Fast and exact implementation of 3-dimensional Tukey depth regions

TL;DR

This paper introduces a new exact and efficient algorithm for computing 3-dimensional Tukey depth regions, significantly improving the speed over existing methods and facilitating practical applications in multivariate data analysis.

Contribution

The paper presents a novel, fast, and exact algorithm specifically designed for 3-dimensional Tukey depth regions, addressing computational inefficiencies of previous methods.

Findings

The new algorithm runs much faster than existing methods.

It provides exact computation of 3D Tukey depth regions.

Data examples demonstrate practical efficiency improvements.

Abstract

Tukey depth regions are important notions in nonparametric multivariate data analysis. A $\tau$-th Tukey depth region $\mathcal{D}_{\tau}$ is the set of all points that have at least depth $\tau$. While the Tukey depth regions are easily defined and interpreted as $p$-variate quantiles, their practical applications is impeded by the lack of efficient computational procedures in dimensions with $p > 2$. Feasible algorithms are available, but practically very slow. In this paper we present a new exact algorithm for 3-dimensional data. An efficient implementation is also provided. Data examples indicate that the proposed algorithm runs much faster than the existing ones.