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<li><a class="reference internal" href="#">Linear and Quadratic Discriminant Analysis with covariance ellipsoid</a><ul>
<li><a class="reference internal" href="#data-generation">Data generation</a></li>
<li><a class="reference internal" href="#plotting-functions">Plotting Functions</a></li>
<li><a class="reference internal" href="#comparison-of-lda-and-qda">Comparison of LDA and QDA</a></li>
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<section class="sphx-glr-example-title" id="linear-and-quadratic-discriminant-analysis-with-covariance-ellipsoid">
<span id="sphx-glr-auto-examples-classification-plot-lda-qda-py"></span><h1>Linear and Quadratic Discriminant Analysis with covariance ellipsoid<a class="headerlink" href="#linear-and-quadratic-discriminant-analysis-with-covariance-ellipsoid" title="Link to this heading">¶</a></h1>
<p>This example plots the covariance ellipsoids of each class and the decision boundary
learned by <a class="reference internal" href="../../modules/generated/sklearn.discriminant_analysis.LinearDiscriminantAnalysis.html#sklearn.discriminant_analysis.LinearDiscriminantAnalysis" title="sklearn.discriminant_analysis.LinearDiscriminantAnalysis"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearDiscriminantAnalysis</span></code></a> (LDA) and
<a class="reference internal" href="../../modules/generated/sklearn.discriminant_analysis.QuadraticDiscriminantAnalysis.html#sklearn.discriminant_analysis.QuadraticDiscriminantAnalysis" title="sklearn.discriminant_analysis.QuadraticDiscriminantAnalysis"><code class="xref py py-class docutils literal notranslate"><span class="pre">QuadraticDiscriminantAnalysis</span></code></a> (QDA). The
ellipsoids display the double standard deviation for each class. With LDA, the standard
deviation is the same for all the classes, while each class has its own standard
deviation with QDA.</p>
<section id="data-generation">
<h2>Data generation<a class="headerlink" href="#data-generation" title="Link to this heading">¶</a></h2>
<p>First, we define a function to generate synthetic data. It creates two blobs centered
at <code class="docutils literal notranslate"><span class="pre">(0,</span> <span class="pre">0)</span></code> and <code class="docutils literal notranslate"><span class="pre">(1,</span> <span class="pre">1)</span></code>. Each blob is assigned a specific class. The dispersion of
the blob is controlled by the parameters <code class="docutils literal notranslate"><span class="pre">cov_class_1</span></code> and <code class="docutils literal notranslate"><span class="pre">cov_class_2</span></code>, that are the
covariance matrices used when generating the samples from the Gaussian distributions.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="k">def</span> <span class="nf">make_data</span><span class="p">(</span><span class="n">n_samples</span><span class="p">,</span> <span class="n">n_features</span><span class="p">,</span> <span class="n">cov_class_1</span><span class="p">,</span> <span class="n">cov_class_2</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
<span class="n">rng</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/random/legacy.html#numpy.random.RandomState" title="numpy.random.RandomState" class="sphx-glr-backref-module-numpy-random sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">RandomState</span></a><span class="p">(</span><span class="n">seed</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.concatenate.html#numpy.concatenate" title="numpy.concatenate" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">concatenate</span></a><span class="p">(</span>
<span class="p">[</span>
<span class="n">rng</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_samples</span><span class="p">,</span> <span class="n">n_features</span><span class="p">)</span> <span class="o">@</span> <span class="n">cov_class_1</span><span class="p">,</span>
<span class="n">rng</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_samples</span><span class="p">,</span> <span class="n">n_features</span><span class="p">)</span> <span class="o">@</span> <span class="n">cov_class_2</span> <span class="o">+</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.array.html#numpy.array" title="numpy.array" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">array</span></a><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]),</span>
<span class="p">]</span>
<span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.concatenate.html#numpy.concatenate" title="numpy.concatenate" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">concatenate</span></a><span class="p">([</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.zeros.html#numpy.zeros" title="numpy.zeros" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">zeros</span></a><span class="p">(</span><span class="n">n_samples</span><span class="p">),</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.ones.html#numpy.ones" title="numpy.ones" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">ones</span></a><span class="p">(</span><span class="n">n_samples</span><span class="p">)])</span>
<span class="k">return</span> <span class="n">X</span><span class="p">,</span> <span class="n">y</span>
</pre></div>
</div>
<p>We generate three datasets. In the first dataset, the two classes share the same
covariance matrix, and this covariance matrix has the specificity of being spherical
(isotropic). The second dataset is similar to the first one but does not enforce the
covariance to be spherical. Finally, the third dataset has a non-spherical covariance
matrix for each class.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="n">covariance</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.array.html#numpy.array" title="numpy.array" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">array</span></a><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
<span class="n">X_isotropic_covariance</span><span class="p">,</span> <span class="n">y_isotropic_covariance</span> <span class="o">=</span> <span class="n">make_data</span><span class="p">(</span>
<span class="n">n_samples</span><span class="o">=</span><span class="mi">1_000</span><span class="p">,</span>
<span class="n">n_features</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span>
<span class="n">cov_class_1</span><span class="o">=</span><span class="n">covariance</span><span class="p">,</span>
<span class="n">cov_class_2</span><span class="o">=</span><span class="n">covariance</span><span class="p">,</span>
<span class="n">seed</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span>
<span class="p">)</span>
<span class="n">covariance</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.array.html#numpy.array" title="numpy.array" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">array</span></a><span class="p">([[</span><span class="mf">0.0</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.23</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.83</span><span class="p">,</span> <span class="mf">0.23</span><span class="p">]])</span>
<span class="n">X_shared_covariance</span><span class="p">,</span> <span class="n">y_shared_covariance</span> <span class="o">=</span> <span class="n">make_data</span><span class="p">(</span>
<span class="n">n_samples</span><span class="o">=</span><span class="mi">300</span><span class="p">,</span>
<span class="n">n_features</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span>
<span class="n">cov_class_1</span><span class="o">=</span><span class="n">covariance</span><span class="p">,</span>
<span class="n">cov_class_2</span><span class="o">=</span><span class="n">covariance</span><span class="p">,</span>
<span class="n">seed</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span>
<span class="p">)</span>
<span class="n">cov_class_1</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.array.html#numpy.array" title="numpy.array" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">array</span></a><span class="p">([[</span><span class="mf">0.0</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.0</span><span class="p">],</span> <span class="p">[</span><span class="mf">2.5</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">]])</span> <span class="o">*</span> <span class="mf">2.0</span>
<span class="n">cov_class_2</span> <span class="o">=</span> <span class="n">cov_class_1</span><span class="o">.</span><span class="n">T</span>
<span class="n">X_different_covariance</span><span class="p">,</span> <span class="n">y_different_covariance</span> <span class="o">=</span> <span class="n">make_data</span><span class="p">(</span>
<span class="n">n_samples</span><span class="o">=</span><span class="mi">300</span><span class="p">,</span>
<span class="n">n_features</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span>
<span class="n">cov_class_1</span><span class="o">=</span><span class="n">cov_class_1</span><span class="p">,</span>
<span class="n">cov_class_2</span><span class="o">=</span><span class="n">cov_class_2</span><span class="p">,</span>
<span class="n">seed</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span>
<span class="p">)</span>
</pre></div>
</div>
</section>
<section id="plotting-functions">
<h2>Plotting Functions<a class="headerlink" href="#plotting-functions" title="Link to this heading">¶</a></h2>
<p>The code below is used to plot several pieces of information from the estimators used,
i.e., <a class="reference internal" href="../../modules/generated/sklearn.discriminant_analysis.LinearDiscriminantAnalysis.html#sklearn.discriminant_analysis.LinearDiscriminantAnalysis" title="sklearn.discriminant_analysis.LinearDiscriminantAnalysis"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearDiscriminantAnalysis</span></code></a> (LDA) and
<a class="reference internal" href="../../modules/generated/sklearn.discriminant_analysis.QuadraticDiscriminantAnalysis.html#sklearn.discriminant_analysis.QuadraticDiscriminantAnalysis" title="sklearn.discriminant_analysis.QuadraticDiscriminantAnalysis"><code class="xref py py-class docutils literal notranslate"><span class="pre">QuadraticDiscriminantAnalysis</span></code></a> (QDA). The
displayed information includes:</p>
<ul class="simple">
<li><p>the decision boundary based on the probability estimate of the estimator;</p></li>
<li><p>a scatter plot with circles representing the well-classified samples;</p></li>
<li><p>a scatter plot with crosses representing the misclassified samples;</p></li>
<li><p>the mean of each class, estimated by the estimator, marked with a star;</p></li>
<li><p>the estimated covariance represented by an ellipse at 2 standard deviations from the
mean.</p></li>
</ul>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib</span> <span class="k">as</span> <span class="nn">mpl</span>
<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">colors</span>
<span class="kn">from</span> <span class="nn">sklearn.inspection</span> <span class="kn">import</span> <span class="n">DecisionBoundaryDisplay</span>
<span class="k">def</span> <span class="nf">plot_ellipse</span><span class="p">(</span><span class="n">mean</span><span class="p">,</span> <span class="n">cov</span><span class="p">,</span> <span class="n">color</span><span class="p">,</span> <span class="n">ax</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://numpy.org/doc/stable/reference/generated/numpy.linalg.eigh.html#numpy.linalg.eigh" title="numpy.linalg.eigh" class="sphx-glr-backref-module-numpy-linalg sphx-glr-backref-type-py-function"><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">cov</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://numpy.org/doc/stable/reference/generated/numpy.linalg.norm.html#numpy.linalg.norm" title="numpy.linalg.norm" class="sphx-glr-backref-module-numpy-linalg sphx-glr-backref-type-py-function"><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://numpy.org/doc/stable/reference/generated/numpy.arctan.html#numpy.arctan" title="numpy.arctan" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">arctan</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="o">/</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://numpy.org/doc/stable/reference/constants.html#numpy.pi" title="numpy.pi" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">pi</span></a> <span class="c1"># convert to degrees</span>
<span class="c1"># filled Gaussian at 2 standard deviation</span>
<span class="n">ell</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.patches.Ellipse.html#matplotlib.patches.Ellipse" title="matplotlib.patches.Ellipse" class="sphx-glr-backref-module-matplotlib-patches sphx-glr-backref-type-py-class sphx-glr-backref-instance"><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">mean</span><span class="p">,</span>
<span class="mi">2</span> <span class="o">*</span> <span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">**</span> <span class="mf">0.5</span><span class="p">,</span>
<span class="mi">2</span> <span class="o">*</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">**</span> <span class="mf">0.5</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="p">,</span>
<span class="n">facecolor</span><span class="o">=</span><span class="n">color</span><span class="p">,</span>
<span class="n">edgecolor</span><span class="o">=</span><span class="s2">"black"</span><span class="p">,</span>
<span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</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">0.4</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="k">def</span> <span class="nf">plot_result</span><span class="p">(</span><span class="n">estimator</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">ax</span><span class="p">):</span>
<span class="n">cmap</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.colors.ListedColormap.html#matplotlib.colors.ListedColormap" title="matplotlib.colors.ListedColormap" class="sphx-glr-backref-module-matplotlib-colors sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">colors</span><span class="o">.</span><span class="n">ListedColormap</span></a><span class="p">([</span><span class="s2">"tab:red"</span><span class="p">,</span> <span class="s2">"tab:blue"</span><span class="p">])</span>
<a href="../../modules/generated/sklearn.inspection.DecisionBoundaryDisplay.html#sklearn.inspection.DecisionBoundaryDisplay.from_estimator" title="sklearn.inspection.DecisionBoundaryDisplay.from_estimator" class="sphx-glr-backref-module-sklearn-inspection-DecisionBoundaryDisplay sphx-glr-backref-type-py-method"><span class="n">DecisionBoundaryDisplay</span><span class="o">.</span><span class="n">from_estimator</span></a><span class="p">(</span>
<span class="n">estimator</span><span class="p">,</span>
<span class="n">X</span><span class="p">,</span>
<span class="n">response_method</span><span class="o">=</span><span class="s2">"predict_proba"</span><span class="p">,</span>
<span class="n">plot_method</span><span class="o">=</span><span class="s2">"pcolormesh"</span><span class="p">,</span>
<span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span>
<span class="n">cmap</span><span class="o">=</span><span class="s2">"RdBu"</span><span class="p">,</span>
<span class="n">alpha</span><span class="o">=</span><span class="mf">0.3</span><span class="p">,</span>
<span class="p">)</span>
<a href="../../modules/generated/sklearn.inspection.DecisionBoundaryDisplay.html#sklearn.inspection.DecisionBoundaryDisplay.from_estimator" title="sklearn.inspection.DecisionBoundaryDisplay.from_estimator" class="sphx-glr-backref-module-sklearn-inspection-DecisionBoundaryDisplay sphx-glr-backref-type-py-method"><span class="n">DecisionBoundaryDisplay</span><span class="o">.</span><span class="n">from_estimator</span></a><span class="p">(</span>
<span class="n">estimator</span><span class="p">,</span>
<span class="n">X</span><span class="p">,</span>
<span class="n">response_method</span><span class="o">=</span><span class="s2">"predict_proba"</span><span class="p">,</span>
<span class="n">plot_method</span><span class="o">=</span><span class="s2">"contour"</span><span class="p">,</span>
<span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span>
<span class="n">alpha</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span>
<span class="n">levels</span><span class="o">=</span><span class="p">[</span><span class="mf">0.5</span><span class="p">],</span>
<span class="p">)</span>
<span class="n">y_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</span><span class="p">)</span>
<span class="n">X_right</span><span class="p">,</span> <span class="n">y_right</span> <span class="o">=</span> <span class="n">X</span><span class="p">[</span><span class="n">y</span> <span class="o">==</span> <span class="n">y_pred</span><span class="p">],</span> <span class="n">y</span><span class="p">[</span><span class="n">y</span> <span class="o">==</span> <span class="n">y_pred</span><span class="p">]</span>
<span class="n">X_wrong</span><span class="p">,</span> <span class="n">y_wrong</span> <span class="o">=</span> <span class="n">X</span><span class="p">[</span><span class="n">y</span> <span class="o">!=</span> <span class="n">y_pred</span><span class="p">],</span> <span class="n">y</span><span class="p">[</span><span class="n">y</span> <span class="o">!=</span> <span class="n">y_pred</span><span class="p">]</span>
<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X_right</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X_right</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="n">y_right</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">cmap</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span>
<span class="n">X_wrong</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span>
<span class="n">X_wrong</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span>
<span class="n">c</span><span class="o">=</span><span class="n">y_wrong</span><span class="p">,</span>
<span class="n">s</span><span class="o">=</span><span class="mi">30</span><span class="p">,</span>
<span class="n">cmap</span><span class="o">=</span><span class="n">cmap</span><span class="p">,</span>
<span class="n">alpha</span><span class="o">=</span><span class="mf">0.9</span><span class="p">,</span>
<span class="n">marker</span><span class="o">=</span><span class="s2">"x"</span><span class="p">,</span>
<span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span>
<span class="n">estimator</span><span class="o">.</span><span class="n">means_</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span>
<span class="n">estimator</span><span class="o">.</span><span class="n">means_</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="s2">"yellow"</span><span class="p">,</span>
<span class="n">s</span><span class="o">=</span><span class="mi">200</span><span class="p">,</span>
<span class="n">marker</span><span class="o">=</span><span class="s2">"*"</span><span class="p">,</span>
<span class="n">edgecolor</span><span class="o">=</span><span class="s2">"black"</span><span class="p">,</span>
<span class="p">)</span>
<span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">estimator</span><span class="p">,</span> <a href="../../modules/generated/sklearn.discriminant_analysis.LinearDiscriminantAnalysis.html#sklearn.discriminant_analysis.LinearDiscriminantAnalysis" title="sklearn.discriminant_analysis.LinearDiscriminantAnalysis" class="sphx-glr-backref-module-sklearn-discriminant_analysis sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">LinearDiscriminantAnalysis</span></a><span class="p">):</span>
<span class="n">covariance</span> <span class="o">=</span> <span class="p">[</span><span class="n">estimator</span><span class="o">.</span><span class="n">covariance_</span><span class="p">]</span> <span class="o">*</span> <span class="mi">2</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">covariance</span> <span class="o">=</span> <span class="n">estimator</span><span class="o">.</span><span class="n">covariance_</span>
<span class="n">plot_ellipse</span><span class="p">(</span><span class="n">estimator</span><span class="o">.</span><span class="n">means_</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">covariance</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="s2">"tab:red"</span><span class="p">,</span> <span class="n">ax</span><span class="p">)</span>
<span class="n">plot_ellipse</span><span class="p">(</span><span class="n">estimator</span><span class="o">.</span><span class="n">means_</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">covariance</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="s2">"tab:blue"</span><span class="p">,</span> <span class="n">ax</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_box_aspect</span><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">spines</span><span class="p">[</span><span class="s2">"top"</span><span class="p">]</span><span class="o">.</span><span class="n">set_visible</span><span class="p">(</span><span class="kc">False</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s2">"bottom"</span><span class="p">]</span><span class="o">.</span><span class="n">set_visible</span><span class="p">(</span><span class="kc">False</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s2">"left"</span><span class="p">]</span><span class="o">.</span><span class="n">set_visible</span><span class="p">(</span><span class="kc">False</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s2">"right"</span><span class="p">]</span><span class="o">.</span><span class="n">set_visible</span><span class="p">(</span><span class="kc">False</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">xticks</span><span class="o">=</span><span class="p">[],</span> <span class="n">yticks</span><span class="o">=</span><span class="p">[])</span>
</pre></div>
</div>
</section>
<section id="comparison-of-lda-and-qda">
<h2>Comparison of LDA and QDA<a class="headerlink" href="#comparison-of-lda-and-qda" title="Link to this heading">¶</a></h2>
<p>We compare the two estimators LDA and QDA on all three datasets.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">sklearn.discriminant_analysis</span> <span class="kn">import</span> <span class="p">(</span>
<a href="../../modules/generated/sklearn.discriminant_analysis.LinearDiscriminantAnalysis.html#sklearn.discriminant_analysis.LinearDiscriminantAnalysis" title="sklearn.discriminant_analysis.LinearDiscriminantAnalysis" class="sphx-glr-backref-module-sklearn-discriminant_analysis sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">LinearDiscriminantAnalysis</span></a><span class="p">,</span>
<a href="../../modules/generated/sklearn.discriminant_analysis.QuadraticDiscriminantAnalysis.html#sklearn.discriminant_analysis.QuadraticDiscriminantAnalysis" title="sklearn.discriminant_analysis.QuadraticDiscriminantAnalysis" class="sphx-glr-backref-module-sklearn-discriminant_analysis sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">QuadraticDiscriminantAnalysis</span></a><span class="p">,</span>
<span class="p">)</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">axs</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplots.html#matplotlib.pyplot.subplots" title="matplotlib.pyplot.subplots" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">subplots</span></a><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">sharex</span><span class="o">=</span><span class="s2">"row"</span><span class="p">,</span> <span class="n">sharey</span><span class="o">=</span><span class="s2">"row"</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">12</span><span class="p">))</span>
<span class="n">lda</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.discriminant_analysis.LinearDiscriminantAnalysis.html#sklearn.discriminant_analysis.LinearDiscriminantAnalysis" title="sklearn.discriminant_analysis.LinearDiscriminantAnalysis" class="sphx-glr-backref-module-sklearn-discriminant_analysis sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">LinearDiscriminantAnalysis</span></a><span class="p">(</span><span class="n">solver</span><span class="o">=</span><span class="s2">"svd"</span><span class="p">,</span> <span class="n">store_covariance</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">qda</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.discriminant_analysis.QuadraticDiscriminantAnalysis.html#sklearn.discriminant_analysis.QuadraticDiscriminantAnalysis" title="sklearn.discriminant_analysis.QuadraticDiscriminantAnalysis" class="sphx-glr-backref-module-sklearn-discriminant_analysis sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">QuadraticDiscriminantAnalysis</span></a><span class="p">(</span><span class="n">store_covariance</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="k">for</span> <span class="n">ax_row</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span>
<span class="n">axs</span><span class="p">,</span>
<span class="p">(</span><span class="n">X_isotropic_covariance</span><span class="p">,</span> <span class="n">X_shared_covariance</span><span class="p">,</span> <span class="n">X_different_covariance</span><span class="p">),</span>
<span class="p">(</span><span class="n">y_isotropic_covariance</span><span class="p">,</span> <span class="n">y_shared_covariance</span><span class="p">,</span> <span class="n">y_different_covariance</span><span class="p">),</span>
<span class="p">):</span>
<span class="n">lda</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">plot_result</span><span class="p">(</span><span class="n">lda</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">ax_row</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">qda</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">plot_result</span><span class="p">(</span><span class="n">qda</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">ax_row</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
<span class="n">axs</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="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Linear Discriminant Analysis"</span><span class="p">)</span>
<span class="n">axs</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="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">"Data with fixed and spherical covariance"</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">"Data with fixed covariance"</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Quadratic Discriminant Analysis"</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">"Data with varying covariances"</span><span class="p">)</span>
<span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span>
<span class="s2">"Linear Discriminant Analysis vs Quadratic Discriminant Analysis"</span><span class="p">,</span>
<span class="n">y</span><span class="o">=</span><span class="mf">0.94</span><span class="p">,</span>
<span class="n">fontsize</span><span class="o">=</span><span class="mi">15</span><span class="p">,</span>
<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_lda_qda_001.png" srcset="../../_images/sphx_glr_plot_lda_qda_001.png" alt="Linear Discriminant Analysis vs Quadratic Discriminant Analysis, Linear Discriminant Analysis, Quadratic Discriminant Analysis" class = "sphx-glr-single-img"/><p>The first important thing to notice is that LDA and QDA are equivalent for the
first and second datasets. Indeed, the major difference is that LDA assumes
that the covariance matrix of each class is equal, while QDA estimates a
covariance matrix per class. Since in these cases the data generative process
has the same covariance matrix for both classes, QDA estimates two covariance
matrices that are (almost) equal and therefore equivalent to the covariance
matrix estimated by LDA.</p>
<p>In the first dataset the covariance matrix used to generate the dataset is
spherical, which results in a discriminant boundary that aligns with the
perpendicular bisector between the two means. This is no longer the case for
the second dataset. The discriminant boundary only passes through the middle
of the two means.</p>
<p>Finally, in the third dataset, we observe the real difference between LDA and
QDA. QDA fits two covariance matrices and provides a non-linear discriminant
boundary, whereas LDA underfits since it assumes that both classes share a
single covariance matrix.</p>
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