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<title>Methods — AIM-CU 1.0.0 documentation</title> |
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<section id="methods"> |
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<h1>Methods<a class="headerlink" href="#methods" title="Link to this heading">¶</a></h1> |
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<section id="cusum-parameters"> |
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<h2>CUSUM parameters<a class="headerlink" href="#cusum-parameters" title="Link to this heading">¶</a></h2> |
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<table class="docutils align-default" id="id1"> |
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<caption><span class="caption-text">CUSUM parameters</span><a class="headerlink" href="#id1" title="Link to this table">¶</a></caption> |
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<thead> |
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<tr class="row-odd"><th class="head"><p>Parameter</p></th> |
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<th class="head"><p>Description</p></th> |
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</thead> |
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<tbody> |
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<tr class="row-even"><td><p>μ_in</p></td> |
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<td><p>The mean of the performance metric when the process is in-control, i.e., when there is no performance drift</p></td> |
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</tr> |
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<tr class="row-odd"><td><p>ARL_0</p></td> |
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<td><p>Number of observations before the control chart signals a false detection</p></td> |
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</tr> |
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<tr class="row-even"><td><p>σ_in</p></td> |
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<td><p>The in-control standard deviation of the metric</p></td> |
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</tr> |
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<tr class="row-odd"><td><p>ARL_1</p></td> |
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<td><p>Number of observations before the control chart signals a true detection</p></td> |
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</tr> |
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<tr class="row-even"><td><p>k</p></td> |
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<td><p>The normalized reference value, which is related to the magnitude of change that one is interested in detecting. k = 0.5 is the default choice for detecting a unit standard deviation change</p></td> |
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</tr> |
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<tr class="row-odd"><td><p>S_hi</p></td> |
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<td><p>Cumulative sum of positive changes in the metric</p></td> |
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</tr> |
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<tr class="row-even"><td><p>h</p></td> |
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<td><p>The normalized threshold or control limit (default =4). This threshold determines when the control chart signals a detection</p></td> |
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</tr> |
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<tr class="row-odd"><td><p>S_lo</p></td> |
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<td><p>Cumulative sum of negative changes in the metric</p></td> |
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</tr> |
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</tbody> |
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</table> |
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</section> |
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<section id="cusum-chart"> |
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<h2>CUSUM chart<a class="headerlink" href="#cusum-chart" title="Link to this heading">¶</a></h2> |
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<p>A two-sided CUSUM control chart computes the cumulative differences or |
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deviations of individual observations from the target mean (or |
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in-control mean, <span class="math notranslate nohighlight">\(\mu_{in}\)</span>). The positive and negative cumulative |
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sums are calculated:</p> |
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<div class="math notranslate nohighlight"> |
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\[\begin{split}\\ S_{hi}(d) = max(0, S_{hi}(d-1) + x_d - \hat{\mu}_{in} - K) |
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\\ S_{lo}(d) = max(0, S_{lo}(d-1) - x_d + \hat{\mu}_{in} - K)\end{split}\]</div> |
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<p>where <em>d</em> denotes a unit of time, <span class="math notranslate nohighlight">\(x_d\)</span> is the value of quantity |
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being monitored at time <span class="math notranslate nohighlight">\(d\)</span>, <span class="math notranslate nohighlight">\(\hat{\mu}_{in}\)</span> is the |
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in-control mean of <span class="math notranslate nohighlight">\(x_d\)</span>, and <span class="math notranslate nohighlight">\(K\)</span> is a “reference value” |
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related to the magnitude of change that one is interested in detecting. |
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<span class="math notranslate nohighlight">\(S_{hi}\)</span> and <span class="math notranslate nohighlight">\(S_{lo}\)</span> are the cumulative sum of positive and |
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negative changes. To detect a change in the observed values from the |
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in-control mean, the CUSUM scheme accumulates deviations that are |
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<span class="math notranslate nohighlight">\(K\)</span> units away from the in-control mean. Let <span class="math notranslate nohighlight">\(\sigma_{in}\)</span> |
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denote the in-control standard deviation of <span class="math notranslate nohighlight">\(x_d\)</span>.</p> |
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<p class="caption" role="heading"><span class="caption-text">Contents:</span></p> |
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<ul class="current"> |
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<li class="toctree-l1 current"><a class="current reference internal" href="#">Methods</a><ul> |
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<li class="toctree-l2"><a class="reference internal" href="#cusum-parameters">CUSUM parameters</a></li> |
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<li class="toctree-l2"><a class="reference internal" href="#cusum-chart">CUSUM chart</a></li> |
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<li class="toctree-l1"><a class="reference internal" href="ref_cusum.html">CUSUM</a></li> |
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<li class="toctree-l1"><a class="reference internal" href="ref_theoretical.html">ARLTheoretical</a></li> |
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<li class="toctree-l1"><a class="reference internal" href="ref_utils.html">Utils</a></li> |
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