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<li><a class="reference internal" href="#">1.4. Support Vector Machines</a><ul>
<li><a class="reference internal" href="#classification">1.4.1. Classification</a><ul>
<li><a class="reference internal" href="#multi-class-classification">1.4.1.1. Multi-class classification</a></li>
<li><a class="reference internal" href="#scores-and-probabilities">1.4.1.2. Scores and probabilities</a></li>
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<section id="support-vector-machines">
<span id="svm"></span><h1><span class="section-number">1.4. </span>Support Vector Machines<a class="headerlink" href="#support-vector-machines" title="Permalink to this heading">¶</a></h1>
<p><strong>Support vector machines (SVMs)</strong> are a set of supervised learning
methods used for <a class="reference internal" href="#svm-classification"><span class="std std-ref">classification</span></a>,
<a class="reference internal" href="#svm-regression"><span class="std std-ref">regression</span></a> and <a class="reference internal" href="#svm-outlier-detection"><span class="std std-ref">outliers detection</span></a>.</p>
<p>The advantages of support vector machines are:</p>
<blockquote>
<div><ul class="simple">
<li><p>Effective in high dimensional spaces.</p></li>
<li><p>Still effective in cases where number of dimensions is greater
than the number of samples.</p></li>
<li><p>Uses a subset of training points in the decision function (called
support vectors), so it is also memory efficient.</p></li>
<li><p>Versatile: different <a class="reference internal" href="#svm-kernels"><span class="std std-ref">Kernel functions</span></a> can be
specified for the decision function. Common kernels are
provided, but it is also possible to specify custom kernels.</p></li>
</ul>
</div></blockquote>
<p>The disadvantages of support vector machines include:</p>
<blockquote>
<div><ul class="simple">
<li><p>If the number of features is much greater than the number of
samples, avoid over-fitting in choosing <a class="reference internal" href="#svm-kernels"><span class="std std-ref">Kernel functions</span></a> and regularization
term is crucial.</p></li>
<li><p>SVMs do not directly provide probability estimates, these are
calculated using an expensive five-fold cross-validation
(see <a class="reference internal" href="#scores-probabilities"><span class="std std-ref">Scores and probabilities</span></a>, below).</p></li>
</ul>
</div></blockquote>
<p>The support vector machines in scikit-learn support both dense
(<code class="docutils literal notranslate"><span class="pre">numpy.ndarray</span></code> and convertible to that by <code class="docutils literal notranslate"><span class="pre">numpy.asarray</span></code>) and
sparse (any <code class="docutils literal notranslate"><span class="pre">scipy.sparse</span></code>) sample vectors as input. However, to use
an SVM to make predictions for sparse data, it must have been fit on such
data. For optimal performance, use C-ordered <code class="docutils literal notranslate"><span class="pre">numpy.ndarray</span></code> (dense) or
<code class="docutils literal notranslate"><span class="pre">scipy.sparse.csr_matrix</span></code> (sparse) with <code class="docutils literal notranslate"><span class="pre">dtype=float64</span></code>.</p>
<section id="classification">
<span id="svm-classification"></span><h2><span class="section-number">1.4.1. </span>Classification<a class="headerlink" href="#classification" title="Permalink to this heading">¶</a></h2>
<p><a class="reference internal" href="generated/sklearn.svm.SVC.html#sklearn.svm.SVC" title="sklearn.svm.SVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">SVC</span></code></a>, <a class="reference internal" href="generated/sklearn.svm.NuSVC.html#sklearn.svm.NuSVC" title="sklearn.svm.NuSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">NuSVC</span></code></a> and <a class="reference internal" href="generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearSVC</span></code></a> are classes
capable of performing binary and multi-class classification on a dataset.</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/svm/plot_iris_svc.html"><img alt="../_images/sphx_glr_plot_iris_svc_001.png" src="../_images/sphx_glr_plot_iris_svc_001.png" /></a>
</figure>
<p><a class="reference internal" href="generated/sklearn.svm.SVC.html#sklearn.svm.SVC" title="sklearn.svm.SVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">SVC</span></code></a> and <a class="reference internal" href="generated/sklearn.svm.NuSVC.html#sklearn.svm.NuSVC" title="sklearn.svm.NuSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">NuSVC</span></code></a> are similar methods, but accept
slightly different sets of parameters and have different mathematical
formulations (see section <a class="reference internal" href="#svm-mathematical-formulation"><span class="std std-ref">Mathematical formulation</span></a>). On the
other hand, <a class="reference internal" href="generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearSVC</span></code></a> is another (faster) implementation of Support
Vector Classification for the case of a linear kernel. Note that
<a class="reference internal" href="generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearSVC</span></code></a> does not accept parameter <code class="docutils literal notranslate"><span class="pre">kernel</span></code>, as this is
assumed to be linear. It also lacks some of the attributes of
<a class="reference internal" href="generated/sklearn.svm.SVC.html#sklearn.svm.SVC" title="sklearn.svm.SVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">SVC</span></code></a> and <a class="reference internal" href="generated/sklearn.svm.NuSVC.html#sklearn.svm.NuSVC" title="sklearn.svm.NuSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">NuSVC</span></code></a>, like <code class="docutils literal notranslate"><span class="pre">support_</span></code>.</p>
<p>As other classifiers, <a class="reference internal" href="generated/sklearn.svm.SVC.html#sklearn.svm.SVC" title="sklearn.svm.SVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">SVC</span></code></a>, <a class="reference internal" href="generated/sklearn.svm.NuSVC.html#sklearn.svm.NuSVC" title="sklearn.svm.NuSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">NuSVC</span></code></a> and
<a class="reference internal" href="generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearSVC</span></code></a> take as input two arrays: an array <code class="docutils literal notranslate"><span class="pre">X</span></code> of shape
<code class="docutils literal notranslate"><span class="pre">(n_samples,</span> <span class="pre">n_features)</span></code> holding the training samples, and an array <code class="docutils literal notranslate"><span class="pre">y</span></code> of
class labels (strings or integers), of shape <code class="docutils literal notranslate"><span class="pre">(n_samples)</span></code>:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">svm</span>
<span class="gp">>>> </span><span class="n">X</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]]</span>
<span class="gp">>>> </span><span class="n">y</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">clf</span> <span class="o">=</span> <span class="n">svm</span><span class="o">.</span><span class="n">SVC</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="go">SVC()</span>
</pre></div>
</div>
<p>After being fitted, the model can then be used to predict new values:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">clf</span><span class="o">.</span><span class="n">predict</span><span class="p">([[</span><span class="mf">2.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">]])</span>
<span class="go">array([1])</span>
</pre></div>
</div>
<p>SVMs decision function (detailed in the <a class="reference internal" href="#svm-mathematical-formulation"><span class="std std-ref">Mathematical formulation</span></a>)
depends on some subset of the training data, called the support vectors. Some
properties of these support vectors can be found in attributes
<code class="docutils literal notranslate"><span class="pre">support_vectors_</span></code>, <code class="docutils literal notranslate"><span class="pre">support_</span></code> and <code class="docutils literal notranslate"><span class="pre">n_support_</span></code>:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="c1"># get support vectors</span>
<span class="gp">>>> </span><span class="n">clf</span><span class="o">.</span><span class="n">support_vectors_</span>
<span class="go">array([[0., 0.],</span>
<span class="go"> [1., 1.]])</span>
<span class="gp">>>> </span><span class="c1"># get indices of support vectors</span>
<span class="gp">>>> </span><span class="n">clf</span><span class="o">.</span><span class="n">support_</span>
<span class="go">array([0, 1]...)</span>
<span class="gp">>>> </span><span class="c1"># get number of support vectors for each class</span>
<span class="gp">>>> </span><span class="n">clf</span><span class="o">.</span><span class="n">n_support_</span>
<span class="go">array([1, 1]...)</span>
</pre></div>
</div>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/svm/plot_separating_hyperplane.html#sphx-glr-auto-examples-svm-plot-separating-hyperplane-py"><span class="std std-ref">SVM: Maximum margin separating hyperplane</span></a>,</p></li>
<li><p><a class="reference internal" href="../auto_examples/svm/plot_svm_nonlinear.html#sphx-glr-auto-examples-svm-plot-svm-nonlinear-py"><span class="std std-ref">Non-linear SVM</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/svm/plot_svm_anova.html#sphx-glr-auto-examples-svm-plot-svm-anova-py"><span class="std std-ref">SVM-Anova: SVM with univariate feature selection</span></a>,</p></li>
</ul>
</aside>
<section id="multi-class-classification">
<span id="svm-multi-class"></span><h3><span class="section-number">1.4.1.1. </span>Multi-class classification<a class="headerlink" href="#multi-class-classification" title="Permalink to this heading">¶</a></h3>
<p><a class="reference internal" href="generated/sklearn.svm.SVC.html#sklearn.svm.SVC" title="sklearn.svm.SVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">SVC</span></code></a> and <a class="reference internal" href="generated/sklearn.svm.NuSVC.html#sklearn.svm.NuSVC" title="sklearn.svm.NuSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">NuSVC</span></code></a> implement the “one-versus-one”
approach for multi-class classification. In total,
<code class="docutils literal notranslate"><span class="pre">n_classes</span> <span class="pre">*</span> <span class="pre">(n_classes</span> <span class="pre">-</span> <span class="pre">1)</span> <span class="pre">/</span> <span class="pre">2</span></code>
classifiers are constructed and each one trains data from two classes.
To provide a consistent interface with other classifiers, the
<code class="docutils literal notranslate"><span class="pre">decision_function_shape</span></code> option allows to monotonically transform the
results of the “one-versus-one” classifiers to a “one-vs-rest” decision
function of shape <code class="docutils literal notranslate"><span class="pre">(n_samples,</span> <span class="pre">n_classes)</span></code>.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">X</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">]]</span>
<span class="gp">>>> </span><span class="n">Y</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">clf</span> <span class="o">=</span> <span class="n">svm</span><span class="o">.</span><span class="n">SVC</span><span class="p">(</span><span class="n">decision_function_shape</span><span class="o">=</span><span class="s1">'ovo'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>
<span class="go">SVC(decision_function_shape='ovo')</span>
<span class="gp">>>> </span><span class="n">dec</span> <span class="o">=</span> <span class="n">clf</span><span class="o">.</span><span class="n">decision_function</span><span class="p">([[</span><span class="mi">1</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="n">dec</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="c1"># 4 classes: 4*3/2 = 6</span>
<span class="go">6</span>
<span class="gp">>>> </span><span class="n">clf</span><span class="o">.</span><span class="n">decision_function_shape</span> <span class="o">=</span> <span class="s2">"ovr"</span>
<span class="gp">>>> </span><span class="n">dec</span> <span class="o">=</span> <span class="n">clf</span><span class="o">.</span><span class="n">decision_function</span><span class="p">([[</span><span class="mi">1</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="n">dec</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="c1"># 4 classes</span>
<span class="go">4</span>
</pre></div>
</div>
<p>On the other hand, <a class="reference internal" href="generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearSVC</span></code></a> implements “one-vs-the-rest”
multi-class strategy, thus training <code class="docutils literal notranslate"><span class="pre">n_classes</span></code> models.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">lin_clf</span> <span class="o">=</span> <span class="n">svm</span><span class="o">.</span><span class="n">LinearSVC</span><span class="p">(</span><span class="n">dual</span><span class="o">=</span><span class="s2">"auto"</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">lin_clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>
<span class="go">LinearSVC(dual='auto')</span>
<span class="gp">>>> </span><span class="n">dec</span> <span class="o">=</span> <span class="n">lin_clf</span><span class="o">.</span><span class="n">decision_function</span><span class="p">([[</span><span class="mi">1</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="n">dec</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
<span class="go">4</span>
</pre></div>
</div>
<p>See <a class="reference internal" href="#svm-mathematical-formulation"><span class="std std-ref">Mathematical formulation</span></a> for a complete description of
the decision function.</p>
<p><details>
<summary class="btn btn-light">
<strong>Details on multi-class strategies</strong>
<span class="tooltiptext">Click for more details</span>
</summary>
<div class="card"></p>
<p>Note that the <a class="reference internal" href="generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearSVC</span></code></a> also implements an alternative multi-class
strategy, the so-called multi-class SVM formulated by Crammer and Singer
<a class="footnote-reference brackets" href="#id18" id="id1" role="doc-noteref"><span class="fn-bracket">[</span>16<span class="fn-bracket">]</span></a>, by using the option <code class="docutils literal notranslate"><span class="pre">multi_class='crammer_singer'</span></code>. In practice,
one-vs-rest classification is usually preferred, since the results are mostly
similar, but the runtime is significantly less.</p>
<p>For “one-vs-rest” <a class="reference internal" href="generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearSVC</span></code></a> the attributes <code class="docutils literal notranslate"><span class="pre">coef_</span></code> and <code class="docutils literal notranslate"><span class="pre">intercept_</span></code>
have the shape <code class="docutils literal notranslate"><span class="pre">(n_classes,</span> <span class="pre">n_features)</span></code> and <code class="docutils literal notranslate"><span class="pre">(n_classes,)</span></code> respectively.
Each row of the coefficients corresponds to one of the <code class="docutils literal notranslate"><span class="pre">n_classes</span></code>
“one-vs-rest” classifiers and similar for the intercepts, in the
order of the “one” class.</p>
<p>In the case of “one-vs-one” <a class="reference internal" href="generated/sklearn.svm.SVC.html#sklearn.svm.SVC" title="sklearn.svm.SVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">SVC</span></code></a> and <a class="reference internal" href="generated/sklearn.svm.NuSVC.html#sklearn.svm.NuSVC" title="sklearn.svm.NuSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">NuSVC</span></code></a>, the layout of
the attributes is a little more involved. In the case of a linear
kernel, the attributes <code class="docutils literal notranslate"><span class="pre">coef_</span></code> and <code class="docutils literal notranslate"><span class="pre">intercept_</span></code> have the shape
<code class="docutils literal notranslate"><span class="pre">(n_classes</span> <span class="pre">*</span> <span class="pre">(n_classes</span> <span class="pre">-</span> <span class="pre">1)</span> <span class="pre">/</span> <span class="pre">2,</span> <span class="pre">n_features)</span></code> and <code class="docutils literal notranslate"><span class="pre">(n_classes</span> <span class="pre">*</span>
<span class="pre">(n_classes</span> <span class="pre">-</span> <span class="pre">1)</span> <span class="pre">/</span> <span class="pre">2)</span></code> respectively. This is similar to the layout for
<a class="reference internal" href="generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearSVC</span></code></a> described above, with each row now corresponding
to a binary classifier. The order for classes
0 to n is “0 vs 1”, “0 vs 2” , … “0 vs n”, “1 vs 2”, “1 vs 3”, “1 vs n”, . .
. “n-1 vs n”.</p>
<p>The shape of <code class="docutils literal notranslate"><span class="pre">dual_coef_</span></code> is <code class="docutils literal notranslate"><span class="pre">(n_classes-1,</span> <span class="pre">n_SV)</span></code> with
a somewhat hard to grasp layout.
The columns correspond to the support vectors involved in any
of the <code class="docutils literal notranslate"><span class="pre">n_classes</span> <span class="pre">*</span> <span class="pre">(n_classes</span> <span class="pre">-</span> <span class="pre">1)</span> <span class="pre">/</span> <span class="pre">2</span></code> “one-vs-one” classifiers.
Each support vector <code class="docutils literal notranslate"><span class="pre">v</span></code> has a dual coefficient in each of the
<code class="docutils literal notranslate"><span class="pre">n_classes</span> <span class="pre">-</span> <span class="pre">1</span></code> classifiers comparing the class of <code class="docutils literal notranslate"><span class="pre">v</span></code> against another class.
Note that some, but not all, of these dual coefficients, may be zero.
The <code class="docutils literal notranslate"><span class="pre">n_classes</span> <span class="pre">-</span> <span class="pre">1</span></code> entries in each column are these dual coefficients,
ordered by the opposing class.</p>
<p>This might be clearer with an example: consider a three class problem with
class 0 having three support vectors
<span class="math notranslate nohighlight">\(v^{0}_0, v^{1}_0, v^{2}_0\)</span> and class 1 and 2 having two support vectors
<span class="math notranslate nohighlight">\(v^{0}_1, v^{1}_1\)</span> and <span class="math notranslate nohighlight">\(v^{0}_2, v^{1}_2\)</span> respectively. For each
support vector <span class="math notranslate nohighlight">\(v^{j}_i\)</span>, there are two dual coefficients. Let’s call
the coefficient of support vector <span class="math notranslate nohighlight">\(v^{j}_i\)</span> in the classifier between
classes <span class="math notranslate nohighlight">\(i\)</span> and <span class="math notranslate nohighlight">\(k\)</span> <span class="math notranslate nohighlight">\(\alpha^{j}_{i,k}\)</span>.
Then <code class="docutils literal notranslate"><span class="pre">dual_coef_</span></code> looks like this:</p>
<table class="docutils align-default">
<tbody>
<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(\alpha^{0}_{0,1}\)</span></p></td>
<td><p><span class="math notranslate nohighlight">\(\alpha^{1}_{0,1}\)</span></p></td>
<td><p><span class="math notranslate nohighlight">\(\alpha^{2}_{0,1}\)</span></p></td>
<td><p><span class="math notranslate nohighlight">\(\alpha^{0}_{1,0}\)</span></p></td>
<td><p><span class="math notranslate nohighlight">\(\alpha^{1}_{1,0}\)</span></p></td>
<td><p><span class="math notranslate nohighlight">\(\alpha^{0}_{2,0}\)</span></p></td>
<td><p><span class="math notranslate nohighlight">\(\alpha^{1}_{2,0}\)</span></p></td>
</tr>
<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(\alpha^{0}_{0,2}\)</span></p></td>
<td><p><span class="math notranslate nohighlight">\(\alpha^{1}_{0,2}\)</span></p></td>
<td><p><span class="math notranslate nohighlight">\(\alpha^{2}_{0,2}\)</span></p></td>
<td><p><span class="math notranslate nohighlight">\(\alpha^{0}_{1,2}\)</span></p></td>
<td><p><span class="math notranslate nohighlight">\(\alpha^{1}_{1,2}\)</span></p></td>
<td><p><span class="math notranslate nohighlight">\(\alpha^{0}_{2,1}\)</span></p></td>
<td><p><span class="math notranslate nohighlight">\(\alpha^{1}_{2,1}\)</span></p></td>
</tr>
<tr class="row-odd"><td colspan="3"><p>Coefficients
for SVs of class 0</p></td>
<td colspan="2"><p>Coefficients
for SVs of class 1</p></td>
<td colspan="2"><p>Coefficients
for SVs of class 2</p></td>
</tr>
</tbody>
</table>
<p></div>
</details></p>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/svm/plot_iris_svc.html#sphx-glr-auto-examples-svm-plot-iris-svc-py"><span class="std std-ref">Plot different SVM classifiers in the iris dataset</span></a>,</p></li>
</ul>
</aside>
</section>
<section id="scores-and-probabilities">
<span id="scores-probabilities"></span><h3><span class="section-number">1.4.1.2. </span>Scores and probabilities<a class="headerlink" href="#scores-and-probabilities" title="Permalink to this heading">¶</a></h3>
<p>The <code class="docutils literal notranslate"><span class="pre">decision_function</span></code> method of <a class="reference internal" href="generated/sklearn.svm.SVC.html#sklearn.svm.SVC" title="sklearn.svm.SVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">SVC</span></code></a> and <a class="reference internal" href="generated/sklearn.svm.NuSVC.html#sklearn.svm.NuSVC" title="sklearn.svm.NuSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">NuSVC</span></code></a> gives
per-class scores for each sample (or a single score per sample in the binary
case). When the constructor option <code class="docutils literal notranslate"><span class="pre">probability</span></code> is set to <code class="docutils literal notranslate"><span class="pre">True</span></code>,
class membership probability estimates (from the methods <code class="docutils literal notranslate"><span class="pre">predict_proba</span></code> and
<code class="docutils literal notranslate"><span class="pre">predict_log_proba</span></code>) are enabled. In the binary case, the probabilities are
calibrated using Platt scaling <a class="footnote-reference brackets" href="#id11" id="id2" role="doc-noteref"><span class="fn-bracket">[</span>9<span class="fn-bracket">]</span></a>: logistic regression on the SVM’s scores,
fit by an additional cross-validation on the training data.
In the multiclass case, this is extended as per <a class="footnote-reference brackets" href="#id12" id="id3" role="doc-noteref"><span class="fn-bracket">[</span>10<span class="fn-bracket">]</span></a>.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The same probability calibration procedure is available for all estimators
via the <a class="reference internal" href="generated/sklearn.calibration.CalibratedClassifierCV.html#sklearn.calibration.CalibratedClassifierCV" title="sklearn.calibration.CalibratedClassifierCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">CalibratedClassifierCV</span></code></a> (see
<a class="reference internal" href="calibration.html#calibration"><span class="std std-ref">Probability calibration</span></a>). In the case of <a class="reference internal" href="generated/sklearn.svm.SVC.html#sklearn.svm.SVC" title="sklearn.svm.SVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">SVC</span></code></a> and <a class="reference internal" href="generated/sklearn.svm.NuSVC.html#sklearn.svm.NuSVC" title="sklearn.svm.NuSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">NuSVC</span></code></a>, this
procedure is builtin in <a class="reference external" href="https://www.csie.ntu.edu.tw/~cjlin/libsvm/">libsvm</a> which is used under the hood, so it does
not rely on scikit-learn’s
<a class="reference internal" href="generated/sklearn.calibration.CalibratedClassifierCV.html#sklearn.calibration.CalibratedClassifierCV" title="sklearn.calibration.CalibratedClassifierCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">CalibratedClassifierCV</span></code></a>.</p>
</div>
<p>The cross-validation involved in Platt scaling
is an expensive operation for large datasets.
In addition, the probability estimates may be inconsistent with the scores:</p>
<ul class="simple">
<li><p>the “argmax” of the scores may not be the argmax of the probabilities</p></li>
<li><p>in binary classification, a sample may be labeled by <code class="docutils literal notranslate"><span class="pre">predict</span></code> as
belonging to the positive class even if the output of <code class="docutils literal notranslate"><span class="pre">predict_proba</span></code> is
less than 0.5; and similarly, it could be labeled as negative even if the
output of <code class="docutils literal notranslate"><span class="pre">predict_proba</span></code> is more than 0.5.</p></li>
</ul>
<p>Platt’s method is also known to have theoretical issues.
If confidence scores are required, but these do not have to be probabilities,
then it is advisable to set <code class="docutils literal notranslate"><span class="pre">probability=False</span></code>
and use <code class="docutils literal notranslate"><span class="pre">decision_function</span></code> instead of <code class="docutils literal notranslate"><span class="pre">predict_proba</span></code>.</p>
<p>Please note that when <code class="docutils literal notranslate"><span class="pre">decision_function_shape='ovr'</span></code> and <code class="docutils literal notranslate"><span class="pre">n_classes</span> <span class="pre">></span> <span class="pre">2</span></code>,
unlike <code class="docutils literal notranslate"><span class="pre">decision_function</span></code>, the <code class="docutils literal notranslate"><span class="pre">predict</span></code> method does not try to break ties
by default. You can set <code class="docutils literal notranslate"><span class="pre">break_ties=True</span></code> for the output of <code class="docutils literal notranslate"><span class="pre">predict</span></code> to be
the same as <code class="docutils literal notranslate"><span class="pre">np.argmax(clf.decision_function(...),</span> <span class="pre">axis=1)</span></code>, otherwise the
first class among the tied classes will always be returned; but have in mind
that it comes with a computational cost. See
<a class="reference internal" href="../auto_examples/svm/plot_svm_tie_breaking.html#sphx-glr-auto-examples-svm-plot-svm-tie-breaking-py"><span class="std std-ref">SVM Tie Breaking Example</span></a> for an example on
tie breaking.</p>
</section>
<section id="unbalanced-problems">
<h3><span class="section-number">1.4.1.3. </span>Unbalanced problems<a class="headerlink" href="#unbalanced-problems" title="Permalink to this heading">¶</a></h3>
<p>In problems where it is desired to give more importance to certain
classes or certain individual samples, the parameters <code class="docutils literal notranslate"><span class="pre">class_weight</span></code> and
<code class="docutils literal notranslate"><span class="pre">sample_weight</span></code> can be used.</p>
<p><a class="reference internal" href="generated/sklearn.svm.SVC.html#sklearn.svm.SVC" title="sklearn.svm.SVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">SVC</span></code></a> (but not <a class="reference internal" href="generated/sklearn.svm.NuSVC.html#sklearn.svm.NuSVC" title="sklearn.svm.NuSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">NuSVC</span></code></a>) implements the parameter
<code class="docutils literal notranslate"><span class="pre">class_weight</span></code> in the <code class="docutils literal notranslate"><span class="pre">fit</span></code> method. It’s a dictionary of the form
<code class="docutils literal notranslate"><span class="pre">{class_label</span> <span class="pre">:</span> <span class="pre">value}</span></code>, where value is a floating point number > 0
that sets the parameter <code class="docutils literal notranslate"><span class="pre">C</span></code> of class <code class="docutils literal notranslate"><span class="pre">class_label</span></code> to <code class="docutils literal notranslate"><span class="pre">C</span> <span class="pre">*</span> <span class="pre">value</span></code>.
The figure below illustrates the decision boundary of an unbalanced problem,
with and without weight correction.</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/svm/plot_separating_hyperplane_unbalanced.html"><img alt="../_images/sphx_glr_plot_separating_hyperplane_unbalanced_001.png" src="../_images/sphx_glr_plot_separating_hyperplane_unbalanced_001.png" style="width: 480.0px; height: 360.0px;" /></a>
</figure>
<p><a class="reference internal" href="generated/sklearn.svm.SVC.html#sklearn.svm.SVC" title="sklearn.svm.SVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">SVC</span></code></a>, <a class="reference internal" href="generated/sklearn.svm.NuSVC.html#sklearn.svm.NuSVC" title="sklearn.svm.NuSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">NuSVC</span></code></a>, <a class="reference internal" href="generated/sklearn.svm.SVR.html#sklearn.svm.SVR" title="sklearn.svm.SVR"><code class="xref py py-class docutils literal notranslate"><span class="pre">SVR</span></code></a>, <a class="reference internal" href="generated/sklearn.svm.NuSVR.html#sklearn.svm.NuSVR" title="sklearn.svm.NuSVR"><code class="xref py py-class docutils literal notranslate"><span class="pre">NuSVR</span></code></a>, <a class="reference internal" href="generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearSVC</span></code></a>,
<a class="reference internal" href="generated/sklearn.svm.LinearSVR.html#sklearn.svm.LinearSVR" title="sklearn.svm.LinearSVR"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearSVR</span></code></a> and <a class="reference internal" href="generated/sklearn.svm.OneClassSVM.html#sklearn.svm.OneClassSVM" title="sklearn.svm.OneClassSVM"><code class="xref py py-class docutils literal notranslate"><span class="pre">OneClassSVM</span></code></a> implement also weights for
individual samples in the <code class="docutils literal notranslate"><span class="pre">fit</span></code> method through the <code class="docutils literal notranslate"><span class="pre">sample_weight</span></code> parameter.
Similar to <code class="docutils literal notranslate"><span class="pre">class_weight</span></code>, this sets the parameter <code class="docutils literal notranslate"><span class="pre">C</span></code> for the i-th
example to <code class="docutils literal notranslate"><span class="pre">C</span> <span class="pre">*</span> <span class="pre">sample_weight[i]</span></code>, which will encourage the classifier to
get these samples right. The figure below illustrates the effect of sample
weighting on the decision boundary. The size of the circles is proportional
to the sample weights:</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/svm/plot_weighted_samples.html"><img alt="../_images/sphx_glr_plot_weighted_samples_001.png" src="../_images/sphx_glr_plot_weighted_samples_001.png" style="width: 1050.0px; height: 450.0px;" /></a>
</figure>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/svm/plot_separating_hyperplane_unbalanced.html#sphx-glr-auto-examples-svm-plot-separating-hyperplane-unbalanced-py"><span class="std std-ref">SVM: Separating hyperplane for unbalanced classes</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/svm/plot_weighted_samples.html#sphx-glr-auto-examples-svm-plot-weighted-samples-py"><span class="std std-ref">SVM: Weighted samples</span></a>,</p></li>
</ul>
</aside>
</section>
</section>
<section id="regression">
<span id="svm-regression"></span><h2><span class="section-number">1.4.2. </span>Regression<a class="headerlink" href="#regression" title="Permalink to this heading">¶</a></h2>
<p>The method of Support Vector Classification can be extended to solve
regression problems. This method is called Support Vector Regression.</p>
<p>The model produced by support vector classification (as described
above) depends only on a subset of the training data, because the cost
function for building the model does not care about training points
that lie beyond the margin. Analogously, the model produced by Support
Vector Regression depends only on a subset of the training data,
because the cost function ignores samples whose prediction is close to their
target.</p>
<p>There are three different implementations of Support Vector Regression:
<a class="reference internal" href="generated/sklearn.svm.SVR.html#sklearn.svm.SVR" title="sklearn.svm.SVR"><code class="xref py py-class docutils literal notranslate"><span class="pre">SVR</span></code></a>, <a class="reference internal" href="generated/sklearn.svm.NuSVR.html#sklearn.svm.NuSVR" title="sklearn.svm.NuSVR"><code class="xref py py-class docutils literal notranslate"><span class="pre">NuSVR</span></code></a> and <a class="reference internal" href="generated/sklearn.svm.LinearSVR.html#sklearn.svm.LinearSVR" title="sklearn.svm.LinearSVR"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearSVR</span></code></a>. <a class="reference internal" href="generated/sklearn.svm.LinearSVR.html#sklearn.svm.LinearSVR" title="sklearn.svm.LinearSVR"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearSVR</span></code></a>
provides a faster implementation than <a class="reference internal" href="generated/sklearn.svm.SVR.html#sklearn.svm.SVR" title="sklearn.svm.SVR"><code class="xref py py-class docutils literal notranslate"><span class="pre">SVR</span></code></a> but only considers
the linear kernel, while <a class="reference internal" href="generated/sklearn.svm.NuSVR.html#sklearn.svm.NuSVR" title="sklearn.svm.NuSVR"><code class="xref py py-class docutils literal notranslate"><span class="pre">NuSVR</span></code></a> implements a slightly different
formulation than <a class="reference internal" href="generated/sklearn.svm.SVR.html#sklearn.svm.SVR" title="sklearn.svm.SVR"><code class="xref py py-class docutils literal notranslate"><span class="pre">SVR</span></code></a> and <a class="reference internal" href="generated/sklearn.svm.LinearSVR.html#sklearn.svm.LinearSVR" title="sklearn.svm.LinearSVR"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearSVR</span></code></a>. See
<a class="reference internal" href="#svm-implementation-details"><span class="std std-ref">Implementation details</span></a> for further details.</p>
<p>As with classification classes, the fit method will take as
argument vectors X, y, only that in this case y is expected to have
floating point values instead of integer values:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">svm</span>
<span class="gp">>>> </span><span class="n">X</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">]]</span>
<span class="gp">>>> </span><span class="n">y</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">regr</span> <span class="o">=</span> <span class="n">svm</span><span class="o">.</span><span class="n">SVR</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">regr</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="go">SVR()</span>
<span class="gp">>>> </span><span class="n">regr</span><span class="o">.</span><span class="n">predict</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
<span class="go">array([1.5])</span>
</pre></div>
</div>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/svm/plot_svm_regression.html#sphx-glr-auto-examples-svm-plot-svm-regression-py"><span class="std std-ref">Support Vector Regression (SVR) using linear and non-linear kernels</span></a></p></li>
</ul>
</aside>
</section>
<section id="density-estimation-novelty-detection">
<span id="svm-outlier-detection"></span><h2><span class="section-number">1.4.3. </span>Density estimation, novelty detection<a class="headerlink" href="#density-estimation-novelty-detection" title="Permalink to this heading">¶</a></h2>
<p>The class <a class="reference internal" href="generated/sklearn.svm.OneClassSVM.html#sklearn.svm.OneClassSVM" title="sklearn.svm.OneClassSVM"><code class="xref py py-class docutils literal notranslate"><span class="pre">OneClassSVM</span></code></a> implements a One-Class SVM which is used in
outlier detection.</p>
<p>See <a class="reference internal" href="outlier_detection.html#outlier-detection"><span class="std std-ref">Novelty and Outlier Detection</span></a> for the description and usage of OneClassSVM.</p>
</section>
<section id="complexity">
<h2><span class="section-number">1.4.4. </span>Complexity<a class="headerlink" href="#complexity" title="Permalink to this heading">¶</a></h2>
<p>Support Vector Machines are powerful tools, but their compute and
storage requirements increase rapidly with the number of training
vectors. The core of an SVM is a quadratic programming problem (QP),
separating support vectors from the rest of the training data. The QP
solver used by the <a class="reference external" href="https://www.csie.ntu.edu.tw/~cjlin/libsvm/">libsvm</a>-based implementation scales between
<span class="math notranslate nohighlight">\(O(n_{features} \times n_{samples}^2)\)</span> and
<span class="math notranslate nohighlight">\(O(n_{features} \times n_{samples}^3)\)</span> depending on how efficiently
the <a class="reference external" href="https://www.csie.ntu.edu.tw/~cjlin/libsvm/">libsvm</a> cache is used in practice (dataset dependent). If the data
is very sparse <span class="math notranslate nohighlight">\(n_{features}\)</span> should be replaced by the average number
of non-zero features in a sample vector.</p>
<p>For the linear case, the algorithm used in
<a class="reference internal" href="generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearSVC</span></code></a> by the <a class="reference external" href="https://www.csie.ntu.edu.tw/~cjlin/liblinear/">liblinear</a> implementation is much more
efficient than its <a class="reference external" href="https://www.csie.ntu.edu.tw/~cjlin/libsvm/">libsvm</a>-based <a class="reference internal" href="generated/sklearn.svm.SVC.html#sklearn.svm.SVC" title="sklearn.svm.SVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">SVC</span></code></a> counterpart and can
scale almost linearly to millions of samples and/or features.</p>
</section>
<section id="tips-on-practical-use">
<h2><span class="section-number">1.4.5. </span>Tips on Practical Use<a class="headerlink" href="#tips-on-practical-use" title="Permalink to this heading">¶</a></h2>
<blockquote>
<div><ul>
<li><p><strong>Avoiding data copy</strong>: For <a class="reference internal" href="generated/sklearn.svm.SVC.html#sklearn.svm.SVC" title="sklearn.svm.SVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">SVC</span></code></a>, <a class="reference internal" href="generated/sklearn.svm.SVR.html#sklearn.svm.SVR" title="sklearn.svm.SVR"><code class="xref py py-class docutils literal notranslate"><span class="pre">SVR</span></code></a>, <a class="reference internal" href="generated/sklearn.svm.NuSVC.html#sklearn.svm.NuSVC" title="sklearn.svm.NuSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">NuSVC</span></code></a> and
<a class="reference internal" href="generated/sklearn.svm.NuSVR.html#sklearn.svm.NuSVR" title="sklearn.svm.NuSVR"><code class="xref py py-class docutils literal notranslate"><span class="pre">NuSVR</span></code></a>, if the data passed to certain methods is not C-ordered
contiguous and double precision, it will be copied before calling the
underlying C implementation. You can check whether a given numpy array is
C-contiguous by inspecting its <code class="docutils literal notranslate"><span class="pre">flags</span></code> attribute.</p>
<p>For <a class="reference internal" href="generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearSVC</span></code></a> (and <a class="reference internal" href="generated/sklearn.linear_model.LogisticRegression.html#sklearn.linear_model.LogisticRegression" title="sklearn.linear_model.LogisticRegression"><code class="xref py py-class docutils literal notranslate"><span class="pre">LogisticRegression</span></code></a>) any input passed as a numpy
array will be copied and converted to the <a class="reference external" href="https://www.csie.ntu.edu.tw/~cjlin/liblinear/">liblinear</a> internal sparse data
representation (double precision floats and int32 indices of non-zero
components). If you want to fit a large-scale linear classifier without
copying a dense numpy C-contiguous double precision array as input, we
suggest to use the <a class="reference internal" href="generated/sklearn.linear_model.SGDClassifier.html#sklearn.linear_model.SGDClassifier" title="sklearn.linear_model.SGDClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">SGDClassifier</span></code></a> class instead. The objective
function can be configured to be almost the same as the <a class="reference internal" href="generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearSVC</span></code></a>
model.</p>
</li>
<li><p><strong>Kernel cache size</strong>: For <a class="reference internal" href="generated/sklearn.svm.SVC.html#sklearn.svm.SVC" title="sklearn.svm.SVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">SVC</span></code></a>, <a class="reference internal" href="generated/sklearn.svm.SVR.html#sklearn.svm.SVR" title="sklearn.svm.SVR"><code class="xref py py-class docutils literal notranslate"><span class="pre">SVR</span></code></a>, <a class="reference internal" href="generated/sklearn.svm.NuSVC.html#sklearn.svm.NuSVC" title="sklearn.svm.NuSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">NuSVC</span></code></a> and
<a class="reference internal" href="generated/sklearn.svm.NuSVR.html#sklearn.svm.NuSVR" title="sklearn.svm.NuSVR"><code class="xref py py-class docutils literal notranslate"><span class="pre">NuSVR</span></code></a>, the size of the kernel cache has a strong impact on run
times for larger problems. If you have enough RAM available, it is
recommended to set <code class="docutils literal notranslate"><span class="pre">cache_size</span></code> to a higher value than the default of
200(MB), such as 500(MB) or 1000(MB).</p></li>
<li><p><strong>Setting C</strong>: <code class="docutils literal notranslate"><span class="pre">C</span></code> is <code class="docutils literal notranslate"><span class="pre">1</span></code> by default and it’s a reasonable default
choice. If you have a lot of noisy observations you should decrease it:
decreasing C corresponds to more regularization.</p>
<p><a class="reference internal" href="generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearSVC</span></code></a> and <a class="reference internal" href="generated/sklearn.svm.LinearSVR.html#sklearn.svm.LinearSVR" title="sklearn.svm.LinearSVR"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearSVR</span></code></a> are less sensitive to <code class="docutils literal notranslate"><span class="pre">C</span></code> when
it becomes large, and prediction results stop improving after a certain
threshold. Meanwhile, larger <code class="docutils literal notranslate"><span class="pre">C</span></code> values will take more time to train,
sometimes up to 10 times longer, as shown in <a class="footnote-reference brackets" href="#id13" id="id4" role="doc-noteref"><span class="fn-bracket">[</span>11<span class="fn-bracket">]</span></a>.</p>
</li>
<li><p>Support Vector Machine algorithms are not scale invariant, so <strong>it
is highly recommended to scale your data</strong>. For example, scale each
attribute on the input vector X to [0,1] or [-1,+1], or standardize it
to have mean 0 and variance 1. Note that the <em>same</em> scaling must be
applied to the test vector to obtain meaningful results. This can be done
easily by using a <a class="reference internal" href="generated/sklearn.pipeline.Pipeline.html#sklearn.pipeline.Pipeline" title="sklearn.pipeline.Pipeline"><code class="xref py py-class docutils literal notranslate"><span class="pre">Pipeline</span></code></a>:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.pipeline</span> <span class="kn">import</span> <span class="n">make_pipeline</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">StandardScaler</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.svm</span> <span class="kn">import</span> <span class="n">SVC</span>
<span class="gp">>>> </span><span class="n">clf</span> <span class="o">=</span> <span class="n">make_pipeline</span><span class="p">(</span><span class="n">StandardScaler</span><span class="p">(),</span> <span class="n">SVC</span><span class="p">())</span>
</pre></div>
</div>
<p>See section <a class="reference internal" href="preprocessing.html#preprocessing"><span class="std std-ref">Preprocessing data</span></a> for more details on scaling and
normalization.</p>
</li>
</ul>
<ul id="shrinking-svm">
<li><p>Regarding the <code class="docutils literal notranslate"><span class="pre">shrinking</span></code> parameter, quoting <a class="footnote-reference brackets" href="#id14" id="id5" role="doc-noteref"><span class="fn-bracket">[</span>12<span class="fn-bracket">]</span></a>: <em>We found that if the
number of iterations is large, then shrinking can shorten the training
time. However, if we loosely solve the optimization problem (e.g., by
using a large stopping tolerance), the code without using shrinking may
be much faster</em></p></li>
<li><p>Parameter <code class="docutils literal notranslate"><span class="pre">nu</span></code> in <a class="reference internal" href="generated/sklearn.svm.NuSVC.html#sklearn.svm.NuSVC" title="sklearn.svm.NuSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">NuSVC</span></code></a>/<a class="reference internal" href="generated/sklearn.svm.OneClassSVM.html#sklearn.svm.OneClassSVM" title="sklearn.svm.OneClassSVM"><code class="xref py py-class docutils literal notranslate"><span class="pre">OneClassSVM</span></code></a>/<a class="reference internal" href="generated/sklearn.svm.NuSVR.html#sklearn.svm.NuSVR" title="sklearn.svm.NuSVR"><code class="xref py py-class docutils literal notranslate"><span class="pre">NuSVR</span></code></a>
approximates the fraction of training errors and support vectors.</p></li>
<li><p>In <a class="reference internal" href="generated/sklearn.svm.SVC.html#sklearn.svm.SVC" title="sklearn.svm.SVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">SVC</span></code></a>, if the data is unbalanced (e.g. many
positive and few negative), set <code class="docutils literal notranslate"><span class="pre">class_weight='balanced'</span></code> and/or try
different penalty parameters <code class="docutils literal notranslate"><span class="pre">C</span></code>.</p></li>
<li><p><strong>Randomness of the underlying implementations</strong>: The underlying
implementations of <a class="reference internal" href="generated/sklearn.svm.SVC.html#sklearn.svm.SVC" title="sklearn.svm.SVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">SVC</span></code></a> and <a class="reference internal" href="generated/sklearn.svm.NuSVC.html#sklearn.svm.NuSVC" title="sklearn.svm.NuSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">NuSVC</span></code></a> use a random number
generator only to shuffle the data for probability estimation (when
<code class="docutils literal notranslate"><span class="pre">probability</span></code> is set to <code class="docutils literal notranslate"><span class="pre">True</span></code>). This randomness can be controlled
with the <code class="docutils literal notranslate"><span class="pre">random_state</span></code> parameter. If <code class="docutils literal notranslate"><span class="pre">probability</span></code> is set to <code class="docutils literal notranslate"><span class="pre">False</span></code>
these estimators are not random and <code class="docutils literal notranslate"><span class="pre">random_state</span></code> has no effect on the
results. The underlying <a class="reference internal" href="generated/sklearn.svm.OneClassSVM.html#sklearn.svm.OneClassSVM" title="sklearn.svm.OneClassSVM"><code class="xref py py-class docutils literal notranslate"><span class="pre">OneClassSVM</span></code></a> implementation is similar to
the ones of <a class="reference internal" href="generated/sklearn.svm.SVC.html#sklearn.svm.SVC" title="sklearn.svm.SVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">SVC</span></code></a> and <a class="reference internal" href="generated/sklearn.svm.NuSVC.html#sklearn.svm.NuSVC" title="sklearn.svm.NuSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">NuSVC</span></code></a>. As no probability estimation
is provided for <a class="reference internal" href="generated/sklearn.svm.OneClassSVM.html#sklearn.svm.OneClassSVM" title="sklearn.svm.OneClassSVM"><code class="xref py py-class docutils literal notranslate"><span class="pre">OneClassSVM</span></code></a>, it is not random.</p>
<p>The underlying <a class="reference internal" href="generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearSVC</span></code></a> implementation uses a random number
generator to select features when fitting the model with a dual coordinate
descent (i.e when <code class="docutils literal notranslate"><span class="pre">dual</span></code> is set to <code class="docutils literal notranslate"><span class="pre">True</span></code>). It is thus not uncommon
to have slightly different results for the same input data. If that
happens, try with a smaller <code class="docutils literal notranslate"><span class="pre">tol</span></code> parameter. This randomness can also be
controlled with the <code class="docutils literal notranslate"><span class="pre">random_state</span></code> parameter. When <code class="docutils literal notranslate"><span class="pre">dual</span></code> is
set to <code class="docutils literal notranslate"><span class="pre">False</span></code> the underlying implementation of <a class="reference internal" href="generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearSVC</span></code></a> is
not random and <code class="docutils literal notranslate"><span class="pre">random_state</span></code> has no effect on the results.</p>
</li>
<li><p>Using L1 penalization as provided by <code class="docutils literal notranslate"><span class="pre">LinearSVC(penalty='l1',</span>
<span class="pre">dual=False)</span></code> yields a sparse solution, i.e. only a subset of feature
weights is different from zero and contribute to the decision function.
Increasing <code class="docutils literal notranslate"><span class="pre">C</span></code> yields a more complex model (more features are selected).
The <code class="docutils literal notranslate"><span class="pre">C</span></code> value that yields a “null” model (all weights equal to zero) can
be calculated using <a class="reference internal" href="generated/sklearn.svm.l1_min_c.html#sklearn.svm.l1_min_c" title="sklearn.svm.l1_min_c"><code class="xref py py-func docutils literal notranslate"><span class="pre">l1_min_c</span></code></a>.</p></li>
</ul>
</div></blockquote>
</section>
<section id="kernel-functions">
<span id="svm-kernels"></span><h2><span class="section-number">1.4.6. </span>Kernel functions<a class="headerlink" href="#kernel-functions" title="Permalink to this heading">¶</a></h2>
<p>The <em>kernel function</em> can be any of the following:</p>
<blockquote>
<div><ul class="simple">
<li><p>linear: <span class="math notranslate nohighlight">\(\langle x, x'\rangle\)</span>.</p></li>
<li><p>polynomial: <span class="math notranslate nohighlight">\((\gamma \langle x, x'\rangle + r)^d\)</span>, where
<span class="math notranslate nohighlight">\(d\)</span> is specified by parameter <code class="docutils literal notranslate"><span class="pre">degree</span></code>, <span class="math notranslate nohighlight">\(r\)</span> by <code class="docutils literal notranslate"><span class="pre">coef0</span></code>.</p></li>
<li><p>rbf: <span class="math notranslate nohighlight">\(\exp(-\gamma \|x-x'\|^2)\)</span>, where <span class="math notranslate nohighlight">\(\gamma\)</span> is
specified by parameter <code class="docutils literal notranslate"><span class="pre">gamma</span></code>, must be greater than 0.</p></li>
<li><p>sigmoid <span class="math notranslate nohighlight">\(\tanh(\gamma \langle x,x'\rangle + r)\)</span>,
where <span class="math notranslate nohighlight">\(r\)</span> is specified by <code class="docutils literal notranslate"><span class="pre">coef0</span></code>.</p></li>
</ul>
</div></blockquote>
<p>Different kernels are specified by the <code class="docutils literal notranslate"><span class="pre">kernel</span></code> parameter:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">linear_svc</span> <span class="o">=</span> <span class="n">svm</span><span class="o">.</span><span class="n">SVC</span><span class="p">(</span><span class="n">kernel</span><span class="o">=</span><span class="s1">'linear'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">linear_svc</span><span class="o">.</span><span class="n">kernel</span>
<span class="go">'linear'</span>
<span class="gp">>>> </span><span class="n">rbf_svc</span> <span class="o">=</span> <span class="n">svm</span><span class="o">.</span><span class="n">SVC</span><span class="p">(</span><span class="n">kernel</span><span class="o">=</span><span class="s1">'rbf'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">rbf_svc</span><span class="o">.</span><span class="n">kernel</span>
<span class="go">'rbf'</span>
</pre></div>
</div>
<p>See also <a class="reference internal" href="kernel_approximation.html#kernel-approximation"><span class="std std-ref">Kernel Approximation</span></a> for a solution to use RBF kernels that is much faster and more scalable.</p>
<section id="parameters-of-the-rbf-kernel">
<h3><span class="section-number">1.4.6.1. </span>Parameters of the RBF Kernel<a class="headerlink" href="#parameters-of-the-rbf-kernel" title="Permalink to this heading">¶</a></h3>
<p>When training an SVM with the <em>Radial Basis Function</em> (RBF) kernel, two
parameters must be considered: <code class="docutils literal notranslate"><span class="pre">C</span></code> and <code class="docutils literal notranslate"><span class="pre">gamma</span></code>. The parameter <code class="docutils literal notranslate"><span class="pre">C</span></code>,
common to all SVM kernels, trades off misclassification of training examples
against simplicity of the decision surface. A low <code class="docutils literal notranslate"><span class="pre">C</span></code> makes the decision
surface smooth, while a high <code class="docutils literal notranslate"><span class="pre">C</span></code> aims at classifying all training examples
correctly. <code class="docutils literal notranslate"><span class="pre">gamma</span></code> defines how much influence a single training example has.
The larger <code class="docutils literal notranslate"><span class="pre">gamma</span></code> is, the closer other examples must be to be affected.</p>
<p>Proper choice of <code class="docutils literal notranslate"><span class="pre">C</span></code> and <code class="docutils literal notranslate"><span class="pre">gamma</span></code> is critical to the SVM’s performance. One
is advised to use <a class="reference internal" href="generated/sklearn.model_selection.GridSearchCV.html#sklearn.model_selection.GridSearchCV" title="sklearn.model_selection.GridSearchCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">GridSearchCV</span></code></a> with
<code class="docutils literal notranslate"><span class="pre">C</span></code> and <code class="docutils literal notranslate"><span class="pre">gamma</span></code> spaced exponentially far apart to choose good values.</p>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/svm/plot_rbf_parameters.html#sphx-glr-auto-examples-svm-plot-rbf-parameters-py"><span class="std std-ref">RBF SVM parameters</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/svm/plot_svm_nonlinear.html#sphx-glr-auto-examples-svm-plot-svm-nonlinear-py"><span class="std std-ref">Non-linear SVM</span></a></p></li>
</ul>
</aside>
<p><details>
<summary class="btn btn-light">
<strong>Custom Kernels</strong>
<span class="tooltiptext">Click for more details</span>
</summary>
<div class="card"></p>
<p>You can define your own kernels by either giving the kernel as a
python function or by precomputing the Gram matrix.</p>
<p>Classifiers with custom kernels behave the same way as any other
classifiers, except that:</p>
<blockquote>
<div><ul class="simple">
<li><p>Field <code class="docutils literal notranslate"><span class="pre">support_vectors_</span></code> is now empty, only indices of support
vectors are stored in <code class="docutils literal notranslate"><span class="pre">support_</span></code></p></li>
<li><p>A reference (and not a copy) of the first argument in the <code class="docutils literal notranslate"><span class="pre">fit()</span></code>
method is stored for future reference. If that array changes between the
use of <code class="docutils literal notranslate"><span class="pre">fit()</span></code> and <code class="docutils literal notranslate"><span class="pre">predict()</span></code> you will have unexpected results.</p></li>
</ul>
</div></blockquote>
<section id="using-python-functions-as-kernels">
<h4><span class="section-number">1.4.6.1.1. </span>Using Python functions as kernels<a class="headerlink" href="#using-python-functions-as-kernels" title="Permalink to this heading">¶</a></h4>
<p>You can use your own defined kernels by passing a function to the
<code class="docutils literal notranslate"><span class="pre">kernel</span></code> parameter.</p>
<p>Your kernel must take as arguments two matrices of shape
<code class="docutils literal notranslate"><span class="pre">(n_samples_1,</span> <span class="pre">n_features)</span></code>, <code class="docutils literal notranslate"><span class="pre">(n_samples_2,</span> <span class="pre">n_features)</span></code>
and return a kernel matrix of shape <code class="docutils literal notranslate"><span class="pre">(n_samples_1,</span> <span class="pre">n_samples_2)</span></code>.</p>
<p>The following code defines a linear kernel and creates a classifier
instance that will use that kernel:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">svm</span>
<span class="gp">>>> </span><span class="k">def</span> <span class="nf">my_kernel</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">):</span>
<span class="gp">... </span> <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="o">.</span><span class="n">T</span><span class="p">)</span>
<span class="gp">...</span>
<span class="gp">>>> </span><span class="n">clf</span> <span class="o">=</span> <span class="n">svm</span><span class="o">.</span><span class="n">SVC</span><span class="p">(</span><span class="n">kernel</span><span class="o">=</span><span class="n">my_kernel</span><span class="p">)</span>
</pre></div>
</div>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/svm/plot_custom_kernel.html#sphx-glr-auto-examples-svm-plot-custom-kernel-py"><span class="std std-ref">SVM with custom kernel</span></a>.</p></li>
</ul>
</aside>
</section>
<section id="using-the-gram-matrix">
<h4><span class="section-number">1.4.6.1.2. </span>Using the Gram matrix<a class="headerlink" href="#using-the-gram-matrix" title="Permalink to this heading">¶</a></h4>
<p>You can pass pre-computed kernels by using the <code class="docutils literal notranslate"><span class="pre">kernel='precomputed'</span></code>
option. You should then pass Gram matrix instead of X to the <code class="docutils literal notranslate"><span class="pre">fit</span></code> and
<code class="docutils literal notranslate"><span class="pre">predict</span></code> methods. The kernel values between <em>all</em> training vectors and the
test vectors must be provided:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">make_classification</span>
<span class="gp">>>> </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="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">svm</span>
<span class="gp">>>> </span><span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">make_classification</span><span class="p">(</span><span class="n">n_samples</span><span class="o">=</span><span class="mi">10</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="gp">>>> </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">X</span><span class="p">,</span> <span class="n">y</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="gp">>>> </span><span class="n">clf</span> <span class="o">=</span> <span class="n">svm</span><span class="o">.</span><span class="n">SVC</span><span class="p">(</span><span class="n">kernel</span><span class="o">=</span><span class="s1">'precomputed'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="c1"># linear kernel computation</span>
<span class="gp">>>> </span><span class="n">gram_train</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">X_train</span><span class="o">.</span><span class="n">T</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">gram_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="go">SVC(kernel='precomputed')</span>
<span class="gp">>>> </span><span class="c1"># predict on training examples</span>
<span class="gp">>>> </span><span class="n">gram_test</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">X_train</span><span class="o">.</span><span class="n">T</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">clf</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">gram_test</span><span class="p">)</span>
<span class="go">array([0, 1, 0])</span>
</pre></div>
</div>
<p></div>
</details></p>
</section>
</section>
</section>
<section id="mathematical-formulation">
<span id="svm-mathematical-formulation"></span><h2><span class="section-number">1.4.7. </span>Mathematical formulation<a class="headerlink" href="#mathematical-formulation" title="Permalink to this heading">¶</a></h2>
<p>A support vector machine constructs a hyper-plane or set of hyper-planes in a
high or infinite dimensional space, which can be used for
classification, regression or other tasks. Intuitively, a good
separation is achieved by the hyper-plane that has the largest distance
to the nearest training data points of any class (so-called functional
margin), since in general the larger the margin the lower the
generalization error of the classifier. The figure below shows the decision
function for a linearly separable problem, with three samples on the
margin boundaries, called “support vectors”:</p>
<figure class="align-center">
<a class="reference internal image-reference" href="../_images/sphx_glr_plot_separating_hyperplane_001.png"><img alt="../_images/sphx_glr_plot_separating_hyperplane_001.png" src="../_images/sphx_glr_plot_separating_hyperplane_001.png" style="width: 480.0px; height: 360.0px;" /></a>
</figure>
<p>In general, when the problem isn’t linearly separable, the support vectors
are the samples <em>within</em> the margin boundaries.</p>
<p>We recommend <a class="footnote-reference brackets" href="#id15" id="id6" role="doc-noteref"><span class="fn-bracket">[</span>13<span class="fn-bracket">]</span></a> and <a class="footnote-reference brackets" href="#id16" id="id7" role="doc-noteref"><span class="fn-bracket">[</span>14<span class="fn-bracket">]</span></a> as good references for the theory and
practicalities of SVMs.</p>
<section id="svc">
<h3><span class="section-number">1.4.7.1. </span>SVC<a class="headerlink" href="#svc" title="Permalink to this heading">¶</a></h3>
<p>Given training vectors <span class="math notranslate nohighlight">\(x_i \in \mathbb{R}^p\)</span>, i=1,…, n, in two classes, and a
vector <span class="math notranslate nohighlight">\(y \in \{1, -1\}^n\)</span>, our goal is to find <span class="math notranslate nohighlight">\(w \in
\mathbb{R}^p\)</span> and <span class="math notranslate nohighlight">\(b \in \mathbb{R}\)</span> such that the prediction given by
<span class="math notranslate nohighlight">\(\text{sign} (w^T\phi(x) + b)\)</span> is correct for most samples.</p>
<p>SVC solves the following primal problem:</p>
<div class="math notranslate nohighlight">
\[ \begin{align}\begin{aligned}\min_ {w, b, \zeta} \frac{1}{2} w^T w + C \sum_{i=1}^{n} \zeta_i\\\begin{split}\textrm {subject to } & y_i (w^T \phi (x_i) + b) \geq 1 - \zeta_i,\\
& \zeta_i \geq 0, i=1, ..., n\end{split}\end{aligned}\end{align} \]</div>
<p>Intuitively, we’re trying to maximize the margin (by minimizing
<span class="math notranslate nohighlight">\(||w||^2 = w^Tw\)</span>), while incurring a penalty when a sample is
misclassified or within the margin boundary. Ideally, the value <span class="math notranslate nohighlight">\(y_i
(w^T \phi (x_i) + b)\)</span> would be <span class="math notranslate nohighlight">\(\geq 1\)</span> for all samples, which
indicates a perfect prediction. But problems are usually not always perfectly
separable with a hyperplane, so we allow some samples to be at a distance <span class="math notranslate nohighlight">\(\zeta_i\)</span> from
their correct margin boundary. The penalty term <code class="docutils literal notranslate"><span class="pre">C</span></code> controls the strength of
this penalty, and as a result, acts as an inverse regularization parameter
(see note below).</p>
<p>The dual problem to the primal is</p>
<div class="math notranslate nohighlight">
\[ \begin{align}\begin{aligned}\min_{\alpha} \frac{1}{2} \alpha^T Q \alpha - e^T \alpha\\\begin{split}
\textrm {subject to } & y^T \alpha = 0\\
& 0 \leq \alpha_i \leq C, i=1, ..., n\end{split}\end{aligned}\end{align} \]</div>
<p>where <span class="math notranslate nohighlight">\(e\)</span> is the vector of all ones,
and <span class="math notranslate nohighlight">\(Q\)</span> is an <span class="math notranslate nohighlight">\(n\)</span> by <span class="math notranslate nohighlight">\(n\)</span> positive semidefinite matrix,
<span class="math notranslate nohighlight">\(Q_{ij} \equiv y_i y_j K(x_i, x_j)\)</span>, where <span class="math notranslate nohighlight">\(K(x_i, x_j) = \phi (x_i)^T \phi (x_j)\)</span>
is the kernel. The terms <span class="math notranslate nohighlight">\(\alpha_i\)</span> are called the dual coefficients,
and they are upper-bounded by <span class="math notranslate nohighlight">\(C\)</span>.
This dual representation highlights the fact that training vectors are
implicitly mapped into a higher (maybe infinite)
dimensional space by the function <span class="math notranslate nohighlight">\(\phi\)</span>: see <a class="reference external" href="https://en.wikipedia.org/wiki/Kernel_method">kernel trick</a>.</p>
<p>Once the optimization problem is solved, the output of
<a class="reference internal" href="../glossary.html#term-decision_function"><span class="xref std std-term">decision_function</span></a> for a given sample <span class="math notranslate nohighlight">\(x\)</span> becomes:</p>
<div class="math notranslate nohighlight">
\[\sum_{i\in SV} y_i \alpha_i K(x_i, x) + b,\]</div>
<p>and the predicted class correspond to its sign. We only need to sum over the
support vectors (i.e. the samples that lie within the margin) because the
dual coefficients <span class="math notranslate nohighlight">\(\alpha_i\)</span> are zero for the other samples.</p>
<p>These parameters can be accessed through the attributes <code class="docutils literal notranslate"><span class="pre">dual_coef_</span></code>
which holds the product <span class="math notranslate nohighlight">\(y_i \alpha_i\)</span>, <code class="docutils literal notranslate"><span class="pre">support_vectors_</span></code> which
holds the support vectors, and <code class="docutils literal notranslate"><span class="pre">intercept_</span></code> which holds the independent
term <span class="math notranslate nohighlight">\(b\)</span></p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>While SVM models derived from <a class="reference external" href="https://www.csie.ntu.edu.tw/~cjlin/libsvm/">libsvm</a> and <a class="reference external" href="https://www.csie.ntu.edu.tw/~cjlin/liblinear/">liblinear</a> use <code class="docutils literal notranslate"><span class="pre">C</span></code> as
regularization parameter, most other estimators use <code class="docutils literal notranslate"><span class="pre">alpha</span></code>. The exact
equivalence between the amount of regularization of two models depends on
the exact objective function optimized by the model. For example, when the
estimator used is <a class="reference internal" href="generated/sklearn.linear_model.Ridge.html#sklearn.linear_model.Ridge" title="sklearn.linear_model.Ridge"><code class="xref py py-class docutils literal notranslate"><span class="pre">Ridge</span></code></a> regression,
the relation between them is given as <span class="math notranslate nohighlight">\(C = \frac{1}{alpha}\)</span>.</p>
</div>
<p><details>
<summary class="btn btn-light">
<strong>LinearSVC</strong>
<span class="tooltiptext">Click for more details</span>
</summary>
<div class="card"></p>
<p>The primal problem can be equivalently formulated as</p>
<div class="math notranslate nohighlight">
\[\min_ {w, b} \frac{1}{2} w^T w + C \sum_{i=1}^{n}\max(0, 1 - y_i (w^T \phi(x_i) + b)),\]</div>
<p>where we make use of the <a class="reference external" href="https://en.wikipedia.org/wiki/Hinge_loss">hinge loss</a>. This is the form that is
directly optimized by <a class="reference internal" href="generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearSVC</span></code></a>, but unlike the dual form, this one
does not involve inner products between samples, so the famous kernel trick
cannot be applied. This is why only the linear kernel is supported by
<a class="reference internal" href="generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearSVC</span></code></a> (<span class="math notranslate nohighlight">\(\phi\)</span> is the identity function).</p>
<p></div>
</details></p>
<p id="nu-svc"><details>
<summary class="btn btn-light">
<strong>NuSVC</strong>
<span class="tooltiptext">Click for more details</span>
</summary>
<div class="card"></p>
<p>The <span class="math notranslate nohighlight">\(\nu\)</span>-SVC formulation <a class="footnote-reference brackets" href="#id17" id="id8" role="doc-noteref"><span class="fn-bracket">[</span>15<span class="fn-bracket">]</span></a> is a reparameterization of the
<span class="math notranslate nohighlight">\(C\)</span>-SVC and therefore mathematically equivalent.</p>
<p>We introduce a new parameter <span class="math notranslate nohighlight">\(\nu\)</span> (instead of <span class="math notranslate nohighlight">\(C\)</span>) which
controls the number of support vectors and <em>margin errors</em>:
<span class="math notranslate nohighlight">\(\nu \in (0, 1]\)</span> is an upper bound on the fraction of margin errors and
a lower bound of the fraction of support vectors. A margin error corresponds
to a sample that lies on the wrong side of its margin boundary: it is either
misclassified, or it is correctly classified but does not lie beyond the
margin.</p>
<p></div>
</details></p>
</section>
<section id="svr">
<h3><span class="section-number">1.4.7.2. </span>SVR<a class="headerlink" href="#svr" title="Permalink to this heading">¶</a></h3>
<p>Given training vectors <span class="math notranslate nohighlight">\(x_i \in \mathbb{R}^p\)</span>, i=1,…, n, and a
vector <span class="math notranslate nohighlight">\(y \in \mathbb{R}^n\)</span> <span class="math notranslate nohighlight">\(\varepsilon\)</span>-SVR solves the following primal problem:</p>
<div class="math notranslate nohighlight">
\[ \begin{align}\begin{aligned}\min_ {w, b, \zeta, \zeta^*} \frac{1}{2} w^T w + C \sum_{i=1}^{n} (\zeta_i + \zeta_i^*)\\\begin{split}\textrm {subject to } & y_i - w^T \phi (x_i) - b \leq \varepsilon + \zeta_i,\\
& w^T \phi (x_i) + b - y_i \leq \varepsilon + \zeta_i^*,\\
& \zeta_i, \zeta_i^* \geq 0, i=1, ..., n\end{split}\end{aligned}\end{align} \]</div>
<p>Here, we are penalizing samples whose prediction is at least <span class="math notranslate nohighlight">\(\varepsilon\)</span>
away from their true target. These samples penalize the objective by
<span class="math notranslate nohighlight">\(\zeta_i\)</span> or <span class="math notranslate nohighlight">\(\zeta_i^*\)</span>, depending on whether their predictions
lie above or below the <span class="math notranslate nohighlight">\(\varepsilon\)</span> tube.</p>
<p>The dual problem is</p>
<div class="math notranslate nohighlight">
\[ \begin{align}\begin{aligned}\min_{\alpha, \alpha^*} \frac{1}{2} (\alpha - \alpha^*)^T Q (\alpha - \alpha^*) + \varepsilon e^T (\alpha + \alpha^*) - y^T (\alpha - \alpha^*)\\\begin{split}
\textrm {subject to } & e^T (\alpha - \alpha^*) = 0\\
& 0 \leq \alpha_i, \alpha_i^* \leq C, i=1, ..., n\end{split}\end{aligned}\end{align} \]</div>
<p>where <span class="math notranslate nohighlight">\(e\)</span> is the vector of all ones,
<span class="math notranslate nohighlight">\(Q\)</span> is an <span class="math notranslate nohighlight">\(n\)</span> by <span class="math notranslate nohighlight">\(n\)</span> positive semidefinite matrix,
<span class="math notranslate nohighlight">\(Q_{ij} \equiv K(x_i, x_j) = \phi (x_i)^T \phi (x_j)\)</span>
is the kernel. Here training vectors are implicitly mapped into a higher
(maybe infinite) dimensional space by the function <span class="math notranslate nohighlight">\(\phi\)</span>.</p>
<p>The prediction is:</p>
<div class="math notranslate nohighlight">
\[\sum_{i \in SV}(\alpha_i - \alpha_i^*) K(x_i, x) + b\]</div>
<p>These parameters can be accessed through the attributes <code class="docutils literal notranslate"><span class="pre">dual_coef_</span></code>
which holds the difference <span class="math notranslate nohighlight">\(\alpha_i - \alpha_i^*\)</span>, <code class="docutils literal notranslate"><span class="pre">support_vectors_</span></code> which
holds the support vectors, and <code class="docutils literal notranslate"><span class="pre">intercept_</span></code> which holds the independent
term <span class="math notranslate nohighlight">\(b\)</span></p>
<p><details>
<summary class="btn btn-light">
<strong>LinearSVR</strong>
<span class="tooltiptext">Click for more details</span>
</summary>
<div class="card"></p>
<p>The primal problem can be equivalently formulated as</p>
<div class="math notranslate nohighlight">
\[\min_ {w, b} \frac{1}{2} w^T w + C \sum_{i=1}^{n}\max(0, |y_i - (w^T \phi(x_i) + b)| - \varepsilon),\]</div>
<p>where we make use of the epsilon-insensitive loss, i.e. errors of less than
<span class="math notranslate nohighlight">\(\varepsilon\)</span> are ignored. This is the form that is directly optimized
by <a class="reference internal" href="generated/sklearn.svm.LinearSVR.html#sklearn.svm.LinearSVR" title="sklearn.svm.LinearSVR"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearSVR</span></code></a>.</p>
<p></div>
</details></p>
</section>
</section>
<section id="implementation-details">
<span id="svm-implementation-details"></span><h2><span class="section-number">1.4.8. </span>Implementation details<a class="headerlink" href="#implementation-details" title="Permalink to this heading">¶</a></h2>
<p>Internally, we use <a class="reference external" href="https://www.csie.ntu.edu.tw/~cjlin/libsvm/">libsvm</a> <a class="footnote-reference brackets" href="#id14" id="id9" role="doc-noteref"><span class="fn-bracket">[</span>12<span class="fn-bracket">]</span></a> and <a class="reference external" href="https://www.csie.ntu.edu.tw/~cjlin/liblinear/">liblinear</a> <a class="footnote-reference brackets" href="#id13" id="id10" role="doc-noteref"><span class="fn-bracket">[</span>11<span class="fn-bracket">]</span></a> to handle all
computations. These libraries are wrapped using C and Cython.
For a description of the implementation and details of the algorithms
used, please refer to their respective papers.</p>
<aside class="topic">
<p class="topic-title">References:</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id11" role="note">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id2">9</a><span class="fn-bracket">]</span></span>
<p>Platt <a class="reference external" href="https://www.cs.colorado.edu/~mozer/Teaching/syllabi/6622/papers/Platt1999.pdf">“Probabilistic outputs for SVMs and comparisons to
regularized likelihood methods”</a>.</p>
</aside>
<aside class="footnote brackets" id="id12" role="note">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id3">10</a><span class="fn-bracket">]</span></span>
<p>Wu, Lin and Weng, <a class="reference external" href="https://www.csie.ntu.edu.tw/~cjlin/papers/svmprob/svmprob.pdf">“Probability estimates for multi-class
classification by pairwise coupling”</a>, JMLR
5:975-1005, 2004.</p>
</aside>
<aside class="footnote brackets" id="id13" role="note">
<span class="label"><span class="fn-bracket">[</span>11<span class="fn-bracket">]</span></span>
<span class="backrefs">(<a role="doc-backlink" href="#id4">1</a>,<a role="doc-backlink" href="#id10">2</a>)</span>
<p>Fan, Rong-En, et al.,
<a class="reference external" href="https://www.csie.ntu.edu.tw/~cjlin/papers/liblinear.pdf">“LIBLINEAR: A library for large linear classification.”</a>,
Journal of machine learning research 9.Aug (2008): 1871-1874.</p>
</aside>
<aside class="footnote brackets" id="id14" role="note">
<span class="label"><span class="fn-bracket">[</span>12<span class="fn-bracket">]</span></span>
<span class="backrefs">(<a role="doc-backlink" href="#id5">1</a>,<a role="doc-backlink" href="#id9">2</a>)</span>
<p>Chang and Lin, <a class="reference external" href="https://www.csie.ntu.edu.tw/~cjlin/papers/libsvm.pdf">LIBSVM: A Library for Support Vector Machines</a>.</p>
</aside>
<aside class="footnote brackets" id="id15" role="note">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id6">13</a><span class="fn-bracket">]</span></span>
<p>Bishop, <a class="reference external" href="https://www.microsoft.com/en-us/research/uploads/prod/2006/01/Bishop-Pattern-Recognition-and-Machine-Learning-2006.pdf">Pattern recognition and machine learning</a>,
chapter 7 Sparse Kernel Machines</p>
</aside>
<aside class="footnote brackets" id="id16" role="note">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id7">14</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://doi.org/10.1023/B:STCO.0000035301.49549.88">“A Tutorial on Support Vector Regression”</a>
Alex J. Smola, Bernhard Schölkopf - Statistics and Computing archive
Volume 14 Issue 3, August 2004, p. 199-222.</p>
</aside>
<aside class="footnote brackets" id="id17" role="note">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id8">15</a><span class="fn-bracket">]</span></span>
<p>Schölkopf et. al <a class="reference external" href="https://www.stat.purdue.edu/~yuzhu/stat598m3/Papers/NewSVM.pdf">New Support Vector Algorithms</a></p>
</aside>
<aside class="footnote brackets" id="id18" role="note">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id1">16</a><span class="fn-bracket">]</span></span>
<p>Crammer and Singer <a class="reference external" href="http://jmlr.csail.mit.edu/papers/volume2/crammer01a/crammer01a.pdf">On the Algorithmic Implementation ofMulticlass
Kernel-based Vector Machines</a>,
JMLR 2001.</p>
</aside>
</aside>
</aside>
</section>
</section>
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