A fully data-driven method for estimating the shape of a point cloud

Department of Statistics and Operations Research, University of Santiago de Compostela, Spain.
paula.saavedra@usc.es

Received: 11 August 2015
Revised: 11 April 2016
Accepted: 23 May 2016

Abstract

Given a random sample of points from some unknown distribution, we propose a new data-driven method for estimating its probability support S. Under the mild assumption that S is r-convex, the smallest r-convex set which contains the sample points is the natural estimator. The main problem for using this estimator in practice is that r is an unknown geometric characteristic of the set S. A stochastic algorithm is proposed for selecting its optimal value from the data under the hypothesis that the sample is uniformly generated. The new data-driven reconstruction of S is able to achieve the same convergence rates as the convex hull for estimating convex sets, but under a much more flexible smoothness shape condition.