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<section id="common-derivations">
<h1>common derivations</h1>
<blockquote>
<div></div></blockquote>
<ol class="arabic" id="time-dependent-from-time-independent-variable">
<li><p>time dependent from time independent variable</p>
<div class="math notranslate nohighlight">
\[\forall t: x(t,:) = x(:)\]</div>
</li>
<li id="derivation-total-uncertainty-from-random-and-systematic-uncertainty"><p>total uncertainty from random and systematic uncertainty</p>
<table class="docutils align-default">
<thead>
<tr class="row-odd"><th class="head"><p>symbol</p></th>
<th class="head"><p>description</p></th>
<th class="head"><p>unit</p></th>
<th class="head"><p>variable name</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(\mu_{x}\)</span></p></td>
<td><p>total uncertainty for a variable x</p></td>
<td><p>same as that of <span class="math notranslate nohighlight">\(x\)</span></p></td>
<td><p><cite><variable>_uncertainty {:}</cite></p></td>
</tr>
<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(\mu^{r}_{x}\)</span></p></td>
<td><p>uncertainty due to random effects
for a variable x</p></td>
<td><p>same as that of <span class="math notranslate nohighlight">\(x\)</span></p></td>
<td><p><cite><variable>_uncertainty_random {:}</cite></p></td>
</tr>
<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(\mu^{s}_{x}\)</span></p></td>
<td><p>uncertainty due to systematic
effects for a variable x</p></td>
<td><p>same as that of <span class="math notranslate nohighlight">\(x\)</span></p></td>
<td><p><cite><variable>_uncertainty_systematic {:}</cite></p></td>
</tr>
</tbody>
</table>
<p>The pattern <cite>:</cite> for the first dimensions can represent any combination of dimensions for which <cite>x {:}</cite> exists.</p>
<div class="math notranslate nohighlight">
\[\mu_{x} = \sqrt{{\mu^{r}_{x}}^2 + {\mu^{s}_{x}}^2}\]</div>
</li>
<li id="derivation-total-uncertainty-from-vertical-covariance"><p>total uncertainty from vertical covariance</p>
<table class="docutils align-default">
<thead>
<tr class="row-odd"><th class="head"><p>symbol</p></th>
<th class="head"><p>description</p></th>
<th class="head"><p>unit</p></th>
<th class="head"><p>variable name</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(S_{x}(i,j)\)</span></p></td>
<td><p>covariance for a variable x</p></td>
<td><p>square of that of <span class="math notranslate nohighlight">\(x\)</span></p></td>
<td><p><cite><variable>_covariance {:,vertical,vertical}</cite></p></td>
</tr>
<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(\mu_{x}(i)\)</span></p></td>
<td><p>total uncertainty for a variable x</p></td>
<td><p>same as that of <span class="math notranslate nohighlight">\(x\)</span></p></td>
<td><p><cite><variable>_uncertainty {:,vertical}</cite></p></td>
</tr>
</tbody>
</table>
<p>The pattern <cite>:</cite> for the dimensions can represent any combination of dimensions for which <cite>x {:,vertical}</cite> exists.</p>
<div class="math notranslate nohighlight">
\[\forall i: \mu_{x}(i) = \sqrt{S(i,i)}\]</div>
</li>
</ol>
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