Approximate Data Depth Revisited
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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 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 spherical depth and 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 fitting function and 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/ε, n)), β≥ 1. Regarding the β-skeleton depth functions, it is also proved that this family of depth functions converge when β→∞. Finally, some experimental results are provided to support the proposed method of approximation and convergence of β-skeleton depth functions.
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