Morten Hjorth-Jensen
322 lines
20 KiB
HTML
322 lines
20 KiB
HTML
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<title>Week 46: Support Vector Machines and Project 3. Start Principal Component Analysis</title>
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'sections': [('Overview of week 46', 2, None, 'overview-of-week-46'),
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('Support Vector Machines, overarching aims',
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2,
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None,
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('A $p$-dimensional space of features',
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('First attempt at a minimization approach',
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('Adding the Multiplier', 2, None, 'adding-the-multiplier'),
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('Setting up the Problem', 2, None, 'setting-up-the-problem'),
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('The problem to solve', 2, None, 'the-problem-to-solve'),
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('The last steps', 2, None, 'the-last-steps'),
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('A soft classifier', 2, None, 'a-soft-classifier'),
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('Soft optmization problem', 2, None, 'soft-optmization-problem'),
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('Kernels and non-linearity',
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('The equations', 2, None, 'the-equations'),
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("Different kernels and Mercer's theorem",
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('The moons example', 2, None, 'the-moons-example'),
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('Mathematical optimization of convex functions',
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('A simple example', 2, None, 'a-simple-example'),
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('Back to the more realistic cases',
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'back-to-the-more-realistic-cases'),
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('Basic ideas of the Principal Component Analysis (PCA)',
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('Writing our own PCA code', 2, None, 'writing-our-own-pca-code'),
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('Collecting all Steps', 2, None, 'collecting-all-steps'),
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('Classical PCA Theorem', 2, None, 'classical-pca-theorem'),
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('The PCA Theorem', 2, None, 'the-pca-theorem'),
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('Geometric Interpretation and link with Singular Value '
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('Incremental PCA', 2, None, 'incremental-pca'),
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('Randomized PCA', 3, None, 'randomized-pca'),
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('Kernel PCA', 3, None, 'kernel-pca'),
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('Other techniques', 2, None, 'other-techniques')]}
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<a class="navbar-brand" href="week46-bs.html">Week 46: Support Vector Machines and Project 3. Start Principal Component Analysis</a>
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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%;"><b>Overview of week 46</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs002.html#support-vector-machines-overarching-aims" style="font-size: 80%;"><b>Support Vector Machines, overarching aims</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs003.html#hyperplanes-and-all-that" style="font-size: 80%;"><b>Hyperplanes and all that</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs004.html#what-is-a-hyperplane" style="font-size: 80%;"><b>What is a hyperplane?</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs005.html#a-p-dimensional-space-of-features" style="font-size: 80%;"><b>A \( p \)-dimensional space of features</b></a></li>
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<!-- navigation toc: --> <li><a href="#the-two-dimensional-case" style="font-size: 80%;"><b>The two-dimensional case</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs007.html#getting-into-the-details" style="font-size: 80%;"><b>Getting into the details</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs008.html#first-attempt-at-a-minimization-approach" style="font-size: 80%;"><b>First attempt at a minimization approach</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs009.html#solving-the-equations" style="font-size: 80%;"><b>Solving the equations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs010.html#code-example" style="font-size: 80%;"><b>Code Example</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs011.html#problems-with-the-simpler-approach" style="font-size: 80%;"><b>Problems with the Simpler Approach</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs012.html#a-better-approach" style="font-size: 80%;"><b>A better approach</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs013.html#a-quick-reminder-on-lagrangian-multipliers" style="font-size: 80%;"><b>A quick Reminder on Lagrangian Multipliers</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs014.html#adding-the-multiplier" style="font-size: 80%;"><b>Adding the Multiplier</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs015.html#setting-up-the-problem" style="font-size: 80%;"><b>Setting up the Problem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs022.html#the-problem-to-solve" style="font-size: 80%;"><b>The problem to solve</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs017.html#the-last-steps" style="font-size: 80%;"><b>The last steps</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs018.html#a-soft-classifier" style="font-size: 80%;"><b>A soft classifier</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs019.html#soft-optmization-problem" style="font-size: 80%;"><b>Soft optmization problem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs020.html#kernels-and-non-linearity" style="font-size: 80%;"><b>Kernels and non-linearity</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs021.html#the-equations" style="font-size: 80%;"><b>The equations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs022.html#the-problem-to-solve" style="font-size: 80%;"><b>The problem to solve</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs023.html#different-kernels-and-mercer-s-theorem" style="font-size: 80%;"><b>Different kernels and Mercer's theorem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs024.html#the-moons-example" style="font-size: 80%;"><b>The moons example</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs025.html#mathematical-optimization-of-convex-functions" style="font-size: 80%;"><b>Mathematical optimization of convex functions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs026.html#how-do-we-solve-these-problems" style="font-size: 80%;"><b>How do we solve these problems?</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs027.html#a-simple-example" style="font-size: 80%;"><b>A simple example</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs028.html#back-to-the-more-realistic-cases" style="font-size: 80%;"><b>Back to the more realistic cases</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs029.html#basic-ideas-of-the-principal-component-analysis-pca" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs030.html#introducing-the-covariance-and-correlation-functions" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs031.html#more-on-the-covariance" style="font-size: 80%;"><b>More on the covariance</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs032.html#reminding-ourselves-about-linear-regression" style="font-size: 80%;"><b>Reminding ourselves about Linear Regression</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs033.html#simple-example" style="font-size: 80%;"><b>Simple Example</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs034.html#the-correlation-matrix" style="font-size: 80%;"><b>The Correlation Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs035.html#numpy-functionality" style="font-size: 80%;"><b>Numpy Functionality</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs036.html#correlation-matrix-again" style="font-size: 80%;"><b>Correlation Matrix again</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs037.html#using-pandas" style="font-size: 80%;"><b>Using Pandas</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs038.html#and-then-the-franke-function" style="font-size: 80%;"><b>And then the Franke Function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs039.html#links-with-the-design-matrix" style="font-size: 80%;"><b>Links with the Design Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs040.html#computing-the-expectation-values" style="font-size: 80%;"><b>Computing the Expectation Values</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs041.html#towards-the-pca-theorem" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs042.html#more-on-the-pca-theorem" style="font-size: 80%;"><b>More on the PCA Theorem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs043.html#writing-our-own-pca-code" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs044.html#implementing-it" style="font-size: 80%;"><b>Implementing it</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs045.html#first-step" style="font-size: 80%;"><b>First Step</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs046.html#scaling" style="font-size: 80%;"><b>Scaling</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs047.html#centered-data" style="font-size: 80%;"><b>Centered Data</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs048.html#exploring" style="font-size: 80%;"><b>Exploring</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs049.html#diagonalize-the-sample-covariance-matrix-to-obtain-the-principal-components" style="font-size: 80%;"><b>Diagonalize the sample covariance matrix to obtain the principal components</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs050.html#collecting-all-steps" style="font-size: 80%;"><b>Collecting all Steps</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs051.html#classical-pca-theorem" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs052.html#the-pca-theorem" style="font-size: 80%;"><b>The PCA Theorem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs053.html#geometric-interpretation-and-link-with-singular-value-decomposition" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs054.html#pca-and-scikit-learn" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs055.html#back-to-the-cancer-data" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs056.html#incremental-pca" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs056.html#randomized-pca" style="font-size: 80%;"> Randomized PCA</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs056.html#kernel-pca" style="font-size: 80%;"> Kernel PCA</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs057.html#other-techniques" style="font-size: 80%;"><b>Other techniques</b></a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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</div> <!-- end of navigation bar -->
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<div class="container">
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0006"></a>
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<!-- !split -->
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<h2 id="the-two-dimensional-case" class="anchor">The two-dimensional case </h2>
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<p>Let us try to develop our intuition about SVMs by limiting ourselves to a two-dimensional
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plane. To separate the two classes of data points, there are many
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possible lines (hyperplanes if you prefer a more strict naming)
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that could be chosen. Our objective is to find a
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plane that has the maximum margin, i.e the maximum distance between
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data points of both classes. Maximizing the margin distance provides
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some reinforcement so that future data points can be classified with
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more confidence.
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</p>
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<p>What a linear classifier attempts to accomplish is to split the
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feature space into two half spaces by placing a hyperplane between the
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data points. This hyperplane will be our decision boundary. All
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points on one side of the plane will belong to class one and all points
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on the other side of the plane will belong to the second class two.
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</p>
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<p>Unfortunately there are many ways in which we can place a hyperplane
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to divide the data. Below is an example of two candidate hyperplanes
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for our data sample.
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</p>
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<p>
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<!-- navigation buttons at the bottom of the page -->
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<ul class="pagination">
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<li><a href="._week46-bs005.html">«</a></li>
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<li><a href="._week46-bs002.html">3</a></li>
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<li><a href="._week46-bs011.html">12</a></li>
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<li><a href="._week46-bs012.html">13</a></li>
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<li><a href="._week46-bs013.html">14</a></li>
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