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<span id="sphx-glr-auto-examples-gaussian-process-plot-gpc-isoprobability-py"></span><h1>Iso-probability lines for Gaussian Processes classification (GPC)<a class="headerlink" href="#iso-probability-lines-for-gaussian-processes-classification-gpc" title="Permalink to this headline">¶</a></h1>
<p>A two-dimensional classification example showing iso-probability lines for
the predicted probabilities.</p>
<img alt="../../_images/sphx_glr_plot_gpc_isoprobability_001.png" class="sphx-glr-single-img" src="../../_images/sphx_glr_plot_gpc_isoprobability_001.png" />
<p class="sphx-glr-script-out">Out:</p>
<div class="sphx-glr-script-out highlight-none"><div class="highlight"><pre><span></span>Learned kernel: 0.0256**2 * DotProduct(sigma_0=5.72) ** 2
</pre></div>
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<div class="highlight-python"><div class="highlight"><pre><span></span><span class="k">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</span>
<span class="c1"># Author: Vincent Dubourg <[email protected]></span>
<span class="c1"># Adapted to GaussianProcessClassifier:</span>
<span class="c1"># Jan Hendrik Metzen <[email protected]></span>
<span class="c1"># License: BSD 3 clause</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">pyplot</span> <span class="k">as</span> <span class="n">plt</span>
<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">cm</span>
<span class="kn">from</span> <span class="nn">sklearn.gaussian_process</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.gaussian_process.GaussianProcessClassifier.html#sklearn.gaussian_process.GaussianProcessClassifier" title="View documentation for sklearn.gaussian_process.GaussianProcessClassifier"><span class="n">GaussianProcessClassifier</span></a>
<span class="kn">from</span> <span class="nn">sklearn.gaussian_process.kernels</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.gaussian_process.kernels.DotProduct.html#sklearn.gaussian_process.kernels.DotProduct" title="View documentation for sklearn.gaussian_process.kernels.DotProduct"><span class="n">DotProduct</span></a><span class="p">,</span> <span class="n">ConstantKernel</span> <span class="k">as</span> <a href="../../modules/generated/sklearn.gaussian_process.kernels.ConstantKernel.html#sklearn.gaussian_process.kernels.ConstantKernel" title="View documentation for sklearn.gaussian_process.kernels.ConstantKernel"><span class="n">C</span></a>
<span class="c1"># A few constants</span>
<span class="n">lim</span> <span class="o">=</span> <span class="mi">8</span>
<span class="k">def</span> <span class="nf">g</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="sd">"""The function to predict (classification will then consist in predicting</span>
<span class="sd"> whether g(x) <= 0 or not)"""</span>
<span class="k">return</span> <span class="mf">5.</span> <span class="o">-</span> <span class="n">x</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="o">.</span><span class="mi">5</span> <span class="o">*</span> <span class="n">x</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">**</span> <span class="mf">2.</span>
<span class="c1"># Design of experiments</span>
<span class="n">X</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.array.html#numpy.array" title="View documentation for numpy.array"><span class="n">np</span><span class="o">.</span><span class="n">array</span></a><span class="p">([[</span><span class="o">-</span><span class="mf">4.61611719</span><span class="p">,</span> <span class="o">-</span><span class="mf">6.00099547</span><span class="p">],</span>
<span class="p">[</span><span class="mf">4.10469096</span><span class="p">,</span> <span class="mf">5.32782448</span><span class="p">],</span>
<span class="p">[</span><span class="mf">0.00000000</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.50000000</span><span class="p">],</span>
<span class="p">[</span><span class="o">-</span><span class="mf">6.17289014</span><span class="p">,</span> <span class="o">-</span><span class="mf">4.6984743</span><span class="p">],</span>
<span class="p">[</span><span class="mf">1.3109306</span><span class="p">,</span> <span class="o">-</span><span class="mf">6.93271427</span><span class="p">],</span>
<span class="p">[</span><span class="o">-</span><span class="mf">5.03823144</span><span class="p">,</span> <span class="mf">3.10584743</span><span class="p">],</span>
<span class="p">[</span><span class="o">-</span><span class="mf">2.87600388</span><span class="p">,</span> <span class="mf">6.74310541</span><span class="p">],</span>
<span class="p">[</span><span class="mf">5.21301203</span><span class="p">,</span> <span class="mf">4.26386883</span><span class="p">]])</span>
<span class="c1"># Observations</span>
<span class="n">y</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.array.html#numpy.array" title="View documentation for numpy.array"><span class="n">np</span><span class="o">.</span><span class="n">array</span></a><span class="p">(</span><span class="n">g</span><span class="p">(</span><span class="n">X</span><span class="p">)</span> <span class="o">></span> <span class="mi">0</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">int</span><span class="p">)</span>
<span class="c1"># Instantiate and fit Gaussian Process Model</span>
<span class="n">kernel</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.gaussian_process.kernels.ConstantKernel.html#sklearn.gaussian_process.kernels.ConstantKernel" title="View documentation for sklearn.gaussian_process.kernels.ConstantKernel"><span class="n">C</span></a><span class="p">(</span><span class="mf">0.1</span><span class="p">,</span> <span class="p">(</span><span class="mf">1e-5</span><span class="p">,</span> <a href="https://docs.scipy.org/doc/numpy/reference/constants.html#numpy.inf" title="View documentation for numpy.inf"><span class="n">np</span><span class="o">.</span><span class="n">inf</span></a><span class="p">))</span> <span class="o">*</span> <a href="../../modules/generated/sklearn.gaussian_process.kernels.DotProduct.html#sklearn.gaussian_process.kernels.DotProduct" title="View documentation for sklearn.gaussian_process.kernels.DotProduct"><span class="n">DotProduct</span></a><span class="p">(</span><span class="n">sigma_0</span><span class="o">=</span><span class="mf">0.1</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span>
<span class="n">gp</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.gaussian_process.GaussianProcessClassifier.html#sklearn.gaussian_process.GaussianProcessClassifier" title="View documentation for sklearn.gaussian_process.GaussianProcessClassifier"><span class="n">GaussianProcessClassifier</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">gp</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="k">print</span><span class="p">(</span><span class="s2">"Learned kernel: </span><span class="si">%s</span><span class="s2"> "</span> <span class="o">%</span> <span class="n">gp</span><span class="o">.</span><span class="n">kernel_</span><span class="p">)</span>
<span class="c1"># Evaluate real function and the predicted probability</span>
<span class="n">res</span> <span class="o">=</span> <span class="mi">50</span>
<span class="n">x1</span><span class="p">,</span> <span class="n">x2</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.meshgrid.html#numpy.meshgrid" title="View documentation for numpy.meshgrid"><span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span></a><span class="p">(</span><a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.linspace.html#numpy.linspace" title="View documentation for 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="n">lim</span><span class="p">,</span> <span class="n">lim</span><span class="p">,</span> <span class="n">res</span><span class="p">),</span>
<a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.linspace.html#numpy.linspace" title="View documentation for 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="n">lim</span><span class="p">,</span> <span class="n">lim</span><span class="p">,</span> <span class="n">res</span><span class="p">))</span>
<span class="n">xx</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.vstack.html#numpy.vstack" title="View documentation for 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="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">x1</span><span class="o">.</span><span class="n">size</span><span class="p">),</span> <span class="n">x2</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">x2</span><span class="o">.</span><span class="n">size</span><span class="p">)])</span><span class="o">.</span><span class="n">T</span>
<span class="n">y_true</span> <span class="o">=</span> <span class="n">g</span><span class="p">(</span><span class="n">xx</span><span class="p">)</span>
<span class="n">y_prob</span> <span class="o">=</span> <span class="n">gp</span><span class="o">.</span><span class="n">predict_proba</span><span class="p">(</span><span class="n">xx</span><span class="p">)[:,</span> <span class="mi">1</span><span class="p">]</span>
<span class="n">y_true</span> <span class="o">=</span> <span class="n">y_true</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="n">res</span><span class="p">,</span> <span class="n">res</span><span class="p">))</span>
<span class="n">y_prob</span> <span class="o">=</span> <span class="n">y_prob</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="n">res</span><span class="p">,</span> <span class="n">res</span><span class="p">))</span>
<span class="c1"># Plot the probabilistic classification iso-values</span>
<span class="n">fig</span> <span class="o">=</span> <a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.figure.html#matplotlib.pyplot.figure" title="View documentation for matplotlib.pyplot.figure"><span class="n">plt</span><span class="o">.</span><span class="n">figure</span></a><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">gca</span><span class="p">()</span>
<span class="n">ax</span><span class="o">.</span><span class="n">axes</span><span class="o">.</span><span class="n">set_aspect</span><span class="p">(</span><span class="s1">'equal'</span><span class="p">)</span>
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<span class="n">ax</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">([])</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_yticklabels</span><span class="p">([])</span>
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<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.ylabel.html#matplotlib.pyplot.ylabel" title="View documentation for matplotlib.pyplot.ylabel"><span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span></a><span class="p">(</span><span class="s1">'$x_2$'</span><span class="p">)</span>
<span class="n">cax</span> <span class="o">=</span> <a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.imshow.html#matplotlib.pyplot.imshow" title="View documentation for matplotlib.pyplot.imshow"><span class="n">plt</span><span class="o">.</span><span class="n">imshow</span></a><span class="p">(</span><span class="n">y_prob</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">cm</span><span class="o">.</span><span class="n">gray_r</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.8</span><span class="p">,</span>
<span class="n">extent</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="n">lim</span><span class="p">,</span> <span class="n">lim</span><span class="p">,</span> <span class="o">-</span><span class="n">lim</span><span class="p">,</span> <span class="n">lim</span><span class="p">))</span>
<span class="n">norm</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">matplotlib</span><span class="o">.</span><span class="n">colors</span><span class="o">.</span><span class="n">Normalize</span><span class="p">(</span><span class="n">vmin</span><span class="o">=</span><span class="mf">0.</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="mf">0.9</span><span class="p">)</span>
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<span class="n">cb</span><span class="o">.</span><span class="n">set_label</span><span class="p">(</span><span class="sa">r</span><span class="s1">'${\rm \mathbb{P}}\left[\widehat{G}(\mathbf{x}) \leq 0\right]$'</span><span class="p">)</span>
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<span class="n">cs</span> <span class="o">=</span> <a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.contour.html#matplotlib.pyplot.contour" title="View documentation for matplotlib.pyplot.contour"><span class="n">plt</span><span class="o">.</span><span class="n">contour</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_prob</span><span class="p">,</span> <span class="p">[</span><span class="mf">0.666</span><span class="p">],</span> <span class="n">colors</span><span class="o">=</span><span class="s1">'b'</span><span class="p">,</span>
<span class="n">linestyles</span><span class="o">=</span><span class="s1">'solid'</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.clabel.html#matplotlib.pyplot.clabel" title="View documentation for matplotlib.pyplot.clabel"><span class="n">plt</span><span class="o">.</span><span class="n">clabel</span></a><span class="p">(</span><span class="n">cs</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">11</span><span class="p">)</span>
<span class="n">cs</span> <span class="o">=</span> <a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.contour.html#matplotlib.pyplot.contour" title="View documentation for matplotlib.pyplot.contour"><span class="n">plt</span><span class="o">.</span><span class="n">contour</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_prob</span><span class="p">,</span> <span class="p">[</span><span class="mf">0.5</span><span class="p">],</span> <span class="n">colors</span><span class="o">=</span><span class="s1">'k'</span><span class="p">,</span>
<span class="n">linestyles</span><span class="o">=</span><span class="s1">'dashed'</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.clabel.html#matplotlib.pyplot.clabel" title="View documentation for matplotlib.pyplot.clabel"><span class="n">plt</span><span class="o">.</span><span class="n">clabel</span></a><span class="p">(</span><span class="n">cs</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">11</span><span class="p">)</span>
<span class="n">cs</span> <span class="o">=</span> <a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.contour.html#matplotlib.pyplot.contour" title="View documentation for matplotlib.pyplot.contour"><span class="n">plt</span><span class="o">.</span><span class="n">contour</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_prob</span><span class="p">,</span> <span class="p">[</span><span class="mf">0.334</span><span class="p">],</span> <span class="n">colors</span><span class="o">=</span><span class="s1">'r'</span><span class="p">,</span>
<span class="n">linestyles</span><span class="o">=</span><span class="s1">'solid'</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.clabel.html#matplotlib.pyplot.clabel" title="View documentation for matplotlib.pyplot.clabel"><span class="n">plt</span><span class="o">.</span><span class="n">clabel</span></a><span class="p">(</span><span class="n">cs</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">11</span><span class="p">)</span>
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