Robust vs Empirical covariance estimate

The usual covariance maximum likelihood estimate is very sensitive to the presence of outliers in the data set. In such a case, 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.

Minimum Covariance Determinant Estimator

The Minimum Covariance Determinant estimator is a robust, high-breakdown point (i.e. it can be used to estimate the covariance matrix of highly contaminated datasets, up to \(\frac{n_\text{samples} - n_\text{features}-1}{2}\) outliers) estimator of covariance. The idea is to find \(\frac{n_\text{samples} + n_\text{features}+1}{2}\) observations whose empirical covariance has the smallest determinant, yielding a “pure” subset of observations from which to compute standards estimates of location and covariance. After a correction step aiming at compensating the fact that the estimates were learned from only a portion of the initial data, we end up with robust estimates of the data set location and covariance.

The Minimum Covariance Determinant estimator (MCD) has been introduced by P.J.Rousseuw in [1].

Evaluation

In this example, we compare the estimation errors that are made when using various types of location and covariance estimates on contaminated Gaussian distributed data sets:

• The mean and the empirical covariance of the full dataset, which break down as soon as there are outliers in the data set
• The robust MCD, that has a low error provided \(n_\text{samples} > 5n_\text{features}\)
• The mean and the empirical covariance of the observations that are known to be good ones. This can be considered as a “perfect” MCD estimation, so one can trust our implementation by comparing to this case.

References

[1]	P. J. Rousseeuw. Least median of squares regression. Journal of American Statistical Ass., 79:871, 1984.

[2]	Johanna Hardin, David M Rocke. The distribution of robust distances. Journal of Computational and Graphical Statistics. December 1, 2005, 14(4): 928-946.

[3]	Zoubir A., Koivunen V., Chakhchoukh Y. and Muma M. (2012). Robust estimation in signal processing: A tutorial-style treatment of fundamental concepts. IEEE Signal Processing Magazine 29(4), 61-80.

Out:

print(__doc__)

import numpy as np
import matplotlib.pyplot as plt
import matplotlib.font_manager

from sklearn.covariance import EmpiricalCovariance, MinCovDet

# example settings
n_samples = 80
n_features = 5
repeat = 10

range_n_outliers = np.concatenate(
    (np.linspace(0, n_samples / 8, 5),
     np.linspace(n_samples / 8, n_samples / 2, 5)[1:-1])).astype(np.int)

# definition of arrays to store results
err_loc_mcd = np.zeros((range_n_outliers.size, repeat))
err_cov_mcd = np.zeros((range_n_outliers.size, repeat))
err_loc_emp_full = np.zeros((range_n_outliers.size, repeat))
err_cov_emp_full = np.zeros((range_n_outliers.size, repeat))
err_loc_emp_pure = np.zeros((range_n_outliers.size, repeat))
err_cov_emp_pure = np.zeros((range_n_outliers.size, repeat))

# computation
for i, n_outliers in enumerate(range_n_outliers):
    for j in range(repeat):

        rng = np.random.RandomState(i * j)

        # generate data
        X = rng.randn(n_samples, n_features)
        # add some outliers
        outliers_index = rng.permutation(n_samples)[:n_outliers]
        outliers_offset = 10. * \
            (np.random.randint(2, size=(n_outliers, n_features)) - 0.5)
        X[outliers_index] += outliers_offset
        inliers_mask = np.ones(n_samples).astype(bool)
        inliers_mask[outliers_index] = False

        # fit a Minimum Covariance Determinant (MCD) robust estimator to data
        mcd = MinCovDet().fit(X)
        # compare raw robust estimates with the true location and covariance
        err_loc_mcd[i, j] = np.sum(mcd.location_ ** 2)
        err_cov_mcd[i, j] = mcd.error_norm(np.eye(n_features))

        # compare estimators learned from the full data set with true
        # parameters
        err_loc_emp_full[i, j] = np.sum(X.mean(0) ** 2)
        err_cov_emp_full[i, j] = EmpiricalCovariance().fit(X).error_norm(
            np.eye(n_features))

        # compare with an empirical covariance learned from a pure data set
        # (i.e. "perfect" mcd)
        pure_X = X[inliers_mask]
        pure_location = pure_X.mean(0)
        pure_emp_cov = EmpiricalCovariance().fit(pure_X)
        err_loc_emp_pure[i, j] = np.sum(pure_location ** 2)
        err_cov_emp_pure[i, j] = pure_emp_cov.error_norm(np.eye(n_features))

# Display results
font_prop = matplotlib.font_manager.FontProperties(size=11)
plt.subplot(2, 1, 1)
lw = 2
plt.errorbar(range_n_outliers, err_loc_mcd.mean(1),
             yerr=err_loc_mcd.std(1) / np.sqrt(repeat),
             label="Robust location", lw=lw, color='m')
plt.errorbar(range_n_outliers, err_loc_emp_full.mean(1),
             yerr=err_loc_emp_full.std(1) / np.sqrt(repeat),
             label="Full data set mean", lw=lw, color='green')
plt.errorbar(range_n_outliers, err_loc_emp_pure.mean(1),
             yerr=err_loc_emp_pure.std(1) / np.sqrt(repeat),
             label="Pure data set mean", lw=lw, color='black')
plt.title("Influence of outliers on the location estimation")
plt.ylabel(r"Error ($||\mu - \hat{\mu}||_2^2$)")
plt.legend(loc="upper left", prop=font_prop)

plt.subplot(2, 1, 2)
x_size = range_n_outliers.size
plt.errorbar(range_n_outliers, err_cov_mcd.mean(1),
             yerr=err_cov_mcd.std(1),
             label="Robust covariance (mcd)", color='m')
plt.errorbar(range_n_outliers[:(x_size // 5 + 1)],
             err_cov_emp_full.mean(1)[:(x_size // 5 + 1)],
             yerr=err_cov_emp_full.std(1)[:(x_size // 5 + 1)],
             label="Full data set empirical covariance", color='green')
plt.plot(range_n_outliers[(x_size // 5):(x_size // 2 - 1)],
         err_cov_emp_full.mean(1)[(x_size // 5):(x_size // 2 - 1)],
         color='green', ls='--')
plt.errorbar(range_n_outliers, err_cov_emp_pure.mean(1),
             yerr=err_cov_emp_pure.std(1),
             label="Pure data set empirical covariance", color='black')
plt.title("Influence of outliers on the covariance estimation")
plt.xlabel("Amount of contamination (%)")
plt.ylabel("RMSE")
plt.legend(loc="upper center", prop=font_prop)

plt.show()