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2. Resampling Methods
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<div class="section" id="resampling-methods">
<h1><span class="section-number">2. </span>Resampling Methods<a class="headerlink" href="#resampling-methods" title="Permalink to this headline"></a></h1>
<p><a class="reference external" href="https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureSept3.mp4?vrtx=view-as-webpage">Video of Lecture</a></p>
<div class="section" id="introduction">
<h2><span class="section-number">2.1. </span>Introduction<a class="headerlink" href="#introduction" title="Permalink to this headline"></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="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>
<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>
<ul class="simple">
<li><p>Our simulations can be treated as <em>computer experiments</em>. This is particularly the case for Monte Carlo methods</p></li>
<li><p>The results can be analysed with the same statistical tools as we would use 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>
</div>
<div class="section" id="reminder-on-statistics">
<h2><span class="section-number">2.2. </span>Reminder on Statistics<a class="headerlink" href="#reminder-on-statistics" title="Permalink to this headline"></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>
<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>
<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>
<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>
<p>With the OLS expressions for the parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> we can evaluate the expectation value</p>
<div class="math notranslate nohighlight">
\[
\mathbb{E}(\boldsymbol{\beta}) = \mathbb{E}[ (\mathbf{X}^{\top} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbb{E}[ \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1} \mathbf{X}^{T}\mathbf{X}\boldsymbol{\beta}=\boldsymbol{\beta}.
\]</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 <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> is</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\begin{eqnarray*}
\mbox{Var}(\boldsymbol{\beta}) &amp; = &amp; \mathbb{E} \{ [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})] [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})]^{T} \}
\\
&amp; = &amp; \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \boldsymbol{\beta}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \boldsymbol{\beta}]^{T} \}
\\
% &amp; = &amp; \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}]^{T} \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
% \\
% &amp; = &amp; \mathbb{E} \{ (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} \, \mathbf{Y}^{T} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
% \\
&amp; = &amp; (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \mathbb{E} \{ \mathbf{Y} \, \mathbf{Y}^{T} \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
\\
&amp; = &amp; (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \{ \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \, \mathbf{X}^{T} + \sigma^2 \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
% \\
% &amp; = &amp; (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^T \, \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T % \mathbf{X})^{-1}
% \\
% &amp; &amp; + \, \, \sigma^2 \, (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T \mathbf{X})^{-1} - \boldsymbol{\beta} \boldsymbol{\beta}^T
\\
&amp; = &amp; \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} + \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
\, \, \, = \, \, \, \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1},
\end{eqnarray*}
\end{split}\]</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">\(\boldsymbol{\sigma}^2 (\boldsymbol{\beta}_j ) = \boldsymbol{\sigma}^2 \sqrt{
[(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj} }\)</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">
\[
\mathbb{E} \big[ \boldsymbol{\beta}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\boldsymbol{\beta}^{\mathrm{OLS}}.
\]</div>
<p>We see clearly that
<span class="math notranslate nohighlight">\(\mathbb{E} \big[ \boldsymbol{\beta}^{\mathrm{Ridge}} \big] \not= \boldsymbol{\beta}^{\mathrm{OLS}}\)</span> for any <span class="math notranslate nohighlight">\(\lambda &gt; 0\)</span>. We say then that the ridge estimator is biased.</p>
<p>We can also compute the variance as</p>
<div class="math notranslate nohighlight">
\[
\mbox{Var}[\boldsymbol{\beta}^{\mathrm{Ridge}}]=\sigma^2[ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1} \mathbf{X}^{T} \mathbf{X} \{ [ \mathbf{X}^{\top} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T},
\]</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">
\[
\mbox{Var}[\boldsymbol{\beta}^{\mathrm{OLS}}]-\mbox{Var}(\boldsymbol{\beta}^{\mathrm{Ridge}})=\sigma^2 [ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}[ 2\lambda\mathbf{I} + \lambda^2 (\mathbf{X}^{T} \mathbf{X})^{-1} ] \{ [ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T}.
\]</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>
</div>
<div class="section" id="id1">
<h2><span class="section-number">2.3. </span>Resampling methods<a class="headerlink" href="#id1" title="Permalink to this headline"></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 us 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="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>
<p>Two famous
resampling methods are the <strong>independent bootstrap</strong> and <strong>the jackknife</strong>.</p>
<p>The jackknife is a special case of the independent bootstrap. Still, the jackknife was made
popular prior to the independent bootstrap. And as the popularity of
the independent bootstrap soared, new variants, such as <strong>the dependent bootstrap</strong>.</p>
<p>The Jackknife and independent bootstrap work for
independent, identically distributed random variables.
If these conditions are not
satisfied, the methods will fail. Yet, it should be said that if the data are
independent, identically distributed, and we only want to estimate the
variance of <span class="math notranslate nohighlight">\(\overline{X}\)</span> (which often is the case), then there is no
need for bootstrapping.</p>
<p>The Jackknife works by making many replicas of the estimator <span class="math notranslate nohighlight">\(\widehat{\theta}\)</span>.
The jackknife is a resampling method where we systematically leave out one observation from the vector of observed values <span class="math notranslate nohighlight">\(\boldsymbol{x} = (x_1,x_2,\cdots,X_n)\)</span>.
Let <span class="math notranslate nohighlight">\(\boldsymbol{x}_i\)</span> denote the vector</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{x}_i = (x_1,x_2,\cdots,x_{i-1},x_{i+1},\cdots,x_n),
\]</div>
<p>which equals the vector <span class="math notranslate nohighlight">\(\boldsymbol{x}\)</span> with the exception that observation
number <span class="math notranslate nohighlight">\(i\)</span> is left out. Using this notation, define
<span class="math notranslate nohighlight">\(\widehat{\theta}_i\)</span> to be the estimator
<span class="math notranslate nohighlight">\(\widehat{\theta}\)</span> computed using <span class="math notranslate nohighlight">\(\vec{X}_i\)</span>.</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="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="o">*</span>
<span class="kn">from</span> <span class="nn">numpy.random</span> <span class="kn">import</span> <span class="n">randint</span><span class="p">,</span> <span class="n">randn</span>
<span class="kn">from</span> <span class="nn">time</span> <span class="kn">import</span> <span class="n">time</span>
<span class="k">def</span> <span class="nf">jackknife</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">stat</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="n">t</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="p">);</span> <span class="n">inds</span> <span class="o">=</span> <span class="n">arange</span><span class="p">(</span><span class="n">n</span><span class="p">);</span> <span class="n">t0</span> <span class="o">=</span> <span class="n">time</span><span class="p">()</span>
<span class="c1">## &#39;jackknifing&#39; by leaving out an observation for each i </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</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">stat</span><span class="p">(</span><span class="n">delete</span><span class="p">(</span><span class="n">data</span><span class="p">,</span><span class="n">i</span><span class="p">)</span> <span class="p">)</span>
<span class="c1"># analysis </span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Runtime: </span><span class="si">%g</span><span class="s2"> sec&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">time</span><span class="p">()</span><span class="o">-</span><span class="n">t0</span><span class="p">));</span> <span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Jackknife 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">%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">stat</span><span class="p">(</span><span class="n">data</span><span class="p">),(</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">)</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="o">/</span><span class="n">n</span><span class="p">,</span> <span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="n">var</span><span class="p">(</span><span class="n">t</span><span class="p">))</span><span class="o">**.</span><span class="mi">5</span><span class="p">))</span>
<span class="k">return</span> <span class="n">t</span>
<span class="c1"># Returns mean of data samples </span>
<span class="k">def</span> <span class="nf">stat</span><span class="p">(</span><span class="n">data</span><span class="p">):</span>
<span class="k">return</span> <span class="n">mean</span><span class="p">(</span><span class="n">data</span><span class="p">)</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="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">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"># jackknife returns the data sample </span>
<span class="n">t</span> <span class="o">=</span> <span class="n">jackknife</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">stat</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>Runtime: 0.136239 sec
Jackknife Statistics :
original bias std. error
99.9154 99.9054 0.150467
</pre></div>
</div>
</div>
</div>
<div class="section" id="bootstrap">
<h3><span class="section-number">2.3.1. </span>Bootstrap<a class="headerlink" href="#bootstrap" title="Permalink to this headline"></a></h3>
<p>Bootstrapping is a nonparametric approach to statistical inference
that substitutes computation for more traditional distributional
assumptions and asymptotic results. Bootstrapping offers a number of
advantages:</p>
<ol class="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>Since <span class="math notranslate nohighlight">\(\widehat{\theta} = \widehat{\theta}(\boldsymbol{X})\)</span> is a function of random variables,
<span class="math notranslate nohighlight">\(\widehat{\theta}\)</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">\(\widehat{\theta}\)</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>
<p>In the case that <span class="math notranslate nohighlight">\(\widehat{\theta}\)</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 straight forward to do this by:</p>
<ol class="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">\(\widehat{\theta}\)</span> called <span class="math notranslate nohighlight">\(\widehat{\theta}^*\)</span>.</p></li>
</ol>
<p>By repeated use of (1) and (2), many
estimates of <span class="math notranslate nohighlight">\(\widehat{\theta}\)</span> could have been obtained. The
idea is to use the relative frequency of <span class="math notranslate nohighlight">\(\widehat{\theta}^*\)</span>
(think of a histogram) as an estimate of <span class="math notranslate nohighlight">\(p(\boldsymbol{t})\)</span>.</p>
<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>; 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>
<p>Instead of generating the histogram for the relative
frequency of the observation <span class="math notranslate nohighlight">\(X_i\)</span>, just draw the values
<span class="math notranslate nohighlight">\((X_1^*,X_2^*,\cdots,X_n^*)\)</span> with replacement from the vector
<span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span>.</p>
<p>The independent bootstrap works like this:</p>
<ol class="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">\(\widehat{\theta}^*\)</span> by evaluating <span class="math notranslate nohighlight">\(\widehat \theta\)</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">\(\widehat \theta^*\)</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">\(\widehat{\theta}^*\)</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">\(\widehat
\theta\)</span>, apply the etsimator <span class="math notranslate nohighlight">\(\widehat \sigma^2\)</span> to the values
<span class="math notranslate nohighlight">\(\widehat \theta ^*\)</span>.</p>
<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>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="o">%</span><span class="k">matplotlib</span> inline
<span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="o">*</span>
<span class="kn">from</span> <span class="nn">numpy.random</span> <span class="kn">import</span> <span class="n">randint</span><span class="p">,</span> <span class="n">randn</span>
<span class="kn">from</span> <span class="nn">time</span> <span class="kn">import</span> <span class="n">time</span>
<span class="kn">import</span> <span class="nn">matplotlib.mlab</span> <span class="k">as</span> <span class="nn">mlab</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="k">def</span> <span class="nf">stat</span><span class="p">(</span><span class="n">data</span><span class="p">):</span>
<span class="k">return</span> <span class="n">mean</span><span class="p">(</span><span class="n">data</span><span class="p">)</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">statistic</span><span class="p">,</span> <span class="n">R</span><span class="p">):</span>
<span class="n">t</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="n">R</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="n">inds</span> <span class="o">=</span> <span class="n">arange</span><span class="p">(</span><span class="n">n</span><span class="p">);</span> <span class="n">t0</span> <span class="o">=</span> <span class="n">time</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">R</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">statistic</span><span class="p">(</span><span class="n">data</span><span class="p">[</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;Runtime: </span><span class="si">%g</span><span class="s2"> sec&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">time</span><span class="p">()</span><span class="o">-</span><span class="n">t0</span><span class="p">));</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">statistic</span><span class="p">(</span><span class="n">data</span><span class="p">),</span> <span class="n">std</span><span class="p">(</span><span class="n">data</span><span class="p">),</span><span class="n">mean</span><span class="p">(</span><span class="n">t</span><span class="p">),</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="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="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">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">stat</span><span class="p">,</span> <span class="n">datapoints</span><span class="p">)</span>
<span class="c1"># the histogram of the bootstrapped data </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">normed</span><span class="o">=</span><span class="mi">1</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">mlab</span><span class="o">.</span><span class="n">normpdf</span><span class="p">(</span> <span class="n">binsboot</span><span class="p">,</span> <span class="n">mean</span><span class="p">(</span><span class="n">t</span><span class="p">),</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;r--&#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;Smarts&#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">axis</span><span class="p">([</span><span class="mf">99.5</span><span class="p">,</span> <span class="mf">100.6</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mf">3.0</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>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Runtime: 1.77593 sec
Bootstrap Statistics :
original bias std. error
100.207 14.9646 100.209 0.150205
</pre></div>
</div>
<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="gt">---------------------------------------------------------------------------</span>
<span class="ne">AttributeError</span><span class="g g-Whitespace"> </span>Traceback (most recent call last)
<span class="o">&lt;</span><span class="n">ipython</span><span class="o">-</span><span class="nb">input</span><span class="o">-</span><span class="mi">2</span><span class="o">-</span><span class="mi">772</span><span class="n">b904ae9cb</span><span class="o">&gt;</span> <span class="ow">in</span> <span class="o">&lt;</span><span class="n">module</span><span class="o">&gt;</span>
<span class="g g-Whitespace"> </span><span class="mi">31</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">stat</span><span class="p">,</span> <span class="n">datapoints</span><span class="p">)</span>
<span class="g g-Whitespace"> </span><span class="mi">32</span> <span class="c1"># the histogram of the bootstrapped data</span>
<span class="ne">---&gt; </span><span class="mi">33</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">normed</span><span class="o">=</span><span class="mi">1</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="g g-Whitespace"> </span><span class="mi">34</span>
<span class="g g-Whitespace"> </span><span class="mi">35</span> <span class="c1"># add a &#39;best fit&#39; line</span>
<span class="nn">~/opt/anaconda3/lib/python3.8/site-packages/matplotlib/pyplot.py</span> in <span class="ni">hist</span><span class="nt">(x, bins, range, density, weights, cumulative, bottom, histtype, align, orientation, rwidth, log, color, label, stacked, data, **kwargs)</span>
<span class="g g-Whitespace"> </span><span class="mi">2683</span> <span class="n">orientation</span><span class="o">=</span><span class="s1">&#39;vertical&#39;</span><span class="p">,</span> <span class="n">rwidth</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">log</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
<span class="g g-Whitespace"> </span><span class="mi">2684</span> <span class="n">label</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">stacked</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="o">*</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
<span class="ne">-&gt; </span><span class="mi">2685</span> <span class="k">return</span> <span class="n">gca</span><span class="p">()</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span>
<span class="g g-Whitespace"> </span><span class="mi">2686</span> <span class="n">x</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="n">bins</span><span class="p">,</span> <span class="nb">range</span><span class="o">=</span><span class="nb">range</span><span class="p">,</span> <span class="n">density</span><span class="o">=</span><span class="n">density</span><span class="p">,</span> <span class="n">weights</span><span class="o">=</span><span class="n">weights</span><span class="p">,</span>
<span class="g g-Whitespace"> </span><span class="mi">2687</span> <span class="n">cumulative</span><span class="o">=</span><span class="n">cumulative</span><span class="p">,</span> <span class="n">bottom</span><span class="o">=</span><span class="n">bottom</span><span class="p">,</span> <span class="n">histtype</span><span class="o">=</span><span class="n">histtype</span><span class="p">,</span>
<span class="nn">~/opt/anaconda3/lib/python3.8/site-packages/matplotlib/__init__.py</span> in <span class="ni">inner</span><span class="nt">(ax, data, *args, **kwargs)</span>
<span class="g g-Whitespace"> </span><span class="mi">1445</span> <span class="k">def</span> <span class="nf">inner</span><span class="p">(</span><span class="n">ax</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
<span class="g g-Whitespace"> </span><span class="mi">1446</span> <span class="k">if</span> <span class="n">data</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="ne">-&gt; </span><span class="mi">1447</span> <span class="k">return</span> <span class="n">func</span><span class="p">(</span><span class="n">ax</span><span class="p">,</span> <span class="o">*</span><span class="nb">map</span><span class="p">(</span><span class="n">sanitize_sequence</span><span class="p">,</span> <span class="n">args</span><span class="p">),</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
<span class="g g-Whitespace"> </span><span class="mi">1448</span>
<span class="g g-Whitespace"> </span><span class="mi">1449</span> <span class="n">bound</span> <span class="o">=</span> <span class="n">new_sig</span><span class="o">.</span><span class="n">bind</span><span class="p">(</span><span class="n">ax</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
<span class="nn">~/opt/anaconda3/lib/python3.8/site-packages/matplotlib/axes/_axes.py</span> in <span class="ni">hist</span><span class="nt">(self, x, bins, range, density, weights, cumulative, bottom, histtype, align, orientation, rwidth, log, color, label, stacked, **kwargs)</span>
<span class="g g-Whitespace"> </span><span class="mi">6813</span> <span class="k">if</span> <span class="n">patch</span><span class="p">:</span>
<span class="g g-Whitespace"> </span><span class="mi">6814</span> <span class="n">p</span> <span class="o">=</span> <span class="n">patch</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="ne">-&gt; </span><span class="mi">6815</span> <span class="n">p</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">kwargs</span><span class="p">)</span>
<span class="g g-Whitespace"> </span><span class="mi">6816</span> <span class="k">if</span> <span class="n">lbl</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
<span class="g g-Whitespace"> </span><span class="mi">6817</span> <span class="n">p</span><span class="o">.</span><span class="n">set_label</span><span class="p">(</span><span class="n">lbl</span><span class="p">)</span>
<span class="nn">~/opt/anaconda3/lib/python3.8/site-packages/matplotlib/artist.py</span> in <span class="ni">update</span><span class="nt">(self, props)</span>
<span class="g g-Whitespace"> </span><span class="mi">994</span> <span class="n">func</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="sa">f</span><span class="s2">&quot;set_</span><span class="si">{</span><span class="n">k</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
<span class="g g-Whitespace"> </span><span class="mi">995</span> <span class="k">if</span> <span class="ow">not</span> <span class="n">callable</span><span class="p">(</span><span class="n">func</span><span class="p">):</span>
<span class="ne">--&gt; </span><span class="mi">996</span> <span class="k">raise</span> <span class="ne">AttributeError</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;</span><span class="si">{</span><span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="vm">__name__</span><span class="si">!r}</span><span class="s2"> object &quot;</span>
<span class="g g-Whitespace"> </span><span class="mi">997</span> <span class="sa">f</span><span class="s2">&quot;has no property </span><span class="si">{</span><span class="n">k</span><span class="si">!r}</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="g g-Whitespace"> </span><span class="mi">998</span> <span class="n">ret</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">func</span><span class="p">(</span><span class="n">v</span><span class="p">))</span>
<span class="ne">AttributeError</span>: &#39;Rectangle&#39; object has no property &#39;normed&#39;
</pre></div>
</div>
<img alt="_images/chapter2_25_2.png" src="_images/chapter2_25_2.png" />
</div>
</div>
</div>
</div>
<div class="section" id="various-steps-in-cross-validation">
<h2><span class="section-number">2.4. </span>Various steps in cross-validation<a class="headerlink" href="#various-steps-in-cross-validation" title="Permalink to this headline"></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>
<ul class="simple">
<li><p>Define a range of interest for the penalty parameter.</p></li>
<li><p>Divide the data set into training and test set comprising samples <span class="math notranslate nohighlight">\(\{1, \ldots, n\} \setminus i\)</span> and <span class="math notranslate nohighlight">\(\{ i \}\)</span>, respectively.</p></li>
<li><p>Fit the linear regression model by means of ridge estimation for each <span class="math notranslate nohighlight">\(\lambda\)</span> in the grid using the training set, and the corresponding estimate of the error variance <span class="math notranslate nohighlight">\(\boldsymbol{\sigma}_{-i}^2(\lambda)\)</span>, as</p></li>
</ul>
<div class="math notranslate nohighlight">
\[
\begin{align*}
\boldsymbol{\beta}_{-i}(\lambda) &amp; = ( \boldsymbol{X}_{-i, \ast}^{T}
\boldsymbol{X}_{-i, \ast} + \lambda \boldsymbol{I}_{pp})^{-1}
\boldsymbol{X}_{-i, \ast}^{T} \boldsymbol{y}_{-i}
\end{align*}
\]</div>
<ul class="simple">
<li><p>Evaluate the prediction performance of these models on the test set by <span class="math notranslate nohighlight">\(\log\{L[y_i, \boldsymbol{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)]\}\)</span>. Or, by the prediction error <span class="math notranslate nohighlight">\(|y_i - \boldsymbol{X}_{i, \ast} \boldsymbol{\beta}_{-i}(\lambda)|\)</span>, the relative error, the error squared or the R2 score function.</p></li>
<li><p>Repeat the first three steps such that each sample plays the role of the test set once.</p></li>
<li><p>Average the prediction performances of the test sets at each grid point of the penalty bias/parameter. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data. It is defined as</p></li>
</ul>
<div class="math notranslate nohighlight">
\[
\begin{align*}
\frac{1}{n} \sum_{i = 1}^n \log\{L[y_i, \mathbf{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)]\}.
\end{align*}
\]</div>
<p>For the various values of <span class="math notranslate nohighlight">\(k\)</span></p>
<ol class="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="simple">
<li><p>Summarize the model using the sample of model evaluation scores</p></li>
</ol>
<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>
<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">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>
</div>
</div>
<div class="section" id="the-bias-variance-tradeoff">
<h2><span class="section-number">2.5. </span>The bias-variance tradeoff<a class="headerlink" href="#the-bias-variance-tradeoff" title="Permalink to this headline"></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">\(\mathcal{L}\)</span> consisting of the data
<span class="math notranslate nohighlight">\(\mathbf{X}_\mathcal{L}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\}\)</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>
<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>
<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>
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<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>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="sd">&quot;&quot;&quot;</span>
<span class="sd">============================</span>
<span class="sd">Underfitting vs. Overfitting</span>
<span class="sd">============================</span>
<span class="sd">This example demonstrates the problems of underfitting and overfitting and</span>
<span class="sd">how we can use linear regression with polynomial features to approximate</span>
<span class="sd">nonlinear functions. The plot shows the function that we want to approximate,</span>
<span class="sd">which is a part of the cosine function. In addition, the samples from the</span>
<span class="sd">real function and the approximations of different models are displayed. The</span>
<span class="sd">models have polynomial features of different degrees. We can see that a</span>
<span class="sd">linear function (polynomial with degree 1) is not sufficient to fit the</span>
<span class="sd">training samples. This is called **underfitting**. A polynomial of degree 4</span>
<span class="sd">approximates the true function almost perfectly. However, for higher degrees</span>
<span class="sd">the model will **overfit** the training data, i.e. it learns the noise of the</span>
<span class="sd">training data.</span>
<span class="sd">We evaluate quantitatively **overfitting** / **underfitting** by using</span>
<span class="sd">cross-validation. We calculate the mean squared error (MSE) on the validation</span>
<span class="sd">set, the higher, the less likely the model generalizes correctly from the</span>
<span class="sd">training data.</span>
<span class="sd">&quot;&quot;&quot;</span>
<span class="nb">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</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>
</div>
</div>
</div>
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<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">True</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>
</div>
<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="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>
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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="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">n</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">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="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">10</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="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">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">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="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="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">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>
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