mhjensen
451 lines
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HTML
451 lines
22 KiB
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<section>
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<!-- ------------------- main content ---------------------- -->
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<center><h1 style="text-align: center;">Data Analysis and Machine Learning: Support Vector Machines</h1></center> <!-- document title -->
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<p>
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<!-- author(s): Morten Hjorth-Jensen -->
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<center>
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<b>Morten Hjorth-Jensen</b> [1, 2]
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</center>
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<p> <br>
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<!-- institution(s) -->
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<center>[1] <b>Department of Physics, University of Oslo</b></center>
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<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
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<br>
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<p> <br>
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<center><h4>Nov 2, 2018</h4></center> <!-- date -->
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<br>
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<p>
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<center style="font-size:80%">
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<!-- copyright --> © 1999-2018, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
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</center>
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</section>
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<section>
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<h2 id="___sec0">Support Vector Machines, overarching aims </h2>
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<p>
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A Support Vector Machine (SVM) is a very powerful and versatile
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Machine Learning model, capable of performing linear or nonlinear
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classification, regression, and even outlier detection. It is one of
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the most popular models in Machine Learning, and anyone interested in
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Machine Learning should have it in their toolbox. SVMs are
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particularly well suited for classification of complex but small-sized or
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medium-sized datasets.
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<p>
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The basic mathematics relies on the definition of hyperplanes and the definition of a <b>margin</b> which separates
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classes (in case of classification problems) of variables. It is also used for regression problems.
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<p>
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With SVMs we distinguish between hard margin and soft margins. The latter introduces a so-called softening parameter to be discussed below.
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We distringuish also between linearn and non-linear approaches.
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<p>
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<b>These notes will be updated shortly with more material</b>
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</section>
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<section>
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<h2 id="___sec1">Strength and weakness </h2>
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When we implement a linear support vector machine, the main parameter is the constant \( C \). Small values of \( C \) mean simple models.
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These models are fast to train and also fast to predict and scale to very large data sets and work well with sparse data. Linear support vector machines make it easy to understand how a prediction is made, however it is often not easy to understand why coefficients are the way they are. These models work also well in higer dimensions.
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</section>
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<section>
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<h2 id="___sec2">Examples with kernels </h2>
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
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<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span><span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">IPython.display</span> <span style="color: #8B008B; font-weight: bold">import</span> set_matplotlib_formats, display
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<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">pandas</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">pd</span>
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<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
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<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
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<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">mglearn</span>
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<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">cycler</span> <span style="color: #8B008B; font-weight: bold">import</span> cycler
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<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.linear_model</span> <span style="color: #8B008B; font-weight: bold">import</span> LogisticRegression
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<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.svm</span> <span style="color: #8B008B; font-weight: bold">import</span> LinearSVC
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<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.datasets</span> <span style="color: #8B008B; font-weight: bold">import</span> make_blobs
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X, y = make_blobs(centers=<span style="color: #B452CD">4</span>, random_state=<span style="color: #B452CD">8</span>)
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y = y % <span style="color: #B452CD">2</span>
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mglearn.discrete_scatter(X[:, <span style="color: #B452CD">0</span>], X[:, <span style="color: #B452CD">1</span>], y)
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plt.xlabel(<span style="color: #CD5555">"Feature 0"</span>)
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plt.ylabel(<span style="color: #CD5555">"Feature 1"</span>)
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plt.show()
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<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.svm</span> <span style="color: #8B008B; font-weight: bold">import</span> LinearSVC
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linear_svm = LinearSVC().fit(X, y)
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mglearn.plots.plot_2d_separator(linear_svm, X)
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mglearn.discrete_scatter(X[:, <span style="color: #B452CD">0</span>], X[:, <span style="color: #B452CD">1</span>], y)
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plt.xlabel(<span style="color: #CD5555">"Feature 0"</span>)
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plt.ylabel(<span style="color: #CD5555">"Feature 1"</span>)
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</pre></div>
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
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<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span><span style="color: #228B22"># add the squared first feature</span>
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X_new = np.hstack([X, X[:, <span style="color: #B452CD">1</span>:] ** <span style="color: #B452CD">2</span>])
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<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">mpl_toolkits.mplot3d</span> <span style="color: #8B008B; font-weight: bold">import</span> Axes3D, axes3d
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figure = plt.figure()
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<span style="color: #228B22"># visualize in 3D</span>
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ax = Axes3D(figure, elev=-<span style="color: #B452CD">152</span>, azim=-<span style="color: #B452CD">26</span>)
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<span style="color: #228B22"># plot first all the points with y==0, then all with y == 1</span>
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mask = y == <span style="color: #B452CD">0</span>
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ax.scatter(X_new[mask, <span style="color: #B452CD">0</span>], X_new[mask, <span style="color: #B452CD">1</span>], X_new[mask, <span style="color: #B452CD">2</span>], c=<span style="color: #CD5555">'b'</span>,
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cmap=mglearn.cm2, s=<span style="color: #B452CD">60</span>, edgecolor=<span style="color: #CD5555">'k'</span>)
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ax.scatter(X_new[~mask, <span style="color: #B452CD">0</span>], X_new[~mask, <span style="color: #B452CD">1</span>], X_new[~mask, <span style="color: #B452CD">2</span>], c=<span style="color: #CD5555">'r'</span>, marker=<span style="color: #CD5555">'^'</span>,
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cmap=mglearn.cm2, s=<span style="color: #B452CD">60</span>, edgecolor=<span style="color: #CD5555">'k'</span>)
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ax.set_xlabel(<span style="color: #CD5555">"feature0"</span>)
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ax.set_ylabel(<span style="color: #CD5555">"feature1"</span>)
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ax.set_zlabel(<span style="color: #CD5555">"feature1 ** 2"</span>)
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</pre></div>
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
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<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span>linear_svm_3d = LinearSVC().fit(X_new, y)
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coef, intercept = linear_svm_3d.coef_.ravel(), linear_svm_3d.intercept_
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<span style="color: #228B22"># show linear decision boundary</span>
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figure = plt.figure()
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ax = Axes3D(figure, elev=-<span style="color: #B452CD">152</span>, azim=-<span style="color: #B452CD">26</span>)
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xx = np.linspace(X_new[:, <span style="color: #B452CD">0</span>].min() - <span style="color: #B452CD">2</span>, X_new[:, <span style="color: #B452CD">0</span>].max() + <span style="color: #B452CD">2</span>, <span style="color: #B452CD">50</span>)
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yy = np.linspace(X_new[:, <span style="color: #B452CD">1</span>].min() - <span style="color: #B452CD">2</span>, X_new[:, <span style="color: #B452CD">1</span>].max() + <span style="color: #B452CD">2</span>, <span style="color: #B452CD">50</span>)
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XX, YY = np.meshgrid(xx, yy)
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ZZ = (coef[<span style="color: #B452CD">0</span>] * XX + coef[<span style="color: #B452CD">1</span>] * YY + intercept) / -coef[<span style="color: #B452CD">2</span>]
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ax.plot_surface(XX, YY, ZZ, rstride=<span style="color: #B452CD">8</span>, cstride=<span style="color: #B452CD">8</span>, alpha=<span style="color: #B452CD">0.3</span>)
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ax.scatter(X_new[mask, <span style="color: #B452CD">0</span>], X_new[mask, <span style="color: #B452CD">1</span>], X_new[mask, <span style="color: #B452CD">2</span>], c=<span style="color: #CD5555">'b'</span>,
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cmap=mglearn.cm2, s=<span style="color: #B452CD">60</span>, edgecolor=<span style="color: #CD5555">'k'</span>)
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ax.scatter(X_new[~mask, <span style="color: #B452CD">0</span>], X_new[~mask, <span style="color: #B452CD">1</span>], X_new[~mask, <span style="color: #B452CD">2</span>], c=<span style="color: #CD5555">'r'</span>, marker=<span style="color: #CD5555">'^'</span>,
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cmap=mglearn.cm2, s=<span style="color: #B452CD">60</span>, edgecolor=<span style="color: #CD5555">'k'</span>)
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ax.set_xlabel(<span style="color: #CD5555">"feature0"</span>)
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ax.set_ylabel(<span style="color: #CD5555">"feature1"</span>)
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ax.set_zlabel(<span style="color: #CD5555">"feature1 ** 2"</span>)
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ZZ = YY ** <span style="color: #B452CD">2</span>
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dec = linear_svm_3d.decision_function(np.c_[XX.ravel(), YY.ravel(), ZZ.ravel()])
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plt.contourf(XX, YY, dec.reshape(XX.shape), levels=[dec.min(), <span style="color: #B452CD">0</span>, dec.max()],
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cmap=mglearn.cm2, alpha=<span style="color: #B452CD">0.5</span>)
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mglearn.discrete_scatter(X[:, <span style="color: #B452CD">0</span>], X[:, <span style="color: #B452CD">1</span>], y)
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plt.xlabel(<span style="color: #CD5555">"Feature 0"</span>)
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plt.ylabel(<span style="color: #CD5555">"Feature 1"</span>)
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</pre></div>
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
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<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span><span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.svm</span> <span style="color: #8B008B; font-weight: bold">import</span> SVC
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X, y = mglearn.tools.make_handcrafted_dataset()
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svm = SVC(kernel=<span style="color: #CD5555">'rbf'</span>, C=<span style="color: #B452CD">10</span>, gamma=<span style="color: #B452CD">0.1</span>).fit(X, y)
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mglearn.plots.plot_2d_separator(svm, X, eps=.<span style="color: #B452CD">5</span>)
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mglearn.discrete_scatter(X[:, <span style="color: #B452CD">0</span>], X[:, <span style="color: #B452CD">1</span>], y)
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<span style="color: #228B22"># plot support vectors</span>
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sv = svm.support_vectors_
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<span style="color: #228B22"># class labels of support vectors are given by the sign of the dual coefficients</span>
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sv_labels = svm.dual_coef_.ravel() > <span style="color: #B452CD">0</span>
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mglearn.discrete_scatter(sv[:, <span style="color: #B452CD">0</span>], sv[:, <span style="color: #B452CD">1</span>], sv_labels, s=<span style="color: #B452CD">15</span>, markeredgewidth=<span style="color: #B452CD">3</span>)
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plt.xlabel(<span style="color: #CD5555">"Feature 0"</span>)
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plt.ylabel(<span style="color: #CD5555">"Feature 1"</span>)
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fig, axes = plt.subplots(<span style="color: #B452CD">3</span>, <span style="color: #B452CD">3</span>, figsize=(<span style="color: #B452CD">15</span>, <span style="color: #B452CD">10</span>))
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<span style="color: #8B008B; font-weight: bold">for</span> ax, C <span style="color: #8B008B">in</span> <span style="color: #658b00">zip</span>(axes, [-<span style="color: #B452CD">1</span>, <span style="color: #B452CD">0</span>, <span style="color: #B452CD">3</span>]):
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<span style="color: #8B008B; font-weight: bold">for</span> a, gamma <span style="color: #8B008B">in</span> <span style="color: #658b00">zip</span>(ax, <span style="color: #658b00">range</span>(-<span style="color: #B452CD">1</span>, <span style="color: #B452CD">2</span>)):
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mglearn.plots.plot_svm(log_C=C, log_gamma=gamma, ax=a)
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axes[<span style="color: #B452CD">0</span>, <span style="color: #B452CD">0</span>].legend([<span style="color: #CD5555">"class 0"</span>, <span style="color: #CD5555">"class 1"</span>, <span style="color: #CD5555">"sv class 0"</span>, <span style="color: #CD5555">"sv class 1"</span>],
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ncol=<span style="color: #B452CD">4</span>, loc=(.<span style="color: #B452CD">9</span>, <span style="color: #B452CD">1.2</span>))
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</pre></div>
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</section>
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</div> <!-- class="slides" -->
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</div> <!-- class="reveal" -->
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<script src="reveal.js/lib/js/head.min.js"></script>
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<script src="reveal.js/js/reveal.js"></script>
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<script>
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// Full list of configuration options available here:
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// https://github.com/hakimel/reveal.js#configuration
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Reveal.initialize({
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// Display navigation controls in the bottom right corner
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controls: true,
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// Display progress bar (below the horiz. slider)
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progress: true,
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// Display the page number of the current slide
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slideNumber: true,
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// Push each slide change to the browser history
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history: false,
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// Enable keyboard shortcuts for navigation
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keyboard: true,
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|
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// Enable the slide overview mode
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overview: true,
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|
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// Vertical centering of slides
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//center: true,
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center: false,
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// Enables touch navigation on devices with touch input
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touch: true,
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// Loop the presentation
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loop: false,
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|
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// Change the presentation direction to be RTL
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rtl: false,
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|
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// Turns fragments on and off globally
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fragments: true,
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|
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// Flags if the presentation is running in an embedded mode,
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// i.e. contained within a limited portion of the screen
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embedded: false,
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|
|
// Number of milliseconds between automatically proceeding to the
|
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// next slide, disabled when set to 0, this value can be overwritten
|
|
// by using a data-autoslide attribute on your slides
|
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autoSlide: 0,
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|
|
|
// Stop auto-sliding after user input
|
|
autoSlideStoppable: true,
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|
|
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// Enable slide navigation via mouse wheel
|
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mouseWheel: false,
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|
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// Hides the address bar on mobile devices
|
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hideAddressBar: true,
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|
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// Opens links in an iframe preview overlay
|
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previewLinks: false,
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|
|
// Transition style
|
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transition: 'default', // default/cube/page/concave/zoom/linear/fade/none
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// Transition speed
|
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transitionSpeed: 'default', // default/fast/slow
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|
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// Transition style for full page slide backgrounds
|
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backgroundTransition: 'default', // default/none/slide/concave/convex/zoom
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|
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// Number of slides away from the current that are visible
|
|
viewDistance: 3,
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|
|
|
// Parallax background image
|
|
//parallaxBackgroundImage: '', // e.g. "'https://s3.amazonaws.com/hakim-static/reveal-js/reveal-parallax-1.jpg'"
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|
|
|
// Parallax background size
|
|
//parallaxBackgroundSize: '' // CSS syntax, e.g. "2100px 900px"
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|
|
|
theme: Reveal.getQueryHash().theme, // available themes are in reveal.js/css/theme
|
|
transition: Reveal.getQueryHash().transition || 'default', // default/cube/page/concave/zoom/linear/none
|
|
|
|
});
|
|
|
|
Reveal.initialize({
|
|
dependencies: [
|
|
// Cross-browser shim that fully implements classList - https://github.com/eligrey/classList.js/
|
|
{ src: 'reveal.js/lib/js/classList.js', condition: function() { return !document.body.classList; } },
|
|
|
|
// Interpret Markdown in <section> elements
|
|
{ src: 'reveal.js/plugin/markdown/marked.js', condition: function() { return !!document.querySelector( '[data-markdown]' ); } },
|
|
{ src: 'reveal.js/plugin/markdown/markdown.js', condition: function() { return !!document.querySelector( '[data-markdown]' ); } },
|
|
|
|
// Syntax highlight for <code> elements
|
|
{ src: 'reveal.js/plugin/highlight/highlight.js', async: true, callback: function() { hljs.initHighlightingOnLoad(); } },
|
|
|
|
// Zoom in and out with Alt+click
|
|
{ src: 'reveal.js/plugin/zoom-js/zoom.js', async: true, condition: function() { return !!document.body.classList; } },
|
|
|
|
// Speaker notes
|
|
{ src: 'reveal.js/plugin/notes/notes.js', async: true, condition: function() { return !!document.body.classList; } },
|
|
|
|
// Remote control your reveal.js presentation using a touch device
|
|
//{ src: 'reveal.js/plugin/remotes/remotes.js', async: true, condition: function() { return !!document.body.classList; } },
|
|
|
|
// MathJax
|
|
//{ src: 'reveal.js/plugin/math/math.js', async: true }
|
|
]
|
|
});
|
|
|
|
Reveal.initialize({
|
|
|
|
// The "normal" size of the presentation, aspect ratio will be preserved
|
|
// when the presentation is scaled to fit different resolutions. Can be
|
|
// specified using percentage units.
|
|
width: 1170, // original: 960,
|
|
height: 700,
|
|
|
|
// Factor of the display size that should remain empty around the content
|
|
margin: 0.1,
|
|
|
|
// Bounds for smallest/largest possible scale to apply to content
|
|
minScale: 0.2,
|
|
maxScale: 1.0
|
|
|
|
});
|
|
</script>
|
|
|
|
<!-- begin footer logo
|
|
<div style="position: absolute; bottom: 0px; left: 0; margin-left: 0px">
|
|
<img src="somelogo.png">
|
|
</div>
|
|
end footer logo -->
|
|
|
|
|
|
|
|
</body>
|
|
</html>
|