Isotropic covariance functions for disparate spatial resolutions

TL;DR

This work introduces the ball-Hausdorff distance, a distance between bounded spatial sets defined via the Hausdorff distance between their minimum enclosing balls. Under length-space assumptions, it has a closed-form expression in terms of Chebyshev centers and radii, and if the base metric is conditionally negative definite, bh is CND, enabling isometric Hilbert-space embeddings and a broad class of valid isotropic covariances for set-indexed random fields. The authors show that common covariance families, including Matérn and powered exponential, remain valid after appropriate transformations, while achieving substantial computational gains by reducing set relationships to centers and radii. Numerical results demonstrate improved covariance validity and drastic speedups over standard Hausdorff-based approaches, especially on non-Euclidean domains like the sphere. The framework supports mixed-resolution spatial data problems (change of support, data fusion, misalignment) and points to future avenues in cross-covariance modeling and differentiability analysis on manifolds.

Abstract

Distances between sets arise naturally in data fusion problems involving both point-referenced and areal observations, as well as in set-indexed stochastic processes more broadly. 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, which implies, via Schoenberg's theorem, an isometric embedding into a Hilbert space. As a consequence, broad classes of isotropic covariance functions, including the Matérn and powered exponential families, are valid for random fields indexed by sets. The resulting construction reduces set-to-set dependence to low-dimensional geometric summaries, leading to substantial computational simplifications in covariance evaluation.

Figures (3)

• Figure 1: Synthetic spatial configurations used in the numerical examples. Rows correspond to the Euclidean plane $(\mathbb{R}^2)$ and the sphere $(\mathbb{S}^2)$, while columns represent gridded, fused, and areal settings. These examples are designed to span increasing spatial complexity across domains commonly encountered in multi-resolution spatial analyses.
• Figure 2: wo-dimensional kernel density estimates comparing pairwise Hausdorff distances (x-axis) and ball--Hausdorff distances (y-axis) across spatial configurations and domains. Columns correspond to gridded, fused, and areal settings, while rows represent the Euclidean plane and the sphere. Distances are rescaled to the unit interval within each scenario. Concentration of mass along the dashed $y = x$ line indicates close empirical agreement between the two distance measures across increasing spatial complexity.
• Figure 3: Eigenvalue diagnostics for covariance matrices constructed using the PEXP family with $\nu = 1$ (top row) and the Matérn family with $\nu = 2.5$ (bottom row). For each panel, the smallest eigenvalue $\lambda_1$ is plotted as a function of the practical range $\rho$ across spatial configurations (columns) and domains (rows). Solid red lines correspond to covariance matrices based on the ball-Hausdorff distance, while dashed blue lines correspond to those based on the Hausdorff distance. For the Matérn family, square-root transformed distances are used as required by Theorem \ref{['th:2']}. The gray shaded region indicates invalid (non-positive definite) covariance structures ($\lambda_1 < 0$).

Theorems & Definitions (16)

• Lemma 2.1: Linear expansion of balls
• Theorem 2.2
• Corollary 2.2.1
• Theorem 2.3
• proof
• Definition 3.1: Chebyshev center
• Definition 3.2
• Theorem 3.3
• proof
• Theorem 3.4