Large Dimensional Analysis of Robust M-Estimators of Covariance With Outliers

Abstract

International audienceA large dimensional characterization of robust M-estimators of covariance (or scatter) is provided under the assumption that the dataset comprises independent (essentially Gaussian) legitimate samples as well as arbitrary deterministic samples, referred to as outliers. Building upon recent random matrix advances in the area of robust statistics, we specifically show that the so-called Maronna M-estimator of scatter asymp-totically behaves similar to well-known random matrices when the population and sample sizes grow together to infinity. The introduction of outliers leads the robust estimator to behave asymptotically as the weighted sum of the sample outer products, with a constant weight for all legitimate samples and different weights for the outliers. A fine analysis of this structure reveals importantly that the propensity of the M-estimator to attenuate (or enhance) the impact of outliers is mostly dictated by the alignment of the outliers with the inverse population covariance matrix of the legitimate samples. Thus, robust M-estimators can bring substantial benefits over more simplistic estimators such as the per-sample normalized version of the sample covariance matrix, which is not capable of differentiating the outlying samples. The analysis shows that, within the class of Maronna's estimators of scatter, the Huber estimator is most favorable for rejecting outliers. On the contrary, estimators more similar to Tyler's scale invariant estimator (often preferred in the literature) run the risk of inadvertently enhancing some outliers

Large-Dimensional Behavior of Regularized Maronna's M-Estimators of Covariance Matrices

“…NON-REGULARIZED CASE In this section, we turn to the case where the data is contaminated by random outliers and study the robustness of M-estimators for distinct u functions. Some initial insight has been previously provided in [19] for the non-regularized case. Specifically, that study focused on the comparison of the weights given by the estimator to outlying and legitimate samples.…”

“…Their structure is non-trivial, involving matrix fixedpoint equations, and their analysis challenging. Nonetheless, significant progress towards understanding these estimators has been made in large-dimensional settings [15][16][17][18][19], motivated by the increasing number of applications where N, n are both large and comparable. Salient messages of these works are: (i) outliers or impulsive data can be handled by these estimators, if appropriately designed (the choice of the specific form of the estimator is important to handle different types of outliers) [19]; (ii) in the absence of outliers, robust M-estimators essentially behave as the SCM and, therefore, are still subject to the data scarcity issue [16].…”

“…Second, we will investigate the effect of the introduction of outliers in the data, under different contamination models. Initial insights were obtained in [19] for non-regularized estimators, focusing on the weights given by the M-estimator to outlying and legitimate data. However, the current study, by proposing an intuitive measure of robustness, takes a more formal approach to qualify the robustness of these estimators.…”

Robust adaptive detection of buried pipes using GPR

“…The Tyler's estimator is known to be robust to the statistic of the noise and the presence in the secondary data of outliers. Nevertheless, in a recent paper [34], the authors proved that the Tyler's estimator is sensitive to data contamination for some specific configurations. In this case, it could be preferable to use the Mestimators of the covariance matrix which are defined by the following implicit function:R…”

“…This estimator is well-known to be robust to strong heterogeneity and outliers. Nevertheless, a recent paper [34] shows that it could be suffering from bad estimation for some specific outliers. The authors recommend to use another estimate of the covariance matrix in the family of the M-estimators [35], [26]: the Huber's estimator [36], [37], [38] which is the combination of the SCM and the Tyler's estimator.…”