.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/covariance/plot_mahalanobis_distances.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note Click :ref:here  to download the full example code or to run this example in your browser via Binder .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_examples_covariance_plot_mahalanobis_distances.py: ================================================================ Robust covariance estimation and Mahalanobis distances relevance ================================================================ This example shows covariance estimation with Mahalanobis distances on Gaussian distributed data. For Gaussian distributed data, the distance of an observation :math:x_i to the mode of the distribution can be computed using its Mahalanobis distance: .. math:: d_{(\mu,\Sigma)}(x_i)^2 = (x_i - \mu)^T\Sigma^{-1}(x_i - \mu) where :math:\mu and :math:\Sigma are the location and the covariance of the underlying Gaussian distributions. In practice, :math:\mu and :math:\Sigma are replaced by some estimates. The standard covariance maximum likelihood estimate (MLE) is very sensitive to the presence of outliers in the data set and therefore, the downstream Mahalanobis distances also are. It would be better to use a robust estimator of covariance to guarantee that the estimation is resistant to "erroneous" observations in the dataset and that the calculated Mahalanobis distances accurately reflect the true organization of the observations. The Minimum Covariance Determinant estimator (MCD) is a robust, high-breakdown point (i.e. it can be used to estimate the covariance matrix of highly contaminated datasets, up to :math:\frac{n_\text{samples}-n_\text{features}-1}{2} outliers) estimator of covariance. The idea behind the MCD is to find :math:\frac{n_\text{samples}+n_\text{features}+1}{2} observations whose empirical covariance has the smallest determinant, yielding a "pure" subset of observations from which to compute standards estimates of location and covariance. The MCD was introduced by P.J.Rousseuw in [1]_. This example illustrates how the Mahalanobis distances are affected by outlying data. Observations drawn from a contaminating distribution are not distinguishable from the observations coming from the real, Gaussian distribution when using standard covariance MLE based Mahalanobis distances. Using MCD-based Mahalanobis distances, the two populations become distinguishable. Associated applications include outlier detection, observation ranking and clustering. .. note:: See also :ref:sphx_glr_auto_examples_covariance_plot_robust_vs_empirical_covariance.py .. topic:: References: .. [1] P. J. Rousseeuw. Least median of squares regression _. J. Am Stat Ass, 79:871, 1984. .. [2] Wilson, E. B., & Hilferty, M. M. (1931). The distribution of chi-square. _ Proceedings of the National Academy of Sciences of the United States of America, 17, 684-688. .. GENERATED FROM PYTHON SOURCE LINES 66-75 Generate data -------------- First, we generate a dataset of 125 samples and 2 features. Both features are Gaussian distributed with mean of 0 but feature 1 has a standard deviation equal to 2 and feature 2 has a standard deviation equal to 1. Next, 25 samples are replaced with Gaussian outlier samples where feature 1 has a standard deviation equal to 1 and feature 2 has a standard deviation equal to 7. .. GENERATED FROM PYTHON SOURCE LINES 75-94 .. code-block:: default import numpy as np # for consistent results np.random.seed(7) n_samples = 125 n_outliers = 25 n_features = 2 # generate Gaussian data of shape (125, 2) gen_cov = np.eye(n_features) gen_cov[0, 0] = 2.0 X = np.dot(np.random.randn(n_samples, n_features), gen_cov) # add some outliers outliers_cov = np.eye(n_features) outliers_cov[np.arange(1, n_features), np.arange(1, n_features)] = 7.0 X[-n_outliers:] = np.dot(np.random.randn(n_outliers, n_features), outliers_cov) .. GENERATED FROM PYTHON SOURCE LINES 95-104 Comparison of results --------------------- Below, we fit MCD and MLE based covariance estimators to our data and print the estimated covariance matrices. Note that the estimated variance of feature 2 is much higher with the MLE based estimator (7.5) than that of the MCD robust estimator (1.2). This shows that the MCD based robust estimator is much more resistant to the outlier samples, which were designed to have a much larger variance in feature 2. .. GENERATED FROM PYTHON SOURCE LINES 104-118 .. code-block:: default import matplotlib.pyplot as plt from sklearn.covariance import EmpiricalCovariance, MinCovDet # fit a MCD robust estimator to data robust_cov = MinCovDet().fit(X) # fit a MLE estimator to data emp_cov = EmpiricalCovariance().fit(X) print( "Estimated covariance matrix:\nMCD (Robust):\n{}\nMLE:\n{}".format( robust_cov.covariance_, emp_cov.covariance_ ) ) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none Estimated covariance matrix: MCD (Robust): [[ 3.26253567e+00 -3.06695631e-03] [-3.06695631e-03 1.22747343e+00]] MLE: [[ 3.23773583 -0.24640578] [-0.24640578 7.51963999]] .. GENERATED FROM PYTHON SOURCE LINES 119-124 To better visualize the difference, we plot contours of the Mahalanobis distances calculated by both methods. Notice that the robust MCD based Mahalanobis distances fit the inlier black points much better, whereas the MLE based distances are more influenced by the outlier red points. .. GENERATED FROM PYTHON SOURCE LINES 124-168 .. code-block:: default fig, ax = plt.subplots(figsize=(10, 5)) # Plot data set inlier_plot = ax.scatter(X[:, 0], X[:, 1], color="black", label="inliers") outlier_plot = ax.scatter( X[:, 0][-n_outliers:], X[:, 1][-n_outliers:], color="red", label="outliers" ) ax.set_xlim(ax.get_xlim()[0], 10.0) ax.set_title("Mahalanobis distances of a contaminated data set") # Create meshgrid of feature 1 and feature 2 values xx, yy = np.meshgrid( np.linspace(plt.xlim()[0], plt.xlim()[1], 100), np.linspace(plt.ylim()[0], plt.ylim()[1], 100), ) zz = np.c_[xx.ravel(), yy.ravel()] # Calculate the MLE based Mahalanobis distances of the meshgrid mahal_emp_cov = emp_cov.mahalanobis(zz) mahal_emp_cov = mahal_emp_cov.reshape(xx.shape) emp_cov_contour = plt.contour( xx, yy, np.sqrt(mahal_emp_cov), cmap=plt.cm.PuBu_r, linestyles="dashed" ) # Calculate the MCD based Mahalanobis distances mahal_robust_cov = robust_cov.mahalanobis(zz) mahal_robust_cov = mahal_robust_cov.reshape(xx.shape) robust_contour = ax.contour( xx, yy, np.sqrt(mahal_robust_cov), cmap=plt.cm.YlOrBr_r, linestyles="dotted" ) # Add legend ax.legend( [ emp_cov_contour.collections[1], robust_contour.collections[1], inlier_plot, outlier_plot, ], ["MLE dist", "MCD dist", "inliers", "outliers"], loc="upper right", borderaxespad=0, ) plt.show() .. image-sg:: /auto_examples/covariance/images/sphx_glr_plot_mahalanobis_distances_001.png :alt: Mahalanobis distances of a contaminated data set :srcset: /auto_examples/covariance/images/sphx_glr_plot_mahalanobis_distances_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 169-175 Finally, we highlight the ability of MCD based Mahalanobis distances to distinguish outliers. We take the cubic root of the Mahalanobis distances, yielding approximately normal distributions (as suggested by Wilson and Hilferty [2]_), then plot the values of inlier and outlier samples with boxplots. The distribution of outlier samples is more separated from the distribution of inlier samples for robust MCD based Mahalanobis distances. .. GENERATED FROM PYTHON SOURCE LINES 175-212 .. code-block:: default fig, (ax1, ax2) = plt.subplots(1, 2) plt.subplots_adjust(wspace=0.6) # Calculate cubic root of MLE Mahalanobis distances for samples emp_mahal = emp_cov.mahalanobis(X - np.mean(X, 0)) ** (0.33) # Plot boxplots ax1.boxplot([emp_mahal[:-n_outliers], emp_mahal[-n_outliers:]], widths=0.25) # Plot individual samples ax1.plot( np.full(n_samples - n_outliers, 1.26), emp_mahal[:-n_outliers], "+k", markeredgewidth=1, ) ax1.plot(np.full(n_outliers, 2.26), emp_mahal[-n_outliers:], "+k", markeredgewidth=1) ax1.axes.set_xticklabels(("inliers", "outliers"), size=15) ax1.set_ylabel(r"$\sqrt[3]{\rm{(Mahal. dist.)}}$", size=16) ax1.set_title("Using non-robust estimates\n(Maximum Likelihood)") # Calculate cubic root of MCD Mahalanobis distances for samples robust_mahal = robust_cov.mahalanobis(X - robust_cov.location_) ** (0.33) # Plot boxplots ax2.boxplot([robust_mahal[:-n_outliers], robust_mahal[-n_outliers:]], widths=0.25) # Plot individual samples ax2.plot( np.full(n_samples - n_outliers, 1.26), robust_mahal[:-n_outliers], "+k", markeredgewidth=1, ) ax2.plot(np.full(n_outliers, 2.26), robust_mahal[-n_outliers:], "+k", markeredgewidth=1) ax2.axes.set_xticklabels(("inliers", "outliers"), size=15) ax2.set_ylabel(r"$\sqrt[3]{\rm{(Mahal. dist.)}}$", size=16) ax2.set_title("Using robust estimates\n(Minimum Covariance Determinant)") plt.show() .. image-sg:: /auto_examples/covariance/images/sphx_glr_plot_mahalanobis_distances_002.png :alt: Using non-robust estimates (Maximum Likelihood), Using robust estimates (Minimum Covariance Determinant) :srcset: /auto_examples/covariance/images/sphx_glr_plot_mahalanobis_distances_002.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 0.289 seconds) .. _sphx_glr_download_auto_examples_covariance_plot_mahalanobis_distances.py: .. only :: html .. container:: sphx-glr-footer :class: sphx-glr-footer-example .. container:: binder-badge .. image:: images/binder_badge_logo.svg :target: https://mybinder.org/v2/gh/scikit-learn/scikit-learn/main?urlpath=lab/tree/notebooks/auto_examples/covariance/plot_mahalanobis_distances.ipynb :alt: Launch binder :width: 150 px .. container:: sphx-glr-download sphx-glr-download-python :download:Download Python source code: plot_mahalanobis_distances.py  .. container:: sphx-glr-download sphx-glr-download-jupyter :download:Download Jupyter notebook: plot_mahalanobis_distances.ipynb  .. only:: html .. rst-class:: sphx-glr-signature Gallery generated by Sphinx-Gallery _