Robust estimation of distance between sets of points

Abstract

This paper proposes a new methodology for computing Hausdorff distances between sets of points in a robust way. In a first step, robust nearest neighbor distance distributions between the two sets of points are obtained by considering reliability measures in the computations through a Monte Carlo scheme. In a second step, the computed distributions are operated using random variables algebra in order to obtain probability distributions of the average, minimum or maximum distances. In the last step, different statistics are computed from these distributions. A statistical test of significance, the nearest neighbor index, in addition to the newly proposed divergence and clustering indices are used to compare the computed measurements with respect to values obtained by chance. Results on synthetic and real data show that the proposed method is more robust than the standard Hausdorff distance. In addition, unlike previously proposed methods based on thresholding, it is appropriate for problems that can be modeled through point processes.

Methodological analysis of personalization in urban recommender systems by distance measures

“…The only geospatial distance measure we encountered in the existing literature on URS's is the centroid distance, which measures the distance between the geographic centroids of two item sets (see Figure 1a; see Smets, Montero, and Ballon 2019). Within the context of URS's, however, this measure has several drawbacks because it completely ignores the geographic scatter of item sets and fails to spot differences in geographic patterns as well as geographic outliers (Moreno, Koppal, and de Muinck 2013).…”

Stochastic Distance Transform

“…A stochastic Monte Carlo-based method and a deterministic method are proposed for computing the SDT. A similar idea is explored in [16], where a robust method for estimating distributions of Hausdorff set distances between sets of points, based on random removal of the points in the observed sets, is proposed. In that work, the authors utilize DT only as a tool for estimation of the Hausdorff set distance by computing weighted distance histograms based on user-provided point-wise reliability coefficients, without exploring how these random sets can increase the robustness and accuracy of the DT itself.…”