Halfspace depths for scatter, concentration and shape matrices

TL;DR

This paper introduces and thoroughly investigates new halfspace depth concepts for scatter, concentration, and shape matrices, exploring their properties, geometries, and practical applications without restrictive assumptions.

Contribution

It proposes novel depth concepts for matrices, analyzes their properties under minimal assumptions, and extends the framework to concentration and shape matrices.

Findings

Depth concepts are applicable beyond elliptical distributions.

Different geometries are needed to understand scatter halfspace depth.

Practical relevance demonstrated through a finance data example.

Abstract

We propose halfspace depth concepts for scatter, concentration and shape matrices. For scatter matrices, our concept is similar to those from Chen, Gao and Ren (2017) and Zhang (2002). Rather than focusing, as in these earlier works, on deepest scatter matrices, we thoroughly investigate the properties of the proposed depth and of the corresponding depth regions. We do so under minimal assumptions and, in particular, we do not restrict to elliptical distributions nor to absolutely continuous distributions. Interestingly, fully understanding scatter halfspace depth requires considering different geometries/topologies on the space of scatter matrices. We also discuss, in the spirit of Zuo and Serfling (2000), the structural properties a scatter depth should satisfy, and investigate whether or not these are met by scatter halfspace depth. Companion concepts of depth for concentration matrices and shape matrices are also proposed and studied. We show the practical relevance of the depth concepts considered in a real-data example from finance.