Lucas da Cunha Godoy | Searcharxiv

arXiv2026

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

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 Matérn 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.

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arXiv2025

Statistical Inferences and Predictions for Areal Data and Spatial Data Fusion with Hausdorff--Gaussian Processes

Accurate modeling of spatial dependence is pivotal in analyzing spatial data, influencing parameter estimation and predictions. The spatial structure of the data significantly impacts valid statistical inference. Existing models for areal data often rely on adjacency matrices, struggling to differentiate between polygons of varying sizes and shapes. Conversely, data fusion models rely on computationally intensive numerical integrals, presenting challenges for moderately large datasets. In response to these issues, we propose the Hausdorff-Gaussian process (HGP), a versatile model utilizing the Hausdorff distance to capture spatial dependence in both point and areal data. Integration into generalized linear mixed-effects models enhances its applicability, particularly in addressing data fusion challenges. We validate our approach through a comprehensive simulation study and application to two real-world scenarios: one involving areal data and another demonstrating its effectiveness in data fusion. The results suggest that the HGP is competitive with specialized models regarding goodness-of-fit and prediction performances. In summary, the HGP offers a flexible and robust solution for modeling spatial data of various types and shapes, with potential applications spanning fields such as public health and climate science.

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