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<div class="section" id="sparse-non-linear-classification">
<span id="example-plot-sparse-non-linear-py"></span><h1>Sparse non-linear classification<a class="headerlink" href="#sparse-non-linear-classification" title="Permalink to this headline">¶</a></h1>
<p>This examples demonstrates how to use <cite>CDClassifier</cite> with L1 penalty to do
sparse non-linear classification. The trick simply consists in fitting the
classifier with a kernel matrix (e.g., using an RBF kernel).</p>
<p>There are a few interesting differences with standard kernel SVMs:</p>
<p>1. the kernel matrix does not need to be positive semi-definite (hence the
expression “kernel matrix” above is an abuse of terminology)</p>
<p>2. the number of “support vectors” will be typically smaller thanks to L1
regularization and can be adjusted by the regularization parameter C (the
smaller C, the fewer the support vectors)</p>
<ol class="arabic simple" start="3">
<li>the “support vectors” need not be located at the margin</li>
</ol>
<ul class="horizontal">
<li><a class="first reference internal image-reference" href="../_images/plot_sparse_non_linear_001.png"><img alt="../_images/plot_sparse_non_linear_001.png" src="../_images/plot_sparse_non_linear_001.png" style="width: 376.0px; height: 282.0px;" /></a>
</li>
<li><a class="first reference internal image-reference" href="../_images/plot_sparse_non_linear_002.png"><img alt="../_images/plot_sparse_non_linear_002.png" src="../_images/plot_sparse_non_linear_002.png" style="width: 376.0px; height: 282.0px;" /></a>
</li>
</ul>
<p><strong>Python source code:</strong> <a class="reference download internal" href="../_downloads/plot_sparse_non_linear.py"><tt class="xref download docutils literal"><span class="pre">plot_sparse_non_linear.py</span></tt></a></p>
<div class="highlight-python"><div class="highlight"><pre><span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">pylab</span> <span class="kn">as</span> <span class="nn">pl</span>
<span class="kn">from</span> <span class="nn">sklearn.metrics.pairwise</span> <span class="kn">import</span> <span class="n">rbf_kernel</span>
<span class="kn">from</span> <span class="nn">lightning.classification</span> <span class="kn">import</span> <span class="n">CDClassifier</span>
<span class="kn">from</span> <span class="nn">lightning.classification</span> <span class="kn">import</span> <span class="n">KernelSVC</span>
<a href="http://docs.scipy.org/doc/numpy-1.6.0/reference/generated/numpy.random.seed.html#numpy.random.seed"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span></a><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="k">class</span> <span class="nc">SparseNonlinearClassifier</span><span class="p">(</span><span class="n">CDClassifier</span><span class="p">):</span>
<span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mf">1e-2</span><span class="p">,</span> <span class="n">C</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">gamma</span> <span class="o">=</span> <span class="n">gamma</span>
<span class="nb">super</span><span class="p">(</span><span class="n">SparseNonlinearClassifier</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="n">C</span><span class="o">=</span><span class="n">C</span><span class="p">,</span>
<span class="n">alpha</span><span class="o">=</span><span class="n">alpha</span><span class="p">,</span>
<span class="n">loss</span><span class="o">=</span><span class="s">"squared_hinge"</span><span class="p">,</span>
<span class="n">penalty</span><span class="o">=</span><span class="s">"l1"</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">fit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
<span class="n">K</span> <span class="o">=</span> <span class="n">rbf_kernel</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">gamma</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">X_train_</span> <span class="o">=</span> <span class="n">X</span>
<span class="nb">super</span><span class="p">(</span><span class="n">SparseNonlinearClassifier</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">K</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="k">return</span> <span class="bp">self</span>
<span class="k">def</span> <span class="nf">decision_function</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
<span class="n">K</span> <span class="o">=</span> <span class="n">rbf_kernel</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">X_train_</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">gamma</span><span class="p">)</span>
<span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">SparseNonlinearClassifier</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">decision_function</span><span class="p">(</span><span class="n">K</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">gen_non_lin_separable_data</span><span class="p">():</span>
<span class="n">mean1</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">2</span><span class="p">]</span>
<span class="n">mean2</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="n">mean3</span> <span class="o">=</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="o">-</span><span class="mi">4</span><span class="p">]</span>
<span class="n">mean4</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">]</span>
<span class="n">cov</span> <span class="o">=</span> <span class="p">[[</span><span class="mf">1.0</span><span class="p">,</span><span class="mf">0.8</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.8</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">]]</span>
<span class="n">X1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">multivariate_normal</span><span class="p">(</span><span class="n">mean1</span><span class="p">,</span> <span class="n">cov</span><span class="p">,</span> <span class="mi">50</span><span class="p">)</span>
<span class="n">X1</span> <span class="o">=</span> <a href="http://docs.scipy.org/doc/numpy-1.6.0/reference/generated/numpy.vstack.html#numpy.vstack"><span class="n">np</span><span class="o">.</span><span class="n">vstack</span></a><span class="p">((</span><span class="n">X1</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">multivariate_normal</span><span class="p">(</span><span class="n">mean3</span><span class="p">,</span> <span class="n">cov</span><span class="p">,</span> <span class="mi">50</span><span class="p">)))</span>
<span class="n">y1</span> <span class="o">=</span> <a href="http://docs.scipy.org/doc/numpy-1.6.0/reference/generated/numpy.ones.html#numpy.ones"><span class="n">np</span><span class="o">.</span><span class="n">ones</span></a><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">X1</span><span class="p">))</span>
<span class="n">X2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">multivariate_normal</span><span class="p">(</span><span class="n">mean2</span><span class="p">,</span> <span class="n">cov</span><span class="p">,</span> <span class="mi">50</span><span class="p">)</span>
<span class="n">X2</span> <span class="o">=</span> <a href="http://docs.scipy.org/doc/numpy-1.6.0/reference/generated/numpy.vstack.html#numpy.vstack"><span class="n">np</span><span class="o">.</span><span class="n">vstack</span></a><span class="p">((</span><span class="n">X2</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">multivariate_normal</span><span class="p">(</span><span class="n">mean4</span><span class="p">,</span> <span class="n">cov</span><span class="p">,</span> <span class="mi">50</span><span class="p">)))</span>
<span class="n">y2</span> <span class="o">=</span> <a href="http://docs.scipy.org/doc/numpy-1.6.0/reference/generated/numpy.ones.html#numpy.ones"><span class="n">np</span><span class="o">.</span><span class="n">ones</span></a><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">X2</span><span class="p">))</span> <span class="o">*</span> <span class="o">-</span><span class="mi">1</span>
<span class="k">return</span> <span class="n">X1</span><span class="p">,</span> <span class="n">y1</span><span class="p">,</span> <span class="n">X2</span><span class="p">,</span> <span class="n">y2</span>
<span class="k">def</span> <span class="nf">plot_contour</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">X1</span><span class="p">,</span> <span class="n">X2</span><span class="p">,</span> <span class="n">clf</span><span class="p">,</span> <span class="n">title</span><span class="p">):</span>
<span class="n">pl</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
<span class="n">pl</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="n">title</span><span class="p">)</span>
<span class="c"># Plot instances of class 1.</span>
<span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X1</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">X1</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s">"ro"</span><span class="p">)</span>
<span class="c"># Plot instances of class 2.</span>
<span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X2</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">X2</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s">"bo"</span><span class="p">)</span>
<span class="c"># Select "support vectors".</span>
<span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">clf</span><span class="p">,</span> <span class="s">"support_vectors_"</span><span class="p">):</span>
<span class="n">sv</span> <span class="o">=</span> <span class="n">clf</span><span class="o">.</span><span class="n">support_vectors_</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">sv</span> <span class="o">=</span> <span class="n">X</span><span class="p">[</span><span class="n">clf</span><span class="o">.</span><span class="n">coef_</span><span class="o">.</span><span class="n">ravel</span><span class="p">()</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">]</span>
<span class="c"># Plot support vectors.</span>
<span class="n">pl</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">sv</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">sv</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">100</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s">"g"</span><span class="p">)</span>
<span class="c"># Plot decision surface.</span>
<span class="n">A</span><span class="p">,</span> <span class="n">B</span> <span class="o">=</span> <a href="http://docs.scipy.org/doc/numpy-1.6.0/reference/generated/numpy.meshgrid.html#numpy.meshgrid"><span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span></a><span class="p">(</span><a href="http://docs.scipy.org/doc/numpy-1.6.0/reference/generated/numpy.linspace.html#numpy.linspace"><span class="n">np</span><span class="o">.</span><span class="n">linspace</span></a><span class="p">(</span><span class="o">-</span><span class="mi">6</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">50</span><span class="p">),</span> <a href="http://docs.scipy.org/doc/numpy-1.6.0/reference/generated/numpy.linspace.html#numpy.linspace"><span class="n">np</span><span class="o">.</span><span class="n">linspace</span></a><span class="p">(</span><span class="o">-</span><span class="mi">6</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">50</span><span class="p">))</span>
<span class="n">C</span> <span class="o">=</span> <a href="http://docs.scipy.org/doc/numpy-1.6.0/reference/generated/numpy.array.html#numpy.array"><span class="n">np</span><span class="o">.</span><span class="n">array</span></a><span class="p">([[</span><span class="n">x1</span><span class="p">,</span> <span class="n">x2</span><span class="p">]</span> <span class="k">for</span> <span class="n">x1</span><span class="p">,</span> <span class="n">x2</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><a href="http://docs.scipy.org/doc/numpy-1.6.0/reference/generated/numpy.ravel.html#numpy.ravel"><span class="n">np</span><span class="o">.</span><span class="n">ravel</span></a><span class="p">(</span><span class="n">A</span><span class="p">),</span> <a href="http://docs.scipy.org/doc/numpy-1.6.0/reference/generated/numpy.ravel.html#numpy.ravel"><span class="n">np</span><span class="o">.</span><span class="n">ravel</span></a><span class="p">(</span><span class="n">B</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">decision_function</span><span class="p">(</span><span class="n">C</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">contour</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">Z</span><span class="p">,</span> <span class="p">[</span><span class="mf">0.0</span><span class="p">],</span> <span class="n">colors</span><span class="o">=</span><span class="s">'k'</span><span class="p">,</span> <span class="n">linewidths</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">origin</span><span class="o">=</span><span class="s">'lower'</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s">"tight"</span><span class="p">)</span>
<span class="c"># Generate synthetic data from 2 classes.</span>
<span class="n">X1</span><span class="p">,</span> <span class="n">y1</span><span class="p">,</span> <span class="n">X2</span><span class="p">,</span> <span class="n">y2</span> <span class="o">=</span> <span class="n">gen_non_lin_separable_data</span><span class="p">()</span>
<span class="c"># Combine them to form a training set.</span>
<span class="n">X</span> <span class="o">=</span> <a href="http://docs.scipy.org/doc/numpy-1.6.0/reference/generated/numpy.vstack.html#numpy.vstack"><span class="n">np</span><span class="o">.</span><span class="n">vstack</span></a><span class="p">((</span><span class="n">X1</span><span class="p">,</span> <span class="n">X2</span><span class="p">))</span>
<span class="n">y</span> <span class="o">=</span> <a href="http://docs.scipy.org/doc/numpy-1.6.0/reference/generated/numpy.hstack.html#numpy.hstack"><span class="n">np</span><span class="o">.</span><span class="n">hstack</span></a><span class="p">((</span><span class="n">y1</span><span class="p">,</span> <span class="n">y2</span><span class="p">))</span>
<span class="c"># Train the classifiers.</span>
<span class="n">clf</span> <span class="o">=</span> <span class="n">SparseNonlinearClassifier</span><span class="p">(</span><span class="n">gamma</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">1.</span><span class="o">/</span><span class="mf">0.05</span><span class="p">)</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="n">clf2</span> <span class="o">=</span> <span class="n">KernelSVC</span><span class="p">(</span><span class="n">gamma</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">kernel</span><span class="o">=</span><span class="s">"rbf"</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">1e-2</span><span class="p">)</span>
<span class="n">clf2</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="c"># Plot contours.</span>
<span class="n">plot_contour</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">X1</span><span class="p">,</span> <span class="n">X2</span><span class="p">,</span> <span class="n">clf</span><span class="p">,</span> <span class="s">"Sparse"</span><span class="p">)</span>
<span class="n">plot_contour</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">X1</span><span class="p">,</span> <span class="n">X2</span><span class="p">,</span> <span class="n">clf2</span><span class="p">,</span> <span class="s">"Kernel SVM"</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
<p><strong>Total running time of the example:</strong> 0.13 seconds
( 0 minutes 0.13 seconds)</p>
</div>
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