Approximate data depth revisited

Halfspace depth and β-skeleton depth are two types of depth functions in nonparametric data analysis. The halfspace depth of a query point q ∈ ℝ^d with respect to S ⊂ ℝ^d is the minimum portion of the elements of S which are contained in a halfspace which passes through q. For β ≥ 1, the β-skeleton depth of q with respect to S is defined to be the total number of β-skeleton influence regions that contain q, where each of these influence regions …