Lipschitz-Killing Curvatures of the Excursion Sets of Skew Student's t Random Fields

Abstract

In many applications related with geostatistics, biological and medical imaging, material science, and engineering surfaces, the real observations have asymmetric and heavy-tailed multivariate distributions. These observations are spatially correlated and they could be modeled by the skew random fields. However, several statistical analysis problems require studying the integral geometry of these random fields in order to detect the local changes between the different samples, via the expected Euler-Poincaré characteristic and the intrinsic volumes (Lipschitz-Killing curvatures) of the excursion set obtained at a given threshold. This article is interested in a class of skew random fields, namely, skew Student's t random field. The goal is to derive an explicit formula of Lipschitz-Killing curvatures and the expected Euler-Poincaré characteristic of the skew Student's t excursion sets on a compact subset S of ℝ2 by extending previous results reported in the literature. The motivation comes from the need to model the roughness of some engineering surfaces in order to detect the local changes of the surface peaks/valleys during a specific physical phenomenon. The analytical and empirical Euler-Poincaré characteristics are compared in order to test the skew Student's t random field on the real surface. Simulation results are also presented in the article for illustration and validation.