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<li><a class="reference internal" href="#">Explicit feature map approximation for RBF kernels</a><ul>
<li><a class="reference internal" href="#python-package-and-dataset-imports-load-dataset">Python package and dataset imports, load dataset</a></li>
<li><a class="reference internal" href="#timing-and-accuracy-plots">Timing and accuracy plots</a></li>
<li><a class="reference internal" href="#decision-surfaces-of-rbf-kernel-svm-and-linear-svm">Decision Surfaces of RBF Kernel SVM and Linear SVM</a></li>
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to download the full example code or to run this example in your browser via JupyterLite or Binder</p>
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<section class="sphx-glr-example-title" id="explicit-feature-map-approximation-for-rbf-kernels">
<span id="sphx-glr-auto-examples-miscellaneous-plot-kernel-approximation-py"></span><h1>Explicit feature map approximation for RBF kernels<a class="headerlink" href="#explicit-feature-map-approximation-for-rbf-kernels" title="Link to this heading">¶</a></h1>
<p>An example illustrating the approximation of the feature map
of an RBF kernel.</p>
<p>It shows how to use <a class="reference internal" href="../../modules/generated/sklearn.kernel_approximation.RBFSampler.html#sklearn.kernel_approximation.RBFSampler" title="sklearn.kernel_approximation.RBFSampler"><code class="xref py py-class docutils literal notranslate"><span class="pre">RBFSampler</span></code></a> and <a class="reference internal" href="../../modules/generated/sklearn.kernel_approximation.Nystroem.html#sklearn.kernel_approximation.Nystroem" title="sklearn.kernel_approximation.Nystroem"><code class="xref py py-class docutils literal notranslate"><span class="pre">Nystroem</span></code></a> to
approximate the feature map of an RBF kernel for classification with an SVM on
the digits dataset. Results using a linear SVM in the original space, a linear
SVM using the approximate mappings and using a kernelized SVM are compared.
Timings and accuracy for varying amounts of Monte Carlo samplings (in the case
of <a class="reference internal" href="../../modules/generated/sklearn.kernel_approximation.RBFSampler.html#sklearn.kernel_approximation.RBFSampler" title="sklearn.kernel_approximation.RBFSampler"><code class="xref py py-class docutils literal notranslate"><span class="pre">RBFSampler</span></code></a>, which uses random Fourier features) and different sized
subsets of the training set (for <a class="reference internal" href="../../modules/generated/sklearn.kernel_approximation.Nystroem.html#sklearn.kernel_approximation.Nystroem" title="sklearn.kernel_approximation.Nystroem"><code class="xref py py-class docutils literal notranslate"><span class="pre">Nystroem</span></code></a>) for the approximate mapping
are shown.</p>
<p>Please note that the dataset here is not large enough to show the benefits
of kernel approximation, as the exact SVM is still reasonably fast.</p>
<p>Sampling more dimensions clearly leads to better classification results, but
comes at a greater cost. This means there is a tradeoff between runtime and
accuracy, given by the parameter n_components. Note that solving the Linear
SVM and also the approximate kernel SVM could be greatly accelerated by using
stochastic gradient descent via <a class="reference internal" href="../../modules/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>.
This is not easily possible for the case of the kernelized SVM.</p>
<section id="python-package-and-dataset-imports-load-dataset">
<h2>Python package and dataset imports, load dataset<a class="headerlink" href="#python-package-and-dataset-imports-load-dataset" title="Link to this heading">¶</a></h2>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="c1"># Author: Gael Varoquaux <gael dot varoquaux at normalesup dot org></span>
<span class="c1"># Andreas Mueller <[email protected]></span>
<span class="c1"># License: BSD 3 clause</span>
<span class="c1"># Standard scientific Python imports</span>
<span class="kn">from</span> <span class="nn">time</span> <span class="kn">import</span> <a href="https://docs.python.org/3/library/time.html#time.time" title="time.time" class="sphx-glr-backref-module-time sphx-glr-backref-type-py-function"><span class="n">time</span></a>
<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="c1"># Import datasets, classifiers and performance metrics</span>
<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">datasets</span><span class="p">,</span> <span class="n">pipeline</span><span class="p">,</span> <span class="n">svm</span>
<span class="kn">from</span> <span class="nn">sklearn.decomposition</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA" class="sphx-glr-backref-module-sklearn-decomposition sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">PCA</span></a>
<span class="kn">from</span> <span class="nn">sklearn.kernel_approximation</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.kernel_approximation.Nystroem.html#sklearn.kernel_approximation.Nystroem" title="sklearn.kernel_approximation.Nystroem" class="sphx-glr-backref-module-sklearn-kernel_approximation sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">Nystroem</span></a><span class="p">,</span> <a href="../../modules/generated/sklearn.kernel_approximation.RBFSampler.html#sklearn.kernel_approximation.RBFSampler" title="sklearn.kernel_approximation.RBFSampler" class="sphx-glr-backref-module-sklearn-kernel_approximation sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">RBFSampler</span></a>
<span class="c1"># The digits dataset</span>
<span class="n">digits</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.datasets.load_digits.html#sklearn.datasets.load_digits" title="sklearn.datasets.load_digits" class="sphx-glr-backref-module-sklearn-datasets sphx-glr-backref-type-py-function"><span class="n">datasets</span><span class="o">.</span><span class="n">load_digits</span></a><span class="p">(</span><span class="n">n_class</span><span class="o">=</span><span class="mi">9</span><span class="p">)</span>
</pre></div>
</div>
</section>
<section id="timing-and-accuracy-plots">
<h2>Timing and accuracy plots<a class="headerlink" href="#timing-and-accuracy-plots" title="Link to this heading">¶</a></h2>
<p>To apply an classifier on this data, we need to flatten the image, to
turn the data in a (samples, feature) matrix:</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="n">n_samples</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">digits</span><span class="o">.</span><span class="n">data</span><span class="p">)</span>
<span class="n">data</span> <span class="o">=</span> <span class="n">digits</span><span class="o">.</span><span class="n">data</span> <span class="o">/</span> <span class="mf">16.0</span>
<span class="n">data</span> <span class="o">-=</span> <span class="n">data</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="c1"># We learn the digits on the first half of the digits</span>
<span class="n">data_train</span><span class="p">,</span> <span class="n">targets_train</span> <span class="o">=</span> <span class="p">(</span><span class="n">data</span><span class="p">[:</span> <span class="n">n_samples</span> <span class="o">//</span> <span class="mi">2</span><span class="p">],</span> <span class="n">digits</span><span class="o">.</span><span class="n">target</span><span class="p">[:</span> <span class="n">n_samples</span> <span class="o">//</span> <span class="mi">2</span><span class="p">])</span>
<span class="c1"># Now predict the value of the digit on the second half:</span>
<span class="n">data_test</span><span class="p">,</span> <span class="n">targets_test</span> <span class="o">=</span> <span class="p">(</span><span class="n">data</span><span class="p">[</span><span class="n">n_samples</span> <span class="o">//</span> <span class="mi">2</span> <span class="p">:],</span> <span class="n">digits</span><span class="o">.</span><span class="n">target</span><span class="p">[</span><span class="n">n_samples</span> <span class="o">//</span> <span class="mi">2</span> <span class="p">:])</span>
<span class="c1"># data_test = scaler.transform(data_test)</span>
<span class="c1"># Create a classifier: a support vector classifier</span>
<span class="n">kernel_svm</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">gamma</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
<span class="n">linear_svm</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC" 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">LinearSVC</span></a><span class="p">(</span><span class="n">dual</span><span class="o">=</span><span class="s2">"auto"</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
<span class="c1"># create pipeline from kernel approximation</span>
<span class="c1"># and linear svm</span>
<span class="n">feature_map_fourier</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.kernel_approximation.RBFSampler.html#sklearn.kernel_approximation.RBFSampler" title="sklearn.kernel_approximation.RBFSampler" class="sphx-glr-backref-module-sklearn-kernel_approximation sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">RBFSampler</span></a><span class="p">(</span><span class="n">gamma</span><span class="o">=</span><span class="mf">0.2</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">feature_map_nystroem</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.kernel_approximation.Nystroem.html#sklearn.kernel_approximation.Nystroem" title="sklearn.kernel_approximation.Nystroem" class="sphx-glr-backref-module-sklearn-kernel_approximation sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">Nystroem</span></a><span class="p">(</span><span class="n">gamma</span><span class="o">=</span><span class="mf">0.2</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">fourier_approx_svm</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.pipeline.Pipeline.html#sklearn.pipeline.Pipeline" title="sklearn.pipeline.Pipeline" class="sphx-glr-backref-module-sklearn-pipeline sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">pipeline</span><span class="o">.</span><span class="n">Pipeline</span></a><span class="p">(</span>
<span class="p">[</span>
<span class="p">(</span><span class="s2">"feature_map"</span><span class="p">,</span> <span class="n">feature_map_fourier</span><span class="p">),</span>
<span class="p">(</span><span class="s2">"svm"</span><span class="p">,</span> <a href="../../modules/generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC" 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">LinearSVC</span></a><span class="p">(</span><span class="n">dual</span><span class="o">=</span><span class="s2">"auto"</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)),</span>
<span class="p">]</span>
<span class="p">)</span>
<span class="n">nystroem_approx_svm</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.pipeline.Pipeline.html#sklearn.pipeline.Pipeline" title="sklearn.pipeline.Pipeline" class="sphx-glr-backref-module-sklearn-pipeline sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">pipeline</span><span class="o">.</span><span class="n">Pipeline</span></a><span class="p">(</span>
<span class="p">[</span>
<span class="p">(</span><span class="s2">"feature_map"</span><span class="p">,</span> <span class="n">feature_map_nystroem</span><span class="p">),</span>
<span class="p">(</span><span class="s2">"svm"</span><span class="p">,</span> <a href="../../modules/generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC" 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">LinearSVC</span></a><span class="p">(</span><span class="n">dual</span><span class="o">=</span><span class="s2">"auto"</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)),</span>
<span class="p">]</span>
<span class="p">)</span>
<span class="c1"># fit and predict using linear and kernel svm:</span>
<span class="n">kernel_svm_time</span> <span class="o">=</span> <a href="https://docs.python.org/3/library/time.html#time.time" title="time.time" class="sphx-glr-backref-module-time sphx-glr-backref-type-py-function"><span class="n">time</span></a><span class="p">()</span>
<span class="n">kernel_svm</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">data_train</span><span class="p">,</span> <span class="n">targets_train</span><span class="p">)</span>
<span class="n">kernel_svm_score</span> <span class="o">=</span> <span class="n">kernel_svm</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">data_test</span><span class="p">,</span> <span class="n">targets_test</span><span class="p">)</span>
<span class="n">kernel_svm_time</span> <span class="o">=</span> <a href="https://docs.python.org/3/library/time.html#time.time" title="time.time" class="sphx-glr-backref-module-time sphx-glr-backref-type-py-function"><span class="n">time</span></a><span class="p">()</span> <span class="o">-</span> <span class="n">kernel_svm_time</span>
<span class="n">linear_svm_time</span> <span class="o">=</span> <a href="https://docs.python.org/3/library/time.html#time.time" title="time.time" class="sphx-glr-backref-module-time sphx-glr-backref-type-py-function"><span class="n">time</span></a><span class="p">()</span>
<span class="n">linear_svm</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">data_train</span><span class="p">,</span> <span class="n">targets_train</span><span class="p">)</span>
<span class="n">linear_svm_score</span> <span class="o">=</span> <span class="n">linear_svm</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">data_test</span><span class="p">,</span> <span class="n">targets_test</span><span class="p">)</span>
<span class="n">linear_svm_time</span> <span class="o">=</span> <a href="https://docs.python.org/3/library/time.html#time.time" title="time.time" class="sphx-glr-backref-module-time sphx-glr-backref-type-py-function"><span class="n">time</span></a><span class="p">()</span> <span class="o">-</span> <span class="n">linear_svm_time</span>
<span class="n">sample_sizes</span> <span class="o">=</span> <span class="mi">30</span> <span class="o">*</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.arange.html#numpy.arange" title="numpy.arange" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">arange</span></a><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
<span class="n">fourier_scores</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">nystroem_scores</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">fourier_times</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">nystroem_times</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">D</span> <span class="ow">in</span> <span class="n">sample_sizes</span><span class="p">:</span>
<span class="n">fourier_approx_svm</span><span class="o">.</span><span class="n">set_params</span><span class="p">(</span><span class="n">feature_map__n_components</span><span class="o">=</span><span class="n">D</span><span class="p">)</span>
<span class="n">nystroem_approx_svm</span><span class="o">.</span><span class="n">set_params</span><span class="p">(</span><span class="n">feature_map__n_components</span><span class="o">=</span><span class="n">D</span><span class="p">)</span>
<span class="n">start</span> <span class="o">=</span> <a href="https://docs.python.org/3/library/time.html#time.time" title="time.time" class="sphx-glr-backref-module-time sphx-glr-backref-type-py-function"><span class="n">time</span></a><span class="p">()</span>
<span class="n">nystroem_approx_svm</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">data_train</span><span class="p">,</span> <span class="n">targets_train</span><span class="p">)</span>
<span class="n">nystroem_times</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><a href="https://docs.python.org/3/library/time.html#time.time" title="time.time" class="sphx-glr-backref-module-time sphx-glr-backref-type-py-function"><span class="n">time</span></a><span class="p">()</span> <span class="o">-</span> <span class="n">start</span><span class="p">)</span>
<span class="n">start</span> <span class="o">=</span> <a href="https://docs.python.org/3/library/time.html#time.time" title="time.time" class="sphx-glr-backref-module-time sphx-glr-backref-type-py-function"><span class="n">time</span></a><span class="p">()</span>
<span class="n">fourier_approx_svm</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">data_train</span><span class="p">,</span> <span class="n">targets_train</span><span class="p">)</span>
<span class="n">fourier_times</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><a href="https://docs.python.org/3/library/time.html#time.time" title="time.time" class="sphx-glr-backref-module-time sphx-glr-backref-type-py-function"><span class="n">time</span></a><span class="p">()</span> <span class="o">-</span> <span class="n">start</span><span class="p">)</span>
<span class="n">fourier_score</span> <span class="o">=</span> <span class="n">fourier_approx_svm</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">data_test</span><span class="p">,</span> <span class="n">targets_test</span><span class="p">)</span>
<span class="n">nystroem_score</span> <span class="o">=</span> <span class="n">nystroem_approx_svm</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">data_test</span><span class="p">,</span> <span class="n">targets_test</span><span class="p">)</span>
<span class="n">nystroem_scores</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">nystroem_score</span><span class="p">)</span>
<span class="n">fourier_scores</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">fourier_score</span><span class="p">)</span>
<span class="c1"># plot the results:</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.figure.html#matplotlib.pyplot.figure" title="matplotlib.pyplot.figure" 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">figure</span></a><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
<span class="n">accuracy</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplot.html#matplotlib.pyplot.subplot" title="matplotlib.pyplot.subplot" 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">subplot</span></a><span class="p">(</span><span class="mi">121</span><span class="p">)</span>
<span class="c1"># second y axis for timings</span>
<span class="n">timescale</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplot.html#matplotlib.pyplot.subplot" title="matplotlib.pyplot.subplot" 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">subplot</span></a><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
<span class="n">accuracy</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">sample_sizes</span><span class="p">,</span> <span class="n">nystroem_scores</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"Nystroem approx. kernel"</span><span class="p">)</span>
<span class="n">timescale</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">sample_sizes</span><span class="p">,</span> <span class="n">nystroem_times</span><span class="p">,</span> <span class="s2">"--"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"Nystroem approx. kernel"</span><span class="p">)</span>
<span class="n">accuracy</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">sample_sizes</span><span class="p">,</span> <span class="n">fourier_scores</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"Fourier approx. kernel"</span><span class="p">)</span>
<span class="n">timescale</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">sample_sizes</span><span class="p">,</span> <span class="n">fourier_times</span><span class="p">,</span> <span class="s2">"--"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"Fourier approx. kernel"</span><span class="p">)</span>
<span class="c1"># horizontal lines for exact rbf and linear kernels:</span>
<span class="n">accuracy</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
<span class="p">[</span><span class="n">sample_sizes</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">sample_sizes</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]],</span>
<span class="p">[</span><span class="n">linear_svm_score</span><span class="p">,</span> <span class="n">linear_svm_score</span><span class="p">],</span>
<span class="n">label</span><span class="o">=</span><span class="s2">"linear svm"</span><span class="p">,</span>
<span class="p">)</span>
<span class="n">timescale</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
<span class="p">[</span><span class="n">sample_sizes</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">sample_sizes</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]],</span>
<span class="p">[</span><span class="n">linear_svm_time</span><span class="p">,</span> <span class="n">linear_svm_time</span><span class="p">],</span>
<span class="s2">"--"</span><span class="p">,</span>
<span class="n">label</span><span class="o">=</span><span class="s2">"linear svm"</span><span class="p">,</span>
<span class="p">)</span>
<span class="n">accuracy</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
<span class="p">[</span><span class="n">sample_sizes</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">sample_sizes</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]],</span>
<span class="p">[</span><span class="n">kernel_svm_score</span><span class="p">,</span> <span class="n">kernel_svm_score</span><span class="p">],</span>
<span class="n">label</span><span class="o">=</span><span class="s2">"rbf svm"</span><span class="p">,</span>
<span class="p">)</span>
<span class="n">timescale</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
<span class="p">[</span><span class="n">sample_sizes</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">sample_sizes</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]],</span>
<span class="p">[</span><span class="n">kernel_svm_time</span><span class="p">,</span> <span class="n">kernel_svm_time</span><span class="p">],</span>
<span class="s2">"--"</span><span class="p">,</span>
<span class="n">label</span><span class="o">=</span><span class="s2">"rbf svm"</span><span class="p">,</span>
<span class="p">)</span>
<span class="c1"># vertical line for dataset dimensionality = 64</span>
<span class="n">accuracy</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="mi">64</span><span class="p">,</span> <span class="mi">64</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.7</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">"n_features"</span><span class="p">)</span>
<span class="c1"># legends and labels</span>
<span class="n">accuracy</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Classification accuracy"</span><span class="p">)</span>
<span class="n">timescale</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Training times"</span><span class="p">)</span>
<span class="n">accuracy</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="n">sample_sizes</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">sample_sizes</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
<span class="n">accuracy</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(())</span>
<span class="n">accuracy</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.min.html#numpy.min" title="numpy.min" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">min</span></a><span class="p">(</span><span class="n">fourier_scores</span><span class="p">),</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">timescale</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">"Sampling steps = transformed feature dimension"</span><span class="p">)</span>
<span class="n">accuracy</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">"Classification accuracy"</span><span class="p">)</span>
<span class="n">timescale</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">"Training time in seconds"</span><span class="p">)</span>
<span class="n">accuracy</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">"best"</span><span class="p">)</span>
<span class="n">timescale</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">"best"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.tight_layout.html#matplotlib.pyplot.tight_layout" title="matplotlib.pyplot.tight_layout" 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">tight_layout</span></a><span class="p">()</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_kernel_approximation_001.png" srcset="../../_images/sphx_glr_plot_kernel_approximation_001.png" alt="Classification accuracy, Training times" class = "sphx-glr-single-img"/></section>
<section id="decision-surfaces-of-rbf-kernel-svm-and-linear-svm">
<h2>Decision Surfaces of RBF Kernel SVM and Linear SVM<a class="headerlink" href="#decision-surfaces-of-rbf-kernel-svm-and-linear-svm" title="Link to this heading">¶</a></h2>
<p>The second plot visualized the decision surfaces of the RBF kernel SVM and
the linear SVM with approximate kernel maps.
The plot shows decision surfaces of the classifiers projected onto
the first two principal components of the data. This visualization should
be taken with a grain of salt since it is just an interesting slice through
the decision surface in 64 dimensions. In particular note that
a datapoint (represented as a dot) does not necessarily be classified
into the region it is lying in, since it will not lie on the plane
that the first two principal components span.
The usage of <a class="reference internal" href="../../modules/generated/sklearn.kernel_approximation.RBFSampler.html#sklearn.kernel_approximation.RBFSampler" title="sklearn.kernel_approximation.RBFSampler"><code class="xref py py-class docutils literal notranslate"><span class="pre">RBFSampler</span></code></a> and <a class="reference internal" href="../../modules/generated/sklearn.kernel_approximation.Nystroem.html#sklearn.kernel_approximation.Nystroem" title="sklearn.kernel_approximation.Nystroem"><code class="xref py py-class docutils literal notranslate"><span class="pre">Nystroem</span></code></a> is described in detail
in <a class="reference internal" href="../../modules/kernel_approximation.html#kernel-approximation"><span class="std std-ref">Kernel Approximation</span></a>.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="c1"># visualize the decision surface, projected down to the first</span>
<span class="c1"># two principal components of the dataset</span>
<span class="n">pca</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA" class="sphx-glr-backref-module-sklearn-decomposition sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">PCA</span></a><span class="p">(</span><span class="n">n_components</span><span class="o">=</span><span class="mi">8</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">data_train</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">pca</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">data_train</span><span class="p">)</span>
<span class="c1"># Generate grid along first two principal components</span>
<span class="n">multiples</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.arange.html#numpy.arange" title="numpy.arange" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">arange</span></a><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">)</span>
<span class="c1"># steps along first component</span>
<span class="n">first</span> <span class="o">=</span> <span class="n">multiples</span><span class="p">[:,</span> <a href="https://numpy.org/doc/stable/reference/constants.html#numpy.newaxis" title="numpy.newaxis" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">newaxis</span></a><span class="p">]</span> <span class="o">*</span> <span class="n">pca</span><span class="o">.</span><span class="n">components_</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:]</span>
<span class="c1"># steps along second component</span>
<span class="n">second</span> <span class="o">=</span> <span class="n">multiples</span><span class="p">[:,</span> <a href="https://numpy.org/doc/stable/reference/constants.html#numpy.newaxis" title="numpy.newaxis" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">newaxis</span></a><span class="p">]</span> <span class="o">*</span> <span class="n">pca</span><span class="o">.</span><span class="n">components_</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]</span>
<span class="c1"># combine</span>
<span class="n">grid</span> <span class="o">=</span> <span class="n">first</span><span class="p">[</span><a href="https://numpy.org/doc/stable/reference/constants.html#numpy.newaxis" title="numpy.newaxis" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">newaxis</span></a><span class="p">,</span> <span class="p">:,</span> <span class="p">:]</span> <span class="o">+</span> <span class="n">second</span><span class="p">[:,</span> <a href="https://numpy.org/doc/stable/reference/constants.html#numpy.newaxis" title="numpy.newaxis" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">newaxis</span></a><span class="p">,</span> <span class="p">:]</span>
<span class="n">flat_grid</span> <span class="o">=</span> <span class="n">grid</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">data</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"># title for the plots</span>
<span class="n">titles</span> <span class="o">=</span> <span class="p">[</span>
<span class="s2">"SVC with rbf kernel"</span><span class="p">,</span>
<span class="s2">"SVC (linear kernel)</span><span class="se">\n</span><span class="s2"> with Fourier rbf feature map</span><span class="se">\n</span><span class="s2">n_components=100"</span><span class="p">,</span>
<span class="s2">"SVC (linear kernel)</span><span class="se">\n</span><span class="s2"> with Nystroem rbf feature map</span><span class="se">\n</span><span class="s2">n_components=100"</span><span class="p">,</span>
<span class="p">]</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.figure.html#matplotlib.pyplot.figure" title="matplotlib.pyplot.figure" 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">figure</span></a><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">18</span><span class="p">,</span> <span class="mf">7.5</span><span class="p">))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">rcParams</span><span class="o">.</span><span class="n">update</span><span class="p">({</span><span class="s2">"font.size"</span><span class="p">:</span> <span class="mi">14</span><span class="p">})</span>
<span class="c1"># predict and plot</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">clf</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">((</span><span class="n">kernel_svm</span><span class="p">,</span> <span class="n">nystroem_approx_svm</span><span class="p">,</span> <span class="n">fourier_approx_svm</span><span class="p">)):</span>
<span class="c1"># Plot the decision boundary. For that, we will assign a color to each</span>
<span class="c1"># point in the mesh [x_min, x_max]x[y_min, y_max].</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplot.html#matplotlib.pyplot.subplot" title="matplotlib.pyplot.subplot" 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">subplot</span></a><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">Z</span> <span class="o">=</span> <span class="n">clf</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">flat_grid</span><span class="p">)</span>
<span class="c1"># Put the result into a color plot</span>
<span class="n">Z</span> <span class="o">=</span> <span class="n">Z</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">grid</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
<span class="n">levels</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.arange.html#numpy.arange" title="numpy.arange" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">arange</span></a><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="n">lv_eps</span> <span class="o">=</span> <span class="mf">0.01</span> <span class="c1"># Adjust a mapping from calculated contour levels to color.</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.contourf.html#matplotlib.pyplot.contourf" title="matplotlib.pyplot.contourf" 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">contourf</span></a><span class="p">(</span>
<span class="n">multiples</span><span class="p">,</span>
<span class="n">multiples</span><span class="p">,</span>
<span class="n">Z</span><span class="p">,</span>
<span class="n">levels</span><span class="o">=</span><span class="n">levels</span> <span class="o">-</span> <span class="n">lv_eps</span><span class="p">,</span>
<span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">tab10</span><span class="p">,</span>
<span class="n">vmin</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span>
<span class="n">vmax</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span>
<span class="n">alpha</span><span class="o">=</span><span class="mf">0.7</span><span class="p">,</span>
<span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.axis.html#matplotlib.pyplot.axis" title="matplotlib.pyplot.axis" 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">axis</span></a><span class="p">(</span><span class="s2">"off"</span><span class="p">)</span>
<span class="c1"># Plot also the training points</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.scatter.html#matplotlib.pyplot.scatter" title="matplotlib.pyplot.scatter" 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">scatter</span></a><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">targets_train</span><span class="p">,</span>
<span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">tab10</span><span class="p">,</span>
<span class="n">edgecolors</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="mi">0</span><span class="p">),</span>
<span class="n">vmin</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span>
<span class="n">vmax</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span>
<span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.title.html#matplotlib.pyplot.title" title="matplotlib.pyplot.title" 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">title</span></a><span class="p">(</span><span class="n">titles</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.tight_layout.html#matplotlib.pyplot.tight_layout" title="matplotlib.pyplot.tight_layout" 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">tight_layout</span></a><span class="p">()</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>
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<p><a class="reference internal" href="../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>
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