Hyperplane Distance Depth

TL;DR

This paper introduces Hyperplane Distance Depth (HDD), a new depth measure for data centrality based on distances to hyperplanes, with efficient algorithms for computation and properties analysis.

Contribution

The paper proposes HDD as a simple, efficient depth measure and provides algorithms for its computation and analysis of its properties.

Findings

HDD is convex, symmetric, and vanishes at infinity.

Algorithms for computing HDD in O(d log n) time after preprocessing.

Method for finding a median point in O(d n^{d^2} log n) time.

Abstract

Depth measures quantify central tendency in the analysis of statistical and geometric data. Selecting a depth measure that is simple and efficiently computable is often important, e.g., when calculating depth for multiple query points or when applied to large sets of data. In this work, we introduce \emph{Hyperplane Distance Depth (HDD)}, which measures the centrality of a query point $q$ relative to a given set $P$ of $n$ points in $\mathbb{R}^d$, defined as the sum of the distances from $q$ to all $\binom{n}{d}$ hyperplanes determined by points in $P$. We present algorithms for calculating the HDD of an arbitrary query point $q$ relative to $P$ in $O(d \log n)$ time after preprocessing $P$, and for finding a median point of $P$ in $O(d n^{d^2} \log n)$ time. We study various properties of hyperplane distance depth and show that it is convex, symmetric, and vanishing at infinity.