
'((100+sqrt(13520))/2)'
>>> (50+26*sqrt(5))**3
1264538.909475140509320227
"""
q = ctx.qfrom(tau=tau, **kwargs)
t2 = ctx.jtheta(2,0,q)
t3 = ctx.jtheta(3,0,q)
t4 = ctx.jtheta(4,0,q)
P = (t2**8 + t3**8 + t4**8)**3
Q = 54*(t2*t3*t4)**8
return P/Q
def RF_calc(ctx, x, y, z, r):
if y == z: return RC_calc(ctx, x, y, r)
if x == z: return RC_calc(ctx, y, x, r)
if x == y: return RC_calc(ctx, z, x, r)
if not (ctx.isnormal(x) and ctx.isnormal(y) and ctx.isnormal(z)):
if ctx.isnan(x) or ctx.isnan(y) or ctx.isnan(z):
return x*y*z
if ctx.isinf(x) or ctx.isinf(y) or ctx.isinf(z):
return ctx.zero
xm,ym,zm = x,y,z
A0 = Am = (x+y+z)/3
Q = ctx.root(3*r, -6) * max(abs(A0-x),abs(A0-y),abs(A0-z))
g = ctx.mpf(0.25)
pow4 = ctx.one
m = 0
while 1:
xs = ctx.sqrt(xm)
ys = ctx.sqrt(ym)
zs = ctx.sqrt(zm)
lm = xs*ys + xs*zs + ys*zs
Am1 = (Am+lm)*g
xm, ym, zm = (xm+lm)*g, (ym+lm)*g, (zm+lm)*g
if pow4 * Q < abs(Am):
break
Am = Am1
m += 1
pow4 *= g
t = pow4/Am
X = (A0-x)*t
Y = (A0-y)*t
Z = -X-Y
E2 = X*Y-Z**2
E3 = X*Y*Z
return ctx.power(Am,-0.5) * (9240-924*E2+385*E2**2+660*E3-630*E2*E3)/9240
def RC_calc(ctx, x, y, r, pv=True):
if not (ctx.isnormal(x) and ctx.isnormal(y)):
if ctx.isinf(x) or ctx.isinf(y):
return 1/(x*y)
if y == 0:
return ctx.inf
if x == 0:
return ctx.pi / ctx.sqrt(y) / 2
raise ValueError
# Cauchy principal value
if pv and ctx._im(y) == 0 and ctx._re(y) < 0:
return ctx.sqrt(x/(x-y)) * RC_calc(ctx, x-y, -y, r)
if x == y:
return 1/ctx.sqrt(x)
extraprec = 2*max(0,-ctx.mag(x-y)+ctx.mag(x))
ctx.prec += extraprec
if ctx._is_real_type(x) and ctx._is_real_type(y):
x = ctx._re(x)
y = ctx._re(y)
a = ctx.sqrt(x/y)
if x < y:
b = ctx.sqrt(y-x)
v = ctx.acos(a)/b
else:
b = ctx.sqrt(x-y)
v = ctx.acosh(a)/b
else:
sx = ctx.sqrt(x)
sy = ctx.sqrt(y)
v = ctx.acos(sx/sy)/(ctx.sqrt((1-x/y))*sy)
ctx.prec -= extraprec
return v
def RJ_calc(ctx, x, y, z, p, r):
if not (ctx.isnormal(x) and ctx.isnormal(y) and \
ctx.isnormal(z) and ctx.isnormal(p)):
if ctx.isnan(x) or ctx.isnan(y) or ctx.isnan(z) or ctx.isnan(p):
return x*y*z
if ctx.isinf(x) or ctx.isinf(y) or ctx.isinf(z) or ctx.isinf(p):
return ctx.zero
if not p:
return ctx.inf
xm,ym,zm,pm = x,y,z,p
A0 = Am = (x + y + z + 2*p)/5
delta = (p-x)*(p-y)*(p-z)
Q = ctx.root(0.25*r, -6) * max(abs(A0-x),abs(A0-y),abs(A0-z),abs(A0-p))
m = 0
g = ctx.mpf(0.25)
pow4 = ctx.one
S = 0
while 1:
sx = ctx.sqrt(xm)
sy = ctx.sqrt(ym)
sz = ctx.sqrt(zm)
sp = ctx.sqrt(pm)
lm = sx*sy + sx*sz + sy*sz
Am1 = (Am+lm)*g
xm = (xm+lm)*g; ym = (ym+lm)*g; zm = (zm+lm)*g; pm = (pm+lm)*g
dm = (sp+sx) * (sp+sy) * (sp+sz)
em = delta * ctx.power(4, -3*m) / dm**2
if pow4 * Q < abs(Am):
break
T = RC_calc(ctx, ctx.one, ctx.one+em, r) * pow4 / dm
S += T
pow4 *= g
m += 1
Am = Am1
t = ctx.ldexp(1,-2*m) / Am
X = (A0-x)*t
Y = (A0-y)*t
Z = (A0-z)*t
P = (-X-Y-Z)/2
E2 = X*Y + X*Z + Y*Z - 3*P**2
E3 = X*Y*Z + 2*E2*P + 4*P**3
E4 = (2*X*Y*Z + E2*P + 3*P**3)*P
E5 = X*Y*Z*P**2
P = 24024 - 5148*E2 + 2457*E2**2 + 4004*E3 - 4158*E2*E3 - 3276*E4 + 2772*E5
Q = 24024
v1 = g**m * ctx.power(Am, -1.5) * P/Q
v2 = 6*S
return v1 + v2
@defun
def elliprf(ctx, x, y, z):
r"""
Evaluates the Carlson symmetric elliptic integral of the first kind
.. math ::
R_F(x,y,z) = \frac{1}{2}
\int_0^{\infty} \frac{dt}{\sqrt{(t+x)(t+y)(t+z)}}
which is defined for `x,y,z \notin (-\infty,0)`, and with
at most one of `x,y,z` being zero.
For real `x,y,z \ge 0`, the principal square root is taken in the integrand.
For complex `x,y,z`, the principal square root is taken as `t \to \infty`
and as `t \to 0` non-principal branches are chosen as necessary so as to
make the integrand continuous.
**Examples**
Some basic values and limits::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> elliprf(0,1,1); pi/2
1.570796326794896619231322
1.570796326794896619231322
>>> elliprf(0,1,inf)
0.0
>>> elliprf(1,1,1)
1.0
>>> elliprf(2,2,2)**2
0.5
>>> elliprf(1,0,0); elliprf(0,0,1); elliprf(0,1,0); elliprf(0,0,0)
+inf
+inf
+inf
+inf
Representing complete elliptic integrals in terms of `R_F`::
>>> m = mpf(0.75)
>>> ellipk(m); elliprf(0,1-m,1)
2.156515647499643235438675
2.156515647499643235438675
>>> ellipe(m); elliprf(0,1-m,1)-m*elliprd(0,1-m,1)/3
1.211056027568459524803563
1.211056027568459524803563
Some symmetries and argument transformations::
>>> x,y,z = 2,3,4
>>> elliprf(x,y,z); elliprf(y,x,z); elliprf(z,y,x)
0.5840828416771517066928492
0.5840828416771517066928492
0.5840828416771517066928492
>>> k = mpf(100000)
>>> elliprf(k*x,k*y,k*z); k**(-0.5) * elliprf(x,y,z)
0.001847032121923321253219284
0.001847032121923321253219284
>>> l = sqrt(x*y) + sqrt(y*z) + sqrt(z*x)
>>> elliprf(x,y,z); 2*elliprf(x+l,y+l,z+l)
0.5840828416771517066928492
0.5840828416771517066928492
>>> elliprf((x+l)/4,(y+l)/4,(z+l)/4)
0.5840828416771517066928492
Comparing with numerical integration::
>>> x,y,z = 2,3,4
>>> elliprf(x,y,z)
0.5840828416771517066928492
>>> f = lambda t: 0.5*((t+x)*(t+y)*(t+z))**(-0.5)
>>> q = extradps(25)(quad)
>>> q(f, [0,inf])
0.5840828416771517066928492
With the following arguments, the square root in the integrand becomes
discontinuous at `t = 1/2` if the principal branch is used. To obtain
the right value, `-\sqrt{r}` must be taken instead of `\sqrt{r}`
on `t \in (0, 1/2)`::
>>> x,y,z = j-1,j,0
>>> elliprf(x,y,z)
(0.7961258658423391329305694 - 1.213856669836495986430094j)
>>> -q(f, [0,0.5]) + q(f, [0.5,inf])
(0.7961258658423391329305694 - 1.213856669836495986430094j)
The so-called *first lemniscate constant*, a transcendental number::
>>> elliprf(0,1,2)
1.31102877714605990523242
>>> extradps(25)(quad)(lambda t: 1/sqrt(1-t**4), [0,1])
1.31102877714605990523242
>>> gamma('1/4')**2/(4*sqrt(2*pi))
1.31102877714605990523242
**References**
1. [Carlson]_
2. [DLMF]_ Chapter 19. Elliptic Integrals
"""
x = ctx.convert(x)
y = ctx.convert(y)
z = ctx.convert(z)
prec = ctx.prec
try:
ctx.prec += 20
tol = ctx.eps * 2**10
v = RF_calc(ctx, x, y, z, tol)
finally:
ctx.prec = prec
return +v
@defun
def elliprc(ctx, x, y, pv=True):
r"""
Evaluates the degenerate Carlson symmetric elliptic integral
of the first kind
.. math ::
R_C(x,y) = R_F(x,y,y) =
\frac{1}{2} \int_0^{\infty} \frac{dt}{(t+y) \sqrt{(t+x)}}.
If `y \in (-\infty,0)`, either a value defined by continuity,
or with *pv=True* the Cauchy principal value, can be computed.
If `x \ge 0, y > 0`, the value can be expressed in terms of
elementary functions as
.. math ::
R_C(x,y) =
\begin{cases}
\dfrac{1}{\sqrt{y-x}}
\cos^{-1}\left(\sqrt{\dfrac{x}{y}}\right), & x < y \\
\dfrac{1}{\sqrt{y}}, & x = y \\
\dfrac{1}{\sqrt{x-y}}
\cosh^{-1}\left(\sqrt{\dfrac{x}{y}}\right), & x > y \\
\end{cases}.
**Examples**
Some special values and limits::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> elliprc(1,2)*4; elliprc(0,1)*2; +pi
3.141592653589793238462643
3.141592653589793238462643
3.141592653589793238462643
>>> elliprc(1,0)
+inf
>>> elliprc(5,5)**2
0.2
>>> elliprc(1,inf); elliprc(inf,1); elliprc(inf,inf)
0.0
0.0
0.0
Comparing with the elementary closed-form solution::
>>> elliprc('1/3', '1/5'); sqrt(7.5)*acosh(sqrt('5/3'))
2.041630778983498390751238
2.041630778983498390751238
>>> elliprc('1/5', '1/3'); sqrt(7.5)*acos(sqrt('3/5'))
1.875180765206547065111085
1.875180765206547065111085
Comparing with numerical integration::
>>> q = extradps(25)(quad)
>>> elliprc(2, -3, pv=True)
0.3333969101113672670749334
>>> elliprc(2, -3, pv=False)
(0.3333969101113672670749334 + 0.7024814731040726393156375j)
>>> 0.5*q(lambda t: 1/(sqrt(t+2)*(t-3)), [0,3-j,6,inf])
(0.3333969101113672670749334 + 0.7024814731040726393156375j)
"""
x = ctx.convert(x)
y = ctx.convert(y)
prec = ctx.prec
try:
ctx.prec += 20
tol = ctx.eps * 2**10
v = RC_calc(ctx, x, y, tol, pv)
finally:
ctx.prec = prec
return +v
@defun
def elliprj(ctx, x, y, z, p):
r"""
Evaluates the Carlson symmetric elliptic integral of the third kind
.. math ::
R_J(x,y,z,p) = \frac{3}{2}
\int_0^{\infty} \frac{dt}{(t+p)\sqrt{(t+x)(t+y)(t+z)}}.
Like :func:`~mpmath.elliprf`, the branch of the square root in the integrand
is defined so as to be continuous along the path of integration for
complex values of the arguments.
**Examples**
Some values and limits::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> elliprj(1,1,1,1)
1.0
>>> elliprj(2,2,2,2); 1/(2*sqrt(2))
0.3535533905932737622004222
0.3535533905932737622004222
>>> elliprj(0,1,2,2)
1.067937989667395702268688
>>> 3*(2*gamma('5/4')**2-pi**2/gamma('1/4')**2)/(sqrt(2*pi))
1.067937989667395702268688
>>> elliprj(0,1,1,2); 3*pi*(2-sqrt(2))/4
1.380226776765915172432054
1.380226776765915172432054
>>> elliprj(1,3,2,0); elliprj(0,1,1,0); elliprj(0,0,0,0)
+inf
+inf
+inf
>>> elliprj(1,inf,1,0); elliprj(1,1,1,inf)
0.0
0.0
>>> chop(elliprj(1+j, 1-j, 1, 1))
0.8505007163686739432927844
Scale transformation::
>>> x,y,z,p = 2,3,4,5
>>> k = mpf(100000)
>>> elliprj(k*x,k*y,k*z,k*p); k**(-1.5)*elliprj(x,y,z,p)
4.521291677592745527851168e-9
4.521291677592745527851168e-9
Comparing with numerical integration::
>>> elliprj(1,2,3,4)
0.2398480997495677621758617
>>> f = lambda t: 1/((t+4)*sqrt((t+1)*(t+2)*(t+3)))
>>> 1.5*quad(f, [0,inf])
0.2398480997495677621758617
>>> elliprj(1,2+1j,3,4-2j)
(0.216888906014633498739952 + 0.04081912627366673332369512j)
>>> f = lambda t: 1/((t+4-2j)*sqrt((t+1)*(t+2+1j)*(t+3)))
>>> 1.5*quad(f, [0,inf])
(0.216888906014633498739952 + 0.04081912627366673332369511j)
"""
x = ctx.convert(x)
y = ctx.convert(y)
z = ctx.convert(z)
p = ctx.convert(p)
prec = ctx.prec
try:
ctx.prec += 20
tol = ctx.eps * 2**10
v = RJ_calc(ctx, x, y, z, p, tol)
finally:
ctx.prec = prec
return +v
@defun
def elliprd(ctx, x, y, z):
r"""
Evaluates the degenerate Carlson symmetric elliptic integral
of the third kind or Carlson elliptic integral of the
second kind `R_D(x,y,z) = R_J(x,y,z,z)`.
See :func:`~mpmath.elliprj` for additional information.
**Examples**
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> elliprd(1,2,3)
0.2904602810289906442326534
>>> elliprj(1,2,3,3)
0.2904602810289906442326534
The so-called *second lemniscate constant*, a transcendental number::
>>> elliprd(0,2,1)/3
0.5990701173677961037199612
>>> extradps(25)(quad)(lambda t: t**2/sqrt(1-t**4), [0,1])
0.5990701173677961037199612
>>> gamma('3/4')**2/sqrt(2*pi)
0.5990701173677961037199612
"""
return ctx.elliprj(x,y,z,z)
@defun
def elliprg(ctx, x, y, z):
r"""
Evaluates the Carlson completely symmetric elliptic integral
of the second kind
.. math ::
R_G(x,y,z) = \frac{1}{4} \int_0^{\infty}
\frac{t}{\sqrt{(t+x)(t+y)(t+z)}}
\left( \frac{x}{t+x} + \frac{y}{t+y} + \frac{z}{t+z}\right) dt.
**Examples**
Evaluation for real and complex arguments::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> elliprg(0,1,1)*4; +pi
3.141592653589793238462643
3.141592653589793238462643
>>> elliprg(0,0.5,1)
0.6753219405238377512600874
>>> chop(elliprg(1+j, 1-j, 2))
1.172431327676416604532822
A double integral that can be evaluated in terms of `R_G`::
>>> x,y,z = 2,3,4
>>> def f(t,u):
... st = fp.sin(t); ct = fp.cos(t)
... su = fp.sin(u); cu = fp.cos(u)
... return (x*(st*cu)**2 + y*(st*su)**2 + z*ct**2)**0.5 * st
...
>>> nprint(mpf(fp.quad(f, [0,fp.pi], [0,2*fp.pi])/(4*fp.pi)), 13)
1.725503028069
>>> nprint(elliprg(x,y,z), 13)
1.725503028069
"""
x = ctx.convert(x)
y = ctx.convert(y)
z = ctx.convert(z)
zeros = (not x) + (not y) + (not z)
if zeros == 3:
return (x+y+z)*0
if zeros == 2:
if x: return 0.5*ctx.sqrt(x)
if y: return 0.5*ctx.sqrt(y)
return 0.5*ctx.sqrt(z)
if zeros == 1:
if not z:
x, z = z, x
def terms():
T1 = 0.5*z*ctx.elliprf(x,y,z)
T2 = -0.5*(x-z)*(y-z)*ctx.elliprd(x,y,z)/3
T3 = 0.5*ctx.sqrt(x)*ctx.sqrt(y)/ctx.sqrt(z)
return T1,T2,T3
return ctx.sum_accurately(terms)
@defun_wrapped
def ellipf(ctx, phi, m):
r"""
Evaluates the Legendre incomplete elliptic integral of the first kind
.. math ::
F(\phi,m) = \int_0^{\phi} \frac{dt}{\sqrt{1-m \sin^2 t}}
or equivalently
.. math ::
F(\phi,m) = \int_0^{\sin \phi}
\frac{dt}{\left(\sqrt{1-t^2}\right)\left(\sqrt{1-mt^2}\right)}.
The function reduces to a complete elliptic integral of the first kind
(see :func:`~mpmath.ellipk`) when `\phi = \frac{\pi}{2}`; that is,
.. math ::
F\left(\frac{\pi}{2}, m\right) = K(m).
In the defining integral, it is assumed that the principal branch
of the square root is taken and that the path of integration avoids
crossing any branch cuts. Outside `-\pi/2 \le \Re(\phi) \le \pi/2`,
the function extends quasi-periodically as
.. math ::
F(\phi + n \pi, m) = 2 n K(m) + F(\phi,m), n \in \mathbb{Z}.
**Plots**
.. literalinclude :: /plots/ellipf.py
.. image :: /plots/ellipf.png
**Examples**
Basic values and limits::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> ellipf(0,1)
0.0
>>> ellipf(0,0)
0.0
>>> ellipf(1,0); ellipf(2+3j,0)
1.0
(2.0 + 3.0j)
>>> ellipf(1,1); log(sec(1)+tan(1))
1.226191170883517070813061
1.226191170883517070813061
>>> ellipf(pi/2, -0.5); ellipk(-0.5)
1.415737208425956198892166
1.415737208425956198892166
>>> ellipf(pi/2+eps, 1); ellipf(-pi/2-eps, 1)
+inf
+inf
>>> ellipf(1.5, 1)
3.340677542798311003320813
Comparing with numerical integration::
>>> z,m = 0.5, 1.25
>>> ellipf(z,m)
0.5287219202206327872978255
>>> quad(lambda t: (1-m*sin(t)**2)**(-0.5), [0,z])
0.5287219202206327872978255
The arguments may be complex numbers::
>>> ellipf(3j, 0.5)
(0.0 + 1.713602407841590234804143j)
>>> ellipf(3+4j, 5-6j)
(1.269131241950351323305741 - 0.3561052815014558335412538j)
>>> z,m = 2+3j, 1.25
>>> k = 1011
>>> ellipf(z+pi*k,m); ellipf(z,m) + 2*k*ellipk(m)
(4086.184383622179764082821 - 3003.003538923749396546871j)
(4086.184383622179764082821 - 3003.003538923749396546871j)
For `|\Re(z)| < \pi/2`, the function can be expressed as a
hypergeometric series of two variables
(see :func:`~mpmath.appellf1`)::
>>> z,m = 0.5, 0.25
>>> ellipf(z,m)
0.5050887275786480788831083
>>> sin(z)*appellf1(0.5,0.5,0.5,1.5,sin(z)**2,m*sin(z)**2)
0.5050887275786480788831083
"""
z = phi
if not (ctx.isnormal(z) and ctx.isnormal(m)):
if m == 0:
return z + m
if z == 0:
return z * m
if m == ctx.inf or m == ctx.ninf: return z/m
raise ValueError
x = z.real
ctx.prec += max(0, ctx.mag(x))
pi = +ctx.pi
away = abs(x) > pi/2
if m == 1:
if away:
return ctx.inf
if away:
d = ctx.nint(x/pi)
z = z-pi*d
P = 2*d*ctx.ellipk(m)
else:
P = 0
c, s = ctx.cos_sin(z)
return s * ctx.elliprf(c**2, 1-m*s**2, 1) + P
@defun_wrapped
def ellipe(ctx, *args):
r"""
Called with a single argument `m`, evaluates the Legendre complete
elliptic integral of the second kind, `E(m)`, defined by
.. math :: E(m) = \int_0^{\pi/2} \sqrt{1-m \sin^2 t} \, dt \,=\,
\frac{\pi}{2}
\,_2F_1\left(\frac{1}{2}, -\frac{1}{2}, 1, m\right).
Called with two arguments `\phi, m`, evaluates the incomplete elliptic
integral of the second kind
.. math ::
E(\phi,m) = \int_0^{\phi} \sqrt{1-m \sin^2 t} \, dt =
\int_0^{\sin z}
\frac{\sqrt{1-mt^2}}{\sqrt{1-t^2}} \, dt.
The incomplete integral reduces to a complete integral when
`\phi = \frac{\pi}{2}`; that is,
.. math ::
E\left(\frac{\pi}{2}, m\right) = E(m).
In the defining integral, it is assumed that the principal branch
of the square root is taken and that the path of integration avoids
crossing any branch cuts. Outside `-\pi/2 \le \Re(z) \le \pi/2`,
the function extends quasi-periodically as
.. math ::
E(\phi + n \pi, m) = 2 n E(m) + F(\phi,m), n \in \mathbb{Z}.
**Plots**
.. literalinclude :: /plots/ellipe.py
.. image :: /plots/ellipe.png
**Examples for the complete integral**
Basic values and limits::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> ellipe(0)
1.570796326794896619231322
>>> ellipe(1)
1.0
>>> ellipe(-1)
1.910098894513856008952381
>>> ellipe(2)
(0.5990701173677961037199612 + 0.5990701173677961037199612j)
>>> ellipe(inf)
(0.0 + +infj)
>>> ellipe(-inf)
+inf
Verifying the defining integral and hypergeometric
representation::
>>> ellipe(0.5)
1.350643881047675502520175
>>> quad(lambda t: sqrt(1-0.5*sin(t)**2), [0, pi/2])
1.350643881047675502520175
>>> pi/2*hyp2f1(0.5,-0.5,1,0.5)
1.350643881047675502520175
Evaluation is supported for arbitrary complex `m`::
>>> ellipe(0.5+0.25j)
(1.360868682163129682716687 - 0.1238733442561786843557315j)
>>> ellipe(3+4j)
(1.499553520933346954333612 - 1.577879007912758274533309j)
A definite integral::
>>> quad(ellipe, [0,1])
1.333333333333333333333333
**Examples for the incomplete integral**
Basic values and limits::
>>> ellipe(0,1)
0.0
>>> ellipe(0,0)
0.0
>>> ellipe(1,0)
1.0
>>> ellipe(2+3j,0)
(2.0 + 3.0j)
>>> ellipe(1,1); sin(1)
0.8414709848078965066525023
0.8414709848078965066525023
>>> ellipe(pi/2, -0.5); ellipe(-0.5)
1.751771275694817862026502
1.751771275694817862026502
>>> ellipe(pi/2, 1); ellipe(-pi/2, 1)
1.0
-1.0
>>> ellipe(1.5, 1)
0.9974949866040544309417234
Comparing with numerical integration::
>>> z,m = 0.5, 1.25
>>> ellipe(z,m)
0.4740152182652628394264449
>>> quad(lambda t: sqrt(1-m*sin(t)**2), [0,z])
0.4740152182652628394264449
The arguments may be complex numbers::
>>> ellipe(3j, 0.5)
(0.0 + 7.551991234890371873502105j)
>>> ellipe(3+4j, 5-6j)
(24.15299022574220502424466 + 75.2503670480325997418156j)
>>> k = 35
>>> z,m = 2+3j, 1.25
>>> ellipe(z+pi*k,m); ellipe(z,m) + 2*k*ellipe(m)
(48.30138799412005235090766 + 17.47255216721987688224357j)
(48.30138799412005235090766 + 17.47255216721987688224357j)
For `|\Re(z)| < \pi/2`, the function can be expressed as a
hypergeometric series of two variables
(see :func:`~mpmath.appellf1`)::
>>> z,m = 0.5, 0.25
>>> ellipe(z,m)
0.4950017030164151928870375
>>> sin(z)*appellf1(0.5,0.5,-0.5,1.5,sin(z)**2,m*sin(z)**2)
0.4950017030164151928870376
"""
if len(args) == 1:
return ctx._ellipe(args[0])
else:
phi, m = args
z = phi
if not (ctx.isnormal(z) and ctx.isnormal(m)):
if m == 0:
return z + m
if z == 0:
return z * m
if m == ctx.inf or m == ctx.ninf:
return ctx.inf
raise ValueError
x = z.real
ctx.prec += max(0, ctx.mag(x))
pi = +ctx.pi
away = abs(x) > pi/2
if away:
d = ctx.nint(x/pi)
z = z-pi*d
P = 2*d*ctx.ellipe(m)
else:
P = 0
def terms():
c, s = ctx.cos_sin(z)
x = c**2
y = 1-m*s**2
RF = ctx.elliprf(x, y, 1)
RD = ctx.elliprd(x, y, 1)
return s*RF, -m*s**3*RD/3
return ctx.sum_accurately(terms) + P
@defun_wrapped
def ellippi(ctx, *args):
r"""
Called with three arguments `n, \phi, m`, evaluates the Legendre
incomplete elliptic integral of the third kind
.. math ::
\Pi(n; \phi, m) = \int_0^{\phi}
\frac{dt}{(1-n \sin^2 t) \sqrt{1-m \sin^2 t}} =
\int_0^{\sin \phi}
\frac{dt}{(1-nt^2) \sqrt{1-t^2} \sqrt{1-mt^2}}.
Called with two arguments `n, m`, evaluates the complete
elliptic integral of the third kind
`\Pi(n,m) = \Pi(n; \frac{\pi}{2},m)`.
In the defining integral, it is assumed that the principal branch
of the square root is taken and that the path of integration avoids
crossing any branch cuts. Outside `-\pi/2 \le \Re(\phi) \le \pi/2`,
the function extends quasi-periodically as
.. math ::
\Pi(n,\phi+k\pi,m) = 2k\Pi(n,m) + \Pi(n,\phi,m), k \in \mathbb{Z}.
**Plots**
.. literalinclude :: /plots/ellippi.py
.. image :: /plots/ellippi.png
**Examples for the complete integral**
Some basic values and limits::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> ellippi(0,-5); ellipk(-5)
0.9555039270640439337379334
0.9555039270640439337379334
>>> ellippi(inf,2)
0.0
>>> ellippi(2,inf)
0.0
>>> abs(ellippi(1,5))
+inf
>>> abs(ellippi(0.25,1))
+inf
Evaluation in terms of simpler functions::
>>> ellippi(0.25,0.25); ellipe(0.25)/(1-0.25)
1.956616279119236207279727
1.956616279119236207279727
>>> ellippi(3,0); pi/(2*sqrt(-2))
(0.0 - 1.11072073453959156175397j)
(0.0 - 1.11072073453959156175397j)
>>> ellippi(-3,0); pi/(2*sqrt(4))
0.7853981633974483096156609
0.7853981633974483096156609
**Examples for the incomplete integral**
Basic values and limits::
>>> ellippi(0.25,-0.5); ellippi(0.25,pi/2,-0.5)
1.622944760954741603710555
1.622944760954741603710555
>>> ellippi(1,0,1)
0.0
>>> ellippi(inf,0,1)
0.0
>>> ellippi(0,0.25,0.5); ellipf(0.25,0.5)
0.2513040086544925794134591
0.2513040086544925794134591
>>> ellippi(1,1,1); (log(sec(1)+tan(1))+sec(1)*tan(1))/2
2.054332933256248668692452
2.054332933256248668692452
>>> ellippi(0.25, 53*pi/2, 0.75); 53*ellippi(0.25,0.75)
135.240868757890840755058
135.240868757890840755058
>>> ellippi(0.5,pi/4,0.5); 2*ellipe(pi/4,0.5)-1/sqrt(3)
0.9190227391656969903987269
0.9190227391656969903987269
Complex arguments are supported::
>>> ellippi(0.5, 5+6j-2*pi, -7-8j)
(-0.3612856620076747660410167 + 0.5217735339984807829755815j)
Some degenerate cases::
>>> ellippi(1,1)
+inf
>>> ellippi(1,0)
+inf
>>> ellippi(1,2,0)
+inf
>>> ellippi(1,2,1)
+inf
>>> ellippi(1,0,1)
0.0
"""
if len(args) == 2:
n, m = args
complete = True
z = phi = ctx.pi/2
else:
n, phi, m = args
complete = False
z = phi
if not (ctx.isnormal(n) and ctx.isnormal(z) and ctx.isnormal(m)):
if ctx.isnan(n) or ctx.isnan(z) or ctx.isnan(m):
raise ValueError
if complete:
if m == 0:
if n == 1:
return ctx.inf
return ctx.pi/(2*ctx.sqrt(1-n))
if n == 0: return ctx.ellipk(m)
if ctx.isinf(n) or ctx.isinf(m): return ctx.zero
else:
if z == 0: return z
if ctx.isinf(n): return ctx.zero
if ctx.isinf(m): return ctx.zero
if ctx.isinf(n) or ctx.isinf(z) or ctx.isinf(m):
raise ValueError
if complete:
if m == 1:
if n == 1:
return ctx.inf
return -ctx.inf/ctx.sign(n-1)
away = False
else:
x = z.real
ctx.prec += max(0, ctx.mag(x))
pi = +ctx.pi
away = abs(x) > pi/2
if away:
d = ctx.nint(x/pi)
z = z-pi*d
P = 2*d*ctx.ellippi(n,m)
if ctx.isinf(P):
return ctx.inf
else:
P = 0
def terms():
if complete:
c, s = ctx.zero, ctx.one
else:
c, s = ctx.cos_sin(z)
x = c**2
y = 1-m*s**2
RF = ctx.elliprf(x, y, 1)
RJ = ctx.elliprj(x, y, 1, 1-n*s**2)
return s*RF, n*s**3*RJ/3
return ctx.sum_accurately(terms) + P
| 39,030 | 27.699265 | 86 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/functions/bessel.py
|
from .functions import defun, defun_wrapped
@defun
def j0(ctx, x):
"""Computes the Bessel function `J_0(x)`. See :func:`~mpmath.besselj`."""
return ctx.besselj(0, x)
@defun
def j1(ctx, x):
"""Computes the Bessel function `J_1(x)`. See :func:`~mpmath.besselj`."""
return ctx.besselj(1, x)
@defun
def besselj(ctx, n, z, derivative=0, **kwargs):
if type(n) is int:
n_isint = True
else:
n = ctx.convert(n)
n_isint = ctx.isint(n)
if n_isint:
n = int(ctx._re(n))
if n_isint and n < 0:
return (-1)**n * ctx.besselj(-n, z, derivative, **kwargs)
z = ctx.convert(z)
M = ctx.mag(z)
if derivative:
d = ctx.convert(derivative)
# TODO: the integer special-casing shouldn't be necessary.
# However, the hypergeometric series gets inaccurate for large d
# because of inaccurate pole cancellation at a pole far from
# zero (needs to be fixed in hypercomb or hypsum)
if ctx.isint(d) and d >= 0:
d = int(d)
orig = ctx.prec
try:
ctx.prec += 15
v = ctx.fsum((-1)**k * ctx.binomial(d,k) * ctx.besselj(2*k+n-d,z)
for k in range(d+1))
finally:
ctx.prec = orig
v *= ctx.mpf(2)**(-d)
else:
def h(n,d):
r = ctx.fmul(ctx.fmul(z, z, prec=ctx.prec+M), -0.25, exact=True)
B = [0.5*(n-d+1), 0.5*(n-d+2)]
T = [([2,ctx.pi,z],[d-2*n,0.5,n-d],[],B,[(n+1)*0.5,(n+2)*0.5],B+[n+1],r)]
return T
v = ctx.hypercomb(h, [n,d], **kwargs)
else:
# Fast case: J_n(x), n int, appropriate magnitude for fixed-point calculation
if (not derivative) and n_isint and abs(M) < 10 and abs(n) < 20:
try:
return ctx._besselj(n, z)
except NotImplementedError:
pass
if not z:
if not n:
v = ctx.one + n+z
elif ctx.re(n) > 0:
v = n*z
else:
v = ctx.inf + z + n
else:
#v = 0
orig = ctx.prec
try:
# XXX: workaround for accuracy in low level hypergeometric series
# when alternating, large arguments
ctx.prec += min(3*abs(M), ctx.prec)
w = ctx.fmul(z, 0.5, exact=True)
def h(n):
r = ctx.fneg(ctx.fmul(w, w, prec=max(0,ctx.prec+M)), exact=True)
return [([w], [n], [], [n+1], [], [n+1], r)]
v = ctx.hypercomb(h, [n], **kwargs)
finally:
ctx.prec = orig
v = +v
return v
@defun
def besseli(ctx, n, z, derivative=0, **kwargs):
n = ctx.convert(n)
z = ctx.convert(z)
if not z:
if derivative:
raise ValueError
if not n:
# I(0,0) = 1
return 1+n+z
if ctx.isint(n):
return 0*(n+z)
r = ctx.re(n)
if r == 0:
return ctx.nan*(n+z)
elif r > 0:
return 0*(n+z)
else:
return ctx.inf+(n+z)
M = ctx.mag(z)
if derivative:
d = ctx.convert(derivative)
def h(n,d):
r = ctx.fmul(ctx.fmul(z, z, prec=ctx.prec+M), 0.25, exact=True)
B = [0.5*(n-d+1), 0.5*(n-d+2), n+1]
T = [([2,ctx.pi,z],[d-2*n,0.5,n-d],[n+1],B,[(n+1)*0.5,(n+2)*0.5],B,r)]
return T
v = ctx.hypercomb(h, [n,d], **kwargs)
else:
def h(n):
w = ctx.fmul(z, 0.5, exact=True)
r = ctx.fmul(w, w, prec=max(0,ctx.prec+M))
return [([w], [n], [], [n+1], [], [n+1], r)]
v = ctx.hypercomb(h, [n], **kwargs)
return v
@defun_wrapped
def bessely(ctx, n, z, derivative=0, **kwargs):
if not z:
if derivative:
# Not implemented
raise ValueError
if not n:
# ~ log(z/2)
return -ctx.inf + (n+z)
if ctx.im(n):
return ctx.nan * (n+z)
r = ctx.re(n)
q = n+0.5
if ctx.isint(q):
if n > 0:
return -ctx.inf + (n+z)
else:
return 0 * (n+z)
if r < 0 and int(ctx.floor(q)) % 2:
return ctx.inf + (n+z)
else:
return ctx.ninf + (n+z)
# XXX: use hypercomb
ctx.prec += 10
m, d = ctx.nint_distance(n)
if d < -ctx.prec:
h = +ctx.eps
ctx.prec *= 2
n += h
elif d < 0:
ctx.prec -= d
# TODO: avoid cancellation for imaginary arguments
cos, sin = ctx.cospi_sinpi(n)
return (ctx.besselj(n,z,derivative,**kwargs)*cos - \
ctx.besselj(-n,z,derivative,**kwargs))/sin
@defun_wrapped
def besselk(ctx, n, z, **kwargs):
if not z:
return ctx.inf
M = ctx.mag(z)
if M < 1:
# Represent as limit definition
def h(n):
r = (z/2)**2
T1 = [z, 2], [-n, n-1], [n], [], [], [1-n], r
T2 = [z, 2], [n, -n-1], [-n], [], [], [1+n], r
return T1, T2
# We could use the limit definition always, but it leads
# to very bad cancellation (of exponentially large terms)
# for large real z
# Instead represent in terms of 2F0
else:
ctx.prec += M
def h(n):
return [([ctx.pi/2, z, ctx.exp(-z)], [0.5,-0.5,1], [], [], \
[n+0.5, 0.5-n], [], -1/(2*z))]
return ctx.hypercomb(h, [n], **kwargs)
@defun_wrapped
def hankel1(ctx,n,x,**kwargs):
return ctx.besselj(n,x,**kwargs) + ctx.j*ctx.bessely(n,x,**kwargs)
@defun_wrapped
def hankel2(ctx,n,x,**kwargs):
return ctx.besselj(n,x,**kwargs) - ctx.j*ctx.bessely(n,x,**kwargs)
@defun_wrapped
def whitm(ctx,k,m,z,**kwargs):
if z == 0:
# M(k,m,z) = 0^(1/2+m)
if ctx.re(m) > -0.5:
return z
elif ctx.re(m) < -0.5:
return ctx.inf + z
else:
return ctx.nan * z
x = ctx.fmul(-0.5, z, exact=True)
y = 0.5+m
return ctx.exp(x) * z**y * ctx.hyp1f1(y-k, 1+2*m, z, **kwargs)
@defun_wrapped
def whitw(ctx,k,m,z,**kwargs):
if z == 0:
g = abs(ctx.re(m))
if g < 0.5:
return z
elif g > 0.5:
return ctx.inf + z
else:
return ctx.nan * z
x = ctx.fmul(-0.5, z, exact=True)
y = 0.5+m
return ctx.exp(x) * z**y * ctx.hyperu(y-k, 1+2*m, z, **kwargs)
@defun
def hyperu(ctx, a, b, z, **kwargs):
a, atype = ctx._convert_param(a)
b, btype = ctx._convert_param(b)
z = ctx.convert(z)
if not z:
if ctx.re(b) <= 1:
return ctx.gammaprod([1-b],[a-b+1])
else:
return ctx.inf + z
bb = 1+a-b
bb, bbtype = ctx._convert_param(bb)
try:
orig = ctx.prec
try:
ctx.prec += 10
v = ctx.hypsum(2, 0, (atype, bbtype), [a, bb], -1/z, maxterms=ctx.prec)
return v / z**a
finally:
ctx.prec = orig
except ctx.NoConvergence:
pass
def h(a,b):
w = ctx.sinpi(b)
T1 = ([ctx.pi,w],[1,-1],[],[a-b+1,b],[a],[b],z)
T2 = ([-ctx.pi,w,z],[1,-1,1-b],[],[a,2-b],[a-b+1],[2-b],z)
return T1, T2
return ctx.hypercomb(h, [a,b], **kwargs)
@defun
def struveh(ctx,n,z, **kwargs):
n = ctx.convert(n)
z = ctx.convert(z)
# http://functions.wolfram.com/Bessel-TypeFunctions/StruveH/26/01/02/
def h(n):
return [([z/2, 0.5*ctx.sqrt(ctx.pi)], [n+1, -1], [], [n+1.5], [1], [1.5, n+1.5], -(z/2)**2)]
return ctx.hypercomb(h, [n], **kwargs)
@defun
def struvel(ctx,n,z, **kwargs):
n = ctx.convert(n)
z = ctx.convert(z)
# http://functions.wolfram.com/Bessel-TypeFunctions/StruveL/26/01/02/
def h(n):
return [([z/2, 0.5*ctx.sqrt(ctx.pi)], [n+1, -1], [], [n+1.5], [1], [1.5, n+1.5], (z/2)**2)]
return ctx.hypercomb(h, [n], **kwargs)
def _anger(ctx,which,v,z,**kwargs):
v = ctx._convert_param(v)[0]
z = ctx.convert(z)
def h(v):
b = ctx.mpq_1_2
u = v*b
m = b*3
a1,a2,b1,b2 = m-u, m+u, 1-u, 1+u
c, s = ctx.cospi_sinpi(u)
if which == 0:
A, B = [b*z, s], [c]
if which == 1:
A, B = [b*z, -c], [s]
w = ctx.square_exp_arg(z, mult=-0.25)
T1 = A, [1, 1], [], [a1,a2], [1], [a1,a2], w
T2 = B, [1], [], [b1,b2], [1], [b1,b2], w
return T1, T2
return ctx.hypercomb(h, [v], **kwargs)
@defun
def angerj(ctx, v, z, **kwargs):
return _anger(ctx, 0, v, z, **kwargs)
@defun
def webere(ctx, v, z, **kwargs):
return _anger(ctx, 1, v, z, **kwargs)
@defun
def lommels1(ctx, u, v, z, **kwargs):
u = ctx._convert_param(u)[0]
v = ctx._convert_param(v)[0]
z = ctx.convert(z)
def h(u,v):
b = ctx.mpq_1_2
w = ctx.square_exp_arg(z, mult=-0.25)
return ([u-v+1, u+v+1, z], [-1, -1, u+1], [], [], [1], \
[b*(u-v+3),b*(u+v+3)], w),
return ctx.hypercomb(h, [u,v], **kwargs)
@defun
def lommels2(ctx, u, v, z, **kwargs):
u = ctx._convert_param(u)[0]
v = ctx._convert_param(v)[0]
z = ctx.convert(z)
# Asymptotic expansion (GR p. 947) -- need to be careful
# not to use for small arguments
# def h(u,v):
# b = ctx.mpq_1_2
# w = -(z/2)**(-2)
# return ([z], [u-1], [], [], [b*(1-u+v)], [b*(1-u-v)], w),
def h(u,v):
b = ctx.mpq_1_2
w = ctx.square_exp_arg(z, mult=-0.25)
T1 = [u-v+1, u+v+1, z], [-1, -1, u+1], [], [], [1], [b*(u-v+3),b*(u+v+3)], w
T2 = [2, z], [u+v-1, -v], [v, b*(u+v+1)], [b*(v-u+1)], [], [1-v], w
T3 = [2, z], [u-v-1, v], [-v, b*(u-v+1)], [b*(1-u-v)], [], [1+v], w
#c1 = ctx.cospi((u-v)*b)
#c2 = ctx.cospi((u+v)*b)
#s = ctx.sinpi(v)
#r1 = (u-v+1)*b
#r2 = (u+v+1)*b
#T2 = [c1, s, z, 2], [1, -1, -v, v], [], [-v+1], [], [-v+1], w
#T3 = [-c2, s, z, 2], [1, -1, v, -v], [], [v+1], [], [v+1], w
#T2 = [c1, s, z, 2], [1, -1, -v, v+u-1], [r1, r2], [-v+1], [], [-v+1], w
#T3 = [-c2, s, z, 2], [1, -1, v, -v+u-1], [r1, r2], [v+1], [], [v+1], w
return T1, T2, T3
return ctx.hypercomb(h, [u,v], **kwargs)
@defun
def ber(ctx, n, z, **kwargs):
n = ctx.convert(n)
z = ctx.convert(z)
# http://functions.wolfram.com/Bessel-TypeFunctions/KelvinBer2/26/01/02/0001/
def h(n):
r = -(z/4)**4
cos, sin = ctx.cospi_sinpi(-0.75*n)
T1 = [cos, z/2], [1, n], [], [n+1], [], [0.5, 0.5*(n+1), 0.5*n+1], r
T2 = [sin, z/2], [1, n+2], [], [n+2], [], [1.5, 0.5*(n+3), 0.5*n+1], r
return T1, T2
return ctx.hypercomb(h, [n], **kwargs)
@defun
def bei(ctx, n, z, **kwargs):
n = ctx.convert(n)
z = ctx.convert(z)
# http://functions.wolfram.com/Bessel-TypeFunctions/KelvinBei2/26/01/02/0001/
def h(n):
r = -(z/4)**4
cos, sin = ctx.cospi_sinpi(0.75*n)
T1 = [cos, z/2], [1, n+2], [], [n+2], [], [1.5, 0.5*(n+3), 0.5*n+1], r
T2 = [sin, z/2], [1, n], [], [n+1], [], [0.5, 0.5*(n+1), 0.5*n+1], r
return T1, T2
return ctx.hypercomb(h, [n], **kwargs)
@defun
def ker(ctx, n, z, **kwargs):
n = ctx.convert(n)
z = ctx.convert(z)
# http://functions.wolfram.com/Bessel-TypeFunctions/KelvinKer2/26/01/02/0001/
def h(n):
r = -(z/4)**4
cos1, sin1 = ctx.cospi_sinpi(0.25*n)
cos2, sin2 = ctx.cospi_sinpi(0.75*n)
T1 = [2, z, 4*cos1], [-n-3, n, 1], [-n], [], [], [0.5, 0.5*(1+n), 0.5*(n+2)], r
T2 = [2, z, -sin1], [-n-3, 2+n, 1], [-n-1], [], [], [1.5, 0.5*(3+n), 0.5*(n+2)], r
T3 = [2, z, 4*cos2], [n-3, -n, 1], [n], [], [], [0.5, 0.5*(1-n), 1-0.5*n], r
T4 = [2, z, -sin2], [n-3, 2-n, 1], [n-1], [], [], [1.5, 0.5*(3-n), 1-0.5*n], r
return T1, T2, T3, T4
return ctx.hypercomb(h, [n], **kwargs)
@defun
def kei(ctx, n, z, **kwargs):
n = ctx.convert(n)
z = ctx.convert(z)
# http://functions.wolfram.com/Bessel-TypeFunctions/KelvinKei2/26/01/02/0001/
def h(n):
r = -(z/4)**4
cos1, sin1 = ctx.cospi_sinpi(0.75*n)
cos2, sin2 = ctx.cospi_sinpi(0.25*n)
T1 = [-cos1, 2, z], [1, n-3, 2-n], [n-1], [], [], [1.5, 0.5*(3-n), 1-0.5*n], r
T2 = [-sin1, 2, z], [1, n-1, -n], [n], [], [], [0.5, 0.5*(1-n), 1-0.5*n], r
T3 = [-sin2, 2, z], [1, -n-1, n], [-n], [], [], [0.5, 0.5*(n+1), 0.5*(n+2)], r
T4 = [-cos2, 2, z], [1, -n-3, n+2], [-n-1], [], [], [1.5, 0.5*(n+3), 0.5*(n+2)], r
return T1, T2, T3, T4
return ctx.hypercomb(h, [n], **kwargs)
# TODO: do this more generically?
def c_memo(f):
name = f.__name__
def f_wrapped(ctx):
cache = ctx._misc_const_cache
prec = ctx.prec
p,v = cache.get(name, (-1,0))
if p >= prec:
return +v
else:
cache[name] = (prec, f(ctx))
return cache[name][1]
return f_wrapped
@c_memo
def _airyai_C1(ctx):
return 1 / (ctx.cbrt(9) * ctx.gamma(ctx.mpf(2)/3))
@c_memo
def _airyai_C2(ctx):
return -1 / (ctx.cbrt(3) * ctx.gamma(ctx.mpf(1)/3))
@c_memo
def _airybi_C1(ctx):
return 1 / (ctx.nthroot(3,6) * ctx.gamma(ctx.mpf(2)/3))
@c_memo
def _airybi_C2(ctx):
return ctx.nthroot(3,6) / ctx.gamma(ctx.mpf(1)/3)
def _airybi_n2_inf(ctx):
prec = ctx.prec
try:
v = ctx.power(3,'2/3')*ctx.gamma('2/3')/(2*ctx.pi)
finally:
ctx.prec = prec
return +v
# Derivatives at z = 0
# TODO: could be expressed more elegantly using triple factorials
def _airyderiv_0(ctx, z, n, ntype, which):
if ntype == 'Z':
if n < 0:
return z
r = ctx.mpq_1_3
prec = ctx.prec
try:
ctx.prec += 10
v = ctx.gamma((n+1)*r) * ctx.power(3,n*r) / ctx.pi
if which == 0:
v *= ctx.sinpi(2*(n+1)*r)
v /= ctx.power(3,'2/3')
else:
v *= abs(ctx.sinpi(2*(n+1)*r))
v /= ctx.power(3,'1/6')
finally:
ctx.prec = prec
return +v + z
else:
# singular (does the limit exist?)
raise NotImplementedError
@defun
def airyai(ctx, z, derivative=0, **kwargs):
z = ctx.convert(z)
if derivative:
n, ntype = ctx._convert_param(derivative)
else:
n = 0
# Values at infinities
if not ctx.isnormal(z) and z:
if n and ntype == 'Z':
if n == -1:
if z == ctx.inf:
return ctx.mpf(1)/3 + 1/z
if z == ctx.ninf:
return ctx.mpf(-2)/3 + 1/z
if n < -1:
if z == ctx.inf:
return z
if z == ctx.ninf:
return (-1)**n * (-z)
if (not n) and z == ctx.inf or z == ctx.ninf:
return 1/z
# TODO: limits
raise ValueError("essential singularity of Ai(z)")
# Account for exponential scaling
if z:
extraprec = max(0, int(1.5*ctx.mag(z)))
else:
extraprec = 0
if n:
if n == 1:
def h():
# http://functions.wolfram.com/03.07.06.0005.01
if ctx._re(z) > 4:
ctx.prec += extraprec
w = z**1.5; r = -0.75/w; u = -2*w/3
ctx.prec -= extraprec
C = -ctx.exp(u)/(2*ctx.sqrt(ctx.pi))*ctx.nthroot(z,4)
return ([C],[1],[],[],[(-1,6),(7,6)],[],r),
# http://functions.wolfram.com/03.07.26.0001.01
else:
ctx.prec += extraprec
w = z**3 / 9
ctx.prec -= extraprec
C1 = _airyai_C1(ctx) * 0.5
C2 = _airyai_C2(ctx)
T1 = [C1,z],[1,2],[],[],[],[ctx.mpq_5_3],w
T2 = [C2],[1],[],[],[],[ctx.mpq_1_3],w
return T1, T2
return ctx.hypercomb(h, [], **kwargs)
else:
if z == 0:
return _airyderiv_0(ctx, z, n, ntype, 0)
# http://functions.wolfram.com/03.05.20.0004.01
def h(n):
ctx.prec += extraprec
w = z**3/9
ctx.prec -= extraprec
q13,q23,q43 = ctx.mpq_1_3, ctx.mpq_2_3, ctx.mpq_4_3
a1=q13; a2=1; b1=(1-n)*q13; b2=(2-n)*q13; b3=1-n*q13
T1 = [3, z], [n-q23, -n], [a1], [b1,b2,b3], \
[a1,a2], [b1,b2,b3], w
a1=q23; b1=(2-n)*q13; b2=1-n*q13; b3=(4-n)*q13
T2 = [3, z, -z], [n-q43, -n, 1], [a1], [b1,b2,b3], \
[a1,a2], [b1,b2,b3], w
return T1, T2
v = ctx.hypercomb(h, [n], **kwargs)
if ctx._is_real_type(z) and ctx.isint(n):
v = ctx._re(v)
return v
else:
def h():
if ctx._re(z) > 4:
# We could use 1F1, but it results in huge cancellation;
# the following expansion is better.
# TODO: asymptotic series for derivatives
ctx.prec += extraprec
w = z**1.5; r = -0.75/w; u = -2*w/3
ctx.prec -= extraprec
C = ctx.exp(u)/(2*ctx.sqrt(ctx.pi)*ctx.nthroot(z,4))
return ([C],[1],[],[],[(1,6),(5,6)],[],r),
else:
ctx.prec += extraprec
w = z**3 / 9
ctx.prec -= extraprec
C1 = _airyai_C1(ctx)
C2 = _airyai_C2(ctx)
T1 = [C1],[1],[],[],[],[ctx.mpq_2_3],w
T2 = [z*C2],[1],[],[],[],[ctx.mpq_4_3],w
return T1, T2
return ctx.hypercomb(h, [], **kwargs)
@defun
def airybi(ctx, z, derivative=0, **kwargs):
z = ctx.convert(z)
if derivative:
n, ntype = ctx._convert_param(derivative)
else:
n = 0
# Values at infinities
if not ctx.isnormal(z) and z:
if n and ntype == 'Z':
if z == ctx.inf:
return z
if z == ctx.ninf:
if n == -1:
return 1/z
if n == -2:
return _airybi_n2_inf(ctx)
if n < -2:
return (-1)**n * (-z)
if not n:
if z == ctx.inf:
return z
if z == ctx.ninf:
return 1/z
# TODO: limits
raise ValueError("essential singularity of Bi(z)")
if z:
extraprec = max(0, int(1.5*ctx.mag(z)))
else:
extraprec = 0
if n:
if n == 1:
# http://functions.wolfram.com/03.08.26.0001.01
def h():
ctx.prec += extraprec
w = z**3 / 9
ctx.prec -= extraprec
C1 = _airybi_C1(ctx)*0.5
C2 = _airybi_C2(ctx)
T1 = [C1,z],[1,2],[],[],[],[ctx.mpq_5_3],w
T2 = [C2],[1],[],[],[],[ctx.mpq_1_3],w
return T1, T2
return ctx.hypercomb(h, [], **kwargs)
else:
if z == 0:
return _airyderiv_0(ctx, z, n, ntype, 1)
def h(n):
ctx.prec += extraprec
w = z**3/9
ctx.prec -= extraprec
q13,q23,q43 = ctx.mpq_1_3, ctx.mpq_2_3, ctx.mpq_4_3
q16 = ctx.mpq_1_6
q56 = ctx.mpq_5_6
a1=q13; a2=1; b1=(1-n)*q13; b2=(2-n)*q13; b3=1-n*q13
T1 = [3, z], [n-q16, -n], [a1], [b1,b2,b3], \
[a1,a2], [b1,b2,b3], w
a1=q23; b1=(2-n)*q13; b2=1-n*q13; b3=(4-n)*q13
T2 = [3, z], [n-q56, 1-n], [a1], [b1,b2,b3], \
[a1,a2], [b1,b2,b3], w
return T1, T2
v = ctx.hypercomb(h, [n], **kwargs)
if ctx._is_real_type(z) and ctx.isint(n):
v = ctx._re(v)
return v
else:
def h():
ctx.prec += extraprec
w = z**3 / 9
ctx.prec -= extraprec
C1 = _airybi_C1(ctx)
C2 = _airybi_C2(ctx)
T1 = [C1],[1],[],[],[],[ctx.mpq_2_3],w
T2 = [z*C2],[1],[],[],[],[ctx.mpq_4_3],w
return T1, T2
return ctx.hypercomb(h, [], **kwargs)
def _airy_zero(ctx, which, k, derivative, complex=False):
# Asymptotic formulas are given in DLMF section 9.9
def U(t): return t**(2/3.)*(1-7/(t**2*48))
def T(t): return t**(2/3.)*(1+5/(t**2*48))
k = int(k)
if k < 1:
raise ValueError("k cannot be less than 1")
if not derivative in (0,1):
raise ValueError("Derivative should lie between 0 and 1")
if which == 0:
if derivative:
return ctx.findroot(lambda z: ctx.airyai(z,1),
-U(3*ctx.pi*(4*k-3)/8))
return ctx.findroot(ctx.airyai, -T(3*ctx.pi*(4*k-1)/8))
if which == 1 and complex == False:
if derivative:
return ctx.findroot(lambda z: ctx.airybi(z,1),
-U(3*ctx.pi*(4*k-1)/8))
return ctx.findroot(ctx.airybi, -T(3*ctx.pi*(4*k-3)/8))
if which == 1 and complex == True:
if derivative:
t = 3*ctx.pi*(4*k-3)/8 + 0.75j*ctx.ln2
s = ctx.expjpi(ctx.mpf(1)/3) * T(t)
return ctx.findroot(lambda z: ctx.airybi(z,1), s)
t = 3*ctx.pi*(4*k-1)/8 + 0.75j*ctx.ln2
s = ctx.expjpi(ctx.mpf(1)/3) * U(t)
return ctx.findroot(ctx.airybi, s)
@defun
def airyaizero(ctx, k, derivative=0):
return _airy_zero(ctx, 0, k, derivative, False)
@defun
def airybizero(ctx, k, derivative=0, complex=False):
return _airy_zero(ctx, 1, k, derivative, complex)
def _scorer(ctx, z, which, kwargs):
z = ctx.convert(z)
if ctx.isinf(z):
if z == ctx.inf:
if which == 0: return 1/z
if which == 1: return z
if z == ctx.ninf:
return 1/z
raise ValueError("essential singularity")
if z:
extraprec = max(0, int(1.5*ctx.mag(z)))
else:
extraprec = 0
if kwargs.get('derivative'):
raise NotImplementedError
# Direct asymptotic expansions, to avoid
# exponentially large cancellation
try:
if ctx.mag(z) > 3:
if which == 0 and abs(ctx.arg(z)) < ctx.pi/3 * 0.999:
def h():
return (([ctx.pi,z],[-1,-1],[],[],[(1,3),(2,3),1],[],9/z**3),)
return ctx.hypercomb(h, [], maxterms=ctx.prec, force_series=True)
if which == 1 and abs(ctx.arg(-z)) < 2*ctx.pi/3 * 0.999:
def h():
return (([-ctx.pi,z],[-1,-1],[],[],[(1,3),(2,3),1],[],9/z**3),)
return ctx.hypercomb(h, [], maxterms=ctx.prec, force_series=True)
except ctx.NoConvergence:
pass
def h():
A = ctx.airybi(z, **kwargs)/3
B = -2*ctx.pi
if which == 1:
A *= 2
B *= -1
ctx.prec += extraprec
w = z**3/9
ctx.prec -= extraprec
T1 = [A], [1], [], [], [], [], 0
T2 = [B,z], [-1,2], [], [], [1], [ctx.mpq_4_3,ctx.mpq_5_3], w
return T1, T2
return ctx.hypercomb(h, [], **kwargs)
@defun
def scorergi(ctx, z, **kwargs):
return _scorer(ctx, z, 0, kwargs)
@defun
def scorerhi(ctx, z, **kwargs):
return _scorer(ctx, z, 1, kwargs)
@defun_wrapped
def coulombc(ctx, l, eta, _cache={}):
if (l, eta) in _cache and _cache[l,eta][0] >= ctx.prec:
return +_cache[l,eta][1]
G3 = ctx.loggamma(2*l+2)
G1 = ctx.loggamma(1+l+ctx.j*eta)
G2 = ctx.loggamma(1+l-ctx.j*eta)
v = 2**l * ctx.exp((-ctx.pi*eta+G1+G2)/2 - G3)
if not (ctx.im(l) or ctx.im(eta)):
v = ctx.re(v)
_cache[l,eta] = (ctx.prec, v)
return v
@defun_wrapped
def coulombf(ctx, l, eta, z, w=1, chop=True, **kwargs):
# Regular Coulomb wave function
# Note: w can be either 1 or -1; the other may be better in some cases
# TODO: check that chop=True chops when and only when it should
#ctx.prec += 10
def h(l, eta):
try:
jw = ctx.j*w
jwz = ctx.fmul(jw, z, exact=True)
jwz2 = ctx.fmul(jwz, -2, exact=True)
C = ctx.coulombc(l, eta)
T1 = [C, z, ctx.exp(jwz)], [1, l+1, 1], [], [], [1+l+jw*eta], \
[2*l+2], jwz2
except ValueError:
T1 = [0], [-1], [], [], [], [], 0
return (T1,)
v = ctx.hypercomb(h, [l,eta], **kwargs)
if chop and (not ctx.im(l)) and (not ctx.im(eta)) and (not ctx.im(z)) and \
(ctx.re(z) >= 0):
v = ctx.re(v)
return v
@defun_wrapped
def _coulomb_chi(ctx, l, eta, _cache={}):
if (l, eta) in _cache and _cache[l,eta][0] >= ctx.prec:
return _cache[l,eta][1]
def terms():
l2 = -l-1
jeta = ctx.j*eta
return [ctx.loggamma(1+l+jeta) * (-0.5j),
ctx.loggamma(1+l-jeta) * (0.5j),
ctx.loggamma(1+l2+jeta) * (0.5j),
ctx.loggamma(1+l2-jeta) * (-0.5j),
-(l+0.5)*ctx.pi]
v = ctx.sum_accurately(terms, 1)
_cache[l,eta] = (ctx.prec, v)
return v
@defun_wrapped
def coulombg(ctx, l, eta, z, w=1, chop=True, **kwargs):
# Irregular Coulomb wave function
# Note: w can be either 1 or -1; the other may be better in some cases
# TODO: check that chop=True chops when and only when it should
if not ctx._im(l):
l = ctx._re(l) # XXX: for isint
def h(l, eta):
# Force perturbation for integers and half-integers
if ctx.isint(l*2):
T1 = [0], [-1], [], [], [], [], 0
return (T1,)
l2 = -l-1
try:
chi = ctx._coulomb_chi(l, eta)
jw = ctx.j*w
s = ctx.sin(chi); c = ctx.cos(chi)
C1 = ctx.coulombc(l,eta)
C2 = ctx.coulombc(l2,eta)
u = ctx.exp(jw*z)
x = -2*jw*z
T1 = [s, C1, z, u, c], [-1, 1, l+1, 1, 1], [], [], \
[1+l+jw*eta], [2*l+2], x
T2 = [-s, C2, z, u], [-1, 1, l2+1, 1], [], [], \
[1+l2+jw*eta], [2*l2+2], x
return T1, T2
except ValueError:
T1 = [0], [-1], [], [], [], [], 0
return (T1,)
v = ctx.hypercomb(h, [l,eta], **kwargs)
if chop and (not ctx._im(l)) and (not ctx._im(eta)) and (not ctx._im(z)) and \
(ctx._re(z) >= 0):
v = ctx._re(v)
return v
def mcmahon(ctx,kind,prime,v,m):
"""
Computes an estimate for the location of the Bessel function zero
j_{v,m}, y_{v,m}, j'_{v,m} or y'_{v,m} using McMahon's asymptotic
expansion (Abramowitz & Stegun 9.5.12-13, DLMF 20.21(vi)).
Returns (r,err) where r is the estimated location of the root
and err is a positive number estimating the error of the
asymptotic expansion.
"""
u = 4*v**2
if kind == 1 and not prime: b = (4*m+2*v-1)*ctx.pi/4
if kind == 2 and not prime: b = (4*m+2*v-3)*ctx.pi/4
if kind == 1 and prime: b = (4*m+2*v-3)*ctx.pi/4
if kind == 2 and prime: b = (4*m+2*v-1)*ctx.pi/4
if not prime:
s1 = b
s2 = -(u-1)/(8*b)
s3 = -4*(u-1)*(7*u-31)/(3*(8*b)**3)
s4 = -32*(u-1)*(83*u**2-982*u+3779)/(15*(8*b)**5)
s5 = -64*(u-1)*(6949*u**3-153855*u**2+1585743*u-6277237)/(105*(8*b)**7)
if prime:
s1 = b
s2 = -(u+3)/(8*b)
s3 = -4*(7*u**2+82*u-9)/(3*(8*b)**3)
s4 = -32*(83*u**3+2075*u**2-3039*u+3537)/(15*(8*b)**5)
s5 = -64*(6949*u**4+296492*u**3-1248002*u**2+7414380*u-5853627)/(105*(8*b)**7)
terms = [s1,s2,s3,s4,s5]
s = s1
err = 0.0
for i in range(1,len(terms)):
if abs(terms[i]) < abs(terms[i-1]):
s += terms[i]
else:
err = abs(terms[i])
if i == len(terms)-1:
err = abs(terms[-1])
return s, err
def generalized_bisection(ctx,f,a,b,n):
"""
Given f known to have exactly n simple roots within [a,b],
return a list of n intervals isolating the roots
and having opposite signs at the endpoints.
TODO: this can be optimized, e.g. by reusing evaluation points.
"""
if n < 1:
raise ValueError("n cannot be less than 1")
N = n+1
points = []
signs = []
while 1:
points = ctx.linspace(a,b,N)
signs = [ctx.sign(f(x)) for x in points]
ok_intervals = [(points[i],points[i+1]) for i in range(N-1) \
if signs[i]*signs[i+1] == -1]
if len(ok_intervals) == n:
return ok_intervals
N = N*2
def find_in_interval(ctx, f, ab):
return ctx.findroot(f, ab, solver='illinois', verify=False)
def bessel_zero(ctx, kind, prime, v, m, isoltol=0.01, _interval_cache={}):
prec = ctx.prec
workprec = max(prec, ctx.mag(v), ctx.mag(m))+10
try:
ctx.prec = workprec
v = ctx.mpf(v)
m = int(m)
prime = int(prime)
if v < 0:
raise ValueError("v cannot be negative")
if m < 1:
raise ValueError("m cannot be less than 1")
if not prime in (0,1):
raise ValueError("prime should lie between 0 and 1")
if kind == 1:
if prime: f = lambda x: ctx.besselj(v,x,derivative=1)
else: f = lambda x: ctx.besselj(v,x)
if kind == 2:
if prime: f = lambda x: ctx.bessely(v,x,derivative=1)
else: f = lambda x: ctx.bessely(v,x)
# The first root of J' is very close to 0 for small
# orders, and this needs to be special-cased
if kind == 1 and prime and m == 1:
if v == 0:
return ctx.zero
if v <= 1:
# TODO: use v <= j'_{v,1} < y_{v,1}?
r = 2*ctx.sqrt(v*(1+v)/(v+2))
return find_in_interval(ctx, f, (r/10, 2*r))
if (kind,prime,v,m) in _interval_cache:
return find_in_interval(ctx, f, _interval_cache[kind,prime,v,m])
r, err = mcmahon(ctx, kind, prime, v, m)
if err < isoltol:
return find_in_interval(ctx, f, (r-isoltol, r+isoltol))
# An x such that 0 < x < r_{v,1}
if kind == 1 and not prime: low = 2.4
if kind == 1 and prime: low = 1.8
if kind == 2 and not prime: low = 0.8
if kind == 2 and prime: low = 2.0
n = m+1
while 1:
r1, err = mcmahon(ctx, kind, prime, v, n)
if err < isoltol:
r2, err2 = mcmahon(ctx, kind, prime, v, n+1)
intervals = generalized_bisection(ctx, f, low, 0.5*(r1+r2), n)
for k, ab in enumerate(intervals):
_interval_cache[kind,prime,v,k+1] = ab
return find_in_interval(ctx, f, intervals[m-1])
else:
n = n*2
finally:
ctx.prec = prec
@defun
def besseljzero(ctx, v, m, derivative=0):
r"""
For a real order `\nu \ge 0` and a positive integer `m`, returns
`j_{\nu,m}`, the `m`-th positive zero of the Bessel function of the
first kind `J_{\nu}(z)` (see :func:`~mpmath.besselj`). Alternatively,
with *derivative=1*, gives the first nonnegative simple zero
`j'_{\nu,m}` of `J'_{\nu}(z)`.
The indexing convention is that used by Abramowitz & Stegun
and the DLMF. Note the special case `j'_{0,1} = 0`, while all other
zeros are positive. In effect, only simple zeros are counted
(all zeros of Bessel functions are simple except possibly `z = 0`)
and `j_{\nu,m}` becomes a monotonic function of both `\nu`
and `m`.
The zeros are interlaced according to the inequalities
.. math ::
j'_{\nu,k} < j_{\nu,k} < j'_{\nu,k+1}
j_{\nu,1} < j_{\nu+1,2} < j_{\nu,2} < j_{\nu+1,2} < j_{\nu,3} < \cdots
**Examples**
Initial zeros of the Bessel functions `J_0(z), J_1(z), J_2(z)`::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> besseljzero(0,1); besseljzero(0,2); besseljzero(0,3)
2.404825557695772768621632
5.520078110286310649596604
8.653727912911012216954199
>>> besseljzero(1,1); besseljzero(1,2); besseljzero(1,3)
3.831705970207512315614436
7.01558666981561875353705
10.17346813506272207718571
>>> besseljzero(2,1); besseljzero(2,2); besseljzero(2,3)
5.135622301840682556301402
8.417244140399864857783614
11.61984117214905942709415
Initial zeros of `J'_0(z), J'_1(z), J'_2(z)`::
0.0
3.831705970207512315614436
7.01558666981561875353705
>>> besseljzero(1,1,1); besseljzero(1,2,1); besseljzero(1,3,1)
1.84118378134065930264363
5.331442773525032636884016
8.536316366346285834358961
>>> besseljzero(2,1,1); besseljzero(2,2,1); besseljzero(2,3,1)
3.054236928227140322755932
6.706133194158459146634394
9.969467823087595793179143
Zeros with large index::
>>> besseljzero(0,100); besseljzero(0,1000); besseljzero(0,10000)
313.3742660775278447196902
3140.807295225078628895545
31415.14114171350798533666
>>> besseljzero(5,100); besseljzero(5,1000); besseljzero(5,10000)
321.1893195676003157339222
3148.657306813047523500494
31422.9947255486291798943
>>> besseljzero(0,100,1); besseljzero(0,1000,1); besseljzero(0,10000,1)
311.8018681873704508125112
3139.236339643802482833973
31413.57032947022399485808
Zeros of functions with large order::
>>> besseljzero(50,1)
57.11689916011917411936228
>>> besseljzero(50,2)
62.80769876483536093435393
>>> besseljzero(50,100)
388.6936600656058834640981
>>> besseljzero(50,1,1)
52.99764038731665010944037
>>> besseljzero(50,2,1)
60.02631933279942589882363
>>> besseljzero(50,100,1)
387.1083151608726181086283
Zeros of functions with fractional order::
>>> besseljzero(0.5,1); besseljzero(1.5,1); besseljzero(2.25,4)
3.141592653589793238462643
4.493409457909064175307881
15.15657692957458622921634
Both `J_{\nu}(z)` and `J'_{\nu}(z)` can be expressed as infinite
products over their zeros::
>>> v,z = 2, mpf(1)
>>> (z/2)**v/gamma(v+1) * \
... nprod(lambda k: 1-(z/besseljzero(v,k))**2, [1,inf])
...
0.1149034849319004804696469
>>> besselj(v,z)
0.1149034849319004804696469
>>> (z/2)**(v-1)/2/gamma(v) * \
... nprod(lambda k: 1-(z/besseljzero(v,k,1))**2, [1,inf])
...
0.2102436158811325550203884
>>> besselj(v,z,1)
0.2102436158811325550203884
"""
return +bessel_zero(ctx, 1, derivative, v, m)
@defun
def besselyzero(ctx, v, m, derivative=0):
r"""
For a real order `\nu \ge 0` and a positive integer `m`, returns
`y_{\nu,m}`, the `m`-th positive zero of the Bessel function of the
second kind `Y_{\nu}(z)` (see :func:`~mpmath.bessely`). Alternatively,
with *derivative=1*, gives the first positive zero `y'_{\nu,m}` of
`Y'_{\nu}(z)`.
The zeros are interlaced according to the inequalities
.. math ::
y_{\nu,k} < y'_{\nu,k} < y_{\nu,k+1}
y_{\nu,1} < y_{\nu+1,2} < y_{\nu,2} < y_{\nu+1,2} < y_{\nu,3} < \cdots
**Examples**
Initial zeros of the Bessel functions `Y_0(z), Y_1(z), Y_2(z)`::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> besselyzero(0,1); besselyzero(0,2); besselyzero(0,3)
0.8935769662791675215848871
3.957678419314857868375677
7.086051060301772697623625
>>> besselyzero(1,1); besselyzero(1,2); besselyzero(1,3)
2.197141326031017035149034
5.429681040794135132772005
8.596005868331168926429606
>>> besselyzero(2,1); besselyzero(2,2); besselyzero(2,3)
3.384241767149593472701426
6.793807513268267538291167
10.02347797936003797850539
Initial zeros of `Y'_0(z), Y'_1(z), Y'_2(z)`::
>>> besselyzero(0,1,1); besselyzero(0,2,1); besselyzero(0,3,1)
2.197141326031017035149034
5.429681040794135132772005
8.596005868331168926429606
>>> besselyzero(1,1,1); besselyzero(1,2,1); besselyzero(1,3,1)
3.683022856585177699898967
6.941499953654175655751944
10.12340465543661307978775
>>> besselyzero(2,1,1); besselyzero(2,2,1); besselyzero(2,3,1)
5.002582931446063945200176
8.350724701413079526349714
11.57419546521764654624265
Zeros with large index::
>>> besselyzero(0,100); besselyzero(0,1000); besselyzero(0,10000)
311.8034717601871549333419
3139.236498918198006794026
31413.57034538691205229188
>>> besselyzero(5,100); besselyzero(5,1000); besselyzero(5,10000)
319.6183338562782156235062
3147.086508524556404473186
31421.42392920214673402828
>>> besselyzero(0,100,1); besselyzero(0,1000,1); besselyzero(0,10000,1)
313.3726705426359345050449
3140.807136030340213610065
31415.14112579761578220175
Zeros of functions with large order::
>>> besselyzero(50,1)
53.50285882040036394680237
>>> besselyzero(50,2)
60.11244442774058114686022
>>> besselyzero(50,100)
387.1096509824943957706835
>>> besselyzero(50,1,1)
56.96290427516751320063605
>>> besselyzero(50,2,1)
62.74888166945933944036623
>>> besselyzero(50,100,1)
388.6923300548309258355475
Zeros of functions with fractional order::
>>> besselyzero(0.5,1); besselyzero(1.5,1); besselyzero(2.25,4)
1.570796326794896619231322
2.798386045783887136720249
13.56721208770735123376018
"""
return +bessel_zero(ctx, 2, derivative, v, m)
| 37,938 | 33.210099 | 100 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/functions/rszeta.py
|
"""
---------------------------------------------------------------------
.. sectionauthor:: Juan Arias de Reyna <arias@us.es>
This module implements zeta-related functions using the Riemann-Siegel
expansion: zeta_offline(s,k=0)
* coef(J, eps): Need in the computation of Rzeta(s,k)
* Rzeta_simul(s, der=0) computes Rzeta^(k)(s) and Rzeta^(k)(1-s) simultaneously
for 0 <= k <= der. Used by zeta_offline and z_offline
* Rzeta_set(s, derivatives) computes Rzeta^(k)(s) for given derivatives, used by
z_half(t,k) and zeta_half
* z_offline(w,k): Z(w) and its derivatives of order k <= 4
* z_half(t,k): Z(t) (Riemann Siegel function) and its derivatives of order k <= 4
* zeta_offline(s): zeta(s) and its derivatives of order k<= 4
* zeta_half(1/2+it,k): zeta(s) and its derivatives of order k<= 4
* rs_zeta(s,k=0) Computes zeta^(k)(s) Unifies zeta_half and zeta_offline
* rs_z(w,k=0) Computes Z^(k)(w) Unifies z_offline and z_half
----------------------------------------------------------------------
This program uses Riemann-Siegel expansion even to compute
zeta(s) on points s = sigma + i t with sigma arbitrary not
necessarily equal to 1/2.
It is founded on a new deduction of the formula, with rigorous
and sharp bounds for the terms and rest of this expansion.
More information on the papers:
J. Arias de Reyna, High Precision Computation of Riemann's
Zeta Function by the Riemann-Siegel Formula I, II
We refer to them as I, II.
In them we shall find detailed explanation of all the
procedure.
The program uses Riemann-Siegel expansion.
This is useful when t is big, ( say t > 10000 ).
The precision is limited, roughly it can compute zeta(sigma+it)
with an error less than exp(-c t) for some constant c depending
on sigma. The program gives an error when the Riemann-Siegel
formula can not compute to the wanted precision.
"""
import math
class RSCache(object):
def __init__(ctx):
ctx._rs_cache = [0, 10, {}, {}]
from .functions import defun
#-------------------------------------------------------------------------------#
# #
# coef(ctx, J, eps, _cache=[0, 10, {} ] ) #
# #
#-------------------------------------------------------------------------------#
# This function computes the coefficients c[n] defined on (I, equation (47))
# but see also (II, section 3.14).
#
# Since these coefficients are very difficult to compute we save the values
# in a cache. So if we compute several values of the functions Rzeta(s) for
# near values of s, we do not recompute these coefficients.
#
# c[n] are the Taylor coefficients of the function:
#
# F(z):= (exp(pi*j*(z*z/2+3/8))-j* sqrt(2) cos(pi*z/2))/(2*cos(pi *z))
#
#
def _coef(ctx, J, eps):
r"""
Computes the coefficients `c_n` for `0\le n\le 2J` with error less than eps
**Definition**
The coefficients c_n are defined by
.. math ::
\begin{equation}
F(z)=\frac{e^{\pi i
\bigl(\frac{z^2}{2}+\frac38\bigr)}-i\sqrt{2}\cos\frac{\pi}{2}z}{2\cos\pi
z}=\sum_{n=0}^\infty c_{2n} z^{2n}
\end{equation}
they are computed applying the relation
.. math ::
\begin{multline}
c_{2n}=-\frac{i}{\sqrt{2}}\Bigl(\frac{\pi}{2}\Bigr)^{2n}
\sum_{k=0}^n\frac{(-1)^k}{(2k)!}
2^{2n-2k}\frac{(-1)^{n-k}E_{2n-2k}}{(2n-2k)!}+\\
+e^{3\pi i/8}\sum_{j=0}^n(-1)^j\frac{
E_{2j}}{(2j)!}\frac{i^{n-j}\pi^{n+j}}{(n-j)!2^{n-j+1}}.
\end{multline}
"""
newJ = J+2 # compute more coefficients that are needed
neweps6 = eps/2. # compute with a slight more precision that are needed
# PREPARATION FOR THE COMPUTATION OF V(N) AND W(N)
# See II Section 3.16
#
# Computing the exponent wpvw of the error II equation (81)
wpvw = max(ctx.mag(10*(newJ+3)), 4*newJ+5-ctx.mag(neweps6))
# Preparation of Euler numbers (we need until the 2*RS_NEWJ)
E = ctx._eulernum(2*newJ)
# Now we have in the cache all the needed Euler numbers.
#
# Computing the powers of pi
#
# We need to compute the powers pi**n for 1<= n <= 2*J
# with relative error less than 2**(-wpvw)
# it is easy to show that this is obtained
# taking wppi as the least d with
# 2**d>40*J and 2**d> 4.24 *newJ + 2**wpvw
# In II Section 3.9 we need also that
# wppi > wptcoef[0], and that the powers
# here computed 0<= k <= 2*newJ are more
# than those needed there that are 2*L-2.
# so we need J >= L this will be checked
# before computing tcoef[]
wppi = max(ctx.mag(40*newJ), ctx.mag(newJ)+3 +wpvw)
ctx.prec = wppi
pipower = {}
pipower[0] = ctx.one
pipower[1] = ctx.pi
for n in range(2,2*newJ+1):
pipower[n] = pipower[n-1]*ctx.pi
# COMPUTING THE COEFFICIENTS v(n) AND w(n)
# see II equation (61) and equations (81) and (82)
ctx.prec = wpvw+2
v={}
w={}
for n in range(0,newJ+1):
va = (-1)**n * ctx._eulernum(2*n)
va = ctx.mpf(va)/ctx.fac(2*n)
v[n]=va*pipower[2*n]
for n in range(0,2*newJ+1):
wa = ctx.one/ctx.fac(n)
wa=wa/(2**n)
w[n]=wa*pipower[n]
# COMPUTATION OF THE CONVOLUTIONS RS_P1 AND RS_P2
# See II Section 3.16
ctx.prec = 15
wpp1a = 9 - ctx.mag(neweps6)
P1 = {}
for n in range(0,newJ+1):
ctx.prec = 15
wpp1 = max(ctx.mag(10*(n+4)),4*n+wpp1a)
ctx.prec = wpp1
sump = 0
for k in range(0,n+1):
sump += ((-1)**k) * v[k]*w[2*n-2*k]
P1[n]=((-1)**(n+1))*ctx.j*sump
P2={}
for n in range(0,newJ+1):
ctx.prec = 15
wpp2 = max(ctx.mag(10*(n+4)),4*n+wpp1a)
ctx.prec = wpp2
sump = 0
for k in range(0,n+1):
sump += (ctx.j**(n-k)) * v[k]*w[n-k]
P2[n]=sump
# COMPUTING THE COEFFICIENTS c[2n]
# See II Section 3.14
ctx.prec = 15
wpc0 = 5 - ctx.mag(neweps6)
wpc = max(6,4*newJ+wpc0)
ctx.prec = wpc
mu = ctx.sqrt(ctx.mpf('2'))/2
nu = ctx.expjpi(3./8)/2
c={}
for n in range(0,newJ):
ctx.prec = 15
wpc = max(6,4*n+wpc0)
ctx.prec = wpc
c[2*n] = mu*P1[n]+nu*P2[n]
for n in range(1,2*newJ,2):
c[n] = 0
return [newJ, neweps6, c, pipower]
def coef(ctx, J, eps):
_cache = ctx._rs_cache
if J <= _cache[0] and eps >= _cache[1]:
return _cache[2], _cache[3]
orig = ctx._mp.prec
try:
data = _coef(ctx._mp, J, eps)
finally:
ctx._mp.prec = orig
if ctx is not ctx._mp:
data[2] = dict((k,ctx.convert(v)) for (k,v) in data[2].items())
data[3] = dict((k,ctx.convert(v)) for (k,v) in data[3].items())
ctx._rs_cache[:] = data
return ctx._rs_cache[2], ctx._rs_cache[3]
#-------------------------------------------------------------------------------#
# #
# Rzeta_simul(s,k=0) #
# #
#-------------------------------------------------------------------------------#
# This function return a list with the values:
# Rzeta(sigma+it), conj(Rzeta(1-sigma+it)),Rzeta'(sigma+it), conj(Rzeta'(1-sigma+it)),
# .... , Rzeta^{(k)}(sigma+it), conj(Rzeta^{(k)}(1-sigma+it))
#
# Useful to compute the function zeta(s) and Z(w) or its derivatives.
#
def aux_M_Fp(ctx, xA, xeps4, a, xB1, xL):
# COMPUTING M NUMBER OF DERIVATIVES Fp[m] TO COMPUTE
# See II Section 3.11 equations (47) and (48)
aux1 = 126.0657606*xA/xeps4 # 126.06.. = 316/sqrt(2*pi)
aux1 = ctx.ln(aux1)
aux2 = (2*ctx.ln(ctx.pi)+ctx.ln(xB1)+ctx.ln(a))/3 -ctx.ln(2*ctx.pi)/2
m = 3*xL-3
aux3= (ctx.loggamma(m+1)-ctx.loggamma(m/3.0+2))/2 -ctx.loggamma((m+1)/2.)
while((aux1 < m*aux2+ aux3)and (m>1)):
m = m - 1
aux3 = (ctx.loggamma(m+1)-ctx.loggamma(m/3.0+2))/2 -ctx.loggamma((m+1)/2.)
xM = m
return xM
def aux_J_needed(ctx, xA, xeps4, a, xB1, xM):
# DETERMINATION OF J THE NUMBER OF TERMS NEEDED
# IN THE TAYLOR SERIES OF F.
# See II Section 3.11 equation (49))
# Only determine one
h1 = xeps4/(632*xA)
h2 = xB1*a * 126.31337419529260248 # = pi^2*e^2*sqrt(3)
h2 = h1 * ctx.power((h2/xM**2),(xM-1)/3) / xM
h3 = min(h1,h2)
return h3
def Rzeta_simul(ctx, s, der=0):
# First we take the value of ctx.prec
wpinitial = ctx.prec
# INITIALIZATION
# Take the real and imaginary part of s
t = ctx._im(s)
xsigma = ctx._re(s)
ysigma = 1 - xsigma
# Now compute several parameter that appear on the program
ctx.prec = 15
a = ctx.sqrt(t/(2*ctx.pi))
xasigma = a ** xsigma
yasigma = a ** ysigma
# We need a simple bound A1 < asigma (see II Section 3.1 and 3.3)
xA1=ctx.power(2, ctx.mag(xasigma)-1)
yA1=ctx.power(2, ctx.mag(yasigma)-1)
# We compute various epsilon's (see II end of Section 3.1)
eps = ctx.power(2, -wpinitial)
eps1 = eps/6.
xeps2 = eps * xA1/3.
yeps2 = eps * yA1/3.
# COMPUTING SOME COEFFICIENTS THAT DEPENDS
# ON sigma
# constant b and c (see I Theorem 2 formula (26) )
# coefficients A and B1 (see I Section 6.1 equation (50))
#
# here we not need high precision
ctx.prec = 15
if xsigma > 0:
xb = 2.
xc = math.pow(9,xsigma)/4.44288
# 4.44288 =(math.sqrt(2)*math.pi)
xA = math.pow(9,xsigma)
xB1 = 1
else:
xb = 2.25158 # math.sqrt( (3-2* math.log(2))*math.pi )
xc = math.pow(2,-xsigma)/4.44288
xA = math.pow(2,-xsigma)
xB1 = 1.10789 # = 2*sqrt(1-log(2))
if(ysigma > 0):
yb = 2.
yc = math.pow(9,ysigma)/4.44288
# 4.44288 =(math.sqrt(2)*math.pi)
yA = math.pow(9,ysigma)
yB1 = 1
else:
yb = 2.25158 # math.sqrt( (3-2* math.log(2))*math.pi )
yc = math.pow(2,-ysigma)/4.44288
yA = math.pow(2,-ysigma)
yB1 = 1.10789 # = 2*sqrt(1-log(2))
# COMPUTING L THE NUMBER OF TERMS NEEDED IN THE RIEMANN-SIEGEL
# CORRECTION
# See II Section 3.2
ctx.prec = 15
xL = 1
while 3*xc*ctx.gamma(xL*0.5) * ctx.power(xb*a,-xL) >= xeps2:
xL = xL+1
xL = max(2,xL)
yL = 1
while 3*yc*ctx.gamma(yL*0.5) * ctx.power(yb*a,-yL) >= yeps2:
yL = yL+1
yL = max(2,yL)
# The number L has to satify some conditions.
# If not RS can not compute Rzeta(s) with the prescribed precision
# (see II, Section 3.2 condition (20) ) and
# (II, Section 3.3 condition (22) ). Also we have added
# an additional technical condition in Section 3.17 Proposition 17
if ((3*xL >= 2*a*a/25.) or (3*xL+2+xsigma<0) or (abs(xsigma) > a/2.) or \
(3*yL >= 2*a*a/25.) or (3*yL+2+ysigma<0) or (abs(ysigma) > a/2.)):
ctx.prec = wpinitial
raise NotImplementedError("Riemann-Siegel can not compute with such precision")
# We take the maximum of the two values
L = max(xL, yL)
# INITIALIZATION (CONTINUATION)
#
# eps3 is the constant defined on (II, Section 3.5 equation (27) )
# each term of the RS correction must be computed with error <= eps3
xeps3 = xeps2/(4*xL)
yeps3 = yeps2/(4*yL)
# eps4 is defined on (II Section 3.6 equation (30) )
# each component of the formula (II Section 3.6 equation (29) )
# must be computed with error <= eps4
xeps4 = xeps3/(3*xL)
yeps4 = yeps3/(3*yL)
# COMPUTING M NUMBER OF DERIVATIVES Fp[m] TO COMPUTE
xM = aux_M_Fp(ctx, xA, xeps4, a, xB1, xL)
yM = aux_M_Fp(ctx, yA, yeps4, a, yB1, yL)
M = max(xM, yM)
# COMPUTING NUMBER OF TERMS J NEEDED
h3 = aux_J_needed(ctx, xA, xeps4, a, xB1, xM)
h4 = aux_J_needed(ctx, yA, yeps4, a, yB1, yM)
h3 = min(h3,h4)
J = 12
jvalue = (2*ctx.pi)**J / ctx.gamma(J+1)
while jvalue > h3:
J = J+1
jvalue = (2*ctx.pi)*jvalue/J
# COMPUTING eps5[m] for 1 <= m <= 21
# See II Section 10 equation (43)
# We choose the minimum of the two possibilities
eps5={}
xforeps5 = math.pi*math.pi*xB1*a
yforeps5 = math.pi*math.pi*yB1*a
for m in range(0,22):
xaux1 = math.pow(xforeps5, m/3)/(316.*xA)
yaux1 = math.pow(yforeps5, m/3)/(316.*yA)
aux1 = min(xaux1, yaux1)
aux2 = ctx.gamma(m+1)/ctx.gamma(m/3.0+0.5)
aux2 = math.sqrt(aux2)
eps5[m] = (aux1*aux2*min(xeps4,yeps4))
# COMPUTING wpfp
# See II Section 3.13 equation (59)
twenty = min(3*L-3, 21)+1
aux = 6812*J
wpfp = ctx.mag(44*J)
for m in range(0,twenty):
wpfp = max(wpfp, ctx.mag(aux*ctx.gamma(m+1)/eps5[m]))
# COMPUTING N AND p
# See II Section
ctx.prec = wpfp + ctx.mag(t)+20
a = ctx.sqrt(t/(2*ctx.pi))
N = ctx.floor(a)
p = 1-2*(a-N)
# now we get a rounded version of p
# to the precision wpfp
# this possibly is not necessary
num=ctx.floor(p*(ctx.mpf('2')**wpfp))
difference = p * (ctx.mpf('2')**wpfp)-num
if (difference < 0.5):
num = num
else:
num = num+1
p = ctx.convert(num * (ctx.mpf('2')**(-wpfp)))
# COMPUTING THE COEFFICIENTS c[n] = cc[n]
# We shall use the notation cc[n], since there is
# a constant that is called c
# See II Section 3.14
# We compute the coefficients and also save then in a
# cache. The bulk of the computation is passed to
# the function coef()
#
# eps6 is defined in II Section 3.13 equation (58)
eps6 = ctx.power(ctx.convert(2*ctx.pi), J)/(ctx.gamma(J+1)*3*J)
# Now we compute the coefficients
cc = {}
cont = {}
cont, pipowers = coef(ctx, J, eps6)
cc=cont.copy() # we need a copy since we have to change his values.
Fp={} # this is the adequate locus of this
for n in range(M, 3*L-2):
Fp[n] = 0
Fp={}
ctx.prec = wpfp
for m in range(0,M+1):
sumP = 0
for k in range(2*J-m-1,-1,-1):
sumP = (sumP * p)+ cc[k]
Fp[m] = sumP
# preparation of the new coefficients
for k in range(0,2*J-m-1):
cc[k] = (k+1)* cc[k+1]
# COMPUTING THE NUMBERS xd[u,n,k], yd[u,n,k]
# See II Section 3.17
#
# First we compute the working precisions xwpd[k]
# Se II equation (92)
xwpd={}
d1 = max(6,ctx.mag(40*L*L))
xd2 = 13+ctx.mag((1+abs(xsigma))*xA)-ctx.mag(xeps4)-1
xconst = ctx.ln(8/(ctx.pi*ctx.pi*a*a*xB1*xB1)) /2
for n in range(0,L):
xd3 = ctx.mag(ctx.sqrt(ctx.gamma(n-0.5)))-ctx.floor(n*xconst)+xd2
xwpd[n]=max(xd3,d1)
# procedure of II Section 3.17
ctx.prec = xwpd[1]+10
xpsigma = 1-(2*xsigma)
xd = {}
xd[0,0,-2]=0; xd[0,0,-1]=0; xd[0,0,0]=1; xd[0,0,1]=0
xd[0,-1,-2]=0; xd[0,-1,-1]=0; xd[0,-1,0]=1; xd[0,-1,1]=0
for n in range(1,L):
ctx.prec = xwpd[n]+10
for k in range(0,3*n//2+1):
m = 3*n-2*k
if(m!=0):
m1 = ctx.one/m
c1= m1/4
c2=(xpsigma*m1)/2
c3=-(m+1)
xd[0,n,k]=c3*xd[0,n-1,k-2]+c1*xd[0,n-1,k]+c2*xd[0,n-1,k-1]
else:
xd[0,n,k]=0
for r in range(0,k):
add=xd[0,n,r]*(ctx.mpf('1.0')*ctx.fac(2*k-2*r)/ctx.fac(k-r))
xd[0,n,k] -= ((-1)**(k-r))*add
xd[0,n,-2]=0; xd[0,n,-1]=0; xd[0,n,3*n//2+1]=0
for mu in range(-2,der+1):
for n in range(-2,L):
for k in range(-3,max(1,3*n//2+2)):
if( (mu<0)or (n<0) or(k<0)or (k>3*n//2)):
xd[mu,n,k] = 0
for mu in range(1,der+1):
for n in range(0,L):
ctx.prec = xwpd[n]+10
for k in range(0,3*n//2+1):
aux=(2*mu-2)*xd[mu-2,n-2,k-3]+2*(xsigma+n-2)*xd[mu-1,n-2,k-3]
xd[mu,n,k] = aux - xd[mu-1,n-1,k-1]
# Now we compute the working precisions ywpd[k]
# Se II equation (92)
ywpd={}
d1 = max(6,ctx.mag(40*L*L))
yd2 = 13+ctx.mag((1+abs(ysigma))*yA)-ctx.mag(yeps4)-1
yconst = ctx.ln(8/(ctx.pi*ctx.pi*a*a*yB1*yB1)) /2
for n in range(0,L):
yd3 = ctx.mag(ctx.sqrt(ctx.gamma(n-0.5)))-ctx.floor(n*yconst)+yd2
ywpd[n]=max(yd3,d1)
# procedure of II Section 3.17
ctx.prec = ywpd[1]+10
ypsigma = 1-(2*ysigma)
yd = {}
yd[0,0,-2]=0; yd[0,0,-1]=0; yd[0,0,0]=1; yd[0,0,1]=0
yd[0,-1,-2]=0; yd[0,-1,-1]=0; yd[0,-1,0]=1; yd[0,-1,1]=0
for n in range(1,L):
ctx.prec = ywpd[n]+10
for k in range(0,3*n//2+1):
m = 3*n-2*k
if(m!=0):
m1 = ctx.one/m
c1= m1/4
c2=(ypsigma*m1)/2
c3=-(m+1)
yd[0,n,k]=c3*yd[0,n-1,k-2]+c1*yd[0,n-1,k]+c2*yd[0,n-1,k-1]
else:
yd[0,n,k]=0
for r in range(0,k):
add=yd[0,n,r]*(ctx.mpf('1.0')*ctx.fac(2*k-2*r)/ctx.fac(k-r))
yd[0,n,k] -= ((-1)**(k-r))*add
yd[0,n,-2]=0; yd[0,n,-1]=0; yd[0,n,3*n//2+1]=0
for mu in range(-2,der+1):
for n in range(-2,L):
for k in range(-3,max(1,3*n//2+2)):
if( (mu<0)or (n<0) or(k<0)or (k>3*n//2)):
yd[mu,n,k] = 0
for mu in range(1,der+1):
for n in range(0,L):
ctx.prec = ywpd[n]+10
for k in range(0,3*n//2+1):
aux=(2*mu-2)*yd[mu-2,n-2,k-3]+2*(ysigma+n-2)*yd[mu-1,n-2,k-3]
yd[mu,n,k] = aux - yd[mu-1,n-1,k-1]
# COMPUTING THE COEFFICIENTS xtcoef[k,l]
# See II Section 3.9
#
# computing the needed wp
xwptcoef={}
xwpterm={}
ctx.prec = 15
c1 = ctx.mag(40*(L+2))
xc2 = ctx.mag(68*(L+2)*xA)
xc4 = ctx.mag(xB1*a*math.sqrt(ctx.pi))-1
for k in range(0,L):
xc3 = xc2 - k*xc4+ctx.mag(ctx.fac(k+0.5))/2.
xwptcoef[k] = (max(c1,xc3-ctx.mag(xeps4)+1)+1 +20)*1.5
xwpterm[k] = (max(c1,ctx.mag(L+2)+xc3-ctx.mag(xeps3)+1)+1 +20)
ywptcoef={}
ywpterm={}
ctx.prec = 15
c1 = ctx.mag(40*(L+2))
yc2 = ctx.mag(68*(L+2)*yA)
yc4 = ctx.mag(yB1*a*math.sqrt(ctx.pi))-1
for k in range(0,L):
yc3 = yc2 - k*yc4+ctx.mag(ctx.fac(k+0.5))/2.
ywptcoef[k] = ((max(c1,yc3-ctx.mag(yeps4)+1))+10)*1.5
ywpterm[k] = (max(c1,ctx.mag(L+2)+yc3-ctx.mag(yeps3)+1)+1)+10
# check of power of pi
# computing the fortcoef[mu,k,ell]
xfortcoef={}
for mu in range(0,der+1):
for k in range(0,L):
for ell in range(-2,3*k//2+1):
xfortcoef[mu,k,ell]=0
for mu in range(0,der+1):
for k in range(0,L):
ctx.prec = xwptcoef[k]
for ell in range(0,3*k//2+1):
xfortcoef[mu,k,ell]=xd[mu,k,ell]*Fp[3*k-2*ell]/pipowers[2*k-ell]
xfortcoef[mu,k,ell]=xfortcoef[mu,k,ell]/((2*ctx.j)**ell)
def trunc_a(t):
wp = ctx.prec
ctx.prec = wp + 2
aa = ctx.sqrt(t/(2*ctx.pi))
ctx.prec = wp
return aa
# computing the tcoef[k,ell]
xtcoef={}
for mu in range(0,der+1):
for k in range(0,L):
for ell in range(-2,3*k//2+1):
xtcoef[mu,k,ell]=0
ctx.prec = max(xwptcoef[0],ywptcoef[0])+3
aa= trunc_a(t)
la = -ctx.ln(aa)
for chi in range(0,der+1):
for k in range(0,L):
ctx.prec = xwptcoef[k]
for ell in range(0,3*k//2+1):
xtcoef[chi,k,ell] =0
for mu in range(0, chi+1):
tcoefter=ctx.binomial(chi,mu)*ctx.power(la,mu)*xfortcoef[chi-mu,k,ell]
xtcoef[chi,k,ell] += tcoefter
# COMPUTING THE COEFFICIENTS ytcoef[k,l]
# See II Section 3.9
#
# computing the needed wp
# check of power of pi
# computing the fortcoef[mu,k,ell]
yfortcoef={}
for mu in range(0,der+1):
for k in range(0,L):
for ell in range(-2,3*k//2+1):
yfortcoef[mu,k,ell]=0
for mu in range(0,der+1):
for k in range(0,L):
ctx.prec = ywptcoef[k]
for ell in range(0,3*k//2+1):
yfortcoef[mu,k,ell]=yd[mu,k,ell]*Fp[3*k-2*ell]/pipowers[2*k-ell]
yfortcoef[mu,k,ell]=yfortcoef[mu,k,ell]/((2*ctx.j)**ell)
# computing the tcoef[k,ell]
ytcoef={}
for chi in range(0,der+1):
for k in range(0,L):
for ell in range(-2,3*k//2+1):
ytcoef[chi,k,ell]=0
for chi in range(0,der+1):
for k in range(0,L):
ctx.prec = ywptcoef[k]
for ell in range(0,3*k//2+1):
ytcoef[chi,k,ell] =0
for mu in range(0, chi+1):
tcoefter=ctx.binomial(chi,mu)*ctx.power(la,mu)*yfortcoef[chi-mu,k,ell]
ytcoef[chi,k,ell] += tcoefter
# COMPUTING tv[k,ell]
# See II Section 3.8
#
# a has a good value
ctx.prec = max(xwptcoef[0], ywptcoef[0])+2
av = {}
av[0] = 1
av[1] = av[0]/a
ctx.prec = max(xwptcoef[0],ywptcoef[0])
for k in range(2,L):
av[k] = av[k-1] * av[1]
# Computing the quotients
xtv = {}
for chi in range(0,der+1):
for k in range(0,L):
ctx.prec = xwptcoef[k]
for ell in range(0,3*k//2+1):
xtv[chi,k,ell] = xtcoef[chi,k,ell]* av[k]
# Computing the quotients
ytv = {}
for chi in range(0,der+1):
for k in range(0,L):
ctx.prec = ywptcoef[k]
for ell in range(0,3*k//2+1):
ytv[chi,k,ell] = ytcoef[chi,k,ell]* av[k]
# COMPUTING THE TERMS xterm[k]
# See II Section 3.6
xterm = {}
for chi in range(0,der+1):
for n in range(0,L):
ctx.prec = xwpterm[n]
te = 0
for k in range(0, 3*n//2+1):
te += xtv[chi,n,k]
xterm[chi,n] = te
# COMPUTING THE TERMS yterm[k]
# See II Section 3.6
yterm = {}
for chi in range(0,der+1):
for n in range(0,L):
ctx.prec = ywpterm[n]
te = 0
for k in range(0, 3*n//2+1):
te += ytv[chi,n,k]
yterm[chi,n] = te
# COMPUTING rssum
# See II Section 3.5
xrssum={}
ctx.prec=15
xrsbound = math.sqrt(ctx.pi) * xc /(xb*a)
ctx.prec=15
xwprssum = ctx.mag(4.4*((L+3)**2)*xrsbound / xeps2)
xwprssum = max(xwprssum, ctx.mag(10*(L+1)))
ctx.prec = xwprssum
for chi in range(0,der+1):
xrssum[chi] = 0
for k in range(1,L+1):
xrssum[chi] += xterm[chi,L-k]
yrssum={}
ctx.prec=15
yrsbound = math.sqrt(ctx.pi) * yc /(yb*a)
ctx.prec=15
ywprssum = ctx.mag(4.4*((L+3)**2)*yrsbound / yeps2)
ywprssum = max(ywprssum, ctx.mag(10*(L+1)))
ctx.prec = ywprssum
for chi in range(0,der+1):
yrssum[chi] = 0
for k in range(1,L+1):
yrssum[chi] += yterm[chi,L-k]
# COMPUTING S3
# See II Section 3.19
ctx.prec = 15
A2 = 2**(max(ctx.mag(abs(xrssum[0])), ctx.mag(abs(yrssum[0]))))
eps8 = eps/(3*A2)
T = t *ctx.ln(t/(2*ctx.pi))
xwps3 = 5 + ctx.mag((1+(2/eps8)*ctx.power(a,-xsigma))*T)
ywps3 = 5 + ctx.mag((1+(2/eps8)*ctx.power(a,-ysigma))*T)
ctx.prec = max(xwps3, ywps3)
tpi = t/(2*ctx.pi)
arg = (t/2)*ctx.ln(tpi)-(t/2)-ctx.pi/8
U = ctx.expj(-arg)
a = trunc_a(t)
xasigma = ctx.power(a, -xsigma)
yasigma = ctx.power(a, -ysigma)
xS3 = ((-1)**(N-1)) * xasigma * U
yS3 = ((-1)**(N-1)) * yasigma * U
# COMPUTING S1 the zetasum
# See II Section 3.18
ctx.prec = 15
xwpsum = 4+ ctx.mag((N+ctx.power(N,1-xsigma))*ctx.ln(N) /eps1)
ywpsum = 4+ ctx.mag((N+ctx.power(N,1-ysigma))*ctx.ln(N) /eps1)
wpsum = max(xwpsum, ywpsum)
ctx.prec = wpsum +10
'''
# This can be improved
xS1={}
yS1={}
for chi in range(0,der+1):
xS1[chi] = 0
yS1[chi] = 0
for n in range(1,int(N)+1):
ln = ctx.ln(n)
xexpn = ctx.exp(-ln*(xsigma+ctx.j*t))
yexpn = ctx.conj(1/(n*xexpn))
for chi in range(0,der+1):
pown = ctx.power(-ln, chi)
xterm = pown*xexpn
yterm = pown*yexpn
xS1[chi] += xterm
yS1[chi] += yterm
'''
xS1, yS1 = ctx._zetasum(s, 1, int(N)-1, range(0,der+1), True)
# END OF COMPUTATION of xrz, yrz
# See II Section 3.1
ctx.prec = 15
xabsS1 = abs(xS1[der])
xabsS2 = abs(xrssum[der] * xS3)
xwpend = max(6, wpinitial+ctx.mag(6*(3*xabsS1+7*xabsS2) ) )
ctx.prec = xwpend
xrz={}
for chi in range(0,der+1):
xrz[chi] = xS1[chi]+xrssum[chi]*xS3
ctx.prec = 15
yabsS1 = abs(yS1[der])
yabsS2 = abs(yrssum[der] * yS3)
ywpend = max(6, wpinitial+ctx.mag(6*(3*yabsS1+7*yabsS2) ) )
ctx.prec = ywpend
yrz={}
for chi in range(0,der+1):
yrz[chi] = yS1[chi]+yrssum[chi]*yS3
yrz[chi] = ctx.conj(yrz[chi])
ctx.prec = wpinitial
return xrz, yrz
def Rzeta_set(ctx, s, derivatives=[0]):
r"""
Computes several derivatives of the auxiliary function of Riemann `R(s)`.
**Definition**
The function is defined by
.. math ::
\begin{equation}
{\mathop{\mathcal R }\nolimits}(s)=
\int_{0\swarrow1}\frac{x^{-s} e^{\pi i x^2}}{e^{\pi i x}-
e^{-\pi i x}}\,dx
\end{equation}
To this function we apply the Riemann-Siegel expansion.
"""
der = max(derivatives)
# First we take the value of ctx.prec
# During the computation we will change ctx.prec, and finally we will
# restaurate the initial value
wpinitial = ctx.prec
# Take the real and imaginary part of s
t = ctx._im(s)
sigma = ctx._re(s)
# Now compute several parameter that appear on the program
ctx.prec = 15
a = ctx.sqrt(t/(2*ctx.pi)) # Careful
asigma = ctx.power(a, sigma) # Careful
# We need a simple bound A1 < asigma (see II Section 3.1 and 3.3)
A1 = ctx.power(2, ctx.mag(asigma)-1)
# We compute various epsilon's (see II end of Section 3.1)
eps = ctx.power(2, -wpinitial)
eps1 = eps/6.
eps2 = eps * A1/3.
# COMPUTING SOME COEFFICIENTS THAT DEPENDS
# ON sigma
# constant b and c (see I Theorem 2 formula (26) )
# coefficients A and B1 (see I Section 6.1 equation (50))
# here we not need high precision
ctx.prec = 15
if sigma > 0:
b = 2.
c = math.pow(9,sigma)/4.44288
# 4.44288 =(math.sqrt(2)*math.pi)
A = math.pow(9,sigma)
B1 = 1
else:
b = 2.25158 # math.sqrt( (3-2* math.log(2))*math.pi )
c = math.pow(2,-sigma)/4.44288
A = math.pow(2,-sigma)
B1 = 1.10789 # = 2*sqrt(1-log(2))
# COMPUTING L THE NUMBER OF TERMS NEEDED IN THE RIEMANN-SIEGEL
# CORRECTION
# See II Section 3.2
ctx.prec = 15
L = 1
while 3*c*ctx.gamma(L*0.5) * ctx.power(b*a,-L) >= eps2:
L = L+1
L = max(2,L)
# The number L has to satify some conditions.
# If not RS can not compute Rzeta(s) with the prescribed precision
# (see II, Section 3.2 condition (20) ) and
# (II, Section 3.3 condition (22) ). Also we have added
# an additional technical condition in Section 3.17 Proposition 17
if ((3*L >= 2*a*a/25.) or (3*L+2+sigma<0) or (abs(sigma)> a/2.)):
#print 'Error Riemann-Siegel can not compute with such precision'
ctx.prec = wpinitial
raise NotImplementedError("Riemann-Siegel can not compute with such precision")
# INITIALIZATION (CONTINUATION)
#
# eps3 is the constant defined on (II, Section 3.5 equation (27) )
# each term of the RS correction must be computed with error <= eps3
eps3 = eps2/(4*L)
# eps4 is defined on (II Section 3.6 equation (30) )
# each component of the formula (II Section 3.6 equation (29) )
# must be computed with error <= eps4
eps4 = eps3/(3*L)
# COMPUTING M. NUMBER OF DERIVATIVES Fp[m] TO COMPUTE
M = aux_M_Fp(ctx, A, eps4, a, B1, L)
Fp = {}
for n in range(M, 3*L-2):
Fp[n] = 0
# But I have not seen an instance of M != 3*L-3
#
# DETERMINATION OF J THE NUMBER OF TERMS NEEDED
# IN THE TAYLOR SERIES OF F.
# See II Section 3.11 equation (49))
h1 = eps4/(632*A)
h2 = ctx.pi*ctx.pi*B1*a *ctx.sqrt(3)*math.e*math.e
h2 = h1 * ctx.power((h2/M**2),(M-1)/3) / M
h3 = min(h1,h2)
J=12
jvalue = (2*ctx.pi)**J / ctx.gamma(J+1)
while jvalue > h3:
J = J+1
jvalue = (2*ctx.pi)*jvalue/J
# COMPUTING eps5[m] for 1 <= m <= 21
# See II Section 10 equation (43)
eps5={}
foreps5 = math.pi*math.pi*B1*a
for m in range(0,22):
aux1 = math.pow(foreps5, m/3)/(316.*A)
aux2 = ctx.gamma(m+1)/ctx.gamma(m/3.0+0.5)
aux2 = math.sqrt(aux2)
eps5[m] = aux1*aux2*eps4
# COMPUTING wpfp
# See II Section 3.13 equation (59)
twenty = min(3*L-3, 21)+1
aux = 6812*J
wpfp = ctx.mag(44*J)
for m in range(0, twenty):
wpfp = max(wpfp, ctx.mag(aux*ctx.gamma(m+1)/eps5[m]))
# COMPUTING N AND p
# See II Section
ctx.prec = wpfp + ctx.mag(t) + 20
a = ctx.sqrt(t/(2*ctx.pi))
N = ctx.floor(a)
p = 1-2*(a-N)
# now we get a rounded version of p to the precision wpfp
# this possibly is not necessary
num = ctx.floor(p*(ctx.mpf(2)**wpfp))
difference = p * (ctx.mpf(2)**wpfp)-num
if difference < 0.5:
num = num
else:
num = num+1
p = ctx.convert(num * (ctx.mpf(2)**(-wpfp)))
# COMPUTING THE COEFFICIENTS c[n] = cc[n]
# We shall use the notation cc[n], since there is
# a constant that is called c
# See II Section 3.14
# We compute the coefficients and also save then in a
# cache. The bulk of the computation is passed to
# the function coef()
#
# eps6 is defined in II Section 3.13 equation (58)
eps6 = ctx.power(2*ctx.pi, J)/(ctx.gamma(J+1)*3*J)
# Now we compute the coefficients
cc={}
cont={}
cont, pipowers = coef(ctx, J, eps6)
cc = cont.copy() # we need a copy since we have
Fp={}
for n in range(M, 3*L-2):
Fp[n] = 0
ctx.prec = wpfp
for m in range(0,M+1):
sumP = 0
for k in range(2*J-m-1,-1,-1):
sumP = (sumP * p) + cc[k]
Fp[m] = sumP
# preparation of the new coefficients
for k in range(0, 2*J-m-1):
cc[k] = (k+1) * cc[k+1]
# COMPUTING THE NUMBERS d[n,k]
# See II Section 3.17
# First we compute the working precisions wpd[k]
# Se II equation (92)
wpd = {}
d1 = max(6, ctx.mag(40*L*L))
d2 = 13+ctx.mag((1+abs(sigma))*A)-ctx.mag(eps4)-1
const = ctx.ln(8/(ctx.pi*ctx.pi*a*a*B1*B1)) /2
for n in range(0,L):
d3 = ctx.mag(ctx.sqrt(ctx.gamma(n-0.5)))-ctx.floor(n*const)+d2
wpd[n] = max(d3,d1)
# procedure of II Section 3.17
ctx.prec = wpd[1]+10
psigma = 1-(2*sigma)
d = {}
d[0,0,-2]=0; d[0,0,-1]=0; d[0,0,0]=1; d[0,0,1]=0
d[0,-1,-2]=0; d[0,-1,-1]=0; d[0,-1,0]=1; d[0,-1,1]=0
for n in range(1,L):
ctx.prec = wpd[n]+10
for k in range(0,3*n//2+1):
m = 3*n-2*k
if (m!=0):
m1 = ctx.one/m
c1 = m1/4
c2 = (psigma*m1)/2
c3 = -(m+1)
d[0,n,k] = c3*d[0,n-1,k-2]+c1*d[0,n-1,k]+c2*d[0,n-1,k-1]
else:
d[0,n,k]=0
for r in range(0,k):
add = d[0,n,r]*(ctx.one*ctx.fac(2*k-2*r)/ctx.fac(k-r))
d[0,n,k] -= ((-1)**(k-r))*add
d[0,n,-2]=0; d[0,n,-1]=0; d[0,n,3*n//2+1]=0
for mu in range(-2,der+1):
for n in range(-2,L):
for k in range(-3,max(1,3*n//2+2)):
if ((mu<0)or (n<0) or(k<0)or (k>3*n//2)):
d[mu,n,k] = 0
for mu in range(1,der+1):
for n in range(0,L):
ctx.prec = wpd[n]+10
for k in range(0,3*n//2+1):
aux=(2*mu-2)*d[mu-2,n-2,k-3]+2*(sigma+n-2)*d[mu-1,n-2,k-3]
d[mu,n,k] = aux - d[mu-1,n-1,k-1]
# COMPUTING THE COEFFICIENTS t[k,l]
# See II Section 3.9
#
# computing the needed wp
wptcoef = {}
wpterm = {}
ctx.prec = 15
c1 = ctx.mag(40*(L+2))
c2 = ctx.mag(68*(L+2)*A)
c4 = ctx.mag(B1*a*math.sqrt(ctx.pi))-1
for k in range(0,L):
c3 = c2 - k*c4+ctx.mag(ctx.fac(k+0.5))/2.
wptcoef[k] = max(c1,c3-ctx.mag(eps4)+1)+1 +10
wpterm[k] = max(c1,ctx.mag(L+2)+c3-ctx.mag(eps3)+1)+1 +10
# check of power of pi
# computing the fortcoef[mu,k,ell]
fortcoef={}
for mu in derivatives:
for k in range(0,L):
for ell in range(-2,3*k//2+1):
fortcoef[mu,k,ell]=0
for mu in derivatives:
for k in range(0,L):
ctx.prec = wptcoef[k]
for ell in range(0,3*k//2+1):
fortcoef[mu,k,ell]=d[mu,k,ell]*Fp[3*k-2*ell]/pipowers[2*k-ell]
fortcoef[mu,k,ell]=fortcoef[mu,k,ell]/((2*ctx.j)**ell)
def trunc_a(t):
wp = ctx.prec
ctx.prec = wp + 2
aa = ctx.sqrt(t/(2*ctx.pi))
ctx.prec = wp
return aa
# computing the tcoef[chi,k,ell]
tcoef={}
for chi in derivatives:
for k in range(0,L):
for ell in range(-2,3*k//2+1):
tcoef[chi,k,ell]=0
ctx.prec = wptcoef[0]+3
aa = trunc_a(t)
la = -ctx.ln(aa)
for chi in derivatives:
for k in range(0,L):
ctx.prec = wptcoef[k]
for ell in range(0,3*k//2+1):
tcoef[chi,k,ell] = 0
for mu in range(0, chi+1):
tcoefter = ctx.binomial(chi,mu) * la**mu * \
fortcoef[chi-mu,k,ell]
tcoef[chi,k,ell] += tcoefter
# COMPUTING tv[k,ell]
# See II Section 3.8
# Computing the powers av[k] = a**(-k)
ctx.prec = wptcoef[0] + 2
# a has a good value of a.
# See II Section 3.6
av = {}
av[0] = 1
av[1] = av[0]/a
ctx.prec = wptcoef[0]
for k in range(2,L):
av[k] = av[k-1] * av[1]
# Computing the quotients
tv = {}
for chi in derivatives:
for k in range(0,L):
ctx.prec = wptcoef[k]
for ell in range(0,3*k//2+1):
tv[chi,k,ell] = tcoef[chi,k,ell]* av[k]
# COMPUTING THE TERMS term[k]
# See II Section 3.6
term = {}
for chi in derivatives:
for n in range(0,L):
ctx.prec = wpterm[n]
te = 0
for k in range(0, 3*n//2+1):
te += tv[chi,n,k]
term[chi,n] = te
# COMPUTING rssum
# See II Section 3.5
rssum={}
ctx.prec=15
rsbound = math.sqrt(ctx.pi) * c /(b*a)
ctx.prec=15
wprssum = ctx.mag(4.4*((L+3)**2)*rsbound / eps2)
wprssum = max(wprssum, ctx.mag(10*(L+1)))
ctx.prec = wprssum
for chi in derivatives:
rssum[chi] = 0
for k in range(1,L+1):
rssum[chi] += term[chi,L-k]
# COMPUTING S3
# See II Section 3.19
ctx.prec = 15
A2 = 2**(ctx.mag(rssum[0]))
eps8 = eps/(3* A2)
T = t * ctx.ln(t/(2*ctx.pi))
wps3 = 5 + ctx.mag((1+(2/eps8)*ctx.power(a,-sigma))*T)
ctx.prec = wps3
tpi = t/(2*ctx.pi)
arg = (t/2)*ctx.ln(tpi)-(t/2)-ctx.pi/8
U = ctx.expj(-arg)
a = trunc_a(t)
asigma = ctx.power(a, -sigma)
S3 = ((-1)**(N-1)) * asigma * U
# COMPUTING S1 the zetasum
# See II Section 3.18
ctx.prec = 15
wpsum = 4 + ctx.mag((N+ctx.power(N,1-sigma))*ctx.ln(N)/eps1)
ctx.prec = wpsum + 10
'''
# This can be improved
S1 = {}
for chi in derivatives:
S1[chi] = 0
for n in range(1,int(N)+1):
ln = ctx.ln(n)
expn = ctx.exp(-ln*(sigma+ctx.j*t))
for chi in derivatives:
term = ctx.power(-ln, chi)*expn
S1[chi] += term
'''
S1 = ctx._zetasum(s, 1, int(N)-1, derivatives)[0]
# END OF COMPUTATION
# See II Section 3.1
ctx.prec = 15
absS1 = abs(S1[der])
absS2 = abs(rssum[der] * S3)
wpend = max(6, wpinitial + ctx.mag(6*(3*absS1+7*absS2)))
ctx.prec = wpend
rz = {}
for chi in derivatives:
rz[chi] = S1[chi]+rssum[chi]*S3
ctx.prec = wpinitial
return rz
def z_half(ctx,t,der=0):
r"""
z_half(t,der=0) Computes Z^(der)(t)
"""
s=ctx.mpf('0.5')+ctx.j*t
wpinitial = ctx.prec
ctx.prec = 15
tt = t/(2*ctx.pi)
wptheta = wpinitial +1 + ctx.mag(3*(tt**1.5)*ctx.ln(tt))
wpz = wpinitial + 1 + ctx.mag(12*tt*ctx.ln(tt))
ctx.prec = wptheta
theta = ctx.siegeltheta(t)
ctx.prec = wpz
rz = Rzeta_set(ctx,s, range(der+1))
if der > 0: ps1 = ctx._re(ctx.psi(0,s/2)/2 - ctx.ln(ctx.pi)/2)
if der > 1: ps2 = ctx._re(ctx.j*ctx.psi(1,s/2)/4)
if der > 2: ps3 = ctx._re(-ctx.psi(2,s/2)/8)
if der > 3: ps4 = ctx._re(-ctx.j*ctx.psi(3,s/2)/16)
exptheta = ctx.expj(theta)
if der == 0:
z = 2*exptheta*rz[0]
if der == 1:
zf = 2j*exptheta
z = zf*(ps1*rz[0]+rz[1])
if der == 2:
zf = 2 * exptheta
z = -zf*(2*rz[1]*ps1+rz[0]*ps1**2+rz[2]-ctx.j*rz[0]*ps2)
if der == 3:
zf = -2j*exptheta
z = 3*rz[1]*ps1**2+rz[0]*ps1**3+3*ps1*rz[2]
z = zf*(z-3j*rz[1]*ps2-3j*rz[0]*ps1*ps2+rz[3]-rz[0]*ps3)
if der == 4:
zf = 2*exptheta
z = 4*rz[1]*ps1**3+rz[0]*ps1**4+6*ps1**2*rz[2]
z = z-12j*rz[1]*ps1*ps2-6j*rz[0]*ps1**2*ps2-6j*rz[2]*ps2-3*rz[0]*ps2*ps2
z = z + 4*ps1*rz[3]-4*rz[1]*ps3-4*rz[0]*ps1*ps3+rz[4]+ctx.j*rz[0]*ps4
z = zf*z
ctx.prec = wpinitial
return ctx._re(z)
def zeta_half(ctx, s, k=0):
"""
zeta_half(s,k=0) Computes zeta^(k)(s) when Re s = 0.5
"""
wpinitial = ctx.prec
sigma = ctx._re(s)
t = ctx._im(s)
#--- compute wptheta, wpR, wpbasic ---
ctx.prec = 53
# X see II Section 3.21 (109) and (110)
if sigma > 0:
X = ctx.sqrt(abs(s))
else:
X = (2*ctx.pi)**(sigma-1) * abs(1-s)**(0.5-sigma)
# M1 see II Section 3.21 (111) and (112)
if sigma > 0:
M1 = 2*ctx.sqrt(t/(2*ctx.pi))
else:
M1 = 4 * t * X
# T see II Section 3.21 (113)
abst = abs(0.5-s)
T = 2* abst*math.log(abst)
# computing wpbasic, wptheta, wpR see II Section 3.21
wpbasic = max(6,3+ctx.mag(t))
wpbasic2 = 2+ctx.mag(2.12*M1+21.2*M1*X+1.3*M1*X*T)+wpinitial+1
wpbasic = max(wpbasic, wpbasic2)
wptheta = max(4, 3+ctx.mag(2.7*M1*X)+wpinitial+1)
wpR = 3+ctx.mag(1.1+2*X)+wpinitial+1
ctx.prec = wptheta
theta = ctx.siegeltheta(t-ctx.j*(sigma-ctx.mpf('0.5')))
if k > 0: ps1 = (ctx._re(ctx.psi(0,s/2)))/2 - ctx.ln(ctx.pi)/2
if k > 1: ps2 = -(ctx._im(ctx.psi(1,s/2)))/4
if k > 2: ps3 = -(ctx._re(ctx.psi(2,s/2)))/8
if k > 3: ps4 = (ctx._im(ctx.psi(3,s/2)))/16
ctx.prec = wpR
xrz = Rzeta_set(ctx,s,range(k+1))
yrz={}
for chi in range(0,k+1):
yrz[chi] = ctx.conj(xrz[chi])
ctx.prec = wpbasic
exptheta = ctx.expj(-2*theta)
if k==0:
zv = xrz[0]+exptheta*yrz[0]
if k==1:
zv1 = -yrz[1] - 2*yrz[0]*ps1
zv = xrz[1] + exptheta*zv1
if k==2:
zv1 = 4*yrz[1]*ps1+4*yrz[0]*(ps1**2)+yrz[2]+2j*yrz[0]*ps2
zv = xrz[2]+exptheta*zv1
if k==3:
zv1 = -12*yrz[1]*ps1**2-8*yrz[0]*ps1**3-6*yrz[2]*ps1-6j*yrz[1]*ps2
zv1 = zv1 - 12j*yrz[0]*ps1*ps2-yrz[3]+2*yrz[0]*ps3
zv = xrz[3]+exptheta*zv1
if k == 4:
zv1 = 32*yrz[1]*ps1**3 +16*yrz[0]*ps1**4+24*yrz[2]*ps1**2
zv1 = zv1 +48j*yrz[1]*ps1*ps2+48j*yrz[0]*(ps1**2)*ps2
zv1 = zv1+12j*yrz[2]*ps2-12*yrz[0]*ps2**2+8*yrz[3]*ps1-8*yrz[1]*ps3
zv1 = zv1-16*yrz[0]*ps1*ps3+yrz[4]-2j*yrz[0]*ps4
zv = xrz[4]+exptheta*zv1
ctx.prec = wpinitial
return zv
def zeta_offline(ctx, s, k=0):
"""
Computes zeta^(k)(s) off the line
"""
wpinitial = ctx.prec
sigma = ctx._re(s)
t = ctx._im(s)
#--- compute wptheta, wpR, wpbasic ---
ctx.prec = 53
# X see II Section 3.21 (109) and (110)
if sigma > 0:
X = ctx.power(abs(s), 0.5)
else:
X = ctx.power(2*ctx.pi, sigma-1)*ctx.power(abs(1-s),0.5-sigma)
# M1 see II Section 3.21 (111) and (112)
if (sigma > 0):
M1 = 2*ctx.sqrt(t/(2*ctx.pi))
else:
M1 = 4 * t * X
# M2 see II Section 3.21 (111) and (112)
if (1-sigma > 0):
M2 = 2*ctx.sqrt(t/(2*ctx.pi))
else:
M2 = 4*t*ctx.power(2*ctx.pi, -sigma)*ctx.power(abs(s),sigma-0.5)
# T see II Section 3.21 (113)
abst = abs(0.5-s)
T = 2* abst*math.log(abst)
# computing wpbasic, wptheta, wpR see II Section 3.21
wpbasic = max(6,3+ctx.mag(t))
wpbasic2 = 2+ctx.mag(2.12*M1+21.2*M2*X+1.3*M2*X*T)+wpinitial+1
wpbasic = max(wpbasic, wpbasic2)
wptheta = max(4, 3+ctx.mag(2.7*M2*X)+wpinitial+1)
wpR = 3+ctx.mag(1.1+2*X)+wpinitial+1
ctx.prec = wptheta
theta = ctx.siegeltheta(t-ctx.j*(sigma-ctx.mpf('0.5')))
s1 = s
s2 = ctx.conj(1-s1)
ctx.prec = wpR
xrz, yrz = Rzeta_simul(ctx, s, k)
if k > 0: ps1 = (ctx.psi(0,s1/2)+ctx.psi(0,(1-s1)/2))/4 - ctx.ln(ctx.pi)/2
if k > 1: ps2 = ctx.j*(ctx.psi(1,s1/2)-ctx.psi(1,(1-s1)/2))/8
if k > 2: ps3 = -(ctx.psi(2,s1/2)+ctx.psi(2,(1-s1)/2))/16
if k > 3: ps4 = -ctx.j*(ctx.psi(3,s1/2)-ctx.psi(3,(1-s1)/2))/32
ctx.prec = wpbasic
exptheta = ctx.expj(-2*theta)
if k == 0:
zv = xrz[0]+exptheta*yrz[0]
if k == 1:
zv1 = -yrz[1]-2*yrz[0]*ps1
zv = xrz[1]+exptheta*zv1
if k == 2:
zv1 = 4*yrz[1]*ps1+4*yrz[0]*(ps1**2) +yrz[2]+2j*yrz[0]*ps2
zv = xrz[2]+exptheta*zv1
if k == 3:
zv1 = -12*yrz[1]*ps1**2 -8*yrz[0]*ps1**3-6*yrz[2]*ps1-6j*yrz[1]*ps2
zv1 = zv1 - 12j*yrz[0]*ps1*ps2-yrz[3]+2*yrz[0]*ps3
zv = xrz[3]+exptheta*zv1
if k == 4:
zv1 = 32*yrz[1]*ps1**3 +16*yrz[0]*ps1**4+24*yrz[2]*ps1**2
zv1 = zv1 +48j*yrz[1]*ps1*ps2+48j*yrz[0]*(ps1**2)*ps2
zv1 = zv1+12j*yrz[2]*ps2-12*yrz[0]*ps2**2+8*yrz[3]*ps1-8*yrz[1]*ps3
zv1 = zv1-16*yrz[0]*ps1*ps3+yrz[4]-2j*yrz[0]*ps4
zv = xrz[4]+exptheta*zv1
ctx.prec = wpinitial
return zv
def z_offline(ctx, w, k=0):
r"""
Computes Z(w) and its derivatives off the line
"""
s = ctx.mpf('0.5')+ctx.j*w
s1 = s
s2 = ctx.conj(1-s1)
wpinitial = ctx.prec
ctx.prec = 35
# X see II Section 3.21 (109) and (110)
# M1 see II Section 3.21 (111) and (112)
if (ctx._re(s1) >= 0):
M1 = 2*ctx.sqrt(ctx._im(s1)/(2 * ctx.pi))
X = ctx.sqrt(abs(s1))
else:
X = (2*ctx.pi)**(ctx._re(s1)-1) * abs(1-s1)**(0.5-ctx._re(s1))
M1 = 4 * ctx._im(s1)*X
# M2 see II Section 3.21 (111) and (112)
if (ctx._re(s2) >= 0):
M2 = 2*ctx.sqrt(ctx._im(s2)/(2 * ctx.pi))
else:
M2 = 4 * ctx._im(s2)*(2*ctx.pi)**(ctx._re(s2)-1)*abs(1-s2)**(0.5-ctx._re(s2))
# T see II Section 3.21 Prop. 27
T = 2*abs(ctx.siegeltheta(w))
# defining some precisions
# see II Section 3.22 (115), (116), (117)
aux1 = ctx.sqrt(X)
aux2 = aux1*(M1+M2)
aux3 = 3 +wpinitial
wpbasic = max(6, 3+ctx.mag(T), ctx.mag(aux2*(26+2*T))+aux3)
wptheta = max(4,ctx.mag(2.04*aux2)+aux3)
wpR = ctx.mag(4*aux1)+aux3
# now the computations
ctx.prec = wptheta
theta = ctx.siegeltheta(w)
ctx.prec = wpR
xrz, yrz = Rzeta_simul(ctx,s,k)
pta = 0.25 + 0.5j*w
ptb = 0.25 - 0.5j*w
if k > 0: ps1 = 0.25*(ctx.psi(0,pta)+ctx.psi(0,ptb)) - ctx.ln(ctx.pi)/2
if k > 1: ps2 = (1j/8)*(ctx.psi(1,pta)-ctx.psi(1,ptb))
if k > 2: ps3 = (-1./16)*(ctx.psi(2,pta)+ctx.psi(2,ptb))
if k > 3: ps4 = (-1j/32)*(ctx.psi(3,pta)-ctx.psi(3,ptb))
ctx.prec = wpbasic
exptheta = ctx.expj(theta)
if k == 0:
zv = exptheta*xrz[0]+yrz[0]/exptheta
j = ctx.j
if k == 1:
zv = j*exptheta*(xrz[1]+xrz[0]*ps1)-j*(yrz[1]+yrz[0]*ps1)/exptheta
if k == 2:
zv = exptheta*(-2*xrz[1]*ps1-xrz[0]*ps1**2-xrz[2]+j*xrz[0]*ps2)
zv =zv + (-2*yrz[1]*ps1-yrz[0]*ps1**2-yrz[2]-j*yrz[0]*ps2)/exptheta
if k == 3:
zv1 = -3*xrz[1]*ps1**2-xrz[0]*ps1**3-3*xrz[2]*ps1+j*3*xrz[1]*ps2
zv1 = (zv1+ 3j*xrz[0]*ps1*ps2-xrz[3]+xrz[0]*ps3)*j*exptheta
zv2 = 3*yrz[1]*ps1**2+yrz[0]*ps1**3+3*yrz[2]*ps1+j*3*yrz[1]*ps2
zv2 = j*(zv2 + 3j*yrz[0]*ps1*ps2+ yrz[3]-yrz[0]*ps3)/exptheta
zv = zv1+zv2
if k == 4:
zv1 = 4*xrz[1]*ps1**3+xrz[0]*ps1**4 + 6*xrz[2]*ps1**2
zv1 = zv1-12j*xrz[1]*ps1*ps2-6j*xrz[0]*ps1**2*ps2-6j*xrz[2]*ps2
zv1 = zv1-3*xrz[0]*ps2*ps2+4*xrz[3]*ps1-4*xrz[1]*ps3-4*xrz[0]*ps1*ps3
zv1 = zv1+xrz[4]+j*xrz[0]*ps4
zv2 = 4*yrz[1]*ps1**3+yrz[0]*ps1**4 + 6*yrz[2]*ps1**2
zv2 = zv2+12j*yrz[1]*ps1*ps2+6j*yrz[0]*ps1**2*ps2+6j*yrz[2]*ps2
zv2 = zv2-3*yrz[0]*ps2*ps2+4*yrz[3]*ps1-4*yrz[1]*ps3-4*yrz[0]*ps1*ps3
zv2 = zv2+yrz[4]-j*yrz[0]*ps4
zv = exptheta*zv1+zv2/exptheta
ctx.prec = wpinitial
return zv
@defun
def rs_zeta(ctx, s, derivative=0, **kwargs):
if derivative > 4:
raise NotImplementedError
s = ctx.convert(s)
re = ctx._re(s); im = ctx._im(s)
if im < 0:
z = ctx.conj(ctx.rs_zeta(ctx.conj(s), derivative))
return z
critical_line = (re == 0.5)
if critical_line:
return zeta_half(ctx, s, derivative)
else:
return zeta_offline(ctx, s, derivative)
@defun
def rs_z(ctx, w, derivative=0):
w = ctx.convert(w)
re = ctx._re(w); im = ctx._im(w)
if re < 0:
return rs_z(ctx, -w, derivative)
critical_line = (im == 0)
if critical_line :
return z_half(ctx, w, derivative)
else:
return z_offline(ctx, w, derivative)
| 46,184 | 31.895299 | 90 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/functions/zeta.py
|
from ..libmp.backend import xrange, print_
from .functions import defun, defun_wrapped, defun_static
@defun
def stieltjes(ctx, n, a=1):
n = ctx.convert(n)
a = ctx.convert(a)
if n < 0:
return ctx.bad_domain("Stieltjes constants defined for n >= 0")
if hasattr(ctx, "stieltjes_cache"):
stieltjes_cache = ctx.stieltjes_cache
else:
stieltjes_cache = ctx.stieltjes_cache = {}
if a == 1:
if n == 0:
return +ctx.euler
if n in stieltjes_cache:
prec, s = stieltjes_cache[n]
if prec >= ctx.prec:
return +s
mag = 1
def f(x):
xa = x/a
v = (xa-ctx.j)*ctx.ln(a-ctx.j*x)**n/(1+xa**2)/(ctx.exp(2*ctx.pi*x)-1)
return ctx._re(v) / mag
orig = ctx.prec
try:
# Normalize integrand by approx. magnitude to
# speed up quadrature (which uses absolute error)
if n > 50:
ctx.prec = 20
mag = ctx.quad(f, [0,ctx.inf], maxdegree=3)
ctx.prec = orig + 10 + int(n**0.5)
s = ctx.quad(f, [0,ctx.inf], maxdegree=20)
v = ctx.ln(a)**n/(2*a) - ctx.ln(a)**(n+1)/(n+1) + 2*s/a*mag
finally:
ctx.prec = orig
if a == 1 and ctx.isint(n):
stieltjes_cache[n] = (ctx.prec, v)
return +v
@defun_wrapped
def siegeltheta(ctx, t, derivative=0):
d = int(derivative)
if (t == ctx.inf or t == ctx.ninf):
if d < 2:
if t == ctx.ninf and d == 0:
return ctx.ninf
return ctx.inf
else:
return ctx.zero
if d == 0:
if ctx._im(t):
# XXX: cancellation occurs
a = ctx.loggamma(0.25+0.5j*t)
b = ctx.loggamma(0.25-0.5j*t)
return -ctx.ln(ctx.pi)/2*t - 0.5j*(a-b)
else:
if ctx.isinf(t):
return t
return ctx._im(ctx.loggamma(0.25+0.5j*t)) - ctx.ln(ctx.pi)/2*t
if d > 0:
a = (-0.5j)**(d-1)*ctx.polygamma(d-1, 0.25-0.5j*t)
b = (0.5j)**(d-1)*ctx.polygamma(d-1, 0.25+0.5j*t)
if ctx._im(t):
if d == 1:
return -0.5*ctx.log(ctx.pi)+0.25*(a+b)
else:
return 0.25*(a+b)
else:
if d == 1:
return ctx._re(-0.5*ctx.log(ctx.pi)+0.25*(a+b))
else:
return ctx._re(0.25*(a+b))
@defun_wrapped
def grampoint(ctx, n):
# asymptotic expansion, from
# http://mathworld.wolfram.com/GramPoint.html
g = 2*ctx.pi*ctx.exp(1+ctx.lambertw((8*n+1)/(8*ctx.e)))
return ctx.findroot(lambda t: ctx.siegeltheta(t)-ctx.pi*n, g)
@defun_wrapped
def siegelz(ctx, t, **kwargs):
d = int(kwargs.get("derivative", 0))
t = ctx.convert(t)
t1 = ctx._re(t)
t2 = ctx._im(t)
prec = ctx.prec
try:
if abs(t1) > 500*prec and t2**2 < t1:
v = ctx.rs_z(t, d)
if ctx._is_real_type(t):
return ctx._re(v)
return v
except NotImplementedError:
pass
ctx.prec += 21
e1 = ctx.expj(ctx.siegeltheta(t))
z = ctx.zeta(0.5+ctx.j*t)
if d == 0:
v = e1*z
ctx.prec=prec
if ctx._is_real_type(t):
return ctx._re(v)
return +v
z1 = ctx.zeta(0.5+ctx.j*t, derivative=1)
theta1 = ctx.siegeltheta(t, derivative=1)
if d == 1:
v = ctx.j*e1*(z1+z*theta1)
ctx.prec=prec
if ctx._is_real_type(t):
return ctx._re(v)
return +v
z2 = ctx.zeta(0.5+ctx.j*t, derivative=2)
theta2 = ctx.siegeltheta(t, derivative=2)
comb1 = theta1**2-ctx.j*theta2
if d == 2:
def terms():
return [2*z1*theta1, z2, z*comb1]
v = ctx.sum_accurately(terms, 1)
v = -e1*v
ctx.prec = prec
if ctx._is_real_type(t):
return ctx._re(v)
return +v
ctx.prec += 10
z3 = ctx.zeta(0.5+ctx.j*t, derivative=3)
theta3 = ctx.siegeltheta(t, derivative=3)
comb2 = theta1**3-3*ctx.j*theta1*theta2-theta3
if d == 3:
def terms():
return [3*theta1*z2, 3*z1*comb1, z3+z*comb2]
v = ctx.sum_accurately(terms, 1)
v = -ctx.j*e1*v
ctx.prec = prec
if ctx._is_real_type(t):
return ctx._re(v)
return +v
z4 = ctx.zeta(0.5+ctx.j*t, derivative=4)
theta4 = ctx.siegeltheta(t, derivative=4)
def terms():
return [theta1**4, -6*ctx.j*theta1**2*theta2, -3*theta2**2,
-4*theta1*theta3, ctx.j*theta4]
comb3 = ctx.sum_accurately(terms, 1)
if d == 4:
def terms():
return [6*theta1**2*z2, -6*ctx.j*z2*theta2, 4*theta1*z3,
4*z1*comb2, z4, z*comb3]
v = ctx.sum_accurately(terms, 1)
v = e1*v
ctx.prec = prec
if ctx._is_real_type(t):
return ctx._re(v)
return +v
if d > 4:
h = lambda x: ctx.siegelz(x, derivative=4)
return ctx.diff(h, t, n=d-4)
_zeta_zeros = [
14.134725142,21.022039639,25.010857580,30.424876126,32.935061588,
37.586178159,40.918719012,43.327073281,48.005150881,49.773832478,
52.970321478,56.446247697,59.347044003,60.831778525,65.112544048,
67.079810529,69.546401711,72.067157674,75.704690699,77.144840069,
79.337375020,82.910380854,84.735492981,87.425274613,88.809111208,
92.491899271,94.651344041,95.870634228,98.831194218,101.317851006,
103.725538040,105.446623052,107.168611184,111.029535543,111.874659177,
114.320220915,116.226680321,118.790782866,121.370125002,122.946829294,
124.256818554,127.516683880,129.578704200,131.087688531,133.497737203,
134.756509753,138.116042055,139.736208952,141.123707404,143.111845808,
146.000982487,147.422765343,150.053520421,150.925257612,153.024693811,
156.112909294,157.597591818,158.849988171,161.188964138,163.030709687,
165.537069188,167.184439978,169.094515416,169.911976479,173.411536520,
174.754191523,176.441434298,178.377407776,179.916484020,182.207078484,
184.874467848,185.598783678,187.228922584,189.416158656,192.026656361,
193.079726604,195.265396680,196.876481841,198.015309676,201.264751944,
202.493594514,204.189671803,205.394697202,207.906258888,209.576509717,
211.690862595,213.347919360,214.547044783,216.169538508,219.067596349,
220.714918839,221.430705555,224.007000255,224.983324670,227.421444280,
229.337413306,231.250188700,231.987235253,233.693404179,236.524229666,
]
def _load_zeta_zeros(url):
import urllib
d = urllib.urlopen(url)
L = [float(x) for x in d.readlines()]
# Sanity check
assert round(L[0]) == 14
_zeta_zeros[:] = L
@defun
def oldzetazero(ctx, n, url='http://www.dtc.umn.edu/~odlyzko/zeta_tables/zeros1'):
n = int(n)
if n < 0:
return ctx.zetazero(-n).conjugate()
if n == 0:
raise ValueError("n must be nonzero")
if n > len(_zeta_zeros) and n <= 100000:
_load_zeta_zeros(url)
if n > len(_zeta_zeros):
raise NotImplementedError("n too large for zetazeros")
return ctx.mpc(0.5, ctx.findroot(ctx.siegelz, _zeta_zeros[n-1]))
@defun_wrapped
def riemannr(ctx, x):
if x == 0:
return ctx.zero
# Check if a simple asymptotic estimate is accurate enough
if abs(x) > 1000:
a = ctx.li(x)
b = 0.5*ctx.li(ctx.sqrt(x))
if abs(b) < abs(a)*ctx.eps:
return a
if abs(x) < 0.01:
# XXX
ctx.prec += int(-ctx.log(abs(x),2))
# Sum Gram's series
s = t = ctx.one
u = ctx.ln(x)
k = 1
while abs(t) > abs(s)*ctx.eps:
t = t * u / k
s += t / (k * ctx._zeta_int(k+1))
k += 1
return s
@defun_static
def primepi(ctx, x):
x = int(x)
if x < 2:
return 0
return len(ctx.list_primes(x))
# TODO: fix the interface wrt contexts
@defun_wrapped
def primepi2(ctx, x):
x = int(x)
if x < 2:
return ctx._iv.zero
if x < 2657:
return ctx._iv.mpf(ctx.primepi(x))
mid = ctx.li(x)
# Schoenfeld's estimate for x >= 2657, assuming RH
err = ctx.sqrt(x,rounding='u')*ctx.ln(x,rounding='u')/8/ctx.pi(rounding='d')
a = ctx.floor((ctx._iv.mpf(mid)-err).a, rounding='d')
b = ctx.ceil((ctx._iv.mpf(mid)+err).b, rounding='u')
return ctx._iv.mpf([a,b])
@defun_wrapped
def primezeta(ctx, s):
if ctx.isnan(s):
return s
if ctx.re(s) <= 0:
raise ValueError("prime zeta function defined only for re(s) > 0")
if s == 1:
return ctx.inf
if s == 0.5:
return ctx.mpc(ctx.ninf, ctx.pi)
r = ctx.re(s)
if r > ctx.prec:
return 0.5**s
else:
wp = ctx.prec + int(r)
def terms():
orig = ctx.prec
# zeta ~ 1+eps; need to set precision
# to get logarithm accurately
k = 0
while 1:
k += 1
u = ctx.moebius(k)
if not u:
continue
ctx.prec = wp
t = u*ctx.ln(ctx.zeta(k*s))/k
if not t:
return
#print ctx.prec, ctx.nstr(t)
ctx.prec = orig
yield t
return ctx.sum_accurately(terms)
# TODO: for bernpoly and eulerpoly, ensure that all exact zeros are covered
@defun_wrapped
def bernpoly(ctx, n, z):
# Slow implementation:
#return sum(ctx.binomial(n,k)*ctx.bernoulli(k)*z**(n-k) for k in xrange(0,n+1))
n = int(n)
if n < 0:
raise ValueError("Bernoulli polynomials only defined for n >= 0")
if z == 0 or (z == 1 and n > 1):
return ctx.bernoulli(n)
if z == 0.5:
return (ctx.ldexp(1,1-n)-1)*ctx.bernoulli(n)
if n <= 3:
if n == 0: return z ** 0
if n == 1: return z - 0.5
if n == 2: return (6*z*(z-1)+1)/6
if n == 3: return z*(z*(z-1.5)+0.5)
if ctx.isinf(z):
return z ** n
if ctx.isnan(z):
return z
if abs(z) > 2:
def terms():
t = ctx.one
yield t
r = ctx.one/z
k = 1
while k <= n:
t = t*(n+1-k)/k*r
if not (k > 2 and k & 1):
yield t*ctx.bernoulli(k)
k += 1
return ctx.sum_accurately(terms) * z**n
else:
def terms():
yield ctx.bernoulli(n)
t = ctx.one
k = 1
while k <= n:
t = t*(n+1-k)/k * z
m = n-k
if not (m > 2 and m & 1):
yield t*ctx.bernoulli(m)
k += 1
return ctx.sum_accurately(terms)
@defun_wrapped
def eulerpoly(ctx, n, z):
n = int(n)
if n < 0:
raise ValueError("Euler polynomials only defined for n >= 0")
if n <= 2:
if n == 0: return z ** 0
if n == 1: return z - 0.5
if n == 2: return z*(z-1)
if ctx.isinf(z):
return z**n
if ctx.isnan(z):
return z
m = n+1
if z == 0:
return -2*(ctx.ldexp(1,m)-1)*ctx.bernoulli(m)/m * z**0
if z == 1:
return 2*(ctx.ldexp(1,m)-1)*ctx.bernoulli(m)/m * z**0
if z == 0.5:
if n % 2:
return ctx.zero
# Use exact code for Euler numbers
if n < 100 or n*ctx.mag(0.46839865*n) < ctx.prec*0.25:
return ctx.ldexp(ctx._eulernum(n), -n)
# http://functions.wolfram.com/Polynomials/EulerE2/06/01/02/01/0002/
def terms():
t = ctx.one
k = 0
w = ctx.ldexp(1,n+2)
while 1:
v = n-k+1
if not (v > 2 and v & 1):
yield (2-w)*ctx.bernoulli(v)*t
k += 1
if k > n:
break
t = t*z*(n-k+2)/k
w *= 0.5
return ctx.sum_accurately(terms) / m
@defun
def eulernum(ctx, n, exact=False):
n = int(n)
if exact:
return int(ctx._eulernum(n))
if n < 100:
return ctx.mpf(ctx._eulernum(n))
if n % 2:
return ctx.zero
return ctx.ldexp(ctx.eulerpoly(n,0.5), n)
# TODO: this should be implemented low-level
def polylog_series(ctx, s, z):
tol = +ctx.eps
l = ctx.zero
k = 1
zk = z
while 1:
term = zk / k**s
l += term
if abs(term) < tol:
break
zk *= z
k += 1
return l
def polylog_continuation(ctx, n, z):
if n < 0:
return z*0
twopij = 2j * ctx.pi
a = -twopij**n/ctx.fac(n) * ctx.bernpoly(n, ctx.ln(z)/twopij)
if ctx._is_real_type(z) and z < 0:
a = ctx._re(a)
if ctx._im(z) < 0 or (ctx._im(z) == 0 and ctx._re(z) >= 1):
a -= twopij*ctx.ln(z)**(n-1)/ctx.fac(n-1)
return a
def polylog_unitcircle(ctx, n, z):
tol = +ctx.eps
if n > 1:
l = ctx.zero
logz = ctx.ln(z)
logmz = ctx.one
m = 0
while 1:
if (n-m) != 1:
term = ctx.zeta(n-m) * logmz / ctx.fac(m)
if term and abs(term) < tol:
break
l += term
logmz *= logz
m += 1
l += ctx.ln(z)**(n-1)/ctx.fac(n-1)*(ctx.harmonic(n-1)-ctx.ln(-ctx.ln(z)))
elif n < 1: # else
l = ctx.fac(-n)*(-ctx.ln(z))**(n-1)
logz = ctx.ln(z)
logkz = ctx.one
k = 0
while 1:
b = ctx.bernoulli(k-n+1)
if b:
term = b*logkz/(ctx.fac(k)*(k-n+1))
if abs(term) < tol:
break
l -= term
logkz *= logz
k += 1
else:
raise ValueError
if ctx._is_real_type(z) and z < 0:
l = ctx._re(l)
return l
def polylog_general(ctx, s, z):
v = ctx.zero
u = ctx.ln(z)
if not abs(u) < 5: # theoretically |u| < 2*pi
raise NotImplementedError("polylog for arbitrary s and z")
t = 1
k = 0
while 1:
term = ctx.zeta(s-k) * t
if abs(term) < ctx.eps:
break
v += term
k += 1
t *= u
t /= k
return ctx.gamma(1-s)*(-u)**(s-1) + v
@defun_wrapped
def polylog(ctx, s, z):
s = ctx.convert(s)
z = ctx.convert(z)
if z == 1:
return ctx.zeta(s)
if z == -1:
return -ctx.altzeta(s)
if s == 0:
return z/(1-z)
if s == 1:
return -ctx.ln(1-z)
if s == -1:
return z/(1-z)**2
if abs(z) <= 0.75 or (not ctx.isint(s) and abs(z) < 0.9):
return polylog_series(ctx, s, z)
if abs(z) >= 1.4 and ctx.isint(s):
return (-1)**(s+1)*polylog_series(ctx, s, 1/z) + polylog_continuation(ctx, s, z)
if ctx.isint(s):
return polylog_unitcircle(ctx, int(s), z)
return polylog_general(ctx, s, z)
#raise NotImplementedError("polylog for arbitrary s and z")
# This could perhaps be used in some cases
#from quadrature import quad
#return quad(lambda t: t**(s-1)/(exp(t)/z-1),[0,inf])/gamma(s)
@defun_wrapped
def clsin(ctx, s, z, pi=False):
if ctx.isint(s) and s < 0 and int(s) % 2 == 1:
return z*0
if pi:
a = ctx.expjpi(z)
else:
a = ctx.expj(z)
if ctx._is_real_type(z) and ctx._is_real_type(s):
return ctx.im(ctx.polylog(s,a))
b = 1/a
return (-0.5j)*(ctx.polylog(s,a) - ctx.polylog(s,b))
@defun_wrapped
def clcos(ctx, s, z, pi=False):
if ctx.isint(s) and s < 0 and int(s) % 2 == 0:
return z*0
if pi:
a = ctx.expjpi(z)
else:
a = ctx.expj(z)
if ctx._is_real_type(z) and ctx._is_real_type(s):
return ctx.re(ctx.polylog(s,a))
b = 1/a
return 0.5*(ctx.polylog(s,a) + ctx.polylog(s,b))
@defun
def altzeta(ctx, s, **kwargs):
try:
return ctx._altzeta(s, **kwargs)
except NotImplementedError:
return ctx._altzeta_generic(s)
@defun_wrapped
def _altzeta_generic(ctx, s):
if s == 1:
return ctx.ln2 + 0*s
return -ctx.powm1(2, 1-s) * ctx.zeta(s)
@defun
def zeta(ctx, s, a=1, derivative=0, method=None, **kwargs):
d = int(derivative)
if a == 1 and not (d or method):
try:
return ctx._zeta(s, **kwargs)
except NotImplementedError:
pass
s = ctx.convert(s)
prec = ctx.prec
method = kwargs.get('method')
verbose = kwargs.get('verbose')
if (not s) and (not derivative):
return ctx.mpf(0.5) - ctx._convert_param(a)[0]
if a == 1 and method != 'euler-maclaurin':
im = abs(ctx._im(s))
re = abs(ctx._re(s))
#if (im < prec or method == 'borwein') and not derivative:
# try:
# if verbose:
# print "zeta: Attempting to use the Borwein algorithm"
# return ctx._zeta(s, **kwargs)
# except NotImplementedError:
# if verbose:
# print "zeta: Could not use the Borwein algorithm"
# pass
if abs(im) > 500*prec and 10*re < prec and derivative <= 4 or \
method == 'riemann-siegel':
try: # py2.4 compatible try block
try:
if verbose:
print("zeta: Attempting to use the Riemann-Siegel algorithm")
return ctx.rs_zeta(s, derivative, **kwargs)
except NotImplementedError:
if verbose:
print("zeta: Could not use the Riemann-Siegel algorithm")
pass
finally:
ctx.prec = prec
if s == 1:
return ctx.inf
abss = abs(s)
if abss == ctx.inf:
if ctx.re(s) == ctx.inf:
if d == 0:
return ctx.one
return ctx.zero
return s*0
elif ctx.isnan(abss):
return 1/s
if ctx.re(s) > 2*ctx.prec and a == 1 and not derivative:
return ctx.one + ctx.power(2, -s)
return +ctx._hurwitz(s, a, d, **kwargs)
@defun
def _hurwitz(ctx, s, a=1, d=0, **kwargs):
prec = ctx.prec
verbose = kwargs.get('verbose')
try:
extraprec = 10
ctx.prec += extraprec
# We strongly want to special-case rational a
a, atype = ctx._convert_param(a)
if ctx.re(s) < 0:
if verbose:
print("zeta: Attempting reflection formula")
try:
return _hurwitz_reflection(ctx, s, a, d, atype)
except NotImplementedError:
pass
if verbose:
print("zeta: Reflection formula failed")
if verbose:
print("zeta: Using the Euler-Maclaurin algorithm")
while 1:
ctx.prec = prec + extraprec
T1, T2 = _hurwitz_em(ctx, s, a, d, prec+10, verbose)
cancellation = ctx.mag(T1) - ctx.mag(T1+T2)
if verbose:
print_("Term 1:", T1)
print_("Term 2:", T2)
print_("Cancellation:", cancellation, "bits")
if cancellation < extraprec:
return T1 + T2
else:
extraprec = max(2*extraprec, min(cancellation + 5, 100*prec))
if extraprec > kwargs.get('maxprec', 100*prec):
raise ctx.NoConvergence("zeta: too much cancellation")
finally:
ctx.prec = prec
def _hurwitz_reflection(ctx, s, a, d, atype):
# TODO: implement for derivatives
if d != 0:
raise NotImplementedError
res = ctx.re(s)
negs = -s
# Integer reflection formula
if ctx.isnpint(s):
n = int(res)
if n <= 0:
return ctx.bernpoly(1-n, a) / (n-1)
t = 1-s
# We now require a to be standardized
v = 0
shift = 0
b = a
while ctx.re(b) > 1:
b -= 1
v -= b**negs
shift -= 1
while ctx.re(b) <= 0:
v += b**negs
b += 1
shift += 1
# Rational reflection formula
if atype == 'Q' or atype == 'Z':
try:
p, q = a._mpq_
except:
assert a == int(a)
p = int(a)
q = 1
p += shift*q
assert 1 <= p <= q
g = ctx.fsum(ctx.cospi(t/2-2*k*b)*ctx._hurwitz(t,(k,q)) \
for k in range(1,q+1))
g *= 2*ctx.gamma(t)/(2*ctx.pi*q)**t
v += g
return v
# General reflection formula
# Note: clcos/clsin can raise NotImplementedError
else:
C1, C2 = ctx.cospi_sinpi(0.5*t)
# Clausen functions; could maybe use polylog directly
if C1: C1 *= ctx.clcos(t, 2*a, pi=True)
if C2: C2 *= ctx.clsin(t, 2*a, pi=True)
v += 2*ctx.gamma(t)/(2*ctx.pi)**t*(C1+C2)
return v
def _hurwitz_em(ctx, s, a, d, prec, verbose):
# May not be converted at this point
a = ctx.convert(a)
tol = -prec
# Estimate number of terms for Euler-Maclaurin summation; could be improved
M1 = 0
M2 = prec // 3
N = M2
lsum = 0
# This speeds up the recurrence for derivatives
if ctx.isint(s):
s = int(ctx._re(s))
s1 = s-1
while 1:
# Truncated L-series
l = ctx._zetasum(s, M1+a, M2-M1-1, [d])[0][0]
#if d:
# l = ctx.fsum((-ctx.ln(n+a))**d * (n+a)**negs for n in range(M1,M2))
#else:
# l = ctx.fsum((n+a)**negs for n in range(M1,M2))
lsum += l
M2a = M2+a
logM2a = ctx.ln(M2a)
logM2ad = logM2a**d
logs = [logM2ad]
logr = 1/logM2a
rM2a = 1/M2a
M2as = rM2a**s
if d:
tailsum = ctx.gammainc(d+1, s1*logM2a) / s1**(d+1)
else:
tailsum = 1/((s1)*(M2a)**s1)
tailsum += 0.5 * logM2ad * M2as
U = [1]
r = M2as
fact = 2
for j in range(1, N+1):
# TODO: the following could perhaps be tidied a bit
j2 = 2*j
if j == 1:
upds = [1]
else:
upds = [j2-2, j2-1]
for m in upds:
D = min(m,d+1)
if m <= d:
logs.append(logs[-1] * logr)
Un = [0]*(D+1)
for i in xrange(D): Un[i] = (1-m-s)*U[i]
for i in xrange(1,D+1): Un[i] += (d-(i-1))*U[i-1]
U = Un
r *= rM2a
t = ctx.fdot(U, logs) * r * ctx.bernoulli(j2)/(-fact)
tailsum += t
if ctx.mag(t) < tol:
return lsum, (-1)**d * tailsum
fact *= (j2+1)*(j2+2)
if verbose:
print_("Sum range:", M1, M2, "term magnitude", ctx.mag(t), "tolerance", tol)
M1, M2 = M2, M2*2
if ctx.re(s) < 0:
N += N//2
@defun
def _zetasum(ctx, s, a, n, derivatives=[0], reflect=False):
"""
Returns [xd0,xd1,...,xdr], [yd0,yd1,...ydr] where
xdk = D^k ( 1/a^s + 1/(a+1)^s + ... + 1/(a+n)^s )
ydk = D^k conj( 1/a^(1-s) + 1/(a+1)^(1-s) + ... + 1/(a+n)^(1-s) )
D^k = kth derivative with respect to s, k ranges over the given list of
derivatives (which should consist of either a single element
or a range 0,1,...r). If reflect=False, the ydks are not computed.
"""
#print "zetasum", s, a, n
# don't use the fixed-point code if there are large exponentials
if abs(ctx.re(s)) < 0.5 * ctx.prec:
try:
return ctx._zetasum_fast(s, a, n, derivatives, reflect)
except NotImplementedError:
pass
negs = ctx.fneg(s, exact=True)
have_derivatives = derivatives != [0]
have_one_derivative = len(derivatives) == 1
if not reflect:
if not have_derivatives:
return [ctx.fsum((a+k)**negs for k in xrange(n+1))], []
if have_one_derivative:
d = derivatives[0]
x = ctx.fsum(ctx.ln(a+k)**d * (a+k)**negs for k in xrange(n+1))
return [(-1)**d * x], []
maxd = max(derivatives)
if not have_one_derivative:
derivatives = range(maxd+1)
xs = [ctx.zero for d in derivatives]
if reflect:
ys = [ctx.zero for d in derivatives]
else:
ys = []
for k in xrange(n+1):
w = a + k
xterm = w ** negs
if reflect:
yterm = ctx.conj(ctx.one / (w * xterm))
if have_derivatives:
logw = -ctx.ln(w)
if have_one_derivative:
logw = logw ** maxd
xs[0] += xterm * logw
if reflect:
ys[0] += yterm * logw
else:
t = ctx.one
for d in derivatives:
xs[d] += xterm * t
if reflect:
ys[d] += yterm * t
t *= logw
else:
xs[0] += xterm
if reflect:
ys[0] += yterm
return xs, ys
@defun
def dirichlet(ctx, s, chi=[1], derivative=0):
s = ctx.convert(s)
q = len(chi)
d = int(derivative)
if d > 2:
raise NotImplementedError("arbitrary order derivatives")
prec = ctx.prec
try:
ctx.prec += 10
if s == 1:
have_pole = True
for x in chi:
if x and x != 1:
have_pole = False
h = +ctx.eps
ctx.prec *= 2*(d+1)
s += h
if have_pole:
return +ctx.inf
z = ctx.zero
for p in range(1,q+1):
if chi[p%q]:
if d == 1:
z += chi[p%q] * (ctx.zeta(s, (p,q), 1) - \
ctx.zeta(s, (p,q))*ctx.log(q))
else:
z += chi[p%q] * ctx.zeta(s, (p,q))
z /= q**s
finally:
ctx.prec = prec
return +z
def secondzeta_main_term(ctx, s, a, **kwargs):
tol = ctx.eps
f = lambda n: ctx.gammainc(0.5*s, a*gamm**2, regularized=True)*gamm**(-s)
totsum = term = ctx.zero
mg = ctx.inf
n = 0
while mg > tol:
totsum += term
n += 1
gamm = ctx.im(ctx.zetazero_memoized(n))
term = f(n)
mg = abs(term)
err = 0
if kwargs.get("error"):
sg = ctx.re(s)
err = 0.5*ctx.pi**(-1)*max(1,sg)*a**(sg-0.5)*ctx.log(gamm/(2*ctx.pi))*\
ctx.gammainc(-0.5, a*gamm**2)/abs(ctx.gamma(s/2))
err = abs(err)
return +totsum, err, n
def secondzeta_prime_term(ctx, s, a, **kwargs):
tol = ctx.eps
f = lambda n: ctx.gammainc(0.5*(1-s),0.25*ctx.log(n)**2 * a**(-1))*\
((0.5*ctx.log(n))**(s-1))*ctx.mangoldt(n)/ctx.sqrt(n)/\
(2*ctx.gamma(0.5*s)*ctx.sqrt(ctx.pi))
totsum = term = ctx.zero
mg = ctx.inf
n = 1
while mg > tol or n < 9:
totsum += term
n += 1
term = f(n)
if term == 0:
mg = ctx.inf
else:
mg = abs(term)
if kwargs.get("error"):
err = mg
return +totsum, err, n
def secondzeta_exp_term(ctx, s, a):
if ctx.isint(s) and ctx.re(s) <= 0:
m = int(round(ctx.re(s)))
if not m & 1:
return ctx.mpf('-0.25')**(-m//2)
tol = ctx.eps
f = lambda n: (0.25*a)**n/((n+0.5*s)*ctx.fac(n))
totsum = ctx.zero
term = f(0)
mg = ctx.inf
n = 0
while mg > tol:
totsum += term
n += 1
term = f(n)
mg = abs(term)
v = a**(0.5*s)*totsum/ctx.gamma(0.5*s)
return v
def secondzeta_singular_term(ctx, s, a, **kwargs):
factor = a**(0.5*(s-1))/(4*ctx.sqrt(ctx.pi)*ctx.gamma(0.5*s))
extraprec = ctx.mag(factor)
ctx.prec += extraprec
factor = a**(0.5*(s-1))/(4*ctx.sqrt(ctx.pi)*ctx.gamma(0.5*s))
tol = ctx.eps
f = lambda n: ctx.bernpoly(n,0.75)*(4*ctx.sqrt(a))**n*\
ctx.gamma(0.5*n)/((s+n-1)*ctx.fac(n))
totsum = ctx.zero
mg1 = ctx.inf
n = 1
term = f(n)
mg2 = abs(term)
while mg2 > tol and mg2 <= mg1:
totsum += term
n += 1
term = f(n)
totsum += term
n +=1
term = f(n)
mg1 = mg2
mg2 = abs(term)
totsum += term
pole = -2*(s-1)**(-2)+(ctx.euler+ctx.log(16*ctx.pi**2*a))*(s-1)**(-1)
st = factor*(pole+totsum)
err = 0
if kwargs.get("error"):
if not ((mg2 > tol) and (mg2 <= mg1)):
if mg2 <= tol:
err = ctx.mpf(10)**int(ctx.log(abs(factor*tol),10))
if mg2 > mg1:
err = ctx.mpf(10)**int(ctx.log(abs(factor*mg1),10))
err = max(err, ctx.eps*1.)
ctx.prec -= extraprec
return +st, err
@defun
def secondzeta(ctx, s, a = 0.015, **kwargs):
r"""
Evaluates the secondary zeta function `Z(s)`, defined for
`\mathrm{Re}(s)>1` by
.. math ::
Z(s) = \sum_{n=1}^{\infty} \frac{1}{\tau_n^s}
where `\frac12+i\tau_n` runs through the zeros of `\zeta(s)` with
imaginary part positive.
`Z(s)` extends to a meromorphic function on `\mathbb{C}` with a
double pole at `s=1` and simple poles at the points `-2n` for
`n=0`, 1, 2, ...
**Examples**
>>> from mpmath import *
>>> mp.pretty = True; mp.dps = 15
>>> secondzeta(2)
0.023104993115419
>>> xi = lambda s: 0.5*s*(s-1)*pi**(-0.5*s)*gamma(0.5*s)*zeta(s)
>>> Xi = lambda t: xi(0.5+t*j)
>>> -0.5*diff(Xi,0,n=2)/Xi(0)
(0.023104993115419 + 0.0j)
We may ask for an approximate error value::
>>> secondzeta(0.5+100j, error=True)
((-0.216272011276718 - 0.844952708937228j), 2.22044604925031e-16)
The function has poles at the negative odd integers,
and dyadic rational values at the negative even integers::
>>> mp.dps = 30
>>> secondzeta(-8)
-0.67236328125
>>> secondzeta(-7)
+inf
**Implementation notes**
The function is computed as sum of four terms `Z(s)=A(s)-P(s)+E(s)-S(s)`
respectively main, prime, exponential and singular terms.
The main term `A(s)` is computed from the zeros of zeta.
The prime term depends on the von Mangoldt function.
The singular term is responsible for the poles of the function.
The four terms depends on a small parameter `a`. We may change the
value of `a`. Theoretically this has no effect on the sum of the four
terms, but in practice may be important.
A smaller value of the parameter `a` makes `A(s)` depend on
a smaller number of zeros of zeta, but `P(s)` uses more values of
von Mangoldt function.
We may also add a verbose option to obtain data about the
values of the four terms.
>>> mp.dps = 10
>>> secondzeta(0.5 + 40j, error=True, verbose=True)
main term = (-30190318549.138656312556 - 13964804384.624622876523j)
computed using 19 zeros of zeta
prime term = (132717176.89212754625045 + 188980555.17563978290601j)
computed using 9 values of the von Mangoldt function
exponential term = (542447428666.07179812536 + 362434922978.80192435203j)
singular term = (512124392939.98154322355 + 348281138038.65531023921j)
((0.059471043 + 0.3463514534j), 1.455191523e-11)
>>> secondzeta(0.5 + 40j, a=0.04, error=True, verbose=True)
main term = (-151962888.19606243907725 - 217930683.90210294051982j)
computed using 9 zeros of zeta
prime term = (2476659342.3038722372461 + 28711581821.921627163136j)
computed using 37 values of the von Mangoldt function
exponential term = (178506047114.7838188264 + 819674143244.45677330576j)
singular term = (175877424884.22441310708 + 790744630738.28669174871j)
((0.059471043 + 0.3463514534j), 1.455191523e-11)
Notice the great cancellation between the four terms. Changing `a`, the
four terms are very different numbers but the cancellation gives
the good value of Z(s).
**References**
A. Voros, Zeta functions for the Riemann zeros, Ann. Institute Fourier,
53, (2003) 665--699.
A. Voros, Zeta functions over Zeros of Zeta Functions, Lecture Notes
of the Unione Matematica Italiana, Springer, 2009.
"""
s = ctx.convert(s)
a = ctx.convert(a)
tol = ctx.eps
if ctx.isint(s) and ctx.re(s) <= 1:
if abs(s-1) < tol*1000:
return ctx.inf
m = int(round(ctx.re(s)))
if m & 1:
return ctx.inf
else:
return ((-1)**(-m//2)*\
ctx.fraction(8-ctx.eulernum(-m,exact=True),2**(-m+3)))
prec = ctx.prec
try:
t3 = secondzeta_exp_term(ctx, s, a)
extraprec = max(ctx.mag(t3),0)
ctx.prec += extraprec + 3
t1, r1, gt = secondzeta_main_term(ctx,s,a,error='True', verbose='True')
t2, r2, pt = secondzeta_prime_term(ctx,s,a,error='True', verbose='True')
t4, r4 = secondzeta_singular_term(ctx,s,a,error='True')
t3 = secondzeta_exp_term(ctx, s, a)
err = r1+r2+r4
t = t1-t2+t3-t4
if kwargs.get("verbose"):
print_('main term =', t1)
print_(' computed using', gt, 'zeros of zeta')
print_('prime term =', t2)
print_(' computed using', pt, 'values of the von Mangoldt function')
print_('exponential term =', t3)
print_('singular term =', t4)
finally:
ctx.prec = prec
if kwargs.get("error"):
w = max(ctx.mag(abs(t)),0)
err = max(err*2**w, ctx.eps*1.*2**w)
return +t, err
return +t
@defun_wrapped
def lerchphi(ctx, z, s, a):
r"""
Gives the Lerch transcendent, defined for `|z| < 1` and
`\Re{a} > 0` by
.. math ::
\Phi(z,s,a) = \sum_{k=0}^{\infty} \frac{z^k}{(a+k)^s}
and generally by the recurrence `\Phi(z,s,a) = z \Phi(z,s,a+1) + a^{-s}`
along with the integral representation valid for `\Re{a} > 0`
.. math ::
\Phi(z,s,a) = \frac{1}{2 a^s} +
\int_0^{\infty} \frac{z^t}{(a+t)^s} dt -
2 \int_0^{\infty} \frac{\sin(t \log z - s
\operatorname{arctan}(t/a)}{(a^2 + t^2)^{s/2}
(e^{2 \pi t}-1)} dt.
The Lerch transcendent generalizes the Hurwitz zeta function :func:`zeta`
(`z = 1`) and the polylogarithm :func:`polylog` (`a = 1`).
**Examples**
Several evaluations in terms of simpler functions::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> lerchphi(-1,2,0.5); 4*catalan
3.663862376708876060218414
3.663862376708876060218414
>>> diff(lerchphi, (-1,-2,1), (0,1,0)); 7*zeta(3)/(4*pi**2)
0.2131391994087528954617607
0.2131391994087528954617607
>>> lerchphi(-4,1,1); log(5)/4
0.4023594781085250936501898
0.4023594781085250936501898
>>> lerchphi(-3+2j,1,0.5); 2*atanh(sqrt(-3+2j))/sqrt(-3+2j)
(1.142423447120257137774002 + 0.2118232380980201350495795j)
(1.142423447120257137774002 + 0.2118232380980201350495795j)
Evaluation works for complex arguments and `|z| \ge 1`::
>>> lerchphi(1+2j, 3-j, 4+2j)
(0.002025009957009908600539469 + 0.003327897536813558807438089j)
>>> lerchphi(-2,2,-2.5)
-12.28676272353094275265944
>>> lerchphi(10,10,10)
(-4.462130727102185701817349e-11 - 1.575172198981096218823481e-12j)
>>> lerchphi(10,10,-10.5)
(112658784011940.5605789002 - 498113185.5756221777743631j)
Some degenerate cases::
>>> lerchphi(0,1,2)
0.5
>>> lerchphi(0,1,-2)
-0.5
Reduction to simpler functions::
>>> lerchphi(1, 4.25+1j, 1)
(1.044674457556746668033975 - 0.04674508654012658932271226j)
>>> zeta(4.25+1j)
(1.044674457556746668033975 - 0.04674508654012658932271226j)
>>> lerchphi(1 - 0.5**10, 4.25+1j, 1)
(1.044629338021507546737197 - 0.04667768813963388181708101j)
>>> lerchphi(3, 4, 1)
(1.249503297023366545192592 - 0.2314252413375664776474462j)
>>> polylog(4, 3) / 3
(1.249503297023366545192592 - 0.2314252413375664776474462j)
>>> lerchphi(3, 4, 1 - 0.5**10)
(1.253978063946663945672674 - 0.2316736622836535468765376j)
**References**
1. [DLMF]_ section 25.14
"""
if z == 0:
return a ** (-s)
# Faster, but these cases are useful for testing right now
if z == 1:
return ctx.zeta(s, a)
if a == 1:
return ctx.polylog(s, z) / z
if ctx.re(a) < 1:
if ctx.isnpint(a):
raise ValueError("Lerch transcendent complex infinity")
m = int(ctx.ceil(1-ctx.re(a)))
v = ctx.zero
zpow = ctx.one
for n in xrange(m):
v += zpow / (a+n)**s
zpow *= z
return zpow * ctx.lerchphi(z,s, a+m) + v
g = ctx.ln(z)
v = 1/(2*a**s) + ctx.gammainc(1-s, -a*g) * (-g)**(s-1) / z**a
h = s / 2
r = 2*ctx.pi
f = lambda t: ctx.sin(s*ctx.atan(t/a)-t*g) / \
((a**2+t**2)**h * ctx.expm1(r*t))
v += 2*ctx.quad(f, [0, ctx.inf])
if not ctx.im(z) and not ctx.im(s) and not ctx.im(a) and ctx.re(z) < 1:
v = ctx.chop(v)
return v
| 36,859 | 30.693895 | 88 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/functions/orthogonal.py
|
from .functions import defun, defun_wrapped
def _hermite_param(ctx, n, z, parabolic_cylinder):
"""
Combined calculation of the Hermite polynomial H_n(z) (and its
generalization to complex n) and the parabolic cylinder
function D.
"""
n, ntyp = ctx._convert_param(n)
z = ctx.convert(z)
q = -ctx.mpq_1_2
# For re(z) > 0, 2F0 -- http://functions.wolfram.com/
# HypergeometricFunctions/HermiteHGeneral/06/02/0009/
# Otherwise, there is a reflection formula
# 2F0 + http://functions.wolfram.com/HypergeometricFunctions/
# HermiteHGeneral/16/01/01/0006/
#
# TODO:
# An alternative would be to use
# http://functions.wolfram.com/HypergeometricFunctions/
# HermiteHGeneral/06/02/0006/
#
# Also, the 1F1 expansion
# http://functions.wolfram.com/HypergeometricFunctions/
# HermiteHGeneral/26/01/02/0001/
# should probably be used for tiny z
if not z:
T1 = [2, ctx.pi], [n, 0.5], [], [q*(n-1)], [], [], 0
if parabolic_cylinder:
T1[1][0] += q*n
return T1,
can_use_2f0 = ctx.isnpint(-n) or ctx.re(z) > 0 or \
(ctx.re(z) == 0 and ctx.im(z) > 0)
expprec = ctx.prec*4 + 20
if parabolic_cylinder:
u = ctx.fmul(ctx.fmul(z,z,prec=expprec), -0.25, exact=True)
w = ctx.fmul(z, ctx.sqrt(0.5,prec=expprec), prec=expprec)
else:
w = z
w2 = ctx.fmul(w, w, prec=expprec)
rw2 = ctx.fdiv(1, w2, prec=expprec)
nrw2 = ctx.fneg(rw2, exact=True)
nw = ctx.fneg(w, exact=True)
if can_use_2f0:
T1 = [2, w], [n, n], [], [], [q*n, q*(n-1)], [], nrw2
terms = [T1]
else:
T1 = [2, nw], [n, n], [], [], [q*n, q*(n-1)], [], nrw2
T2 = [2, ctx.pi, nw], [n+2, 0.5, 1], [], [q*n], [q*(n-1)], [1-q], w2
terms = [T1,T2]
# Multiply by prefactor for D_n
if parabolic_cylinder:
expu = ctx.exp(u)
for i in range(len(terms)):
terms[i][1][0] += q*n
terms[i][0].append(expu)
terms[i][1].append(1)
return tuple(terms)
@defun
def hermite(ctx, n, z, **kwargs):
return ctx.hypercomb(lambda: _hermite_param(ctx, n, z, 0), [], **kwargs)
@defun
def pcfd(ctx, n, z, **kwargs):
r"""
Gives the parabolic cylinder function in Whittaker's notation
`D_n(z) = U(-n-1/2, z)` (see :func:`~mpmath.pcfu`).
It solves the differential equation
.. math ::
y'' + \left(n + \frac{1}{2} - \frac{1}{4} z^2\right) y = 0.
and can be represented in terms of Hermite polynomials
(see :func:`~mpmath.hermite`) as
.. math ::
D_n(z) = 2^{-n/2} e^{-z^2/4} H_n\left(\frac{z}{\sqrt{2}}\right).
**Plots**
.. literalinclude :: /plots/pcfd.py
.. image :: /plots/pcfd.png
**Examples**
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> pcfd(0,0); pcfd(1,0); pcfd(2,0); pcfd(3,0)
1.0
0.0
-1.0
0.0
>>> pcfd(4,0); pcfd(-3,0)
3.0
0.6266570686577501256039413
>>> pcfd('1/2', 2+3j)
(-5.363331161232920734849056 - 3.858877821790010714163487j)
>>> pcfd(2, -10)
1.374906442631438038871515e-9
Verifying the differential equation::
>>> n = mpf(2.5)
>>> y = lambda z: pcfd(n,z)
>>> z = 1.75
>>> chop(diff(y,z,2) + (n+0.5-0.25*z**2)*y(z))
0.0
Rational Taylor series expansion when `n` is an integer::
>>> taylor(lambda z: pcfd(5,z), 0, 7)
[0.0, 15.0, 0.0, -13.75, 0.0, 3.96875, 0.0, -0.6015625]
"""
return ctx.hypercomb(lambda: _hermite_param(ctx, n, z, 1), [], **kwargs)
@defun
def pcfu(ctx, a, z, **kwargs):
r"""
Gives the parabolic cylinder function `U(a,z)`, which may be
defined for `\Re(z) > 0` in terms of the confluent
U-function (see :func:`~mpmath.hyperu`) by
.. math ::
U(a,z) = 2^{-\frac{1}{4}-\frac{a}{2}} e^{-\frac{1}{4} z^2}
U\left(\frac{a}{2}+\frac{1}{4},
\frac{1}{2}, \frac{1}{2}z^2\right)
or, for arbitrary `z`,
.. math ::
e^{-\frac{1}{4}z^2} U(a,z) =
U(a,0) \,_1F_1\left(-\tfrac{a}{2}+\tfrac{1}{4};
\tfrac{1}{2}; -\tfrac{1}{2}z^2\right) +
U'(a,0) z \,_1F_1\left(-\tfrac{a}{2}+\tfrac{3}{4};
\tfrac{3}{2}; -\tfrac{1}{2}z^2\right).
**Examples**
Connection to other functions::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> z = mpf(3)
>>> pcfu(0.5,z)
0.03210358129311151450551963
>>> sqrt(pi/2)*exp(z**2/4)*erfc(z/sqrt(2))
0.03210358129311151450551963
>>> pcfu(0.5,-z)
23.75012332835297233711255
>>> sqrt(pi/2)*exp(z**2/4)*erfc(-z/sqrt(2))
23.75012332835297233711255
>>> pcfu(0.5,-z)
23.75012332835297233711255
>>> sqrt(pi/2)*exp(z**2/4)*erfc(-z/sqrt(2))
23.75012332835297233711255
"""
n, _ = ctx._convert_param(a)
return ctx.pcfd(-n-ctx.mpq_1_2, z)
@defun
def pcfv(ctx, a, z, **kwargs):
r"""
Gives the parabolic cylinder function `V(a,z)`, which can be
represented in terms of :func:`~mpmath.pcfu` as
.. math ::
V(a,z) = \frac{\Gamma(a+\tfrac{1}{2}) (U(a,-z)-\sin(\pi a) U(a,z)}{\pi}.
**Examples**
Wronskian relation between `U` and `V`::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> a, z = 2, 3
>>> pcfu(a,z)*diff(pcfv,(a,z),(0,1))-diff(pcfu,(a,z),(0,1))*pcfv(a,z)
0.7978845608028653558798921
>>> sqrt(2/pi)
0.7978845608028653558798921
>>> a, z = 2.5, 3
>>> pcfu(a,z)*diff(pcfv,(a,z),(0,1))-diff(pcfu,(a,z),(0,1))*pcfv(a,z)
0.7978845608028653558798921
>>> a, z = 0.25, -1
>>> pcfu(a,z)*diff(pcfv,(a,z),(0,1))-diff(pcfu,(a,z),(0,1))*pcfv(a,z)
0.7978845608028653558798921
>>> a, z = 2+1j, 2+3j
>>> chop(pcfu(a,z)*diff(pcfv,(a,z),(0,1))-diff(pcfu,(a,z),(0,1))*pcfv(a,z))
0.7978845608028653558798921
"""
n, ntype = ctx._convert_param(a)
z = ctx.convert(z)
q = ctx.mpq_1_2
r = ctx.mpq_1_4
if ntype == 'Q' and ctx.isint(n*2):
# Faster for half-integers
def h():
jz = ctx.fmul(z, -1j, exact=True)
T1terms = _hermite_param(ctx, -n-q, z, 1)
T2terms = _hermite_param(ctx, n-q, jz, 1)
for T in T1terms:
T[0].append(1j)
T[1].append(1)
T[3].append(q-n)
u = ctx.expjpi((q*n-r)) * ctx.sqrt(2/ctx.pi)
for T in T2terms:
T[0].append(u)
T[1].append(1)
return T1terms + T2terms
v = ctx.hypercomb(h, [], **kwargs)
if ctx._is_real_type(n) and ctx._is_real_type(z):
v = ctx._re(v)
return v
else:
def h(n):
w = ctx.square_exp_arg(z, -0.25)
u = ctx.square_exp_arg(z, 0.5)
e = ctx.exp(w)
l = [ctx.pi, q, ctx.exp(w)]
Y1 = l, [-q, n*q+r, 1], [r-q*n], [], [q*n+r], [q], u
Y2 = l + [z], [-q, n*q-r, 1, 1], [1-r-q*n], [], [q*n+1-r], [1+q], u
c, s = ctx.cospi_sinpi(r+q*n)
Y1[0].append(s)
Y2[0].append(c)
for Y in (Y1, Y2):
Y[1].append(1)
Y[3].append(q-n)
return Y1, Y2
return ctx.hypercomb(h, [n], **kwargs)
@defun
def pcfw(ctx, a, z, **kwargs):
r"""
Gives the parabolic cylinder function `W(a,z)` defined in (DLMF 12.14).
**Examples**
Value at the origin::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> a = mpf(0.25)
>>> pcfw(a,0)
0.9722833245718180765617104
>>> power(2,-0.75)*sqrt(abs(gamma(0.25+0.5j*a)/gamma(0.75+0.5j*a)))
0.9722833245718180765617104
>>> diff(pcfw,(a,0),(0,1))
-0.5142533944210078966003624
>>> -power(2,-0.25)*sqrt(abs(gamma(0.75+0.5j*a)/gamma(0.25+0.5j*a)))
-0.5142533944210078966003624
"""
n, _ = ctx._convert_param(a)
z = ctx.convert(z)
def terms():
phi2 = ctx.arg(ctx.gamma(0.5 + ctx.j*n))
phi2 = (ctx.loggamma(0.5+ctx.j*n) - ctx.loggamma(0.5-ctx.j*n))/2j
rho = ctx.pi/8 + 0.5*phi2
# XXX: cancellation computing k
k = ctx.sqrt(1 + ctx.exp(2*ctx.pi*n)) - ctx.exp(ctx.pi*n)
C = ctx.sqrt(k/2) * ctx.exp(0.25*ctx.pi*n)
yield C * ctx.expj(rho) * ctx.pcfu(ctx.j*n, z*ctx.expjpi(-0.25))
yield C * ctx.expj(-rho) * ctx.pcfu(-ctx.j*n, z*ctx.expjpi(0.25))
v = ctx.sum_accurately(terms)
if ctx._is_real_type(n) and ctx._is_real_type(z):
v = ctx._re(v)
return v
"""
Even/odd PCFs. Useful?
@defun
def pcfy1(ctx, a, z, **kwargs):
a, _ = ctx._convert_param(n)
z = ctx.convert(z)
def h():
w = ctx.square_exp_arg(z)
w1 = ctx.fmul(w, -0.25, exact=True)
w2 = ctx.fmul(w, 0.5, exact=True)
e = ctx.exp(w1)
return [e], [1], [], [], [ctx.mpq_1_2*a+ctx.mpq_1_4], [ctx.mpq_1_2], w2
return ctx.hypercomb(h, [], **kwargs)
@defun
def pcfy2(ctx, a, z, **kwargs):
a, _ = ctx._convert_param(n)
z = ctx.convert(z)
def h():
w = ctx.square_exp_arg(z)
w1 = ctx.fmul(w, -0.25, exact=True)
w2 = ctx.fmul(w, 0.5, exact=True)
e = ctx.exp(w1)
return [e, z], [1, 1], [], [], [ctx.mpq_1_2*a+ctx.mpq_3_4], \
[ctx.mpq_3_2], w2
return ctx.hypercomb(h, [], **kwargs)
"""
@defun_wrapped
def gegenbauer(ctx, n, a, z, **kwargs):
# Special cases: a+0.5, a*2 poles
if ctx.isnpint(a):
return 0*(z+n)
if ctx.isnpint(a+0.5):
# TODO: something else is required here
# E.g.: gegenbauer(-2, -0.5, 3) == -12
if ctx.isnpint(n+1):
raise NotImplementedError("Gegenbauer function with two limits")
def h(a):
a2 = 2*a
T = [], [], [n+a2], [n+1, a2], [-n, n+a2], [a+0.5], 0.5*(1-z)
return [T]
return ctx.hypercomb(h, [a], **kwargs)
def h(n):
a2 = 2*a
T = [], [], [n+a2], [n+1, a2], [-n, n+a2], [a+0.5], 0.5*(1-z)
return [T]
return ctx.hypercomb(h, [n], **kwargs)
@defun_wrapped
def jacobi(ctx, n, a, b, x, **kwargs):
if not ctx.isnpint(a):
def h(n):
return (([], [], [a+n+1], [n+1, a+1], [-n, a+b+n+1], [a+1], (1-x)*0.5),)
return ctx.hypercomb(h, [n], **kwargs)
if not ctx.isint(b):
def h(n, a):
return (([], [], [-b], [n+1, -b-n], [-n, a+b+n+1], [b+1], (x+1)*0.5),)
return ctx.hypercomb(h, [n, a], **kwargs)
# XXX: determine appropriate limit
return ctx.binomial(n+a,n) * ctx.hyp2f1(-n,1+n+a+b,a+1,(1-x)/2, **kwargs)
@defun_wrapped
def laguerre(ctx, n, a, z, **kwargs):
# XXX: limits, poles
#if ctx.isnpint(n):
# return 0*(a+z)
def h(a):
return (([], [], [a+n+1], [a+1, n+1], [-n], [a+1], z),)
return ctx.hypercomb(h, [a], **kwargs)
@defun_wrapped
def legendre(ctx, n, x, **kwargs):
if ctx.isint(n):
n = int(n)
# Accuracy near zeros
if (n + (n < 0)) & 1:
if not x:
return x
mag = ctx.mag(x)
if mag < -2*ctx.prec-10:
return x
if mag < -5:
ctx.prec += -mag
return ctx.hyp2f1(-n,n+1,1,(1-x)/2, **kwargs)
@defun
def legenp(ctx, n, m, z, type=2, **kwargs):
# Legendre function, 1st kind
n = ctx.convert(n)
m = ctx.convert(m)
# Faster
if not m:
return ctx.legendre(n, z, **kwargs)
# TODO: correct evaluation at singularities
if type == 2:
def h(n,m):
g = m*0.5
T = [1+z, 1-z], [g, -g], [], [1-m], [-n, n+1], [1-m], 0.5*(1-z)
return (T,)
return ctx.hypercomb(h, [n,m], **kwargs)
if type == 3:
def h(n,m):
g = m*0.5
T = [z+1, z-1], [g, -g], [], [1-m], [-n, n+1], [1-m], 0.5*(1-z)
return (T,)
return ctx.hypercomb(h, [n,m], **kwargs)
raise ValueError("requires type=2 or type=3")
@defun
def legenq(ctx, n, m, z, type=2, **kwargs):
# Legendre function, 2nd kind
n = ctx.convert(n)
m = ctx.convert(m)
z = ctx.convert(z)
if z in (1, -1):
#if ctx.isint(m):
# return ctx.nan
#return ctx.inf # unsigned
return ctx.nan
if type == 2:
def h(n, m):
cos, sin = ctx.cospi_sinpi(m)
s = 2 * sin / ctx.pi
c = cos
a = 1+z
b = 1-z
u = m/2
w = (1-z)/2
T1 = [s, c, a, b], [-1, 1, u, -u], [], [1-m], \
[-n, n+1], [1-m], w
T2 = [-s, a, b], [-1, -u, u], [n+m+1], [n-m+1, m+1], \
[-n, n+1], [m+1], w
return T1, T2
return ctx.hypercomb(h, [n, m], **kwargs)
if type == 3:
# The following is faster when there only is a single series
# Note: not valid for -1 < z < 0 (?)
if abs(z) > 1:
def h(n, m):
T1 = [ctx.expjpi(m), 2, ctx.pi, z, z-1, z+1], \
[1, -n-1, 0.5, -n-m-1, 0.5*m, 0.5*m], \
[n+m+1], [n+1.5], \
[0.5*(2+n+m), 0.5*(1+n+m)], [n+1.5], z**(-2)
return [T1]
return ctx.hypercomb(h, [n, m], **kwargs)
else:
# not valid for 1 < z < inf ?
def h(n, m):
s = 2 * ctx.sinpi(m) / ctx.pi
c = ctx.expjpi(m)
a = 1+z
b = z-1
u = m/2
w = (1-z)/2
T1 = [s, c, a, b], [-1, 1, u, -u], [], [1-m], \
[-n, n+1], [1-m], w
T2 = [-s, c, a, b], [-1, 1, -u, u], [n+m+1], [n-m+1, m+1], \
[-n, n+1], [m+1], w
return T1, T2
return ctx.hypercomb(h, [n, m], **kwargs)
raise ValueError("requires type=2 or type=3")
@defun_wrapped
def chebyt(ctx, n, x, **kwargs):
if (not x) and ctx.isint(n) and int(ctx._re(n)) % 2 == 1:
return x * 0
return ctx.hyp2f1(-n,n,(1,2),(1-x)/2, **kwargs)
@defun_wrapped
def chebyu(ctx, n, x, **kwargs):
if (not x) and ctx.isint(n) and int(ctx._re(n)) % 2 == 1:
return x * 0
return (n+1) * ctx.hyp2f1(-n, n+2, (3,2), (1-x)/2, **kwargs)
@defun
def spherharm(ctx, l, m, theta, phi, **kwargs):
l = ctx.convert(l)
m = ctx.convert(m)
theta = ctx.convert(theta)
phi = ctx.convert(phi)
l_isint = ctx.isint(l)
l_natural = l_isint and l >= 0
m_isint = ctx.isint(m)
if l_isint and l < 0 and m_isint:
return ctx.spherharm(-(l+1), m, theta, phi, **kwargs)
if theta == 0 and m_isint and m < 0:
return ctx.zero * 1j
if l_natural and m_isint:
if abs(m) > l:
return ctx.zero * 1j
# http://functions.wolfram.com/Polynomials/
# SphericalHarmonicY/26/01/02/0004/
def h(l,m):
absm = abs(m)
C = [-1, ctx.expj(m*phi),
(2*l+1)*ctx.fac(l+absm)/ctx.pi/ctx.fac(l-absm),
ctx.sin(theta)**2,
ctx.fac(absm), 2]
P = [0.5*m*(ctx.sign(m)+1), 1, 0.5, 0.5*absm, -1, -absm-1]
return ((C, P, [], [], [absm-l, l+absm+1], [absm+1],
ctx.sin(0.5*theta)**2),)
else:
# http://functions.wolfram.com/HypergeometricFunctions/
# SphericalHarmonicYGeneral/26/01/02/0001/
def h(l,m):
if ctx.isnpint(l-m+1) or ctx.isnpint(l+m+1) or ctx.isnpint(1-m):
return (([0], [-1], [], [], [], [], 0),)
cos, sin = ctx.cos_sin(0.5*theta)
C = [0.5*ctx.expj(m*phi), (2*l+1)/ctx.pi,
ctx.gamma(l-m+1), ctx.gamma(l+m+1),
cos**2, sin**2]
P = [1, 0.5, 0.5, -0.5, 0.5*m, -0.5*m]
return ((C, P, [], [1-m], [-l,l+1], [1-m], sin**2),)
return ctx.hypercomb(h, [l,m], **kwargs)
| 16,097 | 31.587045 | 84 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/functions/expintegrals.py
|
from .functions import defun, defun_wrapped
@defun_wrapped
def _erf_complex(ctx, z):
z2 = ctx.square_exp_arg(z, -1)
#z2 = -z**2
v = (2/ctx.sqrt(ctx.pi))*z * ctx.hyp1f1((1,2),(3,2), z2)
if not ctx._re(z):
v = ctx._im(v)*ctx.j
return v
@defun_wrapped
def _erfc_complex(ctx, z):
if ctx.re(z) > 2:
z2 = ctx.square_exp_arg(z)
nz2 = ctx.fneg(z2, exact=True)
v = ctx.exp(nz2)/ctx.sqrt(ctx.pi) * ctx.hyperu((1,2),(1,2), z2)
else:
v = 1 - ctx._erf_complex(z)
if not ctx._re(z):
v = 1+ctx._im(v)*ctx.j
return v
@defun
def erf(ctx, z):
z = ctx.convert(z)
if ctx._is_real_type(z):
try:
return ctx._erf(z)
except NotImplementedError:
pass
if ctx._is_complex_type(z) and not z.imag:
try:
return type(z)(ctx._erf(z.real))
except NotImplementedError:
pass
return ctx._erf_complex(z)
@defun
def erfc(ctx, z):
z = ctx.convert(z)
if ctx._is_real_type(z):
try:
return ctx._erfc(z)
except NotImplementedError:
pass
if ctx._is_complex_type(z) and not z.imag:
try:
return type(z)(ctx._erfc(z.real))
except NotImplementedError:
pass
return ctx._erfc_complex(z)
@defun
def square_exp_arg(ctx, z, mult=1, reciprocal=False):
prec = ctx.prec*4+20
if reciprocal:
z2 = ctx.fmul(z, z, prec=prec)
z2 = ctx.fdiv(ctx.one, z2, prec=prec)
else:
z2 = ctx.fmul(z, z, prec=prec)
if mult != 1:
z2 = ctx.fmul(z2, mult, exact=True)
return z2
@defun_wrapped
def erfi(ctx, z):
if not z:
return z
z2 = ctx.square_exp_arg(z)
v = (2/ctx.sqrt(ctx.pi)*z) * ctx.hyp1f1((1,2), (3,2), z2)
if not ctx._re(z):
v = ctx._im(v)*ctx.j
return v
@defun_wrapped
def erfinv(ctx, x):
xre = ctx._re(x)
if (xre != x) or (xre < -1) or (xre > 1):
return ctx.bad_domain("erfinv(x) is defined only for -1 <= x <= 1")
x = xre
#if ctx.isnan(x): return x
if not x: return x
if x == 1: return ctx.inf
if x == -1: return ctx.ninf
if abs(x) < 0.9:
a = 0.53728*x**3 + 0.813198*x
else:
# An asymptotic formula
u = ctx.ln(2/ctx.pi/(abs(x)-1)**2)
a = ctx.sign(x) * ctx.sqrt(u - ctx.ln(u))/ctx.sqrt(2)
ctx.prec += 10
return ctx.findroot(lambda t: ctx.erf(t)-x, a)
@defun_wrapped
def npdf(ctx, x, mu=0, sigma=1):
sigma = ctx.convert(sigma)
return ctx.exp(-(x-mu)**2/(2*sigma**2)) / (sigma*ctx.sqrt(2*ctx.pi))
@defun_wrapped
def ncdf(ctx, x, mu=0, sigma=1):
a = (x-mu)/(sigma*ctx.sqrt(2))
if a < 0:
return ctx.erfc(-a)/2
else:
return (1+ctx.erf(a))/2
@defun_wrapped
def betainc(ctx, a, b, x1=0, x2=1, regularized=False):
if x1 == x2:
v = 0
elif not x1:
if x1 == 0 and x2 == 1:
v = ctx.beta(a, b)
else:
v = x2**a * ctx.hyp2f1(a, 1-b, a+1, x2) / a
else:
m, d = ctx.nint_distance(a)
if m <= 0:
if d < -ctx.prec:
h = +ctx.eps
ctx.prec *= 2
a += h
elif d < -4:
ctx.prec -= d
s1 = x2**a * ctx.hyp2f1(a,1-b,a+1,x2)
s2 = x1**a * ctx.hyp2f1(a,1-b,a+1,x1)
v = (s1 - s2) / a
if regularized:
v /= ctx.beta(a,b)
return v
@defun
def gammainc(ctx, z, a=0, b=None, regularized=False):
regularized = bool(regularized)
z = ctx.convert(z)
if a is None:
a = ctx.zero
lower_modified = False
else:
a = ctx.convert(a)
lower_modified = a != ctx.zero
if b is None:
b = ctx.inf
upper_modified = False
else:
b = ctx.convert(b)
upper_modified = b != ctx.inf
# Complete gamma function
if not (upper_modified or lower_modified):
if regularized:
if ctx.re(z) < 0:
return ctx.inf
elif ctx.re(z) > 0:
return ctx.one
else:
return ctx.nan
return ctx.gamma(z)
if a == b:
return ctx.zero
# Standardize
if ctx.re(a) > ctx.re(b):
return -ctx.gammainc(z, b, a, regularized)
# Generalized gamma
if upper_modified and lower_modified:
return +ctx._gamma3(z, a, b, regularized)
# Upper gamma
elif lower_modified:
return ctx._upper_gamma(z, a, regularized)
# Lower gamma
elif upper_modified:
return ctx._lower_gamma(z, b, regularized)
@defun
def _lower_gamma(ctx, z, b, regularized=False):
# Pole
if ctx.isnpint(z):
return type(z)(ctx.inf)
G = [z] * regularized
negb = ctx.fneg(b, exact=True)
def h(z):
T1 = [ctx.exp(negb), b, z], [1, z, -1], [], G, [1], [1+z], b
return (T1,)
return ctx.hypercomb(h, [z])
@defun
def _upper_gamma(ctx, z, a, regularized=False):
# Fast integer case, when available
if ctx.isint(z):
try:
if regularized:
# Gamma pole
if ctx.isnpint(z):
return type(z)(ctx.zero)
orig = ctx.prec
try:
ctx.prec += 10
return ctx._gamma_upper_int(z, a) / ctx.gamma(z)
finally:
ctx.prec = orig
else:
return ctx._gamma_upper_int(z, a)
except NotImplementedError:
pass
# hypercomb is unable to detect the exact zeros, so handle them here
if z == 2 and a == -1:
return (z+a)*0
if z == 3 and (a == -1-1j or a == -1+1j):
return (z+a)*0
nega = ctx.fneg(a, exact=True)
G = [z] * regularized
# Use 2F0 series when possible; fall back to lower gamma representation
try:
def h(z):
r = z-1
return [([ctx.exp(nega), a], [1, r], [], G, [1, -r], [], 1/nega)]
return ctx.hypercomb(h, [z], force_series=True)
except ctx.NoConvergence:
def h(z):
T1 = [], [1, z-1], [z], G, [], [], 0
T2 = [-ctx.exp(nega), a, z], [1, z, -1], [], G, [1], [1+z], a
return T1, T2
return ctx.hypercomb(h, [z])
@defun
def _gamma3(ctx, z, a, b, regularized=False):
pole = ctx.isnpint(z)
if regularized and pole:
return ctx.zero
try:
ctx.prec += 15
# We don't know in advance whether it's better to write as a difference
# of lower or upper gamma functions, so try both
T1 = ctx.gammainc(z, a, regularized=regularized)
T2 = ctx.gammainc(z, b, regularized=regularized)
R = T1 - T2
if ctx.mag(R) - max(ctx.mag(T1), ctx.mag(T2)) > -10:
return R
if not pole:
T1 = ctx.gammainc(z, 0, b, regularized=regularized)
T2 = ctx.gammainc(z, 0, a, regularized=regularized)
R = T1 - T2
# May be ok, but should probably at least print a warning
# about possible cancellation
if 1: #ctx.mag(R) - max(ctx.mag(T1), ctx.mag(T2)) > -10:
return R
finally:
ctx.prec -= 15
raise NotImplementedError
@defun_wrapped
def expint(ctx, n, z):
if ctx.isint(n) and ctx._is_real_type(z):
try:
return ctx._expint_int(n, z)
except NotImplementedError:
pass
if ctx.isnan(n) or ctx.isnan(z):
return z*n
if z == ctx.inf:
return 1/z
if z == 0:
# integral from 1 to infinity of t^n
if ctx.re(n) <= 1:
# TODO: reasonable sign of infinity
return type(z)(ctx.inf)
else:
return ctx.one/(n-1)
if n == 0:
return ctx.exp(-z)/z
if n == -1:
return ctx.exp(-z)*(z+1)/z**2
return z**(n-1) * ctx.gammainc(1-n, z)
@defun_wrapped
def li(ctx, z, offset=False):
if offset:
if z == 2:
return ctx.zero
return ctx.ei(ctx.ln(z)) - ctx.ei(ctx.ln2)
if not z:
return z
if z == 1:
return ctx.ninf
return ctx.ei(ctx.ln(z))
@defun
def ei(ctx, z):
try:
return ctx._ei(z)
except NotImplementedError:
return ctx._ei_generic(z)
@defun_wrapped
def _ei_generic(ctx, z):
# Note: the following is currently untested because mp and fp
# both use special-case ei code
if z == ctx.inf:
return z
if z == ctx.ninf:
return ctx.zero
if ctx.mag(z) > 1:
try:
r = ctx.one/z
v = ctx.exp(z)*ctx.hyper([1,1],[],r,
maxterms=ctx.prec, force_series=True)/z
im = ctx._im(z)
if im > 0:
v += ctx.pi*ctx.j
if im < 0:
v -= ctx.pi*ctx.j
return v
except ctx.NoConvergence:
pass
v = z*ctx.hyp2f2(1,1,2,2,z) + ctx.euler
if ctx._im(z):
v += 0.5*(ctx.log(z) - ctx.log(ctx.one/z))
else:
v += ctx.log(abs(z))
return v
@defun
def e1(ctx, z):
try:
return ctx._e1(z)
except NotImplementedError:
return ctx.expint(1, z)
@defun
def ci(ctx, z):
try:
return ctx._ci(z)
except NotImplementedError:
return ctx._ci_generic(z)
@defun_wrapped
def _ci_generic(ctx, z):
if ctx.isinf(z):
if z == ctx.inf: return ctx.zero
if z == ctx.ninf: return ctx.pi*1j
jz = ctx.fmul(ctx.j,z,exact=True)
njz = ctx.fneg(jz,exact=True)
v = 0.5*(ctx.ei(jz) + ctx.ei(njz))
zreal = ctx._re(z)
zimag = ctx._im(z)
if zreal == 0:
if zimag > 0: v += ctx.pi*0.5j
if zimag < 0: v -= ctx.pi*0.5j
if zreal < 0:
if zimag >= 0: v += ctx.pi*1j
if zimag < 0: v -= ctx.pi*1j
if ctx._is_real_type(z) and zreal > 0:
v = ctx._re(v)
return v
@defun
def si(ctx, z):
try:
return ctx._si(z)
except NotImplementedError:
return ctx._si_generic(z)
@defun_wrapped
def _si_generic(ctx, z):
if ctx.isinf(z):
if z == ctx.inf: return 0.5*ctx.pi
if z == ctx.ninf: return -0.5*ctx.pi
# Suffers from cancellation near 0
if ctx.mag(z) >= -1:
jz = ctx.fmul(ctx.j,z,exact=True)
njz = ctx.fneg(jz,exact=True)
v = (-0.5j)*(ctx.ei(jz) - ctx.ei(njz))
zreal = ctx._re(z)
if zreal > 0:
v -= 0.5*ctx.pi
if zreal < 0:
v += 0.5*ctx.pi
if ctx._is_real_type(z):
v = ctx._re(v)
return v
else:
return z*ctx.hyp1f2((1,2),(3,2),(3,2),-0.25*z*z)
@defun_wrapped
def chi(ctx, z):
nz = ctx.fneg(z, exact=True)
v = 0.5*(ctx.ei(z) + ctx.ei(nz))
zreal = ctx._re(z)
zimag = ctx._im(z)
if zimag > 0:
v += ctx.pi*0.5j
elif zimag < 0:
v -= ctx.pi*0.5j
elif zreal < 0:
v += ctx.pi*1j
return v
@defun_wrapped
def shi(ctx, z):
# Suffers from cancellation near 0
if ctx.mag(z) >= -1:
nz = ctx.fneg(z, exact=True)
v = 0.5*(ctx.ei(z) - ctx.ei(nz))
zimag = ctx._im(z)
if zimag > 0: v -= 0.5j*ctx.pi
if zimag < 0: v += 0.5j*ctx.pi
return v
else:
return z * ctx.hyp1f2((1,2),(3,2),(3,2),0.25*z*z)
@defun_wrapped
def fresnels(ctx, z):
if z == ctx.inf:
return ctx.mpf(0.5)
if z == ctx.ninf:
return ctx.mpf(-0.5)
return ctx.pi*z**3/6*ctx.hyp1f2((3,4),(3,2),(7,4),-ctx.pi**2*z**4/16)
@defun_wrapped
def fresnelc(ctx, z):
if z == ctx.inf:
return ctx.mpf(0.5)
if z == ctx.ninf:
return ctx.mpf(-0.5)
return z*ctx.hyp1f2((1,4),(1,2),(5,4),-ctx.pi**2*z**4/16)
| 11,644 | 26.335681 | 79 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/functions/__init__.py
|
from . import functions
# Hack to update methods
from . import factorials
from . import hypergeometric
from . import expintegrals
from . import bessel
from . import orthogonal
from . import theta
from . import elliptic
from . import zeta
from . import rszeta
from . import zetazeros
from . import qfunctions
| 308 | 21.071429 | 28 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/functions/zetazeros.py
|
"""
The function zetazero(n) computes the n-th nontrivial zero of zeta(s).
The general strategy is to locate a block of Gram intervals B where we
know exactly the number of zeros contained and which of those zeros
is that which we search.
If n <= 400 000 000 we know exactly the Rosser exceptions, contained
in a list in this file. Hence for n<=400 000 000 we simply
look at these list of exceptions. If our zero is implicated in one of
these exceptions we have our block B. In other case we simply locate
the good Rosser block containing our zero.
For n > 400 000 000 we apply the method of Turing, as complemented by
Lehman, Brent and Trudgian to find a suitable B.
"""
from .functions import defun, defun_wrapped
def find_rosser_block_zero(ctx, n):
"""for n<400 000 000 determines a block were one find our zero"""
for k in range(len(_ROSSER_EXCEPTIONS)//2):
a=_ROSSER_EXCEPTIONS[2*k][0]
b=_ROSSER_EXCEPTIONS[2*k][1]
if ((a<= n-2) and (n-1 <= b)):
t0 = ctx.grampoint(a)
t1 = ctx.grampoint(b)
v0 = ctx._fp.siegelz(t0)
v1 = ctx._fp.siegelz(t1)
my_zero_number = n-a-1
zero_number_block = b-a
pattern = _ROSSER_EXCEPTIONS[2*k+1]
return (my_zero_number, [a,b], [t0,t1], [v0,v1])
k = n-2
t,v,b = compute_triple_tvb(ctx, k)
T = [t]
V = [v]
while b < 0:
k -= 1
t,v,b = compute_triple_tvb(ctx, k)
T.insert(0,t)
V.insert(0,v)
my_zero_number = n-k-1
m = n-1
t,v,b = compute_triple_tvb(ctx, m)
T.append(t)
V.append(v)
while b < 0:
m += 1
t,v,b = compute_triple_tvb(ctx, m)
T.append(t)
V.append(v)
return (my_zero_number, [k,m], T, V)
def wpzeros(t):
"""Precision needed to compute higher zeros"""
wp = 53
if t > 3*10**8:
wp = 63
if t > 10**11:
wp = 70
if t > 10**14:
wp = 83
return wp
def separate_zeros_in_block(ctx, zero_number_block, T, V, limitloop=None,
fp_tolerance=None):
"""Separate the zeros contained in the block T, limitloop
determines how long one must search"""
if limitloop is None:
limitloop = ctx.inf
loopnumber = 0
variations = count_variations(V)
while ((variations < zero_number_block) and (loopnumber <limitloop)):
a = T[0]
v = V[0]
newT = [a]
newV = [v]
variations = 0
for n in range(1,len(T)):
b2 = T[n]
u = V[n]
if (u*v>0):
alpha = ctx.sqrt(u/v)
b= (alpha*a+b2)/(alpha+1)
else:
b = (a+b2)/2
if fp_tolerance < 10:
w = ctx._fp.siegelz(b)
if abs(w)<fp_tolerance:
w = ctx.siegelz(b)
else:
w=ctx.siegelz(b)
if v*w<0:
variations += 1
newT.append(b)
newV.append(w)
u = V[n]
if u*w <0:
variations += 1
newT.append(b2)
newV.append(u)
a = b2
v = u
T = newT
V = newV
loopnumber +=1
if (limitloop>ITERATION_LIMIT)and(loopnumber>2)and(variations+2==zero_number_block):
dtMax=0
dtSec=0
kMax = 0
for k1 in range(1,len(T)):
dt = T[k1]-T[k1-1]
if dt > dtMax:
kMax=k1
dtSec = dtMax
dtMax = dt
elif (dt<dtMax) and(dt >dtSec):
dtSec = dt
if dtMax>3*dtSec:
f = lambda x: ctx.rs_z(x,derivative=1)
t0=T[kMax-1]
t1 = T[kMax]
t=ctx.findroot(f, (t0,t1), solver ='illinois',verify=False, verbose=False)
v = ctx.siegelz(t)
if (t0<t) and (t<t1) and (v*V[kMax]<0):
T.insert(kMax,t)
V.insert(kMax,v)
variations = count_variations(V)
if variations == zero_number_block:
separated = True
else:
separated = False
return (T,V, separated)
def separate_my_zero(ctx, my_zero_number, zero_number_block, T, V, prec):
"""If we know which zero of this block is mine,
the function separates the zero"""
variations = 0
v0 = V[0]
for k in range(1,len(V)):
v1 = V[k]
if v0*v1 < 0:
variations +=1
if variations == my_zero_number:
k0 = k
leftv = v0
rightv = v1
v0 = v1
t1 = T[k0]
t0 = T[k0-1]
ctx.prec = prec
wpz = wpzeros(my_zero_number*ctx.log(my_zero_number))
guard = 4*ctx.mag(my_zero_number)
precs = [ctx.prec+4]
index=0
while precs[0] > 2*wpz:
index +=1
precs = [precs[0] // 2 +3+2*index] + precs
ctx.prec = precs[0] + guard
r = ctx.findroot(lambda x:ctx.siegelz(x), (t0,t1), solver ='illinois', verbose=False)
#print "first step at", ctx.dps, "digits"
z=ctx.mpc(0.5,r)
for prec in precs[1:]:
ctx.prec = prec + guard
#print "refining to", ctx.dps, "digits"
znew = z - ctx.zeta(z) / ctx.zeta(z, derivative=1)
#print "difference", ctx.nstr(abs(z-znew))
z=ctx.mpc(0.5,ctx.im(znew))
return ctx.im(z)
def sure_number_block(ctx, n):
"""The number of good Rosser blocks needed to apply
Turing method
References:
R. P. Brent, On the Zeros of the Riemann Zeta Function
in the Critical Strip, Math. Comp. 33 (1979) 1361--1372
T. Trudgian, Improvements to Turing Method, Math. Comp."""
if n < 9*10**5:
return(2)
g = ctx.grampoint(n-100)
lg = ctx._fp.ln(g)
brent = 0.0061 * lg**2 +0.08*lg
trudgian = 0.0031 * lg**2 +0.11*lg
N = ctx.ceil(min(brent,trudgian))
N = int(N)
return N
def compute_triple_tvb(ctx, n):
t = ctx.grampoint(n)
v = ctx._fp.siegelz(t)
if ctx.mag(abs(v))<ctx.mag(t)-45:
v = ctx.siegelz(t)
b = v*(-1)**n
return t,v,b
ITERATION_LIMIT = 4
def search_supergood_block(ctx, n, fp_tolerance):
"""To use for n>400 000 000"""
sb = sure_number_block(ctx, n)
number_goodblocks = 0
m2 = n-1
t, v, b = compute_triple_tvb(ctx, m2)
Tf = [t]
Vf = [v]
while b < 0:
m2 += 1
t,v,b = compute_triple_tvb(ctx, m2)
Tf.append(t)
Vf.append(v)
goodpoints = [m2]
T = [t]
V = [v]
while number_goodblocks < 2*sb:
m2 += 1
t, v, b = compute_triple_tvb(ctx, m2)
T.append(t)
V.append(v)
while b < 0:
m2 += 1
t,v,b = compute_triple_tvb(ctx, m2)
T.append(t)
V.append(v)
goodpoints.append(m2)
zn = len(T)-1
A, B, separated =\
separate_zeros_in_block(ctx, zn, T, V, limitloop=ITERATION_LIMIT,
fp_tolerance=fp_tolerance)
Tf.pop()
Tf.extend(A)
Vf.pop()
Vf.extend(B)
if separated:
number_goodblocks += 1
else:
number_goodblocks = 0
T = [t]
V = [v]
# Now the same procedure to the left
number_goodblocks = 0
m2 = n-2
t, v, b = compute_triple_tvb(ctx, m2)
Tf.insert(0,t)
Vf.insert(0,v)
while b < 0:
m2 -= 1
t,v,b = compute_triple_tvb(ctx, m2)
Tf.insert(0,t)
Vf.insert(0,v)
goodpoints.insert(0,m2)
T = [t]
V = [v]
while number_goodblocks < 2*sb:
m2 -= 1
t, v, b = compute_triple_tvb(ctx, m2)
T.insert(0,t)
V.insert(0,v)
while b < 0:
m2 -= 1
t,v,b = compute_triple_tvb(ctx, m2)
T.insert(0,t)
V.insert(0,v)
goodpoints.insert(0,m2)
zn = len(T)-1
A, B, separated =\
separate_zeros_in_block(ctx, zn, T, V, limitloop=ITERATION_LIMIT, fp_tolerance=fp_tolerance)
A.pop()
Tf = A+Tf
B.pop()
Vf = B+Vf
if separated:
number_goodblocks += 1
else:
number_goodblocks = 0
T = [t]
V = [v]
r = goodpoints[2*sb]
lg = len(goodpoints)
s = goodpoints[lg-2*sb-1]
tr, vr, br = compute_triple_tvb(ctx, r)
ar = Tf.index(tr)
ts, vs, bs = compute_triple_tvb(ctx, s)
as1 = Tf.index(ts)
T = Tf[ar:as1+1]
V = Vf[ar:as1+1]
zn = s-r
A, B, separated =\
separate_zeros_in_block(ctx, zn,T,V,limitloop=ITERATION_LIMIT, fp_tolerance=fp_tolerance)
if separated:
return (n-r-1,[r,s],A,B)
q = goodpoints[sb]
lg = len(goodpoints)
t = goodpoints[lg-sb-1]
tq, vq, bq = compute_triple_tvb(ctx, q)
aq = Tf.index(tq)
tt, vt, bt = compute_triple_tvb(ctx, t)
at = Tf.index(tt)
T = Tf[aq:at+1]
V = Vf[aq:at+1]
return (n-q-1,[q,t],T,V)
def count_variations(V):
count = 0
vold = V[0]
for n in range(1, len(V)):
vnew = V[n]
if vold*vnew < 0:
count +=1
vold = vnew
return count
def pattern_construct(ctx, block, T, V):
pattern = '('
a = block[0]
b = block[1]
t0,v0,b0 = compute_triple_tvb(ctx, a)
k = 0
k0 = 0
for n in range(a+1,b+1):
t1,v1,b1 = compute_triple_tvb(ctx, n)
lgT =len(T)
while (k < lgT) and (T[k] <= t1):
k += 1
L = V[k0:k]
L.append(v1)
L.insert(0,v0)
count = count_variations(L)
pattern = pattern + ("%s" % count)
if b1 > 0:
pattern = pattern + ')('
k0 = k
t0,v0,b0 = t1,v1,b1
pattern = pattern[:-1]
return pattern
@defun
def zetazero(ctx, n, info=False, round=True):
r"""
Computes the `n`-th nontrivial zero of `\zeta(s)` on the critical line,
i.e. returns an approximation of the `n`-th largest complex number
`s = \frac{1}{2} + ti` for which `\zeta(s) = 0`. Equivalently, the
imaginary part `t` is a zero of the Z-function (:func:`~mpmath.siegelz`).
**Examples**
The first few zeros::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> zetazero(1)
(0.5 + 14.13472514173469379045725j)
>>> zetazero(2)
(0.5 + 21.02203963877155499262848j)
>>> zetazero(20)
(0.5 + 77.14484006887480537268266j)
Verifying that the values are zeros::
>>> for n in range(1,5):
... s = zetazero(n)
... chop(zeta(s)), chop(siegelz(s.imag))
...
(0.0, 0.0)
(0.0, 0.0)
(0.0, 0.0)
(0.0, 0.0)
Negative indices give the conjugate zeros (`n = 0` is undefined)::
>>> zetazero(-1)
(0.5 - 14.13472514173469379045725j)
:func:`~mpmath.zetazero` supports arbitrarily large `n` and arbitrary precision::
>>> mp.dps = 15
>>> zetazero(1234567)
(0.5 + 727690.906948208j)
>>> mp.dps = 50
>>> zetazero(1234567)
(0.5 + 727690.9069482075392389420041147142092708393819935j)
>>> chop(zeta(_)/_)
0.0
with *info=True*, :func:`~mpmath.zetazero` gives additional information::
>>> mp.dps = 15
>>> zetazero(542964976,info=True)
((0.5 + 209039046.578535j), [542964969, 542964978], 6, '(013111110)')
This means that the zero is between Gram points 542964969 and 542964978;
it is the 6-th zero between them. Finally (01311110) is the pattern
of zeros in this interval. The numbers indicate the number of zeros
in each Gram interval (Rosser blocks between parenthesis). In this case
there is only one Rosser block of length nine.
"""
n = int(n)
if n < 0:
return ctx.zetazero(-n).conjugate()
if n == 0:
raise ValueError("n must be nonzero")
wpinitial = ctx.prec
try:
wpz, fp_tolerance = comp_fp_tolerance(ctx, n)
ctx.prec = wpz
if n < 400000000:
my_zero_number, block, T, V =\
find_rosser_block_zero(ctx, n)
else:
my_zero_number, block, T, V =\
search_supergood_block(ctx, n, fp_tolerance)
zero_number_block = block[1]-block[0]
T, V, separated = separate_zeros_in_block(ctx, zero_number_block, T, V,
limitloop=ctx.inf, fp_tolerance=fp_tolerance)
if info:
pattern = pattern_construct(ctx,block,T,V)
prec = max(wpinitial, wpz)
t = separate_my_zero(ctx, my_zero_number, zero_number_block,T,V,prec)
v = ctx.mpc(0.5,t)
finally:
ctx.prec = wpinitial
if round:
v =+v
if info:
return (v,block,my_zero_number,pattern)
else:
return v
def gram_index(ctx, t):
if t > 10**13:
wp = 3*ctx.log(t, 10)
else:
wp = 0
prec = ctx.prec
try:
ctx.prec += wp
x0 = (t/(2*ctx.pi))*ctx.log(t/(2*ctx.pi))
h = ctx.findroot(lambda x:ctx.siegeltheta(t)-ctx.pi*x, x0)
h = int(h)
finally:
ctx.prec = prec
return(h)
def count_to(ctx, t, T, V):
count = 0
vold = V[0]
told = T[0]
tnew = T[1]
k = 1
while tnew < t:
vnew = V[k]
if vold*vnew < 0:
count += 1
vold = vnew
k += 1
tnew = T[k]
a = ctx.siegelz(t)
if a*vold < 0:
count += 1
return count
def comp_fp_tolerance(ctx, n):
wpz = wpzeros(n*ctx.log(n))
if n < 15*10**8:
fp_tolerance = 0.0005
elif n <= 10**14:
fp_tolerance = 0.1
else:
fp_tolerance = 100
return wpz, fp_tolerance
@defun
def nzeros(ctx, t):
r"""
Computes the number of zeros of the Riemann zeta function in
`(0,1) \times (0,t]`, usually denoted by `N(t)`.
**Examples**
The first zero has imaginary part between 14 and 15::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> nzeros(14)
0
>>> nzeros(15)
1
>>> zetazero(1)
(0.5 + 14.1347251417347j)
Some closely spaced zeros::
>>> nzeros(10**7)
21136125
>>> zetazero(21136125)
(0.5 + 9999999.32718175j)
>>> zetazero(21136126)
(0.5 + 10000000.2400236j)
>>> nzeros(545439823.215)
1500000001
>>> zetazero(1500000001)
(0.5 + 545439823.201985j)
>>> zetazero(1500000002)
(0.5 + 545439823.325697j)
This confirms the data given by J. van de Lune,
H. J. J. te Riele and D. T. Winter in 1986.
"""
if t < 14.1347251417347:
return 0
x = gram_index(ctx, t)
k = int(ctx.floor(x))
wpinitial = ctx.prec
wpz, fp_tolerance = comp_fp_tolerance(ctx, k)
ctx.prec = wpz
a = ctx.siegelz(t)
if k == -1 and a < 0:
return 0
elif k == -1 and a > 0:
return 1
if k+2 < 400000000:
Rblock = find_rosser_block_zero(ctx, k+2)
else:
Rblock = search_supergood_block(ctx, k+2, fp_tolerance)
n1, n2 = Rblock[1]
if n2-n1 == 1:
b = Rblock[3][0]
if a*b > 0:
ctx.prec = wpinitial
return k+1
else:
ctx.prec = wpinitial
return k+2
my_zero_number,block, T, V = Rblock
zero_number_block = n2-n1
T, V, separated = separate_zeros_in_block(ctx,\
zero_number_block, T, V,\
limitloop=ctx.inf,\
fp_tolerance=fp_tolerance)
n = count_to(ctx, t, T, V)
ctx.prec = wpinitial
return n+n1+1
@defun_wrapped
def backlunds(ctx, t):
r"""
Computes the function
`S(t) = \operatorname{arg} \zeta(\frac{1}{2} + it) / \pi`.
See Titchmarsh Section 9.3 for details of the definition.
**Examples**
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> backlunds(217.3)
0.16302205431184
Generally, the value is a small number. At Gram points it is an integer,
frequently equal to 0::
>>> chop(backlunds(grampoint(200)))
0.0
>>> backlunds(extraprec(10)(grampoint)(211))
1.0
>>> backlunds(extraprec(10)(grampoint)(232))
-1.0
The number of zeros of the Riemann zeta function up to height `t`
satisfies `N(t) = \theta(t)/\pi + 1 + S(t)` (see :func:nzeros` and
:func:`siegeltheta`)::
>>> t = 1234.55
>>> nzeros(t)
842
>>> siegeltheta(t)/pi+1+backlunds(t)
842.0
"""
return ctx.nzeros(t)-1-ctx.siegeltheta(t)/ctx.pi
"""
_ROSSER_EXCEPTIONS is a list of all exceptions to
Rosser's rule for n <= 400 000 000.
Alternately the entry is of type [n,m], or a string.
The string is the zero pattern of the Block and the relevant
adjacent. For example (010)3 corresponds to a block
composed of three Gram intervals, the first ant third without
a zero and the intermediate with a zero. The next Gram interval
contain three zeros. So that in total we have 4 zeros in 4 Gram
blocks. n and m are the indices of the Gram points of this
interval of four Gram intervals. The Rosser exception is therefore
formed by the three Gram intervals that are signaled between
parenthesis.
We have included also some Rosser's exceptions beyond n=400 000 000
that are noted in the literature by some reason.
The list is composed from the data published in the references:
R. P. Brent, J. van de Lune, H. J. J. te Riele, D. T. Winter,
'On the Zeros of the Riemann Zeta Function in the Critical Strip. II',
Math. Comp. 39 (1982) 681--688.
See also Corrigenda in Math. Comp. 46 (1986) 771.
J. van de Lune, H. J. J. te Riele,
'On the Zeros of the Riemann Zeta Function in the Critical Strip. III',
Math. Comp. 41 (1983) 759--767.
See also Corrigenda in Math. Comp. 46 (1986) 771.
J. van de Lune,
'Sums of Equal Powers of Positive Integers',
Dissertation,
Vrije Universiteit te Amsterdam, Centrum voor Wiskunde en Informatica,
Amsterdam, 1984.
Thanks to the authors all this papers and those others that have
contributed to make this possible.
"""
_ROSSER_EXCEPTIONS = \
[[13999525, 13999528], '(00)3',
[30783329, 30783332], '(00)3',
[30930926, 30930929], '3(00)',
[37592215, 37592218], '(00)3',
[40870156, 40870159], '(00)3',
[43628107, 43628110], '(00)3',
[46082042, 46082045], '(00)3',
[46875667, 46875670], '(00)3',
[49624540, 49624543], '3(00)',
[50799238, 50799241], '(00)3',
[55221453, 55221456], '3(00)',
[56948779, 56948782], '3(00)',
[60515663, 60515666], '(00)3',
[61331766, 61331770], '(00)40',
[69784843, 69784846], '3(00)',
[75052114, 75052117], '(00)3',
[79545240, 79545243], '3(00)',
[79652247, 79652250], '3(00)',
[83088043, 83088046], '(00)3',
[83689522, 83689525], '3(00)',
[85348958, 85348961], '(00)3',
[86513820, 86513823], '(00)3',
[87947596, 87947599], '3(00)',
[88600095, 88600098], '(00)3',
[93681183, 93681186], '(00)3',
[100316551, 100316554], '3(00)',
[100788444, 100788447], '(00)3',
[106236172, 106236175], '(00)3',
[106941327, 106941330], '3(00)',
[107287955, 107287958], '(00)3',
[107532016, 107532019], '3(00)',
[110571044, 110571047], '(00)3',
[111885253, 111885256], '3(00)',
[113239783, 113239786], '(00)3',
[120159903, 120159906], '(00)3',
[121424391, 121424394], '3(00)',
[121692931, 121692934], '3(00)',
[121934170, 121934173], '3(00)',
[122612848, 122612851], '3(00)',
[126116567, 126116570], '(00)3',
[127936513, 127936516], '(00)3',
[128710277, 128710280], '3(00)',
[129398902, 129398905], '3(00)',
[130461096, 130461099], '3(00)',
[131331947, 131331950], '3(00)',
[137334071, 137334074], '3(00)',
[137832603, 137832606], '(00)3',
[138799471, 138799474], '3(00)',
[139027791, 139027794], '(00)3',
[141617806, 141617809], '(00)3',
[144454931, 144454934], '(00)3',
[145402379, 145402382], '3(00)',
[146130245, 146130248], '3(00)',
[147059770, 147059773], '(00)3',
[147896099, 147896102], '3(00)',
[151097113, 151097116], '(00)3',
[152539438, 152539441], '(00)3',
[152863168, 152863171], '3(00)',
[153522726, 153522729], '3(00)',
[155171524, 155171527], '3(00)',
[155366607, 155366610], '(00)3',
[157260686, 157260689], '3(00)',
[157269224, 157269227], '(00)3',
[157755123, 157755126], '(00)3',
[158298484, 158298487], '3(00)',
[160369050, 160369053], '3(00)',
[162962787, 162962790], '(00)3',
[163724709, 163724712], '(00)3',
[164198113, 164198116], '3(00)',
[164689301, 164689305], '(00)40',
[164880228, 164880231], '3(00)',
[166201932, 166201935], '(00)3',
[168573836, 168573839], '(00)3',
[169750763, 169750766], '(00)3',
[170375507, 170375510], '(00)3',
[170704879, 170704882], '3(00)',
[172000992, 172000995], '3(00)',
[173289941, 173289944], '(00)3',
[173737613, 173737616], '3(00)',
[174102513, 174102516], '(00)3',
[174284990, 174284993], '(00)3',
[174500513, 174500516], '(00)3',
[175710609, 175710612], '(00)3',
[176870843, 176870846], '3(00)',
[177332732, 177332735], '3(00)',
[177902861, 177902864], '3(00)',
[179979095, 179979098], '(00)3',
[181233726, 181233729], '3(00)',
[181625435, 181625438], '(00)3',
[182105255, 182105259], '22(00)',
[182223559, 182223562], '3(00)',
[191116404, 191116407], '3(00)',
[191165599, 191165602], '3(00)',
[191297535, 191297539], '(00)22',
[192485616, 192485619], '(00)3',
[193264634, 193264638], '22(00)',
[194696968, 194696971], '(00)3',
[195876805, 195876808], '(00)3',
[195916548, 195916551], '3(00)',
[196395160, 196395163], '3(00)',
[196676303, 196676306], '(00)3',
[197889882, 197889885], '3(00)',
[198014122, 198014125], '(00)3',
[199235289, 199235292], '(00)3',
[201007375, 201007378], '(00)3',
[201030605, 201030608], '3(00)',
[201184290, 201184293], '3(00)',
[201685414, 201685418], '(00)22',
[202762875, 202762878], '3(00)',
[202860957, 202860960], '3(00)',
[203832577, 203832580], '3(00)',
[205880544, 205880547], '(00)3',
[206357111, 206357114], '(00)3',
[207159767, 207159770], '3(00)',
[207167343, 207167346], '3(00)',
[207482539, 207482543], '3(010)',
[207669540, 207669543], '3(00)',
[208053426, 208053429], '(00)3',
[208110027, 208110030], '3(00)',
[209513826, 209513829], '3(00)',
[212623522, 212623525], '(00)3',
[213841715, 213841718], '(00)3',
[214012333, 214012336], '(00)3',
[214073567, 214073570], '(00)3',
[215170600, 215170603], '3(00)',
[215881039, 215881042], '3(00)',
[216274604, 216274607], '3(00)',
[216957120, 216957123], '3(00)',
[217323208, 217323211], '(00)3',
[218799264, 218799267], '(00)3',
[218803557, 218803560], '3(00)',
[219735146, 219735149], '(00)3',
[219830062, 219830065], '3(00)',
[219897904, 219897907], '(00)3',
[221205545, 221205548], '(00)3',
[223601929, 223601932], '(00)3',
[223907076, 223907079], '3(00)',
[223970397, 223970400], '(00)3',
[224874044, 224874048], '22(00)',
[225291157, 225291160], '(00)3',
[227481734, 227481737], '(00)3',
[228006442, 228006445], '3(00)',
[228357900, 228357903], '(00)3',
[228386399, 228386402], '(00)3',
[228907446, 228907449], '(00)3',
[228984552, 228984555], '3(00)',
[229140285, 229140288], '3(00)',
[231810024, 231810027], '(00)3',
[232838062, 232838065], '3(00)',
[234389088, 234389091], '3(00)',
[235588194, 235588197], '(00)3',
[236645695, 236645698], '(00)3',
[236962876, 236962879], '3(00)',
[237516723, 237516727], '04(00)',
[240004911, 240004914], '(00)3',
[240221306, 240221309], '3(00)',
[241389213, 241389217], '(010)3',
[241549003, 241549006], '(00)3',
[241729717, 241729720], '(00)3',
[241743684, 241743687], '3(00)',
[243780200, 243780203], '3(00)',
[243801317, 243801320], '(00)3',
[244122072, 244122075], '(00)3',
[244691224, 244691227], '3(00)',
[244841577, 244841580], '(00)3',
[245813461, 245813464], '(00)3',
[246299475, 246299478], '(00)3',
[246450176, 246450179], '3(00)',
[249069349, 249069352], '(00)3',
[250076378, 250076381], '(00)3',
[252442157, 252442160], '3(00)',
[252904231, 252904234], '3(00)',
[255145220, 255145223], '(00)3',
[255285971, 255285974], '3(00)',
[256713230, 256713233], '(00)3',
[257992082, 257992085], '(00)3',
[258447955, 258447959], '22(00)',
[259298045, 259298048], '3(00)',
[262141503, 262141506], '(00)3',
[263681743, 263681746], '3(00)',
[266527881, 266527885], '(010)3',
[266617122, 266617125], '(00)3',
[266628044, 266628047], '3(00)',
[267305763, 267305766], '(00)3',
[267388404, 267388407], '3(00)',
[267441672, 267441675], '3(00)',
[267464886, 267464889], '(00)3',
[267554907, 267554910], '3(00)',
[269787480, 269787483], '(00)3',
[270881434, 270881437], '(00)3',
[270997583, 270997586], '3(00)',
[272096378, 272096381], '3(00)',
[272583009, 272583012], '(00)3',
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[377119945, 377119948], '(00)3',
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[384401786, 384401790], '22(00)',
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[385908194, 385908197], '3(00)',
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[388832142, 388832145], '3(00)',
[390087103, 390087106], '(00)3',
[390190926, 390190930], '(00)22',
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[391951632, 391951636], '(00)22',
[392963986, 392963989], '(00)3',
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[393759572, 393759575], '(00)3',
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[397076433, 397076436], '3(00)',
[397470601, 397470604], '3(00)',
[398289458, 398289461], '3(00)',
#
[368714778, 368714783], '04(010)',
[437953499, 437953504], '04(010)',
[526196233, 526196238], '032(00)',
[744719566, 744719571], '(010)40',
[750375857, 750375862], '032(00)',
[958241932, 958241937], '04(010)',
[983377342, 983377347], '(00)410',
[1003780080, 1003780085], '04(010)',
[1070232754, 1070232759], '(00)230',
[1209834865, 1209834870], '032(00)',
[1257209100, 1257209105], '(00)410',
[1368002233, 1368002238], '(00)230'
]
| 30,951 | 29.315377 | 103 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/functions/theta.py
|
from .functions import defun, defun_wrapped
@defun
def _jacobi_theta2(ctx, z, q):
extra1 = 10
extra2 = 20
# the loops below break when the fixed precision quantities
# a and b go to zero;
# right shifting small negative numbers by wp one obtains -1, not zero,
# so the condition a**2 + b**2 > MIN is used to break the loops.
MIN = 2
if z == ctx.zero:
if (not ctx._im(q)):
wp = ctx.prec + extra1
x = ctx.to_fixed(ctx._re(q), wp)
x2 = (x*x) >> wp
a = b = x2
s = x2
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
s += a
s = (1 << (wp+1)) + (s << 1)
s = ctx.ldexp(s, -wp)
else:
wp = ctx.prec + extra1
xre = ctx.to_fixed(ctx._re(q), wp)
xim = ctx.to_fixed(ctx._im(q), wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp-1)
are = bre = x2re
aim = bim = x2im
sre = (1<<wp) + are
sim = aim
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
sre += are
sim += aim
sre = (sre << 1)
sim = (sim << 1)
sre = ctx.ldexp(sre, -wp)
sim = ctx.ldexp(sim, -wp)
s = ctx.mpc(sre, sim)
else:
if (not ctx._im(q)) and (not ctx._im(z)):
wp = ctx.prec + extra1
x = ctx.to_fixed(ctx._re(q), wp)
x2 = (x*x) >> wp
a = b = x2
c1, s1 = ctx.cos_sin(ctx._re(z), prec=wp)
cn = c1 = ctx.to_fixed(c1, wp)
sn = s1 = ctx.to_fixed(s1, wp)
c2 = (c1*c1 - s1*s1) >> wp
s2 = (c1 * s1) >> (wp - 1)
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
s = c1 + ((a * cn) >> wp)
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
s += (a * cn) >> wp
s = (s << 1)
s = ctx.ldexp(s, -wp)
s *= ctx.nthroot(q, 4)
return s
# case z real, q complex
elif not ctx._im(z):
wp = ctx.prec + extra2
xre = ctx.to_fixed(ctx._re(q), wp)
xim = ctx.to_fixed(ctx._im(q), wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = x2re
aim = bim = x2im
c1, s1 = ctx.cos_sin(ctx._re(z), prec=wp)
cn = c1 = ctx.to_fixed(c1, wp)
sn = s1 = ctx.to_fixed(s1, wp)
c2 = (c1*c1 - s1*s1) >> wp
s2 = (c1 * s1) >> (wp - 1)
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
sre = c1 + ((are * cn) >> wp)
sim = ((aim * cn) >> wp)
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
sre += ((are * cn) >> wp)
sim += ((aim * cn) >> wp)
sre = (sre << 1)
sim = (sim << 1)
sre = ctx.ldexp(sre, -wp)
sim = ctx.ldexp(sim, -wp)
s = ctx.mpc(sre, sim)
#case z complex, q real
elif not ctx._im(q):
wp = ctx.prec + extra2
x = ctx.to_fixed(ctx._re(q), wp)
x2 = (x*x) >> wp
a = b = x2
prec0 = ctx.prec
ctx.prec = wp
c1, s1 = ctx.cos_sin(z)
ctx.prec = prec0
cnre = c1re = ctx.to_fixed(ctx._re(c1), wp)
cnim = c1im = ctx.to_fixed(ctx._im(c1), wp)
snre = s1re = ctx.to_fixed(ctx._re(s1), wp)
snim = s1im = ctx.to_fixed(ctx._im(s1), wp)
#c2 = (c1*c1 - s1*s1) >> wp
c2re = (c1re*c1re - c1im*c1im - s1re*s1re + s1im*s1im) >> wp
c2im = (c1re*c1im - s1re*s1im) >> (wp - 1)
#s2 = (c1 * s1) >> (wp - 1)
s2re = (c1re*s1re - c1im*s1im) >> (wp - 1)
s2im = (c1re*s1im + c1im*s1re) >> (wp - 1)
#cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
sre = c1re + ((a * cnre) >> wp)
sim = c1im + ((a * cnim) >> wp)
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
sre += ((a * cnre) >> wp)
sim += ((a * cnim) >> wp)
sre = (sre << 1)
sim = (sim << 1)
sre = ctx.ldexp(sre, -wp)
sim = ctx.ldexp(sim, -wp)
s = ctx.mpc(sre, sim)
# case z and q complex
else:
wp = ctx.prec + extra2
xre = ctx.to_fixed(ctx._re(q), wp)
xim = ctx.to_fixed(ctx._im(q), wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = x2re
aim = bim = x2im
prec0 = ctx.prec
ctx.prec = wp
# cos(z), sin(z) with z complex
c1, s1 = ctx.cos_sin(z)
ctx.prec = prec0
cnre = c1re = ctx.to_fixed(ctx._re(c1), wp)
cnim = c1im = ctx.to_fixed(ctx._im(c1), wp)
snre = s1re = ctx.to_fixed(ctx._re(s1), wp)
snim = s1im = ctx.to_fixed(ctx._im(s1), wp)
c2re = (c1re*c1re - c1im*c1im - s1re*s1re + s1im*s1im) >> wp
c2im = (c1re*c1im - s1re*s1im) >> (wp - 1)
s2re = (c1re*s1re - c1im*s1im) >> (wp - 1)
s2im = (c1re*s1im + c1im*s1re) >> (wp - 1)
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
n = 1
termre = c1re
termim = c1im
sre = c1re + ((are * cnre - aim * cnim) >> wp)
sim = c1im + ((are * cnim + aim * cnre) >> wp)
n = 3
termre = ((are * cnre - aim * cnim) >> wp)
termim = ((are * cnim + aim * cnre) >> wp)
sre = c1re + ((are * cnre - aim * cnim) >> wp)
sim = c1im + ((are * cnim + aim * cnre) >> wp)
n = 5
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
#cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
termre = ((are * cnre - aim * cnim) >> wp)
termim = ((aim * cnre + are * cnim) >> wp)
sre += ((are * cnre - aim * cnim) >> wp)
sim += ((aim * cnre + are * cnim) >> wp)
n += 2
sre = (sre << 1)
sim = (sim << 1)
sre = ctx.ldexp(sre, -wp)
sim = ctx.ldexp(sim, -wp)
s = ctx.mpc(sre, sim)
s *= ctx.nthroot(q, 4)
return s
@defun
def _djacobi_theta2(ctx, z, q, nd):
MIN = 2
extra1 = 10
extra2 = 20
if (not ctx._im(q)) and (not ctx._im(z)):
wp = ctx.prec + extra1
x = ctx.to_fixed(ctx._re(q), wp)
x2 = (x*x) >> wp
a = b = x2
c1, s1 = ctx.cos_sin(ctx._re(z), prec=wp)
cn = c1 = ctx.to_fixed(c1, wp)
sn = s1 = ctx.to_fixed(s1, wp)
c2 = (c1*c1 - s1*s1) >> wp
s2 = (c1 * s1) >> (wp - 1)
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
if (nd&1):
s = s1 + ((a * sn * 3**nd) >> wp)
else:
s = c1 + ((a * cn * 3**nd) >> wp)
n = 2
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
if nd&1:
s += (a * sn * (2*n+1)**nd) >> wp
else:
s += (a * cn * (2*n+1)**nd) >> wp
n += 1
s = -(s << 1)
s = ctx.ldexp(s, -wp)
# case z real, q complex
elif not ctx._im(z):
wp = ctx.prec + extra2
xre = ctx.to_fixed(ctx._re(q), wp)
xim = ctx.to_fixed(ctx._im(q), wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = x2re
aim = bim = x2im
c1, s1 = ctx.cos_sin(ctx._re(z), prec=wp)
cn = c1 = ctx.to_fixed(c1, wp)
sn = s1 = ctx.to_fixed(s1, wp)
c2 = (c1*c1 - s1*s1) >> wp
s2 = (c1 * s1) >> (wp - 1)
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
if (nd&1):
sre = s1 + ((are * sn * 3**nd) >> wp)
sim = ((aim * sn * 3**nd) >> wp)
else:
sre = c1 + ((are * cn * 3**nd) >> wp)
sim = ((aim * cn * 3**nd) >> wp)
n = 5
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
if (nd&1):
sre += ((are * sn * n**nd) >> wp)
sim += ((aim * sn * n**nd) >> wp)
else:
sre += ((are * cn * n**nd) >> wp)
sim += ((aim * cn * n**nd) >> wp)
n += 2
sre = -(sre << 1)
sim = -(sim << 1)
sre = ctx.ldexp(sre, -wp)
sim = ctx.ldexp(sim, -wp)
s = ctx.mpc(sre, sim)
#case z complex, q real
elif not ctx._im(q):
wp = ctx.prec + extra2
x = ctx.to_fixed(ctx._re(q), wp)
x2 = (x*x) >> wp
a = b = x2
prec0 = ctx.prec
ctx.prec = wp
c1, s1 = ctx.cos_sin(z)
ctx.prec = prec0
cnre = c1re = ctx.to_fixed(ctx._re(c1), wp)
cnim = c1im = ctx.to_fixed(ctx._im(c1), wp)
snre = s1re = ctx.to_fixed(ctx._re(s1), wp)
snim = s1im = ctx.to_fixed(ctx._im(s1), wp)
#c2 = (c1*c1 - s1*s1) >> wp
c2re = (c1re*c1re - c1im*c1im - s1re*s1re + s1im*s1im) >> wp
c2im = (c1re*c1im - s1re*s1im) >> (wp - 1)
#s2 = (c1 * s1) >> (wp - 1)
s2re = (c1re*s1re - c1im*s1im) >> (wp - 1)
s2im = (c1re*s1im + c1im*s1re) >> (wp - 1)
#cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if (nd&1):
sre = s1re + ((a * snre * 3**nd) >> wp)
sim = s1im + ((a * snim * 3**nd) >> wp)
else:
sre = c1re + ((a * cnre * 3**nd) >> wp)
sim = c1im + ((a * cnim * 3**nd) >> wp)
n = 5
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if (nd&1):
sre += ((a * snre * n**nd) >> wp)
sim += ((a * snim * n**nd) >> wp)
else:
sre += ((a * cnre * n**nd) >> wp)
sim += ((a * cnim * n**nd) >> wp)
n += 2
sre = -(sre << 1)
sim = -(sim << 1)
sre = ctx.ldexp(sre, -wp)
sim = ctx.ldexp(sim, -wp)
s = ctx.mpc(sre, sim)
# case z and q complex
else:
wp = ctx.prec + extra2
xre = ctx.to_fixed(ctx._re(q), wp)
xim = ctx.to_fixed(ctx._im(q), wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = x2re
aim = bim = x2im
prec0 = ctx.prec
ctx.prec = wp
# cos(2*z), sin(2*z) with z complex
c1, s1 = ctx.cos_sin(z)
ctx.prec = prec0
cnre = c1re = ctx.to_fixed(ctx._re(c1), wp)
cnim = c1im = ctx.to_fixed(ctx._im(c1), wp)
snre = s1re = ctx.to_fixed(ctx._re(s1), wp)
snim = s1im = ctx.to_fixed(ctx._im(s1), wp)
c2re = (c1re*c1re - c1im*c1im - s1re*s1re + s1im*s1im) >> wp
c2im = (c1re*c1im - s1re*s1im) >> (wp - 1)
s2re = (c1re*s1re - c1im*s1im) >> (wp - 1)
s2im = (c1re*s1im + c1im*s1re) >> (wp - 1)
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if (nd&1):
sre = s1re + (((are * snre - aim * snim) * 3**nd) >> wp)
sim = s1im + (((are * snim + aim * snre)* 3**nd) >> wp)
else:
sre = c1re + (((are * cnre - aim * cnim) * 3**nd) >> wp)
sim = c1im + (((are * cnim + aim * cnre)* 3**nd) >> wp)
n = 5
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
#cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if (nd&1):
sre += (((are * snre - aim * snim) * n**nd) >> wp)
sim += (((aim * snre + are * snim) * n**nd) >> wp)
else:
sre += (((are * cnre - aim * cnim) * n**nd) >> wp)
sim += (((aim * cnre + are * cnim) * n**nd) >> wp)
n += 2
sre = -(sre << 1)
sim = -(sim << 1)
sre = ctx.ldexp(sre, -wp)
sim = ctx.ldexp(sim, -wp)
s = ctx.mpc(sre, sim)
s *= ctx.nthroot(q, 4)
if (nd&1):
return (-1)**(nd//2) * s
else:
return (-1)**(1 + nd//2) * s
@defun
def _jacobi_theta3(ctx, z, q):
extra1 = 10
extra2 = 20
MIN = 2
if z == ctx.zero:
if not ctx._im(q):
wp = ctx.prec + extra1
x = ctx.to_fixed(ctx._re(q), wp)
s = x
a = b = x
x2 = (x*x) >> wp
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
s += a
s = (1 << wp) + (s << 1)
s = ctx.ldexp(s, -wp)
return s
else:
wp = ctx.prec + extra1
xre = ctx.to_fixed(ctx._re(q), wp)
xim = ctx.to_fixed(ctx._im(q), wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
sre = are = bre = xre
sim = aim = bim = xim
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
sre += are
sim += aim
sre = (1 << wp) + (sre << 1)
sim = (sim << 1)
sre = ctx.ldexp(sre, -wp)
sim = ctx.ldexp(sim, -wp)
s = ctx.mpc(sre, sim)
return s
else:
if (not ctx._im(q)) and (not ctx._im(z)):
s = 0
wp = ctx.prec + extra1
x = ctx.to_fixed(ctx._re(q), wp)
a = b = x
x2 = (x*x) >> wp
c1, s1 = ctx.cos_sin(ctx._re(z)*2, prec=wp)
c1 = ctx.to_fixed(c1, wp)
s1 = ctx.to_fixed(s1, wp)
cn = c1
sn = s1
s += (a * cn) >> wp
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp
s += (a * cn) >> wp
s = (1 << wp) + (s << 1)
s = ctx.ldexp(s, -wp)
return s
# case z real, q complex
elif not ctx._im(z):
wp = ctx.prec + extra2
xre = ctx.to_fixed(ctx._re(q), wp)
xim = ctx.to_fixed(ctx._im(q), wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = xre
aim = bim = xim
c1, s1 = ctx.cos_sin(ctx._re(z)*2, prec=wp)
c1 = ctx.to_fixed(c1, wp)
s1 = ctx.to_fixed(s1, wp)
cn = c1
sn = s1
sre = (are * cn) >> wp
sim = (aim * cn) >> wp
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp
sre += (are * cn) >> wp
sim += (aim * cn) >> wp
sre = (1 << wp) + (sre << 1)
sim = (sim << 1)
sre = ctx.ldexp(sre, -wp)
sim = ctx.ldexp(sim, -wp)
s = ctx.mpc(sre, sim)
return s
#case z complex, q real
elif not ctx._im(q):
wp = ctx.prec + extra2
x = ctx.to_fixed(ctx._re(q), wp)
a = b = x
x2 = (x*x) >> wp
prec0 = ctx.prec
ctx.prec = wp
c1, s1 = ctx.cos_sin(2*z)
ctx.prec = prec0
cnre = c1re = ctx.to_fixed(ctx._re(c1), wp)
cnim = c1im = ctx.to_fixed(ctx._im(c1), wp)
snre = s1re = ctx.to_fixed(ctx._re(s1), wp)
snim = s1im = ctx.to_fixed(ctx._im(s1), wp)
sre = (a * cnre) >> wp
sim = (a * cnim) >> wp
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
t1 = (cnre*c1re - cnim*c1im - snre*s1re + snim*s1im) >> wp
t2 = (cnre*c1im + cnim*c1re - snre*s1im - snim*s1re) >> wp
t3 = (snre*c1re - snim*c1im + cnre*s1re - cnim*s1im) >> wp
t4 = (snre*c1im + snim*c1re + cnre*s1im + cnim*s1re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
sre += (a * cnre) >> wp
sim += (a * cnim) >> wp
sre = (1 << wp) + (sre << 1)
sim = (sim << 1)
sre = ctx.ldexp(sre, -wp)
sim = ctx.ldexp(sim, -wp)
s = ctx.mpc(sre, sim)
return s
# case z and q complex
else:
wp = ctx.prec + extra2
xre = ctx.to_fixed(ctx._re(q), wp)
xim = ctx.to_fixed(ctx._im(q), wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = xre
aim = bim = xim
prec0 = ctx.prec
ctx.prec = wp
# cos(2*z), sin(2*z) with z complex
c1, s1 = ctx.cos_sin(2*z)
ctx.prec = prec0
cnre = c1re = ctx.to_fixed(ctx._re(c1), wp)
cnim = c1im = ctx.to_fixed(ctx._im(c1), wp)
snre = s1re = ctx.to_fixed(ctx._re(s1), wp)
snim = s1im = ctx.to_fixed(ctx._im(s1), wp)
sre = (are * cnre - aim * cnim) >> wp
sim = (aim * cnre + are * cnim) >> wp
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
t1 = (cnre*c1re - cnim*c1im - snre*s1re + snim*s1im) >> wp
t2 = (cnre*c1im + cnim*c1re - snre*s1im - snim*s1re) >> wp
t3 = (snre*c1re - snim*c1im + cnre*s1re - cnim*s1im) >> wp
t4 = (snre*c1im + snim*c1re + cnre*s1im + cnim*s1re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
sre += (are * cnre - aim * cnim) >> wp
sim += (aim * cnre + are * cnim) >> wp
sre = (1 << wp) + (sre << 1)
sim = (sim << 1)
sre = ctx.ldexp(sre, -wp)
sim = ctx.ldexp(sim, -wp)
s = ctx.mpc(sre, sim)
return s
@defun
def _djacobi_theta3(ctx, z, q, nd):
"""nd=1,2,3 order of the derivative with respect to z"""
MIN = 2
extra1 = 10
extra2 = 20
if (not ctx._im(q)) and (not ctx._im(z)):
s = 0
wp = ctx.prec + extra1
x = ctx.to_fixed(ctx._re(q), wp)
a = b = x
x2 = (x*x) >> wp
c1, s1 = ctx.cos_sin(ctx._re(z)*2, prec=wp)
c1 = ctx.to_fixed(c1, wp)
s1 = ctx.to_fixed(s1, wp)
cn = c1
sn = s1
if (nd&1):
s += (a * sn) >> wp
else:
s += (a * cn) >> wp
n = 2
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp
if nd&1:
s += (a * sn * n**nd) >> wp
else:
s += (a * cn * n**nd) >> wp
n += 1
s = -(s << (nd+1))
s = ctx.ldexp(s, -wp)
# case z real, q complex
elif not ctx._im(z):
wp = ctx.prec + extra2
xre = ctx.to_fixed(ctx._re(q), wp)
xim = ctx.to_fixed(ctx._im(q), wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = xre
aim = bim = xim
c1, s1 = ctx.cos_sin(ctx._re(z)*2, prec=wp)
c1 = ctx.to_fixed(c1, wp)
s1 = ctx.to_fixed(s1, wp)
cn = c1
sn = s1
if (nd&1):
sre = (are * sn) >> wp
sim = (aim * sn) >> wp
else:
sre = (are * cn) >> wp
sim = (aim * cn) >> wp
n = 2
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp
if nd&1:
sre += (are * sn * n**nd) >> wp
sim += (aim * sn * n**nd) >> wp
else:
sre += (are * cn * n**nd) >> wp
sim += (aim * cn * n**nd) >> wp
n += 1
sre = -(sre << (nd+1))
sim = -(sim << (nd+1))
sre = ctx.ldexp(sre, -wp)
sim = ctx.ldexp(sim, -wp)
s = ctx.mpc(sre, sim)
#case z complex, q real
elif not ctx._im(q):
wp = ctx.prec + extra2
x = ctx.to_fixed(ctx._re(q), wp)
a = b = x
x2 = (x*x) >> wp
prec0 = ctx.prec
ctx.prec = wp
c1, s1 = ctx.cos_sin(2*z)
ctx.prec = prec0
cnre = c1re = ctx.to_fixed(ctx._re(c1), wp)
cnim = c1im = ctx.to_fixed(ctx._im(c1), wp)
snre = s1re = ctx.to_fixed(ctx._re(s1), wp)
snim = s1im = ctx.to_fixed(ctx._im(s1), wp)
if (nd&1):
sre = (a * snre) >> wp
sim = (a * snim) >> wp
else:
sre = (a * cnre) >> wp
sim = (a * cnim) >> wp
n = 2
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
t1 = (cnre*c1re - cnim*c1im - snre*s1re + snim*s1im) >> wp
t2 = (cnre*c1im + cnim*c1re - snre*s1im - snim*s1re) >> wp
t3 = (snre*c1re - snim*c1im + cnre*s1re - cnim*s1im) >> wp
t4 = (snre*c1im + snim*c1re + cnre*s1im + cnim*s1re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if (nd&1):
sre += (a * snre * n**nd) >> wp
sim += (a * snim * n**nd) >> wp
else:
sre += (a * cnre * n**nd) >> wp
sim += (a * cnim * n**nd) >> wp
n += 1
sre = -(sre << (nd+1))
sim = -(sim << (nd+1))
sre = ctx.ldexp(sre, -wp)
sim = ctx.ldexp(sim, -wp)
s = ctx.mpc(sre, sim)
# case z and q complex
else:
wp = ctx.prec + extra2
xre = ctx.to_fixed(ctx._re(q), wp)
xim = ctx.to_fixed(ctx._im(q), wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = xre
aim = bim = xim
prec0 = ctx.prec
ctx.prec = wp
# cos(2*z), sin(2*z) with z complex
c1, s1 = ctx.cos_sin(2*z)
ctx.prec = prec0
cnre = c1re = ctx.to_fixed(ctx._re(c1), wp)
cnim = c1im = ctx.to_fixed(ctx._im(c1), wp)
snre = s1re = ctx.to_fixed(ctx._re(s1), wp)
snim = s1im = ctx.to_fixed(ctx._im(s1), wp)
if (nd&1):
sre = (are * snre - aim * snim) >> wp
sim = (aim * snre + are * snim) >> wp
else:
sre = (are * cnre - aim * cnim) >> wp
sim = (aim * cnre + are * cnim) >> wp
n = 2
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
t1 = (cnre*c1re - cnim*c1im - snre*s1re + snim*s1im) >> wp
t2 = (cnre*c1im + cnim*c1re - snre*s1im - snim*s1re) >> wp
t3 = (snre*c1re - snim*c1im + cnre*s1re - cnim*s1im) >> wp
t4 = (snre*c1im + snim*c1re + cnre*s1im + cnim*s1re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if(nd&1):
sre += ((are * snre - aim * snim) * n**nd) >> wp
sim += ((aim * snre + are * snim) * n**nd) >> wp
else:
sre += ((are * cnre - aim * cnim) * n**nd) >> wp
sim += ((aim * cnre + are * cnim) * n**nd) >> wp
n += 1
sre = -(sre << (nd+1))
sim = -(sim << (nd+1))
sre = ctx.ldexp(sre, -wp)
sim = ctx.ldexp(sim, -wp)
s = ctx.mpc(sre, sim)
if (nd&1):
return (-1)**(nd//2) * s
else:
return (-1)**(1 + nd//2) * s
@defun
def _jacobi_theta2a(ctx, z, q):
"""
case ctx._im(z) != 0
theta(2, z, q) =
q**1/4 * Sum(q**(n*n + n) * exp(j*(2*n + 1)*z), n=-inf, inf)
max term for minimum (2*n+1)*log(q).real - 2* ctx._im(z)
n0 = int(ctx._im(z)/log(q).real - 1/2)
theta(2, z, q) =
q**1/4 * Sum(q**(n*n + n) * exp(j*(2*n + 1)*z), n=n0, inf) +
q**1/4 * Sum(q**(n*n + n) * exp(j*(2*n + 1)*z), n, n0-1, -inf)
"""
n = n0 = int(ctx._im(z)/ctx._re(ctx.log(q)) - 1/2)
e2 = ctx.expj(2*z)
e = e0 = ctx.expj((2*n+1)*z)
a = q**(n*n + n)
# leading term
term = a * e
s = term
eps1 = ctx.eps*abs(term)
while 1:
n += 1
e = e * e2
term = q**(n*n + n) * e
if abs(term) < eps1:
break
s += term
e = e0
e2 = ctx.expj(-2*z)
n = n0
while 1:
n -= 1
e = e * e2
term = q**(n*n + n) * e
if abs(term) < eps1:
break
s += term
s = s * ctx.nthroot(q, 4)
return s
@defun
def _jacobi_theta3a(ctx, z, q):
"""
case ctx._im(z) != 0
theta3(z, q) = Sum(q**(n*n) * exp(j*2*n*z), n, -inf, inf)
max term for n*abs(log(q).real) + ctx._im(z) ~= 0
n0 = int(- ctx._im(z)/abs(log(q).real))
"""
n = n0 = int(-ctx._im(z)/abs(ctx._re(ctx.log(q))))
e2 = ctx.expj(2*z)
e = e0 = ctx.expj(2*n*z)
s = term = q**(n*n) * e
eps1 = ctx.eps*abs(term)
while 1:
n += 1
e = e * e2
term = q**(n*n) * e
if abs(term) < eps1:
break
s += term
e = e0
e2 = ctx.expj(-2*z)
n = n0
while 1:
n -= 1
e = e * e2
term = q**(n*n) * e
if abs(term) < eps1:
break
s += term
return s
@defun
def _djacobi_theta2a(ctx, z, q, nd):
"""
case ctx._im(z) != 0
dtheta(2, z, q, nd) =
j* q**1/4 * Sum(q**(n*n + n) * (2*n+1)*exp(j*(2*n + 1)*z), n=-inf, inf)
max term for (2*n0+1)*log(q).real - 2* ctx._im(z) ~= 0
n0 = int(ctx._im(z)/log(q).real - 1/2)
"""
n = n0 = int(ctx._im(z)/ctx._re(ctx.log(q)) - 1/2)
e2 = ctx.expj(2*z)
e = e0 = ctx.expj((2*n + 1)*z)
a = q**(n*n + n)
# leading term
term = (2*n+1)**nd * a * e
s = term
eps1 = ctx.eps*abs(term)
while 1:
n += 1
e = e * e2
term = (2*n+1)**nd * q**(n*n + n) * e
if abs(term) < eps1:
break
s += term
e = e0
e2 = ctx.expj(-2*z)
n = n0
while 1:
n -= 1
e = e * e2
term = (2*n+1)**nd * q**(n*n + n) * e
if abs(term) < eps1:
break
s += term
return ctx.j**nd * s * ctx.nthroot(q, 4)
@defun
def _djacobi_theta3a(ctx, z, q, nd):
"""
case ctx._im(z) != 0
djtheta3(z, q, nd) = (2*j)**nd *
Sum(q**(n*n) * n**nd * exp(j*2*n*z), n, -inf, inf)
max term for minimum n*abs(log(q).real) + ctx._im(z)
"""
n = n0 = int(-ctx._im(z)/abs(ctx._re(ctx.log(q))))
e2 = ctx.expj(2*z)
e = e0 = ctx.expj(2*n*z)
a = q**(n*n) * e
s = term = n**nd * a
if n != 0:
eps1 = ctx.eps*abs(term)
else:
eps1 = ctx.eps*abs(a)
while 1:
n += 1
e = e * e2
a = q**(n*n) * e
term = n**nd * a
if n != 0:
aterm = abs(term)
else:
aterm = abs(a)
if aterm < eps1:
break
s += term
e = e0
e2 = ctx.expj(-2*z)
n = n0
while 1:
n -= 1
e = e * e2
a = q**(n*n) * e
term = n**nd * a
if n != 0:
aterm = abs(term)
else:
aterm = abs(a)
if aterm < eps1:
break
s += term
return (2*ctx.j)**nd * s
@defun
def jtheta(ctx, n, z, q, derivative=0):
if derivative:
return ctx._djtheta(n, z, q, derivative)
z = ctx.convert(z)
q = ctx.convert(q)
# Implementation note
# If ctx._im(z) is close to zero, _jacobi_theta2 and _jacobi_theta3
# are used,
# which compute the series starting from n=0 using fixed precision
# numbers;
# otherwise _jacobi_theta2a and _jacobi_theta3a are used, which compute
# the series starting from n=n0, which is the largest term.
# TODO: write _jacobi_theta2a and _jacobi_theta3a using fixed-point
if abs(q) > ctx.THETA_Q_LIM:
raise ValueError('abs(q) > THETA_Q_LIM = %f' % ctx.THETA_Q_LIM)
extra = 10
if z:
M = ctx.mag(z)
if M > 5 or (n == 1 and M < -5):
extra += 2*abs(M)
cz = 0.5
extra2 = 50
prec0 = ctx.prec
try:
ctx.prec += extra
if n == 1:
if ctx._im(z):
if abs(ctx._im(z)) < cz * abs(ctx._re(ctx.log(q))):
ctx.dps += extra2
res = ctx._jacobi_theta2(z - ctx.pi/2, q)
else:
ctx.dps += 10
res = ctx._jacobi_theta2a(z - ctx.pi/2, q)
else:
res = ctx._jacobi_theta2(z - ctx.pi/2, q)
elif n == 2:
if ctx._im(z):
if abs(ctx._im(z)) < cz * abs(ctx._re(ctx.log(q))):
ctx.dps += extra2
res = ctx._jacobi_theta2(z, q)
else:
ctx.dps += 10
res = ctx._jacobi_theta2a(z, q)
else:
res = ctx._jacobi_theta2(z, q)
elif n == 3:
if ctx._im(z):
if abs(ctx._im(z)) < cz * abs(ctx._re(ctx.log(q))):
ctx.dps += extra2
res = ctx._jacobi_theta3(z, q)
else:
ctx.dps += 10
res = ctx._jacobi_theta3a(z, q)
else:
res = ctx._jacobi_theta3(z, q)
elif n == 4:
if ctx._im(z):
if abs(ctx._im(z)) < cz * abs(ctx._re(ctx.log(q))):
ctx.dps += extra2
res = ctx._jacobi_theta3(z, -q)
else:
ctx.dps += 10
res = ctx._jacobi_theta3a(z, -q)
else:
res = ctx._jacobi_theta3(z, -q)
else:
raise ValueError
finally:
ctx.prec = prec0
return res
@defun
def _djtheta(ctx, n, z, q, derivative=1):
z = ctx.convert(z)
q = ctx.convert(q)
nd = int(derivative)
if abs(q) > ctx.THETA_Q_LIM:
raise ValueError('abs(q) > THETA_Q_LIM = %f' % ctx.THETA_Q_LIM)
extra = 10 + ctx.prec * nd // 10
if z:
M = ctx.mag(z)
if M > 5 or (n != 1 and M < -5):
extra += 2*abs(M)
cz = 0.5
extra2 = 50
prec0 = ctx.prec
try:
ctx.prec += extra
if n == 1:
if ctx._im(z):
if abs(ctx._im(z)) < cz * abs(ctx._re(ctx.log(q))):
ctx.dps += extra2
res = ctx._djacobi_theta2(z - ctx.pi/2, q, nd)
else:
ctx.dps += 10
res = ctx._djacobi_theta2a(z - ctx.pi/2, q, nd)
else:
res = ctx._djacobi_theta2(z - ctx.pi/2, q, nd)
elif n == 2:
if ctx._im(z):
if abs(ctx._im(z)) < cz * abs(ctx._re(ctx.log(q))):
ctx.dps += extra2
res = ctx._djacobi_theta2(z, q, nd)
else:
ctx.dps += 10
res = ctx._djacobi_theta2a(z, q, nd)
else:
res = ctx._djacobi_theta2(z, q, nd)
elif n == 3:
if ctx._im(z):
if abs(ctx._im(z)) < cz * abs(ctx._re(ctx.log(q))):
ctx.dps += extra2
res = ctx._djacobi_theta3(z, q, nd)
else:
ctx.dps += 10
res = ctx._djacobi_theta3a(z, q, nd)
else:
res = ctx._djacobi_theta3(z, q, nd)
elif n == 4:
if ctx._im(z):
if abs(ctx._im(z)) < cz * abs(ctx._re(ctx.log(q))):
ctx.dps += extra2
res = ctx._djacobi_theta3(z, -q, nd)
else:
ctx.dps += 10
res = ctx._djacobi_theta3a(z, -q, nd)
else:
res = ctx._djacobi_theta3(z, -q, nd)
else:
raise ValueError
finally:
ctx.prec = prec0
return +res
| 37,320 | 34.54381 | 76 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/matrices/calculus.py
|
from ..libmp.backend import xrange
# TODO: should use diagonalization-based algorithms
class MatrixCalculusMethods(object):
def _exp_pade(ctx, a):
"""
Exponential of a matrix using Pade approximants.
See G. H. Golub, C. F. van Loan 'Matrix Computations',
third Ed., page 572
TODO:
- find a good estimate for q
- reduce the number of matrix multiplications to improve
performance
"""
def eps_pade(p):
return ctx.mpf(2)**(3-2*p) * \
ctx.factorial(p)**2/(ctx.factorial(2*p)**2 * (2*p + 1))
q = 4
extraq = 8
while 1:
if eps_pade(q) < ctx.eps:
break
q += 1
q += extraq
j = int(max(1, ctx.mag(ctx.mnorm(a,'inf'))))
extra = q
prec = ctx.prec
ctx.dps += extra + 3
try:
a = a/2**j
na = a.rows
den = ctx.eye(na)
num = ctx.eye(na)
x = ctx.eye(na)
c = ctx.mpf(1)
for k in range(1, q+1):
c *= ctx.mpf(q - k + 1)/((2*q - k + 1) * k)
x = a*x
cx = c*x
num += cx
den += (-1)**k * cx
f = ctx.lu_solve_mat(den, num)
for k in range(j):
f = f*f
finally:
ctx.prec = prec
return f*1
def expm(ctx, A, method='taylor'):
r"""
Computes the matrix exponential of a square matrix `A`, which is defined
by the power series
.. math ::
\exp(A) = I + A + \frac{A^2}{2!} + \frac{A^3}{3!} + \ldots
With method='taylor', the matrix exponential is computed
using the Taylor series. With method='pade', Pade approximants
are used instead.
**Examples**
Basic examples::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> expm(zeros(3))
[1.0 0.0 0.0]
[0.0 1.0 0.0]
[0.0 0.0 1.0]
>>> expm(eye(3))
[2.71828182845905 0.0 0.0]
[ 0.0 2.71828182845905 0.0]
[ 0.0 0.0 2.71828182845905]
>>> expm([[1,1,0],[1,0,1],[0,1,0]])
[ 3.86814500615414 2.26812870852145 0.841130841230196]
[ 2.26812870852145 2.44114713886289 1.42699786729125]
[0.841130841230196 1.42699786729125 1.6000162976327]
>>> expm([[1,1,0],[1,0,1],[0,1,0]], method='pade')
[ 3.86814500615414 2.26812870852145 0.841130841230196]
[ 2.26812870852145 2.44114713886289 1.42699786729125]
[0.841130841230196 1.42699786729125 1.6000162976327]
>>> expm([[1+j, 0], [1+j,1]])
[(1.46869393991589 + 2.28735528717884j) 0.0]
[ (1.03776739863568 + 3.536943175722j) (2.71828182845905 + 0.0j)]
Matrices with large entries are allowed::
>>> expm(matrix([[1,2],[2,3]])**25)
[5.65024064048415e+2050488462815550 9.14228140091932e+2050488462815550]
[9.14228140091932e+2050488462815550 1.47925220414035e+2050488462815551]
The identity `\exp(A+B) = \exp(A) \exp(B)` does not hold for
noncommuting matrices::
>>> A = hilbert(3)
>>> B = A + eye(3)
>>> chop(mnorm(A*B - B*A))
0.0
>>> chop(mnorm(expm(A+B) - expm(A)*expm(B)))
0.0
>>> B = A + ones(3)
>>> mnorm(A*B - B*A)
1.8
>>> mnorm(expm(A+B) - expm(A)*expm(B))
42.0927851137247
"""
if method == 'pade':
prec = ctx.prec
try:
A = ctx.matrix(A)
ctx.prec += 2*A.rows
res = ctx._exp_pade(A)
finally:
ctx.prec = prec
return res
A = ctx.matrix(A)
prec = ctx.prec
j = int(max(1, ctx.mag(ctx.mnorm(A,'inf'))))
j += int(0.5*prec**0.5)
try:
ctx.prec += 10 + 2*j
tol = +ctx.eps
A = A/2**j
T = A
Y = A**0 + A
k = 2
while 1:
T *= A * (1/ctx.mpf(k))
if ctx.mnorm(T, 'inf') < tol:
break
Y += T
k += 1
for k in xrange(j):
Y = Y*Y
finally:
ctx.prec = prec
Y *= 1
return Y
def cosm(ctx, A):
r"""
Gives the cosine of a square matrix `A`, defined in analogy
with the matrix exponential.
Examples::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> X = eye(3)
>>> cosm(X)
[0.54030230586814 0.0 0.0]
[ 0.0 0.54030230586814 0.0]
[ 0.0 0.0 0.54030230586814]
>>> X = hilbert(3)
>>> cosm(X)
[ 0.424403834569555 -0.316643413047167 -0.221474945949293]
[-0.316643413047167 0.820646708837824 -0.127183694770039]
[-0.221474945949293 -0.127183694770039 0.909236687217541]
>>> X = matrix([[1+j,-2],[0,-j]])
>>> cosm(X)
[(0.833730025131149 - 0.988897705762865j) (1.07485840848393 - 0.17192140544213j)]
[ 0.0 (1.54308063481524 + 0.0j)]
"""
B = 0.5 * (ctx.expm(A*ctx.j) + ctx.expm(A*(-ctx.j)))
if not sum(A.apply(ctx.im).apply(abs)):
B = B.apply(ctx.re)
return B
def sinm(ctx, A):
r"""
Gives the sine of a square matrix `A`, defined in analogy
with the matrix exponential.
Examples::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> X = eye(3)
>>> sinm(X)
[0.841470984807897 0.0 0.0]
[ 0.0 0.841470984807897 0.0]
[ 0.0 0.0 0.841470984807897]
>>> X = hilbert(3)
>>> sinm(X)
[0.711608512150994 0.339783913247439 0.220742837314741]
[0.339783913247439 0.244113865695532 0.187231271174372]
[0.220742837314741 0.187231271174372 0.155816730769635]
>>> X = matrix([[1+j,-2],[0,-j]])
>>> sinm(X)
[(1.29845758141598 + 0.634963914784736j) (-1.96751511930922 + 0.314700021761367j)]
[ 0.0 (0.0 - 1.1752011936438j)]
"""
B = (-0.5j) * (ctx.expm(A*ctx.j) - ctx.expm(A*(-ctx.j)))
if not sum(A.apply(ctx.im).apply(abs)):
B = B.apply(ctx.re)
return B
def _sqrtm_rot(ctx, A, _may_rotate):
# If the iteration fails to converge, cheat by performing
# a rotation by a complex number
u = ctx.j**0.3
return ctx.sqrtm(u*A, _may_rotate) / ctx.sqrt(u)
def sqrtm(ctx, A, _may_rotate=2):
r"""
Computes a square root of the square matrix `A`, i.e. returns
a matrix `B = A^{1/2}` such that `B^2 = A`. The square root
of a matrix, if it exists, is not unique.
**Examples**
Square roots of some simple matrices::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> sqrtm([[1,0], [0,1]])
[1.0 0.0]
[0.0 1.0]
>>> sqrtm([[0,0], [0,0]])
[0.0 0.0]
[0.0 0.0]
>>> sqrtm([[2,0],[0,1]])
[1.4142135623731 0.0]
[ 0.0 1.0]
>>> sqrtm([[1,1],[1,0]])
[ (0.920442065259926 - 0.21728689675164j) (0.568864481005783 + 0.351577584254143j)]
[(0.568864481005783 + 0.351577584254143j) (0.351577584254143 - 0.568864481005783j)]
>>> sqrtm([[1,0],[0,1]])
[1.0 0.0]
[0.0 1.0]
>>> sqrtm([[-1,0],[0,1]])
[(0.0 - 1.0j) 0.0]
[ 0.0 (1.0 + 0.0j)]
>>> sqrtm([[j,0],[0,j]])
[(0.707106781186547 + 0.707106781186547j) 0.0]
[ 0.0 (0.707106781186547 + 0.707106781186547j)]
A square root of a rotation matrix, giving the corresponding
half-angle rotation matrix::
>>> t1 = 0.75
>>> t2 = t1 * 0.5
>>> A1 = matrix([[cos(t1), -sin(t1)], [sin(t1), cos(t1)]])
>>> A2 = matrix([[cos(t2), -sin(t2)], [sin(t2), cos(t2)]])
>>> sqrtm(A1)
[0.930507621912314 -0.366272529086048]
[0.366272529086048 0.930507621912314]
>>> A2
[0.930507621912314 -0.366272529086048]
[0.366272529086048 0.930507621912314]
The identity `(A^2)^{1/2} = A` does not necessarily hold::
>>> A = matrix([[4,1,4],[7,8,9],[10,2,11]])
>>> sqrtm(A**2)
[ 4.0 1.0 4.0]
[ 7.0 8.0 9.0]
[10.0 2.0 11.0]
>>> sqrtm(A)**2
[ 4.0 1.0 4.0]
[ 7.0 8.0 9.0]
[10.0 2.0 11.0]
>>> A = matrix([[-4,1,4],[7,-8,9],[10,2,11]])
>>> sqrtm(A**2)
[ 7.43715112194995 -0.324127569985474 1.8481718827526]
[-0.251549715716942 9.32699765900402 2.48221180985147]
[ 4.11609388833616 0.775751877098258 13.017955697342]
>>> chop(sqrtm(A)**2)
[-4.0 1.0 4.0]
[ 7.0 -8.0 9.0]
[10.0 2.0 11.0]
For some matrices, a square root does not exist::
>>> sqrtm([[0,1], [0,0]])
Traceback (most recent call last):
...
ZeroDivisionError: matrix is numerically singular
Two examples from the documentation for Matlab's ``sqrtm``::
>>> mp.dps = 15; mp.pretty = True
>>> sqrtm([[7,10],[15,22]])
[1.56669890360128 1.74077655955698]
[2.61116483933547 4.17786374293675]
>>>
>>> X = matrix(\
... [[5,-4,1,0,0],
... [-4,6,-4,1,0],
... [1,-4,6,-4,1],
... [0,1,-4,6,-4],
... [0,0,1,-4,5]])
>>> Y = matrix(\
... [[2,-1,-0,-0,-0],
... [-1,2,-1,0,-0],
... [0,-1,2,-1,0],
... [-0,0,-1,2,-1],
... [-0,-0,-0,-1,2]])
>>> mnorm(sqrtm(X) - Y)
4.53155328326114e-19
"""
A = ctx.matrix(A)
# Trivial
if A*0 == A:
return A
prec = ctx.prec
if _may_rotate:
d = ctx.det(A)
if abs(ctx.im(d)) < 16*ctx.eps and ctx.re(d) < 0:
return ctx._sqrtm_rot(A, _may_rotate-1)
try:
ctx.prec += 10
tol = ctx.eps * 128
Y = A
Z = I = A**0
k = 0
# Denman-Beavers iteration
while 1:
Yprev = Y
try:
Y, Z = 0.5*(Y+ctx.inverse(Z)), 0.5*(Z+ctx.inverse(Y))
except ZeroDivisionError:
if _may_rotate:
Y = ctx._sqrtm_rot(A, _may_rotate-1)
break
else:
raise
mag1 = ctx.mnorm(Y-Yprev, 'inf')
mag2 = ctx.mnorm(Y, 'inf')
if mag1 <= mag2*tol:
break
if _may_rotate and k > 6 and not mag1 < mag2 * 0.001:
return ctx._sqrtm_rot(A, _may_rotate-1)
k += 1
if k > ctx.prec:
raise ctx.NoConvergence
finally:
ctx.prec = prec
Y *= 1
return Y
def logm(ctx, A):
r"""
Computes a logarithm of the square matrix `A`, i.e. returns
a matrix `B = \log(A)` such that `\exp(B) = A`. The logarithm
of a matrix, if it exists, is not unique.
**Examples**
Logarithms of some simple matrices::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> X = eye(3)
>>> logm(X)
[0.0 0.0 0.0]
[0.0 0.0 0.0]
[0.0 0.0 0.0]
>>> logm(2*X)
[0.693147180559945 0.0 0.0]
[ 0.0 0.693147180559945 0.0]
[ 0.0 0.0 0.693147180559945]
>>> logm(expm(X))
[1.0 0.0 0.0]
[0.0 1.0 0.0]
[0.0 0.0 1.0]
A logarithm of a complex matrix::
>>> X = matrix([[2+j, 1, 3], [1-j, 1-2*j, 1], [-4, -5, j]])
>>> B = logm(X)
>>> nprint(B)
[ (0.808757 + 0.107759j) (2.20752 + 0.202762j) (1.07376 - 0.773874j)]
[ (0.905709 - 0.107795j) (0.0287395 - 0.824993j) (0.111619 + 0.514272j)]
[(-0.930151 + 0.399512j) (-2.06266 - 0.674397j) (0.791552 + 0.519839j)]
>>> chop(expm(B))
[(2.0 + 1.0j) 1.0 3.0]
[(1.0 - 1.0j) (1.0 - 2.0j) 1.0]
[ -4.0 -5.0 (0.0 + 1.0j)]
A matrix `X` close to the identity matrix, for which
`\log(\exp(X)) = \exp(\log(X)) = X` holds::
>>> X = eye(3) + hilbert(3)/4
>>> X
[ 1.25 0.125 0.0833333333333333]
[ 0.125 1.08333333333333 0.0625]
[0.0833333333333333 0.0625 1.05]
>>> logm(expm(X))
[ 1.25 0.125 0.0833333333333333]
[ 0.125 1.08333333333333 0.0625]
[0.0833333333333333 0.0625 1.05]
>>> expm(logm(X))
[ 1.25 0.125 0.0833333333333333]
[ 0.125 1.08333333333333 0.0625]
[0.0833333333333333 0.0625 1.05]
A logarithm of a rotation matrix, giving back the angle of
the rotation::
>>> t = 3.7
>>> A = matrix([[cos(t),sin(t)],[-sin(t),cos(t)]])
>>> chop(logm(A))
[ 0.0 -2.58318530717959]
[2.58318530717959 0.0]
>>> (2*pi-t)
2.58318530717959
For some matrices, a logarithm does not exist::
>>> logm([[1,0], [0,0]])
Traceback (most recent call last):
...
ZeroDivisionError: matrix is numerically singular
Logarithm of a matrix with large entries::
>>> logm(hilbert(3) * 10**20).apply(re)
[ 45.5597513593433 1.27721006042799 0.317662687717978]
[ 1.27721006042799 42.5222778973542 2.24003708791604]
[0.317662687717978 2.24003708791604 42.395212822267]
"""
A = ctx.matrix(A)
prec = ctx.prec
try:
ctx.prec += 10
tol = ctx.eps * 128
I = A**0
B = A
n = 0
while 1:
B = ctx.sqrtm(B)
n += 1
if ctx.mnorm(B-I, 'inf') < 0.125:
break
T = X = B-I
L = X*0
k = 1
while 1:
if k & 1:
L += T / k
else:
L -= T / k
T *= X
if ctx.mnorm(T, 'inf') < tol:
break
k += 1
if k > ctx.prec:
raise ctx.NoConvergence
finally:
ctx.prec = prec
L *= 2**n
return L
def powm(ctx, A, r):
r"""
Computes `A^r = \exp(A \log r)` for a matrix `A` and complex
number `r`.
**Examples**
Powers and inverse powers of a matrix::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> A = matrix([[4,1,4],[7,8,9],[10,2,11]])
>>> powm(A, 2)
[ 63.0 20.0 69.0]
[174.0 89.0 199.0]
[164.0 48.0 179.0]
>>> chop(powm(powm(A, 4), 1/4.))
[ 4.0 1.0 4.0]
[ 7.0 8.0 9.0]
[10.0 2.0 11.0]
>>> powm(extraprec(20)(powm)(A, -4), -1/4.)
[ 4.0 1.0 4.0]
[ 7.0 8.0 9.0]
[10.0 2.0 11.0]
>>> chop(powm(powm(A, 1+0.5j), 1/(1+0.5j)))
[ 4.0 1.0 4.0]
[ 7.0 8.0 9.0]
[10.0 2.0 11.0]
>>> powm(extraprec(5)(powm)(A, -1.5), -1/(1.5))
[ 4.0 1.0 4.0]
[ 7.0 8.0 9.0]
[10.0 2.0 11.0]
A Fibonacci-generating matrix::
>>> powm([[1,1],[1,0]], 10)
[89.0 55.0]
[55.0 34.0]
>>> fib(10)
55.0
>>> powm([[1,1],[1,0]], 6.5)
[(16.5166626964253 - 0.0121089837381789j) (10.2078589271083 + 0.0195927472575932j)]
[(10.2078589271083 + 0.0195927472575932j) (6.30880376931698 - 0.0317017309957721j)]
>>> (phi**6.5 - (1-phi)**6.5)/sqrt(5)
(10.2078589271083 - 0.0195927472575932j)
>>> powm([[1,1],[1,0]], 6.2)
[ (14.3076953002666 - 0.008222855781077j) (8.81733464837593 + 0.0133048601383712j)]
[(8.81733464837593 + 0.0133048601383712j) (5.49036065189071 - 0.0215277159194482j)]
>>> (phi**6.2 - (1-phi)**6.2)/sqrt(5)
(8.81733464837593 - 0.0133048601383712j)
"""
A = ctx.matrix(A)
r = ctx.convert(r)
prec = ctx.prec
try:
ctx.prec += 10
if ctx.isint(r):
v = A ** int(r)
elif ctx.isint(r*2):
y = int(r*2)
v = ctx.sqrtm(A) ** y
else:
v = ctx.expm(r*ctx.logm(A))
finally:
ctx.prec = prec
v *= 1
return v
| 18,609 | 33.981203 | 96 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/matrices/linalg.py
|
"""
Linear algebra
--------------
Linear equations
................
Basic linear algebra is implemented; you can for example solve the linear
equation system::
x + 2*y = -10
3*x + 4*y = 10
using ``lu_solve``::
>>> from mpmath import *
>>> mp.pretty = False
>>> A = matrix([[1, 2], [3, 4]])
>>> b = matrix([-10, 10])
>>> x = lu_solve(A, b)
>>> x
matrix(
[['30.0'],
['-20.0']])
If you don't trust the result, use ``residual`` to calculate the residual ||A*x-b||::
>>> residual(A, x, b)
matrix(
[['3.46944695195361e-18'],
['3.46944695195361e-18']])
>>> str(eps)
'2.22044604925031e-16'
As you can see, the solution is quite accurate. The error is caused by the
inaccuracy of the internal floating point arithmetic. Though, it's even smaller
than the current machine epsilon, which basically means you can trust the
result.
If you need more speed, use NumPy. Or choose a faster data type using the
keyword ``force_type``::
>>> lu_solve(A, b, force_type=float)
matrix(
[['30.0'],
['-20.0']])
``lu_solve`` accepts overdetermined systems. It is usually not possible to solve
such systems, so the residual is minimized instead. Internally this is done
using Cholesky decomposition to compute a least squares approximation. This means
that that ``lu_solve`` will square the errors. If you can't afford this, use
``qr_solve`` instead. It is twice as slow but more accurate, and it calculates
the residual automatically.
Matrix factorization
....................
The function ``lu`` computes an explicit LU factorization of a matrix::
>>> P, L, U = lu(matrix([[0,2,3],[4,5,6],[7,8,9]]))
>>> print(P)
[0.0 0.0 1.0]
[1.0 0.0 0.0]
[0.0 1.0 0.0]
>>> print(L)
[ 1.0 0.0 0.0]
[ 0.0 1.0 0.0]
[0.571428571428571 0.214285714285714 1.0]
>>> print(U)
[7.0 8.0 9.0]
[0.0 2.0 3.0]
[0.0 0.0 0.214285714285714]
>>> print(P.T*L*U)
[0.0 2.0 3.0]
[4.0 5.0 6.0]
[7.0 8.0 9.0]
Interval matrices
-----------------
Matrices may contain interval elements. This allows one to perform
basic linear algebra operations such as matrix multiplication
and equation solving with rigorous error bounds::
>>> a = iv.matrix([['0.1','0.3','1.0'],
... ['7.1','5.5','4.8'],
... ['3.2','4.4','5.6']], force_type=mpi)
>>>
>>> b = iv.matrix(['4','0.6','0.5'], force_type=mpi)
>>> c = iv.lu_solve(a, b)
>>> print(c)
[ [5.2582327113062568605927528666, 5.25823271130625686059275702219]]
[[-13.1550493962678375411635581388, -13.1550493962678375411635540152]]
[ [7.42069154774972557628979076189, 7.42069154774972557628979190734]]
>>> print(a*c)
[ [3.99999999999999999999999844904, 4.00000000000000000000000155096]]
[[0.599999999999999999999968898009, 0.600000000000000000000031763736]]
[[0.499999999999999999999979320485, 0.500000000000000000000020679515]]
"""
# TODO:
# *implement high-level qr()
# *test unitvector
# *iterative solving
from copy import copy
from ..libmp.backend import xrange
class LinearAlgebraMethods(object):
def LU_decomp(ctx, A, overwrite=False, use_cache=True):
"""
LU-factorization of a n*n matrix using the Gauss algorithm.
Returns L and U in one matrix and the pivot indices.
Use overwrite to specify whether A will be overwritten with L and U.
"""
if not A.rows == A.cols:
raise ValueError('need n*n matrix')
# get from cache if possible
if use_cache and isinstance(A, ctx.matrix) and A._LU:
return A._LU
if not overwrite:
orig = A
A = A.copy()
tol = ctx.absmin(ctx.mnorm(A,1) * ctx.eps) # each pivot element has to be bigger
n = A.rows
p = [None]*(n - 1)
for j in xrange(n - 1):
# pivoting, choose max(abs(reciprocal row sum)*abs(pivot element))
biggest = 0
for k in xrange(j, n):
s = ctx.fsum([ctx.absmin(A[k,l]) for l in xrange(j, n)])
if ctx.absmin(s) <= tol:
raise ZeroDivisionError('matrix is numerically singular')
current = 1/s * ctx.absmin(A[k,j])
if current > biggest: # TODO: what if equal?
biggest = current
p[j] = k
# swap rows according to p
ctx.swap_row(A, j, p[j])
if ctx.absmin(A[j,j]) <= tol:
raise ZeroDivisionError('matrix is numerically singular')
# calculate elimination factors and add rows
for i in xrange(j + 1, n):
A[i,j] /= A[j,j]
for k in xrange(j + 1, n):
A[i,k] -= A[i,j]*A[j,k]
if ctx.absmin(A[n - 1,n - 1]) <= tol:
raise ZeroDivisionError('matrix is numerically singular')
# cache decomposition
if not overwrite and isinstance(orig, ctx.matrix):
orig._LU = (A, p)
return A, p
def L_solve(ctx, L, b, p=None):
"""
Solve the lower part of a LU factorized matrix for y.
"""
if L.rows != L.cols:
raise RuntimeError("need n*n matrix")
n = L.rows
if len(b) != n:
raise ValueError("Value should be equal to n")
b = copy(b)
if p: # swap b according to p
for k in xrange(0, len(p)):
ctx.swap_row(b, k, p[k])
# solve
for i in xrange(1, n):
for j in xrange(i):
b[i] -= L[i,j] * b[j]
return b
def U_solve(ctx, U, y):
"""
Solve the upper part of a LU factorized matrix for x.
"""
if U.rows != U.cols:
raise RuntimeError("need n*n matrix")
n = U.rows
if len(y) != n:
raise ValueError("Value should be equal to n")
x = copy(y)
for i in xrange(n - 1, -1, -1):
for j in xrange(i + 1, n):
x[i] -= U[i,j] * x[j]
x[i] /= U[i,i]
return x
def lu_solve(ctx, A, b, **kwargs):
"""
Ax = b => x
Solve a determined or overdetermined linear equations system.
Fast LU decomposition is used, which is less accurate than QR decomposition
(especially for overdetermined systems), but it's twice as efficient.
Use qr_solve if you want more precision or have to solve a very ill-
conditioned system.
If you specify real=True, it does not check for overdeterminded complex
systems.
"""
prec = ctx.prec
try:
ctx.prec += 10
# do not overwrite A nor b
A, b = ctx.matrix(A, **kwargs).copy(), ctx.matrix(b, **kwargs).copy()
if A.rows < A.cols:
raise ValueError('cannot solve underdetermined system')
if A.rows > A.cols:
# use least-squares method if overdetermined
# (this increases errors)
AH = A.H
A = AH * A
b = AH * b
if (kwargs.get('real', False) or
not sum(type(i) is ctx.mpc for i in A)):
# TODO: necessary to check also b?
x = ctx.cholesky_solve(A, b)
else:
x = ctx.lu_solve(A, b)
else:
# LU factorization
A, p = ctx.LU_decomp(A)
b = ctx.L_solve(A, b, p)
x = ctx.U_solve(A, b)
finally:
ctx.prec = prec
return x
def improve_solution(ctx, A, x, b, maxsteps=1):
"""
Improve a solution to a linear equation system iteratively.
This re-uses the LU decomposition and is thus cheap.
Usually 3 up to 4 iterations are giving the maximal improvement.
"""
if A.rows != A.cols:
raise RuntimeError("need n*n matrix") # TODO: really?
for _ in xrange(maxsteps):
r = ctx.residual(A, x, b)
if ctx.norm(r, 2) < 10*ctx.eps:
break
# this uses cached LU decomposition and is thus cheap
dx = ctx.lu_solve(A, -r)
x += dx
return x
def lu(ctx, A):
"""
A -> P, L, U
LU factorisation of a square matrix A. L is the lower, U the upper part.
P is the permutation matrix indicating the row swaps.
P*A = L*U
If you need efficiency, use the low-level method LU_decomp instead, it's
much more memory efficient.
"""
# get factorization
A, p = ctx.LU_decomp(A)
n = A.rows
L = ctx.matrix(n)
U = ctx.matrix(n)
for i in xrange(n):
for j in xrange(n):
if i > j:
L[i,j] = A[i,j]
elif i == j:
L[i,j] = 1
U[i,j] = A[i,j]
else:
U[i,j] = A[i,j]
# calculate permutation matrix
P = ctx.eye(n)
for k in xrange(len(p)):
ctx.swap_row(P, k, p[k])
return P, L, U
def unitvector(ctx, n, i):
"""
Return the i-th n-dimensional unit vector.
"""
assert 0 < i <= n, 'this unit vector does not exist'
return [ctx.zero]*(i-1) + [ctx.one] + [ctx.zero]*(n-i)
def inverse(ctx, A, **kwargs):
"""
Calculate the inverse of a matrix.
If you want to solve an equation system Ax = b, it's recommended to use
solve(A, b) instead, it's about 3 times more efficient.
"""
prec = ctx.prec
try:
ctx.prec += 10
# do not overwrite A
A = ctx.matrix(A, **kwargs).copy()
n = A.rows
# get LU factorisation
A, p = ctx.LU_decomp(A)
cols = []
# calculate unit vectors and solve corresponding system to get columns
for i in xrange(1, n + 1):
e = ctx.unitvector(n, i)
y = ctx.L_solve(A, e, p)
cols.append(ctx.U_solve(A, y))
# convert columns to matrix
inv = []
for i in xrange(n):
row = []
for j in xrange(n):
row.append(cols[j][i])
inv.append(row)
result = ctx.matrix(inv, **kwargs)
finally:
ctx.prec = prec
return result
def householder(ctx, A):
"""
(A|b) -> H, p, x, res
(A|b) is the coefficient matrix with left hand side of an optionally
overdetermined linear equation system.
H and p contain all information about the transformation matrices.
x is the solution, res the residual.
"""
if not isinstance(A, ctx.matrix):
raise TypeError("A should be a type of ctx.matrix")
m = A.rows
n = A.cols
if m < n - 1:
raise RuntimeError("Columns should not be less than rows")
# calculate Householder matrix
p = []
for j in xrange(0, n - 1):
s = ctx.fsum((A[i,j])**2 for i in xrange(j, m))
if not abs(s) > ctx.eps:
raise ValueError('matrix is numerically singular')
p.append(-ctx.sign(A[j,j]) * ctx.sqrt(s))
kappa = ctx.one / (s - p[j] * A[j,j])
A[j,j] -= p[j]
for k in xrange(j+1, n):
y = ctx.fsum(A[i,j] * A[i,k] for i in xrange(j, m)) * kappa
for i in xrange(j, m):
A[i,k] -= A[i,j] * y
# solve Rx = c1
x = [A[i,n - 1] for i in xrange(n - 1)]
for i in xrange(n - 2, -1, -1):
x[i] -= ctx.fsum(A[i,j] * x[j] for j in xrange(i + 1, n - 1))
x[i] /= p[i]
# calculate residual
if not m == n - 1:
r = [A[m-1-i, n-1] for i in xrange(m - n + 1)]
else:
# determined system, residual should be 0
r = [0]*m # maybe a bad idea, changing r[i] will change all elements
return A, p, x, r
#def qr(ctx, A):
# """
# A -> Q, R
#
# QR factorisation of a square matrix A using Householder decomposition.
# Q is orthogonal, this leads to very few numerical errors.
#
# A = Q*R
# """
# H, p, x, res = householder(A)
# TODO: implement this
def residual(ctx, A, x, b, **kwargs):
"""
Calculate the residual of a solution to a linear equation system.
r = A*x - b for A*x = b
"""
oldprec = ctx.prec
try:
ctx.prec *= 2
A, x, b = ctx.matrix(A, **kwargs), ctx.matrix(x, **kwargs), ctx.matrix(b, **kwargs)
return A*x - b
finally:
ctx.prec = oldprec
def qr_solve(ctx, A, b, norm=None, **kwargs):
"""
Ax = b => x, ||Ax - b||
Solve a determined or overdetermined linear equations system and
calculate the norm of the residual (error).
QR decomposition using Householder factorization is applied, which gives very
accurate results even for ill-conditioned matrices. qr_solve is twice as
efficient.
"""
if norm is None:
norm = ctx.norm
prec = ctx.prec
try:
ctx.prec += 10
# do not overwrite A nor b
A, b = ctx.matrix(A, **kwargs).copy(), ctx.matrix(b, **kwargs).copy()
if A.rows < A.cols:
raise ValueError('cannot solve underdetermined system')
H, p, x, r = ctx.householder(ctx.extend(A, b))
res = ctx.norm(r)
# calculate residual "manually" for determined systems
if res == 0:
res = ctx.norm(ctx.residual(A, x, b))
return ctx.matrix(x, **kwargs), res
finally:
ctx.prec = prec
def cholesky(ctx, A, tol=None):
r"""
Cholesky decomposition of a symmetric positive-definite matrix `A`.
Returns a lower triangular matrix `L` such that `A = L \times L^T`.
More generally, for a complex Hermitian positive-definite matrix,
a Cholesky decomposition satisfying `A = L \times L^H` is returned.
The Cholesky decomposition can be used to solve linear equation
systems twice as efficiently as LU decomposition, or to
test whether `A` is positive-definite.
The optional parameter ``tol`` determines the tolerance for
verifying positive-definiteness.
**Examples**
Cholesky decomposition of a positive-definite symmetric matrix::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> A = eye(3) + hilbert(3)
>>> nprint(A)
[ 2.0 0.5 0.333333]
[ 0.5 1.33333 0.25]
[0.333333 0.25 1.2]
>>> L = cholesky(A)
>>> nprint(L)
[ 1.41421 0.0 0.0]
[0.353553 1.09924 0.0]
[0.235702 0.15162 1.05899]
>>> chop(A - L*L.T)
[0.0 0.0 0.0]
[0.0 0.0 0.0]
[0.0 0.0 0.0]
Cholesky decomposition of a Hermitian matrix::
>>> A = eye(3) + matrix([[0,0.25j,-0.5j],[-0.25j,0,0],[0.5j,0,0]])
>>> L = cholesky(A)
>>> nprint(L)
[ 1.0 0.0 0.0]
[(0.0 - 0.25j) (0.968246 + 0.0j) 0.0]
[ (0.0 + 0.5j) (0.129099 + 0.0j) (0.856349 + 0.0j)]
>>> chop(A - L*L.H)
[0.0 0.0 0.0]
[0.0 0.0 0.0]
[0.0 0.0 0.0]
Attempted Cholesky decomposition of a matrix that is not positive
definite::
>>> A = -eye(3) + hilbert(3)
>>> L = cholesky(A)
Traceback (most recent call last):
...
ValueError: matrix is not positive-definite
**References**
1. [Wikipedia]_ http://en.wikipedia.org/wiki/Cholesky_decomposition
"""
if not isinstance(A, ctx.matrix):
raise RuntimeError("A should be a type of ctx.matrix")
if not A.rows == A.cols:
raise ValueError('need n*n matrix')
if tol is None:
tol = +ctx.eps
n = A.rows
L = ctx.matrix(n)
for j in xrange(n):
c = ctx.re(A[j,j])
if abs(c-A[j,j]) > tol:
raise ValueError('matrix is not Hermitian')
s = c - ctx.fsum((L[j,k] for k in xrange(j)),
absolute=True, squared=True)
if s < tol:
raise ValueError('matrix is not positive-definite')
L[j,j] = ctx.sqrt(s)
for i in xrange(j, n):
it1 = (L[i,k] for k in xrange(j))
it2 = (L[j,k] for k in xrange(j))
t = ctx.fdot(it1, it2, conjugate=True)
L[i,j] = (A[i,j] - t) / L[j,j]
return L
def cholesky_solve(ctx, A, b, **kwargs):
"""
Ax = b => x
Solve a symmetric positive-definite linear equation system.
This is twice as efficient as lu_solve.
Typical use cases:
* A.T*A
* Hessian matrix
* differential equations
"""
prec = ctx.prec
try:
ctx.prec += 10
# do not overwrite A nor b
A, b = ctx.matrix(A, **kwargs).copy(), ctx.matrix(b, **kwargs).copy()
if A.rows != A.cols:
raise ValueError('can only solve determined system')
# Cholesky factorization
L = ctx.cholesky(A)
# solve
n = L.rows
if len(b) != n:
raise ValueError("Value should be equal to n")
for i in xrange(n):
b[i] -= ctx.fsum(L[i,j] * b[j] for j in xrange(i))
b[i] /= L[i,i]
x = ctx.U_solve(L.T, b)
return x
finally:
ctx.prec = prec
def det(ctx, A):
"""
Calculate the determinant of a matrix.
"""
prec = ctx.prec
try:
# do not overwrite A
A = ctx.matrix(A).copy()
# use LU factorization to calculate determinant
try:
R, p = ctx.LU_decomp(A)
except ZeroDivisionError:
return 0
z = 1
for i, e in enumerate(p):
if i != e:
z *= -1
for i in xrange(A.rows):
z *= R[i,i]
return z
finally:
ctx.prec = prec
def cond(ctx, A, norm=None):
"""
Calculate the condition number of a matrix using a specified matrix norm.
The condition number estimates the sensitivity of a matrix to errors.
Example: small input errors for ill-conditioned coefficient matrices
alter the solution of the system dramatically.
For ill-conditioned matrices it's recommended to use qr_solve() instead
of lu_solve(). This does not help with input errors however, it just avoids
to add additional errors.
Definition: cond(A) = ||A|| * ||A**-1||
"""
if norm is None:
norm = lambda x: ctx.mnorm(x,1)
return norm(A) * norm(ctx.inverse(A))
def lu_solve_mat(ctx, a, b):
"""Solve a * x = b where a and b are matrices."""
r = ctx.matrix(a.rows, b.cols)
for i in range(b.cols):
c = ctx.lu_solve(a, b.column(i))
for j in range(len(c)):
r[j, i] = c[j]
return r
def qr(ctx, A, mode = 'full', edps = 10):
"""
Compute a QR factorization $A = QR$ where
A is an m x n matrix of real or complex numbers where m >= n
mode has following meanings:
(1) mode = 'raw' returns two matrixes (A, tau) in the
internal format used by LAPACK
(2) mode = 'skinny' returns the leading n columns of Q
and n rows of R
(3) Any other value returns the leading m columns of Q
and m rows of R
edps is the increase in mp precision used for calculations
**Examples**
>>> from mpmath import *
>>> mp.dps = 15
>>> mp.pretty = True
>>> A = matrix([[1, 2], [3, 4], [1, 1]])
>>> Q, R = qr(A)
>>> Q
[-0.301511344577764 0.861640436855329 0.408248290463863]
[-0.904534033733291 -0.123091490979333 -0.408248290463863]
[-0.301511344577764 -0.492365963917331 0.816496580927726]
>>> R
[-3.3166247903554 -4.52267016866645]
[ 0.0 0.738548945875996]
[ 0.0 0.0]
>>> Q * R
[1.0 2.0]
[3.0 4.0]
[1.0 1.0]
>>> chop(Q.T * Q)
[1.0 0.0 0.0]
[0.0 1.0 0.0]
[0.0 0.0 1.0]
>>> B = matrix([[1+0j, 2-3j], [3+j, 4+5j]])
>>> Q, R = qr(B)
>>> nprint(Q)
[ (-0.301511 + 0.0j) (0.0695795 - 0.95092j)]
[(-0.904534 - 0.301511j) (-0.115966 + 0.278318j)]
>>> nprint(R)
[(-3.31662 + 0.0j) (-5.72872 - 2.41209j)]
[ 0.0 (3.91965 + 0.0j)]
>>> Q * R
[(1.0 + 0.0j) (2.0 - 3.0j)]
[(3.0 + 1.0j) (4.0 + 5.0j)]
>>> chop(Q.T * Q.conjugate())
[1.0 0.0]
[0.0 1.0]
"""
# check values before continuing
assert isinstance(A, ctx.matrix)
m = A.rows
n = A.cols
assert n > 1
assert m >= n
assert edps >= 0
# check for complex data type
cmplx = any(type(x) is ctx.mpc for x in A)
# temporarily increase the precision and initialize
with ctx.extradps(edps):
tau = ctx.matrix(n,1)
A = A.copy()
# ---------------
# FACTOR MATRIX A
# ---------------
if cmplx:
one = ctx.mpc('1.0', '0.0')
zero = ctx.mpc('0.0', '0.0')
rzero = ctx.mpf('0.0')
# main loop to factor A (complex)
for j in xrange(0, n):
alpha = A[j,j]
alphr = ctx.re(alpha)
alphi = ctx.im(alpha)
if (m-j) >= 2:
xnorm = ctx.fsum( A[i,j]*ctx.conj(A[i,j]) for i in xrange(j+1, m) )
xnorm = ctx.re( ctx.sqrt(xnorm) )
else:
xnorm = rzero
if (xnorm == rzero) and (alphi == rzero):
tau[j] = zero
continue
if alphr < rzero:
beta = ctx.sqrt(alphr**2 + alphi**2 + xnorm**2)
else:
beta = -ctx.sqrt(alphr**2 + alphi**2 + xnorm**2)
tau[j] = ctx.mpc( (beta - alphr) / beta, -alphi / beta )
t = -ctx.conj(tau[j])
za = one / (alpha - beta)
for i in xrange(j+1, m):
A[i,j] *= za
A[j,j] = one
for k in xrange(j+1, n):
y = ctx.fsum(A[i,j] * ctx.conj(A[i,k]) for i in xrange(j, m))
temp = t * ctx.conj(y)
for i in xrange(j, m):
A[i,k] += A[i,j] * temp
A[j,j] = ctx.mpc(beta, '0.0')
else:
one = ctx.mpf('1.0')
zero = ctx.mpf('0.0')
# main loop to factor A (real)
for j in xrange(0, n):
alpha = A[j,j]
if (m-j) > 2:
xnorm = ctx.fsum( (A[i,j])**2 for i in xrange(j+1, m) )
xnorm = ctx.sqrt(xnorm)
elif (m-j) == 2:
xnorm = abs( A[m-1,j] )
else:
xnorm = zero
if xnorm == zero:
tau[j] = zero
continue
if alpha < zero:
beta = ctx.sqrt(alpha**2 + xnorm**2)
else:
beta = -ctx.sqrt(alpha**2 + xnorm**2)
tau[j] = (beta - alpha) / beta
t = -tau[j]
da = one / (alpha - beta)
for i in xrange(j+1, m):
A[i,j] *= da
A[j,j] = one
for k in xrange(j+1, n):
y = ctx.fsum( A[i,j] * A[i,k] for i in xrange(j, m) )
temp = t * y
for i in xrange(j,m):
A[i,k] += A[i,j] * temp
A[j,j] = beta
# return factorization in same internal format as LAPACK
if (mode == 'raw') or (mode == 'RAW'):
return A, tau
# ----------------------------------
# FORM Q USING BACKWARD ACCUMULATION
# ----------------------------------
# form R before the values are overwritten
R = A.copy()
for j in xrange(0, n):
for i in xrange(j+1, m):
R[i,j] = zero
# set the value of p (number of columns of Q to return)
p = m
if (mode == 'skinny') or (mode == 'SKINNY'):
p = n
# add columns to A if needed and initialize
A.cols += (p-n)
for j in xrange(0, p):
A[j,j] = one
for i in xrange(0, j):
A[i,j] = zero
# main loop to form Q
for j in xrange(n-1, -1, -1):
t = -tau[j]
A[j,j] += t
for k in xrange(j+1, p):
if cmplx:
y = ctx.fsum(A[i,j] * ctx.conj(A[i,k]) for i in xrange(j+1, m))
temp = t * ctx.conj(y)
else:
y = ctx.fsum(A[i,j] * A[i,k] for i in xrange(j+1, m))
temp = t * y
A[j,k] = temp
for i in xrange(j+1, m):
A[i,k] += A[i,j] * temp
for i in xrange(j+1, m):
A[i, j] *= t
return A, R[0:p,0:n]
# ------------------
# END OF FUNCTION QR
# ------------------
| 26,999 | 33.005038 | 95 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/matrices/eigen.py
|
#!/usr/bin/python
# -*- coding: utf-8 -*-
##################################################################################################
# module for the eigenvalue problem
# Copyright 2013 Timo Hartmann (thartmann15 at gmail.com)
#
# todo:
# - implement balancing
# - agressive early deflation
#
##################################################################################################
"""
The eigenvalue problem
----------------------
This file contains routines for the eigenvalue problem.
high level routines:
hessenberg : reduction of a real or complex square matrix to upper Hessenberg form
schur : reduction of a real or complex square matrix to upper Schur form
eig : eigenvalues and eigenvectors of a real or complex square matrix
low level routines:
hessenberg_reduce_0 : reduction of a real or complex square matrix to upper Hessenberg form
hessenberg_reduce_1 : auxiliary routine to hessenberg_reduce_0
qr_step : a single implicitly shifted QR step for an upper Hessenberg matrix
hessenberg_qr : Schur decomposition of an upper Hessenberg matrix
eig_tr_r : right eigenvectors of an upper triangular matrix
eig_tr_l : left eigenvectors of an upper triangular matrix
"""
from ..libmp.backend import xrange
class Eigen(object):
pass
def defun(f):
setattr(Eigen, f.__name__, f)
def hessenberg_reduce_0(ctx, A, T):
"""
This routine computes the (upper) Hessenberg decomposition of a square matrix A.
Given A, an unitary matrix Q is calculated such that
Q' A Q = H and Q' Q = Q Q' = 1
where H is an upper Hessenberg matrix, meaning that it only contains zeros
below the first subdiagonal. Here ' denotes the hermitian transpose (i.e.
transposition and conjugation).
parameters:
A (input/output) On input, A contains the square matrix A of
dimension (n,n). On output, A contains a compressed representation
of Q and H.
T (output) An array of length n containing the first elements of
the Householder reflectors.
"""
# internally we work with householder reflections from the right.
# let u be a row vector (i.e. u[i]=A[i,:i]). then
# Q is build up by reflectors of the type (1-v'v) where v is a suitable
# modification of u. these reflectors are applyed to A from the right.
# because we work with reflectors from the right we have to start with
# the bottom row of A and work then upwards (this corresponds to
# some kind of RQ decomposition).
# the first part of the vectors v (i.e. A[i,:(i-1)]) are stored as row vectors
# in the lower left part of A (excluding the diagonal and subdiagonal).
# the last entry of v is stored in T.
# the upper right part of A (including diagonal and subdiagonal) becomes H.
n = A.rows
if n <= 2: return
for i in xrange(n-1, 1, -1):
# scale the vector
scale = 0
for k in xrange(0, i):
scale += abs(ctx.re(A[i,k])) + abs(ctx.im(A[i,k]))
scale_inv = 0
if scale != 0:
scale_inv = 1 / scale
if scale == 0 or ctx.isinf(scale_inv):
# sadly there are floating point numbers not equal to zero whose reciprocal is infinity
T[i] = 0
A[i,i-1] = 0
continue
# calculate parameters for housholder transformation
H = 0
for k in xrange(0, i):
A[i,k] *= scale_inv
rr = ctx.re(A[i,k])
ii = ctx.im(A[i,k])
H += rr * rr + ii * ii
F = A[i,i-1]
f = abs(F)
G = ctx.sqrt(H)
A[i,i-1] = - G * scale
if f == 0:
T[i] = G
else:
ff = F / f
T[i] = F + G * ff
A[i,i-1] *= ff
H += G * f
H = 1 / ctx.sqrt(H)
T[i] *= H
for k in xrange(0, i - 1):
A[i,k] *= H
for j in xrange(0, i):
# apply housholder transformation (from right)
G = ctx.conj(T[i]) * A[j,i-1]
for k in xrange(0, i-1):
G += ctx.conj(A[i,k]) * A[j,k]
A[j,i-1] -= G * T[i]
for k in xrange(0, i-1):
A[j,k] -= G * A[i,k]
for j in xrange(0, n):
# apply housholder transformation (from left)
G = T[i] * A[i-1,j]
for k in xrange(0, i-1):
G += A[i,k] * A[k,j]
A[i-1,j] -= G * ctx.conj(T[i])
for k in xrange(0, i-1):
A[k,j] -= G * ctx.conj(A[i,k])
def hessenberg_reduce_1(ctx, A, T):
"""
This routine forms the unitary matrix Q described in hessenberg_reduce_0.
parameters:
A (input/output) On input, A is the same matrix as delivered by
hessenberg_reduce_0. On output, A is set to Q.
T (input) On input, T is the same array as delivered by hessenberg_reduce_0.
"""
n = A.rows
if n == 1:
A[0,0] = 1
return
A[0,0] = A[1,1] = 1
A[0,1] = A[1,0] = 0
for i in xrange(2, n):
if T[i] != 0:
for j in xrange(0, i):
G = T[i] * A[i-1,j]
for k in xrange(0, i-1):
G += A[i,k] * A[k,j]
A[i-1,j] -= G * ctx.conj(T[i])
for k in xrange(0, i-1):
A[k,j] -= G * ctx.conj(A[i,k])
A[i,i] = 1
for j in xrange(0, i):
A[j,i] = A[i,j] = 0
@defun
def hessenberg(ctx, A, overwrite_a = False):
"""
This routine computes the Hessenberg decomposition of a square matrix A.
Given A, an unitary matrix Q is determined such that
Q' A Q = H and Q' Q = Q Q' = 1
where H is an upper right Hessenberg matrix. Here ' denotes the hermitian
transpose (i.e. transposition and conjugation).
input:
A : a real or complex square matrix
overwrite_a : if true, allows modification of A which may improve
performance. if false, A is not modified.
output:
Q : an unitary matrix
H : an upper right Hessenberg matrix
example:
>>> from mpmath import mp
>>> A = mp.matrix([[3, -1, 2], [2, 5, -5], [-2, -3, 7]])
>>> Q, H = mp.hessenberg(A)
>>> mp.nprint(H, 3) # doctest:+SKIP
[ 3.15 2.23 4.44]
[-0.769 4.85 3.05]
[ 0.0 3.61 7.0]
>>> print(mp.chop(A - Q * H * Q.transpose_conj()))
[0.0 0.0 0.0]
[0.0 0.0 0.0]
[0.0 0.0 0.0]
return value: (Q, H)
"""
n = A.rows
if n == 1:
return (ctx.matrix([[1]]), A)
if not overwrite_a:
A = A.copy()
T = ctx.matrix(n, 1)
hessenberg_reduce_0(ctx, A, T)
Q = A.copy()
hessenberg_reduce_1(ctx, Q, T)
for x in xrange(n):
for y in xrange(x+2, n):
A[y,x] = 0
return Q, A
###########################################################################
def qr_step(ctx, n0, n1, A, Q, shift):
"""
This subroutine executes a single implicitly shifted QR step applied to an
upper Hessenberg matrix A. Given A and shift as input, first an QR
decomposition is calculated:
Q R = A - shift * 1 .
The output is then following matrix:
R Q + shift * 1
parameters:
n0, n1 (input) Two integers which specify the submatrix A[n0:n1,n0:n1]
on which this subroutine operators. The subdiagonal elements
to the left and below this submatrix must be deflated (i.e. zero).
following restriction is imposed: n1>=n0+2
A (input/output) On input, A is an upper Hessenberg matrix.
On output, A is replaced by "R Q + shift * 1"
Q (input/output) The parameter Q is multiplied by the unitary matrix
Q arising from the QR decomposition. Q can also be false, in which
case the unitary matrix Q is not computated.
shift (input) a complex number specifying the shift. idealy close to an
eigenvalue of the bottemmost part of the submatrix A[n0:n1,n0:n1].
references:
Stoer, Bulirsch - Introduction to Numerical Analysis.
Kresser : Numerical Methods for General and Structured Eigenvalue Problems
"""
# implicitly shifted and bulge chasing is explained at p.398/399 in "Stoer, Bulirsch - Introduction to Numerical Analysis"
# for bulge chasing see also "Watkins - The Matrix Eigenvalue Problem" sec.4.5,p.173
# the Givens rotation we used is determined as follows: let c,s be two complex
# numbers. then we have following relation:
#
# v = sqrt(|c|^2 + |s|^2)
#
# 1/v [ c~ s~] [c] = [v]
# [-s c ] [s] [0]
#
# the matrix on the left is our Givens rotation.
n = A.rows
# first step
# calculate givens rotation
c = A[n0 ,n0] - shift
s = A[n0+1,n0]
v = ctx.hypot(ctx.hypot(ctx.re(c), ctx.im(c)), ctx.hypot(ctx.re(s), ctx.im(s)))
if v == 0:
v = 1
c = 1
s = 0
else:
c /= v
s /= v
for k in xrange(n0, n):
# apply givens rotation from the left
x = A[n0 ,k]
y = A[n0+1,k]
A[n0 ,k] = ctx.conj(c) * x + ctx.conj(s) * y
A[n0+1,k] = -s * x + c * y
for k in xrange(min(n1, n0+3)):
# apply givens rotation from the right
x = A[k,n0 ]
y = A[k,n0+1]
A[k,n0 ] = c * x + s * y
A[k,n0+1] = -ctx.conj(s) * x + ctx.conj(c) * y
if not isinstance(Q, bool):
for k in xrange(n):
# eigenvectors
x = Q[k,n0 ]
y = Q[k,n0+1]
Q[k,n0 ] = c * x + s * y
Q[k,n0+1] = -ctx.conj(s) * x + ctx.conj(c) * y
# chase the bulge
for j in xrange(n0, n1 - 2):
# calculate givens rotation
c = A[j+1,j]
s = A[j+2,j]
v = ctx.hypot(ctx.hypot(ctx.re(c), ctx.im(c)), ctx.hypot(ctx.re(s), ctx.im(s)))
if v == 0:
A[j+1,j] = 0
v = 1
c = 1
s = 0
else:
A[j+1,j] = v
c /= v
s /= v
A[j+2,j] = 0
for k in xrange(j+1, n):
# apply givens rotation from the left
x = A[j+1,k]
y = A[j+2,k]
A[j+1,k] = ctx.conj(c) * x + ctx.conj(s) * y
A[j+2,k] = -s * x + c * y
for k in xrange(0, min(n1, j+4)):
# apply givens rotation from the right
x = A[k,j+1]
y = A[k,j+2]
A[k,j+1] = c * x + s * y
A[k,j+2] = -ctx.conj(s) * x + ctx.conj(c) * y
if not isinstance(Q, bool):
for k in xrange(0, n):
# eigenvectors
x = Q[k,j+1]
y = Q[k,j+2]
Q[k,j+1] = c * x + s * y
Q[k,j+2] = -ctx.conj(s) * x + ctx.conj(c) * y
def hessenberg_qr(ctx, A, Q):
"""
This routine computes the Schur decomposition of an upper Hessenberg matrix A.
Given A, an unitary matrix Q is determined such that
Q' A Q = R and Q' Q = Q Q' = 1
where R is an upper right triangular matrix. Here ' denotes the hermitian
transpose (i.e. transposition and conjugation).
parameters:
A (input/output) On input, A contains an upper Hessenberg matrix.
On output, A is replace by the upper right triangluar matrix R.
Q (input/output) The parameter Q is multiplied by the unitary
matrix Q arising from the Schur decomposition. Q can also be
false, in which case the unitary matrix Q is not computated.
"""
n = A.rows
norm = 0
for x in xrange(n):
for y in xrange(min(x+2, n)):
norm += ctx.re(A[y,x]) ** 2 + ctx.im(A[y,x]) ** 2
norm = ctx.sqrt(norm) / n
if norm == 0:
return
n0 = 0
n1 = n
eps = ctx.eps / (100 * n)
maxits = ctx.dps * 4
its = totalits = 0
while 1:
# kressner p.32 algo 3
# the active submatrix is A[n0:n1,n0:n1]
k = n0
while k + 1 < n1:
s = abs(ctx.re(A[k,k])) + abs(ctx.im(A[k,k])) + abs(ctx.re(A[k+1,k+1])) + abs(ctx.im(A[k+1,k+1]))
if s < eps * norm:
s = norm
if abs(A[k+1,k]) < eps * s:
break
k += 1
if k + 1 < n1:
# deflation found at position (k+1, k)
A[k+1,k] = 0
n0 = k + 1
its = 0
if n0 + 1 >= n1:
# block of size at most two has converged
n0 = 0
n1 = k + 1
if n1 < 2:
# QR algorithm has converged
return
else:
if (its % 30) == 10:
# exceptional shift
shift = A[n1-1,n1-2]
elif (its % 30) == 20:
# exceptional shift
shift = abs(A[n1-1,n1-2])
elif (its % 30) == 29:
# exceptional shift
shift = norm
else:
# A = [ a b ] det(x-A)=x*x-x*tr(A)+det(A)
# [ c d ]
#
# eigenvalues bad: (tr(A)+sqrt((tr(A))**2-4*det(A)))/2
# bad because of cancellation if |c| is small and |a-d| is small, too.
#
# eigenvalues good: (a+d+sqrt((a-d)**2+4*b*c))/2
t = A[n1-2,n1-2] + A[n1-1,n1-1]
s = (A[n1-1,n1-1] - A[n1-2,n1-2]) ** 2 + 4 * A[n1-1,n1-2] * A[n1-2,n1-1]
if ctx.re(s) > 0:
s = ctx.sqrt(s)
else:
s = ctx.sqrt(-s) * 1j
a = (t + s) / 2
b = (t - s) / 2
if abs(A[n1-1,n1-1] - a) > abs(A[n1-1,n1-1] - b):
shift = b
else:
shift = a
its += 1
totalits += 1
qr_step(ctx, n0, n1, A, Q, shift)
if its > maxits:
raise RuntimeError("qr: failed to converge after %d steps" % its)
@defun
def schur(ctx, A, overwrite_a = False):
"""
This routine computes the Schur decomposition of a square matrix A.
Given A, an unitary matrix Q is determined such that
Q' A Q = R and Q' Q = Q Q' = 1
where R is an upper right triangular matrix. Here ' denotes the
hermitian transpose (i.e. transposition and conjugation).
input:
A : a real or complex square matrix
overwrite_a : if true, allows modification of A which may improve
performance. if false, A is not modified.
output:
Q : an unitary matrix
R : an upper right triangular matrix
return value: (Q, R)
example:
>>> from mpmath import mp
>>> A = mp.matrix([[3, -1, 2], [2, 5, -5], [-2, -3, 7]])
>>> Q, R = mp.schur(A)
>>> mp.nprint(R, 3) # doctest:+SKIP
[2.0 0.417 -2.53]
[0.0 4.0 -4.74]
[0.0 0.0 9.0]
>>> print(mp.chop(A - Q * R * Q.transpose_conj()))
[0.0 0.0 0.0]
[0.0 0.0 0.0]
[0.0 0.0 0.0]
warning: The Schur decomposition is not unique.
"""
n = A.rows
if n == 1:
return (ctx.matrix([[1]]), A)
if not overwrite_a:
A = A.copy()
T = ctx.matrix(n, 1)
hessenberg_reduce_0(ctx, A, T)
Q = A.copy()
hessenberg_reduce_1(ctx, Q, T)
for x in xrange(n):
for y in xrange(x + 2, n):
A[y,x] = 0
hessenberg_qr(ctx, A, Q)
return Q, A
def eig_tr_r(ctx, A):
"""
This routine calculates the right eigenvectors of an upper right triangular matrix.
input:
A an upper right triangular matrix
output:
ER a matrix whose columns form the right eigenvectors of A
return value: ER
"""
# this subroutine is inspired by the lapack routines ctrevc.f,clatrs.f
n = A.rows
ER = ctx.eye(n)
eps = ctx.eps
unfl = ctx.ldexp(ctx.one, -ctx.prec * 30)
# since mpmath effectively has no limits on the exponent, we simply scale doubles up
# original double has prec*20
smlnum = unfl * (n / eps)
simin = 1 / ctx.sqrt(eps)
rmax = 1
for i in xrange(1, n):
s = A[i,i]
smin = max(eps * abs(s), smlnum)
for j in xrange(i - 1, -1, -1):
r = 0
for k in xrange(j + 1, i + 1):
r += A[j,k] * ER[k,i]
t = A[j,j] - s
if abs(t) < smin:
t = smin
r = -r / t
ER[j,i] = r
rmax = max(rmax, abs(r))
if rmax > simin:
for k in xrange(j, i+1):
ER[k,i] /= rmax
rmax = 1
if rmax != 1:
for k in xrange(0, i + 1):
ER[k,i] /= rmax
return ER
def eig_tr_l(ctx, A):
"""
This routine calculates the left eigenvectors of an upper right triangular matrix.
input:
A an upper right triangular matrix
output:
EL a matrix whose rows form the left eigenvectors of A
return value: EL
"""
n = A.rows
EL = ctx.eye(n)
eps = ctx.eps
unfl = ctx.ldexp(ctx.one, -ctx.prec * 30)
# since mpmath effectively has no limits on the exponent, we simply scale doubles up
# original double has prec*20
smlnum = unfl * (n / eps)
simin = 1 / ctx.sqrt(eps)
rmax = 1
for i in xrange(0, n - 1):
s = A[i,i]
smin = max(eps * abs(s), smlnum)
for j in xrange(i + 1, n):
r = 0
for k in xrange(i, j):
r += EL[i,k] * A[k,j]
t = A[j,j] - s
if abs(t) < smin:
t = smin
r = -r / t
EL[i,j] = r
rmax = max(rmax, abs(r))
if rmax > simin:
for k in xrange(i, j + 1):
EL[i,k] /= rmax
rmax = 1
if rmax != 1:
for k in xrange(i, n):
EL[i,k] /= rmax
return EL
@defun
def eig(ctx, A, left = False, right = True, overwrite_a = False):
"""
This routine computes the eigenvalues and optionally the left and right
eigenvectors of a square matrix A. Given A, a vector E and matrices ER
and EL are calculated such that
A ER[:,i] = E[i] ER[:,i]
EL[i,:] A = EL[i,:] E[i]
E contains the eigenvalues of A. The columns of ER contain the right eigenvectors
of A whereas the rows of EL contain the left eigenvectors.
input:
A : a real or complex square matrix of shape (n, n)
left : if true, the left eigenvectors are calulated.
right : if true, the right eigenvectors are calculated.
overwrite_a : if true, allows modification of A which may improve
performance. if false, A is not modified.
output:
E : a list of length n containing the eigenvalues of A.
ER : a matrix whose columns contain the right eigenvectors of A.
EL : a matrix whose rows contain the left eigenvectors of A.
return values:
E if left and right are both false.
(E, ER) if right is true and left is false.
(E, EL) if left is true and right is false.
(E, EL, ER) if left and right are true.
examples:
>>> from mpmath import mp
>>> A = mp.matrix([[3, -1, 2], [2, 5, -5], [-2, -3, 7]])
>>> E, ER = mp.eig(A)
>>> print(mp.chop(A * ER[:,0] - E[0] * ER[:,0]))
[0.0]
[0.0]
[0.0]
>>> E, EL, ER = mp.eig(A,left = True, right = True)
>>> E, EL, ER = mp.eig_sort(E, EL, ER)
>>> mp.nprint(E)
[2.0, 4.0, 9.0]
>>> print(mp.chop(A * ER[:,0] - E[0] * ER[:,0]))
[0.0]
[0.0]
[0.0]
>>> print(mp.chop( EL[0,:] * A - EL[0,:] * E[0]))
[0.0 0.0 0.0]
warning:
- If there are multiple eigenvalues, the eigenvectors do not necessarily
span the whole vectorspace, i.e. ER and EL may have not full rank.
Furthermore in that case the eigenvectors are numerical ill-conditioned.
- In the general case the eigenvalues have no natural order.
see also:
- eigh (or eigsy, eighe) for the symmetric eigenvalue problem.
- eig_sort for sorting of eigenvalues and eigenvectors
"""
n = A.rows
if n == 1:
if left and (not right):
return ([A[0]], ctx.matrix([[1]]))
if right and (not left):
return ([A[0]], ctx.matrix([[1]]))
return ([A[0]], ctx.matrix([[1]]), ctx.matrix([[1]]))
if not overwrite_a:
A = A.copy()
T = ctx.zeros(n, 1)
hessenberg_reduce_0(ctx, A, T)
if left or right:
Q = A.copy()
hessenberg_reduce_1(ctx, Q, T)
else:
Q = False
for x in xrange(n):
for y in xrange(x + 2, n):
A[y,x] = 0
hessenberg_qr(ctx, A, Q)
E = [0 for i in xrange(n)]
for i in xrange(n):
E[i] = A[i,i]
if not (left or right):
return E
if left:
EL = eig_tr_l(ctx, A)
EL = EL * Q.transpose_conj()
if right:
ER = eig_tr_r(ctx, A)
ER = Q * ER
if left and (not right):
return (E, EL)
if right and (not left):
return (E, ER)
return (E, EL, ER)
@defun
def eig_sort(ctx, E, EL = False, ER = False, f = "real"):
"""
This routine sorts the eigenvalues and eigenvectors delivered by ``eig``.
parameters:
E : the eigenvalues as delivered by eig
EL : the left eigenvectors as delivered by eig, or false
ER : the right eigenvectors as delivered by eig, or false
f : either a string ("real" sort by increasing real part, "imag" sort by
increasing imag part, "abs" sort by absolute value) or a function
mapping complexs to the reals, i.e. ``f = lambda x: -mp.re(x) ``
would sort the eigenvalues by decreasing real part.
return values:
E if EL and ER are both false.
(E, ER) if ER is not false and left is false.
(E, EL) if EL is not false and right is false.
(E, EL, ER) if EL and ER are not false.
example:
>>> from mpmath import mp
>>> A = mp.matrix([[3, -1, 2], [2, 5, -5], [-2, -3, 7]])
>>> E, EL, ER = mp.eig(A,left = True, right = True)
>>> E, EL, ER = mp.eig_sort(E, EL, ER)
>>> mp.nprint(E)
[2.0, 4.0, 9.0]
>>> E, EL, ER = mp.eig_sort(E, EL, ER,f = lambda x: -mp.re(x))
>>> mp.nprint(E)
[9.0, 4.0, 2.0]
>>> print(mp.chop(A * ER[:,0] - E[0] * ER[:,0]))
[0.0]
[0.0]
[0.0]
>>> print(mp.chop( EL[0,:] * A - EL[0,:] * E[0]))
[0.0 0.0 0.0]
"""
if isinstance(f, str):
if f == "real":
f = ctx.re
elif f == "imag":
f = ctx.im
elif cmp == "abs":
f = abs
else:
raise RuntimeError("unknown function %s" % f)
n = len(E)
# Sort eigenvalues (bubble-sort)
for i in xrange(n):
imax = i
s = f(E[i]) # s is the current maximal element
for j in xrange(i + 1, n):
c = f(E[j])
if c < s:
s = c
imax = j
if imax != i:
# swap eigenvalues
z = E[i]
E[i] = E[imax]
E[imax] = z
if not isinstance(EL, bool):
for j in xrange(n):
z = EL[i,j]
EL[i,j] = EL[imax,j]
EL[imax,j] = z
if not isinstance(ER, bool):
for j in xrange(n):
z = ER[j,i]
ER[j,i] = ER[j,imax]
ER[j,imax] = z
if isinstance(EL, bool) and isinstance(ER, bool):
return E
if isinstance(EL, bool) and not(isinstance(ER, bool)):
return (E, ER)
if isinstance(ER, bool) and not(isinstance(EL, bool)):
return (E, EL)
return (E, EL, ER)
| 24,524 | 27.15729 | 126 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/matrices/matrices.py
|
from ..libmp.backend import xrange
# TODO: interpret list as vectors (for multiplication)
rowsep = '\n'
colsep = ' '
class _matrix(object):
"""
Numerical matrix.
Specify the dimensions or the data as a nested list.
Elements default to zero.
Use a flat list to create a column vector easily.
By default, only mpf is used to store the data. You can specify another type
using force_type=type. It's possible to specify None.
Make sure force_type(force_type()) is fast.
Creating matrices
-----------------
Matrices in mpmath are implemented using dictionaries. Only non-zero values
are stored, so it is cheap to represent sparse matrices.
The most basic way to create one is to use the ``matrix`` class directly.
You can create an empty matrix specifying the dimensions:
>>> from mpmath import *
>>> mp.dps = 15
>>> matrix(2)
matrix(
[['0.0', '0.0'],
['0.0', '0.0']])
>>> matrix(2, 3)
matrix(
[['0.0', '0.0', '0.0'],
['0.0', '0.0', '0.0']])
Calling ``matrix`` with one dimension will create a square matrix.
To access the dimensions of a matrix, use the ``rows`` or ``cols`` keyword:
>>> A = matrix(3, 2)
>>> A
matrix(
[['0.0', '0.0'],
['0.0', '0.0'],
['0.0', '0.0']])
>>> A.rows
3
>>> A.cols
2
You can also change the dimension of an existing matrix. This will set the
new elements to 0. If the new dimension is smaller than before, the
concerning elements are discarded:
>>> A.rows = 2
>>> A
matrix(
[['0.0', '0.0'],
['0.0', '0.0']])
Internally ``mpmathify`` is used every time an element is set. This
is done using the syntax A[row,column], counting from 0:
>>> A = matrix(2)
>>> A[1,1] = 1 + 1j
>>> A
matrix(
[['0.0', '0.0'],
['0.0', mpc(real='1.0', imag='1.0')]])
You can use the keyword ``force_type`` to change the function which is
called on every new element:
>>> matrix(2, 5, force_type=int) # doctest: +SKIP
matrix(
[[0, 0, 0, 0, 0],
[0, 0, 0, 0, 0]])
A more comfortable way to create a matrix lets you use nested lists:
>>> matrix([[1, 2], [3, 4]])
matrix(
[['1.0', '2.0'],
['3.0', '4.0']])
If you want to preserve the type of the elements you can use
``force_type=None``:
>>> matrix([[1, 2.5], [1j, mpf(2)]], force_type=None)
matrix(
[['1.0', '2.5'],
[mpc(real='0.0', imag='1.0'), '2.0']])
Convenient advanced functions are available for creating various standard
matrices, see ``zeros``, ``ones``, ``diag``, ``eye``, ``randmatrix`` and
``hilbert``.
Vectors
.......
Vectors may also be represented by the ``matrix`` class (with rows = 1 or cols = 1).
For vectors there are some things which make life easier. A column vector can
be created using a flat list, a row vectors using an almost flat nested list::
>>> matrix([1, 2, 3])
matrix(
[['1.0'],
['2.0'],
['3.0']])
>>> matrix([[1, 2, 3]])
matrix(
[['1.0', '2.0', '3.0']])
Optionally vectors can be accessed like lists, using only a single index::
>>> x = matrix([1, 2, 3])
>>> x[1]
mpf('2.0')
>>> x[1,0]
mpf('2.0')
Other
.....
Like you probably expected, matrices can be printed::
>>> print randmatrix(3) # doctest:+SKIP
[ 0.782963853573023 0.802057689719883 0.427895717335467]
[0.0541876859348597 0.708243266653103 0.615134039977379]
[ 0.856151514955773 0.544759264818486 0.686210904770947]
Use ``nstr`` or ``nprint`` to specify the number of digits to print::
>>> nprint(randmatrix(5), 3) # doctest:+SKIP
[2.07e-1 1.66e-1 5.06e-1 1.89e-1 8.29e-1]
[6.62e-1 6.55e-1 4.47e-1 4.82e-1 2.06e-2]
[4.33e-1 7.75e-1 6.93e-2 2.86e-1 5.71e-1]
[1.01e-1 2.53e-1 6.13e-1 3.32e-1 2.59e-1]
[1.56e-1 7.27e-2 6.05e-1 6.67e-2 2.79e-1]
As matrices are mutable, you will need to copy them sometimes::
>>> A = matrix(2)
>>> A
matrix(
[['0.0', '0.0'],
['0.0', '0.0']])
>>> B = A.copy()
>>> B[0,0] = 1
>>> B
matrix(
[['1.0', '0.0'],
['0.0', '0.0']])
>>> A
matrix(
[['0.0', '0.0'],
['0.0', '0.0']])
Finally, it is possible to convert a matrix to a nested list. This is very useful,
as most Python libraries involving matrices or arrays (namely NumPy or SymPy)
support this format::
>>> B.tolist()
[[mpf('1.0'), mpf('0.0')], [mpf('0.0'), mpf('0.0')]]
Matrix operations
-----------------
You can add and subtract matrices of compatible dimensions::
>>> A = matrix([[1, 2], [3, 4]])
>>> B = matrix([[-2, 4], [5, 9]])
>>> A + B
matrix(
[['-1.0', '6.0'],
['8.0', '13.0']])
>>> A - B
matrix(
[['3.0', '-2.0'],
['-2.0', '-5.0']])
>>> A + ones(3) # doctest:+ELLIPSIS
Traceback (most recent call last):
...
ValueError: incompatible dimensions for addition
It is possible to multiply or add matrices and scalars. In the latter case the
operation will be done element-wise::
>>> A * 2
matrix(
[['2.0', '4.0'],
['6.0', '8.0']])
>>> A / 4
matrix(
[['0.25', '0.5'],
['0.75', '1.0']])
>>> A - 1
matrix(
[['0.0', '1.0'],
['2.0', '3.0']])
Of course you can perform matrix multiplication, if the dimensions are
compatible::
>>> A * B
matrix(
[['8.0', '22.0'],
['14.0', '48.0']])
>>> matrix([[1, 2, 3]]) * matrix([[-6], [7], [-2]])
matrix(
[['2.0']])
You can raise powers of square matrices::
>>> A**2
matrix(
[['7.0', '10.0'],
['15.0', '22.0']])
Negative powers will calculate the inverse::
>>> A**-1
matrix(
[['-2.0', '1.0'],
['1.5', '-0.5']])
>>> A * A**-1
matrix(
[['1.0', '1.0842021724855e-19'],
['-2.16840434497101e-19', '1.0']])
Matrix transposition is straightforward::
>>> A = ones(2, 3)
>>> A
matrix(
[['1.0', '1.0', '1.0'],
['1.0', '1.0', '1.0']])
>>> A.T
matrix(
[['1.0', '1.0'],
['1.0', '1.0'],
['1.0', '1.0']])
Norms
.....
Sometimes you need to know how "large" a matrix or vector is. Due to their
multidimensional nature it's not possible to compare them, but there are
several functions to map a matrix or a vector to a positive real number, the
so called norms.
For vectors the p-norm is intended, usually the 1-, the 2- and the oo-norm are
used.
>>> x = matrix([-10, 2, 100])
>>> norm(x, 1)
mpf('112.0')
>>> norm(x, 2)
mpf('100.5186549850325')
>>> norm(x, inf)
mpf('100.0')
Please note that the 2-norm is the most used one, though it is more expensive
to calculate than the 1- or oo-norm.
It is possible to generalize some vector norms to matrix norm::
>>> A = matrix([[1, -1000], [100, 50]])
>>> mnorm(A, 1)
mpf('1050.0')
>>> mnorm(A, inf)
mpf('1001.0')
>>> mnorm(A, 'F')
mpf('1006.2310867787777')
The last norm (the "Frobenius-norm") is an approximation for the 2-norm, which
is hard to calculate and not available. The Frobenius-norm lacks some
mathematical properties you might expect from a norm.
"""
def __init__(self, *args, **kwargs):
self.__data = {}
# LU decompostion cache, this is useful when solving the same system
# multiple times, when calculating the inverse and when calculating the
# determinant
self._LU = None
convert = kwargs.get('force_type', self.ctx.convert)
if not convert:
convert = lambda x: x
if isinstance(args[0], (list, tuple)):
if isinstance(args[0][0], (list, tuple)):
# interpret nested list as matrix
A = args[0]
self.__rows = len(A)
self.__cols = len(A[0])
for i, row in enumerate(A):
for j, a in enumerate(row):
self[i, j] = convert(a)
else:
# interpret list as row vector
v = args[0]
self.__rows = len(v)
self.__cols = 1
for i, e in enumerate(v):
self[i, 0] = e
elif isinstance(args[0], int):
# create empty matrix of given dimensions
if len(args) == 1:
self.__rows = self.__cols = args[0]
else:
if not isinstance(args[1], int):
raise TypeError("expected int")
self.__rows = args[0]
self.__cols = args[1]
elif isinstance(args[0], _matrix):
A = args[0].copy()
self.__data = A._matrix__data
self.__rows = A._matrix__rows
self.__cols = A._matrix__cols
for i in xrange(A.__rows):
for j in xrange(A.__cols):
A[i,j] = convert(A[i,j])
elif hasattr(args[0], 'tolist'):
A = self.ctx.matrix(args[0].tolist())
self.__data = A._matrix__data
self.__rows = A._matrix__rows
self.__cols = A._matrix__cols
else:
raise TypeError('could not interpret given arguments')
def apply(self, f):
"""
Return a copy of self with the function `f` applied elementwise.
"""
new = self.ctx.matrix(self.__rows, self.__cols)
for i in xrange(self.__rows):
for j in xrange(self.__cols):
new[i,j] = f(self[i,j])
return new
def __nstr__(self, n=None, **kwargs):
# Build table of string representations of the elements
res = []
# Track per-column max lengths for pretty alignment
maxlen = [0] * self.cols
for i in range(self.rows):
res.append([])
for j in range(self.cols):
if n:
string = self.ctx.nstr(self[i,j], n, **kwargs)
else:
string = str(self[i,j])
res[-1].append(string)
maxlen[j] = max(len(string), maxlen[j])
# Patch strings together
for i, row in enumerate(res):
for j, elem in enumerate(row):
# Pad each element up to maxlen so the columns line up
row[j] = elem.rjust(maxlen[j])
res[i] = "[" + colsep.join(row) + "]"
return rowsep.join(res)
def __str__(self):
return self.__nstr__()
def _toliststr(self, avoid_type=False):
"""
Create a list string from a matrix.
If avoid_type: avoid multiple 'mpf's.
"""
# XXX: should be something like self.ctx._types
typ = self.ctx.mpf
s = '['
for i in xrange(self.__rows):
s += '['
for j in xrange(self.__cols):
if not avoid_type or not isinstance(self[i,j], typ):
a = repr(self[i,j])
else:
a = "'" + str(self[i,j]) + "'"
s += a + ', '
s = s[:-2]
s += '],\n '
s = s[:-3]
s += ']'
return s
def tolist(self):
"""
Convert the matrix to a nested list.
"""
return [[self[i,j] for j in range(self.__cols)] for i in range(self.__rows)]
def __repr__(self):
if self.ctx.pretty:
return self.__str__()
s = 'matrix(\n'
s += self._toliststr(avoid_type=True) + ')'
return s
def __get_element(self, key):
'''
Fast extraction of the i,j element from the matrix
This function is for private use only because is unsafe:
1. Does not check on the value of key it expects key to be a integer tuple (i,j)
2. Does not check bounds
'''
if key in self.__data:
return self.__data[key]
else:
return self.ctx.zero
def __set_element(self, key, value):
'''
Fast assignment of the i,j element in the matrix
This function is unsafe:
1. Does not check on the value of key it expects key to be a integer tuple (i,j)
2. Does not check bounds
3. Does not check the value type
'''
if value: # only store non-zeros
self.__data[key] = value
elif key in self.__data:
del self.__data[key]
def __getitem__(self, key):
'''
Getitem function for mp matrix class with slice index enabled
it allows the following assingments
scalar to a slice of the matrix
B = A[:,2:6]
'''
# Convert vector to matrix indexing
if isinstance(key, int) or isinstance(key,slice):
# only sufficent for vectors
if self.__rows == 1:
key = (0, key)
elif self.__cols == 1:
key = (key, 0)
else:
raise IndexError('insufficient indices for matrix')
if isinstance(key[0],slice) or isinstance(key[1],slice):
#Rows
if isinstance(key[0],slice):
#Check bounds
if (key[0].start is None or key[0].start >= 0) and \
(key[0].stop is None or key[0].stop <= self.__rows+1):
# Generate indices
rows = xrange(*key[0].indices(self.__rows))
else:
raise IndexError('Row index out of bounds')
else:
# Single row
rows = [key[0]]
# Columns
if isinstance(key[1],slice):
# Check bounds
if (key[1].start is None or key[1].start >= 0) and \
(key[1].stop is None or key[1].stop <= self.__cols+1):
# Generate indices
columns = xrange(*key[1].indices(self.__cols))
else:
raise IndexError('Column index out of bounds')
else:
# Single column
columns = [key[1]]
# Create matrix slice
m = self.ctx.matrix(len(rows),len(columns))
# Assign elements to the output matrix
for i,x in enumerate(rows):
for j,y in enumerate(columns):
m.__set_element((i,j),self.__get_element((x,y)))
return m
else:
# single element extraction
if key[0] >= self.__rows or key[1] >= self.__cols:
raise IndexError('matrix index out of range')
if key in self.__data:
return self.__data[key]
else:
return self.ctx.zero
def __setitem__(self, key, value):
# setitem function for mp matrix class with slice index enabled
# it allows the following assingments
# scalar to a slice of the matrix
# A[:,2:6] = 2.5
# submatrix to matrix (the value matrix should be the same size as the slice size)
# A[3,:] = B where A is n x m and B is n x 1
# Convert vector to matrix indexing
if isinstance(key, int) or isinstance(key,slice):
# only sufficent for vectors
if self.__rows == 1:
key = (0, key)
elif self.__cols == 1:
key = (key, 0)
else:
raise IndexError('insufficient indices for matrix')
# Slice indexing
if isinstance(key[0],slice) or isinstance(key[1],slice):
# Rows
if isinstance(key[0],slice):
# Check bounds
if (key[0].start is None or key[0].start >= 0) and \
(key[0].stop is None or key[0].stop <= self.__rows+1):
# generate row indices
rows = xrange(*key[0].indices(self.__rows))
else:
raise IndexError('Row index out of bounds')
else:
# Single row
rows = [key[0]]
# Columns
if isinstance(key[1],slice):
# Check bounds
if (key[1].start is None or key[1].start >= 0) and \
(key[1].stop is None or key[1].stop <= self.__cols+1):
# Generate column indices
columns = xrange(*key[1].indices(self.__cols))
else:
raise IndexError('Column index out of bounds')
else:
# Single column
columns = [key[1]]
# Assign slice with a scalar
if isinstance(value,self.ctx.matrix):
# Assign elements to matrix if input and output dimensions match
if len(rows) == value.rows and len(columns) == value.cols:
for i,x in enumerate(rows):
for j,y in enumerate(columns):
self.__set_element((x,y), value.__get_element((i,j)))
else:
raise ValueError('Dimensions do not match')
else:
# Assign slice with scalars
value = self.ctx.convert(value)
for i in rows:
for j in columns:
self.__set_element((i,j), value)
else:
# Single element assingment
# Check bounds
if key[0] >= self.__rows or key[1] >= self.__cols:
raise IndexError('matrix index out of range')
# Convert and store value
value = self.ctx.convert(value)
if value: # only store non-zeros
self.__data[key] = value
elif key in self.__data:
del self.__data[key]
if self._LU:
self._LU = None
return
def __iter__(self):
for i in xrange(self.__rows):
for j in xrange(self.__cols):
yield self[i,j]
def __mul__(self, other):
if isinstance(other, self.ctx.matrix):
# dot multiplication TODO: use Strassen's method?
if self.__cols != other.__rows:
raise ValueError('dimensions not compatible for multiplication')
new = self.ctx.matrix(self.__rows, other.__cols)
for i in xrange(self.__rows):
for j in xrange(other.__cols):
new[i, j] = self.ctx.fdot((self[i,k], other[k,j])
for k in xrange(other.__rows))
return new
else:
# try scalar multiplication
new = self.ctx.matrix(self.__rows, self.__cols)
for i in xrange(self.__rows):
for j in xrange(self.__cols):
new[i, j] = other * self[i, j]
return new
def __rmul__(self, other):
# assume other is scalar and thus commutative
if isinstance(other, self.ctx.matrix):
raise TypeError("other should not be type of ctx.matrix")
return self.__mul__(other)
def __pow__(self, other):
# avoid cyclic import problems
#from linalg import inverse
if not isinstance(other, int):
raise ValueError('only integer exponents are supported')
if not self.__rows == self.__cols:
raise ValueError('only powers of square matrices are defined')
n = other
if n == 0:
return self.ctx.eye(self.__rows)
if n < 0:
n = -n
neg = True
else:
neg = False
i = n
y = 1
z = self.copy()
while i != 0:
if i % 2 == 1:
y = y * z
z = z*z
i = i // 2
if neg:
y = self.ctx.inverse(y)
return y
def __div__(self, other):
# assume other is scalar and do element-wise divison
assert not isinstance(other, self.ctx.matrix)
new = self.ctx.matrix(self.__rows, self.__cols)
for i in xrange(self.__rows):
for j in xrange(self.__cols):
new[i,j] = self[i,j] / other
return new
__truediv__ = __div__
def __add__(self, other):
if isinstance(other, self.ctx.matrix):
if not (self.__rows == other.__rows and self.__cols == other.__cols):
raise ValueError('incompatible dimensions for addition')
new = self.ctx.matrix(self.__rows, self.__cols)
for i in xrange(self.__rows):
for j in xrange(self.__cols):
new[i,j] = self[i,j] + other[i,j]
return new
else:
# assume other is scalar and add element-wise
new = self.ctx.matrix(self.__rows, self.__cols)
for i in xrange(self.__rows):
for j in xrange(self.__cols):
new[i,j] += self[i,j] + other
return new
def __radd__(self, other):
return self.__add__(other)
def __sub__(self, other):
if isinstance(other, self.ctx.matrix) and not (self.__rows == other.__rows
and self.__cols == other.__cols):
raise ValueError('incompatible dimensions for substraction')
return self.__add__(other * (-1))
def __neg__(self):
return (-1) * self
def __rsub__(self, other):
return -self + other
def __eq__(self, other):
return self.__rows == other.__rows and self.__cols == other.__cols \
and self.__data == other.__data
def __len__(self):
if self.rows == 1:
return self.cols
elif self.cols == 1:
return self.rows
else:
return self.rows # do it like numpy
def __getrows(self):
return self.__rows
def __setrows(self, value):
for key in self.__data.copy():
if key[0] >= value:
del self.__data[key]
self.__rows = value
rows = property(__getrows, __setrows, doc='number of rows')
def __getcols(self):
return self.__cols
def __setcols(self, value):
for key in self.__data.copy():
if key[1] >= value:
del self.__data[key]
self.__cols = value
cols = property(__getcols, __setcols, doc='number of columns')
def transpose(self):
new = self.ctx.matrix(self.__cols, self.__rows)
for i in xrange(self.__rows):
for j in xrange(self.__cols):
new[j,i] = self[i,j]
return new
T = property(transpose)
def conjugate(self):
return self.apply(self.ctx.conj)
def transpose_conj(self):
return self.conjugate().transpose()
H = property(transpose_conj)
def copy(self):
new = self.ctx.matrix(self.__rows, self.__cols)
new.__data = self.__data.copy()
return new
__copy__ = copy
def column(self, n):
m = self.ctx.matrix(self.rows, 1)
for i in range(self.rows):
m[i] = self[i,n]
return m
class MatrixMethods(object):
def __init__(ctx):
# XXX: subclass
ctx.matrix = type('matrix', (_matrix,), {})
ctx.matrix.ctx = ctx
ctx.matrix.convert = ctx.convert
def eye(ctx, n, **kwargs):
"""
Create square identity matrix n x n.
"""
A = ctx.matrix(n, **kwargs)
for i in xrange(n):
A[i,i] = 1
return A
def diag(ctx, diagonal, **kwargs):
"""
Create square diagonal matrix using given list.
Example:
>>> from mpmath import diag, mp
>>> mp.pretty = False
>>> diag([1, 2, 3])
matrix(
[['1.0', '0.0', '0.0'],
['0.0', '2.0', '0.0'],
['0.0', '0.0', '3.0']])
"""
A = ctx.matrix(len(diagonal), **kwargs)
for i in xrange(len(diagonal)):
A[i,i] = diagonal[i]
return A
def zeros(ctx, *args, **kwargs):
"""
Create matrix m x n filled with zeros.
One given dimension will create square matrix n x n.
Example:
>>> from mpmath import zeros, mp
>>> mp.pretty = False
>>> zeros(2)
matrix(
[['0.0', '0.0'],
['0.0', '0.0']])
"""
if len(args) == 1:
m = n = args[0]
elif len(args) == 2:
m = args[0]
n = args[1]
else:
raise TypeError('zeros expected at most 2 arguments, got %i' % len(args))
A = ctx.matrix(m, n, **kwargs)
for i in xrange(m):
for j in xrange(n):
A[i,j] = 0
return A
def ones(ctx, *args, **kwargs):
"""
Create matrix m x n filled with ones.
One given dimension will create square matrix n x n.
Example:
>>> from mpmath import ones, mp
>>> mp.pretty = False
>>> ones(2)
matrix(
[['1.0', '1.0'],
['1.0', '1.0']])
"""
if len(args) == 1:
m = n = args[0]
elif len(args) == 2:
m = args[0]
n = args[1]
else:
raise TypeError('ones expected at most 2 arguments, got %i' % len(args))
A = ctx.matrix(m, n, **kwargs)
for i in xrange(m):
for j in xrange(n):
A[i,j] = 1
return A
def hilbert(ctx, m, n=None):
"""
Create (pseudo) hilbert matrix m x n.
One given dimension will create hilbert matrix n x n.
The matrix is very ill-conditioned and symmetric, positive definite if
square.
"""
if n is None:
n = m
A = ctx.matrix(m, n)
for i in xrange(m):
for j in xrange(n):
A[i,j] = ctx.one / (i + j + 1)
return A
def randmatrix(ctx, m, n=None, min=0, max=1, **kwargs):
"""
Create a random m x n matrix.
All values are >= min and <max.
n defaults to m.
Example:
>>> from mpmath import randmatrix
>>> randmatrix(2) # doctest:+SKIP
matrix(
[['0.53491598236191806', '0.57195669543302752'],
['0.85589992269513615', '0.82444367501382143']])
"""
if not n:
n = m
A = ctx.matrix(m, n, **kwargs)
for i in xrange(m):
for j in xrange(n):
A[i,j] = ctx.rand() * (max - min) + min
return A
def swap_row(ctx, A, i, j):
"""
Swap row i with row j.
"""
if i == j:
return
if isinstance(A, ctx.matrix):
for k in xrange(A.cols):
A[i,k], A[j,k] = A[j,k], A[i,k]
elif isinstance(A, list):
A[i], A[j] = A[j], A[i]
else:
raise TypeError('could not interpret type')
def extend(ctx, A, b):
"""
Extend matrix A with column b and return result.
"""
if not isinstance(A, ctx.matrix):
raise TypeError("A should be a type of ctx.matrix")
if A.rows != len(b):
raise ValueError("Value should be equal to len(b)")
A = A.copy()
A.cols += 1
for i in xrange(A.rows):
A[i, A.cols-1] = b[i]
return A
def norm(ctx, x, p=2):
r"""
Gives the entrywise `p`-norm of an iterable *x*, i.e. the vector norm
`\left(\sum_k |x_k|^p\right)^{1/p}`, for any given `1 \le p \le \infty`.
Special cases:
If *x* is not iterable, this just returns ``absmax(x)``.
``p=1`` gives the sum of absolute values.
``p=2`` is the standard Euclidean vector norm.
``p=inf`` gives the magnitude of the largest element.
For *x* a matrix, ``p=2`` is the Frobenius norm.
For operator matrix norms, use :func:`~mpmath.mnorm` instead.
You can use the string 'inf' as well as float('inf') or mpf('inf')
to specify the infinity norm.
**Examples**
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = False
>>> x = matrix([-10, 2, 100])
>>> norm(x, 1)
mpf('112.0')
>>> norm(x, 2)
mpf('100.5186549850325')
>>> norm(x, inf)
mpf('100.0')
"""
try:
iter(x)
except TypeError:
return ctx.absmax(x)
if type(p) is not int:
p = ctx.convert(p)
if p == ctx.inf:
return max(ctx.absmax(i) for i in x)
elif p == 1:
return ctx.fsum(x, absolute=1)
elif p == 2:
return ctx.sqrt(ctx.fsum(x, absolute=1, squared=1))
elif p > 1:
return ctx.nthroot(ctx.fsum(abs(i)**p for i in x), p)
else:
raise ValueError('p has to be >= 1')
def mnorm(ctx, A, p=1):
r"""
Gives the matrix (operator) `p`-norm of A. Currently ``p=1`` and ``p=inf``
are supported:
``p=1`` gives the 1-norm (maximal column sum)
``p=inf`` gives the `\infty`-norm (maximal row sum).
You can use the string 'inf' as well as float('inf') or mpf('inf')
``p=2`` (not implemented) for a square matrix is the usual spectral
matrix norm, i.e. the largest singular value.
``p='f'`` (or 'F', 'fro', 'Frobenius, 'frobenius') gives the
Frobenius norm, which is the elementwise 2-norm. The Frobenius norm is an
approximation of the spectral norm and satisfies
.. math ::
\frac{1}{\sqrt{\mathrm{rank}(A)}} \|A\|_F \le \|A\|_2 \le \|A\|_F
The Frobenius norm lacks some mathematical properties that might
be expected of a norm.
For general elementwise `p`-norms, use :func:`~mpmath.norm` instead.
**Examples**
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = False
>>> A = matrix([[1, -1000], [100, 50]])
>>> mnorm(A, 1)
mpf('1050.0')
>>> mnorm(A, inf)
mpf('1001.0')
>>> mnorm(A, 'F')
mpf('1006.2310867787777')
"""
A = ctx.matrix(A)
if type(p) is not int:
if type(p) is str and 'frobenius'.startswith(p.lower()):
return ctx.norm(A, 2)
p = ctx.convert(p)
m, n = A.rows, A.cols
if p == 1:
return max(ctx.fsum((A[i,j] for i in xrange(m)), absolute=1) for j in xrange(n))
elif p == ctx.inf:
return max(ctx.fsum((A[i,j] for j in xrange(n)), absolute=1) for i in xrange(m))
else:
raise NotImplementedError("matrix p-norm for arbitrary p")
if __name__ == '__main__':
import doctest
doctest.testmod()
| 31,596 | 30.787726 | 96 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/matrices/eigen_symmetric.py
|
#!/usr/bin/python
# -*- coding: utf-8 -*-
##################################################################################################
# module for the symmetric eigenvalue problem
# Copyright 2013 Timo Hartmann (thartmann15 at gmail.com)
#
# todo:
# - implement balancing
#
##################################################################################################
"""
The symmetric eigenvalue problem.
---------------------------------
This file contains routines for the symmetric eigenvalue problem.
high level routines:
eigsy : real symmetric (ordinary) eigenvalue problem
eighe : complex hermitian (ordinary) eigenvalue problem
eigh : unified interface for eigsy and eighe
svd_r : singular value decomposition for real matrices
svd_c : singular value decomposition for complex matrices
svd : unified interface for svd_r and svd_c
low level routines:
r_sy_tridiag : reduction of real symmetric matrix to real symmetric tridiagonal matrix
c_he_tridiag_0 : reduction of complex hermitian matrix to real symmetric tridiagonal matrix
c_he_tridiag_1 : auxiliary routine to c_he_tridiag_0
c_he_tridiag_2 : auxiliary routine to c_he_tridiag_0
tridiag_eigen : solves the real symmetric tridiagonal matrix eigenvalue problem
svd_r_raw : raw singular value decomposition for real matrices
svd_c_raw : raw singular value decomposition for complex matrices
"""
from ..libmp.backend import xrange
from .eigen import defun
def r_sy_tridiag(ctx, A, D, E, calc_ev = True):
"""
This routine transforms a real symmetric matrix A to a real symmetric
tridiagonal matrix T using an orthogonal similarity transformation:
Q' * A * Q = T (here ' denotes the matrix transpose).
The orthogonal matrix Q is build up from Householder reflectors.
parameters:
A (input/output) On input, A contains the real symmetric matrix of
dimension (n,n). On output, if calc_ev is true, A contains the
orthogonal matrix Q, otherwise A is destroyed.
D (output) real array of length n, contains the diagonal elements
of the tridiagonal matrix
E (output) real array of length n, contains the offdiagonal elements
of the tridiagonal matrix in E[0:(n-1)] where is the dimension of
the matrix A. E[n-1] is undefined.
calc_ev (input) If calc_ev is true, this routine explicitly calculates the
orthogonal matrix Q which is then returned in A. If calc_ev is
false, Q is not explicitly calculated resulting in a shorter run time.
This routine is a python translation of the fortran routine tred2.f in the
software library EISPACK (see netlib.org) which itself is based on the algol
procedure tred2 described in:
- Num. Math. 11, p.181-195 (1968) by Martin, Reinsch and Wilkonson
- Handbook for auto. comp., Vol II, Linear Algebra, p.212-226 (1971)
For a good introduction to Householder reflections, see also
Stoer, Bulirsch - Introduction to Numerical Analysis.
"""
# note : the vector v of the i-th houshoulder reflector is stored in a[(i+1):,i]
# whereas v/<v,v> is stored in a[i,(i+1):]
n = A.rows
for i in xrange(n - 1, 0, -1):
# scale the vector
scale = 0
for k in xrange(0, i):
scale += abs(A[k,i])
scale_inv = 0
if scale != 0:
scale_inv = 1/scale
# sadly there are floating point numbers not equal to zero whose reciprocal is infinity
if i == 1 or scale == 0 or ctx.isinf(scale_inv):
E[i] = A[i-1,i] # nothing to do
D[i] = 0
continue
# calculate parameters for housholder transformation
H = 0
for k in xrange(0, i):
A[k,i] *= scale_inv
H += A[k,i] * A[k,i]
F = A[i-1,i]
G = ctx.sqrt(H)
if F > 0:
G = -G
E[i] = scale * G
H -= F * G
A[i-1,i] = F - G
F = 0
# apply housholder transformation
for j in xrange(0, i):
if calc_ev:
A[i,j] = A[j,i] / H
G = 0 # calculate A*U
for k in xrange(0, j + 1):
G += A[k,j] * A[k,i]
for k in xrange(j + 1, i):
G += A[j,k] * A[k,i]
E[j] = G / H # calculate P
F += E[j] * A[j,i]
HH = F / (2 * H)
for j in xrange(0, i): # calculate reduced A
F = A[j,i]
G = E[j] - HH * F # calculate Q
E[j] = G
for k in xrange(0, j + 1):
A[k,j] -= F * E[k] + G * A[k,i]
D[i] = H
for i in xrange(1, n): # better for compatibility
E[i-1] = E[i]
E[n-1] = 0
if calc_ev:
D[0] = 0
for i in xrange(0, n):
if D[i] != 0:
for j in xrange(0, i): # accumulate transformation matrices
G = 0
for k in xrange(0, i):
G += A[i,k] * A[k,j]
for k in xrange(0, i):
A[k,j] -= G * A[k,i]
D[i] = A[i,i]
A[i,i] = 1
for j in xrange(0, i):
A[j,i] = A[i,j] = 0
else:
for i in xrange(0, n):
D[i] = A[i,i]
def c_he_tridiag_0(ctx, A, D, E, T):
"""
This routine transforms a complex hermitian matrix A to a real symmetric
tridiagonal matrix T using an unitary similarity transformation:
Q' * A * Q = T (here ' denotes the hermitian matrix transpose,
i.e. transposition und conjugation).
The unitary matrix Q is build up from Householder reflectors and
an unitary diagonal matrix.
parameters:
A (input/output) On input, A contains the complex hermitian matrix
of dimension (n,n). On output, A contains the unitary matrix Q
in compressed form.
D (output) real array of length n, contains the diagonal elements
of the tridiagonal matrix.
E (output) real array of length n, contains the offdiagonal elements
of the tridiagonal matrix in E[0:(n-1)] where is the dimension of
the matrix A. E[n-1] is undefined.
T (output) complex array of length n, contains a unitary diagonal
matrix.
This routine is a python translation (in slightly modified form) of the fortran
routine htridi.f in the software library EISPACK (see netlib.org) which itself
is a complex version of the algol procedure tred1 described in:
- Num. Math. 11, p.181-195 (1968) by Martin, Reinsch and Wilkonson
- Handbook for auto. comp., Vol II, Linear Algebra, p.212-226 (1971)
For a good introduction to Householder reflections, see also
Stoer, Bulirsch - Introduction to Numerical Analysis.
"""
n = A.rows
T[n-1] = 1
for i in xrange(n - 1, 0, -1):
# scale the vector
scale = 0
for k in xrange(0, i):
scale += abs(ctx.re(A[k,i])) + abs(ctx.im(A[k,i]))
scale_inv = 0
if scale != 0:
scale_inv = 1 / scale
# sadly there are floating point numbers not equal to zero whose reciprocal is infinity
if scale == 0 or ctx.isinf(scale_inv):
E[i] = 0
D[i] = 0
T[i-1] = 1
continue
if i == 1:
F = A[i-1,i]
f = abs(F)
E[i] = f
D[i] = 0
if f != 0:
T[i-1] = T[i] * F / f
else:
T[i-1] = T[i]
continue
# calculate parameters for housholder transformation
H = 0
for k in xrange(0, i):
A[k,i] *= scale_inv
rr = ctx.re(A[k,i])
ii = ctx.im(A[k,i])
H += rr * rr + ii * ii
F = A[i-1,i]
f = abs(F)
G = ctx.sqrt(H)
H += G * f
E[i] = scale * G
if f != 0:
F = F / f
TZ = - T[i] * F # T[i-1]=-T[i]*F, but we need T[i-1] as temporary storage
G *= F
else:
TZ = -T[i] # T[i-1]=-T[i]
A[i-1,i] += G
F = 0
# apply housholder transformation
for j in xrange(0, i):
A[i,j] = A[j,i] / H
G = 0 # calculate A*U
for k in xrange(0, j + 1):
G += ctx.conj(A[k,j]) * A[k,i]
for k in xrange(j + 1, i):
G += A[j,k] * A[k,i]
T[j] = G / H # calculate P
F += ctx.conj(T[j]) * A[j,i]
HH = F / (2 * H)
for j in xrange(0, i): # calculate reduced A
F = A[j,i]
G = T[j] - HH * F # calculate Q
T[j] = G
for k in xrange(0, j + 1):
A[k,j] -= ctx.conj(F) * T[k] + ctx.conj(G) * A[k,i]
# as we use the lower left part for storage
# we have to use the transpose of the normal formula
T[i-1] = TZ
D[i] = H
for i in xrange(1, n): # better for compatibility
E[i-1] = E[i]
E[n-1] = 0
D[0] = 0
for i in xrange(0, n):
zw = D[i]
D[i] = ctx.re(A[i,i])
A[i,i] = zw
def c_he_tridiag_1(ctx, A, T):
"""
This routine forms the unitary matrix Q described in c_he_tridiag_0.
parameters:
A (input/output) On input, A is the same matrix as delivered by
c_he_tridiag_0. On output, A is set to Q.
T (input) On input, T is the same array as delivered by c_he_tridiag_0.
"""
n = A.rows
for i in xrange(0, n):
if A[i,i] != 0:
for j in xrange(0, i):
G = 0
for k in xrange(0, i):
G += ctx.conj(A[i,k]) * A[k,j]
for k in xrange(0, i):
A[k,j] -= G * A[k,i]
A[i,i] = 1
for j in xrange(0, i):
A[j,i] = A[i,j] = 0
for i in xrange(0, n):
for k in xrange(0, n):
A[i,k] *= T[k]
def c_he_tridiag_2(ctx, A, T, B):
"""
This routine applied the unitary matrix Q described in c_he_tridiag_0
onto the the matrix B, i.e. it forms Q*B.
parameters:
A (input) On input, A is the same matrix as delivered by c_he_tridiag_0.
T (input) On input, T is the same array as delivered by c_he_tridiag_0.
B (input/output) On input, B is a complex matrix. On output B is replaced
by Q*B.
This routine is a python translation of the fortran routine htribk.f in the
software library EISPACK (see netlib.org). See c_he_tridiag_0 for more
references.
"""
n = A.rows
for i in xrange(0, n):
for k in xrange(0, n):
B[k,i] *= T[k]
for i in xrange(0, n):
if A[i,i] != 0:
for j in xrange(0, n):
G = 0
for k in xrange(0, i):
G += ctx.conj(A[i,k]) * B[k,j]
for k in xrange(0, i):
B[k,j] -= G * A[k,i]
def tridiag_eigen(ctx, d, e, z = False):
"""
This subroutine find the eigenvalues and the first components of the
eigenvectors of a real symmetric tridiagonal matrix using the implicit
QL method.
parameters:
d (input/output) real array of length n. on input, d contains the diagonal
elements of the input matrix. on output, d contains the eigenvalues in
ascending order.
e (input) real array of length n. on input, e contains the offdiagonal
elements of the input matrix in e[0:(n-1)]. On output, e has been
destroyed.
z (input/output) If z is equal to False, no eigenvectors will be computed.
Otherwise on input z should have the format z[0:m,0:n] (i.e. a real or
complex matrix of dimension (m,n) ). On output this matrix will be
multiplied by the matrix of the eigenvectors (i.e. the columns of this
matrix are the eigenvectors): z --> z*EV
That means if z[i,j]={1 if j==j; 0 otherwise} on input, then on output
z will contain the first m components of the eigenvectors. That means
if m is equal to n, the i-th eigenvector will be z[:,i].
This routine is a python translation (in slightly modified form) of the
fortran routine imtql2.f in the software library EISPACK (see netlib.org)
which itself is based on the algol procudure imtql2 desribed in:
- num. math. 12, p. 377-383(1968) by matrin and wilkinson
- modified in num. math. 15, p. 450(1970) by dubrulle
- handbook for auto. comp., vol. II-linear algebra, p. 241-248 (1971)
See also the routine gaussq.f in netlog.org or acm algorithm 726.
"""
n = len(d)
e[n-1] = 0
iterlim = 2 * ctx.dps
for l in xrange(n):
j = 0
while 1:
m = l
while 1:
# look for a small subdiagonal element
if m + 1 == n:
break
if abs(e[m]) <= ctx.eps * (abs(d[m]) + abs(d[m + 1])):
break
m = m + 1
if m == l:
break
if j >= iterlim:
raise RuntimeError("tridiag_eigen: no convergence to an eigenvalue after %d iterations" % iterlim)
j += 1
# form shift
p = d[l]
g = (d[l + 1] - p) / (2 * e[l])
r = ctx.hypot(g, 1)
if g < 0:
s = g - r
else:
s = g + r
g = d[m] - p + e[l] / s
s, c, p = 1, 1, 0
for i in xrange(m - 1, l - 1, -1):
f = s * e[i]
b = c * e[i]
if abs(f) > abs(g): # this here is a slight improvement also used in gaussq.f or acm algorithm 726.
c = g / f
r = ctx.hypot(c, 1)
e[i + 1] = f * r
s = 1 / r
c = c * s
else:
s = f / g
r = ctx.hypot(s, 1)
e[i + 1] = g * r
c = 1 / r
s = s * c
g = d[i + 1] - p
r = (d[i] - g) * s + 2 * c * b
p = s * r
d[i + 1] = g + p
g = c * r - b
if not isinstance(z, bool):
# calculate eigenvectors
for w in xrange(z.rows):
f = z[w,i+1]
z[w,i+1] = s * z[w,i] + c * f
z[w,i ] = c * z[w,i] - s * f
d[l] = d[l] - p
e[l] = g
e[m] = 0
for ii in xrange(1, n):
# sort eigenvalues and eigenvectors (bubble-sort)
i = ii - 1
k = i
p = d[i]
for j in xrange(ii, n):
if d[j] >= p:
continue
k = j
p = d[k]
if k == i:
continue
d[k] = d[i]
d[i] = p
if not isinstance(z, bool):
for w in xrange(z.rows):
p = z[w,i]
z[w,i] = z[w,k]
z[w,k] = p
########################################################################################
@defun
def eigsy(ctx, A, eigvals_only = False, overwrite_a = False):
"""
This routine solves the (ordinary) eigenvalue problem for a real symmetric
square matrix A. Given A, an orthogonal matrix Q is calculated which
diagonalizes A:
Q' A Q = diag(E) and Q Q' = Q' Q = 1
Here diag(E) is a diagonal matrix whose diagonal is E.
' denotes the transpose.
The columns of Q are the eigenvectors of A and E contains the eigenvalues:
A Q[:,i] = E[i] Q[:,i]
input:
A: real matrix of format (n,n) which is symmetric
(i.e. A=A' or A[i,j]=A[j,i])
eigvals_only: if true, calculates only the eigenvalues E.
if false, calculates both eigenvectors and eigenvalues.
overwrite_a: if true, allows modification of A which may improve
performance. if false, A is not modified.
output:
E: vector of format (n). contains the eigenvalues of A in ascending order.
Q: orthogonal matrix of format (n,n). contains the eigenvectors
of A as columns.
return value:
E if eigvals_only is true
(E, Q) if eigvals_only is false
example:
>>> from mpmath import mp
>>> A = mp.matrix([[3, 2], [2, 0]])
>>> E = mp.eigsy(A, eigvals_only = True)
>>> print(E)
[-1.0]
[ 4.0]
>>> A = mp.matrix([[1, 2], [2, 3]])
>>> E, Q = mp.eigsy(A)
>>> print(mp.chop(A * Q[:,0] - E[0] * Q[:,0]))
[0.0]
[0.0]
see also: eighe, eigh, eig
"""
if not overwrite_a:
A = A.copy()
d = ctx.zeros(A.rows, 1)
e = ctx.zeros(A.rows, 1)
if eigvals_only:
r_sy_tridiag(ctx, A, d, e, calc_ev = False)
tridiag_eigen(ctx, d, e, False)
return d
else:
r_sy_tridiag(ctx, A, d, e, calc_ev = True)
tridiag_eigen(ctx, d, e, A)
return (d, A)
@defun
def eighe(ctx, A, eigvals_only = False, overwrite_a = False):
"""
This routine solves the (ordinary) eigenvalue problem for a complex
hermitian square matrix A. Given A, an unitary matrix Q is calculated which
diagonalizes A:
Q' A Q = diag(E) and Q Q' = Q' Q = 1
Here diag(E) a is diagonal matrix whose diagonal is E.
' denotes the hermitian transpose (i.e. ordinary transposition and
complex conjugation).
The columns of Q are the eigenvectors of A and E contains the eigenvalues:
A Q[:,i] = E[i] Q[:,i]
input:
A: complex matrix of format (n,n) which is hermitian
(i.e. A=A' or A[i,j]=conj(A[j,i]))
eigvals_only: if true, calculates only the eigenvalues E.
if false, calculates both eigenvectors and eigenvalues.
overwrite_a: if true, allows modification of A which may improve
performance. if false, A is not modified.
output:
E: vector of format (n). contains the eigenvalues of A in ascending order.
Q: unitary matrix of format (n,n). contains the eigenvectors
of A as columns.
return value:
E if eigvals_only is true
(E, Q) if eigvals_only is false
example:
>>> from mpmath import mp
>>> A = mp.matrix([[1, -3 - 1j], [-3 + 1j, -2]])
>>> E = mp.eighe(A, eigvals_only = True)
>>> print(E)
[-4.0]
[ 3.0]
>>> A = mp.matrix([[1, 2 + 5j], [2 - 5j, 3]])
>>> E, Q = mp.eighe(A)
>>> print(mp.chop(A * Q[:,0] - E[0] * Q[:,0]))
[0.0]
[0.0]
see also: eigsy, eigh, eig
"""
if not overwrite_a:
A = A.copy()
d = ctx.zeros(A.rows, 1)
e = ctx.zeros(A.rows, 1)
t = ctx.zeros(A.rows, 1)
if eigvals_only:
c_he_tridiag_0(ctx, A, d, e, t)
tridiag_eigen(ctx, d, e, False)
return d
else:
c_he_tridiag_0(ctx, A, d, e, t)
B = ctx.eye(A.rows)
tridiag_eigen(ctx, d, e, B)
c_he_tridiag_2(ctx, A, t, B)
return (d, B)
@defun
def eigh(ctx, A, eigvals_only = False, overwrite_a = False):
"""
"eigh" is a unified interface for "eigsy" and "eighe". Depending on
whether A is real or complex the appropriate function is called.
This routine solves the (ordinary) eigenvalue problem for a real symmetric
or complex hermitian square matrix A. Given A, an orthogonal (A real) or
unitary (A complex) matrix Q is calculated which diagonalizes A:
Q' A Q = diag(E) and Q Q' = Q' Q = 1
Here diag(E) a is diagonal matrix whose diagonal is E.
' denotes the hermitian transpose (i.e. ordinary transposition and
complex conjugation).
The columns of Q are the eigenvectors of A and E contains the eigenvalues:
A Q[:,i] = E[i] Q[:,i]
input:
A: a real or complex square matrix of format (n,n) which is symmetric
(i.e. A[i,j]=A[j,i]) or hermitian (i.e. A[i,j]=conj(A[j,i])).
eigvals_only: if true, calculates only the eigenvalues E.
if false, calculates both eigenvectors and eigenvalues.
overwrite_a: if true, allows modification of A which may improve
performance. if false, A is not modified.
output:
E: vector of format (n). contains the eigenvalues of A in ascending order.
Q: an orthogonal or unitary matrix of format (n,n). contains the
eigenvectors of A as columns.
return value:
E if eigvals_only is true
(E, Q) if eigvals_only is false
example:
>>> from mpmath import mp
>>> A = mp.matrix([[3, 2], [2, 0]])
>>> E = mp.eigh(A, eigvals_only = True)
>>> print(E)
[-1.0]
[ 4.0]
>>> A = mp.matrix([[1, 2], [2, 3]])
>>> E, Q = mp.eigh(A)
>>> print(mp.chop(A * Q[:,0] - E[0] * Q[:,0]))
[0.0]
[0.0]
>>> A = mp.matrix([[1, 2 + 5j], [2 - 5j, 3]])
>>> E, Q = mp.eigh(A)
>>> print(mp.chop(A * Q[:,0] - E[0] * Q[:,0]))
[0.0]
[0.0]
see also: eigsy, eighe, eig
"""
iscomplex = any(type(x) is ctx.mpc for x in A)
if iscomplex:
return ctx.eighe(A, eigvals_only = eigvals_only, overwrite_a = overwrite_a)
else:
return ctx.eigsy(A, eigvals_only = eigvals_only, overwrite_a = overwrite_a)
@defun
def gauss_quadrature(ctx, n, qtype = "legendre", alpha = 0, beta = 0):
"""
This routine calulates gaussian quadrature rules for different
families of orthogonal polynomials. Let (a, b) be an interval,
W(x) a positive weight function and n a positive integer.
Then the purpose of this routine is to calculate pairs (x_k, w_k)
for k=0, 1, 2, ... (n-1) which give
int(W(x) * F(x), x = a..b) = sum(w_k * F(x_k),k = 0..(n-1))
exact for all polynomials F(x) of degree (strictly) less than 2*n. For all
integrable functions F(x) the sum is a (more or less) good approximation to
the integral. The x_k are called nodes (which are the zeros of the
related orthogonal polynomials) and the w_k are called the weights.
parameters
n (input) The degree of the quadrature rule, i.e. its number of
nodes.
qtype (input) The family of orthogonal polynmomials for which to
compute the quadrature rule. See the list below.
alpha (input) real number, used as parameter for some orthogonal
polynomials
beta (input) real number, used as parameter for some orthogonal
polynomials.
return value
(X, W) a pair of two real arrays where x_k = X[k] and w_k = W[k].
orthogonal polynomials:
qtype polynomial
----- ----------
"legendre" Legendre polynomials, W(x)=1 on the interval (-1, +1)
"legendre01" shifted Legendre polynomials, W(x)=1 on the interval (0, +1)
"hermite" Hermite polynomials, W(x)=exp(-x*x) on (-infinity,+infinity)
"laguerre" Laguerre polynomials, W(x)=exp(-x) on (0,+infinity)
"glaguerre" generalized Laguerre polynomials, W(x)=exp(-x)*x**alpha
on (0, +infinity)
"chebyshev1" Chebyshev polynomials of the first kind, W(x)=1/sqrt(1-x*x)
on (-1, +1)
"chebyshev2" Chebyshev polynomials of the second kind, W(x)=sqrt(1-x*x)
on (-1, +1)
"jacobi" Jacobi polynomials, W(x)=(1-x)**alpha * (1+x)**beta on (-1, +1)
with alpha>-1 and beta>-1
examples:
>>> from mpmath import mp
>>> f = lambda x: x**8 + 2 * x**6 - 3 * x**4 + 5 * x**2 - 7
>>> X, W = mp.gauss_quadrature(5, "hermite")
>>> A = mp.fdot([(f(x), w) for x, w in zip(X, W)])
>>> B = mp.sqrt(mp.pi) * 57 / 16
>>> C = mp.quad(lambda x: mp.exp(- x * x) * f(x), [-mp.inf, +mp.inf])
>>> print mp.chop(A-B, tol = 1e-10), mp.chop(A-C, tol = 1e-10)
0.0 0.0
>>> f = lambda x: x**5 - 2 * x**4 + 3 * x**3 - 5 * x**2 + 7 * x - 11
>>> X, W = mp.gauss_quadrature(3, "laguerre")
>>> A = mp.fdot([(f(x), w) for x, w in zip(X, W)])
>>> B = 76
>>> C = mp.quad(lambda x: mp.exp(-x) * f(x), [0, +mp.inf])
>>> print mp.chop(A-B, tol = 1e-10), mp.chop(A-C, tol = 1e-10)
0.0 0.0
# orthogonality of the chebyshev polynomials:
>>> f = lambda x: mp.chebyt(3, x) * mp.chebyt(2, x)
>>> X, W = mp.gauss_quadrature(3, "chebyshev1")
>>> A = mp.fdot([(f(x), w) for x, w in zip(X, W)])
>>> print(mp.chop(A, tol = 1e-10))
0.0
references:
- golub and welsch, "calculations of gaussian quadrature rules", mathematics of
computation 23, p. 221-230 (1969)
- golub, "some modified matrix eigenvalue problems", siam review 15, p. 318-334 (1973)
- stroud and secrest, "gaussian quadrature formulas", prentice-hall (1966)
See also the routine gaussq.f in netlog.org or ACM Transactions on
Mathematical Software algorithm 726.
"""
d = ctx.zeros(n, 1)
e = ctx.zeros(n, 1)
z = ctx.zeros(1, n)
z[0,0] = 1
if qtype == "legendre":
# legendre on the range -1 +1 , abramowitz, table 25.4, p.916
w = 2
for i in xrange(n):
j = i + 1
e[i] = ctx.sqrt(j * j / (4 * j * j - ctx.mpf(1)))
elif qtype == "legendre01":
# legendre shifted to 0 1 , abramowitz, table 25.8, p.921
w = 1
for i in xrange(n):
d[i] = 1 / ctx.mpf(2)
j = i + 1
e[i] = ctx.sqrt(j * j / (16 * j * j - ctx.mpf(4)))
elif qtype == "hermite":
# hermite on the range -inf +inf , abramowitz, table 25.10,p.924
w = ctx.sqrt(ctx.pi)
for i in xrange(n):
j = i + 1
e[i] = ctx.sqrt(j / ctx.mpf(2))
elif qtype == "laguerre":
# laguerre on the range 0 +inf , abramowitz, table 25.9, p. 923
w = 1
for i in xrange(n):
j = i + 1
d[i] = 2 * j - 1
e[i] = j
elif qtype=="chebyshev1":
# chebyshev polynimials of the first kind
w = ctx.pi
for i in xrange(n):
e[i] = 1 / ctx.mpf(2)
e[0] = ctx.sqrt(1 / ctx.mpf(2))
elif qtype == "chebyshev2":
# chebyshev polynimials of the second kind
w = ctx.pi / 2
for i in xrange(n):
e[i] = 1 / ctx.mpf(2)
elif qtype == "glaguerre":
# generalized laguerre on the range 0 +inf
w = ctx.gamma(1 + alpha)
for i in xrange(n):
j = i + 1
d[i] = 2 * j - 1 + alpha
e[i] = ctx.sqrt(j * (j + alpha))
elif qtype == "jacobi":
# jacobi polynomials
alpha = ctx.mpf(alpha)
beta = ctx.mpf(beta)
ab = alpha + beta
abi = ab + 2
w = (2**(ab+1)) * ctx.gamma(alpha + 1) * ctx.gamma(beta + 1) / ctx.gamma(abi)
d[0] = (beta - alpha) / abi
e[0] = ctx.sqrt(4 * (1 + alpha) * (1 + beta) / ((abi + 1) * (abi * abi)))
a2b2 = beta * beta - alpha * alpha
for i in xrange(1, n):
j = i + 1
abi = 2 * j + ab
d[i] = a2b2 / ((abi - 2) * abi)
e[i] = ctx.sqrt(4 * j * (j + alpha) * (j + beta) * (j + ab) / ((abi * abi - 1) * abi * abi))
elif isinstance(qtype, str):
raise ValueError("unknown quadrature rule \"%s\"" % qtype)
elif not isinstance(qtype, str):
w = qtype(d, e)
else:
assert 0
tridiag_eigen(ctx, d, e, z)
for i in xrange(len(z)):
z[i] *= z[i]
z = z.transpose()
return (d, w * z)
##################################################################################################
##################################################################################################
##################################################################################################
def svd_r_raw(ctx, A, V = False, calc_u = False):
"""
This routine computes the singular value decomposition of a matrix A.
Given A, two orthogonal matrices U and V are calculated such that
A = U S V
where S is a suitable shaped matrix whose off-diagonal elements are zero.
The diagonal elements of S are the singular values of A, i.e. the
squareroots of the eigenvalues of A' A or A A'. Here ' denotes the transpose.
Householder bidiagonalization and a variant of the QR algorithm is used.
overview of the matrices :
A : m*n A gets replaced by U
U : m*n U replaces A. If n>m then only the first m*m block of U is
non-zero. column-orthogonal: U' U = B
here B is a n*n matrix whose first min(m,n) diagonal
elements are 1 and all other elements are zero.
S : n*n diagonal matrix, only the diagonal elements are stored in
the array S. only the first min(m,n) diagonal elements are non-zero.
V : n*n orthogonal: V V' = V' V = 1
parameters:
A (input/output) On input, A contains a real matrix of shape m*n.
On output, if calc_u is true A contains the column-orthogonal
matrix U; otherwise A is simply used as workspace and thus destroyed.
V (input/output) if false, the matrix V is not calculated. otherwise
V must be a matrix of shape n*n.
calc_u (input) If true, the matrix U is calculated and replaces A.
if false, U is not calculated and A is simply destroyed
return value:
S an array of length n containing the singular values of A sorted by
decreasing magnitude. only the first min(m,n) elements are non-zero.
This routine is a python translation of the fortran routine svd.f in the
software library EISPACK (see netlib.org) which itself is based on the
algol procedure svd described in:
- num. math. 14, 403-420(1970) by golub and reinsch.
- wilkinson/reinsch: handbook for auto. comp., vol ii-linear algebra, 134-151(1971).
"""
m, n = A.rows, A.cols
S = ctx.zeros(n, 1)
# work is a temporary array of size n
work = ctx.zeros(n, 1)
g = scale = anorm = 0
maxits = 3 * ctx.dps
for i in xrange(n): # householder reduction to bidiagonal form
work[i] = scale*g
g = s = scale = 0
if i < m:
for k in xrange(i, m):
scale += ctx.fabs(A[k,i])
if scale != 0:
for k in xrange(i, m):
A[k,i] /= scale
s += A[k,i] * A[k,i]
f = A[i,i]
g = -ctx.sqrt(s)
if f < 0:
g = -g
h = f * g - s
A[i,i] = f - g
for j in xrange(i+1, n):
s = 0
for k in xrange(i, m):
s += A[k,i] * A[k,j]
f = s / h
for k in xrange(i, m):
A[k,j] += f * A[k,i]
for k in xrange(i,m):
A[k,i] *= scale
S[i] = scale * g
g = s = scale = 0
if i < m and i != n - 1:
for k in xrange(i+1, n):
scale += ctx.fabs(A[i,k])
if scale:
for k in xrange(i+1, n):
A[i,k] /= scale
s += A[i,k] * A[i,k]
f = A[i,i+1]
g = -ctx.sqrt(s)
if f < 0:
g = -g
h = f * g - s
A[i,i+1] = f - g
for k in xrange(i+1, n):
work[k] = A[i,k] / h
for j in xrange(i+1, m):
s = 0
for k in xrange(i+1, n):
s += A[j,k] * A[i,k]
for k in xrange(i+1, n):
A[j,k] += s * work[k]
for k in xrange(i+1, n):
A[i,k] *= scale
anorm = max(anorm, ctx.fabs(S[i]) + ctx.fabs(work[i]))
if not isinstance(V, bool):
for i in xrange(n-2, -1, -1): # accumulation of right hand transformations
V[i+1,i+1] = 1
if work[i+1] != 0:
for j in xrange(i+1, n):
V[i,j] = (A[i,j] / A[i,i+1]) / work[i+1]
for j in xrange(i+1, n):
s = 0
for k in xrange(i+1, n):
s += A[i,k] * V[j,k]
for k in xrange(i+1, n):
V[j,k] += s * V[i,k]
for j in xrange(i+1, n):
V[j,i] = V[i,j] = 0
V[0,0] = 1
if m<n : minnm = m
else : minnm = n
if calc_u:
for i in xrange(minnm-1, -1, -1): # accumulation of left hand transformations
g = S[i]
for j in xrange(i+1, n):
A[i,j] = 0
if g != 0:
g = 1 / g
for j in xrange(i+1, n):
s = 0
for k in xrange(i+1, m):
s += A[k,i] * A[k,j]
f = (s / A[i,i]) * g
for k in xrange(i, m):
A[k,j] += f * A[k,i]
for j in xrange(i, m):
A[j,i] *= g
else:
for j in xrange(i, m):
A[j,i] = 0
A[i,i] += 1
for k in xrange(n - 1, -1, -1):
# diagonalization of the bidiagonal form:
# loop over singular values, and over allowed itations
its = 0
while 1:
its += 1
flag = True
for l in xrange(k, -1, -1):
nm = l-1
if ctx.fabs(work[l]) + anorm == anorm:
flag = False
break
if ctx.fabs(S[nm]) + anorm == anorm:
break
if flag:
c = 0
s = 1
for i in xrange(l, k + 1):
f = s * work[i]
work[i] *= c
if ctx.fabs(f) + anorm == anorm:
break
g = S[i]
h = ctx.hypot(f, g)
S[i] = h
h = 1 / h
c = g * h
s = - f * h
if calc_u:
for j in xrange(m):
y = A[j,nm]
z = A[j,i]
A[j,nm] = y * c + z * s
A[j,i] = z * c - y * s
z = S[k]
if l == k: # convergence
if z < 0: # singular value is made nonnegative
S[k] = -z
if not isinstance(V, bool):
for j in xrange(n):
V[k,j] = -V[k,j]
break
if its >= maxits:
raise RuntimeError("svd: no convergence to an eigenvalue after %d iterations" % its)
x = S[l] # shift from bottom 2 by 2 minor
nm = k-1
y = S[nm]
g = work[nm]
h = work[k]
f = ((y - z) * (y + z) + (g - h) * (g + h))/(2 * h * y)
g = ctx.hypot(f, 1)
if f >= 0: f = ((x - z) * (x + z) + h * ((y / (f + g)) - h)) / x
else: f = ((x - z) * (x + z) + h * ((y / (f - g)) - h)) / x
c = s = 1 # next qt transformation
for j in xrange(l, nm + 1):
g = work[j+1]
y = S[j+1]
h = s * g
g = c * g
z = ctx.hypot(f, h)
work[j] = z
c = f / z
s = h / z
f = x * c + g * s
g = g * c - x * s
h = y * s
y *= c
if not isinstance(V, bool):
for jj in xrange(n):
x = V[j ,jj]
z = V[j+1,jj]
V[j ,jj]= x * c + z * s
V[j+1 ,jj]= z * c - x * s
z = ctx.hypot(f, h)
S[j] = z
if z != 0: # rotation can be arbitray if z=0
z = 1 / z
c = f * z
s = h * z
f = c * g + s * y
x = c * y - s * g
if calc_u:
for jj in xrange(m):
y = A[jj,j ]
z = A[jj,j+1]
A[jj,j ] = y * c + z * s
A[jj,j+1 ] = z * c - y * s
work[l] = 0
work[k] = f
S[k] = x
##########################
# Sort singular values into decreasing order (bubble-sort)
for i in xrange(n):
imax = i
s = ctx.fabs(S[i]) # s is the current maximal element
for j in xrange(i + 1, n):
c = ctx.fabs(S[j])
if c > s:
s = c
imax = j
if imax != i:
# swap singular values
z = S[i]
S[i] = S[imax]
S[imax] = z
if calc_u:
for j in xrange(m):
z = A[j,i]
A[j,i] = A[j,imax]
A[j,imax] = z
if not isinstance(V, bool):
for j in xrange(n):
z = V[i,j]
V[i,j] = V[imax,j]
V[imax,j] = z
return S
#######################
def svd_c_raw(ctx, A, V = False, calc_u = False):
"""
This routine computes the singular value decomposition of a matrix A.
Given A, two unitary matrices U and V are calculated such that
A = U S V
where S is a suitable shaped matrix whose off-diagonal elements are zero.
The diagonal elements of S are the singular values of A, i.e. the
squareroots of the eigenvalues of A' A or A A'. Here ' denotes the hermitian
transpose (i.e. transposition and conjugation). Householder bidiagonalization
and a variant of the QR algorithm is used.
overview of the matrices :
A : m*n A gets replaced by U
U : m*n U replaces A. If n>m then only the first m*m block of U is
non-zero. column-unitary: U' U = B
here B is a n*n matrix whose first min(m,n) diagonal
elements are 1 and all other elements are zero.
S : n*n diagonal matrix, only the diagonal elements are stored in
the array S. only the first min(m,n) diagonal elements are non-zero.
V : n*n unitary: V V' = V' V = 1
parameters:
A (input/output) On input, A contains a complex matrix of shape m*n.
On output, if calc_u is true A contains the column-unitary
matrix U; otherwise A is simply used as workspace and thus destroyed.
V (input/output) if false, the matrix V is not calculated. otherwise
V must be a matrix of shape n*n.
calc_u (input) If true, the matrix U is calculated and replaces A.
if false, U is not calculated and A is simply destroyed
return value:
S an array of length n containing the singular values of A sorted by
decreasing magnitude. only the first min(m,n) elements are non-zero.
This routine is a python translation of the fortran routine svd.f in the
software library EISPACK (see netlib.org) which itself is based on the
algol procedure svd described in:
- num. math. 14, 403-420(1970) by golub and reinsch.
- wilkinson/reinsch: handbook for auto. comp., vol ii-linear algebra, 134-151(1971).
"""
m, n = A.rows, A.cols
S = ctx.zeros(n, 1)
# work is a temporary array of size n
work = ctx.zeros(n, 1)
lbeta = ctx.zeros(n, 1)
rbeta = ctx.zeros(n, 1)
dwork = ctx.zeros(n, 1)
g = scale = anorm = 0
maxits = 3 * ctx.dps
for i in xrange(n): # householder reduction to bidiagonal form
dwork[i] = scale * g # dwork are the side-diagonal elements
g = s = scale = 0
if i < m:
for k in xrange(i, m):
scale += ctx.fabs(ctx.re(A[k,i])) + ctx.fabs(ctx.im(A[k,i]))
if scale != 0:
for k in xrange(i, m):
A[k,i] /= scale
ar = ctx.re(A[k,i])
ai = ctx.im(A[k,i])
s += ar * ar + ai * ai
f = A[i,i]
g = -ctx.sqrt(s)
if ctx.re(f) < 0:
beta = -g - ctx.conj(f)
g = -g
else:
beta = -g + ctx.conj(f)
beta /= ctx.conj(beta)
beta += 1
h = 2 * (ctx.re(f) * g - s)
A[i,i] = f - g
beta /= h
lbeta[i] = (beta / scale) / scale
for j in xrange(i+1, n):
s = 0
for k in xrange(i, m):
s += ctx.conj(A[k,i]) * A[k,j]
f = beta * s
for k in xrange(i, m):
A[k,j] += f * A[k,i]
for k in xrange(i, m):
A[k,i] *= scale
S[i] = scale * g # S are the diagonal elements
g = s = scale = 0
if i < m and i != n - 1:
for k in xrange(i+1, n):
scale += ctx.fabs(ctx.re(A[i,k])) + ctx.fabs(ctx.im(A[i,k]))
if scale:
for k in xrange(i+1, n):
A[i,k] /= scale
ar = ctx.re(A[i,k])
ai = ctx.im(A[i,k])
s += ar * ar + ai * ai
f = A[i,i+1]
g = -ctx.sqrt(s)
if ctx.re(f) < 0:
beta = -g - ctx.conj(f)
g = -g
else:
beta = -g + ctx.conj(f)
beta /= ctx.conj(beta)
beta += 1
h = 2 * (ctx.re(f) * g - s)
A[i,i+1] = f - g
beta /= h
rbeta[i] = (beta / scale) / scale
for k in xrange(i+1, n):
work[k] = A[i, k]
for j in xrange(i+1, m):
s = 0
for k in xrange(i+1, n):
s += ctx.conj(A[i,k]) * A[j,k]
f = s * beta
for k in xrange(i+1,n):
A[j,k] += f * work[k]
for k in xrange(i+1, n):
A[i,k] *= scale
anorm = max(anorm,ctx.fabs(S[i]) + ctx.fabs(dwork[i]))
if not isinstance(V, bool):
for i in xrange(n-2, -1, -1): # accumulation of right hand transformations
V[i+1,i+1] = 1
if dwork[i+1] != 0:
f = ctx.conj(rbeta[i])
for j in xrange(i+1, n):
V[i,j] = A[i,j] * f
for j in xrange(i+1, n):
s = 0
for k in xrange(i+1, n):
s += ctx.conj(A[i,k]) * V[j,k]
for k in xrange(i+1, n):
V[j,k] += s * V[i,k]
for j in xrange(i+1,n):
V[j,i] = V[i,j] = 0
V[0,0] = 1
if m < n : minnm = m
else : minnm = n
if calc_u:
for i in xrange(minnm-1, -1, -1): # accumulation of left hand transformations
g = S[i]
for j in xrange(i+1, n):
A[i,j] = 0
if g != 0:
g = 1 / g
for j in xrange(i+1, n):
s = 0
for k in xrange(i+1, m):
s += ctx.conj(A[k,i]) * A[k,j]
f = s * ctx.conj(lbeta[i])
for k in xrange(i, m):
A[k,j] += f * A[k,i]
for j in xrange(i, m):
A[j,i] *= g
else:
for j in xrange(i, m):
A[j,i] = 0
A[i,i] += 1
for k in xrange(n-1, -1, -1):
# diagonalization of the bidiagonal form:
# loop over singular values, and over allowed itations
its = 0
while 1:
its += 1
flag = True
for l in xrange(k, -1, -1):
nm = l - 1
if ctx.fabs(dwork[l]) + anorm == anorm:
flag = False
break
if ctx.fabs(S[nm]) + anorm == anorm:
break
if flag:
c = 0
s = 1
for i in xrange(l, k+1):
f = s * dwork[i]
dwork[i] *= c
if ctx.fabs(f) + anorm == anorm:
break
g = S[i]
h = ctx.hypot(f, g)
S[i] = h
h = 1 / h
c = g * h
s = -f * h
if calc_u:
for j in xrange(m):
y = A[j,nm]
z = A[j,i]
A[j,nm]= y * c + z * s
A[j,i] = z * c - y * s
z = S[k]
if l == k: # convergence
if z < 0: # singular value is made nonnegative
S[k] = -z
if not isinstance(V, bool):
for j in xrange(n):
V[k,j] = -V[k,j]
break
if its >= maxits:
raise RuntimeError("svd: no convergence to an eigenvalue after %d iterations" % its)
x = S[l] # shift from bottom 2 by 2 minor
nm = k-1
y = S[nm]
g = dwork[nm]
h = dwork[k]
f = ((y - z) * (y + z) + (g - h) * (g + h)) / (2 * h * y)
g = ctx.hypot(f, 1)
if f >=0: f = (( x - z) *( x + z) + h *((y / (f + g)) - h)) / x
else: f = (( x - z) *( x + z) + h *((y / (f - g)) - h)) / x
c = s = 1 # next qt transformation
for j in xrange(l, nm + 1):
g = dwork[j+1]
y = S[j+1]
h = s * g
g = c * g
z = ctx.hypot(f, h)
dwork[j] = z
c = f / z
s = h / z
f = x * c + g * s
g = g * c - x * s
h = y * s
y *= c
if not isinstance(V, bool):
for jj in xrange(n):
x = V[j ,jj]
z = V[j+1,jj]
V[j ,jj]= x * c + z * s
V[j+1,jj ]= z * c - x * s
z = ctx.hypot(f, h)
S[j] = z
if z != 0: # rotation can be arbitray if z=0
z = 1 / z
c = f * z
s = h * z
f = c * g + s * y
x = c * y - s * g
if calc_u:
for jj in xrange(m):
y = A[jj,j ]
z = A[jj,j+1]
A[jj,j ]= y * c + z * s
A[jj,j+1 ]= z * c - y * s
dwork[l] = 0
dwork[k] = f
S[k] = x
##########################
# Sort singular values into decreasing order (bubble-sort)
for i in xrange(n):
imax = i
s = ctx.fabs(S[i]) # s is the current maximal element
for j in xrange(i + 1, n):
c = ctx.fabs(S[j])
if c > s:
s = c
imax = j
if imax != i:
# swap singular values
z = S[i]
S[i] = S[imax]
S[imax] = z
if calc_u:
for j in xrange(m):
z = A[j,i]
A[j,i] = A[j,imax]
A[j,imax] = z
if not isinstance(V, bool):
for j in xrange(n):
z = V[i,j]
V[i,j] = V[imax,j]
V[imax,j] = z
return S
##################################################################################################
@defun
def svd_r(ctx, A, full_matrices = False, compute_uv = True, overwrite_a = False):
"""
This routine computes the singular value decomposition of a matrix A.
Given A, two orthogonal matrices U and V are calculated such that
A = U S V and U' U = 1 and V V' = 1
where S is a suitable shaped matrix whose off-diagonal elements are zero.
Here ' denotes the transpose. The diagonal elements of S are the singular
values of A, i.e. the squareroots of the eigenvalues of A' A or A A'.
input:
A : a real matrix of shape (m, n)
full_matrices : if true, U and V are of shape (m, m) and (n, n).
if false, U and V are of shape (m, min(m, n)) and (min(m, n), n).
compute_uv : if true, U and V are calculated. if false, only S is calculated.
overwrite_a : if true, allows modification of A which may improve
performance. if false, A is not modified.
output:
U : an orthogonal matrix: U' U = 1. if full_matrices is true, U is of
shape (m, m). ortherwise it is of shape (m, min(m, n)).
S : an array of length min(m, n) containing the singular values of A sorted by
decreasing magnitude.
V : an orthogonal matrix: V V' = 1. if full_matrices is true, V is of
shape (n, n). ortherwise it is of shape (min(m, n), n).
return value:
S if compute_uv is false
(U, S, V) if compute_uv is true
overview of the matrices:
full_matrices true:
A : m*n
U : m*m U' U = 1
S as matrix : m*n
V : n*n V V' = 1
full_matrices false:
A : m*n
U : m*min(n,m) U' U = 1
S as matrix : min(m,n)*min(m,n)
V : min(m,n)*n V V' = 1
examples:
>>> from mpmath import mp
>>> A = mp.matrix([[2, -2, -1], [3, 4, -2], [-2, -2, 0]])
>>> S = mp.svd_r(A, compute_uv = False)
>>> print(S)
[6.0]
[3.0]
[1.0]
>>> U, S, V = mp.svd_r(A)
>>> print(mp.chop(A - U * mp.diag(S) * V))
[0.0 0.0 0.0]
[0.0 0.0 0.0]
[0.0 0.0 0.0]
see also: svd, svd_c
"""
m, n = A.rows, A.cols
if not compute_uv:
if not overwrite_a:
A = A.copy()
S = svd_r_raw(ctx, A, V = False, calc_u = False)
S = S[:min(m,n)]
return S
if full_matrices and n < m:
V = ctx.zeros(m, m)
A0 = ctx.zeros(m, m)
A0[:,:n] = A
S = svd_r_raw(ctx, A0, V, calc_u = True)
S = S[:n]
V = V[:n,:n]
return (A0, S, V)
else:
if not overwrite_a:
A = A.copy()
V = ctx.zeros(n, n)
S = svd_r_raw(ctx, A, V, calc_u = True)
if n > m:
if full_matrices == False:
V = V[:m,:]
S = S[:m]
A = A[:,:m]
return (A, S, V)
##############################
@defun
def svd_c(ctx, A, full_matrices = False, compute_uv = True, overwrite_a = False):
"""
This routine computes the singular value decomposition of a matrix A.
Given A, two unitary matrices U and V are calculated such that
A = U S V and U' U = 1 and V V' = 1
where S is a suitable shaped matrix whose off-diagonal elements are zero.
Here ' denotes the hermitian transpose (i.e. transposition and complex
conjugation). The diagonal elements of S are the singular values of A,
i.e. the squareroots of the eigenvalues of A' A or A A'.
input:
A : a complex matrix of shape (m, n)
full_matrices : if true, U and V are of shape (m, m) and (n, n).
if false, U and V are of shape (m, min(m, n)) and (min(m, n), n).
compute_uv : if true, U and V are calculated. if false, only S is calculated.
overwrite_a : if true, allows modification of A which may improve
performance. if false, A is not modified.
output:
U : an unitary matrix: U' U = 1. if full_matrices is true, U is of
shape (m, m). ortherwise it is of shape (m, min(m, n)).
S : an array of length min(m, n) containing the singular values of A sorted by
decreasing magnitude.
V : an unitary matrix: V V' = 1. if full_matrices is true, V is of
shape (n, n). ortherwise it is of shape (min(m, n), n).
return value:
S if compute_uv is false
(U, S, V) if compute_uv is true
overview of the matrices:
full_matrices true:
A : m*n
U : m*m U' U = 1
S as matrix : m*n
V : n*n V V' = 1
full_matrices false:
A : m*n
U : m*min(n,m) U' U = 1
S as matrix : min(m,n)*min(m,n)
V : min(m,n)*n V V' = 1
example:
>>> from mpmath import mp
>>> A = mp.matrix([[-2j, -1-3j, -2+2j], [2-2j, -1-3j, 1], [-3+1j,-2j,0]])
>>> S = mp.svd_c(A, compute_uv = False)
>>> print(mp.chop(S - mp.matrix([mp.sqrt(34), mp.sqrt(15), mp.sqrt(6)])))
[0.0]
[0.0]
[0.0]
>>> U, S, V = mp.svd_c(A)
>>> print(mp.chop(A - U * mp.diag(S) * V))
[0.0 0.0 0.0]
[0.0 0.0 0.0]
[0.0 0.0 0.0]
see also: svd, svd_r
"""
m, n = A.rows, A.cols
if not compute_uv:
if not overwrite_a:
A = A.copy()
S = svd_c_raw(ctx, A, V = False, calc_u = False)
S = S[:min(m,n)]
return S
if full_matrices and n < m:
V = ctx.zeros(m, m)
A0 = ctx.zeros(m, m)
A0[:,:n] = A
S = svd_c_raw(ctx, A0, V, calc_u = True)
S = S[:n]
V = V[:n,:n]
return (A0, S, V)
else:
if not overwrite_a:
A = A.copy()
V = ctx.zeros(n, n)
S = svd_c_raw(ctx, A, V, calc_u = True)
if n > m:
if full_matrices == False:
V = V[:m,:]
S = S[:m]
A = A[:,:m]
return (A, S, V)
@defun
def svd(ctx, A, full_matrices = False, compute_uv = True, overwrite_a = False):
"""
"svd" is a unified interface for "svd_r" and "svd_c". Depending on
whether A is real or complex the appropriate function is called.
This routine computes the singular value decomposition of a matrix A.
Given A, two orthogonal (A real) or unitary (A complex) matrices U and V
are calculated such that
A = U S V and U' U = 1 and V V' = 1
where S is a suitable shaped matrix whose off-diagonal elements are zero.
Here ' denotes the hermitian transpose (i.e. transposition and complex
conjugation). The diagonal elements of S are the singular values of A,
i.e. the squareroots of the eigenvalues of A' A or A A'.
input:
A : a real or complex matrix of shape (m, n)
full_matrices : if true, U and V are of shape (m, m) and (n, n).
if false, U and V are of shape (m, min(m, n)) and (min(m, n), n).
compute_uv : if true, U and V are calculated. if false, only S is calculated.
overwrite_a : if true, allows modification of A which may improve
performance. if false, A is not modified.
output:
U : an orthogonal or unitary matrix: U' U = 1. if full_matrices is true, U is of
shape (m, m). ortherwise it is of shape (m, min(m, n)).
S : an array of length min(m, n) containing the singular values of A sorted by
decreasing magnitude.
V : an orthogonal or unitary matrix: V V' = 1. if full_matrices is true, V is of
shape (n, n). ortherwise it is of shape (min(m, n), n).
return value:
S if compute_uv is false
(U, S, V) if compute_uv is true
overview of the matrices:
full_matrices true:
A : m*n
U : m*m U' U = 1
S as matrix : m*n
V : n*n V V' = 1
full_matrices false:
A : m*n
U : m*min(n,m) U' U = 1
S as matrix : min(m,n)*min(m,n)
V : min(m,n)*n V V' = 1
examples:
>>> from mpmath import mp
>>> A = mp.matrix([[2, -2, -1], [3, 4, -2], [-2, -2, 0]])
>>> S = mp.svd(A, compute_uv = False)
>>> print(S)
[6.0]
[3.0]
[1.0]
>>> U, S, V = mp.svd(A)
>>> print(mp.chop(A - U * mp.diag(S) * V))
[0.0 0.0 0.0]
[0.0 0.0 0.0]
[0.0 0.0 0.0]
see also: svd_r, svd_c
"""
iscomplex = any(type(x) is ctx.mpc for x in A)
if iscomplex:
return ctx.svd_c(A, full_matrices = full_matrices, compute_uv = compute_uv, overwrite_a = overwrite_a)
else:
return ctx.svd_r(A, full_matrices = full_matrices, compute_uv = compute_uv, overwrite_a = overwrite_a)
| 58,524 | 31.370022 | 127 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/matrices/__init__.py
|
from . import eigen # to set methods
from . import eigen_symmetric # to set methods
| 94 | 30.666667 | 46 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/calculus/optimization.py
|
from copy import copy
from ..libmp.backend import xrange, print_
class OptimizationMethods(object):
def __init__(ctx):
pass
##############
# 1D-SOLVERS #
##############
class Newton:
"""
1d-solver generating pairs of approximative root and error.
Needs starting points x0 close to the root.
Pro:
* converges fast
* sometimes more robust than secant with bad second starting point
Contra:
* converges slowly for multiple roots
* needs first derivative
* 2 function evaluations per iteration
"""
maxsteps = 20
def __init__(self, ctx, f, x0, **kwargs):
self.ctx = ctx
if len(x0) == 1:
self.x0 = x0[0]
else:
raise ValueError('expected 1 starting point, got %i' % len(x0))
self.f = f
if not 'df' in kwargs:
def df(x):
return self.ctx.diff(f, x)
else:
df = kwargs['df']
self.df = df
def __iter__(self):
f = self.f
df = self.df
x0 = self.x0
while True:
x1 = x0 - f(x0) / df(x0)
error = abs(x1 - x0)
x0 = x1
yield (x1, error)
class Secant:
"""
1d-solver generating pairs of approximative root and error.
Needs starting points x0 and x1 close to the root.
x1 defaults to x0 + 0.25.
Pro:
* converges fast
Contra:
* converges slowly for multiple roots
"""
maxsteps = 30
def __init__(self, ctx, f, x0, **kwargs):
self.ctx = ctx
if len(x0) == 1:
self.x0 = x0[0]
self.x1 = self.x0 + 0.25
elif len(x0) == 2:
self.x0 = x0[0]
self.x1 = x0[1]
else:
raise ValueError('expected 1 or 2 starting points, got %i' % len(x0))
self.f = f
def __iter__(self):
f = self.f
x0 = self.x0
x1 = self.x1
f0 = f(x0)
while True:
f1 = f(x1)
l = x1 - x0
if not l:
break
s = (f1 - f0) / l
if not s:
break
x0, x1 = x1, x1 - f1/s
f0 = f1
yield x1, abs(l)
class MNewton:
"""
1d-solver generating pairs of approximative root and error.
Needs starting point x0 close to the root.
Uses modified Newton's method that converges fast regardless of the
multiplicity of the root.
Pro:
* converges fast for multiple roots
Contra:
* needs first and second derivative of f
* 3 function evaluations per iteration
"""
maxsteps = 20
def __init__(self, ctx, f, x0, **kwargs):
self.ctx = ctx
if not len(x0) == 1:
raise ValueError('expected 1 starting point, got %i' % len(x0))
self.x0 = x0[0]
self.f = f
if not 'df' in kwargs:
def df(x):
return self.ctx.diff(f, x)
else:
df = kwargs['df']
self.df = df
if not 'd2f' in kwargs:
def d2f(x):
return self.ctx.diff(df, x)
else:
d2f = kwargs['df']
self.d2f = d2f
def __iter__(self):
x = self.x0
f = self.f
df = self.df
d2f = self.d2f
while True:
prevx = x
fx = f(x)
if fx == 0:
break
dfx = df(x)
d2fx = d2f(x)
# x = x - F(x)/F'(x) with F(x) = f(x)/f'(x)
x -= fx / (dfx - fx * d2fx / dfx)
error = abs(x - prevx)
yield x, error
class Halley:
"""
1d-solver generating pairs of approximative root and error.
Needs a starting point x0 close to the root.
Uses Halley's method with cubic convergence rate.
Pro:
* converges even faster the Newton's method
* useful when computing with *many* digits
Contra:
* needs first and second derivative of f
* 3 function evaluations per iteration
* converges slowly for multiple roots
"""
maxsteps = 20
def __init__(self, ctx, f, x0, **kwargs):
self.ctx = ctx
if not len(x0) == 1:
raise ValueError('expected 1 starting point, got %i' % len(x0))
self.x0 = x0[0]
self.f = f
if not 'df' in kwargs:
def df(x):
return self.ctx.diff(f, x)
else:
df = kwargs['df']
self.df = df
if not 'd2f' in kwargs:
def d2f(x):
return self.ctx.diff(df, x)
else:
d2f = kwargs['df']
self.d2f = d2f
def __iter__(self):
x = self.x0
f = self.f
df = self.df
d2f = self.d2f
while True:
prevx = x
fx = f(x)
dfx = df(x)
d2fx = d2f(x)
x -= 2*fx*dfx / (2*dfx**2 - fx*d2fx)
error = abs(x - prevx)
yield x, error
class Muller:
"""
1d-solver generating pairs of approximative root and error.
Needs starting points x0, x1 and x2 close to the root.
x1 defaults to x0 + 0.25; x2 to x1 + 0.25.
Uses Muller's method that converges towards complex roots.
Pro:
* converges fast (somewhat faster than secant)
* can find complex roots
Contra:
* converges slowly for multiple roots
* may have complex values for real starting points and real roots
http://en.wikipedia.org/wiki/Muller's_method
"""
maxsteps = 30
def __init__(self, ctx, f, x0, **kwargs):
self.ctx = ctx
if len(x0) == 1:
self.x0 = x0[0]
self.x1 = self.x0 + 0.25
self.x2 = self.x1 + 0.25
elif len(x0) == 2:
self.x0 = x0[0]
self.x1 = x0[1]
self.x2 = self.x1 + 0.25
elif len(x0) == 3:
self.x0 = x0[0]
self.x1 = x0[1]
self.x2 = x0[2]
else:
raise ValueError('expected 1, 2 or 3 starting points, got %i'
% len(x0))
self.f = f
self.verbose = kwargs['verbose']
def __iter__(self):
f = self.f
x0 = self.x0
x1 = self.x1
x2 = self.x2
fx0 = f(x0)
fx1 = f(x1)
fx2 = f(x2)
while True:
# TODO: maybe refactoring with function for divided differences
# calculate divided differences
fx2x1 = (fx1 - fx2) / (x1 - x2)
fx2x0 = (fx0 - fx2) / (x0 - x2)
fx1x0 = (fx0 - fx1) / (x0 - x1)
w = fx2x1 + fx2x0 - fx1x0
fx2x1x0 = (fx1x0 - fx2x1) / (x0 - x2)
if w == 0 and fx2x1x0 == 0:
if self.verbose:
print_('canceled with')
print_('x0 =', x0, ', x1 =', x1, 'and x2 =', x2)
break
x0 = x1
fx0 = fx1
x1 = x2
fx1 = fx2
# denominator should be as large as possible => choose sign
r = self.ctx.sqrt(w**2 - 4*fx2*fx2x1x0)
if abs(w - r) > abs(w + r):
r = -r
x2 -= 2*fx2 / (w + r)
fx2 = f(x2)
error = abs(x2 - x1)
yield x2, error
# TODO: consider raising a ValueError when there's no sign change in a and b
class Bisection:
"""
1d-solver generating pairs of approximative root and error.
Uses bisection method to find a root of f in [a, b].
Might fail for multiple roots (needs sign change).
Pro:
* robust and reliable
Contra:
* converges slowly
* needs sign change
"""
maxsteps = 100
def __init__(self, ctx, f, x0, **kwargs):
self.ctx = ctx
if len(x0) != 2:
raise ValueError('expected interval of 2 points, got %i' % len(x0))
self.f = f
self.a = x0[0]
self.b = x0[1]
def __iter__(self):
f = self.f
a = self.a
b = self.b
l = b - a
fb = f(b)
while True:
m = self.ctx.ldexp(a + b, -1)
fm = f(m)
sign = fm * fb
if sign < 0:
a = m
elif sign > 0:
b = m
fb = fm
else:
yield m, self.ctx.zero
l /= 2
yield (a + b)/2, abs(l)
def _getm(method):
"""
Return a function to calculate m for Illinois-like methods.
"""
if method == 'illinois':
def getm(fz, fb):
return 0.5
elif method == 'pegasus':
def getm(fz, fb):
return fb/(fb + fz)
elif method == 'anderson':
def getm(fz, fb):
m = 1 - fz/fb
if m > 0:
return m
else:
return 0.5
else:
raise ValueError("method '%s' not recognized" % method)
return getm
class Illinois:
"""
1d-solver generating pairs of approximative root and error.
Uses Illinois method or similar to find a root of f in [a, b].
Might fail for multiple roots (needs sign change).
Combines bisect with secant (improved regula falsi).
The only difference between the methods is the scaling factor m, which is
used to ensure convergence (you can choose one using the 'method' keyword):
Illinois method ('illinois'):
m = 0.5
Pegasus method ('pegasus'):
m = fb/(fb + fz)
Anderson-Bjoerk method ('anderson'):
m = 1 - fz/fb if positive else 0.5
Pro:
* converges very fast
Contra:
* has problems with multiple roots
* needs sign change
"""
maxsteps = 30
def __init__(self, ctx, f, x0, **kwargs):
self.ctx = ctx
if len(x0) != 2:
raise ValueError('expected interval of 2 points, got %i' % len(x0))
self.a = x0[0]
self.b = x0[1]
self.f = f
self.tol = kwargs['tol']
self.verbose = kwargs['verbose']
self.method = kwargs.get('method', 'illinois')
self.getm = _getm(self.method)
if self.verbose:
print_('using %s method' % self.method)
def __iter__(self):
method = self.method
f = self.f
a = self.a
b = self.b
fa = f(a)
fb = f(b)
m = None
while True:
l = b - a
if l == 0:
break
s = (fb - fa) / l
z = a - fa/s
fz = f(z)
if abs(fz) < self.tol:
# TODO: better condition (when f is very flat)
if self.verbose:
print_('canceled with z =', z)
yield z, l
break
if fz * fb < 0: # root in [z, b]
a = b
fa = fb
b = z
fb = fz
else: # root in [a, z]
m = self.getm(fz, fb)
b = z
fb = fz
fa = m*fa # scale down to ensure convergence
if self.verbose and m and not method == 'illinois':
print_('m:', m)
yield (a + b)/2, abs(l)
def Pegasus(*args, **kwargs):
"""
1d-solver generating pairs of approximative root and error.
Uses Pegasus method to find a root of f in [a, b].
Wrapper for illinois to use method='pegasus'.
"""
kwargs['method'] = 'pegasus'
return Illinois(*args, **kwargs)
def Anderson(*args, **kwargs):
"""
1d-solver generating pairs of approximative root and error.
Uses Anderson-Bjoerk method to find a root of f in [a, b].
Wrapper for illinois to use method='pegasus'.
"""
kwargs['method'] = 'anderson'
return Illinois(*args, **kwargs)
# TODO: check whether it's possible to combine it with Illinois stuff
class Ridder:
"""
1d-solver generating pairs of approximative root and error.
Ridders' method to find a root of f in [a, b].
Is told to perform as well as Brent's method while being simpler.
Pro:
* very fast
* simpler than Brent's method
Contra:
* two function evaluations per step
* has problems with multiple roots
* needs sign change
http://en.wikipedia.org/wiki/Ridders'_method
"""
maxsteps = 30
def __init__(self, ctx, f, x0, **kwargs):
self.ctx = ctx
self.f = f
if len(x0) != 2:
raise ValueError('expected interval of 2 points, got %i' % len(x0))
self.x1 = x0[0]
self.x2 = x0[1]
self.verbose = kwargs['verbose']
self.tol = kwargs['tol']
def __iter__(self):
ctx = self.ctx
f = self.f
x1 = self.x1
fx1 = f(x1)
x2 = self.x2
fx2 = f(x2)
while True:
x3 = 0.5*(x1 + x2)
fx3 = f(x3)
x4 = x3 + (x3 - x1) * ctx.sign(fx1 - fx2) * fx3 / ctx.sqrt(fx3**2 - fx1*fx2)
fx4 = f(x4)
if abs(fx4) < self.tol:
# TODO: better condition (when f is very flat)
if self.verbose:
print_('canceled with f(x4) =', fx4)
yield x4, abs(x1 - x2)
break
if fx4 * fx2 < 0: # root in [x4, x2]
x1 = x4
fx1 = fx4
else: # root in [x1, x4]
x2 = x4
fx2 = fx4
error = abs(x1 - x2)
yield (x1 + x2)/2, error
class ANewton:
"""
EXPERIMENTAL 1d-solver generating pairs of approximative root and error.
Uses Newton's method modified to use Steffensens method when convergence is
slow. (I.e. for multiple roots.)
"""
maxsteps = 20
def __init__(self, ctx, f, x0, **kwargs):
self.ctx = ctx
if not len(x0) == 1:
raise ValueError('expected 1 starting point, got %i' % len(x0))
self.x0 = x0[0]
self.f = f
if not 'df' in kwargs:
def df(x):
return self.ctx.diff(f, x)
else:
df = kwargs['df']
self.df = df
def phi(x):
return x - f(x) / df(x)
self.phi = phi
self.verbose = kwargs['verbose']
def __iter__(self):
x0 = self.x0
f = self.f
df = self.df
phi = self.phi
error = 0
counter = 0
while True:
prevx = x0
try:
x0 = phi(x0)
except ZeroDivisionError:
if self.verbose:
print_('ZeroDivisionError: canceled with x =', x0)
break
preverror = error
error = abs(prevx - x0)
# TODO: decide not to use convergence acceleration
if error and abs(error - preverror) / error < 1:
if self.verbose:
print_('converging slowly')
counter += 1
if counter >= 3:
# accelerate convergence
phi = steffensen(phi)
counter = 0
if self.verbose:
print_('accelerating convergence')
yield x0, error
# TODO: add Brent
############################
# MULTIDIMENSIONAL SOLVERS #
############################
def jacobian(ctx, f, x):
"""
Calculate the Jacobian matrix of a function at the point x0.
This is the first derivative of a vectorial function:
f : R^m -> R^n with m >= n
"""
x = ctx.matrix(x)
h = ctx.sqrt(ctx.eps)
fx = ctx.matrix(f(*x))
m = len(fx)
n = len(x)
J = ctx.matrix(m, n)
for j in xrange(n):
xj = x.copy()
xj[j] += h
Jj = (ctx.matrix(f(*xj)) - fx) / h
for i in xrange(m):
J[i,j] = Jj[i]
return J
# TODO: test with user-specified jacobian matrix, support force_type
class MDNewton:
"""
Find the root of a vector function numerically using Newton's method.
f is a vector function representing a nonlinear equation system.
x0 is the starting point close to the root.
J is a function returning the Jacobian matrix for a point.
Supports overdetermined systems.
Use the 'norm' keyword to specify which norm to use. Defaults to max-norm.
The function to calculate the Jacobian matrix can be given using the
keyword 'J'. Otherwise it will be calculated numerically.
Please note that this method converges only locally. Especially for high-
dimensional systems it is not trivial to find a good starting point being
close enough to the root.
It is recommended to use a faster, low-precision solver from SciPy [1] or
OpenOpt [2] to get an initial guess. Afterwards you can use this method for
root-polishing to any precision.
[1] http://scipy.org
[2] http://openopt.org/Welcome
"""
maxsteps = 10
def __init__(self, ctx, f, x0, **kwargs):
self.ctx = ctx
self.f = f
if isinstance(x0, (tuple, list)):
x0 = ctx.matrix(x0)
assert x0.cols == 1, 'need a vector'
self.x0 = x0
if 'J' in kwargs:
self.J = kwargs['J']
else:
def J(*x):
return ctx.jacobian(f, x)
self.J = J
self.norm = kwargs['norm']
self.verbose = kwargs['verbose']
def __iter__(self):
f = self.f
x0 = self.x0
norm = self.norm
J = self.J
fx = self.ctx.matrix(f(*x0))
fxnorm = norm(fx)
cancel = False
while not cancel:
# get direction of descent
fxn = -fx
Jx = J(*x0)
s = self.ctx.lu_solve(Jx, fxn)
if self.verbose:
print_('Jx:')
print_(Jx)
print_('s:', s)
# damping step size TODO: better strategy (hard task)
l = self.ctx.one
x1 = x0 + s
while True:
if x1 == x0:
if self.verbose:
print_("canceled, won't get more excact")
cancel = True
break
fx = self.ctx.matrix(f(*x1))
newnorm = norm(fx)
if newnorm < fxnorm:
# new x accepted
fxnorm = newnorm
x0 = x1
break
l /= 2
x1 = x0 + l*s
yield (x0, fxnorm)
#############
# UTILITIES #
#############
str2solver = {'newton':Newton, 'secant':Secant, 'mnewton':MNewton,
'halley':Halley, 'muller':Muller, 'bisect':Bisection,
'illinois':Illinois, 'pegasus':Pegasus, 'anderson':Anderson,
'ridder':Ridder, 'anewton':ANewton, 'mdnewton':MDNewton}
def findroot(ctx, f, x0, solver='secant', tol=None, verbose=False, verify=True, **kwargs):
r"""
Find a solution to `f(x) = 0`, using *x0* as starting point or
interval for *x*.
Multidimensional overdetermined systems are supported.
You can specify them using a function or a list of functions.
If the found root does not satisfy `|f(x)|^2 \leq \mathrm{tol}`,
an exception is raised (this can be disabled with *verify=False*).
**Arguments**
*f*
one dimensional function
*x0*
starting point, several starting points or interval (depends on solver)
*tol*
the returned solution has an error smaller than this
*verbose*
print additional information for each iteration if true
*verify*
verify the solution and raise a ValueError if `|f(x)|^2 > \mathrm{tol}`
*solver*
a generator for *f* and *x0* returning approximative solution and error
*maxsteps*
after how many steps the solver will cancel
*df*
first derivative of *f* (used by some solvers)
*d2f*
second derivative of *f* (used by some solvers)
*multidimensional*
force multidimensional solving
*J*
Jacobian matrix of *f* (used by multidimensional solvers)
*norm*
used vector norm (used by multidimensional solvers)
solver has to be callable with ``(f, x0, **kwargs)`` and return an generator
yielding pairs of approximative solution and estimated error (which is
expected to be positive).
You can use the following string aliases:
'secant', 'mnewton', 'halley', 'muller', 'illinois', 'pegasus', 'anderson',
'ridder', 'anewton', 'bisect'
See mpmath.calculus.optimization for their documentation.
**Examples**
The function :func:`~mpmath.findroot` locates a root of a given function using the
secant method by default. A simple example use of the secant method is to
compute `\pi` as the root of `\sin x` closest to `x_0 = 3`::
>>> from mpmath import *
>>> mp.dps = 30; mp.pretty = True
>>> findroot(sin, 3)
3.14159265358979323846264338328
The secant method can be used to find complex roots of analytic functions,
although it must in that case generally be given a nonreal starting value
(or else it will never leave the real line)::
>>> mp.dps = 15
>>> findroot(lambda x: x**3 + 2*x + 1, j)
(0.226698825758202 + 1.46771150871022j)
A nice application is to compute nontrivial roots of the Riemann zeta
function with many digits (good initial values are needed for convergence)::
>>> mp.dps = 30
>>> findroot(zeta, 0.5+14j)
(0.5 + 14.1347251417346937904572519836j)
The secant method can also be used as an optimization algorithm, by passing
it a derivative of a function. The following example locates the positive
minimum of the gamma function::
>>> mp.dps = 20
>>> findroot(lambda x: diff(gamma, x), 1)
1.4616321449683623413
Finally, a useful application is to compute inverse functions, such as the
Lambert W function which is the inverse of `w e^w`, given the first
term of the solution's asymptotic expansion as the initial value. In basic
cases, this gives identical results to mpmath's built-in ``lambertw``
function::
>>> def lambert(x):
... return findroot(lambda w: w*exp(w) - x, log(1+x))
...
>>> mp.dps = 15
>>> lambert(1); lambertw(1)
0.567143290409784
0.567143290409784
>>> lambert(1000); lambert(1000)
5.2496028524016
5.2496028524016
Multidimensional functions are also supported::
>>> f = [lambda x1, x2: x1**2 + x2,
... lambda x1, x2: 5*x1**2 - 3*x1 + 2*x2 - 3]
>>> findroot(f, (0, 0))
[-0.618033988749895]
[-0.381966011250105]
>>> findroot(f, (10, 10))
[ 1.61803398874989]
[-2.61803398874989]
You can verify this by solving the system manually.
Please note that the following (more general) syntax also works::
>>> def f(x1, x2):
... return x1**2 + x2, 5*x1**2 - 3*x1 + 2*x2 - 3
...
>>> findroot(f, (0, 0))
[-0.618033988749895]
[-0.381966011250105]
**Multiple roots**
For multiple roots all methods of the Newtonian family (including secant)
converge slowly. Consider this example::
>>> f = lambda x: (x - 1)**99
>>> findroot(f, 0.9, verify=False)
0.918073542444929
Even for a very close starting point the secant method converges very
slowly. Use ``verbose=True`` to illustrate this.
It is possible to modify Newton's method to make it converge regardless of
the root's multiplicity::
>>> findroot(f, -10, solver='mnewton')
1.0
This variant uses the first and second derivative of the function, which is
not very efficient.
Alternatively you can use an experimental Newtonian solver that keeps track
of the speed of convergence and accelerates it using Steffensen's method if
necessary::
>>> findroot(f, -10, solver='anewton', verbose=True)
x: -9.88888888888888888889
error: 0.111111111111111111111
converging slowly
x: -9.77890011223344556678
error: 0.10998877665544332211
converging slowly
x: -9.67002233332199662166
error: 0.108877778911448945119
converging slowly
accelerating convergence
x: -9.5622443299551077669
error: 0.107778003366888854764
converging slowly
x: 0.99999999999999999214
error: 10.562244329955107759
x: 1.0
error: 7.8598304758094664213e-18
ZeroDivisionError: canceled with x = 1.0
1.0
**Complex roots**
For complex roots it's recommended to use Muller's method as it converges
even for real starting points very fast::
>>> findroot(lambda x: x**4 + x + 1, (0, 1, 2), solver='muller')
(0.727136084491197 + 0.934099289460529j)
**Intersection methods**
When you need to find a root in a known interval, it's highly recommended to
use an intersection-based solver like ``'anderson'`` or ``'ridder'``.
Usually they converge faster and more reliable. They have however problems
with multiple roots and usually need a sign change to find a root::
>>> findroot(lambda x: x**3, (-1, 1), solver='anderson')
0.0
Be careful with symmetric functions::
>>> findroot(lambda x: x**2, (-1, 1), solver='anderson') #doctest:+ELLIPSIS
Traceback (most recent call last):
...
ZeroDivisionError
It fails even for better starting points, because there is no sign change::
>>> findroot(lambda x: x**2, (-1, .5), solver='anderson')
Traceback (most recent call last):
...
ValueError: Could not find root within given tolerance. (1.0 > 2.16840434497100886801e-19)
Try another starting point or tweak arguments.
"""
prec = ctx.prec
try:
ctx.prec += 20
# initialize arguments
if tol is None:
tol = ctx.eps * 2**10
kwargs['verbose'] = kwargs.get('verbose', verbose)
if 'd1f' in kwargs:
kwargs['df'] = kwargs['d1f']
kwargs['tol'] = tol
if isinstance(x0, (list, tuple)):
x0 = [ctx.convert(x) for x in x0]
else:
x0 = [ctx.convert(x0)]
if isinstance(solver, str):
try:
solver = str2solver[solver]
except KeyError:
raise ValueError('could not recognize solver')
# accept list of functions
if isinstance(f, (list, tuple)):
f2 = copy(f)
def tmp(*args):
return [fn(*args) for fn in f2]
f = tmp
# detect multidimensional functions
try:
fx = f(*x0)
multidimensional = isinstance(fx, (list, tuple, ctx.matrix))
except TypeError:
fx = f(x0[0])
multidimensional = False
if 'multidimensional' in kwargs:
multidimensional = kwargs['multidimensional']
if multidimensional:
# only one multidimensional solver available at the moment
solver = MDNewton
if not 'norm' in kwargs:
norm = lambda x: ctx.norm(x, 'inf')
kwargs['norm'] = norm
else:
norm = kwargs['norm']
else:
norm = abs
# happily return starting point if it's a root
if norm(fx) == 0:
if multidimensional:
return ctx.matrix(x0)
else:
return x0[0]
# use solver
iterations = solver(ctx, f, x0, **kwargs)
if 'maxsteps' in kwargs:
maxsteps = kwargs['maxsteps']
else:
maxsteps = iterations.maxsteps
i = 0
for x, error in iterations:
if verbose:
print_('x: ', x)
print_('error:', error)
i += 1
if error < tol * max(1, norm(x)) or i >= maxsteps:
break
if not isinstance(x, (list, tuple, ctx.matrix)):
xl = [x]
else:
xl = x
if verify and norm(f(*xl))**2 > tol: # TODO: better condition?
raise ValueError('Could not find root within given tolerance. '
'(%s > %s)\n'
'Try another starting point or tweak arguments.'
% (norm(f(*xl))**2, tol))
return x
finally:
ctx.prec = prec
def multiplicity(ctx, f, root, tol=None, maxsteps=10, **kwargs):
"""
Return the multiplicity of a given root of f.
Internally, numerical derivatives are used. This might be inefficient for
higher order derviatives. Due to this, ``multiplicity`` cancels after
evaluating 10 derivatives by default. You can be specify the n-th derivative
using the dnf keyword.
>>> from mpmath import *
>>> multiplicity(lambda x: sin(x) - 1, pi/2)
2
"""
if tol is None:
tol = ctx.eps ** 0.8
kwargs['d0f'] = f
for i in xrange(maxsteps):
dfstr = 'd' + str(i) + 'f'
if dfstr in kwargs:
df = kwargs[dfstr]
else:
df = lambda x: ctx.diff(f, x, i)
if not abs(df(root)) < tol:
break
return i
def steffensen(f):
"""
linear convergent function -> quadratic convergent function
Steffensen's method for quadratic convergence of a linear converging
sequence.
Don not use it for higher rates of convergence.
It may even work for divergent sequences.
Definition:
F(x) = (x*f(f(x)) - f(x)**2) / (f(f(x)) - 2*f(x) + x)
Example
.......
You can use Steffensen's method to accelerate a fixpoint iteration of linear
(or less) convergence.
x* is a fixpoint of the iteration x_{k+1} = phi(x_k) if x* = phi(x*). For
phi(x) = x**2 there are two fixpoints: 0 and 1.
Let's try Steffensen's method:
>>> f = lambda x: x**2
>>> from mpmath.calculus.optimization import steffensen
>>> F = steffensen(f)
>>> for x in [0.5, 0.9, 2.0]:
... fx = Fx = x
... for i in xrange(9):
... try:
... fx = f(fx)
... except OverflowError:
... pass
... try:
... Fx = F(Fx)
... except ZeroDivisionError:
... pass
... print('%20g %20g' % (fx, Fx))
0.25 -0.5
0.0625 0.1
0.00390625 -0.0011236
1.52588e-05 1.41691e-09
2.32831e-10 -2.84465e-27
5.42101e-20 2.30189e-80
2.93874e-39 -1.2197e-239
8.63617e-78 0
7.45834e-155 0
0.81 1.02676
0.6561 1.00134
0.430467 1
0.185302 1
0.0343368 1
0.00117902 1
1.39008e-06 1
1.93233e-12 1
3.73392e-24 1
4 1.6
16 1.2962
256 1.10194
65536 1.01659
4.29497e+09 1.00053
1.84467e+19 1
3.40282e+38 1
1.15792e+77 1
1.34078e+154 1
Unmodified, the iteration converges only towards 0. Modified it converges
not only much faster, it converges even to the repelling fixpoint 1.
"""
def F(x):
fx = f(x)
ffx = f(fx)
return (x*ffx - fx**2) / (ffx - 2*fx + x)
return F
OptimizationMethods.jacobian = jacobian
OptimizationMethods.findroot = findroot
OptimizationMethods.multiplicity = multiplicity
if __name__ == '__main__':
import doctest
doctest.testmod()
| 32,219 | 28.559633 | 98 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/calculus/polynomials.py
|
from ..libmp.backend import xrange
from .calculus import defun
#----------------------------------------------------------------------------#
# Polynomials #
#----------------------------------------------------------------------------#
# XXX: extra precision
@defun
def polyval(ctx, coeffs, x, derivative=False):
r"""
Given coefficients `[c_n, \ldots, c_2, c_1, c_0]` and a number `x`,
:func:`~mpmath.polyval` evaluates the polynomial
.. math ::
P(x) = c_n x^n + \ldots + c_2 x^2 + c_1 x + c_0.
If *derivative=True* is set, :func:`~mpmath.polyval` simultaneously
evaluates `P(x)` with the derivative, `P'(x)`, and returns the
tuple `(P(x), P'(x))`.
>>> from mpmath import *
>>> mp.pretty = True
>>> polyval([3, 0, 2], 0.5)
2.75
>>> polyval([3, 0, 2], 0.5, derivative=True)
(2.75, 3.0)
The coefficients and the evaluation point may be any combination
of real or complex numbers.
"""
if not coeffs:
return ctx.zero
p = ctx.convert(coeffs[0])
q = ctx.zero
for c in coeffs[1:]:
if derivative:
q = p + x*q
p = c + x*p
if derivative:
return p, q
else:
return p
@defun
def polyroots(ctx, coeffs, maxsteps=50, cleanup=True, extraprec=10,
error=False, roots_init=None):
"""
Computes all roots (real or complex) of a given polynomial.
The roots are returned as a sorted list, where real roots appear first
followed by complex conjugate roots as adjacent elements. The polynomial
should be given as a list of coefficients, in the format used by
:func:`~mpmath.polyval`. The leading coefficient must be nonzero.
With *error=True*, :func:`~mpmath.polyroots` returns a tuple *(roots, err)*
where *err* is an estimate of the maximum error among the computed roots.
**Examples**
Finding the three real roots of `x^3 - x^2 - 14x + 24`::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> nprint(polyroots([1,-1,-14,24]), 4)
[-4.0, 2.0, 3.0]
Finding the two complex conjugate roots of `4x^2 + 3x + 2`, with an
error estimate::
>>> roots, err = polyroots([4,3,2], error=True)
>>> for r in roots:
... print(r)
...
(-0.375 + 0.59947894041409j)
(-0.375 - 0.59947894041409j)
>>>
>>> err
2.22044604925031e-16
>>>
>>> polyval([4,3,2], roots[0])
(2.22044604925031e-16 + 0.0j)
>>> polyval([4,3,2], roots[1])
(2.22044604925031e-16 + 0.0j)
The following example computes all the 5th roots of unity; that is,
the roots of `x^5 - 1`::
>>> mp.dps = 20
>>> for r in polyroots([1, 0, 0, 0, 0, -1]):
... print(r)
...
1.0
(-0.8090169943749474241 + 0.58778525229247312917j)
(-0.8090169943749474241 - 0.58778525229247312917j)
(0.3090169943749474241 + 0.95105651629515357212j)
(0.3090169943749474241 - 0.95105651629515357212j)
**Precision and conditioning**
The roots are computed to the current working precision accuracy. If this
accuracy cannot be achieved in `maxsteps` steps, then a `NoConvergence`
exception is raised. The algorithm internally is using the current working
precision extended by `extraprec`. If `NoConvergence` was raised, that is
caused either by not having enough extra precision to achieve convergence
(in which case increasing `extraprec` should fix the problem) or too low
`maxsteps` (in which case increasing `maxsteps` should fix the problem), or
a combination of both.
The user should always do a convergence study with regards to `extraprec`
to ensure accurate results. It is possible to get convergence to a wrong
answer with too low `extraprec`.
Provided there are no repeated roots, :func:`~mpmath.polyroots` can
typically compute all roots of an arbitrary polynomial to high precision::
>>> mp.dps = 60
>>> for r in polyroots([1, 0, -10, 0, 1]):
... print r
...
-3.14626436994197234232913506571557044551247712918732870123249
-0.317837245195782244725757617296174288373133378433432554879127
0.317837245195782244725757617296174288373133378433432554879127
3.14626436994197234232913506571557044551247712918732870123249
>>>
>>> sqrt(3) + sqrt(2)
3.14626436994197234232913506571557044551247712918732870123249
>>> sqrt(3) - sqrt(2)
0.317837245195782244725757617296174288373133378433432554879127
**Algorithm**
:func:`~mpmath.polyroots` implements the Durand-Kerner method [1], which
uses complex arithmetic to locate all roots simultaneously.
The Durand-Kerner method can be viewed as approximately performing
simultaneous Newton iteration for all the roots. In particular,
the convergence to simple roots is quadratic, just like Newton's
method.
Although all roots are internally calculated using complex arithmetic, any
root found to have an imaginary part smaller than the estimated numerical
error is truncated to a real number (small real parts are also chopped).
Real roots are placed first in the returned list, sorted by value. The
remaining complex roots are sorted by their real parts so that conjugate
roots end up next to each other.
**References**
1. http://en.wikipedia.org/wiki/Durand-Kerner_method
"""
if len(coeffs) <= 1:
if not coeffs or not coeffs[0]:
raise ValueError("Input to polyroots must not be the zero polynomial")
# Constant polynomial with no roots
return []
orig = ctx.prec
tol = +ctx.eps
with ctx.extraprec(extraprec):
deg = len(coeffs) - 1
# Must be monic
lead = ctx.convert(coeffs[0])
if lead == 1:
coeffs = [ctx.convert(c) for c in coeffs]
else:
coeffs = [c/lead for c in coeffs]
f = lambda x: ctx.polyval(coeffs, x)
if roots_init is None:
roots = [ctx.mpc((0.4+0.9j)**n) for n in xrange(deg)]
else:
roots = [None]*deg;
deg_init = min(deg, len(roots_init))
roots[:deg_init] = list(roots_init[:deg_init])
roots[deg_init:] = [ctx.mpc((0.4+0.9j)**n) for n
in xrange(deg_init,deg)]
err = [ctx.one for n in xrange(deg)]
# Durand-Kerner iteration until convergence
for step in xrange(maxsteps):
if abs(max(err)) < tol:
break
for i in xrange(deg):
p = roots[i]
x = f(p)
for j in range(deg):
if i != j:
try:
x /= (p-roots[j])
except ZeroDivisionError:
continue
roots[i] = p - x
err[i] = abs(x)
if abs(max(err)) >= tol:
raise ctx.NoConvergence("Didn't converge in maxsteps=%d steps." \
% maxsteps)
# Remove small real or imaginary parts
if cleanup:
for i in xrange(deg):
if abs(roots[i]) < tol:
roots[i] = ctx.zero
elif abs(ctx._im(roots[i])) < tol:
roots[i] = roots[i].real
elif abs(ctx._re(roots[i])) < tol:
roots[i] = roots[i].imag * 1j
roots.sort(key=lambda x: (abs(ctx._im(x)), ctx._re(x)))
if error:
err = max(err)
err = max(err, ctx.ldexp(1, -orig+1))
return [+r for r in roots], +err
else:
return [+r for r in roots]
| 7,854 | 35.877934 | 82 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/calculus/approximation.py
|
from ..libmp.backend import xrange
from .calculus import defun
#----------------------------------------------------------------------------#
# Approximation methods #
#----------------------------------------------------------------------------#
# The Chebyshev approximation formula is given at:
# http://mathworld.wolfram.com/ChebyshevApproximationFormula.html
# The only major changes in the following code is that we return the
# expanded polynomial coefficients instead of Chebyshev coefficients,
# and that we automatically transform [a,b] -> [-1,1] and back
# for convenience.
# Coefficient in Chebyshev approximation
def chebcoeff(ctx,f,a,b,j,N):
s = ctx.mpf(0)
h = ctx.mpf(0.5)
for k in range(1, N+1):
t = ctx.cospi((k-h)/N)
s += f(t*(b-a)*h + (b+a)*h) * ctx.cospi(j*(k-h)/N)
return 2*s/N
# Generate Chebyshev polynomials T_n(ax+b) in expanded form
def chebT(ctx, a=1, b=0):
Tb = [1]
yield Tb
Ta = [b, a]
while 1:
yield Ta
# Recurrence: T[n+1](ax+b) = 2*(ax+b)*T[n](ax+b) - T[n-1](ax+b)
Tmp = [0] + [2*a*t for t in Ta]
for i, c in enumerate(Ta): Tmp[i] += 2*b*c
for i, c in enumerate(Tb): Tmp[i] -= c
Ta, Tb = Tmp, Ta
@defun
def chebyfit(ctx, f, interval, N, error=False):
r"""
Computes a polynomial of degree `N-1` that approximates the
given function `f` on the interval `[a, b]`. With ``error=True``,
:func:`~mpmath.chebyfit` also returns an accurate estimate of the
maximum absolute error; that is, the maximum value of
`|f(x) - P(x)|` for `x \in [a, b]`.
:func:`~mpmath.chebyfit` uses the Chebyshev approximation formula,
which gives a nearly optimal solution: that is, the maximum
error of the approximating polynomial is very close to
the smallest possible for any polynomial of the same degree.
Chebyshev approximation is very useful if one needs repeated
evaluation of an expensive function, such as function defined
implicitly by an integral or a differential equation. (For
example, it could be used to turn a slow mpmath function
into a fast machine-precision version of the same.)
**Examples**
Here we use :func:`~mpmath.chebyfit` to generate a low-degree approximation
of `f(x) = \cos(x)`, valid on the interval `[1, 2]`::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> poly, err = chebyfit(cos, [1, 2], 5, error=True)
>>> nprint(poly)
[0.00291682, 0.146166, -0.732491, 0.174141, 0.949553]
>>> nprint(err, 12)
1.61351758081e-5
The polynomial can be evaluated using ``polyval``::
>>> nprint(polyval(poly, 1.6), 12)
-0.0291858904138
>>> nprint(cos(1.6), 12)
-0.0291995223013
Sampling the true error at 1000 points shows that the error
estimate generated by ``chebyfit`` is remarkably good::
>>> error = lambda x: abs(cos(x) - polyval(poly, x))
>>> nprint(max([error(1+n/1000.) for n in range(1000)]), 12)
1.61349954245e-5
**Choice of degree**
The degree `N` can be set arbitrarily high, to obtain an
arbitrarily good approximation. As a rule of thumb, an
`N`-term Chebyshev approximation is good to `N/(b-a)` decimal
places on a unit interval (although this depends on how
well-behaved `f` is). The cost grows accordingly: ``chebyfit``
evaluates the function `(N^2)/2` times to compute the
coefficients and an additional `N` times to estimate the error.
**Possible issues**
One should be careful to use a sufficiently high working
precision both when calling ``chebyfit`` and when evaluating
the resulting polynomial, as the polynomial is sometimes
ill-conditioned. It is for example difficult to reach
15-digit accuracy when evaluating the polynomial using
machine precision floats, no matter the theoretical
accuracy of the polynomial. (The option to return the
coefficients in Chebyshev form should be made available
in the future.)
It is important to note the Chebyshev approximation works
poorly if `f` is not smooth. A function containing singularities,
rapid oscillation, etc can be approximated more effectively by
multiplying it by a weight function that cancels out the
nonsmooth features, or by dividing the interval into several
segments.
"""
a, b = ctx._as_points(interval)
orig = ctx.prec
try:
ctx.prec = orig + int(N**0.5) + 20
c = [chebcoeff(ctx,f,a,b,k,N) for k in range(N)]
d = [ctx.zero] * N
d[0] = -c[0]/2
h = ctx.mpf(0.5)
T = chebT(ctx, ctx.mpf(2)/(b-a), ctx.mpf(-1)*(b+a)/(b-a))
for (k, Tk) in zip(range(N), T):
for i in range(len(Tk)):
d[i] += c[k]*Tk[i]
d = d[::-1]
# Estimate maximum error
err = ctx.zero
for k in range(N):
x = ctx.cos(ctx.pi*k/N) * (b-a)*h + (b+a)*h
err = max(err, abs(f(x) - ctx.polyval(d, x)))
finally:
ctx.prec = orig
if error:
return d, +err
else:
return d
@defun
def fourier(ctx, f, interval, N):
r"""
Computes the Fourier series of degree `N` of the given function
on the interval `[a, b]`. More precisely, :func:`~mpmath.fourier` returns
two lists `(c, s)` of coefficients (the cosine series and sine
series, respectively), such that
.. math ::
f(x) \sim \sum_{k=0}^N
c_k \cos(k m x) + s_k \sin(k m x)
where `m = 2 \pi / (b-a)`.
Note that many texts define the first coefficient as `2 c_0` instead
of `c_0`. The easiest way to evaluate the computed series correctly
is to pass it to :func:`~mpmath.fourierval`.
**Examples**
The function `f(x) = x` has a simple Fourier series on the standard
interval `[-\pi, \pi]`. The cosine coefficients are all zero (because
the function has odd symmetry), and the sine coefficients are
rational numbers::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> c, s = fourier(lambda x: x, [-pi, pi], 5)
>>> nprint(c)
[0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
>>> nprint(s)
[0.0, 2.0, -1.0, 0.666667, -0.5, 0.4]
This computes a Fourier series of a nonsymmetric function on
a nonstandard interval::
>>> I = [-1, 1.5]
>>> f = lambda x: x**2 - 4*x + 1
>>> cs = fourier(f, I, 4)
>>> nprint(cs[0])
[0.583333, 1.12479, -1.27552, 0.904708, -0.441296]
>>> nprint(cs[1])
[0.0, -2.6255, 0.580905, 0.219974, -0.540057]
It is instructive to plot a function along with its truncated
Fourier series::
>>> plot([f, lambda x: fourierval(cs, I, x)], I) #doctest: +SKIP
Fourier series generally converge slowly (and may not converge
pointwise). For example, if `f(x) = \cosh(x)`, a 10-term Fourier
series gives an `L^2` error corresponding to 2-digit accuracy::
>>> I = [-1, 1]
>>> cs = fourier(cosh, I, 9)
>>> g = lambda x: (cosh(x) - fourierval(cs, I, x))**2
>>> nprint(sqrt(quad(g, I)))
0.00467963
:func:`~mpmath.fourier` uses numerical quadrature. For nonsmooth functions,
the accuracy (and speed) can be improved by including all singular
points in the interval specification::
>>> nprint(fourier(abs, [-1, 1], 0), 10)
([0.5000441648], [0.0])
>>> nprint(fourier(abs, [-1, 0, 1], 0), 10)
([0.5], [0.0])
"""
interval = ctx._as_points(interval)
a = interval[0]
b = interval[-1]
L = b-a
cos_series = []
sin_series = []
cutoff = ctx.eps*10
for n in xrange(N+1):
m = 2*n*ctx.pi/L
an = 2*ctx.quadgl(lambda t: f(t)*ctx.cos(m*t), interval)/L
bn = 2*ctx.quadgl(lambda t: f(t)*ctx.sin(m*t), interval)/L
if n == 0:
an /= 2
if abs(an) < cutoff: an = ctx.zero
if abs(bn) < cutoff: bn = ctx.zero
cos_series.append(an)
sin_series.append(bn)
return cos_series, sin_series
@defun
def fourierval(ctx, series, interval, x):
"""
Evaluates a Fourier series (in the format computed by
by :func:`~mpmath.fourier` for the given interval) at the point `x`.
The series should be a pair `(c, s)` where `c` is the
cosine series and `s` is the sine series. The two lists
need not have the same length.
"""
cs, ss = series
ab = ctx._as_points(interval)
a = interval[0]
b = interval[-1]
m = 2*ctx.pi/(ab[-1]-ab[0])
s = ctx.zero
s += ctx.fsum(cs[n]*ctx.cos(m*n*x) for n in xrange(len(cs)) if cs[n])
s += ctx.fsum(ss[n]*ctx.sin(m*n*x) for n in xrange(len(ss)) if ss[n])
return s
| 8,817 | 34.700405 | 79 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/calculus/calculus.py
|
class CalculusMethods(object):
pass
def defun(f):
setattr(CalculusMethods, f.__name__, f)
| 99 | 15.666667 | 43 |
py
|
cba-pipeline-public
|
cba-pipeline-public-master/containernet/ndn-containers/ndn_headless-player/bandits/venv/lib/python3.6/site-packages/mpmath/calculus/extrapolation.py
|
try:
from itertools import izip
except ImportError:
izip = zip
from ..libmp.backend import xrange
from .calculus import defun
try:
next = next
except NameError:
next = lambda _: _.next()
@defun
def richardson(ctx, seq):
r"""
Given a list ``seq`` of the first `N` elements of a slowly convergent
infinite sequence, :func:`~mpmath.richardson` computes the `N`-term
Richardson extrapolate for the limit.
:func:`~mpmath.richardson` returns `(v, c)` where `v` is the estimated
limit and `c` is the magnitude of the largest weight used during the
computation. The weight provides an estimate of the precision
lost to cancellation. Due to cancellation effects, the sequence must
be typically be computed at a much higher precision than the target
accuracy of the extrapolation.
**Applicability and issues**
The `N`-step Richardson extrapolation algorithm used by
:func:`~mpmath.richardson` is described in [1].
Richardson extrapolation only works for a specific type of sequence,
namely one converging like partial sums of
`P(1)/Q(1) + P(2)/Q(2) + \ldots` where `P` and `Q` are polynomials.
When the sequence does not convergence at such a rate
:func:`~mpmath.richardson` generally produces garbage.
Richardson extrapolation has the advantage of being fast: the `N`-term
extrapolate requires only `O(N)` arithmetic operations, and usually
produces an estimate that is accurate to `O(N)` digits. Contrast with
the Shanks transformation (see :func:`~mpmath.shanks`), which requires
`O(N^2)` operations.
:func:`~mpmath.richardson` is unable to produce an estimate for the
approximation error. One way to estimate the error is to perform
two extrapolations with slightly different `N` and comparing the
results.
Richardson extrapolation does not work for oscillating sequences.
As a simple workaround, :func:`~mpmath.richardson` detects if the last
three elements do not differ monotonically, and in that case
applies extrapolation only to the even-index elements.
**Example**
Applying Richardson extrapolation to the Leibniz series for `\pi`::
>>> from mpmath import *
>>> mp.dps = 30; mp.pretty = True
>>> S = [4*sum(mpf(-1)**n/(2*n+1) for n in range(m))
... for m in range(1,30)]
>>> v, c = richardson(S[:10])
>>> v
3.2126984126984126984126984127
>>> nprint([v-pi, c])
[0.0711058, 2.0]
>>> v, c = richardson(S[:30])
>>> v
3.14159265468624052829954206226
>>> nprint([v-pi, c])
[1.09645e-9, 20833.3]
**References**
1. [BenderOrszag]_ pp. 375-376
"""
if len(seq) < 3:
raise ValueError("seq should be of minimum length 3")
if ctx.sign(seq[-1]-seq[-2]) != ctx.sign(seq[-2]-seq[-3]):
seq = seq[::2]
N = len(seq)//2-1
s = ctx.zero
# The general weight is c[k] = (N+k)**N * (-1)**(k+N) / k! / (N-k)!
# To avoid repeated factorials, we simplify the quotient
# of successive weights to obtain a recurrence relation
c = (-1)**N * N**N / ctx.mpf(ctx._ifac(N))
maxc = 1
for k in xrange(N+1):
s += c * seq[N+k]
maxc = max(abs(c), maxc)
c *= (k-N)*ctx.mpf(k+N+1)**N
c /= ((1+k)*ctx.mpf(k+N)**N)
return s, maxc
@defun
def shanks(ctx, seq, table=None, randomized=False):
r"""
Given a list ``seq`` of the first `N` elements of a slowly
convergent infinite sequence `(A_k)`, :func:`~mpmath.shanks` computes the iterated
Shanks transformation `S(A), S(S(A)), \ldots, S^{N/2}(A)`. The Shanks
transformation often provides strong convergence acceleration,
especially if the sequence is oscillating.
The iterated Shanks transformation is computed using the Wynn
epsilon algorithm (see [1]). :func:`~mpmath.shanks` returns the full
epsilon table generated by Wynn's algorithm, which can be read
off as follows:
* The table is a list of lists forming a lower triangular matrix,
where higher row and column indices correspond to more accurate
values.
* The columns with even index hold dummy entries (required for the
computation) and the columns with odd index hold the actual
extrapolates.
* The last element in the last row is typically the most
accurate estimate of the limit.
* The difference to the third last element in the last row
provides an estimate of the approximation error.
* The magnitude of the second last element provides an estimate
of the numerical accuracy lost to cancellation.
For convenience, so the extrapolation is stopped at an odd index
so that ``shanks(seq)[-1][-1]`` always gives an estimate of the
limit.
Optionally, an existing table can be passed to :func:`~mpmath.shanks`.
This can be used to efficiently extend a previous computation after
new elements have been appended to the sequence. The table will
then be updated in-place.
**The Shanks transformation**
The Shanks transformation is defined as follows (see [2]): given
the input sequence `(A_0, A_1, \ldots)`, the transformed sequence is
given by
.. math ::
S(A_k) = \frac{A_{k+1}A_{k-1}-A_k^2}{A_{k+1}+A_{k-1}-2 A_k}
The Shanks transformation gives the exact limit `A_{\infty}` in a
single step if `A_k = A + a q^k`. Note in particular that it
extrapolates the exact sum of a geometric series in a single step.
Applying the Shanks transformation once often improves convergence
substantially for an arbitrary sequence, but the optimal effect is
obtained by applying it iteratively:
`S(S(A_k)), S(S(S(A_k))), \ldots`.
Wynn's epsilon algorithm provides an efficient way to generate
the table of iterated Shanks transformations. It reduces the
computation of each element to essentially a single division, at
the cost of requiring dummy elements in the table. See [1] for
details.
**Precision issues**
Due to cancellation effects, the sequence must be typically be
computed at a much higher precision than the target accuracy
of the extrapolation.
If the Shanks transformation converges to the exact limit (such
as if the sequence is a geometric series), then a division by
zero occurs. By default, :func:`~mpmath.shanks` handles this case by
terminating the iteration and returning the table it has
generated so far. With *randomized=True*, it will instead
replace the zero by a pseudorandom number close to zero.
(TODO: find a better solution to this problem.)
**Examples**
We illustrate by applying Shanks transformation to the Leibniz
series for `\pi`::
>>> from mpmath import *
>>> mp.dps = 50
>>> S = [4*sum(mpf(-1)**n/(2*n+1) for n in range(m))
... for m in range(1,30)]
>>>
>>> T = shanks(S[:7])
>>> for row in T:
... nprint(row)
...
[-0.75]
[1.25, 3.16667]
[-1.75, 3.13333, -28.75]
[2.25, 3.14524, 82.25, 3.14234]
[-2.75, 3.13968, -177.75, 3.14139, -969.937]
[3.25, 3.14271, 327.25, 3.14166, 3515.06, 3.14161]
The extrapolated accuracy is about 4 digits, and about 4 digits
may have been lost due to cancellation::
>>> L = T[-1]
>>> nprint([abs(L[-1] - pi), abs(L[-1] - L[-3]), abs(L[-2])])
[2.22532e-5, 4.78309e-5, 3515.06]
Now we extend the computation::
>>> T = shanks(S[:25], T)
>>> L = T[-1]
>>> nprint([abs(L[-1] - pi), abs(L[-1] - L[-3]), abs(L[-2])])
[3.75527e-19, 1.48478e-19, 2.96014e+17]
The value for pi is now accurate to 18 digits. About 18 digits may
also have been lost to cancellation.
Here is an example with a geometric series, where the convergence
is immediate (the sum is exactly 1)::
>>> mp.dps = 15
>>> for row in shanks([0.5, 0.75, 0.875, 0.9375, 0.96875]):
... nprint(row)
[4.0]
[8.0, 1.0]
**References**
1. [GravesMorris]_
2. [BenderOrszag]_ pp. 368-375
"""
if len(seq) < 2:
raise ValueError("seq should be of minimum length 2")
if table:
START = len(table)
else:
START = 0
table = []
STOP = len(seq) - 1
if STOP & 1:
STOP -= 1
one = ctx.one
eps = +ctx.eps
if randomized:
from random import Random
rnd = Random()
rnd.seed(START)
for i in xrange(START, STOP):
row = []
for j in xrange(i+1):
if j == 0:
a, b = 0, seq[i+1]-seq[i]
else:
if j == 1:
a = seq[i]
else:
a = table[i-1][j-2]
b = row[j-1] - table[i-1][j-1]
if not b:
if randomized:
b = rnd.getrandbits(10)*eps
elif i & 1:
return table[:-1]
else:
return table
row.append(a + one/b)
table.append(row)
return table
class levin_class:
# levin: Copyright 2013 Timo Hartmann (thartmann15 at gmail.com)
r"""
This interface implements Levin's (nonlinear) sequence transformation for
convergence acceleration and summation of divergent series. It performs
better than the Shanks/Wynn-epsilon algorithm for logarithmic convergent
or alternating divergent series.
Let *A* be the series we want to sum:
.. math ::
A = \sum_{k=0}^{\infty} a_k
Attention: all `a_k` must be non-zero!
Let `s_n` be the partial sums of this series:
.. math ::
s_n = \sum_{k=0}^n a_k.
**Methods**
Calling ``levin`` returns an object with the following methods.
``update(...)`` works with the list of individual terms `a_k` of *A*, and
``update_step(...)`` works with the list of partial sums `s_k` of *A*:
.. code ::
v, e = ...update([a_0, a_1,..., a_k])
v, e = ...update_psum([s_0, s_1,..., s_k])
``step(...)`` works with the individual terms `a_k` and ``step_psum(...)``
works with the partial sums `s_k`:
.. code ::
v, e = ...step(a_k)
v, e = ...step_psum(s_k)
*v* is the current estimate for *A*, and *e* is an error estimate which is
simply the difference between the current estimate and the last estimate.
One should not mix ``update``, ``update_psum``, ``step`` and ``step_psum``.
**A word of caution**
One can only hope for good results (i.e. convergence acceleration or
resummation) if the `s_n` have some well defind asymptotic behavior for
large `n` and are not erratic or random. Furthermore one usually needs very
high working precision because of the numerical cancellation. If the working
precision is insufficient, levin may produce silently numerical garbage.
Furthermore even if the Levin-transformation converges, in the general case
there is no proof that the result is mathematically sound. Only for very
special classes of problems one can prove that the Levin-transformation
converges to the expected result (for example Stieltjes-type integrals).
Furthermore the Levin-transform is quite expensive (i.e. slow) in comparison
to Shanks/Wynn-epsilon, Richardson & co.
In summary one can say that the Levin-transformation is powerful but
unreliable and that it may need a copious amount of working precision.
The Levin transform has several variants differing in the choice of weights.
Some variants are better suited for the possible flavours of convergence
behaviour of *A* than other variants:
.. code ::
convergence behaviour levin-u levin-t levin-v shanks/wynn-epsilon
logarithmic + - + -
linear + + + +
alternating divergent + + + +
"+" means the variant is suitable,"-" means the variant is not suitable;
for comparison the Shanks/Wynn-epsilon transform is listed, too.
The variant is controlled though the variant keyword (i.e. ``variant="u"``,
``variant="t"`` or ``variant="v"``). Overall "u" is probably the best choice.
Finally it is possible to use the Sidi-S transform instead of the Levin transform
by using the keyword ``method='sidi'``. The Sidi-S transform works better than the
Levin transformation for some divergent series (see the examples).
Parameters:
.. code ::
method "levin" or "sidi" chooses either the Levin or the Sidi-S transformation
variant "u","t" or "v" chooses the weight variant.
The Levin transform is also accessible through the nsum interface.
``method="l"`` or ``method="levin"`` select the normal Levin transform while
``method="sidi"``
selects the Sidi-S transform. The variant is in both cases selected through the
levin_variant keyword. The stepsize in :func:`~mpmath.nsum` must not be chosen too large, otherwise
it will miss the point where the Levin transform converges resulting in numerical
overflow/garbage. For highly divergent series a copious amount of working precision
must be chosen.
**Examples**
First we sum the zeta function::
>>> from mpmath import mp
>>> mp.prec = 53
>>> eps = mp.mpf(mp.eps)
>>> with mp.extraprec(2 * mp.prec): # levin needs a high working precision
... L = mp.levin(method = "levin", variant = "u")
... S, s, n = [], 0, 1
... while 1:
... s += mp.one / (n * n)
... n += 1
... S.append(s)
... v, e = L.update_psum(S)
... if e < eps:
... break
... if n > 1000: raise RuntimeError("iteration limit exceeded")
>>> print(mp.chop(v - mp.pi ** 2 / 6))
0.0
>>> w = mp.nsum(lambda n: 1 / (n*n), [1, mp.inf], method = "levin", levin_variant = "u")
>>> print(mp.chop(v - w))
0.0
Now we sum the zeta function outside its range of convergence
(attention: This does not work at the negative integers!)::
>>> eps = mp.mpf(mp.eps)
>>> with mp.extraprec(2 * mp.prec): # levin needs a high working precision
... L = mp.levin(method = "levin", variant = "v")
... A, n = [], 1
... while 1:
... s = mp.mpf(n) ** (2 + 3j)
... n += 1
... A.append(s)
... v, e = L.update(A)
... if e < eps:
... break
... if n > 1000: raise RuntimeError("iteration limit exceeded")
>>> print(mp.chop(v - mp.zeta(-2-3j)))
0.0
>>> w = mp.nsum(lambda n: n ** (2 + 3j), [1, mp.inf], method = "levin", levin_variant = "v")
>>> print(mp.chop(v - w))
0.0
Now we sum the divergent asymptotic expansion of an integral related to the
exponential integral (see also [2] p.373). The Sidi-S transform works best here::
>>> z = mp.mpf(10)
>>> exact = mp.quad(lambda x: mp.exp(-x)/(1+x/z),[0,mp.inf])
>>> # exact = z * mp.exp(z) * mp.expint(1,z) # this is the symbolic expression for the integral
>>> eps = mp.mpf(mp.eps)
>>> with mp.extraprec(2 * mp.prec): # high working precisions are mandatory for divergent resummation
... L = mp.levin(method = "sidi", variant = "t")
... n = 0
... while 1:
... s = (-1)**n * mp.fac(n) * z ** (-n)
... v, e = L.step(s)
... n += 1
... if e < eps:
... break
... if n > 1000: raise RuntimeError("iteration limit exceeded")
>>> print(mp.chop(v - exact))
0.0
>>> w = mp.nsum(lambda n: (-1) ** n * mp.fac(n) * z ** (-n), [0, mp.inf], method = "sidi", levin_variant = "t")
>>> print(mp.chop(v - w))
0.0
Another highly divergent integral is also summable::
>>> z = mp.mpf(2)
>>> eps = mp.mpf(mp.eps)
>>> exact = mp.quad(lambda x: mp.exp( -x * x / 2 - z * x ** 4), [0,mp.inf]) * 2 / mp.sqrt(2 * mp.pi)
>>> # exact = mp.exp(mp.one / (32 * z)) * mp.besselk(mp.one / 4, mp.one / (32 * z)) / (4 * mp.sqrt(z * mp.pi)) # this is the symbolic expression for the integral
>>> with mp.extraprec(7 * mp.prec): # we need copious amount of precision to sum this highly divergent series
... L = mp.levin(method = "levin", variant = "t")
... n, s = 0, 0
... while 1:
... s += (-z)**n * mp.fac(4 * n) / (mp.fac(n) * mp.fac(2 * n) * (4 ** n))
... n += 1
... v, e = L.step_psum(s)
... if e < eps:
... break
... if n > 1000: raise RuntimeError("iteration limit exceeded")
>>> print(mp.chop(v - exact))
0.0
>>> w = mp.nsum(lambda n: (-z)**n * mp.fac(4 * n) / (mp.fac(n) * mp.fac(2 * n) * (4 ** n)),
... [0, mp.inf], method = "levin", levin_variant = "t", workprec = 8*mp.prec, steps = [2] + [1 for x in xrange(1000)])
>>> print(mp.chop(v - w))
0.0
These examples run with 15-20 decimal digits precision. For higher precision the
working precision must be raised.
**Examples for nsum**
Here we calculate Euler's constant as the constant term in the Laurent
expansion of `\zeta(s)` at `s=1`. This sum converges extremly slowly because of
the logarithmic convergence behaviour of the Dirichlet series for zeta::
>>> mp.dps = 30
>>> z = mp.mpf(10) ** (-10)
>>> a = mp.nsum(lambda n: n**(-(1+z)), [1, mp.inf], method = "l") - 1 / z
>>> print(mp.chop(a - mp.euler, tol = 1e-10))
0.0
The Sidi-S transform performs excellently for the alternating series of `\log(2)`::
>>> a = mp.nsum(lambda n: (-1)**(n-1) / n, [1, mp.inf], method = "sidi")
>>> print(mp.chop(a - mp.log(2)))
0.0
Hypergeometric series can also be summed outside their range of convergence.
The stepsize in :func:`~mpmath.nsum` must not be chosen too large, otherwise it will miss the
point where the Levin transform converges resulting in numerical overflow/garbage::
>>> z = 2 + 1j
>>> exact = mp.hyp2f1(2 / mp.mpf(3), 4 / mp.mpf(3), 1 / mp.mpf(3), z)
>>> f = lambda n: mp.rf(2 / mp.mpf(3), n) * mp.rf(4 / mp.mpf(3), n) * z**n / (mp.rf(1 / mp.mpf(3), n) * mp.fac(n))
>>> v = mp.nsum(f, [0, mp.inf], method = "levin", steps = [10 for x in xrange(1000)])
>>> print(mp.chop(exact-v))
0.0
References:
[1] E.J. Weniger - "Nonlinear Sequence Transformations for the Acceleration of
Convergence and the Summation of Divergent Series" arXiv:math/0306302
[2] A. Sidi - "Pratical Extrapolation Methods"
[3] H.H.H. Homeier - "Scalar Levin-Type Sequence Transformations" arXiv:math/0005209
"""
def __init__(self, method = "levin", variant = "u"):
self.variant = variant
self.n = 0
self.a0 = 0
self.theta = 1
self.A = []
self.B = []
self.last = 0
self.last_s = False
if method == "levin":
self.factor = self.factor_levin
elif method == "sidi":
self.factor = self.factor_sidi
else:
raise ValueError("levin: unknown method \"%s\"" % method)
def factor_levin(self, i):
# original levin
# [1] p.50,e.7.5-7 (with n-j replaced by i)
return (self.theta + i) * (self.theta + self.n - 1) ** (self.n - i - 2) / self.ctx.mpf(self.theta + self.n) ** (self.n - i - 1)
def factor_sidi(self, i):
# sidi analogon to levin (factorial series)
# [1] p.59,e.8.3-16 (with n-j replaced by i)
return (self.theta + self.n - 1) * (self.theta + self.n - 2) / self.ctx.mpf((self.theta + 2 * self.n - i - 2) * (self.theta + 2 * self.n - i - 3))
def run(self, s, a0, a1 = 0):
if self.variant=="t":
# levin t
w=a0
elif self.variant=="u":
# levin u
w=a0*(self.theta+self.n)
elif self.variant=="v":
# levin v
w=a0*a1/(a0-a1)
else:
assert False, "unknown variant"
if w==0:
raise ValueError("levin: zero weight")
self.A.append(s/w)
self.B.append(1/w)
for i in range(self.n-1,-1,-1):
if i==self.n-1:
f=1
else:
f=self.factor(i)
self.A[i]=self.A[i+1]-f*self.A[i]
self.B[i]=self.B[i+1]-f*self.B[i]
self.n+=1
###########################################################################
def update_psum(self,S):
"""
This routine applies the convergence acceleration to the list of partial sums.
A = sum(a_k, k = 0..infinity)
s_n = sum(a_k, k = 0..n)
v, e = ...update_psum([s_0, s_1,..., s_k])
output:
v current estimate of the series A
e an error estimate which is simply the difference between the current
estimate and the last estimate.
"""
if self.variant!="v":
if self.n==0:
self.run(S[0],S[0])
while self.n<len(S):
self.run(S[self.n],S[self.n]-S[self.n-1])
else:
if len(S)==1:
self.last=0
return S[0],abs(S[0])
if self.n==0:
self.a1=S[1]-S[0]
self.run(S[0],S[0],self.a1)
while self.n<len(S)-1:
na1=S[self.n+1]-S[self.n]
self.run(S[self.n],self.a1,na1)
self.a1=na1
value=self.A[0]/self.B[0]
err=abs(value-self.last)
self.last=value
return value,err
def update(self,X):
"""
This routine applies the convergence acceleration to the list of individual terms.
A = sum(a_k, k = 0..infinity)
v, e = ...update([a_0, a_1,..., a_k])
output:
v current estimate of the series A
e an error estimate which is simply the difference between the current
estimate and the last estimate.
"""
if self.variant!="v":
if self.n==0:
self.s=X[0]
self.run(self.s,X[0])
while self.n<len(X):
self.s+=X[self.n]
self.run(self.s,X[self.n])
else:
if len(X)==1:
self.last=0
return X[0],abs(X[0])
if self.n==0:
self.s=X[0]
self.run(self.s,X[0],X[1])
while self.n<len(X)-1:
self.s+=X[self.n]
self.run(self.s,X[self.n],X[self.n+1])
value=self.A[0]/self.B[0]
err=abs(value-self.last)
self.last=value
return value,err
###########################################################################
def step_psum(self,s):
"""
This routine applies the convergence acceleration to the partial sums.
A = sum(a_k, k = 0..infinity)
s_n = sum(a_k, k = 0..n)
v, e = ...step_psum(s_k)
output:
v current estimate of the series A
e an error estimate which is simply the difference between the current
estimate and the last estimate.
"""
if self.variant!="v":
if self.n==0:
self.last_s=s
self.run(s,s)
else:
self.run(s,s-self.last_s)
self.last_s=s
else:
if isinstance(self.last_s,bool):
self.last_s=s
self.last_w=s
self.last=0
return s,abs(s)
na1=s-self.last_s
self.run(self.last_s,self.last_w,na1)
self.last_w=na1
self.last_s=s
value=self.A[0]/self.B[0]
err=abs(value-self.last)
self.last=value
return value,err
def step(self,x):
"""
This routine applies the convergence acceleration to the individual terms.
A = sum(a_k, k = 0..infinity)
v, e = ...step(a_k)
output:
v current estimate of the series A
e an error estimate which is simply the difference between the current
estimate and the last estimate.
"""
if self.variant!="v":
if self.n==0:
self.s=x
self.run(self.s,x)
else:
self.s+=x
self.run(self.s,x)
else:
if isinstance(self.last_s,bool):
self.last_s=x
self.s=0
self.last=0
return x,abs(x)
self.s+=self.last_s
self.run(self.s,self.last_s,x)
self.last_s=x
value=self.A[0]/self.B[0]
err=abs(value-self.last)
self.last=value
return value,err
def levin(ctx, method = "levin", variant = "u"):
L = levin_class(method = method, variant = variant)
L.ctx = ctx
return L
levin.__doc__ = levin_class.__doc__
defun(levin)
class cohen_alt_class:
# cohen_alt: Copyright 2013 Timo Hartmann (thartmann15 at gmail.com)
"""
This interface implements the convergence acceleration of alternating series
as described in H. Cohen, F.R. Villegas, D. Zagier - "Convergence Acceleration
of Alternating Series". This series transformation works only well if the
individual terms of the series have an alternating sign. It belongs to the
class of linear series transformations (in contrast to the Shanks/Wynn-epsilon
or Levin transform). This series transformation is also able to sum some types
of divergent series. See the paper under which conditions this resummation is
mathematical sound.
Let *A* be the series we want to sum:
.. math ::
A = \sum_{k=0}^{\infty} a_k
Let `s_n` be the partial sums of this series:
.. math ::
s_n = \sum_{k=0}^n a_k.
**Interface**
Calling ``cohen_alt`` returns an object with the following methods.
Then ``update(...)`` works with the list of individual terms `a_k` and
``update_psum(...)`` works with the list of partial sums `s_k`:
.. code ::
v, e = ...update([a_0, a_1,..., a_k])
v, e = ...update_psum([s_0, s_1,..., s_k])
*v* is the current estimate for *A*, and *e* is an error estimate which is
simply the difference between the current estimate and the last estimate.
**Examples**
Here we compute the alternating zeta function using ``update_psum``::
>>> from mpmath import mp
>>> AC = mp.cohen_alt()
>>> S, s, n = [], 0, 1
>>> while 1:
... s += -((-1) ** n) * mp.one / (n * n)
... n += 1
... S.append(s)
... v, e = AC.update_psum(S)
... if e < mp.eps:
... break
... if n > 1000: raise RuntimeError("iteration limit exceeded")
>>> print(mp.chop(v - mp.pi ** 2 / 12))
0.0
Here we compute the product `\prod_{n=1}^{\infty} \Gamma(1+1/(2n-1)) / \Gamma(1+1/(2n))`::
>>> A = []
>>> AC = mp.cohen_alt()
>>> n = 1
>>> while 1:
... A.append( mp.loggamma(1 + mp.one / (2 * n - 1)))
... A.append(-mp.loggamma(1 + mp.one / (2 * n)))
... n += 1
... v, e = AC.update(A)
... if e < mp.eps:
... break
... if n > 1000: raise RuntimeError("iteration limit exceeded")
>>> v = mp.exp(v)
>>> print(mp.chop(v - 1.06215090557106, tol = 1e-12))
0.0
``cohen_alt`` is also accessible through the :func:`~mpmath.nsum` interface::
>>> v = mp.nsum(lambda n: (-1)**(n-1) / n, [1, mp.inf], method = "a")
>>> print(mp.chop(v - mp.log(2)))
0.0
>>> v = mp.nsum(lambda n: (-1)**n / (2 * n + 1), [0, mp.inf], method = "a")
>>> print(mp.chop(v - mp.pi / 4))
0.0
>>> v = mp.nsum(lambda n: (-1)**n * mp.log(n) * n, [1, mp.inf], method = "a")
>>> print(mp.chop(v - mp.diff(lambda s: mp.altzeta(s), -1)))
0.0
"""
def __init__(self):
self.last=0
def update(self, A):
"""
This routine applies the convergence acceleration to the list of individual terms.
A = sum(a_k, k = 0..infinity)
v, e = ...update([a_0, a_1,..., a_k])
output:
v current estimate of the series A
e an error estimate which is simply the difference between the current
estimate and the last estimate.
"""
n = len(A)
d = (3 + self.ctx.sqrt(8)) ** n
d = (d + 1 / d) / 2
b = -self.ctx.one
c = -d
s = 0
for k in xrange(n):
c = b - c
if k % 2 == 0:
s = s + c * A[k]
else:
s = s - c * A[k]
b = 2 * (k + n) * (k - n) * b / ((2 * k + 1) * (k + self.ctx.one))
value = s / d
err = abs(value - self.last)
self.last = value
return value, err
def update_psum(self, S):
"""
This routine applies the convergence acceleration to the list of partial sums.
A = sum(a_k, k = 0..infinity)
s_n = sum(a_k ,k = 0..n)
v, e = ...update_psum([s_0, s_1,..., s_k])
output:
v current estimate of the series A
e an error estimate which is simply the difference between the current
estimate and the last estimate.
"""
n = len(S)
d = (3 + self.ctx.sqrt(8)) ** n
d = (d + 1 / d) / 2
b = self.ctx.one
s = 0
for k in xrange(n):
b = 2 * (n + k) * (n - k) * b / ((2 * k + 1) * (k + self.ctx.one))
s += b * S[k]
value = s / d
err = abs(value - self.last)
self.last = value
return value, err
def cohen_alt(ctx):
L = cohen_alt_class()
L.ctx = ctx
return L
cohen_alt.__doc__ = cohen_alt_class.__doc__
defun(cohen_alt)
@defun
def sumap(ctx, f, interval, integral=None, error=False):
r"""
Evaluates an infinite series of an analytic summand *f* using the
Abel-Plana formula
.. math ::
\sum_{k=0}^{\infty} f(k) = \int_0^{\infty} f(t) dt + \frac{1}{2} f(0) +
i \int_0^{\infty} \frac{f(it)-f(-it)}{e^{2\pi t}-1} dt.
Unlike the Euler-Maclaurin formula (see :func:`~mpmath.sumem`),
the Abel-Plana formula does not require derivatives. However,
it only works when `|f(it)-f(-it)|` does not
increase too rapidly with `t`.
**Examples**
The Abel-Plana formula is particularly useful when the summand
decreases like a power of `k`; for example when the sum is a pure
zeta function::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> sumap(lambda k: 1/k**2.5, [1,inf])
1.34148725725091717975677
>>> zeta(2.5)
1.34148725725091717975677
>>> sumap(lambda k: 1/(k+1j)**(2.5+2.5j), [1,inf])
(-3.385361068546473342286084 - 0.7432082105196321803869551j)
>>> zeta(2.5+2.5j, 1+1j)
(-3.385361068546473342286084 - 0.7432082105196321803869551j)
If the series is alternating, numerical quadrature along the real
line is likely to give poor results, so it is better to evaluate
the first term symbolically whenever possible:
>>> n=3; z=-0.75
>>> I = expint(n,-log(z))
>>> chop(sumap(lambda k: z**k / k**n, [1,inf], integral=I))
-0.6917036036904594510141448
>>> polylog(n,z)
-0.6917036036904594510141448
"""
prec = ctx.prec
try:
ctx.prec += 10
a, b = interval
if b != ctx.inf:
raise ValueError("b should be equal to ctx.inf")
g = lambda x: f(x+a)
if integral is None:
i1, err1 = ctx.quad(g, [0,ctx.inf], error=True)
else:
i1, err1 = integral, 0
j = ctx.j
p = ctx.pi * 2
if ctx._is_real_type(i1):
h = lambda t: -2 * ctx.im(g(j*t)) / ctx.expm1(p*t)
else:
h = lambda t: j*(g(j*t)-g(-j*t)) / ctx.expm1(p*t)
i2, err2 = ctx.quad(h, [0,ctx.inf], error=True)
err = err1+err2
v = i1+i2+0.5*g(ctx.mpf(0))
finally:
ctx.prec = prec
if error:
return +v, err
return +v
@defun
def sumem(ctx, f, interval, tol=None, reject=10, integral=None,
adiffs=None, bdiffs=None, verbose=False, error=False,
_fast_abort=False):
r"""
Uses the Euler-Maclaurin formula to compute an approximation accurate
to within ``tol`` (which defaults to the present epsilon) of the sum
.. math ::
S = \sum_{k=a}^b f(k)
where `(a,b)` are given by ``interval`` and `a` or `b` may be
infinite. The approximation is
.. math ::
S \sim \int_a^b f(x) \,dx + \frac{f(a)+f(b)}{2} +
\sum_{k=1}^{\infty} \frac{B_{2k}}{(2k)!}
\left(f^{(2k-1)}(b)-f^{(2k-1)}(a)\right).
The last sum in the Euler-Maclaurin formula is not generally
convergent (a notable exception is if `f` is a polynomial, in
which case Euler-Maclaurin actually gives an exact result).
The summation is stopped as soon as the quotient between two
consecutive terms falls below *reject*. That is, by default
(*reject* = 10), the summation is continued as long as each
term adds at least one decimal.
Although not convergent, convergence to a given tolerance can
often be "forced" if `b = \infty` by summing up to `a+N` and then
applying the Euler-Maclaurin formula to the sum over the range
`(a+N+1, \ldots, \infty)`. This procedure is implemented by
:func:`~mpmath.nsum`.
By default numerical quadrature and differentiation is used.
If the symbolic values of the integral and endpoint derivatives
are known, it is more efficient to pass the value of the
integral explicitly as ``integral`` and the derivatives
explicitly as ``adiffs`` and ``bdiffs``. The derivatives
should be given as iterables that yield
`f(a), f'(a), f''(a), \ldots` (and the equivalent for `b`).
**Examples**
Summation of an infinite series, with automatic and symbolic
integral and derivative values (the second should be much faster)::
>>> from mpmath import *
>>> mp.dps = 50; mp.pretty = True
>>> sumem(lambda n: 1/n**2, [32, inf])
0.03174336652030209012658168043874142714132886413417
>>> I = mpf(1)/32
>>> D = adiffs=((-1)**n*fac(n+1)*32**(-2-n) for n in range(999))
>>> sumem(lambda n: 1/n**2, [32, inf], integral=I, adiffs=D)
0.03174336652030209012658168043874142714132886413417
An exact evaluation of a finite polynomial sum::
>>> sumem(lambda n: n**5-12*n**2+3*n, [-100000, 200000])
10500155000624963999742499550000.0
>>> print(sum(n**5-12*n**2+3*n for n in range(-100000, 200001)))
10500155000624963999742499550000
"""
tol = tol or +ctx.eps
interval = ctx._as_points(interval)
a = ctx.convert(interval[0])
b = ctx.convert(interval[-1])
err = ctx.zero
prev = 0
M = 10000
if a == ctx.ninf: adiffs = (0 for n in xrange(M))
else: adiffs = adiffs or ctx.diffs(f, a)
if b == ctx.inf: bdiffs = (0 for n in xrange(M))
else: bdiffs = bdiffs or ctx.diffs(f, b)
orig = ctx.prec
#verbose = 1
try:
ctx.prec += 10
s = ctx.zero
for k, (da, db) in enumerate(izip(adiffs, bdiffs)):
if k & 1:
term = (db-da) * ctx.bernoulli(k+1) / ctx.factorial(k+1)
mag = abs(term)
if verbose:
print("term", k, "magnitude =", ctx.nstr(mag))
if k > 4 and mag < tol:
s += term
break
elif k > 4 and abs(prev) / mag < reject:
err += mag
if _fast_abort:
return [s, (s, err)][error]
if verbose:
print("Failed to converge")
break
else:
s += term
prev = term
# Endpoint correction
if a != ctx.ninf: s += f(a)/2
if b != ctx.inf: s += f(b)/2
# Tail integral
if verbose:
print("Integrating f(x) from x = %s to %s" % (ctx.nstr(a), ctx.nstr(b)))
if integral:
s += integral
else:
integral, ierr = ctx.quad(f, interval, error=True)
if verbose:
print("Integration error:", ierr)
s += integral
err += ierr
finally:
ctx.prec = orig
if error:
return s, err
else:
return s
@defun
def adaptive_extrapolation(ctx, update, emfun, kwargs):
option = kwargs.get
if ctx._fixed_precision:
tol = option('tol', ctx.eps*2**10)
else:
tol = option('tol', ctx.eps/2**10)
verbose = option('verbose', False)
maxterms = option('maxterms', ctx.dps*10)
method = set(option('method', 'r+s').split('+'))
skip = option('skip', 0)
steps = iter(option('steps', xrange(10, 10**9, 10)))
strict = option('strict')
#steps = (10 for i in xrange(1000))
summer=[]
if 'd' in method or 'direct' in method:
TRY_RICHARDSON = TRY_SHANKS = TRY_EULER_MACLAURIN = False
else:
TRY_RICHARDSON = ('r' in method) or ('richardson' in method)
TRY_SHANKS = ('s' in method) or ('shanks' in method)
TRY_EULER_MACLAURIN = ('e' in method) or \
('euler-maclaurin' in method)
def init_levin(m):
variant = kwargs.get("levin_variant", "u")
if isinstance(variant, str):
if variant == "all":
variant = ["u", "v", "t"]
else:
variant = [variant]
for s in variant:
L = levin_class(method = m, variant = s)
L.ctx = ctx
L.name = m + "(" + s + ")"
summer.append(L)
if ('l' in method) or ('levin' in method):
init_levin("levin")
if ('sidi' in method):
init_levin("sidi")
if ('a' in method) or ('alternating' in method):
L = cohen_alt_class()
L.ctx = ctx
L.name = "alternating"
summer.append(L)
last_richardson_value = 0
shanks_table = []
index = 0
step = 10
partial = []
best = ctx.zero
orig = ctx.prec
try:
if 'workprec' in kwargs:
ctx.prec = kwargs['workprec']
elif TRY_RICHARDSON or TRY_SHANKS or len(summer)!=0:
ctx.prec = (ctx.prec+10) * 4
else:
ctx.prec += 30
while 1:
if index >= maxterms:
break
# Get new batch of terms
try:
step = next(steps)
except StopIteration:
pass
if verbose:
print("-"*70)
print("Adding terms #%i-#%i" % (index, index+step))
update(partial, xrange(index, index+step))
index += step
# Check direct error
best = partial[-1]
error = abs(best - partial[-2])
if verbose:
print("Direct error: %s" % ctx.nstr(error))
if error <= tol:
return best
# Check each extrapolation method
if TRY_RICHARDSON:
value, maxc = ctx.richardson(partial)
# Convergence
richardson_error = abs(value - last_richardson_value)
if verbose:
print("Richardson error: %s" % ctx.nstr(richardson_error))
# Convergence
if richardson_error <= tol:
return value
last_richardson_value = value
# Unreliable due to cancellation
if ctx.eps*maxc > tol:
if verbose:
print("Ran out of precision for Richardson")
TRY_RICHARDSON = False
if richardson_error < error:
error = richardson_error
best = value
if TRY_SHANKS:
shanks_table = ctx.shanks(partial, shanks_table, randomized=True)
row = shanks_table[-1]
if len(row) == 2:
est1 = row[-1]
shanks_error = 0
else:
est1, maxc, est2 = row[-1], abs(row[-2]), row[-3]
shanks_error = abs(est1-est2)
if verbose:
print("Shanks error: %s" % ctx.nstr(shanks_error))
if shanks_error <= tol:
return est1
if ctx.eps*maxc > tol:
if verbose:
print("Ran out of precision for Shanks")
TRY_SHANKS = False
if shanks_error < error:
error = shanks_error
best = est1
for L in summer:
est, lerror = L.update_psum(partial)
if verbose:
print("%s error: %s" % (L.name, ctx.nstr(lerror)))
if lerror <= tol:
return est
if lerror < error:
error = lerror
best = est
if TRY_EULER_MACLAURIN:
if ctx.mpc(ctx.sign(partial[-1]) / ctx.sign(partial[-2])).ae(-1):
if verbose:
print ("NOT using Euler-Maclaurin: the series appears"
" to be alternating, so numerical\n quadrature"
" will most likely fail")
TRY_EULER_MACLAURIN = False
else:
value, em_error = emfun(index, tol)
value += partial[-1]
if verbose:
print("Euler-Maclaurin error: %s" % ctx.nstr(em_error))
if em_error <= tol:
return value
if em_error < error:
best = value
finally:
ctx.prec = orig
if strict:
raise ctx.NoConvergence
if verbose:
print("Warning: failed to converge to target accuracy")
return best
@defun
def nsum(ctx, f, *intervals, **options):
r"""
Computes the sum
.. math :: S = \sum_{k=a}^b f(k)
where `(a, b)` = *interval*, and where `a = -\infty` and/or
`b = \infty` are allowed, or more generally
.. math :: S = \sum_{k_1=a_1}^{b_1} \cdots
\sum_{k_n=a_n}^{b_n} f(k_1,\ldots,k_n)
if multiple intervals are given.
Two examples of infinite series that can be summed by :func:`~mpmath.nsum`,
where the first converges rapidly and the second converges slowly,
are::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> nsum(lambda n: 1/fac(n), [0, inf])
2.71828182845905
>>> nsum(lambda n: 1/n**2, [1, inf])
1.64493406684823
When appropriate, :func:`~mpmath.nsum` applies convergence acceleration to
accurately estimate the sums of slowly convergent series. If the series is
finite, :func:`~mpmath.nsum` currently does not attempt to perform any
extrapolation, and simply calls :func:`~mpmath.fsum`.
Multidimensional infinite series are reduced to a single-dimensional
series over expanding hypercubes; if both infinite and finite dimensions
are present, the finite ranges are moved innermost. For more advanced
control over the summation order, use nested calls to :func:`~mpmath.nsum`,
or manually rewrite the sum as a single-dimensional series.
**Options**
*tol*
Desired maximum final error. Defaults roughly to the
epsilon of the working precision.
*method*
Which summation algorithm to use (described below).
Default: ``'richardson+shanks'``.
*maxterms*
Cancel after at most this many terms. Default: 10*dps.
*steps*
An iterable giving the number of terms to add between
each extrapolation attempt. The default sequence is
[10, 20, 30, 40, ...]. For example, if you know that
approximately 100 terms will be required, efficiency might be
improved by setting this to [100, 10]. Then the first
extrapolation will be performed after 100 terms, the second
after 110, etc.
*verbose*
Print details about progress.
*ignore*
If enabled, any term that raises ``ArithmeticError``
or ``ValueError`` (e.g. through division by zero) is replaced
by a zero. This is convenient for lattice sums with
a singular term near the origin.
**Methods**
Unfortunately, an algorithm that can efficiently sum any infinite
series does not exist. :func:`~mpmath.nsum` implements several different
algorithms that each work well in different cases. The *method*
keyword argument selects a method.
The default method is ``'r+s'``, i.e. both Richardson extrapolation
and Shanks transformation is attempted. A slower method that
handles more cases is ``'r+s+e'``. For very high precision
summation, or if the summation needs to be fast (for example if
multiple sums need to be evaluated), it is a good idea to
investigate which one method works best and only use that.
``'richardson'`` / ``'r'``:
Uses Richardson extrapolation. Provides useful extrapolation
when `f(k) \sim P(k)/Q(k)` or when `f(k) \sim (-1)^k P(k)/Q(k)`
for polynomials `P` and `Q`. See :func:`~mpmath.richardson` for
additional information.
``'shanks'`` / ``'s'``:
Uses Shanks transformation. Typically provides useful
extrapolation when `f(k) \sim c^k` or when successive terms
alternate signs. Is able to sum some divergent series.
See :func:`~mpmath.shanks` for additional information.
``'levin'`` / ``'l'``:
Uses the Levin transformation. It performs better than the Shanks
transformation for logarithmic convergent or alternating divergent
series. The ``'levin_variant'``-keyword selects the variant. Valid
choices are "u", "t", "v" and "all" whereby "all" uses all three
u,t and v simultanously (This is good for performance comparison in
conjunction with "verbose=True"). Instead of the Levin transform one can
also use the Sidi-S transform by selecting the method ``'sidi'``.
See :func:`~mpmath.levin` for additional details.
``'alternating'`` / ``'a'``:
This is the convergence acceleration of alternating series developped
by Cohen, Villegras and Zagier.
See :func:`~mpmath.cohen_alt` for additional details.
``'euler-maclaurin'`` / ``'e'``:
Uses the Euler-Maclaurin summation formula to approximate
the remainder sum by an integral. This requires high-order
numerical derivatives and numerical integration. The advantage
of this algorithm is that it works regardless of the
decay rate of `f`, as long as `f` is sufficiently smooth.
See :func:`~mpmath.sumem` for additional information.
``'direct'`` / ``'d'``:
Does not perform any extrapolation. This can be used
(and should only be used for) rapidly convergent series.
The summation automatically stops when the terms
decrease below the target tolerance.
**Basic examples**
A finite sum::
>>> nsum(lambda k: 1/k, [1, 6])
2.45
Summation of a series going to negative infinity and a doubly
infinite series::
>>> nsum(lambda k: 1/k**2, [-inf, -1])
1.64493406684823
>>> nsum(lambda k: 1/(1+k**2), [-inf, inf])
3.15334809493716
:func:`~mpmath.nsum` handles sums of complex numbers::
>>> nsum(lambda k: (0.5+0.25j)**k, [0, inf])
(1.6 + 0.8j)
The following sum converges very rapidly, so it is most
efficient to sum it by disabling convergence acceleration::
>>> mp.dps = 1000
>>> a = nsum(lambda k: -(-1)**k * k**2 / fac(2*k), [1, inf],
... method='direct')
>>> b = (cos(1)+sin(1))/4
>>> abs(a-b) < mpf('1e-998')
True
**Examples with Richardson extrapolation**
Richardson extrapolation works well for sums over rational
functions, as well as their alternating counterparts::
>>> mp.dps = 50
>>> nsum(lambda k: 1 / k**3, [1, inf],
... method='richardson')
1.2020569031595942853997381615114499907649862923405
>>> zeta(3)
1.2020569031595942853997381615114499907649862923405
>>> nsum(lambda n: (n + 3)/(n**3 + n**2), [1, inf],
... method='richardson')
2.9348022005446793094172454999380755676568497036204
>>> pi**2/2-2
2.9348022005446793094172454999380755676568497036204
>>> nsum(lambda k: (-1)**k / k**3, [1, inf],
... method='richardson')
-0.90154267736969571404980362113358749307373971925537
>>> -3*zeta(3)/4
-0.90154267736969571404980362113358749307373971925538
**Examples with Shanks transformation**
The Shanks transformation works well for geometric series
and typically provides excellent acceleration for Taylor
series near the border of their disk of convergence.
Here we apply it to a series for `\log(2)`, which can be
seen as the Taylor series for `\log(1+x)` with `x = 1`::
>>> nsum(lambda k: -(-1)**k/k, [1, inf],
... method='shanks')
0.69314718055994530941723212145817656807550013436025
>>> log(2)
0.69314718055994530941723212145817656807550013436025
Here we apply it to a slowly convergent geometric series::
>>> nsum(lambda k: mpf('0.995')**k, [0, inf],
... method='shanks')
200.0
Finally, Shanks' method works very well for alternating series
where `f(k) = (-1)^k g(k)`, and often does so regardless of
the exact decay rate of `g(k)`::
>>> mp.dps = 15
>>> nsum(lambda k: (-1)**(k+1) / k**1.5, [1, inf],
... method='shanks')
0.765147024625408
>>> (2-sqrt(2))*zeta(1.5)/2
0.765147024625408
The following slowly convergent alternating series has no known
closed-form value. Evaluating the sum a second time at higher
precision indicates that the value is probably correct::
>>> nsum(lambda k: (-1)**k / log(k), [2, inf],
... method='shanks')
0.924299897222939
>>> mp.dps = 30
>>> nsum(lambda k: (-1)**k / log(k), [2, inf],
... method='shanks')
0.92429989722293885595957018136
**Examples with Levin transformation**
The following example calculates Euler's constant as the constant term in
the Laurent expansion of zeta(s) at s=1. This sum converges extremly slow
because of the logarithmic convergence behaviour of the Dirichlet series
for zeta.
>>> mp.dps = 30
>>> z = mp.mpf(10) ** (-10)
>>> a = mp.nsum(lambda n: n**(-(1+z)), [1, mp.inf], method = "levin") - 1 / z
>>> print(mp.chop(a - mp.euler, tol = 1e-10))
0.0
Now we sum the zeta function outside its range of convergence
(attention: This does not work at the negative integers!):
>>> mp.dps = 15
>>> w = mp.nsum(lambda n: n ** (2 + 3j), [1, mp.inf], method = "levin", levin_variant = "v")
>>> print(mp.chop(w - mp.zeta(-2-3j)))
0.0
The next example resummates an asymptotic series expansion of an integral
related to the exponential integral.
>>> mp.dps = 15
>>> z = mp.mpf(10)
>>> # exact = mp.quad(lambda x: mp.exp(-x)/(1+x/z),[0,mp.inf])
>>> exact = z * mp.exp(z) * mp.expint(1,z) # this is the symbolic expression for the integral
>>> w = mp.nsum(lambda n: (-1) ** n * mp.fac(n) * z ** (-n), [0, mp.inf], method = "sidi", levin_variant = "t")
>>> print(mp.chop(w - exact))
0.0
Following highly divergent asymptotic expansion needs some care. Firstly we
need copious amount of working precision. Secondly the stepsize must not be
chosen to large, otherwise nsum may miss the point where the Levin transform
converges and reach the point where only numerical garbage is produced due to
numerical cancellation.
>>> mp.dps = 15
>>> z = mp.mpf(2)
>>> # exact = mp.quad(lambda x: mp.exp( -x * x / 2 - z * x ** 4), [0,mp.inf]) * 2 / mp.sqrt(2 * mp.pi)
>>> exact = mp.exp(mp.one / (32 * z)) * mp.besselk(mp.one / 4, mp.one / (32 * z)) / (4 * mp.sqrt(z * mp.pi)) # this is the symbolic expression for the integral
>>> w = mp.nsum(lambda n: (-z)**n * mp.fac(4 * n) / (mp.fac(n) * mp.fac(2 * n) * (4 ** n)),
... [0, mp.inf], method = "levin", levin_variant = "t", workprec = 8*mp.prec, steps = [2] + [1 for x in xrange(1000)])
>>> print(mp.chop(w - exact))
0.0
The hypergeoemtric function can also be summed outside its range of convergence:
>>> mp.dps = 15
>>> z = 2 + 1j
>>> exact = mp.hyp2f1(2 / mp.mpf(3), 4 / mp.mpf(3), 1 / mp.mpf(3), z)
>>> f = lambda n: mp.rf(2 / mp.mpf(3), n) * mp.rf(4 / mp.mpf(3), n) * z**n / (mp.rf(1 / mp.mpf(3), n) * mp.fac(n))
>>> v = mp.nsum(f, [0, mp.inf], method = "levin", steps = [10 for x in xrange(1000)])
>>> print(mp.chop(exact-v))
0.0
**Examples with Cohen's alternating series resummation**
The next example sums the alternating zeta function:
>>> v = mp.nsum(lambda n: (-1)**(n-1) / n, [1, mp.inf], method = "a")
>>> print(mp.chop(v - mp.log(2)))
0.0
The derivate of the alternating zeta function outside its range of
convergence:
>>> v = mp.nsum(lambda n: (-1)**n * mp.log(n) * n, [1, mp.inf], method = "a")
>>> print(mp.chop(v - mp.diff(lambda s: mp.altzeta(s), -1)))
0.0
**Examples with Euler-Maclaurin summation**
The sum in the following example has the wrong rate of convergence
for either Richardson or Shanks to be effective.
>>> f = lambda k: log(k)/k**2.5
>>> mp.dps = 15
>>> nsum(f, [1, inf], method='euler-maclaurin')
0.38734195032621
>>> -diff(zeta, 2.5)
0.38734195032621
Increasing ``steps`` improves speed at higher precision::
>>> mp.dps = 50
>>> nsum(f, [1, inf], method='euler-maclaurin', steps=[250])
0.38734195032620997271199237593105101319948228874688
>>> -diff(zeta, 2.5)
0.38734195032620997271199237593105101319948228874688
**Divergent series**
The Shanks transformation is able to sum some *divergent*
series. In particular, it is often able to sum Taylor series
beyond their radius of convergence (this is due to a relation
between the Shanks transformation and Pade approximations;
see :func:`~mpmath.pade` for an alternative way to evaluate divergent
Taylor series). Furthermore the Levin-transform examples above
contain some divergent series resummation.
Here we apply it to `\log(1+x)` far outside the region of
convergence::
>>> mp.dps = 50
>>> nsum(lambda k: -(-9)**k/k, [1, inf],
... method='shanks')
2.3025850929940456840179914546843642076011014886288
>>> log(10)
2.3025850929940456840179914546843642076011014886288
A particular type of divergent series that can be summed
using the Shanks transformation is geometric series.
The result is the same as using the closed-form formula
for an infinite geometric series::
>>> mp.dps = 15
>>> for n in range(-8, 8):
... if n == 1:
... continue
... print("%s %s %s" % (mpf(n), mpf(1)/(1-n),
... nsum(lambda k: n**k, [0, inf], method='shanks')))
...
-8.0 0.111111111111111 0.111111111111111
-7.0 0.125 0.125
-6.0 0.142857142857143 0.142857142857143
-5.0 0.166666666666667 0.166666666666667
-4.0 0.2 0.2
-3.0 0.25 0.25
-2.0 0.333333333333333 0.333333333333333
-1.0 0.5 0.5
0.0 1.0 1.0
2.0 -1.0 -1.0
3.0 -0.5 -0.5
4.0 -0.333333333333333 -0.333333333333333
5.0 -0.25 -0.25
6.0 -0.2 -0.2
7.0 -0.166666666666667 -0.166666666666667
**Multidimensional sums**
Any combination of finite and infinite ranges is allowed for the
summation indices::
>>> mp.dps = 15
>>> nsum(lambda x,y: x+y, [2,3], [4,5])
28.0
>>> nsum(lambda x,y: x/2**y, [1,3], [1,inf])
6.0
>>> nsum(lambda x,y: y/2**x, [1,inf], [1,3])
6.0
>>> nsum(lambda x,y,z: z/(2**x*2**y), [1,inf], [1,inf], [3,4])
7.0
>>> nsum(lambda x,y,z: y/(2**x*2**z), [1,inf], [3,4], [1,inf])
7.0
>>> nsum(lambda x,y,z: x/(2**z*2**y), [3,4], [1,inf], [1,inf])
7.0
Some nice examples of double series with analytic solutions or
reductions to single-dimensional series (see [1])::
>>> nsum(lambda m, n: 1/2**(m*n), [1,inf], [1,inf])
1.60669515241529
>>> nsum(lambda n: 1/(2**n-1), [1,inf])
1.60669515241529
>>> nsum(lambda i,j: (-1)**(i+j)/(i**2+j**2), [1,inf], [1,inf])
0.278070510848213
>>> pi*(pi-3*ln2)/12
0.278070510848213
>>> nsum(lambda i,j: (-1)**(i+j)/(i+j)**2, [1,inf], [1,inf])
0.129319852864168
>>> altzeta(2) - altzeta(1)
0.129319852864168
>>> nsum(lambda i,j: (-1)**(i+j)/(i+j)**3, [1,inf], [1,inf])
0.0790756439455825
>>> altzeta(3) - altzeta(2)
0.0790756439455825
>>> nsum(lambda m,n: m**2*n/(3**m*(n*3**m+m*3**n)),
... [1,inf], [1,inf])
0.28125
>>> mpf(9)/32
0.28125
>>> nsum(lambda i,j: fac(i-1)*fac(j-1)/fac(i+j),
... [1,inf], [1,inf], workprec=400)
1.64493406684823
>>> zeta(2)
1.64493406684823
A hard example of a multidimensional sum is the Madelung constant
in three dimensions (see [2]). The defining sum converges very
slowly and only conditionally, so :func:`~mpmath.nsum` is lucky to
obtain an accurate value through convergence acceleration. The
second evaluation below uses a much more efficient, rapidly
convergent 2D sum::
>>> nsum(lambda x,y,z: (-1)**(x+y+z)/(x*x+y*y+z*z)**0.5,
... [-inf,inf], [-inf,inf], [-inf,inf], ignore=True)
-1.74756459463318
>>> nsum(lambda x,y: -12*pi*sech(0.5*pi * \
... sqrt((2*x+1)**2+(2*y+1)**2))**2, [0,inf], [0,inf])
-1.74756459463318
Another example of a lattice sum in 2D::
>>> nsum(lambda x,y: (-1)**(x+y) / (x**2+y**2), [-inf,inf],
... [-inf,inf], ignore=True)
-2.1775860903036
>>> -pi*ln2
-2.1775860903036
An example of an Eisenstein series::
>>> nsum(lambda m,n: (m+n*1j)**(-4), [-inf,inf], [-inf,inf],
... ignore=True)
(3.1512120021539 + 0.0j)
**References**
1. [Weisstein]_ http://mathworld.wolfram.com/DoubleSeries.html,
2. [Weisstein]_ http://mathworld.wolfram.com/MadelungConstants.html
"""
infinite, g = standardize(ctx, f, intervals, options)
if not infinite:
return +g()
def update(partial_sums, indices):
if partial_sums:
psum = partial_sums[-1]
else:
psum = ctx.zero
for k in indices:
psum = psum + g(ctx.mpf(k))
partial_sums.append(psum)
prec = ctx.prec
def emfun(point, tol):
workprec = ctx.prec
ctx.prec = prec + 10
v = ctx.sumem(g, [point, ctx.inf], tol, error=1)
ctx.prec = workprec
return v
return +ctx.adaptive_extrapolation(update, emfun, options)
def wrapsafe(f):
def g(*args):
try:
return f(*args)
except (ArithmeticError, ValueError):
return 0
return g
def standardize(ctx, f, intervals, options):
if options.get("ignore"):
f = wrapsafe(f)
finite = []
infinite = []
for k, points in enumerate(intervals):
a, b = ctx._as_points(points)
if b < a:
return False, (lambda: ctx.zero)
if a == ctx.ninf or b == ctx.inf:
infinite.append((k, (a,b)))
else:
finite.append((k, (int(a), int(b))))
if finite:
f = fold_finite(ctx, f, finite)
if not infinite:
return False, lambda: f(*([0]*len(intervals)))
if infinite:
f = standardize_infinite(ctx, f, infinite)
f = fold_infinite(ctx, f, infinite)
args = [0] * len(intervals)
d = infinite[0][0]
def g(k):
args[d] = k
return f(*args)
return True, g
# backwards compatible itertools.product
def cartesian_product(args):
pools = map(tuple, args)
result = [[]]
for pool in pools:
result = [x+[y] for x in result for y in pool]
for prod in result:
yield tuple(prod)
def fold_finite(ctx, f, intervals):
if not intervals:
return f
indices = [v[0] for v in intervals]
points = [v[1] for v in intervals]
ranges = [xrange(a, b+1) for (a,b) in points]
def g(*args):
args = list(args)
s = ctx.zero
for xs in cartesian_product(ranges):
for dim, x in zip(indices, xs):
args[dim] = ctx.mpf(x)
s += f(*args)
return s
#print "Folded finite", indices
return g
# Standardize each interval to [0,inf]
def standardize_infinite(ctx, f, intervals):
if not intervals:
return f
dim, [a,b] = intervals[-1]
if a == ctx.ninf:
if b == ctx.inf:
def g(*args):
args = list(args)
k = args[dim]
if k:
s = f(*args)
args[dim] = -k
s += f(*args)
return s
else:
return f(*args)
else:
def g(*args):
args = list(args)
args[dim] = b - args[dim]
return f(*args)
else:
def g(*args):
args = list(args)
args[dim] += a
return f(*args)
#print "Standardized infinity along dimension", dim, a, b
return standardize_infinite(ctx, g, intervals[:-1])
def fold_infinite(ctx, f, intervals):
if len(intervals) < 2:
return f
dim1 = intervals[-2][0]
dim2 = intervals[-1][0]
# Assume intervals are [0,inf] x [0,inf] x ...
def g(*args):
args = list(args)
#args.insert(dim2, None)
n = int(args[dim1])
s = ctx.zero
#y = ctx.mpf(n)
args[dim2] = ctx.mpf(n) #y
for x in xrange(n+1):
args[dim1] = ctx.mpf(x)
s += f(*args)
args[dim1] = ctx.mpf(n) #ctx.mpf(n)
for y in xrange(n):
args[dim2] = ctx.mpf(y)
s += f(*args)
return s
#print "Folded infinite from", len(intervals), "to", (len(intervals)-1)
return fold_infinite(ctx, g, intervals[:-1])
@defun
def nprod(ctx, f, interval, nsum=False, **kwargs):
r"""
Computes the product
.. math ::
P = \prod_{k=a}^b f(k)
where `(a, b)` = *interval*, and where `a = -\infty` and/or
`b = \infty` are allowed.
By default, :func:`~mpmath.nprod` uses the same extrapolation methods as
:func:`~mpmath.nsum`, except applied to the partial products rather than
partial sums, and the same keyword options as for :func:`~mpmath.nsum` are
supported. If ``nsum=True``, the product is instead computed via
:func:`~mpmath.nsum` as
.. math ::
P = \exp\left( \sum_{k=a}^b \log(f(k)) \right).
This is slower, but can sometimes yield better results. It is
also required (and used automatically) when Euler-Maclaurin
summation is requested.
**Examples**
A simple finite product::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> nprod(lambda k: k, [1, 4])
24.0
A large number of infinite products have known exact values,
and can therefore be used as a reference. Most of the following
examples are taken from MathWorld [1].
A few infinite products with simple values are::
>>> 2*nprod(lambda k: (4*k**2)/(4*k**2-1), [1, inf])
3.141592653589793238462643
>>> nprod(lambda k: (1+1/k)**2/(1+2/k), [1, inf])
2.0
>>> nprod(lambda k: (k**3-1)/(k**3+1), [2, inf])
0.6666666666666666666666667
>>> nprod(lambda k: (1-1/k**2), [2, inf])
0.5
Next, several more infinite products with more complicated
values::
>>> nprod(lambda k: exp(1/k**2), [1, inf]); exp(pi**2/6)
5.180668317897115748416626
5.180668317897115748416626
>>> nprod(lambda k: (k**2-1)/(k**2+1), [2, inf]); pi*csch(pi)
0.2720290549821331629502366
0.2720290549821331629502366
>>> nprod(lambda k: (k**4-1)/(k**4+1), [2, inf])
0.8480540493529003921296502
>>> pi*sinh(pi)/(cosh(sqrt(2)*pi)-cos(sqrt(2)*pi))
0.8480540493529003921296502
>>> nprod(lambda k: (1+1/k+1/k**2)**2/(1+2/k+3/k**2), [1, inf])
1.848936182858244485224927
>>> 3*sqrt(2)*cosh(pi*sqrt(3)/2)**2*csch(pi*sqrt(2))/pi
1.848936182858244485224927
>>> nprod(lambda k: (1-1/k**4), [2, inf]); sinh(pi)/(4*pi)
0.9190194775937444301739244
0.9190194775937444301739244
>>> nprod(lambda k: (1-1/k**6), [2, inf])
0.9826842777421925183244759
>>> (1+cosh(pi*sqrt(3)))/(12*pi**2)
0.9826842777421925183244759
>>> nprod(lambda k: (1+1/k**2), [2, inf]); sinh(pi)/(2*pi)
1.838038955187488860347849
1.838038955187488860347849
>>> nprod(lambda n: (1+1/n)**n * exp(1/(2*n)-1), [1, inf])
1.447255926890365298959138
>>> exp(1+euler/2)/sqrt(2*pi)
1.447255926890365298959138
The following two products are equivalent and can be evaluated in
terms of a Jacobi theta function. Pi can be replaced by any value
(as long as convergence is preserved)::
>>> nprod(lambda k: (1-pi**-k)/(1+pi**-k), [1, inf])
0.3838451207481672404778686
>>> nprod(lambda k: tanh(k*log(pi)/2), [1, inf])
0.3838451207481672404778686
>>> jtheta(4,0,1/pi)
0.3838451207481672404778686
This product does not have a known closed form value::
>>> nprod(lambda k: (1-1/2**k), [1, inf])
0.2887880950866024212788997
A product taken from `-\infty`::
>>> nprod(lambda k: 1-k**(-3), [-inf,-2])
0.8093965973662901095786805
>>> cosh(pi*sqrt(3)/2)/(3*pi)
0.8093965973662901095786805
A doubly infinite product::
>>> nprod(lambda k: exp(1/(1+k**2)), [-inf, inf])
23.41432688231864337420035
>>> exp(pi/tanh(pi))
23.41432688231864337420035
A product requiring the use of Euler-Maclaurin summation to compute
an accurate value::
>>> nprod(lambda k: (1-1/k**2.5), [2, inf], method='e')
0.696155111336231052898125
**References**
1. [Weisstein]_ http://mathworld.wolfram.com/InfiniteProduct.html
"""
if nsum or ('e' in kwargs.get('method', '')):
orig = ctx.prec
try:
# TODO: we are evaluating log(1+eps) -> eps, which is
# inaccurate. This currently works because nsum greatly
# increases the working precision. But we should be
# more intelligent and handle the precision here.
ctx.prec += 10
v = ctx.nsum(lambda n: ctx.ln(f(n)), interval, **kwargs)
finally:
ctx.prec = orig
return +ctx.exp(v)
a, b = ctx._as_points(interval)
if a == ctx.ninf:
if b == ctx.inf:
return f(0) * ctx.nprod(lambda k: f(-k) * f(k), [1, ctx.inf], **kwargs)
return ctx.nprod(f, [-b, ctx.inf], **kwargs)
elif b != ctx.inf:
return ctx.fprod(f(ctx.mpf(k)) for k in xrange(int(a), int(b)+1))
a = int(a)
def update(partial_products, indices):
if partial_products:
pprod = partial_products[-1]
else:
pprod = ctx.one
for k in indices:
pprod = pprod * f(a + ctx.mpf(k))
partial_products.append(pprod)
return +ctx.adaptive_extrapolation(update, None, kwargs)
@defun
def limit(ctx, f, x, direction=1, exp=False, **kwargs):
r"""
Computes an estimate of the limit
.. math ::
\lim_{t \to x} f(t)
where `x` may be finite or infinite.
For finite `x`, :func:`~mpmath.limit` evaluates `f(x + d/n)` for
consecutive integer values of `n`, where the approach direction
`d` may be specified using the *direction* keyword argument.
For infinite `x`, :func:`~mpmath.limit` evaluates values of
`f(\mathrm{sign}(x) \cdot n)`.
If the approach to the limit is not sufficiently fast to give
an accurate estimate directly, :func:`~mpmath.limit` attempts to find
the limit using Richardson extrapolation or the Shanks
transformation. You can select between these methods using
the *method* keyword (see documentation of :func:`~mpmath.nsum` for
more information).
**Options**
The following options are available with essentially the
same meaning as for :func:`~mpmath.nsum`: *tol*, *method*, *maxterms*,
*steps*, *verbose*.
If the option *exp=True* is set, `f` will be
sampled at exponentially spaced points `n = 2^1, 2^2, 2^3, \ldots`
instead of the linearly spaced points `n = 1, 2, 3, \ldots`.
This can sometimes improve the rate of convergence so that
:func:`~mpmath.limit` may return a more accurate answer (and faster).
However, do note that this can only be used if `f`
supports fast and accurate evaluation for arguments that
are extremely close to the limit point (or if infinite,
very large arguments).
**Examples**
A basic evaluation of a removable singularity::
>>> from mpmath import *
>>> mp.dps = 30; mp.pretty = True
>>> limit(lambda x: (x-sin(x))/x**3, 0)
0.166666666666666666666666666667
Computing the exponential function using its limit definition::
>>> limit(lambda n: (1+3/n)**n, inf)
20.0855369231876677409285296546
>>> exp(3)
20.0855369231876677409285296546
A limit for `\pi`::
>>> f = lambda n: 2**(4*n+1)*fac(n)**4/(2*n+1)/fac(2*n)**2
>>> limit(f, inf)
3.14159265358979323846264338328
Calculating the coefficient in Stirling's formula::
>>> limit(lambda n: fac(n) / (sqrt(n)*(n/e)**n), inf)
2.50662827463100050241576528481
>>> sqrt(2*pi)
2.50662827463100050241576528481
Evaluating Euler's constant `\gamma` using the limit representation
.. math ::
\gamma = \lim_{n \rightarrow \infty } \left[ \left(
\sum_{k=1}^n \frac{1}{k} \right) - \log(n) \right]
(which converges notoriously slowly)::
>>> f = lambda n: sum([mpf(1)/k for k in range(1,int(n)+1)]) - log(n)
>>> limit(f, inf)
0.577215664901532860606512090082
>>> +euler
0.577215664901532860606512090082
With default settings, the following limit converges too slowly
to be evaluated accurately. Changing to exponential sampling
however gives a perfect result::
>>> f = lambda x: sqrt(x**3+x**2)/(sqrt(x**3)+x)
>>> limit(f, inf)
0.992831158558330281129249686491
>>> limit(f, inf, exp=True)
1.0
"""
if ctx.isinf(x):
direction = ctx.sign(x)
g = lambda k: f(ctx.mpf(k+1)*direction)
else:
direction *= ctx.one
g = lambda k: f(x + direction/(k+1))
if exp:
h = g
g = lambda k: h(2**k)
def update(values, indices):
for k in indices:
values.append(g(k+1))
# XXX: steps used by nsum don't work well
if not 'steps' in kwargs:
kwargs['steps'] = [10]
return +ctx.adaptive_extrapolation(update, None, kwargs)
| 73,288 | 33.635633 | 169 |
py
|
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