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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="#colormap">Colormap</a></li>
<li><a class="reference internal" href="#datasets-generation-functions">Datasets generation functions</a></li>
<li><a class="reference internal" href="#plot-functions">Plot functions</a></li>
<li><a class="reference internal" href="#plot">Plot</a></li>
</ul>
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<p class="admonition-title">Note</p>
<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-classification-plot-lda-qda-py"><span class="std std-ref">here</span></a>
to download the full example code or to run this example in your browser via Binder</p>
</div>
<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="Permalink to this heading">¶</a></h1>
<p>This example plots the covariance ellipsoids of each class and
decision boundary learned by LDA and 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="colormap">
<h2>Colormap<a class="headerlink" href="#colormap" title="Permalink to this heading">¶</a></h2>
<div class="highlight-default 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">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="n">cmap</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.colors.LinearSegmentedColormap.html#matplotlib.colors.LinearSegmentedColormap" title="matplotlib.colors.LinearSegmentedColormap" 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">LinearSegmentedColormap</span></a><span class="p">(</span>
<span class="s2">"red_blue_classes"</span><span class="p">,</span>
<span class="p">{</span>
<span class="s2">"red"</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="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">)],</span>
<span class="s2">"green"</span><span class="p">:</span> <span class="p">[(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">)],</span>
<span class="s2">"blue"</span><span class="p">:</span> <span class="p">[(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><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">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">register_cmap</span><span class="p">(</span><span class="n">cmap</span><span class="o">=</span><span class="n">cmap</span><span class="p">)</span>
</pre></div>
</div>
</section>
<section id="datasets-generation-functions">
<h2>Datasets generation functions<a class="headerlink" href="#datasets-generation-functions" title="Permalink to this heading">¶</a></h2>
<div class="highlight-default 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">dataset_fixed_cov</span><span class="p">():</span>
<span class="w"> </span><span class="sd">"""Generate 2 Gaussians samples with the same covariance matrix"""</span>
<span class="n">n</span><span class="p">,</span> <span class="n">dim</span> <span class="o">=</span> <span class="mi">300</span><span class="p">,</span> <span class="mi">2</span>
<a href="https://numpy.org/doc/stable/reference/random/generated/numpy.random.seed.html#numpy.random.seed" title="numpy.random.seed" class="sphx-glr-backref-module-numpy-random sphx-glr-backref-type-py-function"><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="n">C</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</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.r_.html#numpy.r_" title="numpy.r_" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">r_</span></a><span class="p">[</span>
<a href="https://numpy.org/doc/stable/reference/generated/numpy.dot.html#numpy.dot" title="numpy.dot" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">dot</span></a><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/random/generated/numpy.random.randn.html#numpy.random.randn" title="numpy.random.randn" class="sphx-glr-backref-module-numpy-random sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span></a><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">dim</span><span class="p">),</span> <span class="n">C</span><span class="p">),</span>
<a href="https://numpy.org/doc/stable/reference/generated/numpy.dot.html#numpy.dot" title="numpy.dot" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">dot</span></a><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/random/generated/numpy.random.randn.html#numpy.random.randn" title="numpy.random.randn" class="sphx-glr-backref-module-numpy-random sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span></a><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">dim</span><span class="p">),</span> <span class="n">C</span><span class="p">)</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="n">y</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.hstack.html#numpy.hstack" title="numpy.hstack" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">hstack</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</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</span><span class="p">)))</span>
<span class="k">return</span> <span class="n">X</span><span class="p">,</span> <span class="n">y</span>
<span class="k">def</span> <span class="nf">dataset_cov</span><span class="p">():</span>
<span class="w"> </span><span class="sd">"""Generate 2 Gaussians samples with different covariance matrices"""</span>
<span class="n">n</span><span class="p">,</span> <span class="n">dim</span> <span class="o">=</span> <span class="mi">300</span><span class="p">,</span> <span class="mi">2</span>
<a href="https://numpy.org/doc/stable/reference/random/generated/numpy.random.seed.html#numpy.random.seed" title="numpy.random.seed" class="sphx-glr-backref-module-numpy-random sphx-glr-backref-type-py-function"><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="n">C</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">X</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.r_.html#numpy.r_" title="numpy.r_" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">r_</span></a><span class="p">[</span>
<a href="https://numpy.org/doc/stable/reference/generated/numpy.dot.html#numpy.dot" title="numpy.dot" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">dot</span></a><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/random/generated/numpy.random.randn.html#numpy.random.randn" title="numpy.random.randn" class="sphx-glr-backref-module-numpy-random sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span></a><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">dim</span><span class="p">),</span> <span class="n">C</span><span class="p">),</span>
<a href="https://numpy.org/doc/stable/reference/generated/numpy.dot.html#numpy.dot" title="numpy.dot" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">dot</span></a><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/random/generated/numpy.random.randn.html#numpy.random.randn" title="numpy.random.randn" class="sphx-glr-backref-module-numpy-random sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span></a><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">dim</span><span class="p">),</span> <span class="n">C</span><span class="o">.</span><span class="n">T</span><span class="p">)</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">4</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.hstack.html#numpy.hstack" title="numpy.hstack" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">hstack</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</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</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>
</section>
<section id="plot-functions">
<h2>Plot functions<a class="headerlink" href="#plot-functions" title="Permalink to this heading">¶</a></h2>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">linalg</span>
<span class="k">def</span> <span class="nf">plot_data</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">y_pred</span><span class="p">,</span> <span class="n">fig_index</span><span class="p">):</span>
<span class="n">splot</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplot.html#matplotlib.pyplot.subplot" title="matplotlib.pyplot.subplot" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">subplot</span></a><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">fig_index</span><span class="p">)</span>
<span class="k">if</span> <span class="n">fig_index</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.title.html#matplotlib.pyplot.title" title="matplotlib.pyplot.title" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">title</span></a><span class="p">(</span><span class="s2">"Linear Discriminant Analysis"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.ylabel.html#matplotlib.pyplot.ylabel" title="matplotlib.pyplot.ylabel" 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">ylabel</span></a><span class="p">(</span><span class="s2">"Data with</span><span class="se">\n</span><span class="s2"> fixed covariance"</span><span class="p">)</span>
<span class="k">elif</span> <span class="n">fig_index</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.title.html#matplotlib.pyplot.title" title="matplotlib.pyplot.title" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">title</span></a><span class="p">(</span><span class="s2">"Quadratic Discriminant Analysis"</span><span class="p">)</span>
<span class="k">elif</span> <span class="n">fig_index</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.ylabel.html#matplotlib.pyplot.ylabel" title="matplotlib.pyplot.ylabel" 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">ylabel</span></a><span class="p">(</span><span class="s2">"Data with</span><span class="se">\n</span><span class="s2"> varying covariances"</span><span class="p">)</span>
<span class="n">tp</span> <span class="o">=</span> <span class="n">y</span> <span class="o">==</span> <span class="n">y_pred</span> <span class="c1"># True Positive</span>
<span class="n">tp0</span><span class="p">,</span> <span class="n">tp1</span> <span class="o">=</span> <span class="n">tp</span><span class="p">[</span><span class="n">y</span> <span class="o">==</span> <span class="mi">0</span><span class="p">],</span> <span class="n">tp</span><span class="p">[</span><span class="n">y</span> <span class="o">==</span> <span class="mi">1</span><span class="p">]</span>
<span class="n">X0</span><span class="p">,</span> <span class="n">X1</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="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[</span><span class="n">y</span> <span class="o">==</span> <span class="mi">1</span><span class="p">]</span>
<span class="n">X0_tp</span><span class="p">,</span> <span class="n">X0_fp</span> <span class="o">=</span> <span class="n">X0</span><span class="p">[</span><span class="n">tp0</span><span class="p">],</span> <span class="n">X0</span><span class="p">[</span><span class="o">~</span><span class="n">tp0</span><span class="p">]</span>
<span class="n">X1_tp</span><span class="p">,</span> <span class="n">X1_fp</span> <span class="o">=</span> <span class="n">X1</span><span class="p">[</span><span class="n">tp1</span><span class="p">],</span> <span class="n">X1</span><span class="p">[</span><span class="o">~</span><span class="n">tp1</span><span class="p">]</span>
<span class="c1"># class 0: dots</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.scatter.html#matplotlib.pyplot.scatter" title="matplotlib.pyplot.scatter" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">scatter</span></a><span class="p">(</span><span class="n">X0_tp</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X0_tp</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="s2">"."</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"red"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.scatter.html#matplotlib.pyplot.scatter" title="matplotlib.pyplot.scatter" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">scatter</span></a><span class="p">(</span><span class="n">X0_fp</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X0_fp</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="s2">"x"</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">color</span><span class="o">=</span><span class="s2">"#990000"</span><span class="p">)</span> <span class="c1"># dark red</span>
<span class="c1"># class 1: dots</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.scatter.html#matplotlib.pyplot.scatter" title="matplotlib.pyplot.scatter" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">scatter</span></a><span class="p">(</span><span class="n">X1_tp</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X1_tp</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="s2">"."</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"blue"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.scatter.html#matplotlib.pyplot.scatter" title="matplotlib.pyplot.scatter" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">scatter</span></a><span class="p">(</span>
<span class="n">X1_fp</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X1_fp</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="s2">"x"</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">color</span><span class="o">=</span><span class="s2">"#000099"</span>
<span class="p">)</span> <span class="c1"># dark blue</span>
<span class="c1"># class 0 and 1 : areas</span>
<span class="n">nx</span><span class="p">,</span> <span class="n">ny</span> <span class="o">=</span> <span class="mi">200</span><span class="p">,</span> <span class="mi">100</span>
<span class="n">x_min</span><span class="p">,</span> <span class="n">x_max</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.xlim.html#matplotlib.pyplot.xlim" title="matplotlib.pyplot.xlim" 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">xlim</span></a><span class="p">()</span>
<span class="n">y_min</span><span class="p">,</span> <span class="n">y_max</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.ylim.html#matplotlib.pyplot.ylim" title="matplotlib.pyplot.ylim" 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">ylim</span></a><span class="p">()</span>
<span class="n">xx</span><span class="p">,</span> <span class="n">yy</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.meshgrid.html#numpy.meshgrid" title="numpy.meshgrid" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span></a><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.linspace.html#numpy.linspace" title="numpy.linspace" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">linspace</span></a><span class="p">(</span><span class="n">x_min</span><span class="p">,</span> <span class="n">x_max</span><span class="p">,</span> <span class="n">nx</span><span class="p">),</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.linspace.html#numpy.linspace" title="numpy.linspace" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">linspace</span></a><span class="p">(</span><span class="n">y_min</span><span class="p">,</span> <span class="n">y_max</span><span class="p">,</span> <span class="n">ny</span><span class="p">))</span>
<span class="n">Z</span> <span class="o">=</span> <span class="n">lda</span><span class="o">.</span><span class="n">predict_proba</span><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.c_.html#numpy.c_" title="numpy.c_" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">c_</span></a><span class="p">[</span><span class="n">xx</span><span class="o">.</span><span class="n">ravel</span><span class="p">(),</span> <span class="n">yy</span><span class="o">.</span><span class="n">ravel</span><span class="p">()])</span>
<span class="n">Z</span> <span class="o">=</span> <span class="n">Z</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">xx</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.pcolormesh.html#matplotlib.pyplot.pcolormesh" title="matplotlib.pyplot.pcolormesh" 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">pcolormesh</span></a><span class="p">(</span>
<span class="n">xx</span><span class="p">,</span> <span class="n">yy</span><span class="p">,</span> <span class="n">Z</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s2">"red_blue_classes"</span><span class="p">,</span> <span class="n">norm</span><span class="o">=</span><a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.colors.Normalize.html#matplotlib.colors.Normalize" title="matplotlib.colors.Normalize" 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">Normalize</span></a><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">),</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">0</span>
<span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.contour.html#matplotlib.pyplot.contour" title="matplotlib.pyplot.contour" 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">contour</span></a><span class="p">(</span><span class="n">xx</span><span class="p">,</span> <span class="n">yy</span><span class="p">,</span> <span class="n">Z</span><span class="p">,</span> <span class="p">[</span><span class="mf">0.5</span><span class="p">],</span> <span class="n">linewidths</span><span class="o">=</span><span class="mf">2.0</span><span class="p">,</span> <span class="n">colors</span><span class="o">=</span><span class="s2">"white"</span><span class="p">)</span>
<span class="c1"># means</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.plot.html#matplotlib.pyplot.plot" title="matplotlib.pyplot.plot" 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">plot</span></a><span class="p">(</span>
<span class="n">lda</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="mi">0</span><span class="p">],</span>
<span class="n">lda</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="mi">1</span><span class="p">],</span>
<span class="s2">"*"</span><span class="p">,</span>
<span class="n">color</span><span class="o">=</span><span class="s2">"yellow"</span><span class="p">,</span>
<span class="n">markersize</span><span class="o">=</span><span class="mi">15</span><span class="p">,</span>
<span class="n">markeredgecolor</span><span class="o">=</span><span class="s2">"grey"</span><span class="p">,</span>
<span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.plot.html#matplotlib.pyplot.plot" title="matplotlib.pyplot.plot" 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">plot</span></a><span class="p">(</span>
<span class="n">lda</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="mi">0</span><span class="p">],</span>
<span class="n">lda</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="mi">1</span><span class="p">],</span>
<span class="s2">"*"</span><span class="p">,</span>
<span class="n">color</span><span class="o">=</span><span class="s2">"yellow"</span><span class="p">,</span>
<span class="n">markersize</span><span class="o">=</span><span class="mi">15</span><span class="p">,</span>
<span class="n">markeredgecolor</span><span class="o">=</span><span class="s2">"grey"</span><span class="p">,</span>
<span class="p">)</span>
<span class="k">return</span> <span class="n">splot</span>
<span class="k">def</span> <span class="nf">plot_ellipse</span><span class="p">(</span><span class="n">splot</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">v</span><span class="p">,</span> <span class="n">w</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.eigh.html#scipy.linalg.eigh" title="scipy.linalg.eigh" class="sphx-glr-backref-module-scipy-linalg sphx-glr-backref-type-py-function"><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://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.norm.html#scipy.linalg.norm" title="scipy.linalg.norm" class="sphx-glr-backref-module-scipy-linalg sphx-glr-backref-type-py-function"><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">splot</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">0.2</span><span class="p">)</span>
<span class="n">splot</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">splot</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(())</span>
<span class="n">splot</span><span class="o">.</span><span class="n">set_yticks</span><span class="p">(())</span>
<span class="k">def</span> <span class="nf">plot_lda_cov</span><span class="p">(</span><span class="n">lda</span><span class="p">,</span> <span class="n">splot</span><span class="p">):</span>
<span class="n">plot_ellipse</span><span class="p">(</span><span class="n">splot</span><span class="p">,</span> <span class="n">lda</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">lda</span><span class="o">.</span><span class="n">covariance_</span><span class="p">,</span> <span class="s2">"red"</span><span class="p">)</span>
<span class="n">plot_ellipse</span><span class="p">(</span><span class="n">splot</span><span class="p">,</span> <span class="n">lda</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">lda</span><span class="o">.</span><span class="n">covariance_</span><span class="p">,</span> <span class="s2">"blue"</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">plot_qda_cov</span><span class="p">(</span><span class="n">qda</span><span class="p">,</span> <span class="n">splot</span><span class="p">):</span>
<span class="n">plot_ellipse</span><span class="p">(</span><span class="n">splot</span><span class="p">,</span> <span class="n">qda</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">qda</span><span class="o">.</span><span class="n">covariance_</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="s2">"red"</span><span class="p">)</span>
<span class="n">plot_ellipse</span><span class="p">(</span><span class="n">splot</span><span class="p">,</span> <span class="n">qda</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">qda</span><span class="o">.</span><span class="n">covariance_</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="s2">"blue"</span><span class="p">)</span>
</pre></div>
</div>
</section>
<section id="plot">
<h2>Plot<a class="headerlink" href="#plot" title="Permalink to this heading">¶</a></h2>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.figure.html#matplotlib.pyplot.figure" title="matplotlib.pyplot.figure" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">figure</span></a><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">8</span><span class="p">),</span> <span class="n">facecolor</span><span class="o">=</span><span class="s2">"white"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.suptitle.html#matplotlib.pyplot.suptitle" title="matplotlib.pyplot.suptitle" 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">suptitle</span></a><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">0.98</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>
<span class="kn">from</span> <span class="nn">sklearn.discriminant_analysis</span> <span class="kn">import</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="kn">from</span> <span class="nn">sklearn.discriminant_analysis</span> <span class="kn">import</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="k">for</span> <span class="n">i</span><span class="p">,</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="ow">in</span> <span class="nb">enumerate</span><span class="p">([</span><span class="n">dataset_fixed_cov</span><span class="p">(),</span> <span class="n">dataset_cov</span><span class="p">()]):</span>
<span class="c1"># Linear Discriminant Analysis</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">y_pred</span> <span class="o">=</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="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="n">splot</span> <span class="o">=</span> <span class="n">plot_data</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">y_pred</span><span class="p">,</span> <span class="n">fig_index</span><span class="o">=</span><span class="mi">2</span> <span class="o">*</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">plot_lda_cov</span><span class="p">(</span><span class="n">lda</span><span class="p">,</span> <span class="n">splot</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.axis.html#matplotlib.pyplot.axis" title="matplotlib.pyplot.axis" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">axis</span></a><span class="p">(</span><span class="s2">"tight"</span><span class="p">)</span>
<span class="c1"># Quadratic Discriminant Analysis</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="n">y_pred</span> <span class="o">=</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="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="n">splot</span> <span class="o">=</span> <span class="n">plot_data</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">y_pred</span><span class="p">,</span> <span class="n">fig_index</span><span class="o">=</span><span class="mi">2</span> <span class="o">*</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span>
<span class="n">plot_qda_cov</span><span class="p">(</span><span class="n">qda</span><span class="p">,</span> <span class="n">splot</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.axis.html#matplotlib.pyplot.axis" title="matplotlib.pyplot.axis" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">axis</span></a><span class="p">(</span><span class="s2">"tight"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.tight_layout.html#matplotlib.pyplot.tight_layout" title="matplotlib.pyplot.tight_layout" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span></a><span class="p">()</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplots_adjust.html#matplotlib.pyplot.subplots_adjust" title="matplotlib.pyplot.subplots_adjust" 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_adjust</span></a><span class="p">(</span><span class="n">top</span><span class="o">=</span><span class="mf">0.92</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>
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