Approximate Data Depth Revisited

Abstract

Halfspace depth and $β$-skeleton depth are two types of depth functions in nonparametric data analysis. The halfspace depth of a query point $q\in \mathbb{R}^d$ with respect to $S\subset\mathbb{R}^d$ is the minimum portion of the elements of $S$ which are contained in a halfspace which passes through $q$. For $β\geq 1$, the $β$-skeleton depth of $q$ with respect to $S$ is defined to be the total number of \emph{$β$-skeleton influence regions} that contain $q$, where each of these influence regions is the intersection of two hyperballs obtained from a pair of points in $S$. The $β$-skeleton depth introduces a family of depth functions that contain \emph{spherical depth} and \emph{lens depth} if $β=1$ and $β=2$, respectively. The main results of this paper include approximating the planar halfspace depth and $β$-skeleton depth using two different approximation methods. First, the halfspace depth is approximated by the $β$-skeleton depth values. For this method, two dissimilarity measures based on the concepts of \emph{fitting function} and \emph{Hamming distance} are defined to train the halfspace depth function by the $β$-skeleton depth values obtaining from a given data set. The goodness of this approximation is measured by a function of error values. Secondly, computing the planar $β$-skeleton depth is reduced to a combination of some range counting problems. Using existing results on range counting approximations, the planar $β$-skeleton depth of a query point is approximated in $O(n\;poly(1/\varepsilon,\log n))$, $β\geq 1$. Regarding the $β$-skeleton depth functions, it is also proved that this family of depth functions converge when $β\to \infty$. Finally, some experimental results are provided to support the proposed method of approximation and convergence of $β$-skeleton depth functions.