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<li><a class="reference internal" href="#">Plot classification boundaries with different SVM Kernels</a><ul>
<li><a class="reference internal" href="#creating-a-dataset">Creating a dataset</a></li>
<li><a class="reference internal" href="#training-svc-model-and-plotting-decision-boundaries">Training SVC model and plotting decision boundaries</a><ul>
<li><a class="reference internal" href="#linear-kernel">Linear kernel</a></li>
<li><a class="reference internal" href="#polynomial-kernel">Polynomial kernel</a></li>
<li><a class="reference internal" href="#rbf-kernel">RBF kernel</a></li>
<li><a class="reference internal" href="#sigmoid-kernel">Sigmoid kernel</a></li>
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<section class="sphx-glr-example-title" id="plot-classification-boundaries-with-different-svm-kernels">
<span id="sphx-glr-auto-examples-svm-plot-svm-kernels-py"></span><h1>Plot classification boundaries with different SVM Kernels<a class="headerlink" href="#plot-classification-boundaries-with-different-svm-kernels" title="Link to this heading">¶</a></h1>
<p>This example shows how different kernels in a <a class="reference internal" href="../../modules/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> (Support Vector
Classifier) influence the classification boundaries in a binary, two-dimensional
classification problem.</p>
<p>SVCs aim to find a hyperplane that effectively separates the classes in their training
data by maximizing the margin between the outermost data points of each class. This is
achieved by finding the best weight vector <span class="math notranslate nohighlight">\(w\)</span> that defines the decision boundary
hyperplane and minimizes the sum of hinge losses for misclassified samples, as measured
by the <a class="reference internal" href="../../modules/generated/sklearn.metrics.hinge_loss.html#sklearn.metrics.hinge_loss" title="sklearn.metrics.hinge_loss"><code class="xref py py-func docutils literal notranslate"><span class="pre">hinge_loss</span></code></a> function. By default, regularization is
applied with the parameter <code class="docutils literal notranslate"><span class="pre">C=1</span></code>, which allows for a certain degree of misclassification
tolerance.</p>
<p>If the data is not linearly separable in the original feature space, a non-linear kernel
parameter can be set. Depending on the kernel, the process involves adding new features
or transforming existing features to enrich and potentially add meaning to the data.
When a kernel other than <code class="docutils literal notranslate"><span class="pre">"linear"</span></code> is set, the SVC applies the <a class="reference external" href="https://en.wikipedia.org/wiki/Kernel_method#Mathematics:_the_kernel_trick">kernel trick</a>, which
computes the similarity between pairs of data points using the kernel function without
explicitly transforming the entire dataset. The kernel trick surpasses the otherwise
necessary matrix transformation of the whole dataset by only considering the relations
between all pairs of data points. The kernel function maps two vectors (each pair of
observations) to their similarity using their dot product.</p>
<p>The hyperplane can then be calculated using the kernel function as if the dataset were
represented in a higher-dimensional space. Using a kernel function instead of an
explicit matrix transformation improves performance, as the kernel function has a time
complexity of <span class="math notranslate nohighlight">\(O({n}^2)\)</span>, whereas matrix transformation scales according to the
specific transformation being applied.</p>
<p>In this example, we compare the most common kernel types of Support Vector Machines: the
linear kernel (<code class="docutils literal notranslate"><span class="pre">"linear"</span></code>), the polynomial kernel (<code class="docutils literal notranslate"><span class="pre">"poly"</span></code>), the radial basis function
kernel (<code class="docutils literal notranslate"><span class="pre">"rbf"</span></code>) and the sigmoid kernel (<code class="docutils literal notranslate"><span class="pre">"sigmoid"</span></code>).</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="c1"># Code source: Gaël Varoquaux</span>
<span class="c1"># License: BSD 3 clause</span>
</pre></div>
</div>
<section id="creating-a-dataset">
<h2>Creating a dataset<a class="headerlink" href="#creating-a-dataset" title="Link to this heading">¶</a></h2>
<p>We create a two-dimensional classification dataset with 16 samples and two classes. We
plot the samples with the colors matching their respective targets.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="n">X</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.array.html#numpy.array" title="numpy.array" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">array</span></a><span class="p">(</span>
<span class="p">[</span>
<span class="p">[</span><span class="mf">0.4</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.7</span><span class="p">],</span>
<span class="p">[</span><span class="o">-</span><span class="mf">1.5</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.0</span><span class="p">],</span>
<span class="p">[</span><span class="o">-</span><span class="mf">1.4</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.9</span><span class="p">],</span>
<span class="p">[</span><span class="o">-</span><span class="mf">1.3</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.2</span><span class="p">],</span>
<span class="p">[</span><span class="o">-</span><span class="mf">1.1</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.2</span><span class="p">],</span>
<span class="p">[</span><span class="o">-</span><span class="mf">1.2</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.4</span><span class="p">],</span>
<span class="p">[</span><span class="o">-</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">],</span>
<span class="p">[</span><span class="o">-</span><span class="mf">1.5</span><span class="p">,</span> <span class="mf">2.1</span><span class="p">],</span>
<span class="p">[</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">],</span>
<span class="p">[</span><span class="mf">1.3</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">],</span>
<span class="p">[</span><span class="mf">1.2</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">],</span>
<span class="p">[</span><span class="mf">0.2</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.0</span><span class="p">],</span>
<span class="p">[</span><span class="mf">0.5</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.4</span><span class="p">],</span>
<span class="p">[</span><span class="mf">0.2</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.3</span><span class="p">],</span>
<span class="p">[</span><span class="mf">0.0</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.7</span><span class="p">],</span>
<span class="p">[</span><span class="mf">1.3</span><span class="p">,</span> <span class="mf">2.1</span><span class="p">],</span>
<span class="p">]</span>
<span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.array.html#numpy.array" title="numpy.array" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">array</span></a><span class="p">([</span><span class="mi">0</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="mi">0</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="mi">0</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">1</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="mi">1</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="mi">1</span><span class="p">])</span>
<span class="c1"># Plotting settings</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplots.html#matplotlib.pyplot.subplots" title="matplotlib.pyplot.subplots" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">subplots</span></a><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
<span class="n">x_min</span><span class="p">,</span> <span class="n">x_max</span><span class="p">,</span> <span class="n">y_min</span><span class="p">,</span> <span class="n">y_max</span> <span class="o">=</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">xlim</span><span class="o">=</span><span class="p">(</span><span class="n">x_min</span><span class="p">,</span> <span class="n">x_max</span><span class="p">),</span> <span class="n">ylim</span><span class="o">=</span><span class="p">(</span><span class="n">y_min</span><span class="p">,</span> <span class="n">y_max</span><span class="p">))</span>
<span class="c1"># Plot samples by color and add legend</span>
<span class="n">scatter</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">s</span><span class="o">=</span><span class="mi">150</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="n">y</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">y</span><span class="p">,</span> <span class="n">edgecolors</span><span class="o">=</span><span class="s2">"k"</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="o">*</span><span class="n">scatter</span><span class="o">.</span><span class="n">legend_elements</span><span class="p">(),</span> <span class="n">loc</span><span class="o">=</span><span class="s2">"upper right"</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s2">"Classes"</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Samples in two-dimensional feature space"</span><span class="p">)</span>
<span class="n">_</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.show.html#matplotlib.pyplot.show" title="matplotlib.pyplot.show" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">show</span></a><span class="p">()</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_svm_kernels_001.png" srcset="../../_images/sphx_glr_plot_svm_kernels_001.png" alt="Samples in two-dimensional feature space" class = "sphx-glr-single-img"/><p>We can see that the samples are not clearly separable by a straight line.</p>
</section>
<section id="training-svc-model-and-plotting-decision-boundaries">
<h2>Training SVC model and plotting decision boundaries<a class="headerlink" href="#training-svc-model-and-plotting-decision-boundaries" title="Link to this heading">¶</a></h2>
<p>We define a function that fits a <a class="reference internal" href="../../modules/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> classifier,
allowing the <code class="docutils literal notranslate"><span class="pre">kernel</span></code> parameter as an input, and then plots the decision
boundaries learned by the model using
<a class="reference internal" href="../../modules/generated/sklearn.inspection.DecisionBoundaryDisplay.html#sklearn.inspection.DecisionBoundaryDisplay" title="sklearn.inspection.DecisionBoundaryDisplay"><code class="xref py py-class docutils literal notranslate"><span class="pre">DecisionBoundaryDisplay</span></code></a>.</p>
<p>Notice that for the sake of simplicity, the <code class="docutils literal notranslate"><span class="pre">C</span></code> parameter is set to its
default value (<code class="docutils literal notranslate"><span class="pre">C=1</span></code>) in this example and the <code class="docutils literal notranslate"><span class="pre">gamma</span></code> parameter is set to
<code class="docutils literal notranslate"><span class="pre">gamma=2</span></code> across all kernels, although it is automatically ignored for the
linear kernel. In a real classification task, where performance matters,
parameter tuning (by using <a class="reference internal" href="../../modules/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> for
instance) is highly recommended to capture different structures within the
data.</p>
<p>Setting <code class="docutils literal notranslate"><span class="pre">response_method="predict"</span></code> in
<a class="reference internal" href="../../modules/generated/sklearn.inspection.DecisionBoundaryDisplay.html#sklearn.inspection.DecisionBoundaryDisplay" title="sklearn.inspection.DecisionBoundaryDisplay"><code class="xref py py-class docutils literal notranslate"><span class="pre">DecisionBoundaryDisplay</span></code></a> colors the areas based
on their predicted class. Using <code class="docutils literal notranslate"><span class="pre">response_method="decision_function"</span></code> allows
us to also plot the decision boundary and the margins to both sides of it.
Finally the support vectors used during training (which always lay on the
margins) are identified by means of the <code class="docutils literal notranslate"><span class="pre">support_vectors_</span></code> attribute of
the trained SVCs, and plotted as well.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">svm</span>
<span class="kn">from</span> <span class="nn">sklearn.inspection</span> <span class="kn">import</span> <span class="n">DecisionBoundaryDisplay</span>
<span class="k">def</span> <span class="nf">plot_training_data_with_decision_boundary</span><span class="p">(</span><span class="n">kernel</span><span class="p">):</span>
<span class="c1"># Train the SVC</span>
<span class="n">clf</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.svm.SVC.html#sklearn.svm.SVC" title="sklearn.svm.SVC" class="sphx-glr-backref-module-sklearn-svm sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">svm</span><span class="o">.</span><span class="n">SVC</span></a><span class="p">(</span><span class="n">kernel</span><span class="o">=</span><span class="n">kernel</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mi">2</span><span class="p">)</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="c1"># Settings for plotting</span>
<span class="n">_</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplots.html#matplotlib.pyplot.subplots" title="matplotlib.pyplot.subplots" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">subplots</span></a><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
<span class="n">x_min</span><span class="p">,</span> <span class="n">x_max</span><span class="p">,</span> <span class="n">y_min</span><span class="p">,</span> <span class="n">y_max</span> <span class="o">=</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">xlim</span><span class="o">=</span><span class="p">(</span><span class="n">x_min</span><span class="p">,</span> <span class="n">x_max</span><span class="p">),</span> <span class="n">ylim</span><span class="o">=</span><span class="p">(</span><span class="n">y_min</span><span class="p">,</span> <span class="n">y_max</span><span class="p">))</span>
<span class="c1"># Plot decision boundary and margins</span>
<span class="n">common_params</span> <span class="o">=</span> <span class="p">{</span><span class="s2">"estimator"</span><span class="p">:</span> <span class="n">clf</span><span class="p">,</span> <span class="s2">"X"</span><span class="p">:</span> <span class="n">X</span><span class="p">,</span> <span class="s2">"ax"</span><span class="p">:</span> <span class="n">ax</span><span class="p">}</span>
<a href="../../modules/generated/sklearn.inspection.DecisionBoundaryDisplay.html#sklearn.inspection.DecisionBoundaryDisplay.from_estimator" title="sklearn.inspection.DecisionBoundaryDisplay.from_estimator" class="sphx-glr-backref-module-sklearn-inspection-DecisionBoundaryDisplay sphx-glr-backref-type-py-method"><span class="n">DecisionBoundaryDisplay</span><span class="o">.</span><span class="n">from_estimator</span></a><span class="p">(</span>
<span class="o">**</span><span class="n">common_params</span><span class="p">,</span>
<span class="n">response_method</span><span class="o">=</span><span class="s2">"predict"</span><span class="p">,</span>
<span class="n">plot_method</span><span class="o">=</span><span class="s2">"pcolormesh"</span><span class="p">,</span>
<span class="n">alpha</span><span class="o">=</span><span class="mf">0.3</span><span class="p">,</span>
<span class="p">)</span>
<a href="../../modules/generated/sklearn.inspection.DecisionBoundaryDisplay.html#sklearn.inspection.DecisionBoundaryDisplay.from_estimator" title="sklearn.inspection.DecisionBoundaryDisplay.from_estimator" class="sphx-glr-backref-module-sklearn-inspection-DecisionBoundaryDisplay sphx-glr-backref-type-py-method"><span class="n">DecisionBoundaryDisplay</span><span class="o">.</span><span class="n">from_estimator</span></a><span class="p">(</span>
<span class="o">**</span><span class="n">common_params</span><span class="p">,</span>
<span class="n">response_method</span><span class="o">=</span><span class="s2">"decision_function"</span><span class="p">,</span>
<span class="n">plot_method</span><span class="o">=</span><span class="s2">"contour"</span><span class="p">,</span>
<span class="n">levels</span><span class="o">=</span><span class="p">[</span><span class="o">-</span><span class="mi">1</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="n">colors</span><span class="o">=</span><span class="p">[</span><span class="s2">"k"</span><span class="p">,</span> <span class="s2">"k"</span><span class="p">,</span> <span class="s2">"k"</span><span class="p">],</span>
<span class="n">linestyles</span><span class="o">=</span><span class="p">[</span><span class="s2">"--"</span><span class="p">,</span> <span class="s2">"-"</span><span class="p">,</span> <span class="s2">"--"</span><span class="p">],</span>
<span class="p">)</span>
<span class="c1"># Plot bigger circles around samples that serve as support vectors</span>
<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span>
<span class="n">clf</span><span class="o">.</span><span class="n">support_vectors_</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span>
<span class="n">clf</span><span class="o">.</span><span class="n">support_vectors_</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span>
<span class="n">s</span><span class="o">=</span><span class="mi">250</span><span class="p">,</span>
<span class="n">facecolors</span><span class="o">=</span><span class="s2">"none"</span><span class="p">,</span>
<span class="n">edgecolors</span><span class="o">=</span><span class="s2">"k"</span><span class="p">,</span>
<span class="p">)</span>
<span class="c1"># Plot samples by color and add legend</span>
<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="n">y</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">150</span><span class="p">,</span> <span class="n">edgecolors</span><span class="o">=</span><span class="s2">"k"</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="o">*</span><span class="n">scatter</span><span class="o">.</span><span class="n">legend_elements</span><span class="p">(),</span> <span class="n">loc</span><span class="o">=</span><span class="s2">"upper right"</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s2">"Classes"</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="sa">f</span><span class="s2">" Decision boundaries of </span><span class="si">{</span><span class="n">kernel</span><span class="si">}</span><span class="s2"> kernel in SVC"</span><span class="p">)</span>
<span class="n">_</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.show.html#matplotlib.pyplot.show" title="matplotlib.pyplot.show" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">show</span></a><span class="p">()</span>
</pre></div>
</div>
<section id="linear-kernel">
<h3>Linear kernel<a class="headerlink" href="#linear-kernel" title="Link to this heading">¶</a></h3>
<p>Linear kernel is the dot product of the input samples:</p>
<div class="math notranslate nohighlight">
\[K(\mathbf{x}_1, \mathbf{x}_2) = \mathbf{x}_1^\top \mathbf{x}_2\]</div>
<p>It is then applied to any combination of two data points (samples) in the
dataset. The dot product of the two points determines the
<a class="reference internal" href="../../modules/generated/sklearn.metrics.pairwise.cosine_similarity.html#sklearn.metrics.pairwise.cosine_similarity" title="sklearn.metrics.pairwise.cosine_similarity"><code class="xref py py-func docutils literal notranslate"><span class="pre">cosine_similarity</span></code></a> between both points. The
higher the value, the more similar the points are.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="n">plot_training_data_with_decision_boundary</span><span class="p">(</span><span class="s2">"linear"</span><span class="p">)</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_svm_kernels_002.png" srcset="../../_images/sphx_glr_plot_svm_kernels_002.png" alt="Decision boundaries of linear kernel in SVC" class = "sphx-glr-single-img"/><p>Training a <a class="reference internal" href="../../modules/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> on a linear kernel results in an
untransformed feature space, where the hyperplane and the margins are
straight lines. Due to the lack of expressivity of the linear kernel, the
trained classes do not perfectly capture the training data.</p>
</section>
<section id="polynomial-kernel">
<h3>Polynomial kernel<a class="headerlink" href="#polynomial-kernel" title="Link to this heading">¶</a></h3>
<p>The polynomial kernel changes the notion of similarity. The kernel function
is defined as:</p>
<div class="math notranslate nohighlight">
\[K(\mathbf{x}_1, \mathbf{x}_2) = (\gamma \cdot \
\mathbf{x}_1^\top\mathbf{x}_2 + r)^d\]</div>
<p>where <span class="math notranslate nohighlight">\({d}\)</span> is the degree (<code class="docutils literal notranslate"><span class="pre">degree</span></code>) of the polynomial, <span class="math notranslate nohighlight">\({\gamma}\)</span>
(<code class="docutils literal notranslate"><span class="pre">gamma</span></code>) controls the influence of each individual training sample on the
decision boundary and <span class="math notranslate nohighlight">\({r}\)</span> is the bias term (<code class="docutils literal notranslate"><span class="pre">coef0</span></code>) that shifts the
data up or down. Here, we use the default value for the degree of the
polynomial in the kernel function (<code class="docutils literal notranslate"><span class="pre">degree=3</span></code>). When <code class="docutils literal notranslate"><span class="pre">coef0=0</span></code> (the default),
the data is only transformed, but no additional dimension is added. Using a
polynomial kernel is equivalent to creating
<a class="reference internal" href="../../modules/generated/sklearn.preprocessing.PolynomialFeatures.html#sklearn.preprocessing.PolynomialFeatures" title="sklearn.preprocessing.PolynomialFeatures"><code class="xref py py-class docutils literal notranslate"><span class="pre">PolynomialFeatures</span></code></a> and then fitting a
<a class="reference internal" href="../../modules/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> with a linear kernel on the transformed data,
although this alternative approach would be computationally expensive for most
datasets.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="n">plot_training_data_with_decision_boundary</span><span class="p">(</span><span class="s2">"poly"</span><span class="p">)</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_svm_kernels_003.png" srcset="../../_images/sphx_glr_plot_svm_kernels_003.png" alt="Decision boundaries of poly kernel in SVC" class = "sphx-glr-single-img"/><p>The polynomial kernel with <code class="docutils literal notranslate"><span class="pre">gamma=2`</span></code> adapts well to the training data,
causing the margins on both sides of the hyperplane to bend accordingly.</p>
</section>
<section id="rbf-kernel">
<h3>RBF kernel<a class="headerlink" href="#rbf-kernel" title="Link to this heading">¶</a></h3>
<p>The radial basis function (RBF) kernel, also known as the Gaussian kernel, is
the default kernel for Support Vector Machines in scikit-learn. It measures
similarity between two data points in infinite dimensions and then approaches
classification by majority vote. The kernel function is defined as:</p>
<div class="math notranslate nohighlight">
\[K(\mathbf{x}_1, \mathbf{x}_2) = \exp\left(-\gamma \cdot
{\|\mathbf{x}_1 - \mathbf{x}_2\|^2}\right)\]</div>
<p>where <span class="math notranslate nohighlight">\({\gamma}\)</span> (<code class="docutils literal notranslate"><span class="pre">gamma</span></code>) controls the influence of each individual
training sample on the decision boundary.</p>
<p>The larger the euclidean distance between two points
<span class="math notranslate nohighlight">\(\|\mathbf{x}_1 - \mathbf{x}_2\|^2\)</span>
the closer the kernel function is to zero. This means that two points far away
are more likely to be dissimilar.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="n">plot_training_data_with_decision_boundary</span><span class="p">(</span><span class="s2">"rbf"</span><span class="p">)</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_svm_kernels_004.png" srcset="../../_images/sphx_glr_plot_svm_kernels_004.png" alt="Decision boundaries of rbf kernel in SVC" class = "sphx-glr-single-img"/><p>In the plot we can see how the decision boundaries tend to contract around
data points that are close to each other.</p>
</section>
<section id="sigmoid-kernel">
<h3>Sigmoid kernel<a class="headerlink" href="#sigmoid-kernel" title="Link to this heading">¶</a></h3>
<p>The sigmoid kernel function is defined as:</p>
<div class="math notranslate nohighlight">
\[K(\mathbf{x}_1, \mathbf{x}_2) = \tanh(\gamma \cdot
\mathbf{x}_1^\top\mathbf{x}_2 + r)\]</div>
<p>where the kernel coefficient <span class="math notranslate nohighlight">\({\gamma}\)</span> (<code class="docutils literal notranslate"><span class="pre">gamma</span></code>) controls the influence
of each individual training sample on the decision boundary and <span class="math notranslate nohighlight">\({r}\)</span> is
the bias term (<code class="docutils literal notranslate"><span class="pre">coef0</span></code>) that shifts the data up or down.</p>
<p>In the sigmoid kernel, the similarity between two data points is computed
using the hyperbolic tangent function (<span class="math notranslate nohighlight">\(\tanh\)</span>). The kernel function
scales and possibly shifts the dot product of the two points
(<span class="math notranslate nohighlight">\(\mathbf{x}_1\)</span> and <span class="math notranslate nohighlight">\(\mathbf{x}_2\)</span>).</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="n">plot_training_data_with_decision_boundary</span><span class="p">(</span><span class="s2">"sigmoid"</span><span class="p">)</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_svm_kernels_005.png" srcset="../../_images/sphx_glr_plot_svm_kernels_005.png" alt="Decision boundaries of sigmoid kernel in SVC" class = "sphx-glr-single-img"/><p>We can see that the decision boundaries obtained with the sigmoid kernel
appear curved and irregular. The decision boundary tries to separate the
classes by fitting a sigmoid-shaped curve, resulting in a complex boundary
that may not generalize well to unseen data. From this example it becomes
obvious, that the sigmoid kernel has very specific use cases, when dealing
with data that exhibits a sigmoidal shape. In this example, careful fine
tuning might find more generalizable decision boundaries. Because of it’s
specificity, the sigmoid kernel is less commonly used in practice compared to
other kernels.</p>
</section>
</section>
<section id="conclusion">
<h2>Conclusion<a class="headerlink" href="#conclusion" title="Link to this heading">¶</a></h2>
<p>In this example, we have visualized the decision boundaries trained with the
provided dataset. The plots serve as an intuitive demonstration of how
different kernels utilize the training data to determine the classification
boundaries.</p>
<p>The hyperplanes and margins, although computed indirectly, can be imagined as
planes in the transformed feature space. However, in the plots, they are
represented relative to the original feature space, resulting in curved
decision boundaries for the polynomial, RBF, and sigmoid kernels.</p>
<p>Please note that the plots do not evaluate the individual kernel’s accuracy or
quality. They are intended to provide a visual understanding of how the
different kernels use the training data.</p>
<p>For a comprehensive evaluation, fine-tuning of <a class="reference internal" href="../../modules/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>
parameters using techniques such as
<a class="reference internal" href="../../modules/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> is recommended to capture the
underlying structures within the data.</p>
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<p class="rubric">Related examples</p>
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<p><a class="reference internal" href="plot_rbf_parameters.html#sphx-glr-auto-examples-svm-plot-rbf-parameters-py"><span class="std std-ref">RBF SVM parameters</span></a></p>
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<p><a class="reference internal" href="../ensemble/plot_adaboost_twoclass.html#sphx-glr-auto-examples-ensemble-plot-adaboost-twoclass-py"><span class="std std-ref">Two-class AdaBoost</span></a></p>
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