Morten Hjorth-Jensen
1136 lines
82 KiB
HTML
1136 lines
82 KiB
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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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</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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<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: Linear Regression and Statistical interpretations</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: 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="exercisesweek39.html">Exercises week 39</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week39.html">Week 39: Optimization and Gradient Methods</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="exercisesweek41.html">Exercises week 41</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="exercisesweek42.html">Exercises week 42</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 current active"><a class="current reference internal" href="#">Exercises Week 42: Logistic Regression and Optimization, reminders from week 38 and week 40</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"><a class="reference internal" href="exercisesweek43.html">Exercises week 43</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week44.html">Week 44, Convolutional Neural Networks (CNN)</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week45.html">Week 45, Convolutional Neural Networks (CCNs) and Recurrent Neural Networks (RNNs)</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week46.html">Week 46: Decision Trees, Ensemble methods and Random Forests</a></li>
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<li class="toctree-l1"><a class="reference internal" href="project1.html">Project 1 on Machine Learning, deadline October 7 (midnight), 2024</a></li>
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<li class="toctree-l1"><a class="reference internal" href="project2.html">Project 2 on Machine Learning, deadline November 4 (Midnight)</a></li>
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<li class="toctree-l1"><a class="reference internal" href="project3.html">Project 3 on Machine Learning, deadline December 9 (midnight), 2024</a></li>
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<div id="jb-print-docs-body" class="onlyprint">
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<h1>Exercises Week 42: Logistic Regression and Optimization, reminders from week 38 and week 40</h1>
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<!-- Table of contents -->
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<div id="print-main-content">
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<div id="jb-print-toc">
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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-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-logistic-function">The logistic function</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#examples-of-likelihood-functions-used-in-logistic-regression-and-nueral-networks">Examples of likelihood functions used in logistic regression and nueral networks</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#two-parameters">Two parameters</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-cost-function">The cost function</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#minimizing-the-cross-entropy">Minimizing the cross entropy</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-more-compact-expression">A more compact expression</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#extending-to-more-predictors">Extending to more predictors</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#including-more-classes">Including more classes</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#more-classes">More classes</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#wisconsin-cancer-data">Wisconsin Cancer Data</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#using-the-correlation-matrix">Using the correlation matrix</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discussing-the-correlation-data">Discussing the correlation data</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#other-measures-in-classification-studies-cancer-data-again">Other measures in classification studies: Cancer Data again</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#gradient-descent-and-logistic-regression">Gradient descent and Logistic regression</a></li>
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</ul>
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</nav>
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</div>
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<div id="searchbox"></div>
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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 additionweek42.do.txt --no_mako -->
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<!-- dom:TITLE: Exercises Week 42: Logistic Regression and Optimization, reminders from week 38 and week 40 --><section class="tex2jax_ignore mathjax_ignore" id="exercises-week-42-logistic-regression-and-optimization-reminders-from-week-38-and-week-40">
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<h1>Exercises Week 42: Logistic Regression and Optimization, reminders from week 38 and week 40<a class="headerlink" href="#exercises-week-42-logistic-regression-and-optimization-reminders-from-week-38-and-week-40" title="Link to this heading">#</a></h1>
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<p><strong>Morten Hjorth-Jensen</strong>, Department of Physics and Center for Computing in Science Education, University of Oslo and Department of Physics and Astronomy and Facility for Rare Isotope Beams, Michigan State University</p>
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<p>Date: <strong>October 14-18, 2024</strong></p>
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<section id="the-logistic-function">
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<h2>The logistic function<a class="headerlink" href="#the-logistic-function" title="Link to this heading">#</a></h2>
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<p>A widely studied model, is the
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perceptron model, which is an example of a “hard classification” model. We
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have used this model when we discussed neural networks as
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well. Each datapoint is deterministically assigned to a category (i.e
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<span class="math notranslate nohighlight">\(y_i=0\)</span> or <span class="math notranslate nohighlight">\(y_i=1\)</span>). In many cases it is favorable to have a “soft”
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classifier that outputs the probability of a given category rather
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than a single value. For example, given <span class="math notranslate nohighlight">\(x_i\)</span>, the classifier
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outputs the probability of being in a category <span class="math notranslate nohighlight">\(k\)</span>. Logistic regression
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is the most common example of a so-called soft classifier. In logistic
|
|
regression, the probability that a data point <span class="math notranslate nohighlight">\(x_i\)</span>
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belongs to a category <span class="math notranslate nohighlight">\(y_i=\{0,1\}\)</span> is given by the so-called logit function (or Sigmoid) which is meant to represent the likelihood for a given event,</p>
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<div class="math notranslate nohighlight">
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\[
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p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}.
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\]</div>
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<p>Note that <span class="math notranslate nohighlight">\(1-p(t)= p(-t)\)</span>.</p>
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</section>
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<section id="examples-of-likelihood-functions-used-in-logistic-regression-and-nueral-networks">
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<h2>Examples of likelihood functions used in logistic regression and nueral networks<a class="headerlink" href="#examples-of-likelihood-functions-used-in-logistic-regression-and-nueral-networks" title="Link to this heading">#</a></h2>
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<p>The following code plots the logistic function, the step function and other functions we will encounter from here and on.</p>
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<div class="cell docutils container">
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<div class="cell_input docutils container">
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="o">%</span><span class="k">matplotlib</span> inline
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<span class="sd">"""The sigmoid function (or the logistic curve) is a</span>
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<span class="sd">function that takes any real number, z, and outputs a number (0,1).</span>
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<span class="sd">It is useful in neural networks for assigning weights on a relative scale.</span>
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<span class="sd">The value z is the weighted sum of parameters involved in the learning algorithm."""</span>
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<span class="kn">import</span> <span class="nn">numpy</span>
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<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
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<span class="kn">import</span> <span class="nn">math</span> <span class="k">as</span> <span class="nn">mt</span>
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<span class="n">z</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mf">.1</span><span class="p">)</span>
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<span class="n">sigma_fn</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">vectorize</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">numpy</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="p">)))</span>
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<span class="n">sigma</span> <span class="o">=</span> <span class="n">sigma_fn</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
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<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
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<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">)</span>
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<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">sigma</span><span class="p">)</span>
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<span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="o">-</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">1.1</span><span class="p">])</span>
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|
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">([</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span>
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|
<span class="n">ax</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">)</span>
|
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<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'z'</span><span class="p">)</span>
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<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'sigmoid function'</span><span class="p">)</span>
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<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<span class="sd">"""Step Function"""</span>
|
|
<span class="n">z</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mf">.02</span><span class="p">)</span>
|
|
<span class="n">step_fn</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">vectorize</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="mf">1.0</span> <span class="k">if</span> <span class="n">z</span> <span class="o">>=</span> <span class="mf">0.0</span> <span class="k">else</span> <span class="mf">0.0</span><span class="p">)</span>
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<span class="n">step</span> <span class="o">=</span> <span class="n">step_fn</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
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<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
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|
<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">)</span>
|
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<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">step</span><span class="p">)</span>
|
|
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="o">-</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">])</span>
|
|
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">([</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span>
|
|
<span class="n">ax</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">)</span>
|
|
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'z'</span><span class="p">)</span>
|
|
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'step function'</span><span class="p">)</span>
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<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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|
|
<span class="sd">"""tanh Function"""</span>
|
|
<span class="n">z</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">mt</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">mt</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">)</span>
|
|
<span class="n">t</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">tanh</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
|
|
|
|
<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
|
|
<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">)</span>
|
|
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
|
|
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="o">-</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">])</span>
|
|
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">([</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">mt</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span><span class="mi">2</span><span class="o">*</span><span class="n">mt</span><span class="o">.</span><span class="n">pi</span><span class="p">])</span>
|
|
<span class="n">ax</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">)</span>
|
|
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'z'</span><span class="p">)</span>
|
|
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'tanh function'</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>
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|
</div>
|
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</div>
|
|
<div class="cell_output docutils container">
|
|
<img alt="_images/f595cb240e803bed7c09675e2c828acbc8a18aab73265f5a086bb9936d9ff30d.png" src="_images/f595cb240e803bed7c09675e2c828acbc8a18aab73265f5a086bb9936d9ff30d.png" />
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<img alt="_images/26f4dc8f430dfce96e23f825488d4442c0f3e5d1d2561dba2bb589bbaabae1a5.png" src="_images/26f4dc8f430dfce96e23f825488d4442c0f3e5d1d2561dba2bb589bbaabae1a5.png" />
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<img alt="_images/96415d6c22672d188f5ac64c6f26737caaf104b04f048daa000e424be5c926fb.png" src="_images/96415d6c22672d188f5ac64c6f26737caaf104b04f048daa000e424be5c926fb.png" />
|
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</div>
|
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</div>
|
|
</section>
|
|
<section id="two-parameters">
|
|
<h2>Two parameters<a class="headerlink" href="#two-parameters" title="Link to this heading">#</a></h2>
|
|
<p>We assume now that we have two classes with <span class="math notranslate nohighlight">\(y_i\)</span> either <span class="math notranslate nohighlight">\(0\)</span> or <span class="math notranslate nohighlight">\(1\)</span>. Furthermore we assume also that we have only two parameters <span class="math notranslate nohighlight">\(\beta\)</span> in our fitting of the Sigmoid function, that is we define probabilities</p>
|
|
<div class="math notranslate nohighlight">
|
|
\[\begin{split}
|
|
\begin{align*}
|
|
p(y_i=1|x_i,\boldsymbol{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\
|
|
p(y_i=0|x_i,\boldsymbol{\beta}) &= 1 - p(y_i=1|x_i,\boldsymbol{\beta}),
|
|
\end{align*}
|
|
\end{split}\]</div>
|
|
<p>where <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> are the weights we wish to extract from data, in our case <span class="math notranslate nohighlight">\(\beta_0\)</span> and <span class="math notranslate nohighlight">\(\beta_1\)</span>.</p>
|
|
<p>Note that we used</p>
|
|
<div class="math notranslate nohighlight">
|
|
\[
|
|
p(y_i=0\vert x_i, \boldsymbol{\beta}) = 1-p(y_i=1\vert x_i, \boldsymbol{\beta}).
|
|
\]</div>
|
|
</section>
|
|
<section id="the-cost-function">
|
|
<h2>The cost function<a class="headerlink" href="#the-cost-function" title="Link to this heading">#</a></h2>
|
|
<p>Reordering the logarithms, we can rewrite the <strong>cost/loss</strong> function as</p>
|
|
<div class="math notranslate nohighlight">
|
|
\[
|
|
\mathcal{C}(\boldsymbol{\beta}) = \sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
|
|
\]</div>
|
|
<p>The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to <span class="math notranslate nohighlight">\(\beta\)</span>.
|
|
Since the cost (error) function is just the negative log-likelihood, for logistic regression we have that</p>
|
|
<div class="math notranslate nohighlight">
|
|
\[
|
|
\mathcal{C}(\boldsymbol{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
|
|
\]</div>
|
|
<p>This equation is known in statistics as the <strong>cross entropy</strong>. Finally, we note that just as in linear regression,
|
|
in practice we often supplement the cross-entropy with additional regularization terms, usually <span class="math notranslate nohighlight">\(L_1\)</span> and <span class="math notranslate nohighlight">\(L_2\)</span> regularization as we did for Ridge and Lasso regression.</p>
|
|
</section>
|
|
<section id="minimizing-the-cross-entropy">
|
|
<h2>Minimizing the cross entropy<a class="headerlink" href="#minimizing-the-cross-entropy" title="Link to this heading">#</a></h2>
|
|
<p>The cross entropy is a convex function of the weights <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> and,
|
|
therefore, any local minimizer is a global minimizer.</p>
|
|
<p>Minimizing this
|
|
cost function with respect to the two parameters <span class="math notranslate nohighlight">\(\beta_0\)</span> and <span class="math notranslate nohighlight">\(\beta_1\)</span> we obtain</p>
|
|
<div class="math notranslate nohighlight">
|
|
\[
|
|
\frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \beta_0} = -\sum_{i=1}^n \left(y_i -\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right),
|
|
\]</div>
|
|
<p>and</p>
|
|
<div class="math notranslate nohighlight">
|
|
\[
|
|
\frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \beta_1} = -\sum_{i=1}^n \left(y_ix_i -x_i\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right).
|
|
\]</div>
|
|
</section>
|
|
<section id="a-more-compact-expression">
|
|
<h2>A more compact expression<a class="headerlink" href="#a-more-compact-expression" title="Link to this heading">#</a></h2>
|
|
<p>Let us now define a vector <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> with <span class="math notranslate nohighlight">\(n\)</span> elements <span class="math notranslate nohighlight">\(y_i\)</span>, an
|
|
<span class="math notranslate nohighlight">\(n\times p\)</span> matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> which contains the <span class="math notranslate nohighlight">\(x_i\)</span> values and a
|
|
vector <span class="math notranslate nohighlight">\(\boldsymbol{p}\)</span> of fitted probabilities <span class="math notranslate nohighlight">\(p(y_i\vert x_i,\boldsymbol{\beta})\)</span>. We can rewrite in a more compact form the first
|
|
derivative of cost function as</p>
|
|
<div class="math notranslate nohighlight">
|
|
\[
|
|
\frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = -\boldsymbol{X}^T\left(\boldsymbol{y}-\boldsymbol{p}\right).
|
|
\]</div>
|
|
<p>If we in addition define a diagonal matrix <span class="math notranslate nohighlight">\(\boldsymbol{W}\)</span> with elements
|
|
<span class="math notranslate nohighlight">\(p(y_i\vert x_i,\boldsymbol{\beta})(1-p(y_i\vert x_i,\boldsymbol{\beta})\)</span>, we can obtain a compact expression of the second derivative as</p>
|
|
<div class="math notranslate nohighlight">
|
|
\[
|
|
\frac{\partial^2 \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}\partial \boldsymbol{\beta}^T} = \boldsymbol{X}^T\boldsymbol{W}\boldsymbol{X}.
|
|
\]</div>
|
|
</section>
|
|
<section id="extending-to-more-predictors">
|
|
<h2>Extending to more predictors<a class="headerlink" href="#extending-to-more-predictors" title="Link to this heading">#</a></h2>
|
|
<p>Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with <span class="math notranslate nohighlight">\(p\)</span> predictors</p>
|
|
<div class="math notranslate nohighlight">
|
|
\[
|
|
\log{ \frac{p(\boldsymbol{\beta}\boldsymbol{x})}{1-p(\boldsymbol{\beta}\boldsymbol{x})}} = \beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p.
|
|
\]</div>
|
|
<p>Here we defined <span class="math notranslate nohighlight">\(\boldsymbol{x}=[1,x_1,x_2,\dots,x_p]\)</span> and <span class="math notranslate nohighlight">\(\boldsymbol{\beta}=[\beta_0, \beta_1, \dots, \beta_p]\)</span> leading to</p>
|
|
<div class="math notranslate nohighlight">
|
|
\[
|
|
p(\boldsymbol{\beta}\boldsymbol{x})=\frac{ \exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}{1+\exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}.
|
|
\]</div>
|
|
</section>
|
|
<section id="including-more-classes">
|
|
<h2>Including more classes<a class="headerlink" href="#including-more-classes" title="Link to this heading">#</a></h2>
|
|
<p>Till now we have mainly focused on two classes, the so-called binary
|
|
system. Suppose we wish to extend to <span class="math notranslate nohighlight">\(K\)</span> classes. Let us for the sake
|
|
of simplicity assume we have only two predictors. We have then following model</p>
|
|
<div class="math notranslate nohighlight">
|
|
\[
|
|
\log{\frac{p(C=1\vert x)}{p(K\vert x)}} = \beta_{10}+\beta_{11}x_1,
|
|
\]</div>
|
|
<p>and</p>
|
|
<div class="math notranslate nohighlight">
|
|
\[
|
|
\log{\frac{p(C=2\vert x)}{p(K\vert x)}} = \beta_{20}+\beta_{21}x_1,
|
|
\]</div>
|
|
<p>and so on till the class <span class="math notranslate nohighlight">\(C=K-1\)</span> class</p>
|
|
<div class="math notranslate nohighlight">
|
|
\[
|
|
\log{\frac{p(C=K-1\vert x)}{p(K\vert x)}} = \beta_{(K-1)0}+\beta_{(K-1)1}x_1,
|
|
\]</div>
|
|
<p>and the model is specified in term of <span class="math notranslate nohighlight">\(K-1\)</span> so-called log-odds or
|
|
<strong>logit</strong> transformations.</p>
|
|
</section>
|
|
<section id="more-classes">
|
|
<h2>More classes<a class="headerlink" href="#more-classes" title="Link to this heading">#</a></h2>
|
|
<p>In our discussion of neural networks we will encounter the above again
|
|
in terms of a slightly modified function, the so-called <strong>Softmax</strong> function.</p>
|
|
<p>The softmax function is used in various multiclass classification
|
|
methods, such as multinomial logistic regression (also known as
|
|
softmax regression), multiclass linear discriminant analysis, naive
|
|
Bayes classifiers, and artificial neural networks. Specifically, in
|
|
multinomial logistic regression and linear discriminant analysis, the
|
|
input to the function is the result of <span class="math notranslate nohighlight">\(K\)</span> distinct linear functions,
|
|
and the predicted probability for the <span class="math notranslate nohighlight">\(k\)</span>-th class given a sample
|
|
vector <span class="math notranslate nohighlight">\(\boldsymbol{x}\)</span> and a weighting vector <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> is (with two
|
|
predictors):</p>
|
|
<div class="math notranslate nohighlight">
|
|
\[
|
|
p(C=k\vert \mathbf {x} )=\frac{\exp{(\beta_{k0}+\beta_{k1}x_1)}}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}.
|
|
\]</div>
|
|
<p>It is easy to extend to more predictors. The final class is</p>
|
|
<div class="math notranslate nohighlight">
|
|
\[
|
|
p(C=K\vert \mathbf {x} )=\frac{1}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}},
|
|
\]</div>
|
|
<p>and they sum to one.</p>
|
|
</section>
|
|
<section id="wisconsin-cancer-data">
|
|
<h2>Wisconsin Cancer Data<a class="headerlink" href="#wisconsin-cancer-data" title="Link to this heading">#</a></h2>
|
|
<p>We show here how we can use a simple regression case on the breast
|
|
cancer data using Logistic regression as our algorithm for
|
|
classification.</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">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="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_breast_cancer</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</span>
|
|
<span class="n">cancer</span> <span class="o">=</span> <span class="n">load_breast_cancer</span><span class="p">()</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">cancer</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="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="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>
|
|
</pre></div>
|
|
</div>
|
|
</div>
|
|
<div class="cell_output docutils container">
|
|
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(426, 30)
|
|
(143, 30)
|
|
Test set accuracy with Logistic Regression: 0.94
|
|
</pre></div>
|
|
</div>
|
|
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/linear_model/_logistic.py:460: ConvergenceWarning: lbfgs failed to converge (status=1):
|
|
STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.
|
|
|
|
Increase the number of iterations (max_iter) or scale the data as shown in:
|
|
https://scikit-learn.org/stable/modules/preprocessing.html
|
|
Please also refer to the documentation for alternative solver options:
|
|
https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression
|
|
n_iter_i = _check_optimize_result(
|
|
</pre></div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</section>
|
|
<section id="using-the-correlation-matrix">
|
|
<h2>Using the correlation matrix<a class="headerlink" href="#using-the-correlation-matrix" title="Link to this heading">#</a></h2>
|
|
<p>In addition to the above scores, we could also study the covariance (and the correlation matrix).
|
|
We use <strong>Pandas</strong> to compute the correlation 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="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="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_breast_cancer</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="n">cancer</span> <span class="o">=</span> <span class="n">load_breast_cancer</span><span class="p">()</span>
|
|
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
|
|
<span class="c1"># Making a data frame</span>
|
|
<span class="n">cancerpd</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">cancer</span><span class="o">.</span><span class="n">data</span><span class="p">,</span> <span class="n">columns</span><span class="o">=</span><span class="n">cancer</span><span class="o">.</span><span class="n">feature_names</span><span class="p">)</span>
|
|
|
|
<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">15</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">20</span><span class="p">))</span>
|
|
<span class="n">malignant</span> <span class="o">=</span> <span class="n">cancer</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="o">==</span> <span class="mi">0</span><span class="p">]</span>
|
|
<span class="n">benign</span> <span class="o">=</span> <span class="n">cancer</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="o">==</span> <span class="mi">1</span><span class="p">]</span>
|
|
<span class="n">ax</span> <span class="o">=</span> <span class="n">axes</span><span class="o">.</span><span class="n">ravel</span><span class="p">()</span>
|
|
|
|
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">30</span><span class="p">):</span>
|
|
<span class="n">_</span><span class="p">,</span> <span class="n">bins</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">histogram</span><span class="p">(</span><span class="n">cancer</span><span class="o">.</span><span class="n">data</span><span class="p">[:,</span><span class="n">i</span><span class="p">],</span> <span class="n">bins</span> <span class="o">=</span><span class="mi">50</span><span class="p">)</span>
|
|
<span class="n">ax</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">malignant</span><span class="p">[:,</span><span class="n">i</span><span class="p">],</span> <span class="n">bins</span> <span class="o">=</span> <span class="n">bins</span><span class="p">,</span> <span class="n">alpha</span> <span class="o">=</span> <span class="mf">0.5</span><span class="p">)</span>
|
|
<span class="n">ax</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">benign</span><span class="p">[:,</span><span class="n">i</span><span class="p">],</span> <span class="n">bins</span> <span class="o">=</span> <span class="n">bins</span><span class="p">,</span> <span class="n">alpha</span> <span class="o">=</span> <span class="mf">0.5</span><span class="p">)</span>
|
|
<span class="n">ax</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="n">cancer</span><span class="o">.</span><span class="n">feature_names</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
|
|
<span class="n">ax</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">set_yticks</span><span class="p">(())</span>
|
|
<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">"Feature magnitude"</span><span class="p">)</span>
|
|
<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">"Frequency"</span><span class="p">)</span>
|
|
<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s2">"Malignant"</span><span class="p">,</span> <span class="s2">"Benign"</span><span class="p">],</span> <span class="n">loc</span> <span class="o">=</span><span class="s2">"best"</span><span class="p">)</span>
|
|
<span class="n">fig</span><span class="o">.</span><span class="n">tight_layout</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="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
|
|
<span class="n">correlation_matrix</span> <span class="o">=</span> <span class="n">cancerpd</span><span class="o">.</span><span class="n">corr</span><span class="p">()</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
|
|
<span class="c1"># use the heatmap function from seaborn to plot the correlation matrix</span>
|
|
<span class="c1"># annot = True to print the values inside the square</span>
|
|
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">15</span><span class="p">,</span><span class="mi">8</span><span class="p">))</span>
|
|
<span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="n">correlation_matrix</span><span class="p">,</span> <span class="n">annot</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>
|
|
</pre></div>
|
|
</div>
|
|
</div>
|
|
<div class="cell_output docutils container">
|
|
<img alt="_images/8bccd564c8ef11fa6a628c29733db8aa23d9dce879b7da0104d7ba24f157d2d5.png" src="_images/8bccd564c8ef11fa6a628c29733db8aa23d9dce879b7da0104d7ba24f157d2d5.png" />
|
|
<img alt="_images/43f947d18886acb70f69ec0139f7bb4e05b3971b9df3b0dab14aec06ae48e1c5.png" src="_images/43f947d18886acb70f69ec0139f7bb4e05b3971b9df3b0dab14aec06ae48e1c5.png" />
|
|
</div>
|
|
</div>
|
|
</section>
|
|
<section id="discussing-the-correlation-data">
|
|
<h2>Discussing the correlation data<a class="headerlink" href="#discussing-the-correlation-data" title="Link to this heading">#</a></h2>
|
|
<p>In the above example we note two things. In the first plot we display
|
|
the overlap of benign and malignant tumors as functions of the various
|
|
features in the Wisconsing breast cancer data set. We see that for
|
|
some of the features we can distinguish clearly the benign and
|
|
malignant cases while for other features we cannot. This can point to
|
|
us which features may be of greater interest when we wish to classify
|
|
a benign or not benign tumour.</p>
|
|
<p>In the second figure we have computed the so-called correlation
|
|
matrix, which in our case with thirty features becomes a <span class="math notranslate nohighlight">\(30\times 30\)</span>
|
|
matrix.</p>
|
|
<p>We constructed this matrix using <strong>pandas</strong> via the statements</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="n">cancerpd</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">cancer</span><span class="o">.</span><span class="n">data</span><span class="p">,</span> <span class="n">columns</span><span class="o">=</span><span class="n">cancer</span><span class="o">.</span><span class="n">feature_names</span><span class="p">)</span>
|
|
</pre></div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<p>and then</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="n">correlation_matrix</span> <span class="o">=</span> <span class="n">cancerpd</span><span class="o">.</span><span class="n">corr</span><span class="p">()</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
|
|
</pre></div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<p>Diagonalizing this matrix we can in turn say something about which
|
|
features are of relevance and which are not. This leads us to
|
|
the classical Principal Component Analysis (PCA) theorem with
|
|
applications. This will be discussed later this semester (<a class="reference external" href="https://compphysics.github.io/MachineLearning/doc/pub/week43/html/week43-bs.html">week 43</a>).</p>
|
|
</section>
|
|
<section id="other-measures-in-classification-studies-cancer-data-again">
|
|
<h2>Other measures in classification studies: Cancer Data again<a class="headerlink" href="#other-measures-in-classification-studies-cancer-data-again" title="Link to this heading">#</a></h2>
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<div class="cell docutils container">
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<div class="cell_input docutils container">
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><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>
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<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
|
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<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_breast_cancer</span>
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<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LogisticRegression</span>
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|
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<span class="c1"># Load the data</span>
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<span class="n">cancer</span> <span class="o">=</span> <span class="n">load_breast_cancer</span><span class="p">()</span>
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<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">cancer</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>
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<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>
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<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>
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<span class="c1"># Logistic Regression</span>
|
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<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>
|
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<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>
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<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>
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|
<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>
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<span class="nb">print</span><span class="p">(</span><span class="n">accuracy</span><span class="p">)</span>
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<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>
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|
|
<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>
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|
<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>
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</pre></div>
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</div>
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</div>
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<div class="cell_output docutils container">
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(426, 30)
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(143, 30)
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</pre></div>
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</div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[1. 0.86666667 1. 0.85714286 1. 0.85714286
|
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1. 0.92857143 0.92857143 1. ]
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Test set accuracy with Logistic Regression: 0.94
|
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</pre></div>
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</div>
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<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/linear_model/_logistic.py:460: ConvergenceWarning: lbfgs failed to converge (status=1):
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|
STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.
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|
|
|
Increase the number of iterations (max_iter) or scale the data as shown in:
|
|
https://scikit-learn.org/stable/modules/preprocessing.html
|
|
Please also refer to the documentation for alternative solver options:
|
|
https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression
|
|
n_iter_i = _check_optimize_result(
|
|
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/linear_model/_logistic.py:460: ConvergenceWarning: lbfgs failed to converge (status=1):
|
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STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.
|
|
|
|
Increase the number of iterations (max_iter) or scale the data as shown in:
|
|
https://scikit-learn.org/stable/modules/preprocessing.html
|
|
Please also refer to the documentation for alternative solver options:
|
|
https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression
|
|
n_iter_i = _check_optimize_result(
|
|
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/linear_model/_logistic.py:460: ConvergenceWarning: lbfgs failed to converge (status=1):
|
|
STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.
|
|
|
|
Increase the number of iterations (max_iter) or scale the data as shown in:
|
|
https://scikit-learn.org/stable/modules/preprocessing.html
|
|
Please also refer to the documentation for alternative solver options:
|
|
https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression
|
|
n_iter_i = _check_optimize_result(
|
|
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/linear_model/_logistic.py:460: ConvergenceWarning: lbfgs failed to converge (status=1):
|
|
STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.
|
|
|
|
Increase the number of iterations (max_iter) or scale the data as shown in:
|
|
https://scikit-learn.org/stable/modules/preprocessing.html
|
|
Please also refer to the documentation for alternative solver options:
|
|
https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression
|
|
n_iter_i = _check_optimize_result(
|
|
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/linear_model/_logistic.py:460: ConvergenceWarning: lbfgs failed to converge (status=1):
|
|
STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.
|
|
|
|
Increase the number of iterations (max_iter) or scale the data as shown in:
|
|
https://scikit-learn.org/stable/modules/preprocessing.html
|
|
Please also refer to the documentation for alternative solver options:
|
|
https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression
|
|
n_iter_i = _check_optimize_result(
|
|
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/linear_model/_logistic.py:460: ConvergenceWarning: lbfgs failed to converge (status=1):
|
|
STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.
|
|
|
|
Increase the number of iterations (max_iter) or scale the data as shown in:
|
|
https://scikit-learn.org/stable/modules/preprocessing.html
|
|
Please also refer to the documentation for alternative solver options:
|
|
https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression
|
|
n_iter_i = _check_optimize_result(
|
|
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/linear_model/_logistic.py:460: ConvergenceWarning: lbfgs failed to converge (status=1):
|
|
STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.
|
|
|
|
Increase the number of iterations (max_iter) or scale the data as shown in:
|
|
https://scikit-learn.org/stable/modules/preprocessing.html
|
|
Please also refer to the documentation for alternative solver options:
|
|
https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression
|
|
n_iter_i = _check_optimize_result(
|
|
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/linear_model/_logistic.py:460: ConvergenceWarning: lbfgs failed to converge (status=1):
|
|
STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.
|
|
|
|
Increase the number of iterations (max_iter) or scale the data as shown in:
|
|
https://scikit-learn.org/stable/modules/preprocessing.html
|
|
Please also refer to the documentation for alternative solver options:
|
|
https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression
|
|
n_iter_i = _check_optimize_result(
|
|
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/linear_model/_logistic.py:460: ConvergenceWarning: lbfgs failed to converge (status=1):
|
|
STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.
|
|
|
|
Increase the number of iterations (max_iter) or scale the data as shown in:
|
|
https://scikit-learn.org/stable/modules/preprocessing.html
|
|
Please also refer to the documentation for alternative solver options:
|
|
https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression
|
|
n_iter_i = _check_optimize_result(
|
|
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/linear_model/_logistic.py:460: ConvergenceWarning: lbfgs failed to converge (status=1):
|
|
STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.
|
|
|
|
Increase the number of iterations (max_iter) or scale the data as shown in:
|
|
https://scikit-learn.org/stable/modules/preprocessing.html
|
|
Please also refer to the documentation for alternative solver options:
|
|
https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression
|
|
n_iter_i = _check_optimize_result(
|
|
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/linear_model/_logistic.py:460: ConvergenceWarning: lbfgs failed to converge (status=1):
|
|
STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.
|
|
|
|
Increase the number of iterations (max_iter) or scale the data as shown in:
|
|
https://scikit-learn.org/stable/modules/preprocessing.html
|
|
Please also refer to the documentation for alternative solver options:
|
|
https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression
|
|
n_iter_i = _check_optimize_result(
|
|
</pre></div>
|
|
</div>
|
|
<img alt="_images/44621123676c9bf6a814eacf9cc3e939fd5b4e9a0a03102c83201c84108e1d15.png" src="_images/44621123676c9bf6a814eacf9cc3e939fd5b4e9a0a03102c83201c84108e1d15.png" />
|
|
<img alt="_images/1dee91854541648a44f30f91519de2b4da460b60c30b88b43a9b149fdd24bdde.png" src="_images/1dee91854541648a44f30f91519de2b4da460b60c30b88b43a9b149fdd24bdde.png" />
|
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<img alt="_images/d92d6345dffc138099ca35c3e1668aec61e7f6519f2e607568a66553f9f47fda.png" src="_images/d92d6345dffc138099ca35c3e1668aec61e7f6519f2e607568a66553f9f47fda.png" />
|
|
</div>
|
|
</div>
|
|
</section>
|
|
<section id="gradient-descent-and-logistic-regression">
|
|
<h2>Gradient descent and Logistic regression<a class="headerlink" href="#gradient-descent-and-logistic-regression" title="Link to this heading">#</a></h2>
|
|
<p>We complete these examples by adding a simple code for
|
|
Logistic regression. Note the more general approach with a class for
|
|
the method. Here we use a so-called <strong>AND</strong> gate for our data set.</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">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
|
<span class="k">class</span> <span class="nc">LogisticRegression</span><span class="p">:</span>
|
|
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">learning_rate</span><span class="o">=</span><span class="mf">0.01</span><span class="p">,</span> <span class="n">num_iterations</span><span class="o">=</span><span class="mi">1000</span><span class="p">):</span>
|
|
<span class="bp">self</span><span class="o">.</span><span class="n">learning_rate</span> <span class="o">=</span> <span class="n">learning_rate</span>
|
|
<span class="bp">self</span><span class="o">.</span><span class="n">num_iterations</span> <span class="o">=</span> <span class="n">num_iterations</span>
|
|
<span class="bp">self</span><span class="o">.</span><span class="n">beta_logreg</span> <span class="o">=</span> <span class="kc">None</span>
|
|
<span class="k">def</span> <span class="nf">sigmoid</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
|
|
<span class="k">return</span> <span class="mi">1</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="p">))</span>
|
|
<span class="k">def</span> <span class="nf">GDfit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
|
|
<span class="n">n_data</span><span class="p">,</span> <span class="n">num_features</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span>
|
|
<span class="bp">self</span><span class="o">.</span><span class="n">beta_logreg</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">num_features</span><span class="p">)</span>
|
|
<span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">num_iterations</span><span class="p">):</span>
|
|
<span class="n">linear_model</span> <span class="o">=</span> <span class="n">X</span> <span class="o">@</span> <span class="bp">self</span><span class="o">.</span><span class="n">beta_logreg</span>
|
|
<span class="n">y_predicted</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sigmoid</span><span class="p">(</span><span class="n">linear_model</span><span class="p">)</span>
|
|
<span class="c1"># Gradient calculation</span>
|
|
<span class="n">gradient</span> <span class="o">=</span> <span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="p">(</span><span class="n">y_predicted</span> <span class="o">-</span> <span class="n">y</span><span class="p">))</span><span class="o">/</span><span class="n">n_data</span>
|
|
<span class="c1"># Update beta_logreg</span>
|
|
<span class="bp">self</span><span class="o">.</span><span class="n">beta_logreg</span> <span class="o">-=</span> <span class="bp">self</span><span class="o">.</span><span class="n">learning_rate</span><span class="o">*</span><span class="n">gradient</span>
|
|
<span class="k">def</span> <span class="nf">predict</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
|
|
<span class="n">linear_model</span> <span class="o">=</span> <span class="n">X</span> <span class="o">@</span> <span class="bp">self</span><span class="o">.</span><span class="n">beta_logreg</span>
|
|
<span class="n">y_predicted</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sigmoid</span><span class="p">(</span><span class="n">linear_model</span><span class="p">)</span>
|
|
<span class="k">return</span> <span class="p">[</span><span class="mi">1</span> <span class="k">if</span> <span class="n">i</span> <span class="o">>=</span> <span class="mf">0.5</span> <span class="k">else</span> <span class="mi">0</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">y_predicted</span><span class="p">]</span>
|
|
<span class="c1"># Example usage</span>
|
|
<span class="k">if</span> <span class="vm">__name__</span> <span class="o">==</span> <span class="s2">"__main__"</span><span class="p">:</span>
|
|
<span class="c1"># Sample data</span>
|
|
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
|
|
<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span> <span class="c1"># This is an AND gate</span>
|
|
<span class="n">model</span> <span class="o">=</span> <span class="n">LogisticRegression</span><span class="p">(</span><span class="n">learning_rate</span><span class="o">=</span><span class="mf">0.01</span><span class="p">,</span> <span class="n">num_iterations</span><span class="o">=</span><span class="mi">1000</span><span class="p">)</span>
|
|
<span class="n">model</span><span class="o">.</span><span class="n">GDfit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
|
|
<span class="n">predictions</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
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<span class="nb">print</span><span class="p">(</span><span class="s2">"Predictions:"</span><span class="p">,</span> <span class="n">predictions</span><span class="p">)</span>
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</pre></div>
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<div class="cell_output docutils container">
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Predictions: [1, 1, 1, 1]
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