Proximity graph depth, depth contours, and a new multimodal median

We propose proximity graph depth as a class of depth functions, based on the minimum path length along proximity graph edges to the convex hull of a point set. We analyze the characteristics of several proximity graph depth functions both theoretically and experimentally, define depth contours enclosing regions of increasing depth, and present algorithms for calculating depth values in a point set and depth contours. The contribution of this paper is in the novel approach for analyzing depth in multimodal data sets. Most existing depth function do not cope with multimodality or distributions with more then one center. We define seeds, the multimodal version of the depth median, as an estimator for the centers of the multimodal data sets and present experimental results that demonstrate that, unlike most depth functions, the proximity graph depth can indeed distinguish multimodal data sets.