ABCDepth: efficient algorithm for Tukey depth

TL;DR

This paper introduces ABCDepth, an efficient algorithm for computing Tukey depth level sets in high-dimensional data, utilizing intersections of balls, and demonstrates its superior speed and scalability over existing methods.

Contribution

The paper presents a novel algorithm for Tukey depth level sets based on intersections of balls, with improved computational complexity and scalability for high-dimensional data.

Findings

Algorithm is significantly faster than existing methods.

Can handle thousands of multidimensional observations.

Effective for high-dimensional data analysis.

Abstract

We present a new algorithm for Tukey (halfspace) depth level sets and its implementation. Given $d$-dimensional data set for any $d\geq 2$, the algorithm is based on representation of level sets as intersections of balls in $R^d$, and can be easily adapted to related depths (Type D, Zuo and Serfling (Ann. Stat. {\bf 28} (2000), 461--482)). The algorithm complexity is $O(dn^2 + n^2\log{n})$ where $n$ is the data set size. Examples with real and synthetic data show that the algorithm is much faster than other implemented algorithms and that it can accept thousands of multidimensional observations, while other algorithms are tested with two-dimensional data or with a couple of hundreds multidimensional observations.