Details
Date: Feb 06, 2026
Location: Storrs, CT - USA
Abstract
Many modern spatial analyses involve observations recorded over spatial units of differing locations, sizes, and geometries, creating fundamental challenges for modeling dependence across spatial supports. Existing approaches model dependence by treating observations at different supports as aggregations of a common underlying continuous process, which leads to substantial computational and theoretical limitations. An alternative strategy avoids aggregation by defining isotropic stochastic processes directly on collections of spatial sets. However, ensuring positive definiteness for the resulting covariance functions remains a challenge. To resolve this, we introduce a framework for isotropic set-indexed random fields based on the ball–Hausdorff distance, defined as the Hausdorff distance between the minimum enclosing balls of bounded sets. We derive an explicit formula for this distance in terms of centers and radii. We show that broad classes of isotropic covariance functions, such as the Matérn, yields valid covariance functions when coupled with the proposed distance. Our construction reduces complex set-to-set dependence to low-dimensional geometric summaries, resulting in significant computational simplifications. We conclude with a surface temperature case study, fusing satellite and in situ measurements to reproduce high-resolution fields comparable to a gold-standard physical model.