{
"cells": [
{
"cell_type": "markdown",
"id": "20fce00f",
"metadata": {},
"source": [
"# ML, Data Analysis\n",
"### Probability: covariance matrix\n",
"\n",
"The **covariance matrix** extends the idea of variance and covariance for n-dimensional random vectors. Thus, the vocariance matrix for an n-dimensional random vector $X$ is $n\\times n$ symmetric matrix where:\n",
"- diagonal elements are variance of each component of vector $X$\n",
"- off-diagonal elements are covariance between componets of vector $X$\n",
"\n",
"Formally, the **covariance matrix** $cov(X)$, also called **auto-covariance matrix**, for a random vector $\\boldsymbol{X}=[X_1,X_2,...,X_n]^T$ is computed by:\n",
"\n",
"$\\large cov(\\boldsymbol{X})=E[(\\boldsymbol{X}-E[\\boldsymbol{X}])(\\boldsymbol{X}-E(\\boldsymbol{X}))^T]$\n",
" Or equivalently: \n",
"$\\large cov(\\boldsymbol{X})=E[\\boldsymbol{X}\\boldsymbol{X}^T]-E[\\boldsymbol{X}]E[\\boldsymbol{X}]^T$\n",
"\n",
"\n",
"where $E[\\boldsymbol{X}]$ is the mean vector of $\\boldsymbol{X}$. In fact, it is the vector of means of the components: \n",
"\n",
"$E[\\boldsymbol{X}]=[E[X_1],E[X_2],...,E[X_n]]^T$\n",
"\n",
" Some properties of covariance matrix $cov(\\boldsymbol{X})$:\n",
"1. **Symmetry:** $cov(\\boldsymbol{X})=cov(\\boldsymbol{X})^T$\n",
"2. **Positive semi-definite:** All eigenvalues of matrix $cov(\\boldsymbol{X})$ are nonnegative. In other words, for any vector $v$, we have: \\boldsymbol{v}^Tcov(\\boldsymbol{X})\\boldsymbol{V}>=0.\n",
"3. **Linear transformation:** For a matrix $A$ and random vector $X$, we have: $cov(A\\cdot \\boldsymbol{X})=A\\cdot cov(\\boldsymbol{X})\\cdot A^T$\n",
"\n",
"**Hint:** We can express the covariance of sum of two random vectors $\\boldsymbol{X}$ and $\\boldsymbol{Y}$ by:\n",
"\n",
"$\\large cov(\\boldsymbol{X}+\\boldsymbol{Y})=cov(\\boldsymbol{X})+cov(\\boldsymbol{Y})+\n",
"cov(\\boldsymbol{X},\\boldsymbol{Y})+cov(\\boldsymbol{Y},\\boldsymbol{X})$\n",
"\n",
"\n",
"where $cov(\\boldsymbol{X},\\boldsymbol{Y})$ is the **cross-covariance** of $\\boldsymbol{X}$ and $\\boldsymbol{Y}$ defined by:\n",
"\n",
"$cov(\\boldsymbol{X},\\boldsymbol{Y})=E[(\\boldsymbol{X}-E[\\boldsymbol{X}])(\\boldsymbol{Y}-E(\\boldsymbol{Y}))^T]$\n",
" Or equivalently: \n",
"$cov(\\boldsymbol{X},\\boldsymbol{Y})=E[\\boldsymbol{X}\\boldsymbol{Y}^T]-E[\\boldsymbol{X}]E[\\boldsymbol{Y}]^T$\n",
"\n",
"\n",
"**Hint:** We also can show that: $cov(\\boldsymbol{Y},\\boldsymbol{X})=cov(\\boldsymbol{X},\\boldsymbol{Y})^T$.\n",
"\n",
"**Reminder:** For the topic on the **expected value**, see the relevent post on this repository.\n",
"\n",
"\n",
"In the following, we compute the covariance matrix for a sample of two-dimensional random vector.\n",
"\n",
"\n",
"https://github.com/ostad-ai/Machine-Learning\n",
" Explanation: https://www.pinterest.com/HamedShahHosseini/Machine-Learning/background-knowledge"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "abc3abf4",
"metadata": {},
"outputs": [],
"source": [
"# Import required modules\n",
"import numpy as np\n",
"import matplotlib.pyplot as plt"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "7c7a622d",
"metadata": {},
"outputs": [
{
"data": {
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"text/plain": [
""
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Cov. matrix, Manual calculation:\n",
" [[0.76366245 0.69698309]\n",
" [0.69698309 0.70075292]]\n",
"\n",
"Cov. matrix, NumPy result:\n",
" [[0.76366245 0.69698309]\n",
" [0.69698309 0.70075292]]\n"
]
}
],
"source": [
"# Example dataset of 2D vectors (e.g., [x, y] coordinates)\n",
"data = np.array([[1.0, 2.0],\n",
" [1.5, 2.5],\n",
" [2.0, 3.0],\n",
" [2.5, 3.5],\n",
" [3.0, 4.0]]) # Shape: (5 samples, 2 dimensions)\n",
"noisy_data=data+np.random.normal(0, .15, data.shape)\n",
"\n",
"# Method 1: Manual calculation\n",
"def manual_covariance(data,bias=False):\n",
" n = data.shape[0]\n",
" mean = np.mean(data, axis=0,keepdims=True)\n",
" centered = data - mean\n",
" cov=centered.T @ centered\n",
" if bias: cov/=n # Population covariance\n",
" else: cov/=(n-1) # Sample covariance\n",
" return cov\n",
"\n",
"# Method 2: Using NumPy; rowvar=False means each row is a sample\n",
"# When bias=False, sample cov. is computed (1/(n-1) is used)\n",
"numpy_cov = np.cov(noisy_data, rowvar=False, bias=False)\n",
"\n",
"# Plot the results\n",
"plt.scatter(noisy_data[:, 0], noisy_data[:, 1])\n",
"plt.xlabel('x'); plt.ylabel('y')\n",
"plt.title('Noisy data points $(x_i,y_i)$')\n",
"plt.axis('equal'); plt.show()\n",
"print('Cov. matrix, Manual calculation:\\n', manual_covariance(noisy_data))\n",
"print('\\nCov. matrix, NumPy result:\\n', numpy_cov)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "a6d476cc",
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "code",
"execution_count": null,
"id": "d41ebef9",
"metadata": {},
"outputs": [],
"source": []
}
],
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"kernelspec": {
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"language": "python",
"name": "python3"
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