Morten Hjorth-Jensen
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Applied Data Analysis and Machine Learning
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About the course
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Teachers and Grading
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Review of Statistics with Resampling Techniques and Linear Algebra
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1. Elements of Probability Theory and Statistical Data Analysis
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</a>
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<a class="reference internal" href="linalg.html">
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2. Linear Algebra, Handling of Arrays and more Python Features
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<p aria-level="2" class="caption" role="heading">
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<span class="caption-text">
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From Regression to Support Vector Machines
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</span>
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3. Linear Regression
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</a>
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<li class="toctree-l1">
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<a class="reference internal" href="chapter2.html">
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4. Ridge and Lasso Regression
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</a>
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<a class="reference internal" href="chapter3.html">
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5. Resampling Methods
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</a>
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<li class="toctree-l1">
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<a class="reference internal" href="chapter4.html">
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6. Logistic Regression
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</a>
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<li class="toctree-l1">
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<a class="reference internal" href="chapteroptimization.html">
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7. Optimization, the central part of any Machine Learning algortithm
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</a>
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<li class="toctree-l1">
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<a class="reference internal" href="chapter5.html">
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8. Support Vector Machines, overarching aims
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</a>
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</ul>
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<p aria-level="2" class="caption" role="heading">
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<span class="caption-text">
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Decision Trees, Ensemble Methods and Boosting
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</span>
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9. Decision trees, overarching aims
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<a class="reference internal" href="chapter7.html">
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10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods
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</a>
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</li>
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</ul>
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<p aria-level="2" class="caption" role="heading">
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<span class="caption-text">
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Dimensionality Reduction
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<a class="reference internal" href="chapter8.html">
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11. Basic ideas of the Principal Component Analysis (PCA)
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</a>
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<li class="toctree-l1">
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<a class="reference internal" href="clustering.html">
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12. Clustering and Unsupervised Learning
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</a>
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<p aria-level="2" class="caption" role="heading">
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<span class="caption-text">
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Deep Learning Methods
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13. Neural networks
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<a class="reference internal" href="chapter10.html">
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14. Building a Feed Forward Neural Network
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</a>
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<a class="reference internal" href="chapter11.html">
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15. Solving Differential Equations with Deep Learning
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</a>
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<a class="reference internal" href="chapter12.html">
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16. Convolutional Neural Networks
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</a>
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<li class="toctree-l1">
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<a class="reference internal" href="chapter13.html">
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17. Recurrent neural networks: Overarching view
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</a>
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<p aria-level="2" class="caption" role="heading">
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Weekly material, notes and exercises
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<a class="reference internal" href="exercisesweek34.html">
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Exercises week 34
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Week 34: Introduction to the course, Logistics and Practicalities
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<a class="reference internal" href="exercisesweek35.html">
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Exercises week 35
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Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
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<a class="reference internal" href="exercisesweek36.html">
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Exercises week 36
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Week 36: Linear Regression and Statistical interpretations
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Exercises week 37
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Week 37: Statistical interpretations and Resampling Methods
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<a class="reference internal" href="exercisesweek38.html">
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Exercises week 38
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Week 38: Logistic Regression and Optimization
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Exercises week 39
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Week 39: Optimization and Gradient Methods
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Week 40: Gradient descent methods (continued) and start Neural networks
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Exercises week 41
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Week 41 Neural networks and constructing a neural network code
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<a class="reference internal" href="exercisesweek42.html">
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Exercises week 42
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Week 42 Constructing a Neural Network code with examples
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<a class="reference internal" href="additionweek42.html">
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Exercises Week 42: Logistic Regression and Optimization, reminders from week 38 and week 40
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</a>
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<li class="toctree-l1">
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<a class="reference internal" href="week43.html">
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Week 43: Deep Learning: Constructing a Neural Network code and solving differential equations
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<a class="reference internal" href="exercisesweek43.html">
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Exercises week 43
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Projects
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Project 1 on Machine Learning, deadline October 7 (midnight), 2024
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Project 2 on Machine Learning, deadline November 4 (Midnight)
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Exercise 1: Expectation values for ordinary least squares expressions
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Exercise 2: Expectation values for Ridge regression
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<h1>Exercises week 37</h1>
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<a class="reference internal nav-link" href="#overarching-aims-of-the-exercises-this-week">
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Overarching aims of the exercises this week
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#exercise-1-expectation-values-for-ordinary-least-squares-expressions">
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Exercise 1: Expectation values for ordinary least squares expressions
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</a>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#exercise-2-expectation-values-for-ridge-regression">
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Exercise 2: Expectation values for Ridge regression
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<!-- dom:TITLE: Exercises week 37 --><div class="tex2jax_ignore mathjax_ignore section" id="exercises-week-37">
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<h1>Exercises week 37<a class="headerlink" href="#exercises-week-37" title="Permalink to this headline">¶</a></h1>
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<p><strong>September 9-13, 2024</strong></p>
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<p>Date: <strong>Deadline is Friday September 13 at midnight</strong></p>
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<div class="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 headline">¶</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
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H. Friedman, The Elements of Statistical Learning,
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Springer</a>). The
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exercise is also a part of project 1 and can be reused in the theory
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part of the project.</p>
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<p>For more discussions on Ridge regression and calculation of
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expectation values, <a class="reference external" href="https://arxiv.org/abs/1509.09169">Wessel van
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Wieringen’s</a> article is highly
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recommended.</p>
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<p>The assumption we have made is that there exists a continuous function
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<span class="math notranslate nohighlight">\(f(\boldsymbol{x})\)</span> and a normal distributed error <span class="math notranslate nohighlight">\(\boldsymbol{\varepsilon}\sim N(0,
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\sigma^2)\)</span> which describes our data</p>
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<div class="math notranslate nohighlight">
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\[
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\boldsymbol{y} = f(\boldsymbol{x})+\boldsymbol{\varepsilon}
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\]</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
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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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\[
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\boldsymbol{\tilde{y}} = \boldsymbol{X}\boldsymbol{\beta}.
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\]</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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</div>
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<div class="section" id="exercise-1-expectation-values-for-ordinary-least-squares-expressions">
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<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 headline">¶</a></h2>
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<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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\[
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\mbox{Var}(y_i) = \sigma^2.
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\]</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
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mean value <span class="math notranslate nohighlight">\(\boldsymbol{X}\boldsymbol{\beta}\)</span> and variance <span class="math notranslate nohighlight">\(\sigma^2\)</span>.</p>
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<p>With the OLS expressions for the optimal parameters <span class="math notranslate nohighlight">\(\boldsymbol{\hat{\beta}}\)</span> show that</p>
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<div class="math notranslate nohighlight">
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\[
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\mathbb{E}(\boldsymbol{\hat{\beta}}) = \boldsymbol{\beta}.
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\]</div>
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<p>Show finally that the variance of <span class="math notranslate nohighlight">\(\boldsymbol{\boldsymbol{\beta}}\)</span> is</p>
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<div class="math notranslate nohighlight">
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\[
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\mbox{Var}(\boldsymbol{\hat{\beta}}) = \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1}.
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\]</div>
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<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>.
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A given parameter <span class="math notranslate nohighlight">\(\beta_j\)</span> is given by the diagonal matrix element of the above matrix.</p>
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</div>
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<div class="section" id="exercise-2-expectation-values-for-ridge-regression">
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<h2>Exercise 2: Expectation values for Ridge regression<a class="headerlink" href="#exercise-2-expectation-values-for-ridge-regression" title="Permalink to this headline">¶</a></h2>
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<p>Show that</p>
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<div class="math notranslate nohighlight">
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\[
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\mathbb{E} \big[ \hat{\boldsymbol{\beta}}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\boldsymbol{\beta}.
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\]</div>
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<p>We see clearly that
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<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>
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<p>Show also that the variance is</p>
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<div class="math notranslate nohighlight">
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\[
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\mbox{Var}[\hat{\boldsymbol{\beta}}^{\mathrm{Ridge}}]=\sigma^2[ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1} \mathbf{X}^{T}\mathbf{X} \{ [ \mathbf{X}^{\top} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T},
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\]</div>
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<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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