Random Marked Sets

TL;DR

This paper introduces a new class of stochastic processes called random marked sets, linking random fields and marked point processes, highlighting their covariance properties and implications for geostatistics.

Contribution

It defines a novel class of processes on random sets, analyzing their covariance functions and identifying cases with positive definiteness based on Gaussian fields.

Findings

Mark covariance function of random marked sets is generally not positive definite.

For Gaussian-based processes, the mark covariance and correlation functions are positive definite.

Implications for geostatistical methods are discussed.

Abstract

We aim to link random fields and marked point processes and therefore introduce a new class of stochastic processes which are defined on a random set in R^d. Unlike for random fields, the mark covariance function of a marked random set is in general not positive definite. This implies that in many situations the use of simple geostatistical methods appears to be questionable. Surprisingly, for a special class of processes based on Gaussian random fields, we do have positive definiteness for the corresponding mark covariance function and mark correlation function.