Robust and Resistant Regularized Covariance Matrices

TL;DR

This paper introduces a new class of robust regularized M-estimators for multivariate scatter that have high breakdown points and adapt to data shape, with a median-based cross-validation method for tuning.

Contribution

It proposes a novel class of regularized M-estimators that account for data shape and possess high breakdown points, extending the spatial sign covariance matrix.

Findings

The new estimators have high breakdown points.

The median-based cross-validation effectively tunes the estimators.

The approach results in a high breakdown point affine equivariant scatter statistic.

Abstract

We introduce a class of regularized M-estimators of multivariate scatter and show, analogous to the popular spatial sign covariance matrix (SSCM), that they possess high breakdown points. We also show that the SSCM can be viewed as an extreme member of this class. Unlike the SSCM, this class of estimators takes into account the shape of the contours of the data cloud when down-weighing observations. We also propose a median based cross validation criterion for selecting the tuning parameter for this class of regularized M-estimators. This cross validation criterion helps assure the resulting tuned scatter estimator is a good fit to the data as well as having a high breakdown point. A motivation for this new median based criterion is that when it is optimized over all possible scatter parameters, rather than only over the tuned candidates, it results in a new high breakdown point affine equivariant multivariate scatter statistic.