On Elliptical Possibility Distributions

Abstract

This paper aims to propose two main contributions in the field of multivariate data analysis through the possibility theory. The first proposition is the definition of a generalized family of multivariate elliptical possibility distributions. These distributions have been derived from a consistent probability-possibility transformation over the family of so-called elliptical probability distributions. The second contribution proposed by this paper is the definition of two divergence measures between possibilistic distributions. We prove that a symmetric version of the Kullback-Leibler divergence guarantees all divergence properties when related to the space of possibility distributions. We further derive analytical expressions of the latter divergence and of the Hellinger divergence for certain possibility distributions pertaining to the elliptical family proposed, especially the normal multivariate possibility divergence in two dimensions. Finally, this paper provides an illustration of the developed possibilistic tools in an application of bi-band change detection between optical satellite images.