Morten Hjorth-Jensen
846 lines
47 KiB
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846 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">About the course</span></p>
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<li class="toctree-l1"><a class="reference internal" href="schedule.html">Course setting</a></li>
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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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<ul class="nav bd-sidenav">
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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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</ul>
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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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</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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<ul class="nav bd-sidenav">
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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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</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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<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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</ul>
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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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<ul class="current nav bd-sidenav">
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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"><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: 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="week42.html">Week 42 Constructing a Neural Network code with examples</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek42.html">Exercises week 42</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week43.html">Week 43: Deep Learning: Constructing a Neural Network code and solving differential equations</a></li>
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<li class="toctree-l1 current active"><a class="current reference internal" href="#">Exercises week 43</a></li>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Projects</span></p>
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<ul class="nav bd-sidenav">
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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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<li class="toctree-l1"><a class="reference internal" href="project2.html">Project 2 on Machine Learning, deadline November 10 (Midnight)</a></li>
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<i class="fa-solid fa-magnifying-glass fa-lg"></i>
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<button class="sidebar-toggle secondary-toggle btn btn-sm" title="Toggle secondary sidebar" data-bs-placement="bottom" data-bs-toggle="tooltip">
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|
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<div id="jb-print-docs-body" class="onlyprint">
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<h1>Exercises week 43</h1>
|
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<!-- Table of contents -->
|
||
<div id="print-main-content">
|
||
<div id="jb-print-toc">
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|
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<div>
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<h2> Contents </h2>
|
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</div>
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<nav aria-label="Page">
|
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<ul class="visible nav section-nav flex-column">
|
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<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#">Exercises week 43</a></li>
|
||
<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#overarching-aims-of-the-exercises-for-week-43">Overarching aims of the exercises for week 43</a><ul class="visible nav section-nav flex-column">
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#confusion-matrix">Confusion Matrix</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#roc-curve">ROC Curve</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#cumulative-gain">Cumulative Gain</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#other-measures-precision-recall-and-the-f-1-measure">Other measures: Precision, Recall, and the F<span class="math notranslate nohighlight">\(_1\)</span> Measure</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercises">Exercises</a><ul class="nav section-nav flex-column">
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-a">Exercise a)</a></li>
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-b">Exercise b)</a></li>
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-c-week-43">Exercise c) week 43</a></li>
|
||
</ul>
|
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|
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|
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</ul>
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<article class="bd-article">
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<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)
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doconce format html exercisesweek43.do.txt -->
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<!-- dom:TITLE: Exercises week 43 --><section class="tex2jax_ignore mathjax_ignore" id="exercises-week-43">
|
||
<h1>Exercises week 43<a class="headerlink" href="#exercises-week-43" title="Link to this heading">#</a></h1>
|
||
<p><strong>October 20-24, 2025</strong></p>
|
||
<p>Date: <strong>Deadline Friday October 24 at midnight</strong></p>
|
||
</section>
|
||
<section class="tex2jax_ignore mathjax_ignore" id="overarching-aims-of-the-exercises-for-week-43">
|
||
<h1>Overarching aims of the exercises for week 43<a class="headerlink" href="#overarching-aims-of-the-exercises-for-week-43" title="Link to this heading">#</a></h1>
|
||
<p>The aim of the exercises this week is to gain some confidence with
|
||
ways to visualize the results of a classification problem. We will
|
||
target three ways of setting up the analysis. The first and simplest
|
||
one is the</p>
|
||
<ol class="arabic simple">
|
||
<li><p>so-called confusion matrix. The next one is the so-called</p></li>
|
||
<li><p>ROC curve. Finally we have the</p></li>
|
||
<li><p>Cumulative gain curve.</p></li>
|
||
</ol>
|
||
<p>We will use Logistic Regression as method for the classification in
|
||
this exercise. You can compare these results with those obtained with
|
||
your neural network code from project 2 without a hidden layer.</p>
|
||
<p>In these exercises we will use binary and multi-class data sets
|
||
(the Iris data set from week 41).</p>
|
||
<p>The underlying mathematics is described here.</p>
|
||
<section id="confusion-matrix">
|
||
<h2>Confusion Matrix<a class="headerlink" href="#confusion-matrix" title="Link to this heading">#</a></h2>
|
||
<p>A <strong>confusion matrix</strong> summarizes a classifier’s performance by
|
||
tabulating predictions versus true labels. For binary classification,
|
||
it is a <span class="math notranslate nohighlight">\(2\times2\)</span> table whose entries are counts of outcomes:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\begin{array}{l|cc} & \text{Predicted Positive} & \text{Predicted Negative} \\ \hline \text{Actual Positive} & TP & FN \\ \text{Actual Negative} & FP & TN \end{array}.
|
||
\end{split}\]</div>
|
||
<p>Here TP (true positives) is the number of cases correctly predicted as
|
||
positive, FP (false positives) is the number incorrectly predicted as
|
||
positive, TN (true negatives) is correctly predicted negative, and FN
|
||
(false negatives) is incorrectly predicted negative . In other words,
|
||
“positive” means class 1 and “negative” means class 0; for example, TP
|
||
occurs when the prediction and actual are both positive. Formally:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\text{TPR} = \frac{\text{TP}}{\text{TP} + \text{FN}}, \quad \text{FPR} = \frac{\text{FP}}{\text{FP} + \text{TN}},
|
||
\]</div>
|
||
<p>where TPR and FPR are the true and false positive rates defined below.</p>
|
||
<p>In multiclass classification with <span class="math notranslate nohighlight">\(K\)</span> classes, the confusion matrix
|
||
generalizes to a <span class="math notranslate nohighlight">\(K\times K\)</span> table. Entry <span class="math notranslate nohighlight">\(N_{ij}\)</span> in the table is
|
||
the count of instances whose true class is <span class="math notranslate nohighlight">\(i\)</span> and whose predicted
|
||
class is <span class="math notranslate nohighlight">\(j\)</span>. For example, a three-class confusion matrix can be written
|
||
as:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\begin{array}{c|ccc} & \text{Pred Class 1} & \text{Pred Class 2} & \text{Pred Class 3} \\ \hline \text{Act Class 1} & N_{11} & N_{12} & N_{13} \\ \text{Act Class 2} & N_{21} & N_{22} & N_{23} \\ \text{Act Class 3} & N_{31} & N_{32} & N_{33} \end{array}.
|
||
\end{split}\]</div>
|
||
<p>Here the diagonal entries <span class="math notranslate nohighlight">\(N_{ii}\)</span> are the true positives for each
|
||
class, and off-diagonal entries are misclassifications. This matrix
|
||
allows computation of per-class metrics: e.g. for class <span class="math notranslate nohighlight">\(i\)</span>,
|
||
<span class="math notranslate nohighlight">\(\mathrm{TP}_i=N_{ii}\)</span>, <span class="math notranslate nohighlight">\(\mathrm{FN}_i=\sum_{j\neq i}N_{ij}\)</span>,
|
||
<span class="math notranslate nohighlight">\(\mathrm{FP}_i=\sum_{j\neq i}N_{ji}\)</span>, and <span class="math notranslate nohighlight">\(\mathrm{TN}_i\)</span> is the sum of
|
||
all remaining entries.</p>
|
||
<p>As defined above, TPR and FPR come from the binary case. In binary
|
||
terms with <span class="math notranslate nohighlight">\(P\)</span> actual positives and <span class="math notranslate nohighlight">\(N\)</span> actual negatives, one has</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\text{TPR} = \frac{TP}{P} = \frac{TP}{TP+FN}, \quad \text{FPR} =
|
||
\frac{FP}{N} = \frac{FP}{FP+TN},
|
||
\]</div>
|
||
<p>as used in standard confusion-matrix
|
||
formulations. These rates will be used in constructing ROC curves.</p>
|
||
</section>
|
||
<section id="roc-curve">
|
||
<h2>ROC Curve<a class="headerlink" href="#roc-curve" title="Link to this heading">#</a></h2>
|
||
<p>The Receiver Operating Characteristic (ROC) curve plots the trade-off
|
||
between true positives and false positives as a discrimination
|
||
threshold varies. Specifically, for a binary classifier that outputs
|
||
a score or probability, one varies the threshold <span class="math notranslate nohighlight">\(t\)</span> for declaring
|
||
<strong>positive</strong>, and computes at each <span class="math notranslate nohighlight">\(t\)</span> the true positive rate
|
||
<span class="math notranslate nohighlight">\(\mathrm{TPR}(t)\)</span> and false positive rate <span class="math notranslate nohighlight">\(\mathrm{FPR}(t)\)</span> using the
|
||
confusion matrix at that threshold. The ROC curve is then the graph
|
||
of TPR versus FPR. By definition,</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathrm{TPR} = \frac{TP}{TP+FN}, \qquad \mathrm{FPR} = \frac{FP}{FP+TN},
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(TP,FP,TN,FN\)</span> are counts determined by threshold <span class="math notranslate nohighlight">\(t\)</span>. A perfect
|
||
classifier would reach the point (FPR=0, TPR=1) at some threshold.</p>
|
||
<p>Formally, the ROC curve is obtained by plotting
|
||
<span class="math notranslate nohighlight">\((\mathrm{FPR}(t),\mathrm{TPR}(t))\)</span> for all <span class="math notranslate nohighlight">\(t\in[0,1]\)</span> (or as <span class="math notranslate nohighlight">\(t\)</span>
|
||
sweeps through the sorted scores). The Area Under the ROC Curve (AUC)
|
||
quantifies the average performance over all thresholds. It can be
|
||
interpreted probabilistically: <span class="math notranslate nohighlight">\(\mathrm{AUC} =
|
||
\Pr\bigl(s(X^+)>s(X^-)\bigr)\)</span>, the probability that a random positive
|
||
instance <span class="math notranslate nohighlight">\(X^+\)</span> receives a higher score <span class="math notranslate nohighlight">\(s\)</span> than a random negative
|
||
instance <span class="math notranslate nohighlight">\(X^-\)</span> . Equivalently, the AUC is the integral under the ROC
|
||
curve:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathrm{AUC} \;=\; \int_{0}^{1} \mathrm{TPR}(f)\,df,
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(f\)</span> ranges over FPR (or fraction of negatives). A model that guesses at random yields a diagonal ROC (AUC=0.5), whereas a perfect model yields AUC=1.0.</p>
|
||
</section>
|
||
<section id="cumulative-gain">
|
||
<h2>Cumulative Gain<a class="headerlink" href="#cumulative-gain" title="Link to this heading">#</a></h2>
|
||
<p>The cumulative gain curve (or gains chart) evaluates how many
|
||
positives are captured as one targets an increasing fraction of the
|
||
population, sorted by model confidence. To construct it, sort all
|
||
instances by decreasing predicted probability of the positive class.
|
||
Then, for the top <span class="math notranslate nohighlight">\(\alpha\)</span> fraction of instances, compute the fraction
|
||
of all actual positives that fall in this subset. In formula form, if
|
||
<span class="math notranslate nohighlight">\(P\)</span> is the total number of positive instances and <span class="math notranslate nohighlight">\(P(\alpha)\)</span> is the
|
||
number of positives among the top <span class="math notranslate nohighlight">\(\alpha\)</span> of the data, the cumulative
|
||
gain at level <span class="math notranslate nohighlight">\(\alpha\)</span> is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathrm{Gain}(\alpha) \;=\; \frac{P(\alpha)}{P}.
|
||
\]</div>
|
||
<p>For example, cutting off at the top 10% of predictions yields a gain
|
||
equal to (positives in top 10%) divided by (total positives) .
|
||
Plotting <span class="math notranslate nohighlight">\(\mathrm{Gain}(\alpha)\)</span> versus <span class="math notranslate nohighlight">\(\alpha\)</span> (often in percent)
|
||
gives the gain curve. The baseline (random) curve is the diagonal
|
||
<span class="math notranslate nohighlight">\(\mathrm{Gain}(\alpha)=\alpha\)</span>, while an ideal model has a steep climb
|
||
toward 1.</p>
|
||
<p>A related measure is the {\em lift}, often called the gain ratio. It is the ratio of the model’s capture rate to that of random selection. Equivalently,</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathrm{Lift}(\alpha) \;=\; \frac{\mathrm{Gain}(\alpha)}{\alpha}.
|
||
\]</div>
|
||
<p>A lift <span class="math notranslate nohighlight">\(>1\)</span> indicates better-than-random targeting. In practice, gain
|
||
and lift charts (used e.g.\ in marketing or imbalanced classification)
|
||
show how many positives can be “gained” by focusing on a fraction of
|
||
the population .</p>
|
||
</section>
|
||
<section id="other-measures-precision-recall-and-the-f-1-measure">
|
||
<h2>Other measures: Precision, Recall, and the F<span class="math notranslate nohighlight">\(_1\)</span> Measure<a class="headerlink" href="#other-measures-precision-recall-and-the-f-1-measure" title="Link to this heading">#</a></h2>
|
||
<p>Precision and recall (sensitivity) quantify binary classification
|
||
accuracy in terms of positive predictions. They are defined from the
|
||
confusion matrix as:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\text{Precision} = \frac{TP}{TP + FP}, \qquad \text{Recall} = \frac{TP}{TP + FN}.
|
||
\]</div>
|
||
<p>Precision is the fraction of predicted positives that are correct, and
|
||
recall is the fraction of actual positives that are correctly
|
||
identified . A high-precision classifier makes few false-positive
|
||
errors, while a high-recall classifier makes few false-negative
|
||
errors.</p>
|
||
<p>The F<span class="math notranslate nohighlight">\(_1\)</span> score (balanced F-measure) combines precision and recall into a single metric via their harmonic mean. The usual formula is:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
F_1 =2\frac{\text{Precision}\times\text{Recall}}{\text{Precision} + \text{Recall}}.
|
||
\]</div>
|
||
<p>This can be shown to equal</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{2\,TP}{2\,TP + FP + FN}.
|
||
\]</div>
|
||
<p>The F<span class="math notranslate nohighlight">\(_1\)</span> score ranges from 0 (worst) to 1 (best), and balances the
|
||
trade-off between precision and recall.</p>
|
||
<p>For multi-class classification, one computes per-class
|
||
precision/recall/F<span class="math notranslate nohighlight">\(_1\)</span> (treating each class as “positive” in a
|
||
one-vs-rest manner) and then averages. Common averaging methods are:</p>
|
||
<p>Micro-averaging: Sum all true positives, false positives, and false negatives across classes, then compute precision/recall/F<span class="math notranslate nohighlight">\(_1\)</span> from these totals.
|
||
Macro-averaging: Compute the F<span class="math notranslate nohighlight">\(1\)</span> score <span class="math notranslate nohighlight">\(F{1,i}\)</span> for each class <span class="math notranslate nohighlight">\(i\)</span> separately, then take the unweighted mean: <span class="math notranslate nohighlight">\(F_{1,\mathrm{macro}} = \frac{1}{K}\sum_{i=1}^K F_{1,i}\)</span> . This treats all classes equally regardless of size.
|
||
Weighted-averaging: Like macro-average, but weight each class’s <span class="math notranslate nohighlight">\(F_{1,i}\)</span> by its support <span class="math notranslate nohighlight">\(n_i\)</span> (true count): <span class="math notranslate nohighlight">\(F_{1,\mathrm{weighted}} = \frac{1}{N}\sum_{i=1}^K n_i F_{1,i}\)</span>, where <span class="math notranslate nohighlight">\(N=\sum_i n_i\)</span>. This accounts for class imbalance by giving more weight to larger classes .</p>
|
||
<p>Each of these averages has different use-cases. Micro-average is
|
||
dominated by common classes, macro-average highlights performance on
|
||
rare classes, and weighted-average is a compromise. These formulas
|
||
and concepts allow rigorous evaluation of classifier performance in
|
||
both binary and multi-class settings.</p>
|
||
</section>
|
||
<section id="exercises">
|
||
<h2>Exercises<a class="headerlink" href="#exercises" title="Link to this heading">#</a></h2>
|
||
<p>Here is a simple code example which uses the Logistic regression machinery from <strong>scikit-learn</strong>.
|
||
At the end it sets up the confusion matrix and the ROC and cumulative gain curves.
|
||
Feel free to use these functionalities (we don’t expect you to write your own code for say the confusion matrix).</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="o">%</span><span class="k">matplotlib</span> inline
|
||
|
||
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
|
||
<span class="c1"># from sklearn.datasets import fill in the data set</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LogisticRegression</span>
|
||
|
||
<span class="c1"># Load the data, fill inn</span>
|
||
<span class="n">mydata</span><span class="o">.</span><span class="n">data</span> <span class="o">=</span> <span class="o">?</span>
|
||
|
||
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">mydata</span><span class="o">.</span><span class="n">data</span><span class="p">,</span><span class="n">cancer</span><span class="o">.</span><span class="n">target</span><span class="p">,</span><span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">X_train</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">X_test</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
|
||
<span class="c1"># Logistic Regression</span>
|
||
<span class="c1"># define which type of problem, binary or multiclass</span>
|
||
<span class="n">logreg</span> <span class="o">=</span> <span class="n">LogisticRegression</span><span class="p">(</span><span class="n">solver</span><span class="o">=</span><span class="s1">'lbfgs'</span><span class="p">)</span>
|
||
<span class="n">logreg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
|
||
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">LabelEncoder</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">cross_validate</span>
|
||
<span class="c1">#Cross validation</span>
|
||
<span class="n">accuracy</span> <span class="o">=</span> <span class="n">cross_validate</span><span class="p">(</span><span class="n">logreg</span><span class="p">,</span><span class="n">X_test</span><span class="p">,</span><span class="n">y_test</span><span class="p">,</span><span class="n">cv</span><span class="o">=</span><span class="mi">10</span><span class="p">)[</span><span class="s1">'test_score'</span><span class="p">]</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">accuracy</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Test set accuracy with Logistic Regression: </span><span class="si">{:.2f}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">logreg</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span><span class="n">y_test</span><span class="p">)))</span>
|
||
|
||
<span class="kn">import</span> <span class="nn">scikitplot</span> <span class="k">as</span> <span class="nn">skplt</span>
|
||
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">logreg</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||
<span class="n">skplt</span><span class="o">.</span><span class="n">metrics</span><span class="o">.</span><span class="n">plot_confusion_matrix</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">,</span> <span class="n">normalize</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
<span class="n">y_probas</span> <span class="o">=</span> <span class="n">logreg</span><span class="o">.</span><span class="n">predict_proba</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||
<span class="n">skplt</span><span class="o">.</span><span class="n">metrics</span><span class="o">.</span><span class="n">plot_roc</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_probas</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
<span class="n">skplt</span><span class="o">.</span><span class="n">metrics</span><span class="o">.</span><span class="n">plot_cumulative_gain</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_probas</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<section id="exercise-a">
|
||
<h3>Exercise a)<a class="headerlink" href="#exercise-a" title="Link to this heading">#</a></h3>
|
||
<p>Convince yourself about the mathematics for the confusion matrix, the ROC and the cumlative gain curves for both a binary and a multiclass classification problem.</p>
|
||
</section>
|
||
<section id="exercise-b">
|
||
<h3>Exercise b)<a class="headerlink" href="#exercise-b" title="Link to this heading">#</a></h3>
|
||
<p>Use a binary classification data available from <strong>scikit-learn</strong>. As an example you can use
|
||
the MNIST data set and just specialize to two numbers. To do so you can use the following code lines</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">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_digits</span>
|
||
<span class="n">digits</span> <span class="o">=</span> <span class="n">load_digits</span><span class="p">(</span><span class="n">n_class</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span> <span class="c1"># Load only two classes, e.g., 0 and 1</span>
|
||
<span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">digits</span><span class="o">.</span><span class="n">data</span><span class="p">,</span> <span class="n">digits</span><span class="o">.</span><span class="n">target</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Alternatively, you can use the <em>make<span class="math notranslate nohighlight">\(\_\)</span>classification</em>
|
||
functionality. This function generates a random <span class="math notranslate nohighlight">\(n\)</span>-class classification
|
||
dataset, which can be configured for binary classification by setting
|
||
n_classes=2. You can also control the number of samples, features,
|
||
informative features, redundant features, and more.</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">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">make_classification</span>
|
||
<span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">make_classification</span><span class="p">(</span><span class="n">n_samples</span><span class="o">=</span><span class="mi">1000</span><span class="p">,</span> <span class="n">n_features</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">n_informative</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">n_redundant</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span> <span class="n">n_classes</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>You can use this option for the multiclass case as well, see the next exercise.
|
||
If you prefer to study other binary classification datasets, feel free
|
||
to replace the above suggestions with your own dataset.</p>
|
||
<p>Make plots of the confusion matrix, the ROC curve and the cumulative gain curve.</p>
|
||
</section>
|
||
<section id="exercise-c-week-43">
|
||
<h3>Exercise c) week 43<a class="headerlink" href="#exercise-c-week-43" title="Link to this heading">#</a></h3>
|
||
<p>As a multiclass problem, we will use the Iris data set discussed in
|
||
the exercises from weeks 41 and 42. This is a three-class data set and
|
||
you can set it up using <strong>scikit-learn</strong>,</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">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_iris</span>
|
||
<span class="n">iris</span> <span class="o">=</span> <span class="n">load_iris</span><span class="p">()</span>
|
||
<span class="n">X</span> <span class="o">=</span> <span class="n">iris</span><span class="o">.</span><span class="n">data</span> <span class="c1"># Features</span>
|
||
<span class="n">y</span> <span class="o">=</span> <span class="n">iris</span><span class="o">.</span><span class="n">target</span> <span class="c1"># Target labels</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Make plots of the confusion matrix, the ROC curve and the cumulative
|
||
gain curve for this (or other) multiclass data set.</p>
|
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|
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<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#">Exercises week 43</a></li>
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<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#overarching-aims-of-the-exercises-for-week-43">Overarching aims of the exercises for week 43</a><ul class="visible nav section-nav flex-column">
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#confusion-matrix">Confusion Matrix</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#roc-curve">ROC Curve</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#cumulative-gain">Cumulative Gain</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#other-measures-precision-recall-and-the-f-1-measure">Other measures: Precision, Recall, and the F<span class="math notranslate nohighlight">\(_1\)</span> Measure</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercises">Exercises</a><ul class="nav section-nav flex-column">
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<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-a">Exercise a)</a></li>
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<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-b">Exercise b)</a></li>
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<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-c-week-43">Exercise c) week 43</a></li>
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