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06 Full example.html
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06 Full example.html
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<!DOCTYPE html>
<html>
<head><meta charset="utf-8" />
<title>06 Full example</title>
<script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
<style type="text/css">
/*!
*
* Twitter Bootstrap
*
*//*! normalize.css v3.0.2 | MIT License | git.io/normalize */html{font-family:sans-serif;-ms-text-size-adjust:100%;-webkit-text-size-adjust:100%;font-size:10px;-webkit-tap-highlight-color:transparent}article,aside,details,figcaption,figure,footer,header,hgroup,main,menu,nav,section,summary{display:block}audio,canvas,progress,video{display:inline-block;vertical-align:baseline}audio:not([controls]){display:none;height:0}[hidden],template{display:none}a{background-color:transparent}a:active,a:hover{outline:0}abbr[title]{border-bottom:1px dotted}b,optgroup,strong{font-weight:700}dfn{font-style:italic}h1{font-size:2em;margin:.67em 0}mark{background:#ff0;color:#000}small{font-size:80%}sub,sup{font-size:75%;line-height:0;position:relative;vertical-align:baseline}sup{top:-.5em}sub{bottom:-.25em}img{border:0;vertical-align:middle}svg:not(:root){overflow:hidden}hr{-moz-box-sizing:content-box;box-sizing:content-box;height:0}pre,textarea{overflow:auto}code,kbd,pre,samp{font-size:1em}button,input,optgroup,select,textarea{color:inherit;font:inherit;margin:0}button{overflow:visible}button,select{text-transform:none}button,html input[type=button],input[type=reset],input[type=submit]{-webkit-appearance:button;cursor:pointer}button[disabled],html input[disabled]{cursor:default}button::-moz-focus-inner,input::-moz-focus-inner{border:0;padding:0}input{line-height:normal}input[type=checkbox],input[type=radio]{box-sizing:border-box;padding:0}input[type=number]::-webkit-inner-spin-button,input[type=number]::-webkit-outer-spin-button{height:auto}input[type=search]::-webkit-search-cancel-button,input[type=search]::-webkit-search-decoration{-webkit-appearance:none}table{border-collapse:collapse;border-spacing:0}td,th{padding:0}/*! Source: https://github.com/h5bp/html5-boilerplate/blob/master/src/css/main.css */@media print{*,:after,:before{background:0 0!important;color:#000!important;box-shadow:none!important;text-shadow:none!important}a,a:visited{text-decoration:underline}a[href]:after{content:" (" attr(href)")"}abbr[title]:after{content:" (" attr(title)")"}a[href^="javascript:"]:after,a[href^="#"]:after{content:""}blockquote,pre{border:1px solid #999;page-break-inside:avoid}thead{display:table-header-group}img,tr{page-break-inside:avoid}img{max-width:100%!important}h2,h3,p{orphans:3;widows:3}h2,h3{page-break-after:avoid}select{background:#fff!important}.navbar{display:none}.btn>.caret,.dropup>.btn>.caret{border-top-color:#000!important}.label{border:1px solid #000}.table{border-collapse:collapse!important}.table td,.table th{background-color:#fff!important}.table-bordered td,.table-bordered th{border:1px solid #ddd!important}}@font-face{font-family:'Glyphicons Halflings';src:url(../components/bootstrap/fonts/glyphicons-halflings-regular.eot);src:url(../components/bootstrap/fonts/glyphicons-halflings-regular.eot?#iefix)format('embedded-opentype'),url(../components/bootstrap/fonts/glyphicons-halflings-regular.woff2)format('woff2'),url(../components/bootstrap/fonts/glyphicons-halflings-regular.woff)format('woff'),url(../components/bootstrap/fonts/glyphicons-halflings-regular.ttf)format('truetype'),url(../components/bootstrap/fonts/glyphicons-halflings-regular.svg#glyphicons_halflingsregular)format('svg')}.glyphicon{position:relative;top:1px;display:inline-block;font-family:'Glyphicons 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<div class=" highlight hl-ipython3"><pre><span class="o">%</span><span class="k">pylab</span> inline
<span class="n">pylab</span><span class="o">.</span><span class="n">rcParams</span><span class="p">[</span><span class="s1">'figure.figsize'</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="mf">16.0</span><span class="p">,</span> <span class="mf">8.0</span><span class="p">)</span>
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<pre>Populating the interactive namespace from numpy and matplotlib
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<h1 id="Final-exercise:-a-full-Monte-Carlo-example">Final exercise: a full Monte Carlo example<a class="anchor-link" href="#Final-exercise:-a-full-Monte-Carlo-example">¶</a></h1>
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<p>The length of a gauge block is to be measured by comparison with a reference block of the same nominal length.
The measurand is the difference $\delta L$ between the measured object and the reference block. The mathematical model for the evaluation of the measurand is<br>
$$ \delta L = L_s + D + d_1 + d_2 - L_s(\delta\alpha(\theta_0 - \Delta)+\alpha_s\delta\theta) - L_{nom} $$</p>
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<thead><tr>
<th style="text-align:left">Quantity</th>
<th>description</th>
<th style="text-align:left">distribution </th>
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<td style="text-align:left"> $L_s$</td>
<td>length of reference block</td>
<td style="text-align:left">t-dist with 18 dof, $\mu = 50000623$, $s=25$ </td>
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<td style="text-align:left"> $D$</td>
<td>average comparator value</td>
<td style="text-align:left">t-dist with 24 dof, $\mu = 215$, $s=6$ </td>
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<td style="text-align:left"> $d_1$</td>
<td>random errors in comparator measurement</td>
<td style="text-align:left">t-dist with 5 dof, $\mu=0$, $s=4$</td>
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<td style="text-align:left"> $d_2$</td>
<td>systematic errors in comparator measurement</td>
<td style="text-align:left">t-dist with 8 dof, $\mu=0$, $s=7$</td>
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<td style="text-align:left"> $\alpha_s$</td>
<td>thermal expansion coefficient of reference</td>
<td style="text-align:left">uniform on $[-9.5e-6,13.5e-6]$</td>
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<td style="text-align:left"> $\theta_0$</td>
<td>average temperature of measured object</td>
<td style="text-align:left">$N(-0.1, 0.2^2)$</td>
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<td style="text-align:left"> $\Delta$</td>
<td>cyclic temperature variation</td>
<td style="text-align:left">$\mathcal{U}(-0.5, 0.5)$</td>
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<td style="text-align:left"> $\delta\theta$</td>
<td>temperature deviation</td>
<td style="text-align:left">curvi-linear $ctrap(a=-0.05, b=0.05, u=0.025)$</td>
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<td style="text-align:left"> $\delta\alpha$</td>
<td>thermal expansion coeff. deviation</td>
<td style="text-align:left">curvi-linear $ctrap(a=-1e-6,b=1e-6,u=0.1e-6)$</td>
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<p>Carry out the corresponding estimation of the measurand $\delta L$ and the evaluation of uncertainties using Monte Carlo simulations with adaptive determination of the number of Monte Carlo trials such that a numerical accuracy of $\delta = 0.5$ is achieved at a 99% level of confidence for the estimate and its associated uncertainty. Also calculate the symmetric interval with 95% coverage probalitity for the measurand.</p>
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<div class=" highlight hl-ipython3"><pre><span class="c1"># function to draw from curvi-linear probability distribution</span>
<span class="c1"># See:</span>
<span class="c1"># R. N. Kacker and J. F. Lawrence, </span>
<span class="c1"># Rectangular distribution whose end points are not exactly known: curvilinear trapezoidal distribution</span>
<span class="c1"># Metrologia, vol. 47, no. 3, pp. 120–126, Mar. 2010.</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="k">class</span> <span class="nc">ctrap</span><span class="p">:</span>
<span class="c1"># Model for rectangular distribution with inexactly known endpoints</span>
<span class="c1"># left end point is R(a,c) and right end point is R(d,b) distributed</span>
<span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">ua</span><span class="p">,</span><span class="n">ub</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">ub</span><span class="p">,</span><span class="nb">float</span><span class="p">):</span>
<span class="n">ub</span> <span class="o">=</span> <span class="n">ua</span>
<span class="bp">self</span><span class="o">.</span><span class="n">a</span> <span class="o">=</span> <span class="n">a</span><span class="o">-</span><span class="n">ua</span>
<span class="bp">self</span><span class="o">.</span><span class="n">b</span> <span class="o">=</span> <span class="n">b</span><span class="o">+</span><span class="n">ub</span>
<span class="bp">self</span><span class="o">.</span><span class="n">c</span> <span class="o">=</span> <span class="n">a</span><span class="o">+</span><span class="n">ua</span>
<span class="bp">self</span><span class="o">.</span><span class="n">d</span> <span class="o">=</span> <span class="n">b</span><span class="o">-</span><span class="n">ub</span>
<span class="k">def</span> <span class="nf">rvs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
<span class="n">u</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">size</span><span class="p">)</span>
<span class="n">v</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">size</span><span class="p">)</span>
<span class="n">w</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">size</span><span class="p">)</span>
<span class="n">z1</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">a</span> <span class="o">+</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">c</span><span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">u</span>
<span class="n">z2</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">d</span> <span class="o">+</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">b</span><span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">d</span><span class="p">)</span><span class="o">*</span><span class="n">v</span>
<span class="k">return</span> <span class="n">z1</span> <span class="o">+</span> <span class="p">(</span><span class="n">z2</span><span class="o">-</span><span class="n">z1</span><span class="p">)</span><span class="o">*</span><span class="n">w</span>
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<div class=" highlight hl-ipython3"><pre><span class="kn">import</span> <span class="nn">scipy.stats</span> <span class="k">as</span> <span class="nn">stats</span>
<span class="k">def</span> <span class="nf">model</span><span class="p">(</span><span class="n">Ls</span><span class="p">,</span> <span class="n">D</span><span class="p">,</span> <span class="n">d1</span><span class="p">,</span> <span class="n">d2</span><span class="p">,</span> <span class="n">alphas</span><span class="p">,</span> <span class="n">theta0</span><span class="p">,</span> <span class="n">Delta</span><span class="p">,</span> <span class="n">dalpha</span><span class="p">,</span> <span class="n">dtheta</span><span class="p">):</span>
<span class="k">return</span> <span class="n">Ls</span> <span class="o">+</span> <span class="n">D</span> <span class="o">+</span> <span class="n">d1</span> <span class="o">+</span> <span class="n">d2</span> <span class="o">-</span> <span class="n">Ls</span><span class="o">*</span><span class="p">(</span><span class="n">dalpha</span><span class="o">*</span><span class="p">(</span><span class="n">theta0</span><span class="o">-</span><span class="n">Delta</span><span class="p">)</span> <span class="o">+</span> <span class="n">alphas</span><span class="o">*</span><span class="n">dtheta</span><span class="p">)</span> <span class="o">-</span> <span class="mi">50000000</span>
<span class="n">Ls</span> <span class="o">=</span> <span class="n">stats</span><span class="o">.</span><span class="n">t</span><span class="p">(</span><span class="mi">18</span><span class="p">,</span> <span class="n">loc</span> <span class="o">=</span> <span class="mi">50000623</span><span class="p">,</span> <span class="n">scale</span> <span class="o">=</span> <span class="mi">25</span><span class="p">)</span>
<span class="n">D</span> <span class="o">=</span> <span class="n">stats</span><span class="o">.</span><span class="n">t</span><span class="p">(</span><span class="mi">24</span><span class="p">,</span> <span class="n">loc</span><span class="o">=</span><span class="mi">215</span><span class="p">,</span> <span class="n">scale</span> <span class="o">=</span> <span class="mi">6</span><span class="p">)</span>
<span class="n">d1</span> <span class="o">=</span> <span class="n">stats</span><span class="o">.</span><span class="n">t</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="n">scale</span> <span class="o">=</span> <span class="mi">4</span><span class="p">)</span>
<span class="n">d2</span> <span class="o">=</span> <span class="n">stats</span><span class="o">.</span><span class="n">t</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="n">scale</span> <span class="o">=</span> <span class="mi">7</span><span class="p">)</span>
<span class="n">alphas</span> <span class="o">=</span> <span class="n">stats</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span> <span class="mf">9.5e-6</span><span class="p">,</span> <span class="n">scale</span> <span class="o">=</span> <span class="mi">4</span><span class="n">e</span><span class="o">-</span><span class="mi">6</span><span class="p">)</span>
<span class="n">theta0</span> <span class="o">=</span> <span class="n">stats</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">loc</span><span class="o">=-</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">scale</span> <span class="o">=</span> <span class="mf">0.2</span><span class="p">)</span>
<span class="n">Delta</span> <span class="o">=</span> <span class="n">stats</span><span class="o">.</span><span class="n">arcsine</span><span class="p">(</span><span class="n">loc</span><span class="o">=-</span><span class="mf">0.5</span><span class="p">)</span>
<span class="n">dalpha</span><span class="o">=</span> <span class="n">ctrap</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="n">e</span><span class="o">-</span><span class="mi">6</span><span class="p">,</span> <span class="mi">1</span><span class="n">e</span><span class="o">-</span><span class="mi">6</span><span class="p">,</span> <span class="mf">0.1e-6</span><span class="p">)</span>
<span class="n">dtheta</span><span class="o">=</span> <span class="n">ctrap</span><span class="p">(</span><span class="o">-</span><span class="mf">0.05</span><span class="p">,</span> <span class="mf">0.05</span><span class="p">,</span> <span class="mf">0.025</span><span class="p">)</span>
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<div class=" highlight hl-ipython3"><pre><span class="n">k</span><span class="o">=</span><span class="mi">1</span>
<span class="k">for</span> <span class="n">dist</span> <span class="ow">in</span> <span class="p">[</span><span class="n">Ls</span><span class="p">,</span> <span class="n">D</span><span class="p">,</span> <span class="n">d1</span><span class="p">,</span> <span class="n">d2</span><span class="p">,</span> <span class="n">alphas</span><span class="p">,</span> <span class="n">theta0</span><span class="p">,</span> <span class="n">Delta</span><span class="p">,</span> <span class="n">dalpha</span><span class="p">,</span> <span class="n">dtheta</span><span class="p">]:</span>
<span class="n">samples</span> <span class="o">=</span> <span class="n">dist</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="mi">100000</span><span class="p">)</span>
<span class="n">figure</span><span class="p">()</span>
<span class="n">hist</span><span class="p">(</span><span class="n">samples</span><span class="p">,</span> <span class="n">bins</span> <span class="o">=</span> <span class="mi">100</span><span class="p">);</span>
</pre></div>
</div>
</div>
</div>
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