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Applied Data Analysis and Machine Learning
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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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<p aria-level="2" class="caption" role="heading"><span class="caption-text">From Regression to Support Vector Machines</span></p>
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Review of Statistics with Resampling Techniques and Linear Algebra
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<li class="toctree-l1"><a class="reference internal" href="chapter1.html">3. Linear Regression</a></li>
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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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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Decision Trees, Ensemble Methods and Boosting</span></p>
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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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7. Optimization, the central part of any Machine Learning algortithm
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|
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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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||||
</li>
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||||
</ul>
|
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Dimensionality Reduction</span></p>
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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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<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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9. Decision trees, overarching aims
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<li class="toctree-l1">
|
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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"><span class="caption-text">Deep Learning Methods</span></p>
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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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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Weekly material, notes and exercises</span></p>
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|
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Deep Learning Methods
|
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|
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|
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13. Neural networks
|
||||
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|
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|
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|
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14. Building a Feed Forward Neural Network
|
||||
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|
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|
||||
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|
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15. Solving Differential Equations with Deep Learning
|
||||
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|
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|
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|
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16. Convolutional Neural Networks
|
||||
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|
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|
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|
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||||
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|
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Weekly material, notes and exercises
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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: Statistical interpretation of Linear Regression and Resampling techniques
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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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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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<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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Exercise 1: Expectation values for ordinary least squares expressions
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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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<h1>Exercises week 37<a class="headerlink" href="#exercises-week-37" title="Permalink to this headline">¶</a></h1>
|
||||
<p><strong>September 11-15, 2023</strong></p>
|
||||
<p>Date: <strong>Deadline is Sunday September 17 at midnight</strong></p>
|
||||
<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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<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>
|
||||
<p>This exercise deals with various mean values and variances in linear
|
||||
regression method (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
|
||||
@@ -448,9 +488,9 @@ function <span class="math notranslate nohighlight">\(f\)</span> is approximated
|
||||
\boldsymbol{\tilde{y}} = \boldsymbol{X}\boldsymbol{\beta}.
|
||||
\]</div>
|
||||
<p>The matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> is the so-called design or feature matrix.</p>
|
||||
</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="Permalink to this heading">#</a></h2>
|
||||
</div>
|
||||
<div class="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 headline">¶</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>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
@@ -476,9 +516,9 @@ mean value <span class="math notranslate nohighlight">\(\boldsymbol{X}\boldsymbo
|
||||
\]</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>
|
||||
</div>
|
||||
<div class="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 headline">¶</a></h2>
|
||||
<p>Show that</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
@@ -492,8 +532,8 @@ A given parameter <span class="math notranslate nohighlight">\(\beta_j\)</span>
|
||||
\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>
|
||||
</section>
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@@ -507,7 +547,7 @@ A given parameter <span class="math notranslate nohighlight">\(\beta_j\)</span>
|
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mode: "python"
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@@ -515,103 +555,44 @@ A given parameter <span class="math notranslate nohighlight">\(\beta_j\)</span>
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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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