On exact computation of Tukey depth central regions

TL;DR

This paper develops and analyzes an exact, efficient algorithm for computing Tukey depth central regions in multivariate data, addressing limitations of previous methods and providing a reliable implementation.

Contribution

It introduces a modified algorithm that guarantees exact computation of Tukey depth regions in higher dimensions and low thresholds, with a dual formulation analysis.

Findings

Algorithm A is exact in 2D and for low thresholds in higher dimensions.

The proposed modification guarantees exact results in all cases.

An efficient C++ implementation is available in the R package TukeyRegion.

Abstract

The Tukey (or halfspace) depth extends nonparametric methods toward multivariate data. The multivariate analogues of the quantiles are the central regions of the Tukey depth, defined as sets of points in the $d$-dimensional space whose Tukey depth exceeds given thresholds $k$. We address the problem of fast and exact computation of those central regions. First, we analyse an efficient Algorithm A from Liu et al. (2019), and prove that it yields exact results in dimension $d=2$, or for a low threshold $k$ in arbitrary dimension. We provide examples where Algorithm A fails to recover the exact Tukey depth region for $d>2$, and propose a modification that is guaranteed to be exact. We express the problem of computing the exact central region in its dual formulation, and use that viewpoint to demonstrate that further substantial improvements to our algorithm are unlikely. An efficient C++ implementation of our exact algorithm is freely available in the R package TukeyRegion.