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04 Summarizing the Monte Carlo output.html
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04 Summarizing the Monte Carlo output.html
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<!DOCTYPE html>
<html>
<head><meta charset="utf-8" />
<title>04 Summarizing the Monte Carlo output</title>
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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="Summarizing-the-Monte-Carlo-output">Summarizing the Monte Carlo output<a class="anchor-link" href="#Summarizing-the-Monte-Carlo-output">¶</a></h1>
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<p>The result of a Monte Carlo simulation is a set of samples from the probability distribution associated with the measurand
$$ \{\mathbf{y}^{(k)},\,k=1,\ldots,M\} $$</p>
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<p>The aim is to derive characteristic information about the measurand from this set:</p>
<p>1) best estimate</p>
<p>2) uncertainty associated with the best estimate</p>
<p>3) intervals/regions with a prescribed coverage probability</p>
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<h2 id="Univariate-measurand">Univariate measurand<a class="anchor-link" href="#Univariate-measurand">¶</a></h2>
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<p>1) best estimate
\begin{align}
y =& mean { y^{(k)}, k=1,\ldots,M} \
=& \frac{1}{M} \sum<em>{k=1}^M y^{(k)}
\end{align}
2) uncertainty associated with the best estimate
\begin{align}
u_y =& std{ y^{(k)}, k=1,\ldots,M} \
=& \frac{1}{M-1} \sum</em>{k=1}^M (y-y^{(k)})^2
\end{align}</p>
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<h3 id="Exercise-4.1">Exercise 4.1<a class="anchor-link" href="#Exercise-4.1">¶</a></h3><p>Draw randomly from the normal distribution with mean $\mu=1.3$ and standard deviation $\sigma=0.4$ and calculate best estimate and its uncertainy using 100, 200, 300, ..., 100 000 draws. Compare estimate and uncertainty with mean and standard deviation of the original distribution for the different number of draws.</p>
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<div class=" highlight hl-ipython3"><pre><span class="kn">from</span> <span class="nn">scipy.stats</span> <span class="k">import</span> <span class="n">norm</span>
<span class="n">Y</span> <span class="o">=</span> <span class="n">norm</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">1.3</span><span class="p">,</span> <span class="n">scale</span><span class="o">=</span><span class="mf">0.4</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="mi">100000</span><span class="p">)</span>
<span class="n">draws</span> <span class="o">=</span> <span class="n">arange</span><span class="p">(</span><span class="mi">100</span><span class="p">,</span><span class="nb">len</span><span class="p">(</span><span class="n">Y</span><span class="p">)</span><span class="o">+</span><span class="mi">100</span><span class="p">,</span><span class="mi">100</span><span class="p">)</span>
<span class="n">means</span> <span class="o">=</span> <span class="n">array</span><span class="p">([</span><span class="n">Y</span><span class="p">[:</span><span class="n">M</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span> <span class="k">for</span> <span class="n">M</span> <span class="ow">in</span> <span class="n">draws</span><span class="p">])</span>
<span class="n">uncs</span> <span class="o">=</span> <span class="n">array</span><span class="p">([</span><span class="n">Y</span><span class="p">[:</span><span class="n">M</span><span class="p">]</span><span class="o">.</span><span class="n">std</span><span class="p">()</span> <span class="k">for</span> <span class="n">M</span> <span class="ow">in</span> <span class="n">draws</span><span class="p">])</span>
<span class="n">figure</span><span class="p">()</span>
<span class="n">subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="n">draws</span><span class="p">,</span> <span class="n">means</span><span class="p">)</span>
<span class="n">axhline</span><span class="p">(</span><span class="mf">1.3</span><span class="p">,</span><span class="n">color</span><span class="o">=</span><span class="s2">"k"</span><span class="p">,</span><span class="n">linestyle</span><span class="o">=</span><span class="s2">"--"</span><span class="p">)</span>
<span class="n">subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="n">draws</span><span class="p">,</span> <span class="n">uncs</span><span class="p">)</span>
<span class="n">axhline</span><span class="p">(</span><span class="mf">0.4</span><span class="p">,</span><span class="n">color</span><span class="o">=</span><span class="s2">"k"</span><span class="p">,</span><span class="n">linestyle</span><span class="o">=</span><span class="s2">"--"</span><span class="p">);</span>
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<p>Intervals with a prescribed coverage probability can be calculated from the Monte Carlo outcome as follows</p>
<p>1) Sort the Monte Carlo outcome in ascending order</p>
<div class="highlight"><pre><span class="n">sort</span><span class="p">(</span><span class="n">Y</span><span class="p">)</span>
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<p>2) For propability $P$ calculate the corresponding fraction of Monte Carlo draws $q=int(PM)$</p>
<p>3) Set lower bound of interval as $r=int(0.5(M-q))$ for a probabilistically symmetrical interval</p>
<p>4) Calculate upper bound of interval as $r+q$</p>
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<h3 id="Exercise-4.2">Exercise 4.2<a class="anchor-link" href="#Exercise-4.2">¶</a></h3><p>Draw randomly from the distribution from Exercise 4.1 and calculate the 95% probabilistally symetrical coverage interval from 100000 draws. Compare the result to the exact 95% interval.</p>
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<div class=" highlight hl-ipython3"><pre><span class="c1"># solution for exercise 4.2</span>
<span class="kn">from</span> <span class="nn">scipy.stats</span> <span class="k">import</span> <span class="n">norm</span>
<span class="n">P</span> <span class="o">=</span> <span class="mf">0.95</span> <span class="c1"># sought probability of coverage interval</span>
<span class="n">draws</span> <span class="o">=</span> <span class="mi">100000</span>
<span class="n">Y</span> <span class="o">=</span> <span class="n">norm</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">1.3</span><span class="p">,</span> <span class="n">scale</span><span class="o">=</span><span class="mf">0.4</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"># step 1 - sort random draws</span>
<span class="n">Ys</span> <span class="o">=</span> <span class="n">sort</span><span class="p">(</span><span class="n">Y</span><span class="p">)</span>
<span class="c1"># step 2 - calculate portion of draws corresponding to P</span>
<span class="n">q</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">draws</span><span class="o">*</span><span class="n">P</span><span class="p">)</span>
<span class="c1"># step 3 - set lower bound of coverage interval</span>
<span class="n">r</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="mf">0.5</span><span class="o">*</span><span class="p">(</span><span class="n">draws</span><span class="o">-</span><span class="n">q</span><span class="p">))</span>
<span class="c1"># interval bounds</span>
<span class="n">ylow</span> <span class="o">=</span> <span class="n">Ys</span><span class="p">[</span><span class="n">r</span><span class="p">]</span>
<span class="n">yhigh</span> <span class="o">=</span> <span class="n">Ys</span><span class="p">[</span><span class="n">r</span><span class="o">+</span><span class="n">q</span><span class="p">]</span>
<span class="n">interval</span> <span class="o">=</span> <span class="p">[</span><span class="n">ylow</span><span class="p">,</span> <span class="n">yhigh</span><span class="p">]</span>
<span class="n">true_interval</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.3</span> <span class="o">-</span> <span class="mf">1.96</span><span class="o">*</span><span class="mf">0.4</span><span class="p">,</span> <span class="mf">1.3</span> <span class="o">+</span> <span class="mf">1.96</span><span class="o">*</span><span class="mf">0.4</span><span class="p">]</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"calculated interval"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">interval</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"true interval"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">true_interval</span><span class="p">)</span>
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<pre>calculated interval
[0.51555327864031619, 2.0847763045218146]
true interval
[0.516, 2.084]
</pre>
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<h2 id="Multivariate-measurand">Multivariate measurand<a class="anchor-link" href="#Multivariate-measurand">¶</a></h2>
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<p>1) best estimate
\begin{align}
\mathbf{y} =& mean { \mathbf{y}^{(k)}, k=1,\ldots,M} \
=& \frac{1}{M} \sum<em>{k=1}^M \mathbf{y}^{(k)}
\end{align}
2) uncertainty associated with the best estimate
\begin{align}
U</em>\mathbf{y} =& cov{ \mathbf{y}^{(k)}, k=1,\ldots,M} \
=& \frac{1}{M-1} \sum_{k=1}^M (\mathbf{y}-\mathbf{y}^{(k)})(\mathbf{y}-\mathbf{y}^{(k)})^T
\end{align}</p>
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<h3 id="Exercise-4.3">Exercise 4.3<a class="anchor-link" href="#Exercise-4.3">¶</a></h3><p>Draw randomly from the normal distribution with mean
$$\mathbf{\mu}=\left( \begin{array}{c}
0.4 \\ -1.5
\end{array}\right)
$$
and covariance
$$
\Sigma=\left(\begin{array}{cc}
0.09 & -0.2 \\ -0.2 & 1.44
\end{array}\right)
$$
and calculate best estimate and its uncertainy using 1000, 10000 and 100 000 draws. Compare estimate and uncertainty with mean and covariance of the original distribution for the different number of draws.</p>
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<div class=" highlight hl-ipython3"><pre><span class="c1"># solution for exercise 4.3</span>
<span class="kn">from</span> <span class="nn">scipy.stats</span> <span class="k">import</span> <span class="n">multivariate_normal</span>
<span class="n">Y</span> <span class="o">=</span> <span class="n">multivariate_normal</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">mean</span><span class="o">=</span><span class="p">[</span><span class="mf">0.4</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.5</span><span class="p">],</span>
<span class="n">cov</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="mf">0.09</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.2</span><span class="p">],[</span><span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">1.44</span><span class="p">]]),</span>
<span class="n">size</span> <span class="o">=</span> <span class="mi">100000</span><span class="p">)</span>
<span class="n">draws</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1000</span><span class="p">,</span> <span class="mi">10000</span><span class="p">,</span> <span class="mi">100000</span><span class="p">]</span>
<span class="n">means</span> <span class="o">=</span> <span class="n">array</span><span class="p">([</span><span class="n">Y</span><span class="p">[:</span><span class="n">M</span><span class="p">]</span><span class="o">.</span><span class="n">mean</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="k">for</span> <span class="n">M</span> <span class="ow">in</span> <span class="n">draws</span><span class="p">])</span>
<span class="n">uncs</span> <span class="o">=</span> <span class="n">array</span><span class="p">([</span><span class="n">cov</span><span class="p">(</span><span class="n">Y</span><span class="p">[:</span><span class="n">M</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="k">for</span> <span class="n">M</span> <span class="ow">in</span> <span class="n">draws</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"calculated estimates"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">means</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"calculated covariances"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">uncs</span><span class="p">)</span>
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<pre>calculated estimates
[[ 0.39823368 -1.47816826]
[ 0.39785645 -1.49430727]
[ 0.39923487 -1.49887348]]
calculated covariances
[[[ 0.0946245 -0.1966798 ]
[-0.1966798 1.37136819]]
[[ 0.09177784 -0.20437818]
[-0.20437818 1.43619558]]
[[ 0.09024844 -0.19873001]
[-0.19873001 1.42682509]]]
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<p>Regions with a prescribed coverage probability can be calculated from the multivariate Monte Carlo outcome as follows</p>
<p>1) Calculate the Cholesky decomposition of the sample covariance matrix $U_{\mathbf{y}}=\mathbf{LL}^T$</p>
<p>2) Transform the Monte Carlo outcomes
$$ \mathbf{y}_{(k)} = \mathbf{L}^{-1}(\mathbf{y}^{(k)}-\mathbf{y})$$
and sort according to the distance measure
$$ d^2_{(k)} = \mathbf{y}_{(k)}^T\mathbf{y}_{(k)} $$</p>
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<p>3) calculate $k_P$ such that a fraction $P$ of all Monte Carlo outcomes satisfies $d_{(k)}<k_P$</p>
<p>This defines the ellipsoidal region $(\mathbf{\eta}-\mathbf{y})^TU_{\mathbf{y}}^{-1}(\mathbf{\eta}-\mathbf{y})<k^2_P$</p>
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<p>For a bivariate normal distribution, the factor for a 95% coverage ellipsoidal region is given as the 95% quantile of the $\chi^2$ distribution with 2 degrees of freedom.</p>
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<div class=" highlight hl-ipython3"><pre><span class="c1">## !! Run this cell before continuing !!</span>
<span class="c1"># helper function to calculate ellipsoidal parameters from mean and covariance</span>
<span class="k">def</span> <span class="nf">calculate_ellipse</span><span class="p">(</span><span class="n">mu</span><span class="p">,</span> <span class="n">Sigma</span><span class="p">,</span> <span class="n">kP</span><span class="p">):</span>
<span class="n">vals</span><span class="p">,</span> <span class="n">vecs</span> <span class="o">=</span> <span class="n">linalg</span><span class="o">.</span><span class="n">eigh</span><span class="p">(</span><span class="n">Sigma</span><span class="p">)</span>
<span class="n">order</span> <span class="o">=</span> <span class="n">vals</span><span class="o">.</span><span class="n">argsort</span><span class="p">()[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="n">vals</span> <span class="o">=</span> <span class="n">vals</span><span class="p">[</span><span class="n">order</span><span class="p">]</span>
<span class="n">vecs</span> <span class="o">=</span> <span class="n">vecs</span><span class="p">[:,</span><span class="n">order</span><span class="p">]</span>
<span class="n">theta</span> <span class="o">=</span> <span class="n">degrees</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arctan2</span><span class="p">(</span><span class="o">*</span><span class="n">vecs</span><span class="p">[:,</span><span class="mi">0</span><span class="p">][::</span><span class="o">-</span><span class="mi">1</span><span class="p">]))</span>
<span class="n">width</span><span class="p">,</span> <span class="n">height</span> <span class="o">=</span> <span class="n">kP</span> <span class="o">*</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">vals</span><span class="p">)</span>
<span class="k">return</span> <span class="n">width</span><span class="p">,</span> <span class="n">height</span><span class="p">,</span> <span class="n">theta</span>
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<h3 id="Exercise-4.4">Exercise 4.4<a class="anchor-link" href="#Exercise-4.4">¶</a></h3><p>Calculate 100 000 random draws from the distribution from Exercise 4.3 and calculate the 95% coverage region. Compare to the true 95% coverage region.</p>
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<div class=" highlight hl-ipython3"><pre><span class="c1"># solution for exercise 4.4</span>
<span class="kn">from</span> <span class="nn">scipy.stats</span> <span class="k">import</span> <span class="n">multivariate_normal</span><span class="p">,</span> <span class="n">chi2</span>
<span class="kn">from</span> <span class="nn">matplotlib.patches</span> <span class="k">import</span> <span class="n">Ellipse</span>
<span class="n">mu</span> <span class="o">=</span> <span class="n">array</span><span class="p">([</span><span class="mf">0.4</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.5</span><span class="p">])</span>
<span class="n">Sigma</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="mf">0.09</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.2</span><span class="p">],[</span><span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">1.44</span><span class="p">]])</span>
<span class="n">P</span> <span class="o">=</span> <span class="mf">0.95</span>
<span class="n">draws</span> <span class="o">=</span> <span class="mi">10000</span>
<span class="n">q</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">P</span><span class="o">*</span><span class="n">draws</span><span class="p">)</span>
<span class="n">Y</span> <span class="o">=</span> <span class="n">multivariate_normal</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">mean</span> <span class="o">=</span> <span class="n">mu</span><span class="p">,</span> <span class="n">cov</span> <span class="o">=</span> <span class="n">Sigma</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">y</span> <span class="o">=</span> <span class="n">Y</span><span class="o">.</span><span class="n">mean</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">Uy</span><span class="o">=</span> <span class="n">cov</span><span class="p">(</span><span class="n">Y</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="c1"># step 1 - Cholesky decomposition of covariance matrix</span>
<span class="n">L</span> <span class="o">=</span> <span class="n">cholesky</span><span class="p">(</span><span class="n">Uy</span><span class="p">)</span>
<span class="c1"># step 2 - Transform the Monte Carlo outcomes</span>
<span class="n">Ys</span> <span class="o">=</span> <span class="n">linalg</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">L</span><span class="p">,</span> <span class="p">(</span><span class="n">Y</span> <span class="o">-</span> <span class="n">tile</span><span class="p">(</span><span class="n">y</span><span class="p">,(</span><span class="n">draws</span><span class="p">,</span><span class="mi">1</span><span class="p">)))</span><span class="o">.</span><span class="n">T</span><span class="p">)</span><span class="o">.</span><span class="n">T</span>
<span class="n">d</span> <span class="o">=</span> <span class="n">diag</span><span class="p">(</span><span class="n">dot</span><span class="p">(</span><span class="n">Ys</span><span class="p">,</span><span class="n">Ys</span><span class="o">.</span><span class="n">T</span><span class="p">))</span>
<span class="c1"># and sort distances</span>
<span class="n">ds</span> <span class="o">=</span> <span class="n">sort</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
<span class="c1"># step 3 - calculate kP as the scale parameter of the ellipse </span>