Morten Hjorth-Jensen
1199 lines
95 KiB
HTML
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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 current active"><a class="current reference internal" href="#">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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<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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<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"><a class="reference internal" href="additionweek42.html">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="week47.html">Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek47.html">Exercise week 47</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week48.html">Week 48: Gradient boosting and summary of course</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek48.html">Exercises week 48</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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</ul>
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</div>
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<button onclick="toggleFullScreen()"
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class="btn btn-sm btn-fullscreen-button"
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title="Fullscreen mode"
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data-bs-placement="bottom" data-bs-toggle="tooltip"
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>
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<span class="btn__icon-container">
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<i class="fas fa-expand"></i>
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</span>
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</button>
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<button class="btn btn-sm nav-link pst-navbar-icon theme-switch-button pst-js-only" aria-label="Color mode" data-bs-title="Color mode" data-bs-placement="bottom" data-bs-toggle="tooltip">
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<i class="theme-switch fa-solid fa-sun fa-lg" data-mode="light" title="Light"></i>
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<i class="theme-switch fa-solid fa-moon fa-lg" data-mode="dark" title="Dark"></i>
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<i class="theme-switch fa-solid fa-circle-half-stroke fa-lg" data-mode="auto" title="System Settings"></i>
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</button>
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<button class="btn btn-sm pst-navbar-icon search-button search-button__button pst-js-only" title="Search" aria-label="Search" data-bs-placement="bottom" data-bs-toggle="tooltip">
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<i class="fa-solid fa-magnifying-glass fa-lg"></i>
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</button>
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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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<span class="fa-solid fa-list"></span>
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</div>
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</div>
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|
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|
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|
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<div id="jb-print-docs-body" class="onlyprint">
|
||
<h1>Logistic Regression</h1>
|
||
<!-- 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>
|
||
</div>
|
||
<nav aria-label="Page">
|
||
<ul class="visible nav section-nav flex-column">
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">6.1. Logistic Regression</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#basics">6.2. Basics</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-logistic-function">6.3. The logistic function</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#examples-of-likelihood-functions-used-in-logistic-regression-and-neural-networks">6.4. Examples of likelihood functions used in logistic regression and neural networks</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#wisconsin-cancer-data">6.5. Wisconsin Cancer Data</a></li>
|
||
</ul>
|
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</nav>
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</div>
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</div>
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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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|
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<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)
|
||
doconce format html chapter4.do.txt --><section class="tex2jax_ignore mathjax_ignore" id="logistic-regression">
|
||
<h1><span class="section-number">6. </span>Logistic Regression<a class="headerlink" href="#logistic-regression" title="Link to this heading">#</a></h1>
|
||
<section id="id1">
|
||
<h2><span class="section-number">6.1. </span>Logistic Regression<a class="headerlink" href="#id1" title="Link to this heading">#</a></h2>
|
||
<p>In linear regression our main interest was centered on learning the
|
||
coefficients of a functional fit (say a polynomial) in order to be
|
||
able to predict the response of a continuous variable on some unseen
|
||
data. The fit to the continuous variable <span class="math notranslate nohighlight">\(y_i\)</span> is based on some
|
||
independent variables <span class="math notranslate nohighlight">\(x_i\)</span>. Linear regression resulted in
|
||
analytical expressions for standard ordinary Least Squares or Ridge
|
||
regression (in terms of matrices to invert) for several quantities,
|
||
ranging from the variance and thereby the confidence intervals of the
|
||
optimal parameters <span class="math notranslate nohighlight">\(\hat{\beta}\)</span> to the mean squared error. If we can invert
|
||
the product of the design matrices, linear regression gives then a
|
||
simple recipe for fitting our data.</p>
|
||
<p>Classification problems, however, are concerned with outcomes taking
|
||
the form of discrete variables (i.e. categories). We may for example,
|
||
on the basis of DNA sequencing for a number of patients, like to find
|
||
out which mutations are important for a certain disease; or based on
|
||
scans of various patients’ brains, figure out if there is a tumor or
|
||
not; or given a specific physical system, we’d like to identify its
|
||
state, say whether it is an ordered or disordered system (typical
|
||
situation in solid state physics); or classify the status of a
|
||
patient, whether she/he has a stroke or not and many other similar
|
||
situations.</p>
|
||
<p>The most common situation we encounter when we apply logistic
|
||
regression is that of two possible outcomes, normally denoted as a
|
||
binary outcome, true or false, positive or negative, success or
|
||
failure etc.</p>
|
||
<p>Logistic regression will also serve as our stepping stone towards
|
||
neural network algorithms and supervised deep learning. For logistic
|
||
learning, the minimization of the cost function leads to a non-linear
|
||
equation in the parameters <span class="math notranslate nohighlight">\(\hat{\beta}\)</span>. The optimization of the
|
||
problem calls therefore for minimization algorithms. This forms the
|
||
bottle neck of all machine learning algorithms, namely how to find
|
||
reliable minima of a multi-variable function. This leads us to the
|
||
family of gradient descent methods. The latter are the working horses
|
||
of basically all modern machine learning algorithms.</p>
|
||
<p>We note also that many of the topics discussed here on logistic
|
||
regression are also commonly used in modern supervised Deep Learning
|
||
models, as we will see later.</p>
|
||
</section>
|
||
<section id="basics">
|
||
<h2><span class="section-number">6.2. </span>Basics<a class="headerlink" href="#basics" title="Link to this heading">#</a></h2>
|
||
<p>We consider the case where the dependent variables, also called the
|
||
responses or the outcomes, <span class="math notranslate nohighlight">\(y_i\)</span> are discrete and only take values
|
||
from <span class="math notranslate nohighlight">\(k=0,\dots,K-1\)</span> (i.e. <span class="math notranslate nohighlight">\(K\)</span> classes).</p>
|
||
<p>The goal is to predict the
|
||
output classes from the design matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\in\mathbb{R}^{n\times p}\)</span>
|
||
made of <span class="math notranslate nohighlight">\(n\)</span> samples, each of which carries <span class="math notranslate nohighlight">\(p\)</span> features or predictors. The
|
||
primary goal is to identify the classes to which new unseen samples
|
||
belong.</p>
|
||
<p>Let us specialize to the case of two classes only, with outputs
|
||
<span class="math notranslate nohighlight">\(y_i=0\)</span> and <span class="math notranslate nohighlight">\(y_i=1\)</span>. Our outcomes could represent the status of a
|
||
credit card user that could default or not on her/his credit card
|
||
debt. That is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
y_i = \begin{bmatrix} 0 & \mathrm{no}\\ 1 & \mathrm{yes} \end{bmatrix}.
|
||
\end{split}\]</div>
|
||
<p>Before moving to the logistic model, let us try to use our linear
|
||
regression model to classify these two outcomes. We could for example
|
||
fit a linear model to the default case if <span class="math notranslate nohighlight">\(y_i > 0.5\)</span> and the no
|
||
default case <span class="math notranslate nohighlight">\(y_i \leq 0.5\)</span>.</p>
|
||
<p>We would then have our
|
||
weighted linear combination, namely</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="_auto1"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{equation}
|
||
\boldsymbol{y} = \boldsymbol{X}^T\boldsymbol{\beta} + \boldsymbol{\epsilon},
|
||
\label{_auto1} \tag{1}
|
||
\end{equation}
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> is a vector representing the possible outcomes, <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> is our
|
||
<span class="math notranslate nohighlight">\(n\times p\)</span> design matrix and <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> represents our estimators/predictors.</p>
|
||
<p>The main problem with our function is that it takes values on the
|
||
entire real axis. In the case of logistic regression, however, the
|
||
labels <span class="math notranslate nohighlight">\(y_i\)</span> are discrete variables. A typical example is the credit
|
||
card data discussed below here, where we can set the state of
|
||
defaulting the debt to <span class="math notranslate nohighlight">\(y_i=1\)</span> and not to <span class="math notranslate nohighlight">\(y_i=0\)</span> for one the persons
|
||
in the data set (see the full example below).</p>
|
||
<p>One simple way to get a discrete output is to have sign
|
||
functions that map the output of a linear regressor to values <span class="math notranslate nohighlight">\(\{0,1\}\)</span>,
|
||
<span class="math notranslate nohighlight">\(f(s_i)=sign(s_i)=1\)</span> if <span class="math notranslate nohighlight">\(s_i\ge 0\)</span> and 0 if otherwise.
|
||
We will encounter this model in our first demonstration of neural networks. Historically it is called the <code class="docutils literal notranslate"><span class="pre">perceptron"</span> <span class="pre">model</span> <span class="pre">in</span> <span class="pre">the</span> <span class="pre">machine</span> <span class="pre">learning</span> <span class="pre">literature.</span> <span class="pre">This</span> <span class="pre">model</span> <span class="pre">is</span> <span class="pre">extremely</span> <span class="pre">simple.</span> <span class="pre">However,</span> <span class="pre">in</span> <span class="pre">many</span> <span class="pre">cases</span> <span class="pre">it</span> <span class="pre">is</span> <span class="pre">more</span> <span class="pre">favorable</span> <span class="pre">to</span> <span class="pre">use</span> <span class="pre">a</span> </code>soft” classifier that outputs
|
||
the probability of a given category. This leads us to the logistic function.</p>
|
||
<p>The following example on data for coronary heart disease (CHD) as function of age may serve as an illustration. In the code here we read and plot whether a person has had CHD (output = 1) or not (output = 0). This ouput is plotted the person’s against age. Clearly, the figure shows that attempting to make a standard linear regression fit may not be very meaningful.</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="c1"># Common imports</span>
|
||
<span class="kn">import</span> <span class="nn">os</span>
|
||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</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">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LinearRegression</span><span class="p">,</span> <span class="n">Ridge</span><span class="p">,</span> <span class="n">Lasso</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.utils</span> <span class="kn">import</span> <span class="n">resample</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">mean_squared_error</span>
|
||
<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">display</span>
|
||
<span class="kn">from</span> <span class="nn">pylab</span> <span class="kn">import</span> <span class="n">plt</span><span class="p">,</span> <span class="n">mpl</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">style</span><span class="o">.</span><span class="n">use</span><span class="p">(</span><span class="s1">'seaborn'</span><span class="p">)</span>
|
||
<span class="n">mpl</span><span class="o">.</span><span class="n">rcParams</span><span class="p">[</span><span class="s1">'font.family'</span><span class="p">]</span> <span class="o">=</span> <span class="s1">'serif'</span>
|
||
|
||
<span class="c1"># Where to save the figures and data files</span>
|
||
<span class="n">PROJECT_ROOT_DIR</span> <span class="o">=</span> <span class="s2">"Results"</span>
|
||
<span class="n">FIGURE_ID</span> <span class="o">=</span> <span class="s2">"Results/FigureFiles"</span>
|
||
<span class="n">DATA_ID</span> <span class="o">=</span> <span class="s2">"DataFiles/"</span>
|
||
|
||
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">PROJECT_ROOT_DIR</span><span class="p">):</span>
|
||
<span class="n">os</span><span class="o">.</span><span class="n">mkdir</span><span class="p">(</span><span class="n">PROJECT_ROOT_DIR</span><span class="p">)</span>
|
||
|
||
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">):</span>
|
||
<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">)</span>
|
||
|
||
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">):</span>
|
||
<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">,</span> <span class="n">fig_id</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">data_path</span><span class="p">(</span><span class="n">dat_id</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">,</span> <span class="n">dat_id</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">save_fig</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">savefig</span><span class="p">(</span><span class="n">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">)</span> <span class="o">+</span> <span class="s2">".png"</span><span class="p">,</span> <span class="nb">format</span><span class="o">=</span><span class="s1">'png'</span><span class="p">)</span>
|
||
|
||
<span class="n">infile</span> <span class="o">=</span> <span class="nb">open</span><span class="p">(</span><span class="n">data_path</span><span class="p">(</span><span class="s2">"chddata.csv"</span><span class="p">),</span><span class="s1">'r'</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Read the chd data as csv file and organize the data into arrays with age group, age, and chd</span>
|
||
<span class="n">chd</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="n">infile</span><span class="p">,</span> <span class="n">names</span><span class="o">=</span><span class="p">(</span><span class="s1">'ID'</span><span class="p">,</span> <span class="s1">'Age'</span><span class="p">,</span> <span class="s1">'Agegroup'</span><span class="p">,</span> <span class="s1">'CHD'</span><span class="p">))</span>
|
||
<span class="n">chd</span><span class="o">.</span><span class="n">columns</span> <span class="o">=</span> <span class="p">[</span><span class="s1">'ID'</span><span class="p">,</span> <span class="s1">'Age'</span><span class="p">,</span> <span class="s1">'Agegroup'</span><span class="p">,</span> <span class="s1">'CHD'</span><span class="p">]</span>
|
||
<span class="n">output</span> <span class="o">=</span> <span class="n">chd</span><span class="p">[</span><span class="s1">'CHD'</span><span class="p">]</span>
|
||
<span class="n">age</span> <span class="o">=</span> <span class="n">chd</span><span class="p">[</span><span class="s1">'Age'</span><span class="p">]</span>
|
||
<span class="n">agegroup</span> <span class="o">=</span> <span class="n">chd</span><span class="p">[</span><span class="s1">'Agegroup'</span><span class="p">]</span>
|
||
<span class="n">numberID</span> <span class="o">=</span> <span class="n">chd</span><span class="p">[</span><span class="s1">'ID'</span><span class="p">]</span>
|
||
<span class="n">display</span><span class="p">(</span><span class="n">chd</span><span class="p">)</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">age</span><span class="p">,</span> <span class="n">output</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">'o'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="mi">18</span><span class="p">,</span><span class="mf">70.0</span><span class="p">,</span><span class="o">-</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">'Age'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">'CHD'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="sa">r</span><span class="s1">'Age distribution and Coronary heart disease'</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">
|
||
<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="gt">---------------------------------------------------------------------------</span>
|
||
<span class="ne">FileNotFoundError</span><span class="g g-Whitespace"> </span>Traceback (most recent call last)
|
||
<span class="nn">File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/matplotlib/style/core.py:137,</span> in <span class="ni">use</span><span class="nt">(style)</span>
|
||
<span class="g g-Whitespace"> </span><span class="mi">136</span> <span class="k">try</span><span class="p">:</span>
|
||
<span class="ne">--> </span><span class="mi">137</span> <span class="n">style</span> <span class="o">=</span> <span class="n">_rc_params_in_file</span><span class="p">(</span><span class="n">style</span><span class="p">)</span>
|
||
<span class="g g-Whitespace"> </span><span class="mi">138</span> <span class="k">except</span> <span class="ne">OSError</span> <span class="k">as</span> <span class="n">err</span><span class="p">:</span>
|
||
|
||
<span class="nn">File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/matplotlib/__init__.py:870,</span> in <span class="ni">_rc_params_in_file</span><span class="nt">(fname, transform, fail_on_error)</span>
|
||
<span class="g g-Whitespace"> </span><span class="mi">869</span> <span class="n">rc_temp</span> <span class="o">=</span> <span class="p">{}</span>
|
||
<span class="ne">--> </span><span class="mi">870</span> <span class="k">with</span> <span class="n">_open_file_or_url</span><span class="p">(</span><span class="n">fname</span><span class="p">)</span> <span class="k">as</span> <span class="n">fd</span><span class="p">:</span>
|
||
<span class="g g-Whitespace"> </span><span class="mi">871</span> <span class="k">try</span><span class="p">:</span>
|
||
|
||
<span class="nn">File ~/miniforge3/envs/myenv/lib/python3.9/contextlib.py:119,</span> in <span class="ni">_GeneratorContextManager.__enter__</span><span class="nt">(self)</span>
|
||
<span class="g g-Whitespace"> </span><span class="mi">118</span> <span class="k">try</span><span class="p">:</span>
|
||
<span class="ne">--> </span><span class="mi">119</span> <span class="k">return</span> <span class="nb">next</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gen</span><span class="p">)</span>
|
||
<span class="g g-Whitespace"> </span><span class="mi">120</span> <span class="k">except</span> <span class="ne">StopIteration</span><span class="p">:</span>
|
||
|
||
<span class="nn">File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/matplotlib/__init__.py:847,</span> in <span class="ni">_open_file_or_url</span><span class="nt">(fname)</span>
|
||
<span class="g g-Whitespace"> </span><span class="mi">846</span> <span class="n">fname</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">expanduser</span><span class="p">(</span><span class="n">fname</span><span class="p">)</span>
|
||
<span class="ne">--> </span><span class="mi">847</span> <span class="k">with</span> <span class="nb">open</span><span class="p">(</span><span class="n">fname</span><span class="p">,</span> <span class="n">encoding</span><span class="o">=</span><span class="s1">'utf-8'</span><span class="p">)</span> <span class="k">as</span> <span class="n">f</span><span class="p">:</span>
|
||
<span class="g g-Whitespace"> </span><span class="mi">848</span> <span class="k">yield</span> <span class="n">f</span>
|
||
|
||
<span class="ne">FileNotFoundError</span>: [Errno 2] No such file or directory: 'seaborn'
|
||
|
||
<span class="n">The</span> <span class="n">above</span> <span class="n">exception</span> <span class="n">was</span> <span class="n">the</span> <span class="n">direct</span> <span class="n">cause</span> <span class="n">of</span> <span class="n">the</span> <span class="n">following</span> <span class="n">exception</span><span class="p">:</span>
|
||
|
||
<span class="ne">OSError</span><span class="g g-Whitespace"> </span>Traceback (most recent call last)
|
||
<span class="n">Cell</span> <span class="n">In</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">line</span> <span class="mi">14</span>
|
||
<span class="g g-Whitespace"> </span><span class="mi">12</span> <span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">display</span>
|
||
<span class="g g-Whitespace"> </span><span class="mi">13</span> <span class="kn">from</span> <span class="nn">pylab</span> <span class="kn">import</span> <span class="n">plt</span><span class="p">,</span> <span class="n">mpl</span>
|
||
<span class="ne">---> </span><span class="mi">14</span> <span class="n">plt</span><span class="o">.</span><span class="n">style</span><span class="o">.</span><span class="n">use</span><span class="p">(</span><span class="s1">'seaborn'</span><span class="p">)</span>
|
||
<span class="g g-Whitespace"> </span><span class="mi">15</span> <span class="n">mpl</span><span class="o">.</span><span class="n">rcParams</span><span class="p">[</span><span class="s1">'font.family'</span><span class="p">]</span> <span class="o">=</span> <span class="s1">'serif'</span>
|
||
<span class="g g-Whitespace"> </span><span class="mi">17</span> <span class="c1"># Where to save the figures and data files</span>
|
||
|
||
<span class="nn">File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/matplotlib/style/core.py:139,</span> in <span class="ni">use</span><span class="nt">(style)</span>
|
||
<span class="g g-Whitespace"> </span><span class="mi">137</span> <span class="n">style</span> <span class="o">=</span> <span class="n">_rc_params_in_file</span><span class="p">(</span><span class="n">style</span><span class="p">)</span>
|
||
<span class="g g-Whitespace"> </span><span class="mi">138</span> <span class="k">except</span> <span class="ne">OSError</span> <span class="k">as</span> <span class="n">err</span><span class="p">:</span>
|
||
<span class="ne">--> </span><span class="mi">139</span> <span class="k">raise</span> <span class="ne">OSError</span><span class="p">(</span>
|
||
<span class="g g-Whitespace"> </span><span class="mi">140</span> <span class="sa">f</span><span class="s2">"</span><span class="si">{</span><span class="n">style</span><span class="si">!r}</span><span class="s2"> is not a valid package style, path of style "</span>
|
||
<span class="g g-Whitespace"> </span><span class="mi">141</span> <span class="sa">f</span><span class="s2">"file, URL of style file, or library style name (library "</span>
|
||
<span class="g g-Whitespace"> </span><span class="mi">142</span> <span class="sa">f</span><span class="s2">"styles are listed in `style.available`)"</span><span class="p">)</span> <span class="kn">from</span> <span class="nn">err</span>
|
||
<span class="g g-Whitespace"> </span><span class="mi">143</span> <span class="n">filtered</span> <span class="o">=</span> <span class="p">{}</span>
|
||
<span class="nn"> 144 for k</span> in <span class="ni">style: # don't trigger RcParams.__getitem__</span><span class="nt">('backend')</span>
|
||
|
||
<span class="ne">OSError</span>: 'seaborn' is not a valid package style, path of style file, URL of style file, or library style name (library styles are listed in `style.available`)
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>What we could attempt however is to plot the mean value for each group.</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">agegroupmean</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="mf">0.1</span><span class="p">,</span> <span class="mf">0.133</span><span class="p">,</span> <span class="mf">0.250</span><span class="p">,</span> <span class="mf">0.333</span><span class="p">,</span> <span class="mf">0.462</span><span class="p">,</span> <span class="mf">0.625</span><span class="p">,</span> <span class="mf">0.765</span><span class="p">,</span> <span class="mf">0.800</span><span class="p">])</span>
|
||
<span class="n">group</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">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">group</span><span class="p">,</span> <span class="n">agegroupmean</span><span class="p">,</span> <span class="s2">"r-"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">'Age group'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">'CHD mean values'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="sa">r</span><span class="s1">'Mean values for each age group'</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>
|
||
<p>We are now trying to find a function <span class="math notranslate nohighlight">\(f(y\vert x)\)</span>, that is a function which gives us an expected value for the output <span class="math notranslate nohighlight">\(y\)</span> with a given input <span class="math notranslate nohighlight">\(x\)</span>.
|
||
In standard linear regression with a linear dependence on <span class="math notranslate nohighlight">\(x\)</span>, we would write this in terms of our model</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
f(y_i\vert x_i)=\beta_0+\beta_1 x_i.
|
||
\]</div>
|
||
<p>This expression implies however that <span class="math notranslate nohighlight">\(f(y_i\vert x_i)\)</span> could take any
|
||
value from minus infinity to plus infinity. If we however let
|
||
<span class="math notranslate nohighlight">\(f(y\vert y)\)</span> be represented by the mean value, the above example
|
||
shows us that we can constrain the function to take values between
|
||
zero and one, that is we have <span class="math notranslate nohighlight">\(0 \le f(y_i\vert x_i) \le 1\)</span>. Looking
|
||
at our last curve we see also that it has an S-shaped form. This leads
|
||
us to a very popular model for the function <span class="math notranslate nohighlight">\(f\)</span>, namely the so-called
|
||
Sigmoid function or logistic model. We will consider this function as
|
||
representing the probability for finding a value of <span class="math notranslate nohighlight">\(y_i\)</span> with a given
|
||
<span class="math notranslate nohighlight">\(x_i\)</span>.</p>
|
||
</section>
|
||
<section id="the-logistic-function">
|
||
<h2><span class="section-number">6.3. </span>The logistic function<a class="headerlink" href="#the-logistic-function" title="Link to this heading">#</a></h2>
|
||
<p>Another widely studied model, is the so-called
|
||
perceptron model, which is an example of a “hard classification” model. We
|
||
will encounter this model when we discuss neural networks as
|
||
well. Each datapoint is deterministically assigned to a category (i.e
|
||
<span class="math notranslate nohighlight">\(y_i=0\)</span> or <span class="math notranslate nohighlight">\(y_i=1\)</span>). In many cases, and the coronary heart disease data forms one of many such examples, it is favorable to have a “soft”
|
||
classifier that outputs the probability of a given category rather
|
||
than a single value. For example, given <span class="math notranslate nohighlight">\(x_i\)</span>, the classifier
|
||
outputs the probability of being in a category <span class="math notranslate nohighlight">\(k\)</span>. Logistic regression
|
||
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>
|
||
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>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}.
|
||
\]</div>
|
||
<p>Note that <span class="math notranslate nohighlight">\(1-p(t)= p(-t)\)</span>.</p>
|
||
</section>
|
||
<section id="examples-of-likelihood-functions-used-in-logistic-regression-and-neural-networks">
|
||
<h2><span class="section-number">6.4. </span>Examples of likelihood functions used in logistic regression and neural networks<a class="headerlink" href="#examples-of-likelihood-functions-used-in-logistic-regression-and-neural-networks" title="Link to this heading">#</a></h2>
|
||
<p>The following code plots the logistic function, the step function and other functions we will encounter from here and on.</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="sd">"""The sigmoid function (or the logistic curve) is a</span>
|
||
<span class="sd">function that takes any real number, z, and outputs a number (0,1).</span>
|
||
<span class="sd">It is useful in neural networks for assigning weights on a relative scale.</span>
|
||
<span class="sd">The value z is the weighted sum of parameters involved in the learning algorithm."""</span>
|
||
|
||
<span class="kn">import</span> <span class="nn">numpy</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">math</span> <span class="k">as</span> <span class="nn">mt</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">.1</span><span class="p">)</span>
|
||
<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>
|
||
<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>
|
||
|
||
<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">sigma</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">0.1</span><span class="p">,</span> <span class="mf">1.1</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">'sigmoid function'</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="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>
|
||
<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>
|
||
|
||
<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">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>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
|
||
<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>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<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>
|
||
<p>In order to define the total likelihood for all possible outcomes from a<br />
|
||
dataset <span class="math notranslate nohighlight">\(\mathcal{D}=\{(y_i,x_i)\}\)</span>, with the binary labels
|
||
<span class="math notranslate nohighlight">\(y_i\in\{0,1\}\)</span> and where the data points are drawn independently, we use the so-called <a class="reference external" href="https://en.wikipedia.org/wiki/Maximum_likelihood_estimation">Maximum Likelihood Estimation</a> (MLE) principle.
|
||
We aim thus at maximizing
|
||
the probability of seeing the observed data. We can then approximate the
|
||
likelihood in terms of the product of the individual probabilities of a specific outcome <span class="math notranslate nohighlight">\(y_i\)</span>, that is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\begin{align*}
|
||
P(\mathcal{D}|\boldsymbol{\beta})& = \prod_{i=1}^n \left[p(y_i=1|x_i,\boldsymbol{\beta})\right]^{y_i}\left[1-p(y_i=1|x_i,\boldsymbol{\beta}))\right]^{1-y_i}\nonumber \\
|
||
\end{align*}
|
||
\end{split}\]</div>
|
||
<p>from which we obtain the log-likelihood and our <strong>cost/loss</strong> function</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathcal{C}(\boldsymbol{\beta}) = \sum_{i=1}^n \left( y_i\log{p(y_i=1|x_i,\boldsymbol{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\boldsymbol{\beta}))\right]\right).
|
||
\]</div>
|
||
<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>
|
||
<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>
|
||
<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>
|
||
<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>
|
||
<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>
|
||
<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. Our earlier discussions were all specialized to
|
||
the case with two classes only. It is easy to see from the above that
|
||
what we derived earlier is compatible with these equations.</p>
|
||
<p>To find the optimal parameters we would typically use a gradient
|
||
descent method. Newton’s method and gradient descent methods are
|
||
discussed in the material on <a class="reference external" href="https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html">optimization
|
||
methods</a>.</p>
|
||
</section>
|
||
<section id="wisconsin-cancer-data">
|
||
<h2><span class="section-number">6.5. </span>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>
|
||
<span class="c1">#now scale the data</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">StandardScaler</span>
|
||
<span class="n">scaler</span> <span class="o">=</span> <span class="n">StandardScaler</span><span class="p">()</span>
|
||
<span class="n">scaler</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">X_train_scaled</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
|
||
<span class="n">X_test_scaled</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||
<span class="c1"># Logistic Regression</span>
|
||
<span class="n">logreg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train_scaled</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 Logistic Regression with scaled data: </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_scaled</span><span class="p">,</span><span class="n">y_test</span><span class="p">)))</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<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>
|
||
<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>
|
||
<p>Here we present a further way to present our results in terms of a so-called <strong>confusion matrix</strong>, the cumulative gain and the <strong>ROC</strong> curve.
|
||
This way of displaying our data are based upon different ways to classify our possible outcomes. Before we proceed we need some definitions.</p>
|
||
<ol class="arabic simple">
|
||
<li><p><strong>TP</strong>: true positive or in other words, something equivalent with a proper classification</p></li>
|
||
<li><p><strong>TN</strong>: true negative, which is equivalent with a correct rejection</p></li>
|
||
<li><p><strong>FP</strong>: false positive, or in simpler words something that is equivalent with a false alarm</p></li>
|
||
<li><p><strong>FN</strong>: false negative, which is mean to be equivalent with a miss.</p></li>
|
||
</ol>
|
||
<p>The total data set is then the sum of the true positive and true negative targets or outputs, labeled by <span class="math notranslate nohighlight">\(n\)</span>.
|
||
Based on this we can then define the accuracy score as the sum of correctly predicted <strong>TP</strong> and <strong>TN</strong> cases divided by the sum of true positive and treue negative events in our data set, or as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathrm{Accuracy} = \frac{\sum_{i=0}^{n-1}I(y_i=\tilde{y}_i)}{n}.
|
||
\]</div>
|
||
<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>
|
||
<span class="c1">#now scale the data</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">StandardScaler</span>
|
||
<span class="n">scaler</span> <span class="o">=</span> <span class="n">StandardScaler</span><span class="p">()</span>
|
||
<span class="n">scaler</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">X_train_scaled</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
|
||
<span class="n">X_test_scaled</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||
<span class="c1"># Logistic Regression</span>
|
||
<span class="n">logreg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train_scaled</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 Logistic Regression with scaled data: </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_scaled</span><span class="p">,</span><span class="n">y_test</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_scaled</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 and scaled data: </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_scaled</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_scaled</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_scaled</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>
|
||
</section>
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<i class="fa-solid fa-list"></i> Contents
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">6.1. Logistic Regression</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#basics">6.2. Basics</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-logistic-function">6.3. 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-neural-networks">6.4. Examples of likelihood functions used in logistic regression and neural networks</a></li>
|
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#wisconsin-cancer-data">6.5. Wisconsin Cancer Data</a></li>
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