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401 lines
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<a class="navbar-brand" href="week37-bs.html">Week 37: Statistical interpretations and Resampling Methods</a>
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._week37-bs001.html#plans-for-week-37" style="font-size: 80%;">Plans for week 37</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs002.html#material-from-last-week-and-relevant-for-the-weekly-exercises" style="font-size: 80%;">Material from last week and relevant for the weekly exercises</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs003.html#linking-the-regression-analysis-with-a-statistical-interpretation" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs004.html#assumptions-made" style="font-size: 80%;">Assumptions made</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs005.html#expectation-value-and-variance" style="font-size: 80%;">Expectation value and variance</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs006.html#expectation-value-and-variance-for-boldsymbol-beta" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs007.html#material-for-lecture-thursday-september-14" style="font-size: 80%;">Material for lecture Thursday September 14</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs008.html#deriving-ols-from-a-probability-distribution" style="font-size: 80%;">Deriving OLS from a probability distribution</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs009.html#independent-and-identically-distrubuted-iid" style="font-size: 80%;">Independent and Identically Distrubuted (iid)</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs010.html#maximum-likelihood-estimation-mle" style="font-size: 80%;">Maximum Likelihood Estimation (MLE)</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs011.html#a-new-cost-function" style="font-size: 80%;">A new Cost Function</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs012.html#more-basic-statistics-and-bayes-theorem" style="font-size: 80%;">More basic Statistics and Bayes' theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs013.html#marginal-probability" style="font-size: 80%;">Marginal Probability</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs014.html#conditional-probability" style="font-size: 80%;">Conditional Probability</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs015.html#bayes-theorem" style="font-size: 80%;">Bayes' Theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs016.html#interpretations-of-bayes-theorem" style="font-size: 80%;">Interpretations of Bayes' Theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs017.html#example-of-usage-of-bayes-theorem" style="font-size: 80%;">Example of Usage of Bayes' theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs018.html#doing-it-correctly" style="font-size: 80%;">Doing it correctly</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs019.html#bayes-theorem-and-ridge-and-lasso-regression" style="font-size: 80%;">Bayes' Theorem and Ridge and Lasso Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs020.html#ridge-and-bayes" style="font-size: 80%;">Ridge and Bayes</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs021.html#lasso-and-bayes" style="font-size: 80%;">Lasso and Bayes</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs025.html#why-resampling-methods" style="font-size: 80%;">Why resampling methods</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods" style="font-size: 80%;">Resampling methods</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-approaches-can-be-computationally-expensive" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs025.html#why-resampling-methods" style="font-size: 80%;">Why resampling methods ?</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs026.html#statistical-analysis" style="font-size: 80%;">Statistical analysis</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods" style="font-size: 80%;">Resampling methods</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs028.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs030.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs031.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs032.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs033.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs034.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs035.html#standard-approach-based-on-the-normal-distribution" style="font-size: 80%;">Standard Approach based on the Normal Distribution</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs036.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs037.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs038.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs039.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
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<!-- navigation toc: --> <li><a href="#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs041.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs042.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs043.html#a-way-to-read-the-bias-variance-tradeoff" style="font-size: 80%;">A way to Read the Bias-Variance Tradeoff</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs044.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs045.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs046.html#summing-up" style="font-size: 80%;">Summing up</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs047.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs048.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs049.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs050.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs051.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs052.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs052.html#notes-on-scaling-with-examples" style="font-size: 80%;">Notes on scaling with examples</a></li>
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</ul>
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</li>
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</div> <!-- end of navigation bar -->
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<div class="container">
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0040"></a>
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<!-- !split -->
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<h2 id="code-example-for-the-bootstrap-method" class="anchor">Code example for the Bootstrap method </h2>
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<p>The following code starts with a Gaussian distribution with mean value
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\( \mu =100 \) and variance \( \sigma=15 \). We use this to generate the data
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used in the bootstrap analysis. The bootstrap analysis returns a data
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set after a given number of bootstrap operations (as many as we have
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data points). This data set consists of estimated mean values for each
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bootstrap operation. The histogram generated by the bootstrap method
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shows that the distribution for these mean values is also a Gaussian,
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centered around the mean value \( \mu=100 \) but with standard deviation
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\( \sigma/\sqrt{n} \), where \( n \) is the number of bootstrap samples (in
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this case the same as the number of original data points). The value
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of the standard deviation is what we expect from the central limit
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theorem.
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</p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="cell border-box-sizing code_cell rendered">
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<div class="input">
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<div class="highlight" style="background: #f8f8f8">
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<pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">time</span> <span style="color: #008000; font-weight: bold">import</span> time
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">scipy.stats</span> <span style="color: #008000; font-weight: bold">import</span> norm
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
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<span style="color: #408080; font-style: italic"># Returns mean of bootstrap samples </span>
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<span style="color: #408080; font-style: italic"># Bootstrap algorithm</span>
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">bootstrap</span>(data, datapoints):
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t <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(datapoints)
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n <span style="color: #666666">=</span> <span style="color: #008000">len</span>(data)
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<span style="color: #408080; font-style: italic"># non-parametric bootstrap </span>
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<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(datapoints):
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t[i] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean(data[np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randint(<span style="color: #666666">0</span>,n,n)])
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<span style="color: #408080; font-style: italic"># analysis </span>
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<span style="color: #008000">print</span>(<span style="color: #BA2121">"Bootstrap Statistics :"</span>)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">"original bias std. error"</span>)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">"</span><span style="color: #BB6688; font-weight: bold">%8g</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">%8g</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">%14g</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">%15g</span><span style="color: #BA2121">"</span> <span style="color: #666666">%</span> (np<span style="color: #666666">.</span>mean(data), np<span style="color: #666666">.</span>std(data),np<span style="color: #666666">.</span>mean(t),np<span style="color: #666666">.</span>std(t)))
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<span style="color: #008000; font-weight: bold">return</span> t
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<span style="color: #408080; font-style: italic"># We set the mean value to 100 and the standard deviation to 15</span>
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mu, sigma <span style="color: #666666">=</span> <span style="color: #666666">100</span>, <span style="color: #666666">15</span>
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datapoints <span style="color: #666666">=</span> <span style="color: #666666">10000</span>
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<span style="color: #408080; font-style: italic"># We generate random numbers according to the normal distribution</span>
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x <span style="color: #666666">=</span> mu <span style="color: #666666">+</span> sigma<span style="color: #666666">*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(datapoints)
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<span style="color: #408080; font-style: italic"># bootstrap returns the data sample </span>
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t <span style="color: #666666">=</span> bootstrap(x, datapoints)
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</pre>
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<p>We see that our new variance and from that the standard deviation, agrees with the central limit theorem.</p>
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