Towards a complete characterization of indicator variograms and madograms

TL;DR

This paper offers new necessary and sufficient conditions for characterizing indicator variograms and madograms, advancing the understanding of their properties and relationships in random fields using distance geometry and Gaussian theory.

Contribution

It provides the first complete characterization of indicator variograms and madograms, including conditions for their equivalence, based on distance geometry and Gaussian random field theory.

Findings

Derived necessary and sufficient conditions for indicator variograms.

Established conditions under which indicator variograms and madograms coincide.

Applied distance geometry and Gaussian theory to characterize these functions.

Abstract

Indicator variograms and madograms are structural tools used in many disciplines of the natural sciences and engineering to describe random sets and random fields. To date, several necessary conditions are known for a function to be a valid indicator variogram but, except for intractable corner-positive inequalities, a complete characterization of indicator variograms is missing. Likewise, only partial characterizations of madograms are known. This paper provides novel necessary and sufficient conditions for a given function to be the variogram of an indicator random field with constant mean value or to be the madogram of a random field, and establishes under which conditions these two families of functions coincide. Our results apply to any set of points where the random field is defined and rely on distance geometry and Gaussian random field theory.