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Morten Hjorth-Jensen
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@@ -159,7 +159,7 @@
</li>
<li class="toctree-l1">
<a class="reference internal" href="chapter5.html">
7. Support Vector Machines, overarching aims
8. Support Vector Machines, overarching aims
</a>
</li>
</ul>
@@ -171,12 +171,12 @@
<ul class="nav bd-sidenav">
<li class="toctree-l1">
<a class="reference internal" href="chapter6.html">
8. Decision trees, overarching aims
9. Decision trees, overarching aims
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="chapter7.html">
9. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods
10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods
</a>
</li>
</ul>
@@ -188,12 +188,12 @@
<ul class="nav bd-sidenav">
<li class="toctree-l1">
<a class="reference internal" href="chapter8.html">
10. Basic ideas of the Principal Component Analysis (PCA)
11. Basic ideas of the Principal Component Analysis (PCA)
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="Clustering.html">
11. Clustering Analysis
12. Clustering Analysis
</a>
</li>
</ul>
@@ -205,12 +205,12 @@
<ul class="nav bd-sidenav">
<li class="toctree-l1">
<a class="reference internal" href="chapter9.html">
12. Neural networks
13. Neural networks
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="chapter10.html">
13. Building a Feed Forward Neural Network
14. Building a Feed Forward Neural Network
</a>
</li>
</ul>
@@ -307,13 +307,13 @@
</ul>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#various-steps-in-cross-validation">
5.4. Various steps in cross-validation
<a class="reference internal nav-link" href="#the-bias-variance-tradeoff">
5.4. The bias-variance tradeoff
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-bias-variance-tradeoff">
5.5. The bias-variance tradeoff
<a class="reference internal nav-link" href="#cross-validation">
5.5. Cross-validation
</a>
</li>
</ul>
@@ -587,10 +587,10 @@ number <span class="math notranslate nohighlight">\(i\)</span> is left out. Usin
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Runtime: 0.136687 sec
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Runtime: 0.138472 sec
Jackknife Statistics :
original bias std. error
99.9196 99.9096 0.149076
100.094 100.084 0.150306
</pre></div>
</div>
</div>
@@ -718,10 +718,10 @@ theorem.</p>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Runtime: 1.78082 sec
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Runtime: 2.06834 sec
Bootstrap Statistics :
original bias std. error
99.9769 14.9969 99.9765 0.149832
99.9933 15.0354 99.9941 0.149739
</pre></div>
</div>
<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="gt">---------------------------------------------------------------------------</span>
@@ -769,158 +769,8 @@ original bias std. error
</div>
</div>
</div>
<div class="section" id="various-steps-in-cross-validation">
<h2><span class="section-number">5.4. </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">5.5. </span>The bias-variance tradeoff<a class="headerlink" href="#the-bias-variance-tradeoff" title="Permalink to this headline"></a></h2>
<h2><span class="section-number">5.4. </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
@@ -1265,6 +1115,157 @@ flexible statistical methods have higher variance.</p>
</div>
</div>
</div>
</div>
<div class="section" id="cross-validation">
<h2><span class="section-number">5.5. </span>Cross-validation<a class="headerlink" href="#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>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">KFold</span>
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">Ridge</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">cross_val_score</span>
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">PolynomialFeatures</span>
<span class="c1"># A seed just to ensure that the random numbers are the same for every run.</span>
<span class="c1"># Useful for eventual debugging.</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">3155</span><span class="p">)</span>
<span class="c1"># Generate the data.</span>
<span class="n">nsamples</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">nsamples</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">nsamples</span><span class="p">)</span>
<span class="c1">## Cross-validation on Ridge regression using KFold only</span>
<span class="c1"># Decide degree on polynomial to fit</span>
<span class="n">poly</span> <span class="o">=</span> <span class="n">PolynomialFeatures</span><span class="p">(</span><span class="n">degree</span> <span class="o">=</span> <span class="mi">6</span><span class="p">)</span>
<span class="c1"># Decide which values of lambda to use</span>
<span class="n">nlambdas</span> <span class="o">=</span> <span class="mi">500</span>
<span class="n">lambdas</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="n">nlambdas</span><span class="p">)</span>
<span class="c1"># Initialize a KFold instance</span>
<span class="n">k</span> <span class="o">=</span> <span class="mi">5</span>
<span class="n">kfold</span> <span class="o">=</span> <span class="n">KFold</span><span class="p">(</span><span class="n">n_splits</span> <span class="o">=</span> <span class="n">k</span><span class="p">)</span>
<span class="c1"># Perform the cross-validation to estimate MSE</span>
<span class="n">scores_KFold</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nlambdas</span><span class="p">,</span> <span class="n">k</span><span class="p">))</span>
<span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">lmb</span> <span class="ow">in</span> <span class="n">lambdas</span><span class="p">:</span>
<span class="n">ridge</span> <span class="o">=</span> <span class="n">Ridge</span><span class="p">(</span><span class="n">alpha</span> <span class="o">=</span> <span class="n">lmb</span><span class="p">)</span>
<span class="n">j</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">train_inds</span><span class="p">,</span> <span class="n">test_inds</span> <span class="ow">in</span> <span class="n">kfold</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="n">xtrain</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="n">train_inds</span><span class="p">]</span>
<span class="n">ytrain</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="n">train_inds</span><span class="p">]</span>
<span class="n">xtest</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="n">test_inds</span><span class="p">]</span>
<span class="n">ytest</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="n">test_inds</span><span class="p">]</span>
<span class="n">Xtrain</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">xtrain</span><span class="p">[:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">])</span>
<span class="n">ridge</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">Xtrain</span><span class="p">,</span> <span class="n">ytrain</span><span class="p">[:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">])</span>
<span class="n">Xtest</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">xtest</span><span class="p">[:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">])</span>
<span class="n">ypred</span> <span class="o">=</span> <span class="n">ridge</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">Xtest</span><span class="p">)</span>
<span class="n">scores_KFold</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">ypred</span> <span class="o">-</span> <span class="n">ytest</span><span class="p">[:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">])</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">ypred</span><span class="p">)</span>
<span class="n">j</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="n">estimated_mse_KFold</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">scores_KFold</span><span class="p">,</span> <span class="n">axis</span> <span class="o">=</span> <span class="mi">1</span><span class="p">)</span>
<span class="c1">## Cross-validation using cross_val_score from sklearn along with KFold</span>
<span class="c1"># kfold is an instance initialized above as:</span>
<span class="c1"># kfold = KFold(n_splits = k)</span>
<span class="n">estimated_mse_sklearn</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">nlambdas</span><span class="p">)</span>
<span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">lmb</span> <span class="ow">in</span> <span class="n">lambdas</span><span class="p">:</span>
<span class="n">ridge</span> <span class="o">=</span> <span class="n">Ridge</span><span class="p">(</span><span class="n">alpha</span> <span class="o">=</span> <span class="n">lmb</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">])</span>
<span class="n">estimated_mse_folds</span> <span class="o">=</span> <span class="n">cross_val_score</span><span class="p">(</span><span class="n">ridge</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">[:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">],</span> <span class="n">scoring</span><span class="o">=</span><span class="s1">&#39;neg_mean_squared_error&#39;</span><span class="p">,</span> <span class="n">cv</span><span class="o">=</span><span class="n">kfold</span><span class="p">)</span>
<span class="c1"># cross_val_score return an array containing the estimated negative mse for every fold.</span>
<span class="c1"># we have to the the mean of every array in order to get an estimate of the mse of the model</span>
<span class="n">estimated_mse_sklearn</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="o">-</span><span class="n">estimated_mse_folds</span><span class="p">)</span>
<span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="c1">## Plot and compare the slightly different ways to perform cross-validation</span>
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">lambdas</span><span class="p">),</span> <span class="n">estimated_mse_sklearn</span><span class="p">,</span> <span class="n">label</span> <span class="o">=</span> <span class="s1">&#39;cross_val_score&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">lambdas</span><span class="p">),</span> <span class="n">estimated_mse_KFold</span><span class="p">,</span> <span class="s1">&#39;r--&#39;</span><span class="p">,</span> <span class="n">label</span> <span class="o">=</span> <span class="s1">&#39;KFold&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;log10(lambda)&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;mse&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
<p>More examples of the application of cross-validation follow here.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Common imports</span>