Exact and approximate computation of the scatter halfspace depth

TL;DR

This paper introduces an exact algorithm for computing the scatter halfspace depth in any dimension and proposes two fast approximate algorithms, with implementations available in the scatterdepth R package.

Contribution

It provides the first exact algorithm for sHD computation in any dimension and offers efficient approximate methods for higher dimensions.

Findings

Exact algorithm implemented in C++ for d ≤ 5

Fast approximate algorithms for higher dimensions

Open-source implementation in the scatterdepth R package

Abstract

The scatter halfspace depth (sHD) is an extension of the location halfspace (also called Tukey) depth that is applicable in the nonparametric analysis of scatter. Using sHD, it is possible to define minimax optimal robust scatter estimators for multivariate data. The problem of exact computation of sHD for data of dimension $d \geq 2$ has, however, not been addressed in the literature. We develop an exact algorithm for the computation of sHD in any dimension $d$ and implement it efficiently using C++ for $d \leq 5$, and in R for any dimension $d \geq 1$. Since the exact computation of sHD is slow especially for higher dimensions, we also propose two fast approximate algorithms. All our programs are freely available in the R package scatterdepth.