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03 Propagating through the model function.html
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<!DOCTYPE html>
<html>
<head><meta charset="utf-8" />
<title>03 Propagating through the model function</title>
<script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
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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="Propagating-samples-through-the-model-function">Propagating samples through the model function<a class="anchor-link" href="#Propagating-samples-through-the-model-function">¶</a></h1>
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<p>Assume the measurement model, i.e. the relation between the input quantities and the measurand, to be given as
$$ Y = f(X_1,X_2,\ldots,X_N) $$</p>
<p>with joint PDF associated with the input quantities given as $g_{\mathbf{X}}(\mathbf{\xi})$.</p>
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<p>The aim of the Monte Carlo method is the propagation of samples from the PDF $g_{\mathbf{X}}(\mathbf{\xi})$ through the measurement model $f()$ as a means to calculate samples from the PDF $g_{Y}(\eta)$.</p>
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<p>Basically, two types of implementation have to be considered: for-loops and vectorisation.</p>
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<h4 id="Monte-Carlo-using-for-loops">Monte Carlo using for-loops<a class="anchor-link" href="#Monte-Carlo-using-for-loops">¶</a></h4><p>This is the most easiest way of implementing Monte Carlo for a given function $f()$.</p>
<div class="highlight"><pre><span class="n">Y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">number_of_runs</span><span class="p">)</span>
<span class="n">X1</span><span class="p">,</span><span class="n">X2</span><span class="p">,</span><span class="o">...</span><span class="p">,</span><span class="n">XN</span> <span class="o">=</span> <span class="n">draw_from_gX</span><span class="p">(</span><span class="n">number_of_runs</span><span class="p">)</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">number_of_runs</span><span class="p">):</span>
<span class="n">Y</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">model</span><span class="p">(</span><span class="n">X1</span><span class="p">[</span><span class="n">k</span><span class="p">],</span><span class="n">X2</span><span class="p">[</span><span class="n">k</span><span class="p">],</span><span class="o">...</span><span class="p">,</span><span class="n">XN</span><span class="p">[</span><span class="n">k</span><span class="p">])</span>
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<h4 id="Monte-Carlo-using-vectorisation">Monte Carlo using vectorisation<a class="anchor-link" href="#Monte-Carlo-using-vectorisation">¶</a></h4><p>For scripting languages, such as Python, Matlab or LabView, a for-loop is often very inefficient. Instead, a vectorisation of the model function should be considered.</p>
<p>For instance, let the model function be
$$ Y = X_1 \sin (2\pi X_2) - X_3^2 $$</p>
<p>The vectorized Monte Carlo method for this model is given as</p>
<div class="highlight"><pre><span class="n">X1</span><span class="p">,</span><span class="n">X2</span><span class="p">,</span><span class="n">X3</span> <span class="o">=</span> <span class="n">draw_from_gX</span><span class="p">(</span><span class="n">number_of_runs</span><span class="p">)</span>
<span class="n">Y</span> <span class="o">=</span> <span class="n">X1</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">X2</span><span class="p">)</span> <span class="o">-</span> <span class="n">X3</span><span class="o">**</span><span class="mi">2</span>
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<h3 id="Exercise-3.1">Exercise 3.1<a class="anchor-link" href="#Exercise-3.1">¶</a></h3><p>Carry out a Monte Carlo propagation of samples through the model
$$ Y = \frac{X_2^2}{\sqrt{1-X_1^2}}$$
with knowledge about the input quantities given as</p>
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<li><p>The value of $X_1$ satisfies $-0.8 \leq x_1 \leq 0.8$</p>
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<li><p>$X_2$ has best estimate $x_2 = 3.4$ with associated standard uncertainty $\sigma=0.5$ determined from 12 repeated measurements</p>
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<div class=" highlight hl-ipython3"><pre><span class="c1"># solution for exercise 3.1</span>
<span class="kn">from</span> <span class="nn">scipy.stats</span> <span class="k">import</span> <span class="n">uniform</span><span class="p">,</span> <span class="n">t</span>
<span class="c1"># random draws from input quantities</span>
<span class="n">draws</span> <span class="o">=</span> <span class="mi">10000</span>
<span class="n">X1</span> <span class="o">=</span> <span class="n">uniform</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">loc</span><span class="o">=-</span><span class="mf">0.8</span><span class="p">,</span> <span class="n">scale</span><span class="o">=</span><span class="mf">1.6</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="n">draws</span><span class="p">)</span>
<span class="n">X2</span> <span class="o">=</span> <span class="n">t</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="n">loc</span><span class="o">=</span><span class="mf">3.4</span><span class="p">,</span> <span class="n">scale</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="n">draws</span><span class="p">)</span>
<span class="c1"># evaluate model function</span>
<span class="n">Y</span> <span class="o">=</span> <span class="n">X2</span><span class="o">**</span><span class="mi">2</span> <span class="o">/</span> <span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">X1</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
<span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span><span class="mi">6</span><span class="p">))</span>
<span class="n">subplot</span><span class="p">(</span><span class="mi">131</span><span class="p">)</span>
<span class="n">hist</span><span class="p">(</span><span class="n">X1</span><span class="p">,</span><span class="n">bins</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">edgecolor</span><span class="o">=</span><span class="s2">"none"</span><span class="p">)</span>
<span class="n">subplot</span><span class="p">(</span><span class="mi">132</span><span class="p">)</span>
<span class="n">hist</span><span class="p">(</span><span class="n">X2</span><span class="p">,</span><span class="n">bins</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">edgecolor</span><span class="o">=</span><span class="s2">"none"</span><span class="p">)</span>
<span class="n">subplot</span><span class="p">(</span><span class="mi">133</span><span class="p">)</span>
<span class="n">hist</span><span class="p">(</span><span class="n">Y</span><span class="p">,</span><span class="n">bins</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">edgecolor</span><span class="o">=</span><span class="s2">"none"</span><span class="p">)</span>
<span class="n">suptitle</span><span class="p">(</span><span class="s2">"PDFs associated with input and output quantities"</span><span class="p">,</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">);</span>
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<h3 id="Exercise-3.2">Exercise 3.2<a class="anchor-link" href="#Exercise-3.2">¶</a></h3><p>Carry out a Monte Carlo propagation for the following parameter estimation problem.</p>
<p>For a measurement device under test, a parametrised calibration curve is to be determined by means of a non-linear least-squares fit to a set of measured pairs $(t_i, x_i)$ assuming as functional relationship
$$ x_i = f(t_i) = a\cdot\exp (-b t_i)\cdot \sin (c t_i) $$
The measurement data is assumed to be disturbed by white noise with a standard deviation of $\sigma=0.3$. The values $x_i$ are all assumed to be obtained independently, i.e. to be uncorrelated.</p>
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<p>For the fitting of the parameters use</p>
<div class="highlight"><pre><span class="kn">from</span> <span class="nn">scipy.optimize</span> <span class="kn">import</span> <span class="n">curve_fit</span>
<span class="n">noise</span> <span class="o">=</span> <span class="n">sigma</span><span class="o">*</span><span class="n">ones_like</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">phat</span><span class="p">,</span><span class="n">cphat</span> <span class="o">=</span> <span class="n">curve_fit</span><span class="p">(</span><span class="n">modelfun</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="n">noise</span><span class="p">)</span>
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<h3 id="Load-data-to-be-fitted">Load data to be fitted<a class="anchor-link" href="#Load-data-to-be-fitted">¶</a></h3>
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<div class=" highlight hl-ipython3"><pre><span class="n">data</span> <span class="o">=</span> <span class="n">loadtxt</span><span class="p">(</span><span class="s2">"test_data.txt"</span><span class="p">,</span> <span class="n">delimiter</span><span class="o">=</span><span class="s2">","</span><span class="p">)</span>
<span class="n">t</span> <span class="o">=</span> <span class="n">data</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">data</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span>
<span class="n">sigma</span> <span class="o">=</span> <span class="mf">0.2</span>
<span class="n">errorbar</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">x</span><span class="p">,</span><span class="n">sigma</span><span class="o">*</span><span class="n">ones_like</span><span class="p">(</span><span class="n">t</span><span class="p">),</span><span class="n">fmt</span><span class="o">=</span><span class="s2">"o"</span><span class="p">);</span>
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<div class=" highlight hl-ipython3"><pre><span class="c1"># solution for exercise 3.2</span>
<span class="kn">from</span> <span class="nn">scipy.optimize</span> <span class="k">import</span> <span class="n">curve_fit</span>
<span class="c1"># model function</span>
<span class="k">def</span> <span class="nf">modelfun</span><span class="p">(</span><span class="n">t</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">c</span><span class="p">):</span>
<span class="k">return</span> <span class="n">a</span> <span class="o">*</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">b</span><span class="o">*</span><span class="n">t</span><span class="p">)</span> <span class="o">*</span> <span class="n">sin</span><span class="p">(</span><span class="n">c</span><span class="o">*</span><span class="n">t</span><span class="p">)</span>
<span class="c1"># GUM-like uncertainty evaluation</span>
<span class="n">noise</span> <span class="o">=</span> <span class="n">sigma</span><span class="o">*</span><span class="n">ones_like</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">phat</span><span class="p">,</span><span class="n">uphat</span> <span class="o">=</span> <span class="n">curve_fit</span><span class="p">(</span><span class="n">modelfun</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="n">noise</span><span class="p">)</span>
<span class="c1"># GUM Monte-Carlo uncertainty evaluation</span>
<span class="n">draws</span> <span class="o">=</span> <span class="mi">1000</span>
<span class="n">pMC</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">((</span><span class="n">draws</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span> <span class="c1"># set up matrix to store MC results</span>
<span class="n">success</span><span class="o">=</span> <span class="p">[]</span> <span class="c1"># list of successfull fits</span>
<span class="n">failed</span> <span class="o">=</span> <span class="p">[]</span> <span class="c1"># list of failed fits</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">draws</span><span class="p">):</span>
<span class="n">xMC</span> <span class="o">=</span> <span class="n">x</span> <span class="o">+</span> <span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">*</span><span class="n">sigma</span> <span class="c1"># draw from input quantities</span>
<span class="c1"># use try-except structure to catch failed fits</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">pMC</span><span class="p">[</span><span class="n">k</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">curve_fit</span><span class="p">(</span><span class="n">modelfun</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">xMC</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="n">noise</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
<span class="n">success</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
<span class="k">except</span> <span class="ne">RuntimeError</span><span class="p">:</span>
<span class="n">failed</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
<span class="c1"># use fancy indexing to select only results where the fit was successful</span>
<span class="n">phatMC</span> <span class="o">=</span> <span class="n">mean</span><span class="p">(</span><span class="n">pMC</span><span class="p">[</span><span class="n">success</span><span class="p">],</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">uphatMC</span><span class="o">=</span> <span class="n">cov</span><span class="p">(</span><span class="n">pMC</span><span class="p">[</span><span class="n">success</span><span class="p">],</span><span class="n">rowvar</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">figure</span><span class="p">()</span>
<span class="n">errorbar</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">),</span> <span class="n">phat</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">diag</span><span class="p">(</span><span class="n">uphat</span><span class="p">)),</span> <span class="n">fmt</span><span class="o">=</span><span class="s2">"o"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"GUM result"</span><span class="p">)</span>
<span class="n">errorbar</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">),</span> <span class="n">phatMC</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">diag</span><span class="p">(</span><span class="n">uphatMC</span><span class="p">)),</span> <span class="n">fmt</span><span class="o">=</span><span class="s2">"s"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"Monte Carlo result"</span><span class="p">)</span>
<span class="n">legend</span><span class="p">()</span>
<span class="n">xlim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">4</span><span class="p">)</span>
<span class="n">xticks</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="s2">"a"</span><span class="p">,</span> <span class="s2">"b"</span><span class="p">,</span> <span class="s2">"c"</span><span class="p">]);</span>
<span class="n">tick_params</span><span class="p">(</span><span class="n">which</span><span class="o">=</span><span class="s2">"both"</span><span class="p">,</span><span class="n">labelsize</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
<span class="c1"># calculate Monte-Carlo model function outputs</span>
<span class="n">t_long</span> <span class="o">=</span> <span class="n">linspace</span><span class="p">(</span><span class="n">t</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">t</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="mi">1000</span><span class="p">)</span>
<span class="n">Y</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">t_long</span><span class="p">),</span> <span class="n">draws</span><span class="p">))</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">success</span><span class="p">)):</span>
<span class="n">pars</span> <span class="o">=</span> <span class="n">pMC</span><span class="p">[</span><span class="n">success</span><span class="p">[</span><span class="n">k</span><span class="p">]]</span>
<span class="n">Y</span><span class="p">[:,</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">modelfun</span><span class="p">(</span><span class="n">t_long</span><span class="p">,</span> <span class="o">*</span><span class="n">pars</span><span class="p">)</span>
<span class="n">figure</span><span class="p">()</span>
<span class="n">errorbar</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">sigma</span><span class="o">*</span><span class="n">ones_like</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="n">fmt</span><span class="o">=</span><span class="s2">"o"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"measured"</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">modelfun</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="o">*</span><span class="n">phat</span><span class="p">),</span> <span class="s2">"gd"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"modelled"</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="n">t_long</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="s2">"g"</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.05</span><span class="p">)</span>
<span class="n">legend</span><span class="p">();</span>
</pre></div>
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