added program
This commit is contained in:
Morten Hjorth-Jensen
@@ -38,6 +38,10 @@ doconce format html week46.do.txt --html_style=bootstrap --pygments_html_style=d
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{'highest level': 2,
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'sections': [('Overview of week 46', 2, None, 'overview-of-week-46'),
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('Friday', 2, None, 'friday'),
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('Workshop plan Friday November 19 and the rest of the lecture',
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2,
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None,
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'workshop-plan-friday-november-19-and-the-rest-of-the-lecture'),
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('Support Vector Machines, overarching aims',
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2,
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None,
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@@ -131,33 +135,34 @@ MathJax.Hub.Config({
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._week46-bs001.html#overview-of-week-46" style="font-size: 80%;">Overview of week 46</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs002.html#friday" style="font-size: 80%;">Friday</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs003.html#support-vector-machines-overarching-aims" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs004.html#hyperplanes-and-all-that" style="font-size: 80%;">Hyperplanes and all that</a></li>
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<!-- navigation toc: --> <li><a href="#what-is-a-hyperplane" style="font-size: 80%;">What is a hyperplane?</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs006.html#a-p-dimensional-space-of-features" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs007.html#the-two-dimensional-case" style="font-size: 80%;">The two-dimensional case</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs008.html#getting-into-the-details" style="font-size: 80%;">Getting into the details</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs009.html#first-attempt-at-a-minimization-approach" style="font-size: 80%;">First attempt at a minimization approach</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs010.html#solving-the-equations" style="font-size: 80%;">Solving the equations</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs011.html#code-example" style="font-size: 80%;">Code Example</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs012.html#problems-with-the-simpler-approach" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs013.html#a-better-approach" style="font-size: 80%;">A better approach</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs014.html#a-quick-reminder-on-lagrangian-multipliers" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs015.html#adding-the-multiplier" style="font-size: 80%;">Adding the Multiplier</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs016.html#setting-up-the-problem" style="font-size: 80%;">Setting up the Problem</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs023.html#the-problem-to-solve" style="font-size: 80%;">The problem to solve</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs018.html#the-last-steps" style="font-size: 80%;">The last steps</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs019.html#a-soft-classifier" style="font-size: 80%;">A soft classifier</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs020.html#soft-optmization-problem" style="font-size: 80%;">Soft optmization problem</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs021.html#kernels-and-non-linearity" style="font-size: 80%;">Kernels and non-linearity</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs022.html#the-equations" style="font-size: 80%;">The equations</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs023.html#the-problem-to-solve" style="font-size: 80%;">The problem to solve</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs024.html#different-kernels-and-mercer-s-theorem" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs025.html#the-moons-example" style="font-size: 80%;">The moons example</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs026.html#mathematical-optimization-of-convex-functions" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs027.html#how-do-we-solve-these-problems" style="font-size: 80%;">How do we solve these problems?</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs028.html#a-simple-example" style="font-size: 80%;">A simple example</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs029.html#back-to-the-more-realistic-cases" style="font-size: 80%;">Back to the more realistic cases</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs003.html#workshop-plan-friday-november-19-and-the-rest-of-the-lecture" style="font-size: 80%;">Workshop plan Friday November 19 and the rest of the lecture</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs004.html#support-vector-machines-overarching-aims" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
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<!-- navigation toc: --> <li><a href="#hyperplanes-and-all-that" style="font-size: 80%;">Hyperplanes and all that</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs006.html#what-is-a-hyperplane" style="font-size: 80%;">What is a hyperplane?</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs007.html#a-p-dimensional-space-of-features" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs008.html#the-two-dimensional-case" style="font-size: 80%;">The two-dimensional case</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs009.html#getting-into-the-details" style="font-size: 80%;">Getting into the details</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs010.html#first-attempt-at-a-minimization-approach" style="font-size: 80%;">First attempt at a minimization approach</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs011.html#solving-the-equations" style="font-size: 80%;">Solving the equations</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs012.html#code-example" style="font-size: 80%;">Code Example</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs013.html#problems-with-the-simpler-approach" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs014.html#a-better-approach" style="font-size: 80%;">A better approach</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs015.html#a-quick-reminder-on-lagrangian-multipliers" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs016.html#adding-the-multiplier" style="font-size: 80%;">Adding the Multiplier</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs017.html#setting-up-the-problem" style="font-size: 80%;">Setting up the Problem</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs024.html#the-problem-to-solve" style="font-size: 80%;">The problem to solve</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs019.html#the-last-steps" style="font-size: 80%;">The last steps</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs020.html#a-soft-classifier" style="font-size: 80%;">A soft classifier</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs021.html#soft-optmization-problem" style="font-size: 80%;">Soft optmization problem</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs022.html#kernels-and-non-linearity" style="font-size: 80%;">Kernels and non-linearity</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs023.html#the-equations" style="font-size: 80%;">The equations</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs024.html#the-problem-to-solve" style="font-size: 80%;">The problem to solve</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs025.html#different-kernels-and-mercer-s-theorem" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs026.html#the-moons-example" style="font-size: 80%;">The moons example</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs027.html#mathematical-optimization-of-convex-functions" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs028.html#how-do-we-solve-these-problems" style="font-size: 80%;">How do we solve these problems?</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs029.html#a-simple-example" style="font-size: 80%;">A simple example</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs030.html#back-to-the-more-realistic-cases" style="font-size: 80%;">Back to the more realistic cases</a></li>
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</ul>
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</li>
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@@ -169,32 +174,107 @@ MathJax.Hub.Config({
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0005"></a>
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<!-- !split -->
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<h2 id="what-is-a-hyperplane" class="anchor">What is a hyperplane? </h2>
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<h2 id="hyperplanes-and-all-that" class="anchor">Hyperplanes and all that </h2>
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<p>The aim of the SVM algorithm is to find a hyperplane in a
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\( p \)-dimensional space, where \( p \) is the number of features that
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distinctly classifies the data points.
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<p>The theory behind support vector machines (SVM hereafter) is based on
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the mathematical description of so-called hyperplanes. Let us start
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with a two-dimensional case. This will also allow us to introduce our
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first SVM examples. These will be tailored to the case of two specific
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classes, as displayed in the figure here based on the usage of the petal data.
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</p>
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<p>In a \( p \)-dimensional space, a hyperplane is what we call an affine subspace of dimension of \( p-1 \).
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As an example, in two dimension, a hyperplane is simply as straight line while in three dimensions it is
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a two-dimensional subspace, or stated simply, a plane.
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<p>We assume here that our data set can be well separated into two
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domains, where a straight line does the job in the separating the two
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classes. Here the two classes are represented by either squares or
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circles.
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</p>
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<p>In two dimensions, with the variables \( x_1 \) and \( x_2 \), the hyperplane is defined as</p>
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$$
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b+w_1x_1+w_2x_2=0,
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$$
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="cell border-box-sizing code_cell rendered">
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<div class="input">
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<div class="inner_cell">
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<div class="input_area">
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<div class="highlight" style="background: #f8f8f8">
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<pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn</span> <span style="color: #008000; font-weight: bold">import</span> datasets
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.svm</span> <span style="color: #008000; font-weight: bold">import</span> SVC, LinearSVC
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> SGDClassifier
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> StandardScaler
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
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plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'axes.labelsize'</span>] <span style="color: #666666">=</span> <span style="color: #666666">14</span>
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plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'xtick.labelsize'</span>] <span style="color: #666666">=</span> <span style="color: #666666">12</span>
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plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'ytick.labelsize'</span>] <span style="color: #666666">=</span> <span style="color: #666666">12</span>
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<p>where \( b \) is the intercept and \( w_1 \) and \( w_2 \) define the elements of a vector orthogonal to the line
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\( b+w_1x_1+w_2x_2=0 \).
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In two dimensions we define the vectors \( \boldsymbol{x} =[x1,x2] \) and \( \boldsymbol{w}=[w1,w2] \).
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We can then rewrite the above equation as
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</p>
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$$
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\boldsymbol{x}^T\boldsymbol{w}+b=0.
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$$
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iris <span style="color: #666666">=</span> datasets<span style="color: #666666">.</span>load_iris()
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X <span style="color: #666666">=</span> iris[<span style="color: #BA2121">"data"</span>][:, (<span style="color: #666666">2</span>, <span style="color: #666666">3</span>)] <span style="color: #408080; font-style: italic"># petal length, petal width</span>
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y <span style="color: #666666">=</span> iris[<span style="color: #BA2121">"target"</span>]
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setosa_or_versicolor <span style="color: #666666">=</span> (y <span style="color: #666666">==</span> <span style="color: #666666">0</span>) <span style="color: #666666">|</span> (y <span style="color: #666666">==</span> <span style="color: #666666">1</span>)
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X <span style="color: #666666">=</span> X[setosa_or_versicolor]
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y <span style="color: #666666">=</span> y[setosa_or_versicolor]
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C <span style="color: #666666">=</span> <span style="color: #666666">5</span>
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alpha <span style="color: #666666">=</span> <span style="color: #666666">1</span> <span style="color: #666666">/</span> (C <span style="color: #666666">*</span> <span style="color: #008000">len</span>(X))
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lin_clf <span style="color: #666666">=</span> LinearSVC(loss<span style="color: #666666">=</span><span style="color: #BA2121">"hinge"</span>, C<span style="color: #666666">=</span>C, random_state<span style="color: #666666">=42</span>)
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svm_clf <span style="color: #666666">=</span> SVC(kernel<span style="color: #666666">=</span><span style="color: #BA2121">"linear"</span>, C<span style="color: #666666">=</span>C)
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sgd_clf <span style="color: #666666">=</span> SGDClassifier(loss<span style="color: #666666">=</span><span style="color: #BA2121">"hinge"</span>, learning_rate<span style="color: #666666">=</span><span style="color: #BA2121">"constant"</span>, eta0<span style="color: #666666">=0.001</span>, alpha<span style="color: #666666">=</span>alpha,
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max_iter<span style="color: #666666">=100000</span>, random_state<span style="color: #666666">=42</span>)
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scaler <span style="color: #666666">=</span> StandardScaler()
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X_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>fit_transform(X)
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lin_clf<span style="color: #666666">.</span>fit(X_scaled, y)
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svm_clf<span style="color: #666666">.</span>fit(X_scaled, y)
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sgd_clf<span style="color: #666666">.</span>fit(X_scaled, y)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">"LinearSVC: "</span>, lin_clf<span style="color: #666666">.</span>intercept_, lin_clf<span style="color: #666666">.</span>coef_)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">"SVC: "</span>, svm_clf<span style="color: #666666">.</span>intercept_, svm_clf<span style="color: #666666">.</span>coef_)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">"SGDClassifier(alpha=</span><span style="color: #BB6688; font-weight: bold">{:.5f}</span><span style="color: #BA2121">):"</span><span style="color: #666666">.</span>format(sgd_clf<span style="color: #666666">.</span>alpha), sgd_clf<span style="color: #666666">.</span>intercept_, sgd_clf<span style="color: #666666">.</span>coef_)
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<span style="color: #408080; font-style: italic"># Compute the slope and bias of each decision boundary</span>
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w1 <span style="color: #666666">=</span> <span style="color: #666666">-</span>lin_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">0</span>]<span style="color: #666666">/</span>lin_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">1</span>]
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b1 <span style="color: #666666">=</span> <span style="color: #666666">-</span>lin_clf<span style="color: #666666">.</span>intercept_[<span style="color: #666666">0</span>]<span style="color: #666666">/</span>lin_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">1</span>]
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w2 <span style="color: #666666">=</span> <span style="color: #666666">-</span>svm_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">0</span>]<span style="color: #666666">/</span>svm_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">1</span>]
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b2 <span style="color: #666666">=</span> <span style="color: #666666">-</span>svm_clf<span style="color: #666666">.</span>intercept_[<span style="color: #666666">0</span>]<span style="color: #666666">/</span>svm_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">1</span>]
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w3 <span style="color: #666666">=</span> <span style="color: #666666">-</span>sgd_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">0</span>]<span style="color: #666666">/</span>sgd_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">1</span>]
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b3 <span style="color: #666666">=</span> <span style="color: #666666">-</span>sgd_clf<span style="color: #666666">.</span>intercept_[<span style="color: #666666">0</span>]<span style="color: #666666">/</span>sgd_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">1</span>]
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<span style="color: #408080; font-style: italic"># Transform the decision boundary lines back to the original scale</span>
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line1 <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>inverse_transform([[<span style="color: #666666">-10</span>, <span style="color: #666666">-10</span> <span style="color: #666666">*</span> w1 <span style="color: #666666">+</span> b1], [<span style="color: #666666">10</span>, <span style="color: #666666">10</span> <span style="color: #666666">*</span> w1 <span style="color: #666666">+</span> b1]])
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line2 <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>inverse_transform([[<span style="color: #666666">-10</span>, <span style="color: #666666">-10</span> <span style="color: #666666">*</span> w2 <span style="color: #666666">+</span> b2], [<span style="color: #666666">10</span>, <span style="color: #666666">10</span> <span style="color: #666666">*</span> w2 <span style="color: #666666">+</span> b2]])
|
||||
line3 <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>inverse_transform([[<span style="color: #666666">-10</span>, <span style="color: #666666">-10</span> <span style="color: #666666">*</span> w3 <span style="color: #666666">+</span> b3], [<span style="color: #666666">10</span>, <span style="color: #666666">10</span> <span style="color: #666666">*</span> w3 <span style="color: #666666">+</span> b3]])
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Plot all three decision boundaries</span>
|
||||
plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">11</span>, <span style="color: #666666">4</span>))
|
||||
plt<span style="color: #666666">.</span>plot(line1[:, <span style="color: #666666">0</span>], line1[:, <span style="color: #666666">1</span>], <span style="color: #BA2121">"k:"</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"LinearSVC"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(line2[:, <span style="color: #666666">0</span>], line2[:, <span style="color: #666666">1</span>], <span style="color: #BA2121">"b--"</span>, linewidth<span style="color: #666666">=2</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"SVC"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(line3[:, <span style="color: #666666">0</span>], line3[:, <span style="color: #666666">1</span>], <span style="color: #BA2121">"r-"</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"SGDClassifier"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(X[:, <span style="color: #666666">0</span>][y<span style="color: #666666">==1</span>], X[:, <span style="color: #666666">1</span>][y<span style="color: #666666">==1</span>], <span style="color: #BA2121">"bs"</span>) <span style="color: #408080; font-style: italic"># label="Iris-Versicolor"</span>
|
||||
plt<span style="color: #666666">.</span>plot(X[:, <span style="color: #666666">0</span>][y<span style="color: #666666">==0</span>], X[:, <span style="color: #666666">1</span>][y<span style="color: #666666">==0</span>], <span style="color: #BA2121">"yo"</span>) <span style="color: #408080; font-style: italic"># label="Iris-Setosa"</span>
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">"Petal length"</span>, fontsize<span style="color: #666666">=14</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">"Petal width"</span>, fontsize<span style="color: #666666">=14</span>)
|
||||
plt<span style="color: #666666">.</span>legend(loc<span style="color: #666666">=</span><span style="color: #BA2121">"upper center"</span>, fontsize<span style="color: #666666">=14</span>)
|
||||
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">0</span>, <span style="color: #666666">5.5</span>, <span style="color: #666666">0</span>, <span style="color: #666666">2</span>])
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
<div class="output_area">
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
|
||||
<p>
|
||||
@@ -217,7 +297,7 @@ $$
|
||||
<li><a href="._week46-bs013.html">14</a></li>
|
||||
<li><a href="._week46-bs014.html">15</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week46-bs029.html">30</a></li>
|
||||
<li><a href="._week46-bs030.html">31</a></li>
|
||||
<li><a href="._week46-bs006.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
Reference in New Issue
Block a user