On the Validity of Isotropic Covariance Functions for Set-indexed Random Fields

TL;DR

This paper introduces the ball--Hausdorff distance for set-indexed random fields, ensuring valid isotropic covariance functions by leveraging conditionally negative definite metrics, simplifying dependence modeling.

Contribution

It proposes the ball--Hausdorff distance, a new set distance that guarantees valid isotropic covariance functions for set-indexed random fields, extending the applicability of common covariance models.

Findings

Ball--Hausdorff distance is conditionally negative definite for length spaces.

Guarantees validity of Matérn and powered exponential covariance functions.

Simplifies covariance evaluation through low-dimensional geometric summaries.

Abstract

Distances between sets arise naturally when modeling stochastic dependence on collections of spatial supports, including settings with point-referenced and areal observations. However, commonly used constructions of distances on sets, including those derived from the Hausdorff distance, generally fail to be conditionally negative definite, precluding their use in isotropic covariance models. We propose the ball--Hausdorff distance, defined as the Hausdorff distance between the minimum enclosing balls of bounded sets in a metric space. For length spaces, we derive an explicit representation of this distance in terms of the associated centers and radii. We show that the ball--Hausdorff distance is conditionally negative definite whenever the underlying metric is conditionally negative definite. By Schoenberg's theorem, this implies an isometric embedding into a Hilbert space and guarantees the validity of broad classes of isotropic covariance functions, including the Mat\'ern and powered exponential families, for set-indexed random fields. The construction reduces dependence between sets to low-dimensional geometric summaries, leading to substantial simplifications in covariance evaluation.