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<li><a class="reference internal" href="#">GMM covariances</a></li>
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<div class="sphx-glr-example-title section" id="gmm-covariances">
<span id="sphx-glr-auto-examples-mixture-plot-gmm-covariances-py"></span><h1>GMM covariances<a class="headerlink" href="#gmm-covariances" title="Permalink to this headline">¶</a></h1>
<p>Demonstration of several covariances types for Gaussian mixture models.</p>
<p>See <a class="reference internal" href="../../modules/mixture.html#gmm"><span class="std std-ref">Gaussian mixture models</span></a> for more information on the estimator.</p>
<p>Although GMM are often used for clustering, we can compare the obtained
clusters with the actual classes from the dataset. We initialize the means
of the Gaussians with the means of the classes from the training set to make
this comparison valid.</p>
<p>We plot predicted labels on both training and held out test data using a
variety of GMM covariance types on the iris dataset.
We compare GMMs with spherical, diagonal, full, and tied covariance
matrices in increasing order of performance. Although one would
expect full covariance to perform best in general, it is prone to
overfitting on small datasets and does not generalize well to held out
test data.</p>
<p>On the plots, train data is shown as dots, while test data is shown as
crosses. The iris dataset is four-dimensional. Only the first two
dimensions are shown here, and thus some points are separated in other
dimensions.</p>
<img alt="../../_images/sphx_glr_plot_gmm_covariances_001.png" class="sphx-glr-single-img" src="../../_images/sphx_glr_plot_gmm_covariances_001.png" />
<p class="sphx-glr-script-out">Out:</p>
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</pre></div>
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<div class="highlight-default"><div class="highlight"><pre><span></span><span class="c1"># Author: Ron Weiss <[email protected]>, Gael Varoquaux</span>
<span class="c1"># Modified by Thierry Guillemot <[email protected]></span>
<span class="c1"># License: BSD 3 clause</span>
<span class="kn">import</span> <span class="nn">matplotlib</span> <span class="k">as</span> <span class="nn">mpl</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="kn">from</span> <span class="nn">sklearn</span> <span class="k">import</span> <span class="n">datasets</span>
<span class="kn">from</span> <span class="nn">sklearn.mixture</span> <span class="k">import</span> <a href="../../modules/generated/sklearn.mixture.GaussianMixture.html#sklearn.mixture.GaussianMixture" title="View documentation for sklearn.mixture.GaussianMixture"><span class="n">GaussianMixture</span></a>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="k">import</span> <a href="../../modules/generated/sklearn.model_selection.StratifiedKFold.html#sklearn.model_selection.StratifiedKFold" title="View documentation for sklearn.model_selection.StratifiedKFold"><span class="n">StratifiedKFold</span></a>
<span class="nb">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</span>
<span class="n">colors</span> <span class="o">=</span> <span class="p">[</span><span class="s1">'navy'</span><span class="p">,</span> <span class="s1">'turquoise'</span><span class="p">,</span> <span class="s1">'darkorange'</span><span class="p">]</span>
<span class="k">def</span> <span class="nf">make_ellipses</span><span class="p">(</span><span class="n">gmm</span><span class="p">,</span> <span class="n">ax</span><span class="p">):</span>
<span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="n">color</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">colors</span><span class="p">):</span>
<span class="k">if</span> <span class="n">gmm</span><span class="o">.</span><span class="n">covariance_type</span> <span class="o">==</span> <span class="s1">'full'</span><span class="p">:</span>
<span class="n">covariances</span> <span class="o">=</span> <span class="n">gmm</span><span class="o">.</span><span class="n">covariances_</span><span class="p">[</span><span class="n">n</span><span class="p">][:</span><span class="mi">2</span><span class="p">,</span> <span class="p">:</span><span class="mi">2</span><span class="p">]</span>
<span class="k">elif</span> <span class="n">gmm</span><span class="o">.</span><span class="n">covariance_type</span> <span class="o">==</span> <span class="s1">'tied'</span><span class="p">:</span>
<span class="n">covariances</span> <span class="o">=</span> <span class="n">gmm</span><span class="o">.</span><span class="n">covariances_</span><span class="p">[:</span><span class="mi">2</span><span class="p">,</span> <span class="p">:</span><span class="mi">2</span><span class="p">]</span>
<span class="k">elif</span> <span class="n">gmm</span><span class="o">.</span><span class="n">covariance_type</span> <span class="o">==</span> <span class="s1">'diag'</span><span class="p">:</span>
<span class="n">covariances</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.diag.html#numpy.diag" title="View documentation for numpy.diag"><span class="n">np</span><span class="o">.</span><span class="n">diag</span></a><span class="p">(</span><span class="n">gmm</span><span class="o">.</span><span class="n">covariances_</span><span class="p">[</span><span class="n">n</span><span class="p">][:</span><span class="mi">2</span><span class="p">])</span>
<span class="k">elif</span> <span class="n">gmm</span><span class="o">.</span><span class="n">covariance_type</span> <span class="o">==</span> <span class="s1">'spherical'</span><span class="p">:</span>
<span class="n">covariances</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.eye.html#numpy.eye" title="View documentation for numpy.eye"><span class="n">np</span><span class="o">.</span><span class="n">eye</span></a><span class="p">(</span><span class="n">gmm</span><span class="o">.</span><span class="n">means_</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="o">*</span> <span class="n">gmm</span><span class="o">.</span><span class="n">covariances_</span><span class="p">[</span><span class="n">n</span><span class="p">]</span>
<span class="n">v</span><span class="p">,</span> <span class="n">w</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.linalg.eigh.html#numpy.linalg.eigh" title="View documentation for numpy.linalg.eigh"><span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eigh</span></a><span class="p">(</span><span class="n">covariances</span><span class="p">)</span>
<span class="n">u</span> <span class="o">=</span> <span class="n">w</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">/</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.linalg.norm.html#numpy.linalg.norm" title="View documentation for numpy.linalg.norm"><span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">norm</span></a><span class="p">(</span><span class="n">w</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">angle</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.arctan2.html#numpy.arctan2" title="View documentation for numpy.arctan2"><span class="n">np</span><span class="o">.</span><span class="n">arctan2</span></a><span class="p">(</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">u</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">angle</span> <span class="o">=</span> <span class="mi">180</span> <span class="o">*</span> <span class="n">angle</span> <span class="o">/</span> <a href="https://docs.scipy.org/doc/numpy/reference/constants.html#numpy.pi" title="View documentation for numpy.pi"><span class="n">np</span><span class="o">.</span><span class="n">pi</span></a> <span class="c1"># convert to degrees</span>
<span class="n">v</span> <span class="o">=</span> <span class="mf">2.</span> <span class="o">*</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.sqrt.html#numpy.sqrt" title="View documentation for numpy.sqrt"><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span></a><span class="p">(</span><span class="mf">2.</span><span class="p">)</span> <span class="o">*</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.sqrt.html#numpy.sqrt" title="View documentation for numpy.sqrt"><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span></a><span class="p">(</span><span class="n">v</span><span class="p">)</span>
<span class="n">ell</span> <span class="o">=</span> <a href="https://matplotlib.org/api/_as_gen/matplotlib.patches.Ellipse.html#matplotlib.patches.Ellipse" title="View documentation for matplotlib.patches.Ellipse"><span class="n">mpl</span><span class="o">.</span><span class="n">patches</span><span class="o">.</span><span class="n">Ellipse</span></a><span class="p">(</span><span class="n">gmm</span><span class="o">.</span><span class="n">means_</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="p">:</span><span class="mi">2</span><span class="p">],</span> <span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span>
<span class="mi">180</span> <span class="o">+</span> <span class="n">angle</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">color</span><span class="p">)</span>
<span class="n">ell</span><span class="o">.</span><span class="n">set_clip_box</span><span class="p">(</span><span class="n">ax</span><span class="o">.</span><span class="n">bbox</span><span class="p">)</span>
<span class="n">ell</span><span class="o">.</span><span class="n">set_alpha</span><span class="p">(</span><span class="mf">0.5</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">add_artist</span><span class="p">(</span><span class="n">ell</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_aspect</span><span class="p">(</span><span class="s1">'equal'</span><span class="p">,</span> <span class="s1">'datalim'</span><span class="p">)</span>
<span class="n">iris</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.datasets.load_iris.html#sklearn.datasets.load_iris" title="View documentation for sklearn.datasets.load_iris"><span class="n">datasets</span><span class="o">.</span><span class="n">load_iris</span></a><span class="p">()</span>
<span class="c1"># Break up the dataset into non-overlapping training (75%) and testing</span>
<span class="c1"># (25%) sets.</span>
<span class="n">skf</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.model_selection.StratifiedKFold.html#sklearn.model_selection.StratifiedKFold" title="View documentation for sklearn.model_selection.StratifiedKFold"><span class="n">StratifiedKFold</span></a><span class="p">(</span><span class="n">n_splits</span><span class="o">=</span><span class="mi">4</span><span class="p">)</span>
<span class="c1"># Only take the first fold.</span>
<span class="n">train_index</span><span class="p">,</span> <span class="n">test_index</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="nb">iter</span><span class="p">(</span><span class="n">skf</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="n">iris</span><span class="o">.</span><span class="n">data</span><span class="p">,</span> <span class="n">iris</span><span class="o">.</span><span class="n">target</span><span class="p">)))</span>
<span class="n">X_train</span> <span class="o">=</span> <span class="n">iris</span><span class="o">.</span><span class="n">data</span><span class="p">[</span><span class="n">train_index</span><span class="p">]</span>
<span class="n">y_train</span> <span class="o">=</span> <span class="n">iris</span><span class="o">.</span><span class="n">target</span><span class="p">[</span><span class="n">train_index</span><span class="p">]</span>
<span class="n">X_test</span> <span class="o">=</span> <span class="n">iris</span><span class="o">.</span><span class="n">data</span><span class="p">[</span><span class="n">test_index</span><span class="p">]</span>
<span class="n">y_test</span> <span class="o">=</span> <span class="n">iris</span><span class="o">.</span><span class="n">target</span><span class="p">[</span><span class="n">test_index</span><span class="p">]</span>
<span class="n">n_classes</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.unique.html#numpy.unique" title="View documentation for numpy.unique"><span class="n">np</span><span class="o">.</span><span class="n">unique</span></a><span class="p">(</span><span class="n">y_train</span><span class="p">))</span>
<span class="c1"># Try GMMs using different types of covariances.</span>
<span class="n">estimators</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">((</span><span class="n">cov_type</span><span class="p">,</span> <a href="../../modules/generated/sklearn.mixture.GaussianMixture.html#sklearn.mixture.GaussianMixture" title="View documentation for sklearn.mixture.GaussianMixture"><span class="n">GaussianMixture</span></a><span class="p">(</span><span class="n">n_components</span><span class="o">=</span><span class="n">n_classes</span><span class="p">,</span>
<span class="n">covariance_type</span><span class="o">=</span><span class="n">cov_type</span><span class="p">,</span> <span class="n">max_iter</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">))</span>
<span class="k">for</span> <span class="n">cov_type</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">'spherical'</span><span class="p">,</span> <span class="s1">'diag'</span><span class="p">,</span> <span class="s1">'tied'</span><span class="p">,</span> <span class="s1">'full'</span><span class="p">])</span>
<span class="n">n_estimators</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">estimators</span><span class="p">)</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="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">3</span> <span class="o">*</span> <span class="n">n_estimators</span> <span class="o">//</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">6</span><span class="p">))</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.subplots_adjust.html#matplotlib.pyplot.subplots_adjust" title="View documentation for matplotlib.pyplot.subplots_adjust"><span class="n">plt</span><span class="o">.</span><span class="n">subplots_adjust</span></a><span class="p">(</span><span class="n">bottom</span><span class="o">=.</span><span class="mi">01</span><span class="p">,</span> <span class="n">top</span><span class="o">=</span><span class="mf">0.95</span><span class="p">,</span> <span class="n">hspace</span><span class="o">=.</span><span class="mi">15</span><span class="p">,</span> <span class="n">wspace</span><span class="o">=.</span><span class="mi">05</span><span class="p">,</span>
<span class="n">left</span><span class="o">=.</span><span class="mi">01</span><span class="p">,</span> <span class="n">right</span><span class="o">=.</span><span class="mi">99</span><span class="p">)</span>
<span class="k">for</span> <span class="n">index</span><span class="p">,</span> <span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">estimator</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">estimators</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
<span class="c1"># Since we have class labels for the training data, we can</span>
<span class="c1"># initialize the GMM parameters in a supervised manner.</span>
<span class="n">estimator</span><span class="o">.</span><span class="n">means_init</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">X_train</span><span class="p">[</span><span class="n">y_train</span> <span class="o">==</span> <span class="n">i</span><span class="p">]</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="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_classes</span><span class="p">)])</span>
<span class="c1"># Train the other parameters using the EM algorithm.</span>
<span class="n">estimator</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
<span class="n">h</span> <span class="o">=</span> <a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.subplot.html#matplotlib.pyplot.subplot" title="View documentation for matplotlib.pyplot.subplot"><span class="n">plt</span><span class="o">.</span><span class="n">subplot</span></a><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">n_estimators</span> <span class="o">//</span> <span class="mi">2</span><span class="p">,</span> <span class="n">index</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">make_ellipses</span><span class="p">(</span><span class="n">estimator</span><span class="p">,</span> <span class="n">h</span><span class="p">)</span>
<span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="n">color</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">colors</span><span class="p">):</span>
<span class="n">data</span> <span class="o">=</span> <span class="n">iris</span><span class="o">.</span><span class="n">data</span><span class="p">[</span><span class="n">iris</span><span class="o">.</span><span class="n">target</span> <span class="o">==</span> <span class="n">n</span><span class="p">]</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.scatter.html#matplotlib.pyplot.scatter" title="View documentation for matplotlib.pyplot.scatter"><span class="n">plt</span><span class="o">.</span><span class="n">scatter</span></a><span class="p">(</span><span class="n">data</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">data</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="mf">0.8</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">color</span><span class="p">,</span>
<span class="n">label</span><span class="o">=</span><span class="n">iris</span><span class="o">.</span><span class="n">target_names</span><span class="p">[</span><span class="n">n</span><span class="p">])</span>
<span class="c1"># Plot the test data with crosses</span>
<span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="n">color</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">colors</span><span class="p">):</span>
<span class="n">data</span> <span class="o">=</span> <span class="n">X_test</span><span class="p">[</span><span class="n">y_test</span> <span class="o">==</span> <span class="n">n</span><span class="p">]</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.scatter.html#matplotlib.pyplot.scatter" title="View documentation for matplotlib.pyplot.scatter"><span class="n">plt</span><span class="o">.</span><span class="n">scatter</span></a><span class="p">(</span><span class="n">data</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">data</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">marker</span><span class="o">=</span><span class="s1">'x'</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">color</span><span class="p">)</span>
<span class="n">y_train_pred</span> <span class="o">=</span> <span class="n">estimator</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
<span class="n">train_accuracy</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.mean.html#numpy.mean" title="View documentation for numpy.mean"><span class="n">np</span><span class="o">.</span><span class="n">mean</span></a><span class="p">(</span><span class="n">y_train_pred</span><span class="o">.</span><span class="n">ravel</span><span class="p">()</span> <span class="o">==</span> <span class="n">y_train</span><span class="o">.</span><span class="n">ravel</span><span class="p">())</span> <span class="o">*</span> <span class="mi">100</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.text.html#matplotlib.pyplot.text" title="View documentation for matplotlib.pyplot.text"><span class="n">plt</span><span class="o">.</span><span class="n">text</span></a><span class="p">(</span><span class="mf">0.05</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="s1">'Train accuracy: </span><span class="si">%.1f</span><span class="s1">'</span> <span class="o">%</span> <span class="n">train_accuracy</span><span class="p">,</span>
<span class="n">transform</span><span class="o">=</span><span class="n">h</span><span class="o">.</span><span class="n">transAxes</span><span class="p">)</span>
<span class="n">y_test_pred</span> <span class="o">=</span> <span class="n">estimator</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="n">test_accuracy</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.mean.html#numpy.mean" title="View documentation for numpy.mean"><span class="n">np</span><span class="o">.</span><span class="n">mean</span></a><span class="p">(</span><span class="n">y_test_pred</span><span class="o">.</span><span class="n">ravel</span><span class="p">()</span> <span class="o">==</span> <span class="n">y_test</span><span class="o">.</span><span class="n">ravel</span><span class="p">())</span> <span class="o">*</span> <span class="mi">100</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.text.html#matplotlib.pyplot.text" title="View documentation for matplotlib.pyplot.text"><span class="n">plt</span><span class="o">.</span><span class="n">text</span></a><span class="p">(</span><span class="mf">0.05</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="s1">'Test accuracy: </span><span class="si">%.1f</span><span class="s1">'</span> <span class="o">%</span> <span class="n">test_accuracy</span><span class="p">,</span>
<span class="n">transform</span><span class="o">=</span><span class="n">h</span><span class="o">.</span><span class="n">transAxes</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.xticks.html#matplotlib.pyplot.xticks" title="View documentation for matplotlib.pyplot.xticks"><span class="n">plt</span><span class="o">.</span><span class="n">xticks</span></a><span class="p">(())</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.yticks.html#matplotlib.pyplot.yticks" title="View documentation for matplotlib.pyplot.yticks"><span class="n">plt</span><span class="o">.</span><span class="n">yticks</span></a><span class="p">(())</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.title.html#matplotlib.pyplot.title" title="View documentation for matplotlib.pyplot.title"><span class="n">plt</span><span class="o">.</span><span class="n">title</span></a><span class="p">(</span><span class="n">name</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.legend.html#matplotlib.pyplot.legend" title="View documentation for matplotlib.pyplot.legend"><span class="n">plt</span><span class="o">.</span><span class="n">legend</span></a><span class="p">(</span><span class="n">scatterpoints</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">loc</span><span class="o">=</span><span class="s1">'lower right'</span><span class="p">,</span> <span class="n">prop</span><span class="o">=</span><span class="nb">dict</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="mi">12</span><span class="p">))</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.show.html#matplotlib.pyplot.show" title="View documentation for matplotlib.pyplot.show"><span class="n">plt</span><span class="o">.</span><span class="n">show</span></a><span class="p">()</span>
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