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</ul> </div> <div class="content"> <div class="documentwrapper"> <div class="bodywrapper"> <div class="body"> <div class="section" id="linear-discriminant-analysis-quadratic-discriminant-analysis"> <span id="example-plot-lda-vs-qda-py"></span><h1>Linear Discriminant Analysis & Quadratic Discriminant Analysis<a class="headerlink" href="#linear-discriminant-analysis-quadratic-discriminant-analysis" title="Permalink to this headline">ΒΆ</a></h1> <p>Plot the confidence ellipsoids of each class and decision boundary</p> <img alt="auto_examples/images/plot_lda_vs_qda.png" class="align-center" src="auto_examples/images/plot_lda_vs_qda.png" /> <p><strong>Python source code:</strong> <a class="reference download internal" href="../_downloads/plot_lda_vs_qda.py"><tt class="xref download docutils literal"><span class="pre">plot_lda_vs_qda.py</span></tt></a></p> <div class="highlight-python"><div class="highlight"><pre><span class="k">print</span> <span class="n">__doc__</span> <span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">linalg</span> <span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span> <span class="kn">import</span> <span class="nn">pylab</span> <span class="kn">as</span> <span class="nn">pl</span> <span class="kn">import</span> <span class="nn">matplotlib</span> <span class="kn">as</span> <span class="nn">mpl</span> <span class="kn">from</span> <span class="nn">scikits.learn.lda</span> <span class="kn">import</span> <span class="n">LDA</span> <span class="kn">from</span> <span class="nn">scikits.learn.qda</span> <span class="kn">import</span> <span class="n">QDA</span> <span class="c">################################################################################</span> <span class="c"># load sample dataset</span> <span class="kn">from</span> <span class="nn">scikits.learn.datasets</span> <span class="kn">import</span> <span class="n">load_iris</span> <span class="n">iris</span> <span class="o">=</span> <span class="n">load_iris</span><span class="p">()</span> <span class="n">X</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="mi">2</span><span class="p">]</span> <span class="c"># Take only 2 dimensions</span> <span class="n">y</span> <span class="o">=</span> <span class="n">iris</span><span class="o">.</span><span class="n">target</span> <span class="n">X</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">y</span> <span class="o">=</span> <span class="n">y</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">y</span> <span class="o">-=</span> <span class="mi">1</span> <span class="n">target_names</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="mi">1</span><span class="p">:]</span> <span class="c">################################################################################</span> <span class="c"># LDA</span> <span class="n">lda</span> <span class="o">=</span> <span class="n">LDA</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="n">store_covariance</span><span class="o">=</span><span class="bp">True</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="c"># QDA</span> <span class="n">qda</span> <span class="o">=</span> <span class="n">QDA</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="n">store_covariances</span><span class="o">=</span><span class="bp">True</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="c">###############################################################################</span> <span class="c"># Plot results</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> <span class="n">linalg</span><span class="o">.</span><span class="n">eigh</span><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> <span class="n">linalg</span><span class="o">.</span><span class="n">norm</span><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> <span class="n">np</span><span class="o">.</span><span class="n">arctan</span><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> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="c"># convert to degrees</span> <span class="c"># filled gaussian at 2 standard deviation</span> <span class="n">ell</span> <span class="o">=</span> <span class="n">mpl</span><span class="o">.</span><span class="n">patches</span><span class="o">.</span><span class="n">Ellipse</span><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="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">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">0.5</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">xx</span><span class="p">,</span> <span class="n">yy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mf">8.5</span><span class="p">,</span> <span class="mi">200</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">1.5</span><span class="p">,</span> <span class="mf">4.5</span><span class="p">,</span> <span class="mi">200</span><span class="p">))</span> <span class="n">X_grid</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><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">zz_lda</span> <span class="o">=</span> <span class="n">lda</span><span class="o">.</span><span class="n">predict_proba</span><span class="p">(</span><span class="n">X_grid</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> <span class="n">zz_qda</span> <span class="o">=</span> <span class="n">qda</span><span class="o">.</span><span class="n">predict_proba</span><span class="p">(</span><span class="n">X_grid</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> <span class="n">pl</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span> <span class="n">splot</span> <span class="o">=</span> <span class="n">pl</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="n">pl</span><span class="o">.</span><span class="n">contourf</span><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">zz_lda</span> <span class="o">></span> <span class="mf">0.5</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">pl</span><span class="o">.</span><span class="n">scatter</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">0</span><span class="p">,</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">0</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="s">'b'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">target_names</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="n">pl</span><span class="o">.</span><span class="n">scatter</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="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="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="s">'r'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">target_names</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="n">pl</span><span class="o">.</span><span class="n">contour</span><span class="p">(</span><span class="n">xx</span><span class="p">,</span> <span class="n">yy</span><span class="p">,</span> <span class="n">zz_lda</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.</span><span class="p">,</span> <span class="n">colors</span><span class="o">=</span><span class="s">'k'</span><span class="p">)</span> <span class="n">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="s">'b'</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="s">'r'</span><span class="p">)</span> <span class="n">pl</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span> <span class="n">pl</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s">'tight'</span><span class="p">)</span> <span class="n">pl</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s">'Linear Discriminant Analysis'</span><span class="p">)</span> <span class="n">splot</span> <span class="o">=</span> <span class="n">pl</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">1</span><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">pl</span><span class="o">.</span><span class="n">contourf</span><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">zz_qda</span> <span class="o">></span> <span class="mf">0.5</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">pl</span><span class="o">.</span><span class="n">scatter</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">0</span><span class="p">,</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">0</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="s">'b'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">target_names</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="n">pl</span><span class="o">.</span><span class="n">scatter</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="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="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="s">'r'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">target_names</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="n">pl</span><span class="o">.</span><span class="n">contour</span><span class="p">(</span><span class="n">xx</span><span class="p">,</span> <span class="n">yy</span><span class="p">,</span> <span class="n">zz_qda</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.</span><span class="p">,</span> <span class="n">colors</span><span class="o">=</span><span class="s">'k'</span><span class="p">)</span> <span class="n">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">covariances_</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="s">'b'</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">covariances_</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="s">'r'</span><span class="p">)</span> <span class="n">pl</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span> <span class="n">pl</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s">'tight'</span><span class="p">)</span> <span class="n">pl</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s">'Quadratic Discriminant Analysis'</span><span class="p">)</span> <span class="n">pl</span><span class="o">.</span><span class="n">show</span><span class="p">()</span> </pre></div> </div> </div> </div> </div> </div> <div class="clearer"></div> </div> </div> <div class="footer"> <p style="text-align: center">This documentation is relative to scikits.learn version 0.6.0<p> © 2010, scikits.learn developers (BSD Lincense). 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