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<li><a class="reference internal" href="#">Outlier detection on a real data set</a><ul>
<li><a class="reference internal" href="#first-example">First example</a></li>
<li><a class="reference internal" href="#second-example">Second example</a></li>
</ul>
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<p><a class="reference internal" href="#sphx-glr-download-auto-examples-applications-plot-outlier-detection-wine-py"><span class="std std-ref">Go to the end</span></a>
to download the full example code or to run this example in your browser via JupyterLite or Binder</p>
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<section class="sphx-glr-example-title" id="outlier-detection-on-a-real-data-set">
<span id="sphx-glr-auto-examples-applications-plot-outlier-detection-wine-py"></span><h1>Outlier detection on a real data set<a class="headerlink" href="#outlier-detection-on-a-real-data-set" title="Link to this heading">¶</a></h1>
<p>This example illustrates the need for robust covariance estimation
on a real data set. It is useful both for outlier detection and for
a better understanding of the data structure.</p>
<p>We selected two sets of two variables from the Wine data set
as an illustration of what kind of analysis can be done with several
outlier detection tools. For the purpose of visualization, we are working
with two-dimensional examples, but one should be aware that things are
not so trivial in high-dimension, as it will be pointed out.</p>
<p>In both examples below, the main result is that the empirical covariance
estimate, as a non-robust one, is highly influenced by the heterogeneous
structure of the observations. Although the robust covariance estimate is
able to focus on the main mode of the data distribution, it sticks to the
assumption that the data should be Gaussian distributed, yielding some biased
estimation of the data structure, but yet accurate to some extent.
The One-Class SVM does not assume any parametric form of the data distribution
and can therefore model the complex shape of the data much better.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="c1"># Author: Virgile Fritsch <[email protected]></span>
<span class="c1"># License: BSD 3 clause</span>
</pre></div>
</div>
<section id="first-example">
<h2>First example<a class="headerlink" href="#first-example" title="Link to this heading">¶</a></h2>
<p>The first example illustrates how the Minimum Covariance Determinant
robust estimator can help concentrate on a relevant cluster when outlying
points exist. Here the empirical covariance estimation is skewed by points
outside of the main cluster. Of course, some screening tools would have pointed
out the presence of two clusters (Support Vector Machines, Gaussian Mixture
Models, univariate outlier detection, …). But had it been a high-dimensional
example, none of these could be applied that easily.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.covariance</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.covariance.EllipticEnvelope.html#sklearn.covariance.EllipticEnvelope" title="sklearn.covariance.EllipticEnvelope" class="sphx-glr-backref-module-sklearn-covariance sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">EllipticEnvelope</span></a>
<span class="kn">from</span> <span class="nn">sklearn.inspection</span> <span class="kn">import</span> <span class="n">DecisionBoundaryDisplay</span>
<span class="kn">from</span> <span class="nn">sklearn.svm</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.svm.OneClassSVM.html#sklearn.svm.OneClassSVM" title="sklearn.svm.OneClassSVM" class="sphx-glr-backref-module-sklearn-svm sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">OneClassSVM</span></a>
<span class="n">estimators</span> <span class="o">=</span> <span class="p">{</span>
<span class="s2">"Empirical Covariance"</span><span class="p">:</span> <a href="../../modules/generated/sklearn.covariance.EllipticEnvelope.html#sklearn.covariance.EllipticEnvelope" title="sklearn.covariance.EllipticEnvelope" class="sphx-glr-backref-module-sklearn-covariance sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">EllipticEnvelope</span></a><span class="p">(</span><span class="n">support_fraction</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">contamination</span><span class="o">=</span><span class="mf">0.25</span><span class="p">),</span>
<span class="s2">"Robust Covariance (Minimum Covariance Determinant)"</span><span class="p">:</span> <a href="../../modules/generated/sklearn.covariance.EllipticEnvelope.html#sklearn.covariance.EllipticEnvelope" title="sklearn.covariance.EllipticEnvelope" class="sphx-glr-backref-module-sklearn-covariance sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">EllipticEnvelope</span></a><span class="p">(</span>
<span class="n">contamination</span><span class="o">=</span><span class="mf">0.25</span>
<span class="p">),</span>
<span class="s2">"OCSVM"</span><span class="p">:</span> <a href="../../modules/generated/sklearn.svm.OneClassSVM.html#sklearn.svm.OneClassSVM" title="sklearn.svm.OneClassSVM" class="sphx-glr-backref-module-sklearn-svm sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">OneClassSVM</span></a><span class="p">(</span><span class="n">nu</span><span class="o">=</span><span class="mf">0.25</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mf">0.35</span><span class="p">),</span>
<span class="p">}</span>
</pre></div>
</div>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.lines</span> <span class="k">as</span> <span class="nn">mlines</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.datasets</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.datasets.load_wine.html#sklearn.datasets.load_wine" title="sklearn.datasets.load_wine" class="sphx-glr-backref-module-sklearn-datasets sphx-glr-backref-type-py-function"><span class="n">load_wine</span></a>
<span class="n">X</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.datasets.load_wine.html#sklearn.datasets.load_wine" title="sklearn.datasets.load_wine" class="sphx-glr-backref-module-sklearn-datasets sphx-glr-backref-type-py-function"><span class="n">load_wine</span></a><span class="p">()[</span><span class="s2">"data"</span><span class="p">][:,</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="c1"># two clusters</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</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">colors</span> <span class="o">=</span> <span class="p">[</span><span class="s2">"tab:blue"</span><span class="p">,</span> <span class="s2">"tab:orange"</span><span class="p">,</span> <span class="s2">"tab:red"</span><span class="p">]</span>
<span class="c1"># Learn a frontier for outlier detection with several classifiers</span>
<span class="n">legend_lines</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">color</span><span class="p">,</span> <span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">estimator</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">colors</span><span class="p">,</span> <span class="n">estimators</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
<span class="n">estimator</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</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">"decision_function"</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">levels</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span>
<span class="n">colors</span><span class="o">=</span><span class="n">color</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="p">)</span>
<span class="n">legend_lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.lines.Line2D.html#matplotlib.lines.Line2D" title="matplotlib.lines.Line2D" class="sphx-glr-backref-module-matplotlib-lines sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">mlines</span><span class="o">.</span><span class="n">Line2D</span></a><span class="p">([],</span> <span class="p">[],</span> <span class="n">color</span><span class="o">=</span><span class="n">color</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">name</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</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="mi">1</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s2">"black"</span><span class="p">)</span>
<span class="n">bbox_args</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">boxstyle</span><span class="o">=</span><span class="s2">"round"</span><span class="p">,</span> <span class="n">fc</span><span class="o">=</span><span class="s2">"0.8"</span><span class="p">)</span>
<span class="n">arrow_args</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">arrowstyle</span><span class="o">=</span><span class="s2">"->"</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">annotate</span><span class="p">(</span>
<span class="s2">"outlying points"</span><span class="p">,</span>
<span class="n">xy</span><span class="o">=</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span>
<span class="n">xycoords</span><span class="o">=</span><span class="s2">"data"</span><span class="p">,</span>
<span class="n">textcoords</span><span class="o">=</span><span class="s2">"data"</span><span class="p">,</span>
<span class="n">xytext</span><span class="o">=</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mf">1.25</span><span class="p">),</span>
<span class="n">bbox</span><span class="o">=</span><span class="n">bbox_args</span><span class="p">,</span>
<span class="n">arrowprops</span><span class="o">=</span><span class="n">arrow_args</span><span class="p">,</span>
<span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">handles</span><span class="o">=</span><span class="n">legend_lines</span><span class="p">,</span> <span class="n">loc</span><span class="o">=</span><span class="s2">"upper center"</span><span class="p">)</span>
<span class="n">_</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">set</span><span class="p">(</span>
<span class="n">xlabel</span><span class="o">=</span><span class="s2">"ash"</span><span class="p">,</span>
<span class="n">ylabel</span><span class="o">=</span><span class="s2">"malic_acid"</span><span class="p">,</span>
<span class="n">title</span><span class="o">=</span><span class="s2">"Outlier detection on a real data set (wine recognition)"</span><span class="p">,</span>
<span class="p">)</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_outlier_detection_wine_001.png" srcset="../../_images/sphx_glr_plot_outlier_detection_wine_001.png" alt="Outlier detection on a real data set (wine recognition)" class = "sphx-glr-single-img"/></section>
<section id="second-example">
<h2>Second example<a class="headerlink" href="#second-example" title="Link to this heading">¶</a></h2>
<p>The second example shows the ability of the Minimum Covariance Determinant
robust estimator of covariance to concentrate on the main mode of the data
distribution: the location seems to be well estimated, although the
covariance is hard to estimate due to the banana-shaped distribution. Anyway,
we can get rid of some outlying observations. The One-Class SVM is able to
capture the real data structure, but the difficulty is to adjust its kernel
bandwidth parameter so as to obtain a good compromise between the shape of
the data scatter matrix and the risk of over-fitting the data.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="n">X</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.datasets.load_wine.html#sklearn.datasets.load_wine" title="sklearn.datasets.load_wine" class="sphx-glr-backref-module-sklearn-datasets sphx-glr-backref-type-py-function"><span class="n">load_wine</span></a><span class="p">()[</span><span class="s2">"data"</span><span class="p">][:,</span> <span class="p">[</span><span class="mi">6</span><span class="p">,</span> <span class="mi">9</span><span class="p">]]</span> <span class="c1"># "banana"-shaped</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</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">colors</span> <span class="o">=</span> <span class="p">[</span><span class="s2">"tab:blue"</span><span class="p">,</span> <span class="s2">"tab:orange"</span><span class="p">,</span> <span class="s2">"tab:red"</span><span class="p">]</span>
<span class="c1"># Learn a frontier for outlier detection with several classifiers</span>
<span class="n">legend_lines</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">color</span><span class="p">,</span> <span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">estimator</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">colors</span><span class="p">,</span> <span class="n">estimators</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
<span class="n">estimator</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</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">"decision_function"</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">levels</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span>
<span class="n">colors</span><span class="o">=</span><span class="n">color</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="p">)</span>
<span class="n">legend_lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.lines.Line2D.html#matplotlib.lines.Line2D" title="matplotlib.lines.Line2D" class="sphx-glr-backref-module-matplotlib-lines sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">mlines</span><span class="o">.</span><span class="n">Line2D</span></a><span class="p">([],</span> <span class="p">[],</span> <span class="n">color</span><span class="o">=</span><span class="n">color</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">name</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</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="mi">1</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s2">"black"</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">handles</span><span class="o">=</span><span class="n">legend_lines</span><span class="p">,</span> <span class="n">loc</span><span class="o">=</span><span class="s2">"upper center"</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">xlabel</span><span class="o">=</span><span class="s2">"flavanoids"</span><span class="p">,</span>
<span class="n">ylabel</span><span class="o">=</span><span class="s2">"color_intensity"</span><span class="p">,</span>
<span class="n">title</span><span class="o">=</span><span class="s2">"Outlier detection on a real data set (wine recognition)"</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_outlier_detection_wine_002.png" srcset="../../_images/sphx_glr_plot_outlier_detection_wine_002.png" alt="Outlier detection on a real data set (wine recognition)" class = "sphx-glr-single-img"/><p class="sphx-glr-timing"><strong>Total running time of the script:</strong> (0 minutes 0.380 seconds)</p>
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<p><a class="reference download internal" download="" href="../../_downloads/609eccf9ab7d476daf68967ce1fce0b7/plot_outlier_detection_wine.py"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Python</span> <span class="pre">source</span> <span class="pre">code:</span> <span class="pre">plot_outlier_detection_wine.py</span></code></a></p>
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<p class="rubric">Related examples</p>
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<p><a class="reference internal" href="../miscellaneous/plot_anomaly_comparison.html#sphx-glr-auto-examples-miscellaneous-plot-anomaly-comparison-py"><span class="std std-ref">Comparing anomaly detection algorithms for outlier detection on toy datasets</span></a></p>
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</div><div class="sphx-glr-thumbcontainer" tooltip="This example plots the covariance ellipsoids of each class and the decision boundary learned by..."><img alt="" src="../../_images/sphx_glr_plot_lda_qda_thumb.png" />
<p><a class="reference internal" href="../classification/plot_lda_qda.html#sphx-glr-auto-examples-classification-plot-lda-qda-py"><span class="std std-ref">Linear and Quadratic Discriminant Analysis with covariance ellipsoid</span></a></p>
<div class="sphx-glr-thumbnail-title">Linear and Quadratic Discriminant Analysis with covariance ellipsoid</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="The usual covariance maximum likelihood estimate is very sensitive to the presence of outliers ..."><img alt="" src="../../_images/sphx_glr_plot_robust_vs_empirical_covariance_thumb.png" />
<p><a class="reference internal" href="../covariance/plot_robust_vs_empirical_covariance.html#sphx-glr-auto-examples-covariance-plot-robust-vs-empirical-covariance-py"><span class="std std-ref">Robust vs Empirical covariance estimate</span></a></p>
<div class="sphx-glr-thumbnail-title">Robust vs Empirical covariance estimate</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="This example shows covariance estimation with Mahalanobis distances on Gaussian distributed dat..."><img alt="" src="../../_images/sphx_glr_plot_mahalanobis_distances_thumb.png" />
<p><a class="reference internal" href="../covariance/plot_mahalanobis_distances.html#sphx-glr-auto-examples-covariance-plot-mahalanobis-distances-py"><span class="std std-ref">Robust covariance estimation and Mahalanobis distances relevance</span></a></p>
<div class="sphx-glr-thumbnail-title">Robust covariance estimation and Mahalanobis distances relevance</div>
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