KarlHenrik
760 lines
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HTML
760 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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</ul>
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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="chapter5.html">8. Support Vector Machines, overarching aims</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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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Dimensionality Reduction</span></p>
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<li class="toctree-l1"><a class="reference internal" href="chapter8.html">11. Basic ideas of the Principal Component Analysis (PCA)</a></li>
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<li class="toctree-l1"><a class="reference internal" href="clustering.html">12. Clustering and Unsupervised Learning</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter9.html">13. Neural networks</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter10.html">14. Building a Feed Forward Neural Network</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"><a class="reference internal" href="exercisesweek36.html">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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<li class="toctree-l1"><a class="reference internal" href="exercisesweek37.html">Exercises week 37</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week37.html">Week 37: Gradient descent methods</a></li>
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<li class="toctree-l1 current active"><a class="current reference internal" href="#">Exercises week 38</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week38.html">Week 38: Statistical analysis, bias-variance tradeoff and resampling methods</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek39.html">Exercises week 39</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week39.html">Week 39: Resampling methods and logistic regression</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week40.html">Week 40: Gradient descent methods (continued) and start Neural networks</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week41.html">Week 41 Neural networks and constructing a neural network code</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek41.html">Exercises week 41</a></li>
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<li class="toctree-l1"><a class="reference internal" href="project1.html">Project 1 on Machine Learning, deadline October 6 (midnight), 2025</a></li>
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<h1>Exercises week 38</h1>
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<h2> Contents </h2>
|
||
</div>
|
||
<nav aria-label="Page">
|
||
<ul class="visible nav section-nav flex-column">
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#september-15-19">September 15-19</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#resampling-and-the-bias-variance-trade-off">Resampling and the Bias-Variance Trade-off</a><ul class="nav section-nav flex-column">
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#learning-goals">Learning goals</a></li>
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#deliverables">Deliverables</a></li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#use-the-books">Use the books!</a><ul class="nav section-nav flex-column">
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#definitions">Definitions</a></li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-1-expectation-values-for-ordinary-least-squares-expressions">Exercise 1: Expectation values for ordinary least squares expressions</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-2-expectation-values-for-ridge-regression">Exercise 2: Expectation values for Ridge regression</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-3-deriving-the-expression-for-the-bias-variance-trade-off">Exercise 3: Deriving the expression for the Bias-Variance Trade-off</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-4-computing-the-bias-and-variance">Exercise 4: Computing the Bias and Variance</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-5-interpretation-of-scaling-and-metrics">Exercise 5: Interpretation of scaling and metrics</a></li>
|
||
</ul>
|
||
</nav>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
|
||
|
||
<div id="searchbox"></div>
|
||
<article class="bd-article">
|
||
|
||
<section class="tex2jax_ignore mathjax_ignore" id="exercises-week-38">
|
||
<h1>Exercises week 38<a class="headerlink" href="#exercises-week-38" title="Link to this heading">#</a></h1>
|
||
<section id="september-15-19">
|
||
<h2>September 15-19<a class="headerlink" href="#september-15-19" title="Link to this heading">#</a></h2>
|
||
</section>
|
||
<section id="resampling-and-the-bias-variance-trade-off">
|
||
<h2>Resampling and the Bias-Variance Trade-off<a class="headerlink" href="#resampling-and-the-bias-variance-trade-off" title="Link to this heading">#</a></h2>
|
||
<section id="learning-goals">
|
||
<h3>Learning goals<a class="headerlink" href="#learning-goals" title="Link to this heading">#</a></h3>
|
||
<p>After completing these exercises, you will know how to</p>
|
||
<ul class="simple">
|
||
<li><p>Derive expectation and variances values related to linear regression</p></li>
|
||
<li><p>Compute expectation and variances values related to linear regression</p></li>
|
||
<li><p>Compute and evaluate the trade-off between bias and variance of a model</p></li>
|
||
</ul>
|
||
</section>
|
||
<section id="deliverables">
|
||
<h3>Deliverables<a class="headerlink" href="#deliverables" title="Link to this heading">#</a></h3>
|
||
<p>Complete the following exercises while working in a jupyter notebook. Then, in canvas, include</p>
|
||
<ul class="simple">
|
||
<li><p>The jupyter notebook with the exercises completed</p></li>
|
||
<li><p>An exported PDF of the notebook (<a class="reference external" href="https://code.visualstudio.com/docs/datascience/jupyter-notebooks#_export-your-jupyter-notebook">https://code.visualstudio.com/docs/datascience/jupyter-notebooks#_export-your-jupyter-notebook</a>)</p></li>
|
||
</ul>
|
||
</section>
|
||
</section>
|
||
<section id="use-the-books">
|
||
<h2>Use the books!<a class="headerlink" href="#use-the-books" title="Link to this heading">#</a></h2>
|
||
<p>This week deals with various mean values and variances in linear regression methods (here it may be useful to look up chapter 3, equation (3.8) of <a class="reference external" href="https://www.springer.com/gp/book/9780387848570">Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer</a>).</p>
|
||
<p>For more discussions on Ridge regression and calculation of expectation values, <a class="reference external" href="https://arxiv.org/abs/1509.09169">Wessel van Wieringen’s</a> article is highly recommended.</p>
|
||
<p>The exercises this week are also a part of project 1 and can be reused in the theory part of the project.</p>
|
||
<section id="definitions">
|
||
<h3>Definitions<a class="headerlink" href="#definitions" title="Link to this heading">#</a></h3>
|
||
<p>We assume that there exists a continuous function <span class="math notranslate nohighlight">\(f(\boldsymbol{x})\)</span> and a normal distributed error <span class="math notranslate nohighlight">\(\boldsymbol{\varepsilon}\sim N(0, \sigma^2)\)</span> which describes our data</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{y} = f(\boldsymbol{x})+\boldsymbol{\varepsilon}
|
||
\]</div>
|
||
<p>We further assume that this continous function can be modeled with a linear model <span class="math notranslate nohighlight">\(\mathbf{\tilde{y}}\)</span> of some features <span class="math notranslate nohighlight">\(\mathbf{X}\)</span>.</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{y} = \boldsymbol{\tilde{y}} + \boldsymbol{\varepsilon} = \boldsymbol{X}\boldsymbol{\beta} +\boldsymbol{\varepsilon}
|
||
\]</div>
|
||
<p>We therefore get that our data <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> has an expectation value <span class="math notranslate nohighlight">\(\boldsymbol{X}\boldsymbol{\beta}\)</span> and variance <span class="math notranslate nohighlight">\(\sigma^2\)</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>.</p>
|
||
</section>
|
||
</section>
|
||
<section id="exercise-1-expectation-values-for-ordinary-least-squares-expressions">
|
||
<h2>Exercise 1: Expectation values for ordinary least squares expressions<a class="headerlink" href="#exercise-1-expectation-values-for-ordinary-least-squares-expressions" title="Link to this heading">#</a></h2>
|
||
<p><strong>a)</strong> With the expressions for the optimal parameters <span class="math notranslate nohighlight">\(\boldsymbol{\hat{\beta}_{OLS}}\)</span> show that</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathbb{E}(\boldsymbol{\hat{\beta}_{OLS}}) = \boldsymbol{\beta}.
|
||
\]</div>
|
||
<p><strong>b)</strong> Show that the variance of <span class="math notranslate nohighlight">\(\boldsymbol{\hat{\beta}_{OLS}}\)</span> is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathbf{Var}(\boldsymbol{\hat{\beta}_{OLS}}) = \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1}.
|
||
\]</div>
|
||
<p>We can use the last expression when we define a <a class="reference external" href="https://en.wikipedia.org/wiki/Confidence_interval">confidence interval</a> for the parameters <span class="math notranslate nohighlight">\(\boldsymbol{\hat{\beta}_{OLS}}\)</span>.
|
||
A given parameter <span class="math notranslate nohighlight">\({\boldsymbol{\hat{\beta}_{OLS}}}_j\)</span> is given by the diagonal matrix element of the above matrix.</p>
|
||
</section>
|
||
<section id="exercise-2-expectation-values-for-ridge-regression">
|
||
<h2>Exercise 2: Expectation values for Ridge regression<a class="headerlink" href="#exercise-2-expectation-values-for-ridge-regression" title="Link to this heading">#</a></h2>
|
||
<p><strong>a)</strong> With the expressions for the optimal parameters <span class="math notranslate nohighlight">\(\boldsymbol{\hat{\beta}_{Ridge}}\)</span> show that</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathbb{E} \big[ \hat{\boldsymbol{\beta}}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\boldsymbol{\beta}
|
||
\]</div>
|
||
<p>We see that <span class="math notranslate nohighlight">\(\mathbb{E} \big[ \hat{\boldsymbol{\beta}}^{\mathrm{Ridge}} \big] \not= \mathbb{E} \big[\hat{\boldsymbol{\beta}}^{\mathrm{OLS}}\big ]\)</span> for any <span class="math notranslate nohighlight">\(\lambda > 0\)</span>.</p>
|
||
<p><strong>b)</strong> Show that the variance is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathbf{Var}[\hat{\boldsymbol{\beta}}^{\mathrm{Ridge}}]=\sigma^2[ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1} \mathbf{X}^{T}\mathbf{X} \{ [ \mathbf{X}^{\top} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T}
|
||
\]</div>
|
||
<p>We see that if the parameter <span class="math notranslate nohighlight">\(\lambda\)</span> goes to infinity then the variance of the Ridge parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> goes to zero.</p>
|
||
</section>
|
||
<section id="exercise-3-deriving-the-expression-for-the-bias-variance-trade-off">
|
||
<h2>Exercise 3: Deriving the expression for the Bias-Variance Trade-off<a class="headerlink" href="#exercise-3-deriving-the-expression-for-the-bias-variance-trade-off" title="Link to this heading">#</a></h2>
|
||
<p>The aim of this exercise is to derive the equations for the bias-variance tradeoff to be used in project 1.</p>
|
||
<p>The parameters <span class="math notranslate nohighlight">\(\boldsymbol{\hat{\beta}_{OLS}}\)</span> are found by optimizing the mean 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><strong>a)</strong> Show that you can rewrite this into an expression which contains</p>
|
||
<ul class="simple">
|
||
<li><p>the variance of the model (the variance term)</p></li>
|
||
<li><p>the expected deviation of the mean of the model from the true data (the bias term)</p></li>
|
||
<li><p>the variance of the noise</p></li>
|
||
</ul>
|
||
<p>In other words, show that:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathrm{Bias}[\tilde{y}]+\mathrm{var}[\tilde{y}]+\sigma^2,
|
||
\]</div>
|
||
<p>with</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathrm{Bias}[\tilde{y}]=\mathbb{E}\left[\left(\boldsymbol{y}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\right)^2\right],
|
||
\]</div>
|
||
<p>and</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathrm{var}[\tilde{y}]=\mathbb{E}\left[\left(\tilde{\boldsymbol{y}}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\right)^2\right]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2.
|
||
\]</div>
|
||
<p>In order to arrive at the equation for the bias, we have to approximate the unknown function <span class="math notranslate nohighlight">\(f\)</span> with the output/target values <span class="math notranslate nohighlight">\(y\)</span>.</p>
|
||
<p><strong>b)</strong> Explain what the terms mean and discuss their interpretations.</p>
|
||
</section>
|
||
<section id="exercise-4-computing-the-bias-and-variance">
|
||
<h2>Exercise 4: Computing the Bias and Variance<a class="headerlink" href="#exercise-4-computing-the-bias-and-variance" title="Link to this heading">#</a></h2>
|
||
<p>Before you compute the bias and variance of a real model for different complexities, let’s for now assume that you have sampled predictions and targets for a single model complexity using bootstrap resampling.</p>
|
||
<p><strong>a)</strong> Using the expression above, compute the mean squared error, bias and variance of the given data. Check that the sum of the bias and variance correctly gives (approximately) the mean squared error.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span><span class="w"> </span><span class="nn">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
|
||
|
||
<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span>
|
||
<span class="n">bootstraps</span> <span class="o">=</span> <span class="mi">1000</span>
|
||
|
||
<span class="n">predictions</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">rand</span><span class="p">(</span><span class="n">bootstraps</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">*</span> <span class="mi">10</span> <span class="o">+</span> <span class="mi">10</span>
|
||
<span class="c1"># The definition of targets has been updated, and was wrong earlier in the week.</span>
|
||
<span class="n">targets</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">rand</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
|
||
|
||
<span class="n">mse</span> <span class="o">=</span> <span class="o">...</span>
|
||
<span class="n">bias</span> <span class="o">=</span> <span class="o">...</span>
|
||
<span class="n">variance</span> <span class="o">=</span> <span class="o">...</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p><strong>b)</strong> Change the prediction values in some way to increase the bias while decreasing the variance.</p>
|
||
<p><strong>c)</strong> Change the prediction values in some way to increase the variance while decreasing the bias.</p>
|
||
<p><strong>d)</strong> Perform a bias-variance analysis of a polynomial OLS model fit to a one-dimensional function by computing and plotting the bias and variances values as a function of the polynomial degree of your model.</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="w"> </span><span class="nn">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
|
||
<span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
|
||
<span class="kn">from</span><span class="w"> </span><span class="nn">sklearn.preprocessing</span><span class="w"> </span><span class="kn">import</span> <span class="p">(</span>
|
||
<span class="n">PolynomialFeatures</span><span class="p">,</span>
|
||
<span class="p">)</span> <span class="c1"># use the fit_transform method of the created object!</span>
|
||
<span class="kn">from</span><span class="w"> </span><span class="nn">sklearn.linear_model</span><span class="w"> </span><span class="kn">import</span> <span class="n">LinearRegression</span>
|
||
<span class="kn">from</span><span class="w"> </span><span class="nn">sklearn.metrics</span><span class="w"> </span><span class="kn">import</span> <span class="n">mean_squared_error</span>
|
||
<span class="kn">from</span><span class="w"> </span><span class="nn">sklearn.model_selection</span><span class="w"> </span><span class="kn">import</span> <span class="n">train_test_split</span>
|
||
<span class="kn">from</span><span class="w"> </span><span class="nn">sklearn.utils</span><span class="w"> </span><span class="kn">import</span> <span class="n">resample</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">bootstraps</span> <span class="o">=</span> <span class="mi">1000</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="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="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">biases</span> <span class="o">=</span> <span class="p">[]</span>
|
||
<span class="n">variances</span> <span class="o">=</span> <span class="p">[]</span>
|
||
<span class="n">mses</span> <span class="o">=</span> <span class="p">[]</span>
|
||
|
||
<span class="c1"># for p in range(1, 5):</span>
|
||
<span class="c1"># predictions = ...</span>
|
||
<span class="c1"># targets = ...</span>
|
||
<span class="c1">#</span>
|
||
<span class="c1"># X = ...</span>
|
||
<span class="c1"># X_train, X_test, y_train, y_test = ...</span>
|
||
<span class="c1"># for b in range(bootstraps):</span>
|
||
<span class="c1"># X_train_re, y_train_re = ...</span>
|
||
<span class="c1">#</span>
|
||
<span class="c1"># # fit your model on the sampled data</span>
|
||
<span class="c1">#</span>
|
||
<span class="c1"># # make predictions on the test data</span>
|
||
<span class="c1"># predictions[b, :] =</span>
|
||
<span class="c1"># targets[b, :] =</span>
|
||
<span class="c1">#</span>
|
||
<span class="c1"># biases.append(...)</span>
|
||
<span class="c1"># variances.append(...)</span>
|
||
<span class="c1"># mses.append(...)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p><strong>e)</strong> Discuss the bias-variance trade-off as function of your model complexity (the degree of the polynomial).</p>
|
||
<p><strong>f)</strong> Compute and discuss the bias and variance as function of the number of data points (choose a suitable polynomial degree to show something interesting).</p>
|
||
</section>
|
||
<section id="exercise-5-interpretation-of-scaling-and-metrics">
|
||
<h2>Exercise 5: Interpretation of scaling and metrics<a class="headerlink" href="#exercise-5-interpretation-of-scaling-and-metrics" title="Link to this heading">#</a></h2>
|
||
<p>In this course, we often ask you to scale data and compute various metrics. Although these practices are “standard” in the field, we will require you to demonstrate an understanding of <em>why</em> you need to scale data and use these metrics. Both so that you can make better arguements about your results, and so that you will hopefully make fewer mistakes.</p>
|
||
<p>First, a few reminders: In this course you should always scale the columns of the feature matrix, and sometimes scale the target data, when it is worth the effort. By scaling, we mean subtracting the mean and dividing by the standard deviation, though there are many other ways to scale data. When scaling either the feature matrix or the target data, the intercept becomes a bit harder to implement and understand, so take care.</p>
|
||
<p>Briefly answer the following:</p>
|
||
<p><strong>a)</strong> Why do we scale data?</p>
|
||
<p><strong>b)</strong> Why does the OLS method give practically equivelent models on scaled and unscaled data?</p>
|
||
<p><strong>c)</strong> Why does the Ridge method <strong>not</strong> give practically equivelent models on scaled and unscaled data? Why do we only consider the model on scaled data correct?</p>
|
||
<p><strong>d)</strong> Why do we say that the Ridge method gives a biased model?</p>
|
||
<p><strong>e)</strong> Is the MSE of the OLS method affected by scaling of the feature matrix? Is it affected by scaling of the target data?</p>
|
||
<p><strong>f)</strong> Read about the R2 score, a metric we will ask you to use a lot later in the course. Is the R2 score of the OLS method affected by scaling of the feature matrix? Is it affected by scaling of the target data?</p>
|
||
<p><strong>g)</strong> Give interpretations of the following R2 scores: 0, 0.5, 1.</p>
|
||
<p><strong>h)</strong> What is an advantage of the R2 score over the MSE?</p>
|
||
</section>
|
||
</section>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#resampling-and-the-bias-variance-trade-off">Resampling and the Bias-Variance Trade-off</a><ul class="nav section-nav flex-column">
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