Morten Hjorth-Jensen
617 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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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Decision Trees, Ensemble Methods and Boosting</span></p>
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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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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Deep Learning Methods</span></p>
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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: Statistical interpretation of Linear Regression and Resampling techniques</a></li>
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<li class="toctree-l1 current active"><a class="current reference internal" href="#">Exercises week 37</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week37.html">Week 37: Statistical interpretations and Resampling Methods</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek38.html">Exercises week 38</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week38.html">Week 38: Logistic Regression and Optimization</a></li>
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<li class="toctree-l1"><a class="reference internal" href="project1.html">Project 1 on Machine Learning, deadline October 9 (midnight), 2023</a></li>
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<h1>Exercises week 37</h1>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#overarching-aims-of-the-exercises-this-week">Overarching aims of the exercises this week</a></li>
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<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>
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<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>
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<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)
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<!-- dom:TITLE: Exercises week 37 --><section class="tex2jax_ignore mathjax_ignore" id="exercises-week-37">
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<h1>Exercises week 37<a class="headerlink" href="#exercises-week-37" title="Permalink to this heading">#</a></h1>
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<p><strong>September 11-15, 2023</strong></p>
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<p>Date: <strong>Deadline is Sunday September 17 at midnight</strong></p>
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<section id="overarching-aims-of-the-exercises-this-week">
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<h2>Overarching aims of the exercises this week<a class="headerlink" href="#overarching-aims-of-the-exercises-this-week" title="Permalink to this heading">#</a></h2>
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<p>This exercise deals with various mean values and variances in linear
|
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regression method (here it may be useful to look up chapter 3,
|
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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,
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||
Springer</a>). The
|
||
exercise is also a part of project 1 and can be reused in the theory
|
||
part of the project.</p>
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<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>
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||
<p>The assumption we have made is 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>
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<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{y} = f(\boldsymbol{x})+\boldsymbol{\varepsilon}
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||
\]</div>
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<p>We then approximate this function <span class="math notranslate nohighlight">\(f(\boldsymbol{x})\)</span> with our model <span class="math notranslate nohighlight">\(\boldsymbol{\tilde{y}}\)</span> from the solution of the linear regression equations (ordinary least squares OLS), that is our
|
||
function <span class="math notranslate nohighlight">\(f\)</span> is approximated by <span class="math notranslate nohighlight">\(\boldsymbol{\tilde{y}}\)</span> where we minimized <span class="math notranslate nohighlight">\((\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\)</span>, with</p>
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<div class="math notranslate nohighlight">
|
||
\[
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||
\boldsymbol{\tilde{y}} = \boldsymbol{X}\boldsymbol{\beta}.
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\]</div>
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<p>The matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> is the so-called design or feature matrix.</p>
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</section>
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<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="Permalink to this heading">#</a></h2>
|
||
<p>Show that the expectation value of <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> for a given element <span class="math notranslate nohighlight">\(i\)</span></p>
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<div class="math notranslate nohighlight">
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||
\[
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\mathbb{E}(y_i) =\sum_{j}x_{ij} \beta_j=\mathbf{X}_{i, \ast} \, \boldsymbol{\beta},
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\]</div>
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<p>and that
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its variance is</p>
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<div class="math notranslate nohighlight">
|
||
\[
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||
\mbox{Var}(y_i) = \sigma^2.
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||
\]</div>
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<p>Hence, <span class="math notranslate nohighlight">\(y_i \sim N( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta}, \sigma^2)\)</span>, that is <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> follows a normal distribution with
|
||
mean value <span class="math notranslate nohighlight">\(\boldsymbol{X}\boldsymbol{\beta}\)</span> and variance <span class="math notranslate nohighlight">\(\sigma^2\)</span>.</p>
|
||
<p>With the OLS expressions for the optimal parameters <span class="math notranslate nohighlight">\(\boldsymbol{\hat{\beta}}\)</span> show that</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathbb{E}(\boldsymbol{\hat{\beta}}) = \boldsymbol{\beta}.
|
||
\]</div>
|
||
<p>Show finally that the variance of <span class="math notranslate nohighlight">\(\boldsymbol{\boldsymbol{\beta}}\)</span> is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mbox{Var}(\boldsymbol{\hat{\beta}}) = \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1}.
|
||
\]</div>
|
||
<p>We can use the last expression when we define a <a class="reference external" href="https://en.wikipedia.org/wiki/Confidence_interval">so-called confidence interval</a> for the parameters <span class="math notranslate nohighlight">\(\beta\)</span>.
|
||
A given parameter <span class="math notranslate nohighlight">\(\beta_j\)</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="Permalink to this heading">#</a></h2>
|
||
<p>Show that</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathbb{E} \big[ \hat{\boldsymbol{\beta}}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\boldsymbol{\beta}.
|
||
\]</div>
|
||
<p>We see clearly that
|
||
<span class="math notranslate nohighlight">\(\mathbb{E} \big[ \hat{\boldsymbol{\beta}}^{\mathrm{Ridge}} \big] \not= \mathbb{E} \big[\hat{\boldsymbol{\beta}}^{\mathrm{OLS}}\big ]\)</span> for any <span class="math notranslate nohighlight">\(\lambda > 0\)</span>.</p>
|
||
<p>Show also that the variance is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mbox{Var}[\hat{\boldsymbol{\beta}}^{\mathrm{Ridge}}]=\sigma^2[ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1} \mathbf{X}^{T}\mathbf{X} \{ [ \mathbf{X}^{\top} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T},
|
||
\]</div>
|
||
<p>and it is easy to see that if the parameter <span class="math notranslate nohighlight">\(\lambda\)</span> goes to infinity then the variance of the Ridge parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> goes to zero.</p>
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</section>
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</section>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#overarching-aims-of-the-exercises-this-week">Overarching aims of the exercises this week</a></li>
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<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>
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<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>
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