An Optimal Algorithm for Computing the Spherical Depth of Points in the Plane

TL;DR

This paper introduces an optimal $O(n \,\log n)$ algorithm for computing the spherical depth of points in the plane, significantly improving over the naive $O(dn^2)$ approach, and explores its geometric properties and relationships with simplicial depth.

Contribution

The paper presents the first optimal algorithm for bivariate spherical depth computation and establishes a lower bound, advancing the efficiency of multivariate data analysis methods.

Findings

The algorithm runs in $O(n \log n)$ time for the plane.

Computing spherical depth has a proven lower bound of $\Omega(n \log n)$ time.

Spherical depth is linearly bounded by simplicial depth, with experimental evidence suggesting a stronger bound.

Abstract

For a distribution function $F$ on $\mathbb{R}^d$ and a point $q\in \mathbb{R}^d$, the \emph{spherical depth} $\SphD(q;F)$ is defined to be the probability that a point $q$ is contained inside a random closed hyper-ball obtained from a pair of points from $F$. The spherical depth $\SphD(q;S)$ is also defined for an arbitrary data set $S\subseteq \mathbb{R}^d$ and $q\in \mathbb{R}^d$. This definition is based on counting all of the closed hyper-balls, obtained from pairs of points in $S$, that contain $q$. The significant advantage of using the spherical depth in multivariate data analysis is related to its complexity of computation. Unlike most other data depths, the time complexity of the spherical depth grows linearly rather than exponentially in the dimension $d$. The straightforward algorithm for computing the spherical depth in dimension $d$ takes $O(dn^2)$. The main result of this paper is an optimal algorithm that we present for computing the bivariate spherical depth. The algorithm takes $O(n \log n)$ time. By reducing the problem of \textit{Element Uniqueness}, we prove that computing the spherical depth requires $\Omega(n \log n)$ time. Some geometric properties of spherical depth are also investigated in this paper. These properties indicate that \emph{simplicial depth} ($\SD$) (Liu, 1990) is linearly bounded by spherical depth (in particular, $\SphD\geq \frac{2}{3}SD$). To illustrate this relationship between the spherical depth and the simplicial depth, some experimental results are provided. The obtained experimental bound ($\SphD\geq 2\SD$) indicates that, perhaps, a stronger theoretical bound can be achieved.