Morten Hjorth-Jensen
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Applied Data Analysis and Machine Learning
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About the course
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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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2. Linear Algebra, Handling of Arrays and more Python Features
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From Regression to Support Vector Machines
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3. Linear Regression
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4. Ridge and Lasso Regression
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5. Resampling Methods
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6. Logistic Regression
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Decision Trees, Ensemble Methods and Boosting
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9. Decision trees, overarching aims
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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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Dimensionality Reduction
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11. Basic ideas of the Principal Component Analysis (PCA)
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12. Clustering and Unsupervised Learning
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Deep Learning Methods
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13. Neural networks
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14. Building a Feed Forward Neural Network
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15. Solving Differential Equations with Deep Learning
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16. Convolutional Neural Networks
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17. Recurrent neural networks: Overarching view
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Exercises week 34
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Week 34: Introduction to the course, Logistics and Practicalities
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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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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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Exercises week 38
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<h1>Exercises week 38</h1>
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<h1>Exercises week 38<a class="headerlink" href="#exercises-week-38" title="Permalink to this headline">¶</a></h1>
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<p><strong>September 16-20, 2024</strong></p>
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<p>Date: <strong>Deadline is Friday September 20 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>The aim of the exercises this week is to derive the equations for the bias-variance tradeoff to be used in project 1 as well as testing this for a simpler function using the bootstrap method. The exercises here can be reused in project 1 as well.</p>
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<p>Consider a
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dataset <span class="math notranslate nohighlight">\(\mathcal{L}\)</span> consisting of the data
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<span class="math notranslate nohighlight">\(\mathbf{X}_\mathcal{L}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\}\)</span>.</p>
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<p>We assume that the true data is generated from a noisy model</p>
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<div class="math notranslate nohighlight">
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\[
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\boldsymbol{y}=f(\boldsymbol{x}) + \boldsymbol{\epsilon}.
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\]</div>
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<p>Here <span class="math notranslate nohighlight">\(\epsilon\)</span> is normally distributed with mean zero and standard
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deviation <span class="math notranslate nohighlight">\(\sigma^2\)</span>.</p>
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<p>In our derivation of the ordinary least squares method we defined
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an approximation to the function <span class="math notranslate nohighlight">\(f\)</span> in terms of the parameters
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<span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> and the design matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> which embody our model,
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that is <span class="math notranslate nohighlight">\(\boldsymbol{\tilde{y}}=\boldsymbol{X}\boldsymbol{\beta}\)</span>.</p>
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<p>The parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> are in turn found by optimizing the mean
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squared error via the so-called cost function</p>
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<div class="math notranslate nohighlight">
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\[
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C(\boldsymbol{X},\boldsymbol{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2=\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right].
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\]</div>
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<p>Here the expected value <span class="math notranslate nohighlight">\(\mathbb{E}\)</span> is the sample value.</p>
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<p>Show that you can rewrite this in terms of a term which contains the variance of the model itself (the so-called variance term), a
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term which measures the deviation from the true data and the mean value of the model (the bias term) and finally the variance of the noise.
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That is, show that</p>
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<div class="math notranslate nohighlight">
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\[
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\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathrm{Bias}[\tilde{y}]+\mathrm{var}[\tilde{y}]+\sigma^2,
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\]</div>
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<p>with</p>
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<div class="math notranslate nohighlight">
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\[
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\mathrm{Bias}[\tilde{y}]=\mathbb{E}\left[\left(\boldsymbol{y}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\right)^2\right],
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\]</div>
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<p>and</p>
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<div class="math notranslate nohighlight">
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\[
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\mathrm{var}[\tilde{y}]=\mathbb{E}\left[\left(\tilde{\boldsymbol{y}}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\right)^2\right]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2.
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\]</div>
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<p>Explain what the terms mean and discuss their interpretations.</p>
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<p>Perform then a bias-variance analysis of a simple one-dimensional (or other models of your choice) function by
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studying the MSE value as function of the complexity of your model. Use ordinary least squares only.</p>
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<p>Discuss the bias and variance trade-off as function
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of your model complexity (the degree of the polynomial) and the number
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of data points, and possibly also your training and test data using the <strong>bootstrap</strong> resampling method.
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You can follow the code example in the jupyter-book at <a class="reference external" href="https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/chapter3.html#the-bias-variance-tradeoff">https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/chapter3.html#the-bias-variance-tradeoff</a>.</p>
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