A general class of mosaic random fields

TL;DR

This paper introduces a versatile framework for mosaic random fields that unifies and extends existing models across different spaces, providing explicit correlation functions and an efficient simulation method.

Contribution

It presents a general class of mosaic random fields that unifies well-known models and extends them to new spaces and distributions, with exact simulation procedures.

Findings

Unified model encompasses Poisson hyperplane, token, and dead leaves models.

Extended models to spaces like the sphere and arbitrary discrete distributions.

Derived explicit and new correlation functions for these models.

Abstract

We present a model of a random field on a topological space $M$ that unifies well-known models such as the Poisson hyperplane tessellation model, the random token model, and the dead leaves model. In addition to generalizing these submodels from $\mathbb{R}^d$ to other spaces such as the $d$-dimensional unit sphere $\mathbb{S}^d$, our construction also extends the classical models themselves, e.g. by replacing the Poisson distribution by an arbitrary discrete distribution. Moreover, the method of construction directly produces an exact and fast simulation procedure. By investigating the covariance structure of the general model we recover various explicit correlation functions on $\mathbb{R}^d$ and $\mathbb{S}^d$ and obtain several new ones.