A generalized spatial sign covariance matrix

TL;DR

This paper extends the spatial sign covariance matrix by exploring generalized radial functions, demonstrating robustness, consistency of eigenvectors, and high breakdown value, with simulation results guiding optimal transformation choices.

Contribution

It introduces a generalized framework for the SSCM using various radial functions, maintaining eigenvector consistency and robustness properties.

Findings

Eigenvectors remain consistent under the generalized SSCM.

The influence function indicates high robustness.

Simulation suggests partial transformation yields best results.

Abstract

The well-known spatial sign covariance matrix (SSCM) carries out a radial transform which moves all data points to a sphere, followed by computing the classical covariance matrix of the transformed data. Its popularity stems from its robustness to outliers, fast computation, and applications to correlation and principal component analysis. In this paper we study more general radial functions. It is shown that the eigenvectors of the generalized SSCM are still consistent and the ranks of the eigenvalues are preserved. The influence function of the resulting scatter matrix is derived, and it is shown that its breakdown value is as high as that of the original SSCM. A simulation study indicates that the best results are obtained when the inner half of the data points are not transformed and points lying far away are moved to the center.