Approximate Data Depth Revisited

Halfspace depth and \(\beta\)-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 \(\beta \geq 1\), the \(\beta\)-skeleton depth of \(q\) with respect to \(S\) is defined to be the total number of \emph{\(\beta\)-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 \(\beta\)-skeleton depth introduces a family of depth functions that contain \emph{spherical depth} and \emph{lens depth} if \(\beta=1\) and \(\beta=2\), respectively. The main results of this paper include approximating the planar halfspace depth and \(\beta\)-skeleton depth using two different approximation methods. First, the halfspace depth is approximated by the \(\beta\)-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 \(\beta\)-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 \(\beta\)-skeleton depth is reduced to a combination of some range counting problems. Using existing results on range counting approximations, the planar \(\beta\)-skeleton depth of a query point is approximated in \(O(n\;poly(1/\varepsilon,\log n))\), \(\beta\geq 1\). Regarding the \(\beta\)-skeleton depth functions, it is also proved that this family of depth functions converge when \(\beta \to \infty\). Finally, some experimental results are provided to support the proposed method of approximation and convergence of \(\beta\)-skeleton depth functions.