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<h1>Week 37: Statistical interpretations and Resampling Methods</h1>
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<h2> Contents </h2>
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<ul class="visible nav section-nav flex-column">
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#plans-for-week-37-lecture-monday">Plans for week 37, lecture Monday</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#plans-for-week-37-lab-sessions">Plans for week 37, lab sessions</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#material-for-lecture-monday-september-9">Material for lecture Monday September 9</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#deriving-ols-from-a-probability-distribution">Deriving OLS from a probability distribution</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#independent-and-identically-distrubuted-iid">Independent and Identically Distrubuted (iid)</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#maximum-likelihood-estimation-mle">Maximum Likelihood Estimation (MLE)</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-new-cost-function">A new Cost Function</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#more-basic-statistics-and-bayes-theorem">More basic Statistics and Bayes’ theorem</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#marginal-probability">Marginal Probability</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#conditional-probability">Conditional Probability</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#bayes-theorem">Bayes’ Theorem</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#interpretations-of-bayes-theorem">Interpretations of Bayes’ Theorem</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#example-of-usage-of-bayes-theorem">Example of Usage of Bayes’ theorem</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#doing-it-correctly">Doing it correctly</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#bayes-theorem-and-ridge-and-lasso-regression">Bayes’ Theorem and Ridge and Lasso Regression</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#ridge-and-bayes">Ridge and Bayes</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#lasso-and-bayes">Lasso and Bayes</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#why-resampling-methods">Why resampling methods</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#resampling-methods">Resampling methods</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#resampling-approaches-can-be-computationally-expensive">Resampling approaches can be computationally expensive</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">Why resampling methods ?</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#statistical-analysis">Statistical analysis</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#id2">Resampling methods</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#resampling-methods-bootstrap">Resampling methods: Bootstrap</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-central-limit-theorem">The Central Limit Theorem</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#finding-the-limit">Finding the Limit</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#rewriting-the-delta-function">Rewriting the <span class="math notranslate nohighlight">\(\delta\)</span>-function</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#identifying-terms">Identifying Terms</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#wrapping-it-up">Wrapping it up</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#confidence-intervals">Confidence Intervals</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#standard-approach-based-on-the-normal-distribution">Standard Approach based on the Normal Distribution</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#resampling-methods-bootstrap-background">Resampling methods: Bootstrap background</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#resampling-methods-more-bootstrap-background">Resampling methods: More Bootstrap background</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#resampling-methods-bootstrap-approach">Resampling methods: Bootstrap approach</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#resampling-methods-bootstrap-steps">Resampling methods: Bootstrap steps</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#code-example-for-the-bootstrap-method">Code example for the Bootstrap method</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#plotting-the-histogram">Plotting the Histogram</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-bias-variance-tradeoff">The bias-variance tradeoff</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-way-to-read-the-bias-variance-tradeoff">A way to Read the Bias-Variance Tradeoff</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#example-code-for-bias-variance-tradeoff">Example code for Bias-Variance tradeoff</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#understanding-what-happens">Understanding what happens</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#summing-up">Summing up</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#another-example-from-scikit-learn-s-repository">Another Example from Scikit-Learn’s Repository</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#various-steps-in-cross-validation">Various steps in cross-validation</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#cross-validation-in-brief">Cross-validation in brief</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#code-example-for-cross-validation-and-k-fold-cross-validation">Code Example for Cross-validation and <span class="math notranslate nohighlight">\(k\)</span>-fold Cross-validation</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#more-examples-on-bootstrap-and-cross-validation-and-errors">More examples on bootstrap and cross-validation and errors</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-same-example-but-now-with-cross-validation">The same example but now with cross-validation</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#material-for-the-lab-sessions">Material for the lab sessions</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#linking-the-regression-analysis-with-a-statistical-interpretation">Linking the regression analysis with a statistical interpretation</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#assumptions-made">Assumptions made</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#expectation-value-and-variance">Expectation value and variance</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#expectation-value-and-variance-for-boldsymbol-beta">Expectation value and variance for <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span></a></li>
</ul>
</nav>
</div>
</div>
</div>
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<article class="bd-article">
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<!-- dom:TITLE: Week 37: Statistical interpretations and Resampling Methods --><section class="tex2jax_ignore mathjax_ignore" id="week-37-statistical-interpretations-and-resampling-methods">
<h1>Week 37: Statistical interpretations and Resampling Methods<a class="headerlink" href="#week-37-statistical-interpretations-and-resampling-methods" title="Link to this heading">#</a></h1>
<p><strong>Morten Hjorth-Jensen</strong>, Department of Physics, University of Oslo, Norway</p>
<p>Date: <strong>September 9, 2024</strong></p>
<!-- todo add link to videos and add link to Van Wieringens notes --><section id="plans-for-week-37-lecture-monday">
<h2>Plans for week 37, lecture Monday<a class="headerlink" href="#plans-for-week-37-lecture-monday" title="Link to this heading">#</a></h2>
<p><strong>Material for the lecture on Monday September 9.</strong></p>
<ul class="simple">
<li><p><a class="reference external" href="https://youtu.be/omLmp_kkie0">Video of Lecture</a></p></li>
<li><p><a class="reference external" href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2024/NotesSeptember9.pdf">Whiteboard notes</a></p></li>
<li><p>Statistical interpretation of Ridge and Lasso regression, see also slides from last week</p></li>
<li><p>Resampling techniques, Bootstrap and cross validation and bias-variance tradeoff (this may partly be discussed during the exercise sessions as well.</p></li>
<li><p>Readings and Videos:</p>
<ul>
<li><p>Raschka et al, pages 175-192</p></li>
<li><p>Hastie et al Chapter 7, here we recommend 7.1-7.5 and 7.10 (cross-validation) and 7.11 (bootstrap). See <a class="reference external" href="https://link.springer.com/book/10.1007/978-0-387-84858-7">https://link.springer.com/book/10.1007/978-0-387-84858-7</a>.</p></li>
<li><p><a class="reference external" href="https://www.youtube.com/watch?v=fSytzGwwBVw">Video on cross validation</a></p></li>
<li><p><a class="reference external" href="https://www.youtube.com/watch?v=Xz0x-8-cgaQ">Video on Bootstrapping</a></p></li>
<li><p><a class="reference external" href="https://www.youtube.com/watch?v=EuBBz3bI-aA">Video on bias-variance tradeoff</a></p></li>
</ul>
</li>
</ul>
</section>
<section id="plans-for-week-37-lab-sessions">
<h2>Plans for week 37, lab sessions<a class="headerlink" href="#plans-for-week-37-lab-sessions" title="Link to this heading">#</a></h2>
<p><strong>Material for the lab sessions on Tuesday and Wednesday.</strong></p>
<ul class="simple">
<li><p>Calculations of expectation values</p></li>
<li><p>Discussion of resampling techniques</p></li>
<li><p>Exercise set for week 37</p></li>
<li><p>Work on project 1</p></li>
<li><p><a class="reference external" href="https://youtu.be/bK4AEcTu-oM">Video of exercise sessions week 37</a></p></li>
<li><p>For more discussions of Ridge regression and calculation of averages, <a class="reference external" href="https://arxiv.org/abs/1509.09169">Wessel van Wieringen’s</a> article is highly recommended.</p></li>
</ul>
</section>
<section id="material-for-lecture-monday-september-9">
<h2>Material for lecture Monday September 9<a class="headerlink" href="#material-for-lecture-monday-september-9" title="Link to this heading">#</a></h2>
</section>
<section id="deriving-ols-from-a-probability-distribution">
<h2>Deriving OLS from a probability distribution<a class="headerlink" href="#deriving-ols-from-a-probability-distribution" title="Link to this heading">#</a></h2>
<p>Our basic assumption when we derived the OLS equations was to assume
that our output is determined by a given continuous function
<span class="math notranslate nohighlight">\(f(\boldsymbol{x})\)</span> and a random noise <span class="math notranslate nohighlight">\(\boldsymbol{\epsilon}\)</span> given by the normal
distribution with zero mean value and an undetermined variance
<span class="math notranslate nohighlight">\(\sigma^2\)</span>.</p>
<p>We found above that the outputs <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> have a mean value given by
<span class="math notranslate nohighlight">\(\boldsymbol{X}\hat{\boldsymbol{\beta}}\)</span> and variance <span class="math notranslate nohighlight">\(\sigma^2\)</span>. Since the entries to
the design matrix are not stochastic variables, we can assume that the
probability distribution of our targets is also a normal distribution
but now with mean value <span class="math notranslate nohighlight">\(\boldsymbol{X}\hat{\boldsymbol{\beta}}\)</span>. This means that a
single output <span class="math notranslate nohighlight">\(y_i\)</span> is given by the Gaussian distribution</p>
<div class="math notranslate nohighlight">
\[
y_i\sim \mathcal{N}(\boldsymbol{X}_{i,*}\boldsymbol{\beta}, \sigma^2)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}.
\]</div>
</section>
<section id="independent-and-identically-distrubuted-iid">
<h2>Independent and Identically Distrubuted (iid)<a class="headerlink" href="#independent-and-identically-distrubuted-iid" title="Link to this heading">#</a></h2>
<p>We assume now that the various <span class="math notranslate nohighlight">\(y_i\)</span> values are stochastically distributed according to the above Gaussian distribution.
We define this distribution as</p>
<div class="math notranslate nohighlight">
\[
p(y_i, \boldsymbol{X}\vert\boldsymbol{\beta})=\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]},
\]</div>
<p>which reads as finding the likelihood of an event <span class="math notranslate nohighlight">\(y_i\)</span> with the input variables <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> given the parameters (to be determined) <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span>.</p>
<p>Since these events are assumed to be independent and identicall distributed we can build the probability distribution function (PDF) for all possible event <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> as the product of the single events, that is we have</p>
<div class="math notranslate nohighlight">
\[
p(\boldsymbol{y},\boldsymbol{X}\vert\boldsymbol{\beta})=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}=\prod_{i=0}^{n-1}p(y_i,\boldsymbol{X}\vert\boldsymbol{\beta}).
\]</div>
<p>We will write this in a more compact form reserving <span class="math notranslate nohighlight">\(\boldsymbol{D}\)</span> for the domain of events, including the ouputs (targets) and the inputs. That is
in case we have a simple one-dimensional input and output case</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{D}=[(x_0,y_0), (x_1,y_1),\dots, (x_{n-1},y_{n-1})].
\]</div>
<p>In the more general case the various inputs should be replaced by the possible features represented by the input data set <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span>.
We can now rewrite the above probability as</p>
<div class="math notranslate nohighlight">
\[
p(\boldsymbol{D}\vert\boldsymbol{\beta})=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}.
\]</div>
<p>It is a conditional probability (see below) and reads as the likelihood of a domain of events <span class="math notranslate nohighlight">\(\boldsymbol{D}\)</span> given a set of parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span>.</p>
</section>
<section id="maximum-likelihood-estimation-mle">
<h2>Maximum Likelihood Estimation (MLE)<a class="headerlink" href="#maximum-likelihood-estimation-mle" title="Link to this heading">#</a></h2>
<p>In statistics, maximum likelihood estimation (MLE) is a method of
estimating the parameters of an assumed probability distribution,
given some observed data. This is achieved by maximizing a likelihood
function so that, under the assumed statistical model, the observed
data is the most probable.</p>
<p>We will assume here that our events are given by the above Gaussian
distribution and we will determine the optimal parameters <span class="math notranslate nohighlight">\(\beta\)</span> by
maximizing the above PDF. However, computing the derivatives of a
product function is cumbersome and can easily lead to overflow and/or
underflowproblems, with potentials for loss of numerical precision.</p>
<p>In practice, it is more convenient to maximize the logarithm of the
PDF because it is a monotonically increasing function of the argument.
Alternatively, and this will be our option, we will minimize the
negative of the logarithm since this is a monotonically decreasing
function.</p>
<p>Note also that maximization/minimization of the logarithm of the PDF
is equivalent to the maximization/minimization of the function itself.</p>
</section>
<section id="a-new-cost-function">
<h2>A new Cost Function<a class="headerlink" href="#a-new-cost-function" title="Link to this heading">#</a></h2>
<p>We could now define a new cost function to minimize, namely the negative logarithm of the above PDF</p>
<div class="math notranslate nohighlight">
\[
C(\boldsymbol{\beta}=-\log{\prod_{i=0}^{n-1}p(y_i,\boldsymbol{X}\vert\boldsymbol{\beta})}=-\sum_{i=0}^{n-1}\log{p(y_i,\boldsymbol{X}\vert\boldsymbol{\beta})},
\]</div>
<p>which becomes</p>
<div class="math notranslate nohighlight">
\[
C(\boldsymbol{\beta}=\frac{n}{2}\log{2\pi\sigma^2}+\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}.
\]</div>
<p>Taking the derivative of the <em>new</em> cost function with respect to the parameters <span class="math notranslate nohighlight">\(\beta\)</span> we recognize our familiar OLS equation, namely</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{X}^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right) =0,
\]</div>
<p>which leads to the well-known OLS equation for the optimal paramters <span class="math notranslate nohighlight">\(\beta\)</span></p>
<div class="math notranslate nohighlight">
\[
\hat{\boldsymbol{\beta}}^{\mathrm{OLS}}=\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}!
\]</div>
<p>Before we make a similar analysis for Ridge and Lasso regression, we need a short reminder on statistics.</p>
</section>
<section id="more-basic-statistics-and-bayes-theorem">
<h2>More basic Statistics and Bayes’ theorem<a class="headerlink" href="#more-basic-statistics-and-bayes-theorem" title="Link to this heading">#</a></h2>
<p>A central theorem in statistics is Bayes’ theorem. This theorem plays a similar role as the good old Pythagoras’ theorem in geometry.
Bayes’ theorem is extremely simple to derive. But to do so we need some basic axioms from statistics.</p>
<p>Assume we have two domains of events <span class="math notranslate nohighlight">\(X=[x_0,x_1,\dots,x_{n-1}]\)</span> and <span class="math notranslate nohighlight">\(Y=[y_0,y_1,\dots,y_{n-1}]\)</span>.</p>
<p>We define also the likelihood for <span class="math notranslate nohighlight">\(X\)</span> and <span class="math notranslate nohighlight">\(Y\)</span> as <span class="math notranslate nohighlight">\(p(X)\)</span> and <span class="math notranslate nohighlight">\(p(Y)\)</span> respectively.
The likelihood of a specific event <span class="math notranslate nohighlight">\(x_i\)</span> (or <span class="math notranslate nohighlight">\(y_i\)</span>) is then written as <span class="math notranslate nohighlight">\(p(X=x_i)\)</span> or just <span class="math notranslate nohighlight">\(p(x_i)=p_i\)</span>.</p>
<p><strong>Union of events is given by.</strong></p>
<div class="math notranslate nohighlight">
\[
p(X \cup Y)= p(X)+p(Y)-p(X \cap Y).
\]</div>
<p><strong>The product rule (aka joint probability) is given by.</strong></p>
<div class="math notranslate nohighlight">
\[
p(X \cup Y)= p(X,Y)= p(X\vert Y)p(Y)=p(Y\vert X)p(X),
\]</div>
<p>where we read <span class="math notranslate nohighlight">\(p(X\vert Y)\)</span> as the likelihood of obtaining <span class="math notranslate nohighlight">\(X\)</span> given <span class="math notranslate nohighlight">\(Y\)</span>.</p>
<p>If we have independent events then <span class="math notranslate nohighlight">\(p(X,Y)=p(X)p(Y)\)</span>.</p>
</section>
<section id="marginal-probability">
<h2>Marginal Probability<a class="headerlink" href="#marginal-probability" title="Link to this heading">#</a></h2>
<p>The marginal probability is defined in terms of only one of the set of variables <span class="math notranslate nohighlight">\(X,Y\)</span>. For a discrete probability we have</p>
<div class="math notranslate nohighlight">
\[
p(X)=\sum_{i=0}^{n-1}p(X,Y=y_i)=\sum_{i=0}^{n-1}p(X\vert Y=y_i)p(Y=y_i)=\sum_{i=0}^{n-1}p(X\vert y_i)p(y_i).
\]</div>
</section>
<section id="conditional-probability">
<h2>Conditional Probability<a class="headerlink" href="#conditional-probability" title="Link to this heading">#</a></h2>
<p>The conditional probability, if <span class="math notranslate nohighlight">\(p(Y) &gt; 0\)</span>, is</p>
<div class="math notranslate nohighlight">
\[
p(X\vert Y)= \frac{p(X,Y)}{p(Y)}=\frac{p(X,Y)}{\sum_{i=0}^{n-1}p(Y\vert X=x_i)p(x_i)}.
\]</div>
</section>
<section id="bayes-theorem">
<h2>Bayes’ Theorem<a class="headerlink" href="#bayes-theorem" title="Link to this heading">#</a></h2>
<p>If we combine the conditional probability with the marginal probability and the standard product rule, we have</p>
<div class="math notranslate nohighlight">
\[
p(X\vert Y)= \frac{p(X,Y)}{p(Y)},
\]</div>
<p>which we can rewrite as</p>
<div class="math notranslate nohighlight">
\[
p(X\vert Y)= \frac{p(X,Y)}{\sum_{i=0}^{n-1}p(Y\vert X=x_i)p(x_i)}=\frac{p(Y\vert X)p(X)}{\sum_{i=0}^{n-1}p(Y\vert X=x_i)p(x_i)},
\]</div>
<p>which is Bayes’ theorem. It allows us to evaluate the uncertainty in in <span class="math notranslate nohighlight">\(X\)</span> after we have observed <span class="math notranslate nohighlight">\(Y\)</span>. We can easily interchange <span class="math notranslate nohighlight">\(X\)</span> with <span class="math notranslate nohighlight">\(Y\)</span>.</p>
</section>
<section id="interpretations-of-bayes-theorem">
<h2>Interpretations of Bayes’ Theorem<a class="headerlink" href="#interpretations-of-bayes-theorem" title="Link to this heading">#</a></h2>
<p>The quantity <span class="math notranslate nohighlight">\(p(Y\vert X)\)</span> on the right-hand side of the theorem is
evaluated for the observed data <span class="math notranslate nohighlight">\(Y\)</span> and can be viewed as a function of
the parameter space represented by <span class="math notranslate nohighlight">\(X\)</span>. This function is not
necesseraly normalized and is normally called the likelihood function.</p>
<p>The function <span class="math notranslate nohighlight">\(p(X)\)</span> on the right hand side is called the prior while the function on the left hand side is the called the posterior probability. The denominator on the right hand side serves as a normalization factor for the posterior distribution.</p>
<p>Let us try to illustrate Bayes’ theorem through an example.</p>
</section>
<section id="example-of-usage-of-bayes-theorem">
<h2>Example of Usage of Bayes’ theorem<a class="headerlink" href="#example-of-usage-of-bayes-theorem" title="Link to this heading">#</a></h2>
<p>Let us suppose that you are undergoing a series of mammography scans in
order to rule out possible breast cancer cases. We define the
sensitivity for a positive event by the variable <span class="math notranslate nohighlight">\(X\)</span>. It takes binary
values with <span class="math notranslate nohighlight">\(X=1\)</span> representing a positive event and <span class="math notranslate nohighlight">\(X=0\)</span> being a
negative event. We reserve <span class="math notranslate nohighlight">\(Y\)</span> as a classification parameter for
either a negative or a positive breast cancer confirmation. (Short note on wordings: positive here means having breast cancer, although none of us would consider this being a positive thing).</p>
<p>We let <span class="math notranslate nohighlight">\(Y=1\)</span> represent the the case of having breast cancer and <span class="math notranslate nohighlight">\(Y=0\)</span> as not.</p>
<p>Let us assume that if you have breast cancer, the test will be positive with a probability of <span class="math notranslate nohighlight">\(0.8\)</span>, that is we have</p>
<div class="math notranslate nohighlight">
\[
p(X=1\vert Y=1) =0.8.
\]</div>
<p>This obviously sounds scary since many would conclude that if the test is positive, there is a likelihood of <span class="math notranslate nohighlight">\(80\%\)</span> for having cancer.
It is however not correct, as the following Bayesian analysis shows.</p>
</section>
<section id="doing-it-correctly">
<h2>Doing it correctly<a class="headerlink" href="#doing-it-correctly" title="Link to this heading">#</a></h2>
<p>If we look at various national surveys on breast cancer, the general likelihood of developing breast cancer is a very small number.
Let us assume that the prior probability in the population as a whole is</p>
<div class="math notranslate nohighlight">
\[
p(Y=1) =0.004.
\]</div>
<p>We need also to account for the fact that the test may produce a false positive result (false alarm). Let us here assume that we have</p>
<div class="math notranslate nohighlight">
\[
p(X=1\vert Y=0) =0.1.
\]</div>
<p>Using Bayes’ theorem we can then find the posterior probability that the person has breast cancer in case of a positive test, that is we can compute</p>
<div class="math notranslate nohighlight">
\[
p(Y=1\vert X=1)=\frac{p(X=1\vert Y=1)p(Y=1)}{p(X=1\vert Y=1)p(Y=1)+p(X=1\vert Y=0)p(Y=0)}=\frac{0.8\times 0.004}{0.8\times 0.004+0.1\times 0.996}=0.031.
\]</div>
<p>That is, in case of a positive test, there is only a <span class="math notranslate nohighlight">\(3\%\)</span> chance of having breast cancer!</p>
</section>
<section id="bayes-theorem-and-ridge-and-lasso-regression">
<h2>Bayes’ Theorem and Ridge and Lasso Regression<a class="headerlink" href="#bayes-theorem-and-ridge-and-lasso-regression" title="Link to this heading">#</a></h2>
<p>Using Bayes’ theorem we can gain a better intuition about Ridge and Lasso regression.</p>
<p>For ordinary least squares we postulated that the maximum likelihood for the doamin of events <span class="math notranslate nohighlight">\(\boldsymbol{D}\)</span> (one-dimensional case)</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{D}=[(x_0,y_0), (x_1,y_1),\dots, (x_{n-1},y_{n-1})],
\]</div>
<p>is given by</p>
<div class="math notranslate nohighlight">
\[
p(\boldsymbol{D}\vert\boldsymbol{\beta})=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}.
\]</div>
<p>In Bayes’ theorem this function plays the role of the so-called likelihood. We could now ask the question what is the posterior probability of a parameter set <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> given a domain of events <span class="math notranslate nohighlight">\(\boldsymbol{D}\)</span>? That is, how can we define the posterior probability</p>
<div class="math notranslate nohighlight">
\[
p(\boldsymbol{\beta}\vert\boldsymbol{D}).
\]</div>
<p>Bayes’ theorem comes to our rescue here since (omitting the normalization constant)</p>
<div class="math notranslate nohighlight">
\[
p(\boldsymbol{\beta}\vert\boldsymbol{D})\propto p(\boldsymbol{D}\vert\boldsymbol{\beta})p(\boldsymbol{\beta}).
\]</div>
<p>We have a model for <span class="math notranslate nohighlight">\(p(\boldsymbol{D}\vert\boldsymbol{\beta})\)</span> but need one for the <strong>prior</strong> <span class="math notranslate nohighlight">\(p(\boldsymbol{\beta})\)</span>!</p>
</section>
<section id="ridge-and-bayes">
<h2>Ridge and Bayes<a class="headerlink" href="#ridge-and-bayes" title="Link to this heading">#</a></h2>
<p>With the posterior probability defined by a likelihood which we have
already modeled and an unknown prior, we are now ready to make
additional models for the prior.</p>
<p>We can, based on our discussions of the variance of <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> and the mean value, assume that the prior for the values <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> is given by a Gaussian with mean value zero and variance <span class="math notranslate nohighlight">\(\tau^2\)</span>, that is</p>
<div class="math notranslate nohighlight">
\[
p(\boldsymbol{\beta})=\prod_{j=0}^{p-1}\exp{\left(-\frac{\beta_j^2}{2\tau^2}\right)}.
\]</div>
<p>Our posterior probability becomes then (omitting the normalization factor which is just a constant)</p>
<div class="math notranslate nohighlight">
\[
p(\boldsymbol{\beta\vert\boldsymbol{D})}=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}\prod_{j=0}^{p-1}\exp{\left(-\frac{\beta_j^2}{2\tau^2}\right)}.
\]</div>
<p>We can now optimize this quantity with respect to <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span>. As we
did for OLS, this is most conveniently done by taking the negative
logarithm of the posterior probability. Doing so and leaving out the
constants terms that do not depend on <span class="math notranslate nohighlight">\(\beta\)</span>, we have</p>
<div class="math notranslate nohighlight">
\[
C(\boldsymbol{\beta})=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\frac{1}{2\tau^2}\vert\vert\boldsymbol{\beta}\vert\vert_2^2,
\]</div>
<p>and replacing <span class="math notranslate nohighlight">\(1/2\tau^2\)</span> with <span class="math notranslate nohighlight">\(\lambda\)</span> we have</p>
<div class="math notranslate nohighlight">
\[
C(\boldsymbol{\beta})=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\lambda\vert\vert\boldsymbol{\beta}\vert\vert_2^2,
\]</div>
<p>which is our Ridge cost function! Nice, isn’t it?</p>
</section>
<section id="lasso-and-bayes">
<h2>Lasso and Bayes<a class="headerlink" href="#lasso-and-bayes" title="Link to this heading">#</a></h2>
<p>To derive the Lasso cost function, we simply replace the Gaussian prior with an exponential distribution (<a class="reference external" href="https://en.wikipedia.org/wiki/Laplace_distribution">Laplace in this case</a>) with zero mean value, that is</p>
<div class="math notranslate nohighlight">
\[
p(\boldsymbol{\beta})=\prod_{j=0}^{p-1}\exp{\left(-\frac{\vert\beta_j\vert}{\tau}\right)}.
\]</div>
<p>Our posterior probability becomes then (omitting the normalization factor which is just a constant)</p>
<div class="math notranslate nohighlight">
\[
p(\boldsymbol{\beta}\vert\boldsymbol{D})=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}\prod_{j=0}^{p-1}\exp{\left(-\frac{\vert\beta_j\vert}{\tau}\right)}.
\]</div>
<p>Taking the negative
logarithm of the posterior probability and leaving out the
constants terms that do not depend on <span class="math notranslate nohighlight">\(\beta\)</span>, we have</p>
<div class="math notranslate nohighlight">
\[
C(\boldsymbol{\beta})=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\frac{1}{\tau}\vert\vert\boldsymbol{\beta}\vert\vert_1,
\]</div>
<p>and replacing <span class="math notranslate nohighlight">\(1/\tau\)</span> with <span class="math notranslate nohighlight">\(\lambda\)</span> we have</p>
<div class="math notranslate nohighlight">
\[
C(\boldsymbol{\beta})=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\lambda\vert\vert\boldsymbol{\beta}\vert\vert_1,
\]</div>
<p>which is our Lasso cost function!</p>
</section>
<section id="why-resampling-methods">
<h2>Why resampling methods<a class="headerlink" href="#why-resampling-methods" title="Link to this heading">#</a></h2>
<p>Before we proceed, we need to rethink what we have been doing. In our
eager to fit the data, we have omitted several important elements in
our regression analysis. In what follows we will</p>
<ol class="arabic simple">
<li><p>look at statistical properties, including a discussion of mean values, variance and the so-called bias-variance tradeoff</p></li>
<li><p>introduce resampling techniques like cross-validation, bootstrapping and jackknife and more</p></li>
</ol>
<p>and discuss how to select a given model (one of the difficult parts in machine learning).</p>
</section>
<section id="resampling-methods">
<h2>Resampling methods<a class="headerlink" href="#resampling-methods" title="Link to this heading">#</a></h2>
<p>Resampling methods are an indispensable tool in modern
statistics. They involve repeatedly drawing samples from a training
set and refitting a model of interest on each sample in order to
obtain additional information about the fitted model. For example, in
order to estimate the variability of a linear regression fit, we can
repeatedly draw different samples from the training data, fit a linear
regression to each new sample, and then examine the extent to which
the resulting fits differ. Such an approach may allow us to obtain
information that would not be available from fitting the model only
once using the original training sample.</p>
<p>Two resampling methods are often used in Machine Learning analyses,</p>
<ol class="arabic simple">
<li><p>The <strong>bootstrap method</strong></p></li>
<li><p>and <strong>Cross-Validation</strong></p></li>
</ol>
<p>In addition there are several other methods such as the Jackknife and the Blocking methods. We will discuss in particular
cross-validation and the bootstrap method.</p>
</section>
<section id="resampling-approaches-can-be-computationally-expensive">
<h2>Resampling approaches can be computationally expensive<a class="headerlink" href="#resampling-approaches-can-be-computationally-expensive" title="Link to this heading">#</a></h2>
<p>Resampling approaches can be computationally expensive, because they
involve fitting the same statistical method multiple times using
different subsets of the training data. However, due to recent
advances in computing power, the computational requirements of
resampling methods generally are not prohibitive. In this chapter, we
discuss two of the most commonly used resampling methods,
cross-validation and the bootstrap. Both methods are important tools
in the practical application of many statistical learning
procedures. For example, cross-validation can be used to estimate the
test error associated with a given statistical learning method in
order to evaluate its performance, or to select the appropriate level
of flexibility. The process of evaluating a model’s performance is
known as model assessment, whereas the process of selecting the proper
level of flexibility for a model is known as model selection. The
bootstrap is widely used.</p>
</section>
<section id="id1">
<h2>Why resampling methods ?<a class="headerlink" href="#id1" title="Link to this heading">#</a></h2>
<p><strong>Statistical analysis.</strong></p>
<ul class="simple">
<li><p>Our simulations can be treated as <em>computer experiments</em>. This is particularly the case for Monte Carlo methods which are widely used in statistical analyses.</p></li>
<li><p>The results can be analysed with the same statistical tools as we would use when analysing experimental data.</p></li>
<li><p>As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors.</p></li>
</ul>
</section>
<section id="statistical-analysis">
<h2>Statistical analysis<a class="headerlink" href="#statistical-analysis" title="Link to this heading">#</a></h2>
<ul class="simple">
<li><p>As in other experiments, many numerical experiments have two classes of errors:</p>
<ul>
<li><p>Statistical errors</p></li>
<li><p>Systematical errors</p></li>
</ul>
</li>
<li><p>Statistical errors can be estimated using standard tools from statistics</p></li>
<li><p>Systematical errors are method specific and must be treated differently from case to case.</p></li>
</ul>
</section>
<section id="id2">
<h2>Resampling methods<a class="headerlink" href="#id2" title="Link to this heading">#</a></h2>
<p>With all these analytical equations for both the OLS and Ridge
regression, we will now outline how to assess a given model. This will
lead to a discussion of the so-called bias-variance tradeoff (see
below) and so-called resampling methods.</p>
<p>One of the quantities we have discussed as a way to measure errors is
the mean-squared error (MSE), mainly used for fitting of continuous
functions. Another choice is the absolute error.</p>
<p>In the discussions below we will focus on the MSE and in particular since we will split the data into test and training data,
we discuss the</p>
<ol class="arabic simple">
<li><p>prediction error or simply the <strong>test error</strong> <span class="math notranslate nohighlight">\(\mathrm{Err_{Test}}\)</span>, where we have a fixed training set and the test error is the MSE arising from the data reserved for testing. We discuss also the</p></li>
<li><p>training error <span class="math notranslate nohighlight">\(\mathrm{Err_{Train}}\)</span>, which is the average loss over the training data.</p></li>
</ol>
<p>As our model becomes more and more complex, more of the training data tends to used. The training may thence adapt to more complicated structures in the data. This may lead to a decrease in the bias (see below for code example) and a slight increase of the variance for the test error.
For a certain level of complexity the test error will reach minimum, before starting to increase again. The
training error reaches a saturation.</p>
</section>
<section id="resampling-methods-bootstrap">
<h2>Resampling methods: Bootstrap<a class="headerlink" href="#resampling-methods-bootstrap" title="Link to this heading">#</a></h2>
<p>Bootstrapping is a <a class="reference external" href="https://en.wikipedia.org/wiki/Nonparametric_statistics">non-parametric approach</a> to statistical inference
that substitutes computation for more traditional distributional
assumptions and asymptotic results. Bootstrapping offers a number of
advantages:</p>
<ol class="arabic simple">
<li><p>The bootstrap is quite general, although there are some cases in which it fails.</p></li>
<li><p>Because it does not require distributional assumptions (such as normally distributed errors), the bootstrap can provide more accurate inferences when the data are not well behaved or when the sample size is small.</p></li>
<li><p>It is possible to apply the bootstrap to statistics with sampling distributions that are difficult to derive, even asymptotically.</p></li>
<li><p>It is relatively simple to apply the bootstrap to complex data-collection plans (such as stratified and clustered samples).</p></li>
</ol>
<p>The textbook by <a class="reference external" href="https://www.cambridge.org/core/books/bootstrap-methods-and-their-application/ED2FD043579F27952363566DC09CBD6A">Davison on the Bootstrap Methods and their Applications</a> provides many more insights and proofs. In this course we will take a more practical approach and use the results and theorems provided in the literature. For those interested in reading more about the bootstrap methods, we recommend the above text and the one by <a class="reference external" href="https://www.routledge.com/An-Introduction-to-the-Bootstrap/Efron-Tibshirani/p/book/9780412042317">Efron and Tibshirani</a>.</p>
<p>Before we proceed however, we need to remind ourselves about a central theorem in statistics, namely the so-called <strong>central limit theorem</strong>.</p>
</section>
<section id="the-central-limit-theorem">
<h2>The Central Limit Theorem<a class="headerlink" href="#the-central-limit-theorem" title="Link to this heading">#</a></h2>
<p>Suppose we have a PDF <span class="math notranslate nohighlight">\(p(x)\)</span> from which we generate a series <span class="math notranslate nohighlight">\(N\)</span>
of averages <span class="math notranslate nohighlight">\(\mathbb{E}[x_i]\)</span>. Each mean value <span class="math notranslate nohighlight">\(\mathbb{E}[x_i]\)</span>
is viewed as the average of a specific measurement, e.g., throwing
dice 100 times and then taking the average value, or producing a certain
amount of random numbers.
For notational ease, we set <span class="math notranslate nohighlight">\(\mathbb{E}[x_i]=x_i\)</span> in the discussion
which follows. We do the same for <span class="math notranslate nohighlight">\(\mathbb{E}[z]=z\)</span>.</p>
<p>If we compute the mean <span class="math notranslate nohighlight">\(z\)</span> of <span class="math notranslate nohighlight">\(m\)</span> such mean values <span class="math notranslate nohighlight">\(x_i\)</span></p>
<div class="math notranslate nohighlight">
\[
z=\frac{x_1+x_2+\dots+x_m}{m},
\]</div>
<p>the question we pose is which is the PDF of the new variable <span class="math notranslate nohighlight">\(z\)</span>.</p>
</section>
<section id="finding-the-limit">
<h2>Finding the Limit<a class="headerlink" href="#finding-the-limit" title="Link to this heading">#</a></h2>
<p>The probability of obtaining an average value <span class="math notranslate nohighlight">\(z\)</span> is the product of the
probabilities of obtaining arbitrary individual mean values <span class="math notranslate nohighlight">\(x_i\)</span>,
but with the constraint that the average is <span class="math notranslate nohighlight">\(z\)</span>. We can express this through
the following expression</p>
<div class="math notranslate nohighlight">
\[
\tilde{p}(z)=\int dx_1p(x_1)\int dx_2p(x_2)\dots\int dx_mp(x_m)
\delta(z-\frac{x_1+x_2+\dots+x_m}{m}),
\]</div>
<p>where the <span class="math notranslate nohighlight">\(\delta\)</span>-function enbodies the constraint that the mean is <span class="math notranslate nohighlight">\(z\)</span>.
All measurements that lead to each individual <span class="math notranslate nohighlight">\(x_i\)</span> are expected to
be independent, which in turn means that we can express <span class="math notranslate nohighlight">\(\tilde{p}\)</span> as the
product of individual <span class="math notranslate nohighlight">\(p(x_i)\)</span>. The independence assumption is important in the derivation of the central limit theorem.</p>
</section>
<section id="rewriting-the-delta-function">
<h2>Rewriting the <span class="math notranslate nohighlight">\(\delta\)</span>-function<a class="headerlink" href="#rewriting-the-delta-function" title="Link to this heading">#</a></h2>
<p>If we use the integral expression for the <span class="math notranslate nohighlight">\(\delta\)</span>-function</p>
<div class="math notranslate nohighlight">
\[
\delta(z-\frac{x_1+x_2+\dots+x_m}{m})=\frac{1}{2\pi}\int_{-\infty}^{\infty}
dq\exp{\left(iq(z-\frac{x_1+x_2+\dots+x_m}{m})\right)},
\]</div>
<p>and inserting <span class="math notranslate nohighlight">\(e^{i\mu q-i\mu q}\)</span> where <span class="math notranslate nohighlight">\(\mu\)</span> is the mean value
we arrive at</p>
<div class="math notranslate nohighlight">
\[
\tilde{p}(z)=\frac{1}{2\pi}\int_{-\infty}^{\infty}
dq\exp{\left(iq(z-\mu)\right)}\left[\int_{-\infty}^{\infty}
dxp(x)\exp{\left(iq(\mu-x)/m\right)}\right]^m,
\]</div>
<p>with the integral over <span class="math notranslate nohighlight">\(x\)</span> resulting in</p>
<div class="math notranslate nohighlight">
\[
\int_{-\infty}^{\infty}dxp(x)\exp{\left(iq(\mu-x)/m\right)}=
\int_{-\infty}^{\infty}dxp(x)
\left[1+\frac{iq(\mu-x)}{m}-\frac{q^2(\mu-x)^2}{2m^2}+\dots\right].
\]</div>
</section>
<section id="identifying-terms">
<h2>Identifying Terms<a class="headerlink" href="#identifying-terms" title="Link to this heading">#</a></h2>
<p>The second term on the rhs disappears since this is just the mean and
employing the definition of <span class="math notranslate nohighlight">\(\sigma^2\)</span> we have</p>
<div class="math notranslate nohighlight">
\[
\int_{-\infty}^{\infty}dxp(x)e^{\left(iq(\mu-x)/m\right)}=
1-\frac{q^2\sigma^2}{2m^2}+\dots,
\]</div>
<p>resulting in</p>
<div class="math notranslate nohighlight">
\[
\left[\int_{-\infty}^{\infty}dxp(x)\exp{\left(iq(\mu-x)/m\right)}\right]^m\approx
\left[1-\frac{q^2\sigma^2}{2m^2}+\dots \right]^m,
\]</div>
<p>and in the limit <span class="math notranslate nohighlight">\(m\rightarrow \infty\)</span> we obtain</p>
<div class="math notranslate nohighlight">
\[
\tilde{p}(z)=\frac{1}{\sqrt{2\pi}(\sigma/\sqrt{m})}
\exp{\left(-\frac{(z-\mu)^2}{2(\sigma/\sqrt{m})^2}\right)},
\]</div>
<p>which is the normal distribution with variance
<span class="math notranslate nohighlight">\(\sigma^2_m=\sigma^2/m\)</span>, where <span class="math notranslate nohighlight">\(\sigma\)</span> is the variance of the PDF <span class="math notranslate nohighlight">\(p(x)\)</span>
and <span class="math notranslate nohighlight">\(\mu\)</span> is also the mean of the PDF <span class="math notranslate nohighlight">\(p(x)\)</span>.</p>
</section>
<section id="wrapping-it-up">
<h2>Wrapping it up<a class="headerlink" href="#wrapping-it-up" title="Link to this heading">#</a></h2>
<p>Thus, the central limit theorem states that the PDF <span class="math notranslate nohighlight">\(\tilde{p}(z)\)</span> of
the average of <span class="math notranslate nohighlight">\(m\)</span> random values corresponding to a PDF <span class="math notranslate nohighlight">\(p(x)\)</span>
is a normal distribution whose mean is the
mean value of the PDF <span class="math notranslate nohighlight">\(p(x)\)</span> and whose variance is the variance
of the PDF <span class="math notranslate nohighlight">\(p(x)\)</span> divided by <span class="math notranslate nohighlight">\(m\)</span>, the number of values used to compute <span class="math notranslate nohighlight">\(z\)</span>.</p>
<p>The central limit theorem leads to the well-known expression for the
standard deviation, given by</p>
<div class="math notranslate nohighlight">
\[
\sigma_m=
\frac{\sigma}{\sqrt{m}}.
\]</div>
<p>The latter is true only if the average value is known exactly. This is obtained in the limit
<span class="math notranslate nohighlight">\(m\rightarrow \infty\)</span> only. Because the mean and the variance are measured quantities we obtain
the familiar expression in statistics (the so-called Bessel correction)</p>
<div class="math notranslate nohighlight">
\[
\sigma_m\approx
\frac{\sigma}{\sqrt{m-1}}.
\]</div>
<p>In many cases however the above estimate for the standard deviation,
in particular if correlations are strong, may be too simplistic. Keep
in mind that we have assumed that the variables <span class="math notranslate nohighlight">\(x\)</span> are independent
and identically distributed. This is obviously not always the
case. For example, the random numbers (or better pseudorandom numbers)
we generate in various calculations do always exhibit some
correlations.</p>
<p>The theorem is satisfied by a large class of PDFs. Note however that for a
finite <span class="math notranslate nohighlight">\(m\)</span>, it is not always possible to find a closed form /analytic expression for
<span class="math notranslate nohighlight">\(\tilde{p}(x)\)</span>.</p>
</section>
<section id="confidence-intervals">
<h2>Confidence Intervals<a class="headerlink" href="#confidence-intervals" title="Link to this heading">#</a></h2>
<p>Confidence intervals are used in statistics and represent a type of estimate
computed from the observed data. This gives a range of values for an
unknown parameter such as the parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> from linear regression.</p>
<p>With the OLS expressions for the parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> we found
<span class="math notranslate nohighlight">\(\mathbb{E}(\boldsymbol{\beta}) = \boldsymbol{\beta}\)</span>, which means that the estimator of the regression parameters is unbiased.</p>
<p>In the exercises this week we show that the variance of the estimate of the <span class="math notranslate nohighlight">\(j\)</span>-th regression coefficient is
<span class="math notranslate nohighlight">\(\boldsymbol{\sigma}^2 (\boldsymbol{\beta}_j ) = \boldsymbol{\sigma}^2 [(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj} \)</span>.</p>
<p>This quantity can be used to
construct a confidence interval for the estimates.</p>
</section>
<section id="standard-approach-based-on-the-normal-distribution">
<h2>Standard Approach based on the Normal Distribution<a class="headerlink" href="#standard-approach-based-on-the-normal-distribution" title="Link to this heading">#</a></h2>
<p>We will assume that the parameters <span class="math notranslate nohighlight">\(\beta\)</span> follow a normal
distribution. We can then define the confidence interval. Here we will be using as
shorthands <span class="math notranslate nohighlight">\(\mu_{\beta}\)</span> for the above mean value and <span class="math notranslate nohighlight">\(\sigma_{\beta}\)</span>
for the standard deviation. We have then a confidence interval</p>
<div class="math notranslate nohighlight">
\[
\left(\mu_{\beta}\pm \frac{z\sigma_{\beta}}{\sqrt{n}}\right),
\]</div>
<p>where <span class="math notranslate nohighlight">\(z\)</span> defines the level of certainty (or confidence). For a normal
distribution typical parameters are <span class="math notranslate nohighlight">\(z=2.576\)</span> which corresponds to a
confidence of <span class="math notranslate nohighlight">\(99\%\)</span> while <span class="math notranslate nohighlight">\(z=1.96\)</span> corresponds to a confidence of
<span class="math notranslate nohighlight">\(95\%\)</span>. A confidence level of <span class="math notranslate nohighlight">\(95\%\)</span> is commonly used and it is
normally referred to as a <em>two-sigmas</em> confidence level, that is we
approximate <span class="math notranslate nohighlight">\(z\approx 2\)</span>.</p>
<p>For more discussions of confidence intervals (and in particular linked with a discussion of the bootstrap method), see chapter 5 of the textbook by <a class="reference external" href="https://www.cambridge.org/core/books/bootstrap-methods-and-their-application/ED2FD043579F27952363566DC09CBD6A">Davison on the Bootstrap Methods and their Applications</a></p>
<p>In this text you will also find an in-depth discussion of the
Bootstrap method, why it works and various theorems related to it.</p>
</section>
<section id="resampling-methods-bootstrap-background">
<h2>Resampling methods: Bootstrap background<a class="headerlink" href="#resampling-methods-bootstrap-background" title="Link to this heading">#</a></h2>
<p>Since <span class="math notranslate nohighlight">\(\widehat{\beta} = \widehat{\beta}(\boldsymbol{X})\)</span> is a function of random variables,
<span class="math notranslate nohighlight">\(\widehat{\beta}\)</span> itself must be a random variable. Thus it has
a pdf, call this function <span class="math notranslate nohighlight">\(p(\boldsymbol{t})\)</span>. The aim of the bootstrap is to
estimate <span class="math notranslate nohighlight">\(p(\boldsymbol{t})\)</span> by the relative frequency of
<span class="math notranslate nohighlight">\(\widehat{\beta}\)</span>. You can think of this as using a histogram
in the place of <span class="math notranslate nohighlight">\(p(\boldsymbol{t})\)</span>. If the relative frequency closely
resembles <span class="math notranslate nohighlight">\(p(\vec{t})\)</span>, then using numerics, it is straight forward to
estimate all the interesting parameters of <span class="math notranslate nohighlight">\(p(\boldsymbol{t})\)</span> using point
estimators.</p>
</section>
<section id="resampling-methods-more-bootstrap-background">
<h2>Resampling methods: More Bootstrap background<a class="headerlink" href="#resampling-methods-more-bootstrap-background" title="Link to this heading">#</a></h2>
<p>In the case that <span class="math notranslate nohighlight">\(\widehat{\beta}\)</span> has
more than one component, and the components are independent, we use the
same estimator on each component separately. If the probability
density function of <span class="math notranslate nohighlight">\(X_i\)</span>, <span class="math notranslate nohighlight">\(p(x)\)</span>, had been known, then it would have
been straightforward to do this by:</p>
<ol class="arabic simple">
<li><p>Drawing lots of numbers from <span class="math notranslate nohighlight">\(p(x)\)</span>, suppose we call one such set of numbers <span class="math notranslate nohighlight">\((X_1^*, X_2^*, \cdots, X_n^*)\)</span>.</p></li>
<li><p>Then using these numbers, we could compute a replica of <span class="math notranslate nohighlight">\(\widehat{\beta}\)</span> called <span class="math notranslate nohighlight">\(\widehat{\beta}^*\)</span>.</p></li>
</ol>
<p>By repeated use of the above two points, many
estimates of <span class="math notranslate nohighlight">\(\widehat{\beta}\)</span> can be obtained. The
idea is to use the relative frequency of <span class="math notranslate nohighlight">\(\widehat{\beta}^*\)</span>
(think of a histogram) as an estimate of <span class="math notranslate nohighlight">\(p(\boldsymbol{t})\)</span>.</p>
</section>
<section id="resampling-methods-bootstrap-approach">
<h2>Resampling methods: Bootstrap approach<a class="headerlink" href="#resampling-methods-bootstrap-approach" title="Link to this heading">#</a></h2>
<p>But
unless there is enough information available about the process that
generated <span class="math notranslate nohighlight">\(X_1,X_2,\cdots,X_n\)</span>, <span class="math notranslate nohighlight">\(p(x)\)</span> is in general
unknown. Therefore, <a class="reference external" href="https://projecteuclid.org/euclid.aos/1176344552">Efron in 1979</a> asked the
question: What if we replace <span class="math notranslate nohighlight">\(p(x)\)</span> by the relative frequency
of the observation <span class="math notranslate nohighlight">\(X_i\)</span>?</p>
<p>If we draw observations in accordance with
the relative frequency of the observations, will we obtain the same
result in some asymptotic sense? The answer is yes.</p>
</section>
<section id="resampling-methods-bootstrap-steps">
<h2>Resampling methods: Bootstrap steps<a class="headerlink" href="#resampling-methods-bootstrap-steps" title="Link to this heading">#</a></h2>
<p>The independent bootstrap works like this:</p>
<ol class="arabic simple">
<li><p>Draw with replacement <span class="math notranslate nohighlight">\(n\)</span> numbers for the observed variables <span class="math notranslate nohighlight">\(\boldsymbol{x} = (x_1,x_2,\cdots,x_n)\)</span>.</p></li>
<li><p>Define a vector <span class="math notranslate nohighlight">\(\boldsymbol{x}^*\)</span> containing the values which were drawn from <span class="math notranslate nohighlight">\(\boldsymbol{x}\)</span>.</p></li>
<li><p>Using the vector <span class="math notranslate nohighlight">\(\boldsymbol{x}^*\)</span> compute <span class="math notranslate nohighlight">\(\widehat{\beta}^*\)</span> by evaluating <span class="math notranslate nohighlight">\(\widehat \beta\)</span> under the observations <span class="math notranslate nohighlight">\(\boldsymbol{x}^*\)</span>.</p></li>
<li><p>Repeat this process <span class="math notranslate nohighlight">\(k\)</span> times.</p></li>
</ol>
<p>When you are done, you can draw a histogram of the relative frequency
of <span class="math notranslate nohighlight">\(\widehat \beta^*\)</span>. This is your estimate of the probability
distribution <span class="math notranslate nohighlight">\(p(t)\)</span>. Using this probability distribution you can
estimate any statistics thereof. In principle you never draw the
histogram of the relative frequency of <span class="math notranslate nohighlight">\(\widehat{\beta}^*\)</span>. Instead
you use the estimators corresponding to the statistic of interest. For
example, if you are interested in estimating the variance of <span class="math notranslate nohighlight">\(\widehat
\beta\)</span>, apply the etsimator <span class="math notranslate nohighlight">\(\widehat \sigma^2\)</span> to the values
<span class="math notranslate nohighlight">\(\widehat \beta^*\)</span>.</p>
</section>
<section id="code-example-for-the-bootstrap-method">
<h2>Code example for the Bootstrap method<a class="headerlink" href="#code-example-for-the-bootstrap-method" title="Link to this heading">#</a></h2>
<p>The following code starts with a Gaussian distribution with mean value
<span class="math notranslate nohighlight">\(\mu =100\)</span> and variance <span class="math notranslate nohighlight">\(\sigma=15\)</span>. We use this to generate the data
used in the bootstrap analysis. The bootstrap analysis returns a data
set after a given number of bootstrap operations (as many as we have
data points). This data set consists of estimated mean values for each
bootstrap operation. The histogram generated by the bootstrap method
shows that the distribution for these mean values is also a Gaussian,
centered around the mean value <span class="math notranslate nohighlight">\(\mu=100\)</span> but with standard deviation
<span class="math notranslate nohighlight">\(\sigma/\sqrt{n}\)</span>, where <span class="math notranslate nohighlight">\(n\)</span> is the number of bootstrap samples (in
this case the same as the number of original data points). The value
of the standard deviation is what we expect from the central limit
theorem.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="o">%</span><span class="k">matplotlib</span> inline
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">time</span> <span class="kn">import</span> <span class="n">time</span>
<span class="kn">from</span> <span class="nn">scipy.stats</span> <span class="kn">import</span> <span class="n">norm</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="c1"># Returns mean of bootstrap samples </span>
<span class="c1"># Bootstrap algorithm</span>
<span class="k">def</span> <span class="nf">bootstrap</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">datapoints</span><span class="p">):</span>
<span class="n">t</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">datapoints</span><span class="p">)</span>
<span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
<span class="c1"># non-parametric bootstrap </span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">datapoints</span><span class="p">):</span>
<span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">data</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">n</span><span class="p">,</span><span class="n">n</span><span class="p">)])</span>
<span class="c1"># analysis </span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Bootstrap Statistics :&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;original bias std. error&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;</span><span class="si">%8g</span><span class="s2"> </span><span class="si">%8g</span><span class="s2"> </span><span class="si">%14g</span><span class="s2"> </span><span class="si">%15g</span><span class="s2">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">data</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">data</span><span class="p">),</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">t</span><span class="p">),</span><span class="n">np</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">t</span><span class="p">)))</span>
<span class="k">return</span> <span class="n">t</span>
<span class="c1"># We set the mean value to 100 and the standard deviation to 15</span>
<span class="n">mu</span><span class="p">,</span> <span class="n">sigma</span> <span class="o">=</span> <span class="mi">100</span><span class="p">,</span> <span class="mi">15</span>
<span class="n">datapoints</span> <span class="o">=</span> <span class="mi">10000</span>
<span class="c1"># We generate random numbers according to the normal distribution</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">mu</span> <span class="o">+</span> <span class="n">sigma</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">randn</span><span class="p">(</span><span class="n">datapoints</span><span class="p">)</span>
<span class="c1"># bootstrap returns the data sample </span>
<span class="n">t</span> <span class="o">=</span> <span class="n">bootstrap</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">datapoints</span><span class="p">)</span>
</pre></div>
</div>
</div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Bootstrap Statistics :
original bias std. error
99.8557 14.9926 99.8545 0.150204
</pre></div>
</div>
</div>
</div>
<p>We see that our new variance and from that the standard deviation, agrees with the central limit theorem.</p>
</section>
<section id="plotting-the-histogram">
<h2>Plotting the Histogram<a class="headerlink" href="#plotting-the-histogram" title="Link to this heading">#</a></h2>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># the histogram of the bootstrapped data (normalized data if density = True)</span>
<span class="n">n</span><span class="p">,</span> <span class="n">binsboot</span><span class="p">,</span> <span class="n">patches</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="mi">50</span><span class="p">,</span> <span class="n">density</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">facecolor</span><span class="o">=</span><span class="s1">&#39;red&#39;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.75</span><span class="p">)</span>
<span class="c1"># add a &#39;best fit&#39; line </span>
<span class="n">y</span> <span class="o">=</span> <span class="n">norm</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">binsboot</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">t</span><span class="p">))</span>
<span class="n">lt</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">binsboot</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="s1">&#39;b&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;x&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;Probability&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
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</div>
</section>
<section id="the-bias-variance-tradeoff">
<h2>The bias-variance tradeoff<a class="headerlink" href="#the-bias-variance-tradeoff" title="Link to this heading">#</a></h2>
<p>We will discuss the bias-variance tradeoff in the context of
continuous predictions such as regression. However, many of the
intuitions and ideas discussed here also carry over to classification
tasks. Consider a dataset <span class="math notranslate nohighlight">\(\mathcal{D}\)</span> consisting of the data
<span class="math notranslate nohighlight">\(\mathbf{X}_\mathcal{D}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\}\)</span>.</p>
<p>Let us assume that the true data is generated from a noisy model</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{y}=f(\boldsymbol{x}) + \boldsymbol{\epsilon}
\]</div>
<p>where <span class="math notranslate nohighlight">\(\epsilon\)</span> is normally distributed with mean zero and standard deviation <span class="math notranslate nohighlight">\(\sigma^2\)</span>.</p>
<p>In our derivation of the ordinary least squares method we defined then
an approximation to the function <span class="math notranslate nohighlight">\(f\)</span> in terms of the parameters
<span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> and the design matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> which embody our model,
that is <span class="math notranslate nohighlight">\(\boldsymbol{\tilde{y}}=\boldsymbol{X}\boldsymbol{\beta}\)</span>.</p>
<p>Thereafter we found the parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> by optimizing the means squared error via the so-called cost function</p>
<div class="math notranslate nohighlight">
\[
C(\boldsymbol{X},\boldsymbol{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2=\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right].
\]</div>
<p>We can rewrite this as</p>
<div class="math notranslate nohighlight">
\[
\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\frac{1}{n}\sum_i(f_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\sigma^2.
\]</div>
<p>The three terms represent the square of the bias of the learning
method, which can be thought of as the error caused by the simplifying
assumptions built into the method. The second term represents the
variance of the chosen model and finally the last terms is variance of
the error <span class="math notranslate nohighlight">\(\boldsymbol{\epsilon}\)</span>.</p>
<p>To derive this equation, we need to recall that the variance of <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> and <span class="math notranslate nohighlight">\(\boldsymbol{\epsilon}\)</span> are both equal to <span class="math notranslate nohighlight">\(\sigma^2\)</span>. The mean value of <span class="math notranslate nohighlight">\(\boldsymbol{\epsilon}\)</span> is by definition equal to zero. Furthermore, the function <span class="math notranslate nohighlight">\(f\)</span> is not a stochastics variable, idem for <span class="math notranslate nohighlight">\(\boldsymbol{\tilde{y}}\)</span>.
We use a more compact notation in terms of the expectation value</p>
<div class="math notranslate nohighlight">
\[
\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathbb{E}\left[(\boldsymbol{f}+\boldsymbol{\epsilon}-\boldsymbol{\tilde{y}})^2\right],
\]</div>
<p>and adding and subtracting <span class="math notranslate nohighlight">\(\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\)</span> we get</p>
<div class="math notranslate nohighlight">
\[
\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathbb{E}\left[(\boldsymbol{f}+\boldsymbol{\epsilon}-\boldsymbol{\tilde{y}}+\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2\right],
\]</div>
<p>which, using the abovementioned expectation values can be rewritten as</p>
<div class="math notranslate nohighlight">
\[
\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathbb{E}\left[(\boldsymbol{y}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2\right]+\mathrm{Var}\left[\boldsymbol{\tilde{y}}\right]+\sigma^2,
\]</div>
<p>that is the rewriting in terms of the so-called bias, the variance of the model <span class="math notranslate nohighlight">\(\boldsymbol{\tilde{y}}\)</span> and the variance of <span class="math notranslate nohighlight">\(\boldsymbol{\epsilon}\)</span>.</p>
</section>
<section id="a-way-to-read-the-bias-variance-tradeoff">
<h2>A way to Read the Bias-Variance Tradeoff<a class="headerlink" href="#a-way-to-read-the-bias-variance-tradeoff" title="Link to this heading">#</a></h2>
<!-- dom:FIGURE: [figures/BiasVariance.png, width=600 frac=0.9] -->
<!-- begin figure -->
<p><img src="figures/BiasVariance.png" width="600"><p style="font-size: 0.9em"><i>Figure 1: </i></p></p>
<!-- end figure --></section>
<section id="example-code-for-bias-variance-tradeoff">
<h2>Example code for Bias-Variance tradeoff<a class="headerlink" href="#example-code-for-bias-variance-tradeoff" title="Link to this heading">#</a></h2>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LinearRegression</span><span class="p">,</span> <span class="n">Ridge</span><span class="p">,</span> <span class="n">Lasso</span>
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">PolynomialFeatures</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
<span class="kn">from</span> <span class="nn">sklearn.pipeline</span> <span class="kn">import</span> <span class="n">make_pipeline</span>
<span class="kn">from</span> <span class="nn">sklearn.utils</span> <span class="kn">import</span> <span class="n">resample</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">2018</span><span class="p">)</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">500</span>
<span class="n">n_boostraps</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">degree</span> <span class="o">=</span> <span class="mi">18</span> <span class="c1"># A quite high value, just to show.</span>
<span class="n">noise</span> <span class="o">=</span> <span class="mf">0.1</span>
<span class="c1"># Make data set.</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mf">1.5</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</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">normal</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="n">x</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="c1"># Hold out some test data that is never used in training.</span>
<span class="n">x_train</span><span class="p">,</span> <span class="n">x_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">test_size</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
<span class="c1"># Combine x transformation and model into one operation.</span>
<span class="c1"># Not neccesary, but convenient.</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">make_pipeline</span><span class="p">(</span><span class="n">PolynomialFeatures</span><span class="p">(</span><span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">),</span> <span class="n">LinearRegression</span><span class="p">(</span><span class="n">fit_intercept</span><span class="o">=</span><span class="kc">False</span><span class="p">))</span>
<span class="c1"># The following (m x n_bootstraps) matrix holds the column vectors y_pred</span>
<span class="c1"># for each bootstrap iteration.</span>
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">((</span><span class="n">y_test</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">n_boostraps</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_boostraps</span><span class="p">):</span>
<span class="n">x_</span><span class="p">,</span> <span class="n">y_</span> <span class="o">=</span> <span class="n">resample</span><span class="p">(</span><span class="n">x_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="c1"># Evaluate the new model on the same test data each time.</span>
<span class="n">y_pred</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">x_</span><span class="p">,</span> <span class="n">y_</span><span class="p">)</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">x_test</span><span class="p">)</span><span class="o">.</span><span class="n">ravel</span><span class="p">()</span>
<span class="c1"># Note: Expectations and variances taken w.r.t. different training</span>
<span class="c1"># data sets, hence the axis=1. Subsequent means are taken across the test data</span>
<span class="c1"># set in order to obtain a total value, but before this we have error/bias/variance</span>
<span class="c1"># calculated per data point in the test set.</span>
<span class="c1"># Note 2: The use of keepdims=True is important in the calculation of bias as this </span>
<span class="c1"># maintains the column vector form. Dropping this yields very unexpected results.</span>
<span class="n">error</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">((</span><span class="n">y_test</span> <span class="o">-</span> <span class="n">y_pred</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span> <span class="p">)</span>
<span class="n">bias</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span> <span class="p">(</span><span class="n">y_test</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">y_pred</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span> <span class="p">)</span>
<span class="n">variance</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">var</span><span class="p">(</span><span class="n">y_pred</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span> <span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Error:&#39;</span><span class="p">,</span> <span class="n">error</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Bias^2:&#39;</span><span class="p">,</span> <span class="n">bias</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Var:&#39;</span><span class="p">,</span> <span class="n">variance</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;</span><span class="si">{}</span><span class="s1"> &gt;= </span><span class="si">{}</span><span class="s1"> + </span><span class="si">{}</span><span class="s1"> = </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">error</span><span class="p">,</span> <span class="n">bias</span><span class="p">,</span> <span class="n">variance</span><span class="p">,</span> <span class="n">bias</span><span class="o">+</span><span class="n">variance</span><span class="p">))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">[::</span><span class="mi">5</span><span class="p">,</span> <span class="p">:],</span> <span class="n">y</span><span class="p">[::</span><span class="mi">5</span><span class="p">,</span> <span class="p">:],</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;f(x)&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">x_test</span><span class="p">,</span> <span class="n">y_test</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Data points&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">x_test</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">y_pred</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Pred&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Error: 0.013121574015585152
Bias^2: 0.012073649446193166
Var: 0.0010479245693919886
0.013121574015585152 &gt;= 0.012073649446193166 + 0.0010479245693919886 = 0.013121574015585155
</pre></div>
</div>
<img alt="_images/7578e78979a07628cc1a76650a7a1d76c14840a37ad8e8e0557d70bb3fd37287.png" src="_images/7578e78979a07628cc1a76650a7a1d76c14840a37ad8e8e0557d70bb3fd37287.png" />
</div>
</div>
</section>
<section id="understanding-what-happens">
<h2>Understanding what happens<a class="headerlink" href="#understanding-what-happens" title="Link to this heading">#</a></h2>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LinearRegression</span><span class="p">,</span> <span class="n">Ridge</span><span class="p">,</span> <span class="n">Lasso</span>
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">PolynomialFeatures</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
<span class="kn">from</span> <span class="nn">sklearn.pipeline</span> <span class="kn">import</span> <span class="n">make_pipeline</span>
<span class="kn">from</span> <span class="nn">sklearn.utils</span> <span class="kn">import</span> <span class="n">resample</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">2018</span><span class="p">)</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">40</span>
<span class="n">n_boostraps</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">maxdegree</span> <span class="o">=</span> <span class="mi">14</span>
<span class="c1"># Make data set.</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mf">1.5</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</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">normal</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="n">x</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="n">error</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">maxdegree</span><span class="p">)</span>
<span class="n">bias</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">maxdegree</span><span class="p">)</span>
<span class="n">variance</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">maxdegree</span><span class="p">)</span>
<span class="n">polydegree</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">maxdegree</span><span class="p">)</span>
<span class="n">x_train</span><span class="p">,</span> <span class="n">x_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">test_size</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
<span class="k">for</span> <span class="n">degree</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">maxdegree</span><span class="p">):</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">make_pipeline</span><span class="p">(</span><span class="n">PolynomialFeatures</span><span class="p">(</span><span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">),</span> <span class="n">LinearRegression</span><span class="p">(</span><span class="n">fit_intercept</span><span class="o">=</span><span class="kc">False</span><span class="p">))</span>
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">((</span><span class="n">y_test</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">n_boostraps</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_boostraps</span><span class="p">):</span>
<span class="n">x_</span><span class="p">,</span> <span class="n">y_</span> <span class="o">=</span> <span class="n">resample</span><span class="p">(</span><span class="n">x_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="n">y_pred</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">x_</span><span class="p">,</span> <span class="n">y_</span><span class="p">)</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">x_test</span><span class="p">)</span><span class="o">.</span><span class="n">ravel</span><span class="p">()</span>
<span class="n">polydegree</span><span class="p">[</span><span class="n">degree</span><span class="p">]</span> <span class="o">=</span> <span class="n">degree</span>
<span class="n">error</span><span class="p">[</span><span class="n">degree</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">((</span><span class="n">y_test</span> <span class="o">-</span> <span class="n">y_pred</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span> <span class="p">)</span>
<span class="n">bias</span><span class="p">[</span><span class="n">degree</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span> <span class="p">(</span><span class="n">y_test</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">y_pred</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span> <span class="p">)</span>
<span class="n">variance</span><span class="p">[</span><span class="n">degree</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">var</span><span class="p">(</span><span class="n">y_pred</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span> <span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Polynomial degree:&#39;</span><span class="p">,</span> <span class="n">degree</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Error:&#39;</span><span class="p">,</span> <span class="n">error</span><span class="p">[</span><span class="n">degree</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Bias^2:&#39;</span><span class="p">,</span> <span class="n">bias</span><span class="p">[</span><span class="n">degree</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Var:&#39;</span><span class="p">,</span> <span class="n">variance</span><span class="p">[</span><span class="n">degree</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;</span><span class="si">{}</span><span class="s1"> &gt;= </span><span class="si">{}</span><span class="s1"> + </span><span class="si">{}</span><span class="s1"> = </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">error</span><span class="p">[</span><span class="n">degree</span><span class="p">],</span> <span class="n">bias</span><span class="p">[</span><span class="n">degree</span><span class="p">],</span> <span class="n">variance</span><span class="p">[</span><span class="n">degree</span><span class="p">],</span> <span class="n">bias</span><span class="p">[</span><span class="n">degree</span><span class="p">]</span><span class="o">+</span><span class="n">variance</span><span class="p">[</span><span class="n">degree</span><span class="p">]))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">polydegree</span><span class="p">,</span> <span class="n">error</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Error&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">polydegree</span><span class="p">,</span> <span class="n">bias</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;bias&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">polydegree</span><span class="p">,</span> <span class="n">variance</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Variance&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 0
Error: 0.32149601703519115
Bias^2: 0.3123314713548606
Var: 0.009164545680330616
0.32149601703519115 &gt;= 0.3123314713548606 + 0.009164545680330616 = 0.3214960170351912
Polynomial degree: 1
Error: 0.08426840630693411
Bias^2: 0.0796891867672603
Var: 0.004579219539673834
0.08426840630693411 &gt;= 0.0796891867672603 + 0.004579219539673834 = 0.08426840630693413
Polynomial degree: 2
Error: 0.10398646080125035
Bias^2: 0.1007711427354898
Var: 0.0032153180657605116
0.10398646080125035 &gt;= 0.1007711427354898 + 0.0032153180657605116 = 0.10398646080125032
Polynomial degree: 3
Error: 0.06547790180152355
Bias^2: 0.06208238634231949
Var: 0.0033955154592040936
0.06547790180152355 &gt;= 0.06208238634231949 + 0.0033955154592040936 = 0.06547790180152359
Polynomial degree: 4
Error: 0.06844519414009445
Bias^2: 0.06453579006728324
Var: 0.003909404072811226
0.06844519414009445 &gt;= 0.06453579006728324 + 0.003909404072811226 = 0.06844519414009446
Polynomial degree: 5
Error: 0.05227921801205686
Bias^2: 0.0481872773043029
Var: 0.004091940707753939
0.05227921801205686 &gt;= 0.0481872773043029 + 0.004091940707753939 = 0.052279218012056844
Polynomial degree: 6
Error: 0.037813671417389005
Bias^2: 0.033657685071527665
Var: 0.00415598634586135
0.037813671417389005 &gt;= 0.033657685071527665 + 0.00415598634586135 = 0.03781367141738902
Polynomial degree: 7
Error: 0.02760977349102253
Bias^2: 0.022999498260366312
Var: 0.004610275230656212
0.02760977349102253 &gt;= 0.022999498260366312 + 0.004610275230656212 = 0.027609773491022525
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 8
Error: 0.017355848195593347
Bias^2: 0.010331721306655127
Var: 0.007024126888938232
0.017355848195593347 &gt;= 0.010331721306655127 + 0.007024126888938232 = 0.01735584819559336
Polynomial degree: 9
Error: 0.02660572763718093
Bias^2: 0.010018312644137363
Var: 0.016587414993043573
0.02660572763718093 &gt;= 0.010018312644137363 + 0.016587414993043573 = 0.026605727637180936
Polynomial degree: 10
Error: 0.021592704588025025
Bias^2: 0.010516485576645508
Var: 0.011076219011379514
0.021592704588025025 &gt;= 0.010516485576645508 + 0.011076219011379514 = 0.021592704588025022
Polynomial degree: 11
Error: 0.07160048164233104
Bias^2: 0.014436800088904942
Var: 0.05716368155342608
0.07160048164233104 &gt;= 0.014436800088904942 + 0.05716368155342608 = 0.07160048164233102
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 12
Error: 0.11547777218872497
Bias^2: 0.01628578269596628
Var: 0.09919198949275869
0.11547777218872497 &gt;= 0.01628578269596628 + 0.09919198949275869 = 0.11547777218872497
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 13
Error: 0.22842468702219465
Bias^2: 0.01975416527185249
Var: 0.20867052175034223
0.22842468702219465 &gt;= 0.01975416527185249 + 0.20867052175034223 = 0.2284246870221947
</pre></div>
</div>
<img alt="_images/f6f0ffafe30c3048dea2ba02ba2e0ebff7fbf16263d3181df8b404ba6d25e198.png" src="_images/f6f0ffafe30c3048dea2ba02ba2e0ebff7fbf16263d3181df8b404ba6d25e198.png" />
</div>
</div>
</section>
<section id="summing-up">
<h2>Summing up<a class="headerlink" href="#summing-up" title="Link to this heading">#</a></h2>
<p>The bias-variance tradeoff summarizes the fundamental tension in
machine learning, particularly supervised learning, between the
complexity of a model and the amount of training data needed to train
it. Since data is often limited, in practice it is often useful to
use a less-complex model with higher bias, that is a model whose asymptotic
performance is worse than another model because it is easier to
train and less sensitive to sampling noise arising from having a
finite-sized training dataset (smaller variance).</p>
<p>The above equations tell us that in
order to minimize the expected test error, we need to select a
statistical learning method that simultaneously achieves low variance
and low bias. Note that variance is inherently a nonnegative quantity,
and squared bias is also nonnegative. Hence, we see that the expected
test MSE can never lie below <span class="math notranslate nohighlight">\(Var(\epsilon)\)</span>, the irreducible error.</p>
<p>What do we mean by the variance and bias of a statistical learning
method? The variance refers to the amount by which our model would change if we
estimated it using a different training data set. Since the training
data are used to fit the statistical learning method, different
training data sets will result in a different estimate. But ideally the
estimate for our model should not vary too much between training
sets. However, if a method has high variance then small changes in
the training data can result in large changes in the model. In general, more
flexible statistical methods have higher variance.</p>
<p>You may also find this recent <a class="reference external" href="https://www.pnas.org/content/116/32/15849">article</a> of interest.</p>
</section>
<section id="another-example-from-scikit-learn-s-repository">
<h2>Another Example from Scikit-Learn’s Repository<a class="headerlink" href="#another-example-from-scikit-learn-s-repository" title="Link to this heading">#</a></h2>
<p>This example demonstrates the problems of underfitting and overfitting and
how we can use linear regression with polynomial features to approximate
nonlinear functions. The plot shows the function that we want to approximate,
which is a part of the cosine function. In addition, the samples from the
real function and the approximations of different models are displayed. The
models have polynomial features of different degrees. We can see that a
linear function (polynomial with degree 1) is not sufficient to fit the
training samples. This is called <strong>underfitting</strong>. A polynomial of degree 4
approximates the true function almost perfectly. However, for higher degrees
the model will <strong>overfit</strong> the training data, i.e. it learns the noise of the
training data.
We evaluate quantitatively overfitting and underfitting by using
cross-validation. We calculate the mean squared error (MSE) on the validation
set, the higher, the less likely the model generalizes correctly from the
training data.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#print(__doc__)</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">sklearn.pipeline</span> <span class="kn">import</span> <span class="n">Pipeline</span>
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">PolynomialFeatures</span>
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LinearRegression</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">cross_val_score</span>
<span class="k">def</span> <span class="nf">true_fun</span><span class="p">(</span><span class="n">X</span><span class="p">):</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="mf">1.5</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">X</span><span class="p">)</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">n_samples</span> <span class="o">=</span> <span class="mi">30</span>
<span class="n">degrees</span> <span class="o">=</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="mi">15</span><span class="p">]</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sort</span><span class="p">(</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">n_samples</span><span class="p">))</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">true_fun</span><span class="p">(</span><span class="n">X</span><span class="p">)</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">randn</span><span class="p">(</span><span class="n">n_samples</span><span class="p">)</span> <span class="o">*</span> <span class="mf">0.1</span>
<span class="n">plt</span><span class="o">.</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">14</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</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">degrees</span><span class="p">)):</span>
<span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">degrees</span><span class="p">),</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">setp</span><span class="p">(</span><span class="n">ax</span><span class="p">,</span> <span class="n">xticks</span><span class="o">=</span><span class="p">(),</span> <span class="n">yticks</span><span class="o">=</span><span class="p">())</span>
<span class="n">polynomial_features</span> <span class="o">=</span> <span class="n">PolynomialFeatures</span><span class="p">(</span><span class="n">degree</span><span class="o">=</span><span class="n">degrees</span><span class="p">[</span><span class="n">i</span><span class="p">],</span>
<span class="n">include_bias</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="n">linear_regression</span> <span class="o">=</span> <span class="n">LinearRegression</span><span class="p">()</span>
<span class="n">pipeline</span> <span class="o">=</span> <span class="n">Pipeline</span><span class="p">([(</span><span class="s2">&quot;polynomial_features&quot;</span><span class="p">,</span> <span class="n">polynomial_features</span><span class="p">),</span>
<span class="p">(</span><span class="s2">&quot;linear_regression&quot;</span><span class="p">,</span> <span class="n">linear_regression</span><span class="p">)])</span>
<span class="n">pipeline</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">[:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">],</span> <span class="n">y</span><span class="p">)</span>
<span class="c1"># Evaluate the models using crossvalidation</span>
<span class="n">scores</span> <span class="o">=</span> <span class="n">cross_val_score</span><span class="p">(</span><span class="n">pipeline</span><span class="p">,</span> <span class="n">X</span><span class="p">[:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">],</span> <span class="n">y</span><span class="p">,</span>
<span class="n">scoring</span><span class="o">=</span><span class="s2">&quot;neg_mean_squared_error&quot;</span><span class="p">,</span> <span class="n">cv</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span>
<span class="n">X_test</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">pipeline</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">[:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">]),</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Model&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">true_fun</span><span class="p">(</span><span class="n">X_test</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;True function&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;b&#39;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Samples&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;x&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;y&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">((</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">&quot;best&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Degree </span><span class="si">{}</span><span class="se">\n</span><span class="s2">MSE = </span><span class="si">{:.2e}</span><span class="s2">(+/- </span><span class="si">{:.2e}</span><span class="s2">)&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span>
<span class="n">degrees</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="o">-</span><span class="n">scores</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span> <span class="n">scores</span><span class="o">.</span><span class="n">std</span><span class="p">()))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
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</section>
<section id="various-steps-in-cross-validation">
<h2>Various steps in cross-validation<a class="headerlink" href="#various-steps-in-cross-validation" title="Link to this heading">#</a></h2>
<p>When the repetitive splitting of the data set is done randomly,
samples may accidently end up in a fast majority of the splits in
either training or test set. Such samples may have an unbalanced
influence on either model building or prediction evaluation. To avoid
this <span class="math notranslate nohighlight">\(k\)</span>-fold cross-validation structures the data splitting. The
samples are divided into <span class="math notranslate nohighlight">\(k\)</span> more or less equally sized exhaustive and
mutually exclusive subsets. In turn (at each split) one of these
subsets plays the role of the test set while the union of the
remaining subsets constitutes the training set. Such a splitting
warrants a balanced representation of each sample in both training and
test set over the splits. Still the division into the <span class="math notranslate nohighlight">\(k\)</span> subsets
involves a degree of randomness. This may be fully excluded when
choosing <span class="math notranslate nohighlight">\(k=n\)</span>. This particular case is referred to as leave-one-out
cross-validation (LOOCV).</p>
</section>
<section id="cross-validation-in-brief">
<h2>Cross-validation in brief<a class="headerlink" href="#cross-validation-in-brief" title="Link to this heading">#</a></h2>
<p>For the various values of <span class="math notranslate nohighlight">\(k\)</span></p>
<ol class="arabic simple">
<li><p>shuffle the dataset randomly.</p></li>
<li><p>Split the dataset into <span class="math notranslate nohighlight">\(k\)</span> groups.</p></li>
<li><p>For each unique group:</p></li>
</ol>
<p>a. Decide which group to use as set for test data</p>
<p>b. Take the remaining groups as a training data set</p>
<p>c. Fit a model on the training set and evaluate it on the test set</p>
<p>d. Retain the evaluation score and discard the model</p>
<ol class="arabic simple" start="5">
<li><p>Summarize the model using the sample of model evaluation scores</p></li>
</ol>
</section>
<section id="code-example-for-cross-validation-and-k-fold-cross-validation">
<h2>Code Example for Cross-validation and <span class="math notranslate nohighlight">\(k\)</span>-fold Cross-validation<a class="headerlink" href="#code-example-for-cross-validation-and-k-fold-cross-validation" title="Link to this heading">#</a></h2>
<p>The code here uses Ridge regression with cross-validation (CV) resampling and <span class="math notranslate nohighlight">\(k\)</span>-fold CV in order to fit a specific polynomial.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">KFold</span>
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">Ridge</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">cross_val_score</span>
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">PolynomialFeatures</span>
<span class="c1"># A seed just to ensure that the random numbers are the same for every run.</span>
<span class="c1"># Useful for eventual debugging.</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">3155</span><span class="p">)</span>
<span class="c1"># Generate the data.</span>
<span class="n">nsamples</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">x</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">randn</span><span class="p">(</span><span class="n">nsamples</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</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">randn</span><span class="p">(</span><span class="n">nsamples</span><span class="p">)</span>
<span class="c1">## Cross-validation on Ridge regression using KFold only</span>
<span class="c1"># Decide degree on polynomial to fit</span>
<span class="n">poly</span> <span class="o">=</span> <span class="n">PolynomialFeatures</span><span class="p">(</span><span class="n">degree</span> <span class="o">=</span> <span class="mi">6</span><span class="p">)</span>
<span class="c1"># Decide which values of lambda to use</span>
<span class="n">nlambdas</span> <span class="o">=</span> <span class="mi">500</span>
<span class="n">lambdas</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="n">nlambdas</span><span class="p">)</span>
<span class="c1"># Initialize a KFold instance</span>
<span class="n">k</span> <span class="o">=</span> <span class="mi">5</span>
<span class="n">kfold</span> <span class="o">=</span> <span class="n">KFold</span><span class="p">(</span><span class="n">n_splits</span> <span class="o">=</span> <span class="n">k</span><span class="p">)</span>
<span class="c1"># Perform the cross-validation to estimate MSE</span>
<span class="n">scores_KFold</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">nlambdas</span><span class="p">,</span> <span class="n">k</span><span class="p">))</span>
<span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">lmb</span> <span class="ow">in</span> <span class="n">lambdas</span><span class="p">:</span>
<span class="n">ridge</span> <span class="o">=</span> <span class="n">Ridge</span><span class="p">(</span><span class="n">alpha</span> <span class="o">=</span> <span class="n">lmb</span><span class="p">)</span>
<span class="n">j</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">train_inds</span><span class="p">,</span> <span class="n">test_inds</span> <span class="ow">in</span> <span class="n">kfold</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="n">xtrain</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="n">train_inds</span><span class="p">]</span>
<span class="n">ytrain</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="n">train_inds</span><span class="p">]</span>
<span class="n">xtest</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="n">test_inds</span><span class="p">]</span>
<span class="n">ytest</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="n">test_inds</span><span class="p">]</span>
<span class="n">Xtrain</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">xtrain</span><span class="p">[:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">])</span>
<span class="n">ridge</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">Xtrain</span><span class="p">,</span> <span class="n">ytrain</span><span class="p">[:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">])</span>
<span class="n">Xtest</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">xtest</span><span class="p">[:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">])</span>
<span class="n">ypred</span> <span class="o">=</span> <span class="n">ridge</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">Xtest</span><span class="p">)</span>
<span class="n">scores_KFold</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">ypred</span> <span class="o">-</span> <span class="n">ytest</span><span class="p">[:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">])</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">ypred</span><span class="p">)</span>
<span class="n">j</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="n">estimated_mse_KFold</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">scores_KFold</span><span class="p">,</span> <span class="n">axis</span> <span class="o">=</span> <span class="mi">1</span><span class="p">)</span>
<span class="c1">## Cross-validation using cross_val_score from sklearn along with KFold</span>
<span class="c1"># kfold is an instance initialized above as:</span>
<span class="c1"># kfold = KFold(n_splits = k)</span>
<span class="n">estimated_mse_sklearn</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">nlambdas</span><span class="p">)</span>
<span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">lmb</span> <span class="ow">in</span> <span class="n">lambdas</span><span class="p">:</span>
<span class="n">ridge</span> <span class="o">=</span> <span class="n">Ridge</span><span class="p">(</span><span class="n">alpha</span> <span class="o">=</span> <span class="n">lmb</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">])</span>
<span class="n">estimated_mse_folds</span> <span class="o">=</span> <span class="n">cross_val_score</span><span class="p">(</span><span class="n">ridge</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">[:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">],</span> <span class="n">scoring</span><span class="o">=</span><span class="s1">&#39;neg_mean_squared_error&#39;</span><span class="p">,</span> <span class="n">cv</span><span class="o">=</span><span class="n">kfold</span><span class="p">)</span>
<span class="c1"># cross_val_score return an array containing the estimated negative mse for every fold.</span>
<span class="c1"># we have to the the mean of every array in order to get an estimate of the mse of the model</span>
<span class="n">estimated_mse_sklearn</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="o">-</span><span class="n">estimated_mse_folds</span><span class="p">)</span>
<span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="c1">## Plot and compare the slightly different ways to perform cross-validation</span>
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">lambdas</span><span class="p">),</span> <span class="n">estimated_mse_sklearn</span><span class="p">,</span> <span class="n">label</span> <span class="o">=</span> <span class="s1">&#39;cross_val_score&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">lambdas</span><span class="p">),</span> <span class="n">estimated_mse_KFold</span><span class="p">,</span> <span class="s1">&#39;r--&#39;</span><span class="p">,</span> <span class="n">label</span> <span class="o">=</span> <span class="s1">&#39;KFold&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;log10(lambda)&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;mse&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
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</div>
</section>
<section id="more-examples-on-bootstrap-and-cross-validation-and-errors">
<h2>More examples on bootstrap and cross-validation and errors<a class="headerlink" href="#more-examples-on-bootstrap-and-cross-validation-and-errors" title="Link to this heading">#</a></h2>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Common imports</span>
<span class="kn">import</span> <span class="nn">os</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LinearRegression</span><span class="p">,</span> <span class="n">Ridge</span><span class="p">,</span> <span class="n">Lasso</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
<span class="kn">from</span> <span class="nn">sklearn.utils</span> <span class="kn">import</span> <span class="n">resample</span>
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">mean_squared_error</span>
<span class="c1"># Where to save the figures and data files</span>
<span class="n">PROJECT_ROOT_DIR</span> <span class="o">=</span> <span class="s2">&quot;Results&quot;</span>
<span class="n">FIGURE_ID</span> <span class="o">=</span> <span class="s2">&quot;Results/FigureFiles&quot;</span>
<span class="n">DATA_ID</span> <span class="o">=</span> <span class="s2">&quot;DataFiles/&quot;</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">PROJECT_ROOT_DIR</span><span class="p">):</span>
<span class="n">os</span><span class="o">.</span><span class="n">mkdir</span><span class="p">(</span><span class="n">PROJECT_ROOT_DIR</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">):</span>
<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">):</span>
<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
<span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">,</span> <span class="n">fig_id</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">data_path</span><span class="p">(</span><span class="n">dat_id</span><span class="p">):</span>
<span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">,</span> <span class="n">dat_id</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">save_fig</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
<span class="n">plt</span><span class="o">.</span><span class="n">savefig</span><span class="p">(</span><span class="n">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">)</span> <span class="o">+</span> <span class="s2">&quot;.png&quot;</span><span class="p">,</span> <span class="nb">format</span><span class="o">=</span><span class="s1">&#39;png&#39;</span><span class="p">)</span>
<span class="n">infile</span> <span class="o">=</span> <span class="nb">open</span><span class="p">(</span><span class="n">data_path</span><span class="p">(</span><span class="s2">&quot;EoS.csv&quot;</span><span class="p">),</span><span class="s1">&#39;r&#39;</span><span class="p">)</span>
<span class="c1"># Read the EoS data as csv file and organize the data into two arrays with density and energies</span>
<span class="n">EoS</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="n">infile</span><span class="p">,</span> <span class="n">names</span><span class="o">=</span><span class="p">(</span><span class="s1">&#39;Density&#39;</span><span class="p">,</span> <span class="s1">&#39;Energy&#39;</span><span class="p">))</span>
<span class="n">EoS</span><span class="p">[</span><span class="s1">&#39;Energy&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">to_numeric</span><span class="p">(</span><span class="n">EoS</span><span class="p">[</span><span class="s1">&#39;Energy&#39;</span><span class="p">],</span> <span class="n">errors</span><span class="o">=</span><span class="s1">&#39;coerce&#39;</span><span class="p">)</span>
<span class="n">EoS</span> <span class="o">=</span> <span class="n">EoS</span><span class="o">.</span><span class="n">dropna</span><span class="p">()</span>
<span class="n">Energies</span> <span class="o">=</span> <span class="n">EoS</span><span class="p">[</span><span class="s1">&#39;Energy&#39;</span><span class="p">]</span>
<span class="n">Density</span> <span class="o">=</span> <span class="n">EoS</span><span class="p">[</span><span class="s1">&#39;Density&#39;</span><span class="p">]</span>
<span class="c1"># The design matrix now as function of various polytrops</span>
<span class="n">Maxpolydegree</span> <span class="o">=</span> <span class="mi">30</span>
<span class="n">X</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="nb">len</span><span class="p">(</span><span class="n">Density</span><span class="p">),</span><span class="n">Maxpolydegree</span><span class="p">))</span>
<span class="n">X</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mf">1.0</span>
<span class="n">testerror</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">Maxpolydegree</span><span class="p">)</span>
<span class="n">trainingerror</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">Maxpolydegree</span><span class="p">)</span>
<span class="n">polynomial</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">Maxpolydegree</span><span class="p">)</span>
<span class="n">trials</span> <span class="o">=</span> <span class="mi">100</span>
<span class="k">for</span> <span class="n">polydegree</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">Maxpolydegree</span><span class="p">):</span>
<span class="n">polynomial</span><span class="p">[</span><span class="n">polydegree</span><span class="p">]</span> <span class="o">=</span> <span class="n">polydegree</span>
<span class="k">for</span> <span class="n">degree</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">polydegree</span><span class="p">):</span>
<span class="n">X</span><span class="p">[:,</span><span class="n">degree</span><span class="p">]</span> <span class="o">=</span> <span class="n">Density</span><span class="o">**</span><span class="p">(</span><span class="n">degree</span><span class="o">/</span><span class="mf">3.0</span><span class="p">)</span>
<span class="c1"># loop over trials in order to estimate the expectation value of the MSE</span>
<span class="n">testerror</span><span class="p">[</span><span class="n">polydegree</span><span class="p">]</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="n">trainingerror</span><span class="p">[</span><span class="n">polydegree</span><span class="p">]</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="k">for</span> <span class="n">samples</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">trials</span><span class="p">):</span>
<span class="n">x_train</span><span class="p">,</span> <span class="n">x_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Energies</span><span class="p">,</span> <span class="n">test_size</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">LinearRegression</span><span class="p">(</span><span class="n">fit_intercept</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">x_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="n">ypred</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">x_train</span><span class="p">)</span>
<span class="n">ytilde</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">x_test</span><span class="p">)</span>
<span class="n">testerror</span><span class="p">[</span><span class="n">polydegree</span><span class="p">]</span> <span class="o">+=</span> <span class="n">mean_squared_error</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">ytilde</span><span class="p">)</span>
<span class="n">trainingerror</span><span class="p">[</span><span class="n">polydegree</span><span class="p">]</span> <span class="o">+=</span> <span class="n">mean_squared_error</span><span class="p">(</span><span class="n">y_train</span><span class="p">,</span> <span class="n">ypred</span><span class="p">)</span>
<span class="n">testerror</span><span class="p">[</span><span class="n">polydegree</span><span class="p">]</span> <span class="o">/=</span> <span class="n">trials</span>
<span class="n">trainingerror</span><span class="p">[</span><span class="n">polydegree</span><span class="p">]</span> <span class="o">/=</span> <span class="n">trials</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Degree of polynomial: </span><span class="si">%3d</span><span class="s2">&quot;</span><span class="o">%</span> <span class="n">polynomial</span><span class="p">[</span><span class="n">polydegree</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Mean squared error on training data: </span><span class="si">%.8f</span><span class="s2">&quot;</span> <span class="o">%</span> <span class="n">trainingerror</span><span class="p">[</span><span class="n">polydegree</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Mean squared error on test data: </span><span class="si">%.8f</span><span class="s2">&quot;</span> <span class="o">%</span> <span class="n">testerror</span><span class="p">[</span><span class="n">polydegree</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">polynomial</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">trainingerror</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Training Error&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">polynomial</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">testerror</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Test Error&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;Polynomial degree&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;log10[MSE]&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 1
Mean squared error on training data: 446033.51374050
Mean squared error on test data: 455173.80460179
Degree of polynomial: 2
Mean squared error on training data: 114550.54637219
Mean squared error on test data: 129963.83146596
Degree of polynomial: 3
Mean squared error on training data: 9054.61775176
Mean squared error on test data: 10572.87627342
Degree of polynomial: 4
Mean squared error on training data: 302.15313054
Mean squared error on test data: 433.26292364
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 5
Mean squared error on training data: 3.64316192
Mean squared error on test data: 7.23528337
Degree of polynomial: 6
Mean squared error on training data: 3.56589683
Mean squared error on test data: 10.50427787
Degree of polynomial: 7
Mean squared error on training data: 0.47313680
Mean squared error on test data: 1.53738247
Degree of polynomial: 8
Mean squared error on training data: 0.04926746
Mean squared error on test data: 0.14629156
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 9
Mean squared error on training data: 0.02546675
Mean squared error on test data: 0.11202337
Degree of polynomial: 10
Mean squared error on training data: 0.02424794
Mean squared error on test data: 0.22467274
Degree of polynomial: 11
Mean squared error on training data: 0.01594452
Mean squared error on test data: 1.07641937
Degree of polynomial: 12
Mean squared error on training data: 0.00805074
Mean squared error on test data: 0.04295757
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 13
Mean squared error on training data: 0.00781918
Mean squared error on test data: 0.56965674
Degree of polynomial: 14
Mean squared error on training data: 0.00465099
Mean squared error on test data: 0.28443039
Degree of polynomial: 15
Mean squared error on training data: 0.00420072
Mean squared error on test data: 568.47051432
Degree of polynomial: 16
Mean squared error on training data: 0.00325450
Mean squared error on test data: 48.97630233
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 17
Mean squared error on training data: 0.00242954
Mean squared error on test data: 2.52780600
Degree of polynomial: 18
Mean squared error on training data: 0.00219195
Mean squared error on test data: 429.25695398
Degree of polynomial: 19
Mean squared error on training data: 0.00154853
Mean squared error on test data: 239.97065359
Degree of polynomial: 20
Mean squared error on training data: 0.00140846
Mean squared error on test data: 1350.24493666
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 21
Mean squared error on training data: 0.00119688
Mean squared error on test data: 1840.50530832
Degree of polynomial: 22
Mean squared error on training data: 0.00092898
Mean squared error on test data: 1184.60929685
Degree of polynomial: 23
Mean squared error on training data: 0.00089193
Mean squared error on test data: 3892.17483760
Degree of polynomial: 24
Mean squared error on training data: 0.00083355
Mean squared error on test data: 1332.46736215
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 25
Mean squared error on training data: 0.00079904
Mean squared error on test data: 7577.76690383
Degree of polynomial: 26
Mean squared error on training data: 0.00075590
Mean squared error on test data: 1079.36895644
Degree of polynomial: 27
Mean squared error on training data: 0.00068091
Mean squared error on test data: 3207.25343155
Degree of polynomial: 28
Mean squared error on training data: 0.00063362
Mean squared error on test data: 674.79633065
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 29
Mean squared error on training data: 0.00063866
Mean squared error on test data: 3099.60342978
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_11671/626635268.py:73: RuntimeWarning: divide by zero encountered in log10
plt.plot(polynomial, np.log10(trainingerror), label=&#39;Training Error&#39;)
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_11671/626635268.py:74: RuntimeWarning: divide by zero encountered in log10
plt.plot(polynomial, np.log10(testerror), label=&#39;Test Error&#39;)
</pre></div>
</div>
<img alt="_images/602b30645a719994c061483bdc090ca3c34867d4bc3828152eaf911336723066.png" src="_images/602b30645a719994c061483bdc090ca3c34867d4bc3828152eaf911336723066.png" />
</div>
</div>
<p>Note that we kept the intercept column in the fitting here. This means that we need to set the <strong>intercept</strong> in the call to the <strong>Scikit-Learn</strong> function as <strong>False</strong>. Alternatively, we could have set up the design matrix <span class="math notranslate nohighlight">\(X\)</span> without the first column of ones.</p>
</section>
<section id="the-same-example-but-now-with-cross-validation">
<h2>The same example but now with cross-validation<a class="headerlink" href="#the-same-example-but-now-with-cross-validation" title="Link to this heading">#</a></h2>
<p>In this example we keep the intercept column again but add cross-validation in order to estimate the best possible value of the means squared error.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Common imports</span>
<span class="kn">import</span> <span class="nn">os</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LinearRegression</span><span class="p">,</span> <span class="n">Ridge</span><span class="p">,</span> <span class="n">Lasso</span>
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">mean_squared_error</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">KFold</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">cross_val_score</span>
<span class="c1"># Where to save the figures and data files</span>
<span class="n">PROJECT_ROOT_DIR</span> <span class="o">=</span> <span class="s2">&quot;Results&quot;</span>
<span class="n">FIGURE_ID</span> <span class="o">=</span> <span class="s2">&quot;Results/FigureFiles&quot;</span>
<span class="n">DATA_ID</span> <span class="o">=</span> <span class="s2">&quot;DataFiles/&quot;</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">PROJECT_ROOT_DIR</span><span class="p">):</span>
<span class="n">os</span><span class="o">.</span><span class="n">mkdir</span><span class="p">(</span><span class="n">PROJECT_ROOT_DIR</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">):</span>
<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">):</span>
<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
<span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">,</span> <span class="n">fig_id</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">data_path</span><span class="p">(</span><span class="n">dat_id</span><span class="p">):</span>
<span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">,</span> <span class="n">dat_id</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">save_fig</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
<span class="n">plt</span><span class="o">.</span><span class="n">savefig</span><span class="p">(</span><span class="n">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">)</span> <span class="o">+</span> <span class="s2">&quot;.png&quot;</span><span class="p">,</span> <span class="nb">format</span><span class="o">=</span><span class="s1">&#39;png&#39;</span><span class="p">)</span>
<span class="n">infile</span> <span class="o">=</span> <span class="nb">open</span><span class="p">(</span><span class="n">data_path</span><span class="p">(</span><span class="s2">&quot;EoS.csv&quot;</span><span class="p">),</span><span class="s1">&#39;r&#39;</span><span class="p">)</span>
<span class="c1"># Read the EoS data as csv file and organize the data into two arrays with density and energies</span>
<span class="n">EoS</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="n">infile</span><span class="p">,</span> <span class="n">names</span><span class="o">=</span><span class="p">(</span><span class="s1">&#39;Density&#39;</span><span class="p">,</span> <span class="s1">&#39;Energy&#39;</span><span class="p">))</span>
<span class="n">EoS</span><span class="p">[</span><span class="s1">&#39;Energy&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">to_numeric</span><span class="p">(</span><span class="n">EoS</span><span class="p">[</span><span class="s1">&#39;Energy&#39;</span><span class="p">],</span> <span class="n">errors</span><span class="o">=</span><span class="s1">&#39;coerce&#39;</span><span class="p">)</span>
<span class="n">EoS</span> <span class="o">=</span> <span class="n">EoS</span><span class="o">.</span><span class="n">dropna</span><span class="p">()</span>
<span class="n">Energies</span> <span class="o">=</span> <span class="n">EoS</span><span class="p">[</span><span class="s1">&#39;Energy&#39;</span><span class="p">]</span>
<span class="n">Density</span> <span class="o">=</span> <span class="n">EoS</span><span class="p">[</span><span class="s1">&#39;Density&#39;</span><span class="p">]</span>
<span class="c1"># The design matrix now as function of various polytrops</span>
<span class="n">Maxpolydegree</span> <span class="o">=</span> <span class="mi">30</span>
<span class="n">X</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="nb">len</span><span class="p">(</span><span class="n">Density</span><span class="p">),</span><span class="n">Maxpolydegree</span><span class="p">))</span>
<span class="n">X</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mf">1.0</span>
<span class="n">estimated_mse_sklearn</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">Maxpolydegree</span><span class="p">)</span>
<span class="n">polynomial</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">Maxpolydegree</span><span class="p">)</span>
<span class="n">k</span> <span class="o">=</span><span class="mi">5</span>
<span class="n">kfold</span> <span class="o">=</span> <span class="n">KFold</span><span class="p">(</span><span class="n">n_splits</span> <span class="o">=</span> <span class="n">k</span><span class="p">)</span>
<span class="k">for</span> <span class="n">polydegree</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">Maxpolydegree</span><span class="p">):</span>
<span class="n">polynomial</span><span class="p">[</span><span class="n">polydegree</span><span class="p">]</span> <span class="o">=</span> <span class="n">polydegree</span>
<span class="k">for</span> <span class="n">degree</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">polydegree</span><span class="p">):</span>
<span class="n">X</span><span class="p">[:,</span><span class="n">degree</span><span class="p">]</span> <span class="o">=</span> <span class="n">Density</span><span class="o">**</span><span class="p">(</span><span class="n">degree</span><span class="o">/</span><span class="mf">3.0</span><span class="p">)</span>
<span class="n">OLS</span> <span class="o">=</span> <span class="n">LinearRegression</span><span class="p">(</span><span class="n">fit_intercept</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="c1"># loop over trials in order to estimate the expectation value of the MSE</span>
<span class="n">estimated_mse_folds</span> <span class="o">=</span> <span class="n">cross_val_score</span><span class="p">(</span><span class="n">OLS</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">Energies</span><span class="p">,</span> <span class="n">scoring</span><span class="o">=</span><span class="s1">&#39;neg_mean_squared_error&#39;</span><span class="p">,</span> <span class="n">cv</span><span class="o">=</span><span class="n">kfold</span><span class="p">)</span>
<span class="c1">#[:, np.newaxis]</span>
<span class="n">estimated_mse_sklearn</span><span class="p">[</span><span class="n">polydegree</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="o">-</span><span class="n">estimated_mse_folds</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">polynomial</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">estimated_mse_sklearn</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Test Error&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;Polynomial degree&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;log10[MSE]&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_11671/3817475779.py:63: RuntimeWarning: divide by zero encountered in log10
plt.plot(polynomial, np.log10(estimated_mse_sklearn), label=&#39;Test Error&#39;)
</pre></div>
</div>
<img alt="_images/b54022c1eebf60e706d1ffd57bb1b9cd2b5e5728cdb8963a8594778239f56b00.png" src="_images/b54022c1eebf60e706d1ffd57bb1b9cd2b5e5728cdb8963a8594778239f56b00.png" />
</div>
</div>
</section>
<section id="material-for-the-lab-sessions">
<h2>Material for the lab sessions<a class="headerlink" href="#material-for-the-lab-sessions" title="Link to this heading">#</a></h2>
</section>
<section id="linking-the-regression-analysis-with-a-statistical-interpretation">
<h2>Linking the regression analysis with a statistical interpretation<a class="headerlink" href="#linking-the-regression-analysis-with-a-statistical-interpretation" title="Link to this heading">#</a></h2>
<p>We will now couple the discussions of ordinary least squares, Ridge
and Lasso regression with a statistical interpretation, that is we
move from a linear algebra analysis to a statistical analysis. In
particular, we will focus on what the regularization terms can result
in. We will amongst other things show that the regularization
parameter can reduce considerably the variance of the parameters
<span class="math notranslate nohighlight">\(\beta\)</span>.</p>
<p>The
advantage of doing linear regression is that we actually end up with
analytical expressions for several statistical quantities.<br />
Standard least squares and Ridge regression allow us to
derive quantities like the variance and other expectation values in a
rather straightforward way.</p>
<p>It is assumed that <span class="math notranslate nohighlight">\(\varepsilon_i
\sim \mathcal{N}(0, \sigma^2)\)</span> and the <span class="math notranslate nohighlight">\(\varepsilon_{i}\)</span> are
independent, i.e.:</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\begin{align*}
\mbox{Cov}(\varepsilon_{i_1},
\varepsilon_{i_2}) &amp; = \left\{ \begin{array}{lcc} \sigma^2 &amp; \mbox{if}
&amp; i_1 = i_2, \\ 0 &amp; \mbox{if} &amp; i_1 \not= i_2. \end{array} \right.
\end{align*}
\end{split}\]</div>
<p>The randomness of <span class="math notranslate nohighlight">\(\varepsilon_i\)</span> implies that
<span class="math notranslate nohighlight">\(\mathbf{y}_i\)</span> is also a random variable. In particular,
<span class="math notranslate nohighlight">\(\mathbf{y}_i\)</span> is normally distributed, because <span class="math notranslate nohighlight">\(\varepsilon_i \sim
\mathcal{N}(0, \sigma^2)\)</span> and <span class="math notranslate nohighlight">\(\mathbf{X}_{i,\ast} \, \boldsymbol{\beta}\)</span> is a
non-random scalar. To specify the parameters of the distribution of
<span class="math notranslate nohighlight">\(\mathbf{y}_i\)</span> we need to calculate its first two moments.</p>
<p>Recall that <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> is a matrix of dimensionality <span class="math notranslate nohighlight">\(n\times p\)</span>. The
notation above <span class="math notranslate nohighlight">\(\mathbf{X}_{i,\ast}\)</span> means that we are looking at the
row number <span class="math notranslate nohighlight">\(i\)</span> and perform a sum over all values <span class="math notranslate nohighlight">\(p\)</span>.</p>
</section>
<section id="assumptions-made">
<h2>Assumptions made<a class="headerlink" href="#assumptions-made" title="Link to this heading">#</a></h2>
<p>The assumption we have made here can be summarized as (and this is going to be useful when we discuss the bias-variance trade off)
that there exists a function <span class="math notranslate nohighlight">\(f(\boldsymbol{x})\)</span> and a normal distributed error <span class="math notranslate nohighlight">\(\boldsymbol{\varepsilon}\sim \mathcal{N}(0, \sigma^2)\)</span>
which describe our data</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{y} = f(\boldsymbol{x})+\boldsymbol{\varepsilon}
\]</div>
<p>We approximate this function with our model from the solution of the linear regression equations, that is our
function <span class="math notranslate nohighlight">\(f\)</span> is approximated by <span class="math notranslate nohighlight">\(\boldsymbol{\tilde{y}}\)</span> where we want to minimize <span class="math notranslate nohighlight">\((\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\)</span>, our MSE, with</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{\tilde{y}} = \boldsymbol{X}\boldsymbol{\beta}.
\]</div>
</section>
<section id="expectation-value-and-variance">
<h2>Expectation value and variance<a class="headerlink" href="#expectation-value-and-variance" title="Link to this heading">#</a></h2>
<p>We can calculate the expectation value of <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> for a given element <span class="math notranslate nohighlight">\(i\)</span></p>
<div class="math notranslate nohighlight">
\[
\begin{align*}
\mathbb{E}(y_i) &amp; =
\mathbb{E}(\mathbf{X}_{i, \ast} \, \boldsymbol{\beta}) + \mathbb{E}(\varepsilon_i)
\, \, \, = \, \, \, \mathbf{X}_{i, \ast} \, \beta,
\end{align*}
\]</div>
<p>while
its variance is</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\begin{align*} \mbox{Var}(y_i) &amp; = \mathbb{E} \{ [y_i
- \mathbb{E}(y_i)]^2 \} \, \, \, = \, \, \, \mathbb{E} ( y_i^2 ) -
[\mathbb{E}(y_i)]^2 \\ &amp; = \mathbb{E} [ ( \mathbf{X}_{i, \ast} \,
\beta + \varepsilon_i )^2] - ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 \\ &amp;
= \mathbb{E} [ ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 + 2 \varepsilon_i
\mathbf{X}_{i, \ast} \, \boldsymbol{\beta} + \varepsilon_i^2 ] - ( \mathbf{X}_{i,
\ast} \, \beta)^2 \\ &amp; = ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 + 2
\mathbb{E}(\varepsilon_i) \mathbf{X}_{i, \ast} \, \boldsymbol{\beta} +
\mathbb{E}(\varepsilon_i^2 ) - ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2
\\ &amp; = \mathbb{E}(\varepsilon_i^2 ) \, \, \, = \, \, \,
\mbox{Var}(\varepsilon_i) \, \, \, = \, \, \, \sigma^2.
\end{align*}
\end{split}\]</div>
<p>Hence, <span class="math notranslate nohighlight">\(y_i \sim \mathcal{N}( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta}, \sigma^2)\)</span>, that is <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> follows a normal distribution with
mean value <span class="math notranslate nohighlight">\(\boldsymbol{X}\boldsymbol{\beta}\)</span> and variance <span class="math notranslate nohighlight">\(\sigma^2\)</span> (not be confused with the singular values of the SVD).</p>
</section>
<section id="expectation-value-and-variance-for-boldsymbol-beta">
<h2>Expectation value and variance for <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span><a class="headerlink" href="#expectation-value-and-variance-for-boldsymbol-beta" title="Link to this heading">#</a></h2>
<p>With the OLS expressions for the optimal parameters <span class="math notranslate nohighlight">\(\boldsymbol{\hat{\beta}}\)</span> we can evaluate the expectation value</p>
<div class="math notranslate nohighlight">
\[
\mathbb{E}(\boldsymbol{\hat{\beta}}) = \mathbb{E}[ (\mathbf{X}^{\top} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbb{E}[ \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1} \mathbf{X}^{T}\mathbf{X}\boldsymbol{\beta}=\boldsymbol{\beta}.
\]</div>
<p>This means that the estimator of the regression parameters is unbiased.</p>
<p>We can also calculate the variance</p>
<p>The variance of the optimal value <span class="math notranslate nohighlight">\(\boldsymbol{\hat{\beta}}\)</span> is</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\begin{eqnarray*}
\mbox{Var}(\boldsymbol{\hat{\beta}}) &amp; = &amp; \mathbb{E} \{ [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})] [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})]^{T} \}
\\
&amp; = &amp; \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{y} - \boldsymbol{\beta}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{y} - \boldsymbol{\beta}]^{T} \}
\\
% &amp; = &amp; \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{y}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{y}]^{T} \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
% \\
% &amp; = &amp; \mathbb{E} \{ (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{y} \, \mathbf{y}^{T} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
% \\
&amp; = &amp; (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \mathbb{E} \{ \mathbf{y} \, \mathbf{y}^{T} \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
\\
&amp; = &amp; (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \{ \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \, \mathbf{X}^{T} + \sigma^2 \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
% \\
% &amp; = &amp; (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^T \, \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T % \mathbf{X})^{-1}
% \\
% &amp; &amp; + \, \, \sigma^2 \, (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T \mathbf{X})^{-1} - \boldsymbol{\beta} \boldsymbol{\beta}^T
\\
&amp; = &amp; \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} + \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
\, \, \, = \, \, \, \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1},
\end{eqnarray*}
\end{split}\]</div>
<p>where we have used that <span class="math notranslate nohighlight">\(\mathbb{E} (\mathbf{y} \mathbf{y}^{T}) =
\mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \, \mathbf{X}^{T} +
\sigma^2 \, \mathbf{I}_{nn}\)</span>. From <span class="math notranslate nohighlight">\(\mbox{Var}(\boldsymbol{\beta}) = \sigma^2
\, (\mathbf{X}^{T} \mathbf{X})^{-1}\)</span>, one obtains an estimate of the
variance of the estimate of the <span class="math notranslate nohighlight">\(j\)</span>-th regression coefficient:
<span class="math notranslate nohighlight">\(\boldsymbol{\sigma}^2 (\boldsymbol{\beta}_j ) = \boldsymbol{\sigma}^2 [(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj} \)</span>. This may be used to
construct a confidence interval for the estimates.</p>
<p>In a similar way, we can obtain analytical expressions for say the
expectation values of the parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> and their variance
when we employ Ridge regression, allowing us again to define a confidence interval.</p>
<p>It is rather straightforward to show that</p>
<div class="math notranslate nohighlight">
\[
\mathbb{E} \big[ \hat{\boldsymbol{\beta}}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\boldsymbol{\beta}.
\]</div>
<p>We see clearly that
<span class="math notranslate nohighlight">\(\mathbb{E} \big[ \hat{\boldsymbol{\beta}}^{\mathrm{Ridge}} \big] \not= \hat{\boldsymbol{\beta}}^{\mathrm{OLS}}\)</span> for any <span class="math notranslate nohighlight">\(\lambda &gt; 0\)</span>.</p>
<p>We can also compute the variance as</p>
<div class="math notranslate nohighlight">
\[
\mbox{Var}[\hat{\boldsymbol{\beta}}^{\mathrm{Ridge}}]=\sigma^2[ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1} \mathbf{X}^{T} \mathbf{X} \{ [ \mathbf{X}^{\top} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T},
\]</div>
<p>and it is easy to see that if the parameter <span class="math notranslate nohighlight">\(\lambda\)</span> goes to infinity then the variance of Ridge parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> goes to zero.</p>
<p>With this, we can compute the difference</p>
<div class="math notranslate nohighlight">
\[
\mbox{Var}[\hat{\boldsymbol{\beta}}^{\mathrm{OLS}}]-\mbox{Var}(\hat{\boldsymbol{\beta}}^{\mathrm{Ridge}})=\sigma^2 [ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}[ 2\lambda\mathbf{I} + \lambda^2 (\mathbf{X}^{T} \mathbf{X})^{-1} ] \{ [ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T}.
\]</div>
<p>The difference is non-negative definite since each component of the
matrix product is non-negative definite.
This means the variance we obtain with the standard OLS will always for <span class="math notranslate nohighlight">\(\lambda &gt; 0\)</span> be larger than the variance of <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> obtained with the Ridge estimator. This has interesting consequences when we discuss the so-called bias-variance trade-off below.</p>
<p>For more discussions of Ridge regression and calculation of averages, <a class="reference external" href="https://arxiv.org/abs/1509.09169">Wessel van Wieringen’s</a> article is highly recommended.</p>
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<i class="fa-solid fa-list"></i> Contents
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#plans-for-week-37-lecture-monday">Plans for week 37, lecture Monday</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#plans-for-week-37-lab-sessions">Plans for week 37, lab sessions</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#material-for-lecture-monday-september-9">Material for lecture Monday September 9</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#deriving-ols-from-a-probability-distribution">Deriving OLS from a probability distribution</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#independent-and-identically-distrubuted-iid">Independent and Identically Distrubuted (iid)</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#maximum-likelihood-estimation-mle">Maximum Likelihood Estimation (MLE)</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-new-cost-function">A new Cost Function</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#more-basic-statistics-and-bayes-theorem">More basic Statistics and Bayes’ theorem</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#marginal-probability">Marginal Probability</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#conditional-probability">Conditional Probability</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#bayes-theorem">Bayes’ Theorem</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#interpretations-of-bayes-theorem">Interpretations of Bayes’ Theorem</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#example-of-usage-of-bayes-theorem">Example of Usage of Bayes’ theorem</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#doing-it-correctly">Doing it correctly</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#bayes-theorem-and-ridge-and-lasso-regression">Bayes’ Theorem and Ridge and Lasso Regression</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#ridge-and-bayes">Ridge and Bayes</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#lasso-and-bayes">Lasso and Bayes</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#why-resampling-methods">Why resampling methods</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#resampling-methods">Resampling methods</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#resampling-approaches-can-be-computationally-expensive">Resampling approaches can be computationally expensive</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">Why resampling methods ?</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#statistical-analysis">Statistical analysis</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#id2">Resampling methods</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#resampling-methods-bootstrap">Resampling methods: Bootstrap</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-central-limit-theorem">The Central Limit Theorem</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#finding-the-limit">Finding the Limit</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#rewriting-the-delta-function">Rewriting the <span class="math notranslate nohighlight">\(\delta\)</span>-function</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#identifying-terms">Identifying Terms</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#wrapping-it-up">Wrapping it up</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#confidence-intervals">Confidence Intervals</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#standard-approach-based-on-the-normal-distribution">Standard Approach based on the Normal Distribution</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#resampling-methods-bootstrap-background">Resampling methods: Bootstrap background</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#resampling-methods-more-bootstrap-background">Resampling methods: More Bootstrap background</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#resampling-methods-bootstrap-approach">Resampling methods: Bootstrap approach</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#resampling-methods-bootstrap-steps">Resampling methods: Bootstrap steps</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#code-example-for-the-bootstrap-method">Code example for the Bootstrap method</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#plotting-the-histogram">Plotting the Histogram</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-bias-variance-tradeoff">The bias-variance tradeoff</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-way-to-read-the-bias-variance-tradeoff">A way to Read the Bias-Variance Tradeoff</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#example-code-for-bias-variance-tradeoff">Example code for Bias-Variance tradeoff</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#understanding-what-happens">Understanding what happens</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#summing-up">Summing up</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#another-example-from-scikit-learn-s-repository">Another Example from Scikit-Learn’s Repository</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#various-steps-in-cross-validation">Various steps in cross-validation</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#cross-validation-in-brief">Cross-validation in brief</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#code-example-for-cross-validation-and-k-fold-cross-validation">Code Example for Cross-validation and <span class="math notranslate nohighlight">\(k\)</span>-fold Cross-validation</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#more-examples-on-bootstrap-and-cross-validation-and-errors">More examples on bootstrap and cross-validation and errors</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-same-example-but-now-with-cross-validation">The same example but now with cross-validation</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#material-for-the-lab-sessions">Material for the lab sessions</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#linking-the-regression-analysis-with-a-statistical-interpretation">Linking the regression analysis with a statistical interpretation</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#assumptions-made">Assumptions made</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#expectation-value-and-variance">Expectation value and variance</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#expectation-value-and-variance-for-boldsymbol-beta">Expectation value and variance for <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span></a></li>
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