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<li class="toctree-l1 current active has-children"><a class="reference internal" href="../supervised_learning.html">1. Supervised learning</a><details open="open"><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul class="current">
<li class="toctree-l2"><a class="reference internal" href="linear_model.html">1.1. Linear Models</a></li>
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<li class="toctree-l2"><a class="reference internal" href="neural_networks_supervised.html">1.17. Neural network models (supervised)</a></li>
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<li class="toctree-l2"><a class="reference internal" href="mixture.html">2.1. Gaussian mixture models</a></li>
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<li class="toctree-l2"><a class="reference internal" href="neural_networks_unsupervised.html">2.9. Neural network models (unsupervised)</a></li>
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<li class="toctree-l2"><a class="reference internal" href="model_evaluation.html">3.4. Metrics and scoring: quantifying the quality of predictions</a></li>
<li class="toctree-l2"><a class="reference internal" href="learning_curve.html">3.5. Validation curves: plotting scores to evaluate models</a></li>
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<li class="toctree-l2"><a class="reference internal" href="partial_dependence.html">4.1. Partial Dependence and Individual Conditional Expectation plots</a></li>
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<li class="toctree-l2"><a class="reference internal" href="compose.html">6.1. Pipelines and composite estimators</a></li>
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<li class="toctree-l2"><a class="reference internal" href="unsupervised_reduction.html">6.5. Unsupervised dimensionality reduction</a></li>
<li class="toctree-l2"><a class="reference internal" href="random_projection.html">6.6. Random Projection</a></li>
<li class="toctree-l2"><a class="reference internal" href="kernel_approximation.html">6.7. Kernel Approximation</a></li>
<li class="toctree-l2"><a class="reference internal" href="metrics.html">6.8. Pairwise metrics, Affinities and Kernels</a></li>
<li class="toctree-l2"><a class="reference internal" href="preprocessing_targets.html">6.9. Transforming the prediction target (<code class="docutils literal notranslate"><span class="pre">y</span></code>)</a></li>
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<li class="toctree-l2"><a class="reference internal" href="../datasets/toy_dataset.html">7.1. Toy datasets</a></li>
<li class="toctree-l2"><a class="reference internal" href="../datasets/real_world.html">7.2. Real world datasets</a></li>
<li class="toctree-l2"><a class="reference internal" href="../datasets/sample_generators.html">7.3. Generated datasets</a></li>
<li class="toctree-l2"><a class="reference internal" href="../datasets/loading_other_datasets.html">7.4. Loading other datasets</a></li>
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<li class="toctree-l2"><a class="reference internal" href="../computing/scaling_strategies.html">8.1. Strategies to scale computationally: bigger data</a></li>
<li class="toctree-l2"><a class="reference internal" href="../computing/computational_performance.html">8.2. Computational Performance</a></li>
<li class="toctree-l2"><a class="reference internal" href="../computing/parallelism.html">8.3. Parallelism, resource management, and configuration</a></li>
</ul>
</details></li>
<li class="toctree-l1"><a class="reference internal" href="../model_persistence.html">9. Model persistence</a></li>
<li class="toctree-l1"><a class="reference internal" href="../common_pitfalls.html">10. Common pitfalls and recommended practices</a></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../dispatching.html">11. Dispatching</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="array_api.html">11.1. Array API support (experimental)</a></li>
</ul>
</details></li>
<li class="toctree-l1"><a class="reference internal" href="../machine_learning_map.html">12. Choosing the right estimator</a></li>
<li class="toctree-l1"><a class="reference internal" href="../presentations.html">13. External Resources, Videos and Talks</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="Link 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>
<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>
<p>The disadvantages of support vector machines include:</p>
<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>
<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="Link 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. 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>. <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> uses <code class="docutils literal notranslate"><span class="pre">squared_hinge</span></code> loss and due to its
implementation in <code class="docutils literal notranslate"><span class="pre">liblinear</span></code> it also regularizes the intercept, if considered.
This effect can however be reduced by carefully fine tuning its
<code class="docutils literal notranslate"><span class="pre">intercept_scaling</span></code> parameter, which allows the intercept term to have a
different regularization behavior compared to the other features. The
classification results and score can therefore differ from the other two
classifiers.</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="w"> </span><span class="nn">sklearn</span><span class="w"> </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>
<p class="rubric">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_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>
<li><p><a class="reference internal" href="../auto_examples/classification/plot_classification_probability.html#sphx-glr-auto-examples-classification-plot-classification-probability-py"><span class="std std-ref">Plot classification probability</span></a></p></li>
</ul>
<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="Link 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>, which is the default setting
of the parameter (default=’ovr’).</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"># 6 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="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()</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>
<details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3" id="details-on-multi-class-strategies">
<summary class="sd-summary-title sd-card-header">
<span class="sd-summary-text">Details on multi-class strategies<a class="headerlink" href="#details-on-multi-class-strategies" title="Link to this dropdown">#</a></span><span class="sd-summary-state-marker sd-summary-chevron-right"><svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-right" viewBox="0 0 24 24" aria-hidden="true"><path d="M8.72 18.78a.75.75 0 0 1 0-1.06L14.44 12 8.72 6.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018l6.25 6.25a.75.75 0 0 1 0 1.06l-6.25 6.25a.75.75 0 0 1-1.06 0Z"></path></svg></span></summary><div class="sd-summary-content sd-card-body docutils">
<p class="sd-card-text">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 class="sd-card-text">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 class="sd-card-text">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 class="sd-card-text">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 class="sd-card-text">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>
<div class="pst-scrollable-table-container"><table class="table">
<tbody>
<tr class="row-odd"><td><p class="sd-card-text"><span class="math notranslate nohighlight">\(\alpha^{0}_{0,1}\)</span></p></td>
<td><p class="sd-card-text"><span class="math notranslate nohighlight">\(\alpha^{1}_{0,1}\)</span></p></td>
<td><p class="sd-card-text"><span class="math notranslate nohighlight">\(\alpha^{2}_{0,1}\)</span></p></td>
<td><p class="sd-card-text"><span class="math notranslate nohighlight">\(\alpha^{0}_{1,0}\)</span></p></td>
<td><p class="sd-card-text"><span class="math notranslate nohighlight">\(\alpha^{1}_{1,0}\)</span></p></td>
<td><p class="sd-card-text"><span class="math notranslate nohighlight">\(\alpha^{0}_{2,0}\)</span></p></td>
<td><p class="sd-card-text"><span class="math notranslate nohighlight">\(\alpha^{1}_{2,0}\)</span></p></td>
</tr>
<tr class="row-even"><td><p class="sd-card-text"><span class="math notranslate nohighlight">\(\alpha^{0}_{0,2}\)</span></p></td>
<td><p class="sd-card-text"><span class="math notranslate nohighlight">\(\alpha^{1}_{0,2}\)</span></p></td>
<td><p class="sd-card-text"><span class="math notranslate nohighlight">\(\alpha^{2}_{0,2}\)</span></p></td>
<td><p class="sd-card-text"><span class="math notranslate nohighlight">\(\alpha^{0}_{1,2}\)</span></p></td>
<td><p class="sd-card-text"><span class="math notranslate nohighlight">\(\alpha^{1}_{1,2}\)</span></p></td>
<td><p class="sd-card-text"><span class="math notranslate nohighlight">\(\alpha^{0}_{2,1}\)</span></p></td>
<td><p class="sd-card-text"><span class="math notranslate nohighlight">\(\alpha^{1}_{2,1}\)</span></p></td>
</tr>
<tr class="row-odd"><td colspan="3"><p class="sd-card-text">Coefficients
for SVs of class 0</p></td>
<td colspan="2"><p class="sd-card-text">Coefficients
for SVs of class 1</p></td>
<td colspan="2"><p class="sd-card-text">Coefficients
for SVs of class 2</p></td>
</tr>
</tbody>
</table>
</div>
</div>
</details><p class="rubric">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>
</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="Link 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 to <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="Link 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>
<p class="rubric">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>
</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="Link 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>. Due to its implementation in
<code class="docutils literal notranslate"><span class="pre">liblinear</span></code> <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> also regularizes the intercept, if considered.
This effect can however be reduced by carefully fine tuning its
<code class="docutils literal notranslate"><span class="pre">intercept_scaling</span></code> parameter, which allows the intercept term to have a
different regularization behavior compared to the other features. The
classification results and score can therefore differ from the other two
classifiers. 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="w"> </span><span class="nn">sklearn</span><span class="w"> </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>