Morten Hjorth-Jensen
702 lines
38 KiB
HTML
702 lines
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Applied Data Analysis and Machine Learning
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Review of Statistics with Resampling Techniques and Linear Algebra</span></p>
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<li class="toctree-l1"><a class="reference internal" href="statistics.html">1. Elements of Probability Theory and Statistical Data Analysis</a></li>
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<li class="toctree-l1"><a class="reference internal" href="linalg.html">2. Linear Algebra, Handling of Arrays and more Python Features</a></li>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">From Regression to Support Vector Machines</span></p>
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<li class="toctree-l1"><a class="reference internal" href="chapter1.html">3. Linear Regression</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter2.html">4. Ridge and Lasso Regression</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter3.html">5. Resampling Methods</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter4.html">6. Logistic Regression</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapteroptimization.html">7. Optimization, the central part of any Machine Learning algortithm</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter6.html">9. Decision trees, overarching aims</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter7.html">10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter11.html">15. Solving Differential Equations with Deep Learning</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter12.html">16. Convolutional Neural Networks</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter13.html">17. Recurrent neural networks: Overarching view</a></li>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Weekly material, notes and exercises</span></p>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek34.html">Exercises week 34</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week34.html">Week 34: Introduction to the course, Logistics and Practicalities</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek35.html">Exercises week 35</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week35.html">Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression</a></li>
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<li class="toctree-l1 current active"><a class="current reference internal" href="#">Exercises week 36</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week36.html">Week 36: Linear Regression and Gradient descent</a></li>
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<h1>Exercises week 36</h1>
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<h2> Contents </h2>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#deriving-and-implementing-ridge-regression">Deriving and Implementing Ridge Regression</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#learning-goals">Learning goals</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-1-choice-of-model-and-degrees-of-freedom">Exercise 1 - Choice of model and degrees of freedom</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-2-deriving-the-expression-for-ridge-regression">Exercise 2 - Deriving the expression for Ridge Regression</a><ul class="nav section-nav flex-column">
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<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#a-expression-for-ridge-regression">a) Expression for Ridge regression</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-3-scaling-data">Exercise 3 - Scaling data</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-4-implementing-ridge-regression">Exercise 4 - Implementing Ridge Regression</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-4-testing-multiple-hyperparameters">Exercise 4 - Testing multiple hyperparameters</a></li>
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</ul>
|
||
</nav>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
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|
||
|
||
<div id="searchbox"></div>
|
||
<article class="bd-article">
|
||
|
||
<section class="tex2jax_ignore mathjax_ignore" id="exercises-week-36">
|
||
<h1>Exercises week 36<a class="headerlink" href="#exercises-week-36" title="Link to this heading">#</a></h1>
|
||
<section id="deriving-and-implementing-ridge-regression">
|
||
<h2>Deriving and Implementing Ridge Regression<a class="headerlink" href="#deriving-and-implementing-ridge-regression" title="Link to this heading">#</a></h2>
|
||
</section>
|
||
<section id="learning-goals">
|
||
<h2>Learning goals<a class="headerlink" href="#learning-goals" title="Link to this heading">#</a></h2>
|
||
<p>After completing these exercises, you will know how to</p>
|
||
<ul class="simple">
|
||
<li><p>Take more derivatives of simple products between vectors and matrices</p></li>
|
||
<li><p>Implement Ridge regression using the analytical expressions</p></li>
|
||
<li><p>Scale data appropriately for linear regression</p></li>
|
||
<li><p>Evaluate a model across two different hyperparameters</p></li>
|
||
</ul>
|
||
</section>
|
||
<section id="exercise-1-choice-of-model-and-degrees-of-freedom">
|
||
<h2>Exercise 1 - Choice of model and degrees of freedom<a class="headerlink" href="#exercise-1-choice-of-model-and-degrees-of-freedom" title="Link to this heading">#</a></h2>
|
||
<p><strong>a)</strong> How many degrees of freedom does an OLS model fit to the features <span class="math notranslate nohighlight">\(x, x^2, x^3\)</span> and the intercept have?</p>
|
||
<p><strong>b)</strong> Why is it bad for a model to have too many degrees of freedom?</p>
|
||
<p><strong>c)</strong> Why is it bad for a model to have too few degrees of freedom?</p>
|
||
<p><strong>d)</strong> Read <a class="reference external" href="https://link.springer.com/book/10.1007/978-0-387-84858-7">chapter 3.4.1 of Hastie et al.’s book</a>. What is the expression for the effective degrees of freedom of the ridge regression fit?</p>
|
||
<p><strong>e)</strong> Why might we want to use Ridge regression instead of OLS?</p>
|
||
<p><strong>f)</strong> Why migth we want to use OLS instead of Ridge regression?</p>
|
||
</section>
|
||
<section id="exercise-2-deriving-the-expression-for-ridge-regression">
|
||
<h2>Exercise 2 - Deriving the expression for Ridge Regression<a class="headerlink" href="#exercise-2-deriving-the-expression-for-ridge-regression" title="Link to this heading">#</a></h2>
|
||
<p>The aim here is to derive the expression for the optimal parameters using Ridge regression.</p>
|
||
<p>The expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, was given by the optimization problem</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
{\displaystyle \min_{\boldsymbol{\beta}\in {\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}.
|
||
\]</div>
|
||
<p>By minimizing the above equation with respect to the parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> we could then obtain an analytical expression for the parameters <span class="math notranslate nohighlight">\(\boldsymbol{\hat\beta_{OLS}}\)</span>.</p>
|
||
<p>We can add a regularization parameter <span class="math notranslate nohighlight">\(\lambda\)</span> by
|
||
defining a new cost function to be optimized, that is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
{\displaystyle \min_{\boldsymbol{\beta}\in
|
||
{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_2^2
|
||
\]</div>
|
||
<p>which leads to the Ridge regression minimization problem. (One can require as part of the optimization problem that <span class="math notranslate nohighlight">\(\vert\vert \boldsymbol{\beta}\vert\vert_2^2\le t\)</span>, where <span class="math notranslate nohighlight">\(t\)</span> is a finite number larger than zero. We will not implement that in this course.)</p>
|
||
<section id="a-expression-for-ridge-regression">
|
||
<h3>a) Expression for Ridge regression<a class="headerlink" href="#a-expression-for-ridge-regression" title="Link to this heading">#</a></h3>
|
||
<p>Show that the optimal parameters</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\hat{\boldsymbol{\beta}}_{\mathrm{Ridge}} = \left(\boldsymbol{X}^T\boldsymbol{X}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y},
|
||
\]</div>
|
||
<p>with <span class="math notranslate nohighlight">\(\boldsymbol{I}\)</span> being a <span class="math notranslate nohighlight">\(p\times p\)</span> identity matrix.</p>
|
||
<p>The ordinary least squares result is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\hat{\boldsymbol{\beta}}_{\mathrm{OLS}} = \left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y},
|
||
\]</div>
|
||
</section>
|
||
</section>
|
||
<section id="exercise-3-scaling-data">
|
||
<h2>Exercise 3 - Scaling data<a class="headerlink" href="#exercise-3-scaling-data" title="Link to this heading">#</a></h2>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">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">train_test_split</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">StandardScaler</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="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="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>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p><strong>a)</strong> Adapt your function from last week to only include the intercept column if the boolean argument <code class="docutils literal notranslate"><span class="pre">intercept</span></code> is set to true.</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="k">def</span> <span class="nf">polynomial_features</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">intercept</span><span class="o">=</span><span class="kc">False</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">x</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">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="n">p</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
|
||
<span class="c1">#X[:, 0] = ...</span>
|
||
<span class="c1">#X[:, 1] = ...</span>
|
||
<span class="c1">#X[:, 2] = ...</span>
|
||
<span class="c1"># could this be a loop?</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="k">def</span> <span class="nf">polynomial_features</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">intercept</span><span class="o">=</span><span class="kc">False</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">x</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">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="n">p</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="n">x</span><span class="p">[:]</span>
|
||
<span class="n">X</span><span class="p">[:,</span> <span class="mi">1</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="n">X</span><span class="p">[:,</span> <span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span>
|
||
<span class="k">return</span> <span class="n">X</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p><strong>b)</strong> Split your data into training and test data(80/20 split)</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="n">X</span> <span class="o">=</span> <span class="n">polynomial_features</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">3</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="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="n">x_train</span> <span class="o">=</span> <span class="n">X_train</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="c1"># These are used for plotting</span>
|
||
<span class="n">x_test</span> <span class="o">=</span> <span class="n">X_test</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="c1"># These are used for plotting</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p><strong>c)</strong> Scale your design matrix with the sklearn standard scaler, though based on the mean and standard deviation of the training data only.</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="n">scaler</span> <span class="o">=</span> <span class="n">StandardScaler</span><span class="p">()</span>
|
||
<span class="n">scaler</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">X_train_s</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
|
||
<span class="n">X_test_s</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||
<span class="n">y_offset</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_train</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="exercise-4-implementing-ridge-regression">
|
||
<h2>Exercise 4 - Implementing Ridge Regression<a class="headerlink" href="#exercise-4-implementing-ridge-regression" title="Link to this heading">#</a></h2>
|
||
<p><strong>a)</strong> Implement a function for computing the optimal Ridge parameters using the expression from <strong>2a)</strong>.</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="k">def</span> <span class="nf">Ridge_parameters</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="c1"># Assumes X is scaled and has no intercept column</span>
|
||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">X</span><span class="p">)</span> <span class="o">@</span> <span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">y</span>
|
||
|
||
<span class="n">beta</span> <span class="o">=</span> <span class="n">Ridge_parameters</span><span class="p">(</span><span class="n">X_train_s</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p><strong>b)</strong> Fit a model to the data, and plot the prediction using both the training and test x-values extracted before scaling, and the y_offset.</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="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="n">y</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_train</span><span class="p">,</span> <span class="n">X_train_s</span> <span class="o">@</span> <span class="n">beta</span> <span class="o">+</span> <span class="n">y_offset</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">X_test_s</span> <span class="o">@</span> <span class="n">beta</span> <span class="o">+</span> <span class="n">y_offset</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span><matplotlib.collections.PathCollection at 0x113e21950>
|
||
</pre></div>
|
||
</div>
|
||
<img alt="_images/cb756715680fbe8cbf1d070371bf3950695ddf24b29ecb476c48fb9e559af08e.png" src="_images/cb756715680fbe8cbf1d070371bf3950695ddf24b29ecb476c48fb9e559af08e.png" />
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="exercise-4-testing-multiple-hyperparameters">
|
||
<h2>Exercise 4 - Testing multiple hyperparameters<a class="headerlink" href="#exercise-4-testing-multiple-hyperparameters" title="Link to this heading">#</a></h2>
|
||
<p><strong>a)</strong> Compute the MSE of your ridge model for polynomials of degrees 1 to 5 with lambda set to 0.01. Plot the MSE as a function of polynomial degree.</p>
|
||
<p><strong>b)</strong> Compute the MSE of your ridge model for a polynomial with degree 3, and with lambdas from <span class="math notranslate nohighlight">\(10^{-1}\)</span> to <span class="math notranslate nohighlight">\(10^{-5}\)</span> on a logarithmic scale. Plot the MSE as a function of lambda.</p>
|
||
<p><strong>c)</strong> Compute the MSE of your ridge model for polynomials of degrees 1 to 5, and with lambdas from <span class="math notranslate nohighlight">\(10^{-1}\)</span> to <span class="math notranslate nohighlight">\(10^{-5}\)</span> on a logarithmic scale. Plot the MSE as a function of polynomial degree and lambda using a <a class="reference external" href="https://matplotlib.org/stable/gallery/images_contours_and_fields/image_annotated_heatmap.html">heatmap</a>.</p>
|
||
</section>
|
||
</section>
|
||
|
||
<script type="text/x-thebe-config">
|
||
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|
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|
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|
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<i class="fa-solid fa-list"></i> Contents
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<ul class="visible nav section-nav flex-column">
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#deriving-and-implementing-ridge-regression">Deriving and Implementing Ridge Regression</a></li>
|
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#learning-goals">Learning goals</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-1-choice-of-model-and-degrees-of-freedom">Exercise 1 - Choice of model and degrees of freedom</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-2-deriving-the-expression-for-ridge-regression">Exercise 2 - Deriving the expression for Ridge Regression</a><ul class="nav section-nav flex-column">
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<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#a-expression-for-ridge-regression">a) Expression for Ridge regression</a></li>
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</ul>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-3-scaling-data">Exercise 3 - Scaling data</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-4-implementing-ridge-regression">Exercise 4 - Implementing Ridge Regression</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-4-testing-multiple-hyperparameters">Exercise 4 - Testing multiple hyperparameters</a></li>
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