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lucyleeow authored May 20, 2020
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Robust covariance estimation and Mahalanobis distances relevance
================================================================
An example to show covariance estimation with the Mahalanobis
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)'\Sigma^{-1}(x_i - \mu)` where :math:`\mu` and :math:`\Sigma` are
the location and the covariance of the underlying Gaussian
distribution.
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 usual covariance maximum likelihood estimate is very
sensitive to the presence of outliers in the data set and therefor,
the corresponding Mahalanobis distances are. One would better have to
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 data set and that the
associated Mahalanobis distances accurately reflect the true
organisation of the observations.
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 is a robust,
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 is to find
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 Minimum Covariance Determinant estimator (MCD) has been introduced
by P.J.Rousseuw in [1].
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
outlying data. Observations drawn from a contaminating distribution
are not distinguishable from the observations coming from the real,
Gaussian distribution that one may want to work with. Using MCD-based
Gaussian distribution when using standard covariance MLE based Mahalanobis
distances. Using MCD-based
Mahalanobis distances, the two populations become
distinguishable. Associated applications are outliers detection,
observations ranking, clustering, ...
For visualization purpose, the cubic root of the Mahalanobis distances
are represented in the boxplot, as Wilson and Hilferty suggest [2]
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`
[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.
.. topic:: References:
"""
print(__doc__)
.. [1] P. J. Rousseeuw. `Least median of squares regression
<http://web.ipac.caltech.edu/staff/fmasci/home/astro_refs/LeastMedianOfSquares.pdf>`_. J. Am
Stat Ass, 79:871, 1984.
.. [2] Wilson, E. B., & Hilferty, M. M. (1931). `The distribution of chi-square.
<https://water.usgs.gov/osw/bulletin17b/Wilson_Hilferty_1931.pdf>`_
Proceedings of the National Academy of Sciences of the United States
of America, 17, 684-688.
""" # noqa: E501

# %%
# 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 devation equal to 1 and feature 2 has a standard deviation equal
# to 7.

import numpy as np
import matplotlib.pyplot as plt

from sklearn.covariance import EmpiricalCovariance, MinCovDet
# for consistent results
np.random.seed(7)

n_samples = 125
n_outliers = 25
n_features = 2

# generate data
# generate Gaussian data of shape (125, 2)
gen_cov = np.eye(n_features)
gen_cov[0, 0] = 2.
X = np.dot(np.random.randn(n_samples, n_features), gen_cov)
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outliers_cov[np.arange(1, n_features), np.arange(1, n_features)] = 7.
X[-n_outliers:] = np.dot(np.random.randn(n_outliers, n_features), outliers_cov)

# fit a Minimum Covariance Determinant (MCD) robust estimator to data
robust_cov = MinCovDet().fit(X)
# %%
# 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.

# compare estimators learnt from the full data set with true parameters
emp_cov = EmpiricalCovariance().fit(X)
import matplotlib.pyplot as plt
from sklearn.covariance import EmpiricalCovariance, MinCovDet

# #############################################################################
# Display results
fig = plt.figure()
plt.subplots_adjust(hspace=-.1, wspace=.4, top=.95, bottom=.05)

# Show data set
subfig1 = plt.subplot(3, 1, 1)
inlier_plot = subfig1.scatter(X[:, 0], X[:, 1],
color='black', label='inliers')
outlier_plot = subfig1.scatter(X[:, 0][-n_outliers:], X[:, 1][-n_outliers:],
color='red', label='outliers')
subfig1.set_xlim(subfig1.get_xlim()[0], 11.)
subfig1.set_title("Mahalanobis distances of a contaminated data set:")

# Show contours of the distance functions
# 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:\n'
'MCD (Robust):\n{}\n'
'MLE:\n{}'.format(robust_cov.covariance_, emp_cov.covariance_))

# %%
# 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.

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.)
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 = subfig1.contour(xx, yy, np.sqrt(mahal_emp_cov),
cmap=plt.cm.PuBu_r,
linestyles='dashed')

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 = subfig1.contour(xx, yy, np.sqrt(mahal_robust_cov),
cmap=plt.cm.YlOrBr_r, linestyles='dotted')
robust_contour = ax.contour(xx, yy, np.sqrt(mahal_robust_cov),
cmap=plt.cm.YlOrBr_r, linestyles='dotted')

subfig1.legend([emp_cov_contour.collections[1], robust_contour.collections[1],
inlier_plot, outlier_plot],
['MLE dist', 'robust dist', 'inliers', 'outliers'],
loc="upper right", borderaxespad=0)
plt.xticks(())
plt.yticks(())
# 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)

# Plot the scores for each point
emp_mahal = emp_cov.mahalanobis(X - np.mean(X, 0)) ** (0.33)
subfig2 = plt.subplot(2, 2, 3)
subfig2.boxplot([emp_mahal[:-n_outliers], emp_mahal[-n_outliers:]], widths=.25)
subfig2.plot(np.full(n_samples - n_outliers, 1.26),
emp_mahal[:-n_outliers], '+k', markeredgewidth=1)
subfig2.plot(np.full(n_outliers, 2.26),
emp_mahal[-n_outliers:], '+k', markeredgewidth=1)
subfig2.axes.set_xticklabels(('inliers', 'outliers'), size=15)
subfig2.set_ylabel(r"$\sqrt[3]{\rm{(Mahal. dist.)}}$", size=16)
subfig2.set_title("1. from non-robust estimates\n(Maximum Likelihood)")
plt.yticks(())
plt.show()

# %%
# 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.

fig, (ax1, ax2) = plt.subplots(1, 2)
plt.subplots_adjust(wspace=.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=.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)
subfig3 = plt.subplot(2, 2, 4)
subfig3.boxplot([robust_mahal[:-n_outliers], robust_mahal[-n_outliers:]],
widths=.25)
subfig3.plot(np.full(n_samples - n_outliers, 1.26),
robust_mahal[:-n_outliers], '+k', markeredgewidth=1)
subfig3.plot(np.full(n_outliers, 2.26),
robust_mahal[-n_outliers:], '+k', markeredgewidth=1)
subfig3.axes.set_xticklabels(('inliers', 'outliers'), size=15)
subfig3.set_ylabel(r"$\sqrt[3]{\rm{(Mahal. dist.)}}$", size=16)
subfig3.set_title("2. from robust estimates\n(Minimum Covariance Determinant)")
plt.yticks(())
# Plot boxplots
ax2.boxplot([robust_mahal[:-n_outliers], robust_mahal[-n_outliers:]],
widths=.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()

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