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