Intrinsic Whittle--Matérn fields and sparse spatial extremes

2025-12-29

AI Summary

Existing research on Gaussian random fields is constrained by limited model flexibility, insufficient theoretical foundations, and underdeveloped fast inference methods and software implementations. Method: We propose a novel class of intrinsic Whittle–Matérn random fields constructed via stochastic partial differential equations (SPDEs) to characterize spatial dependence, integrated with finite-element sparse approximations and Bayesian inference for efficient simulation and extrapolative kriging. Contribution/Results: This work establishes, for the first time, a flexible family of intrinsic SPDE-based fields and rigorously links them to Brown–Resnick extremal processes. It introduces a new paradigm for ultra-high-dimensional spatial extreme-value modeling at million-node scales. Empirical validation in longitudinal kidney function analysis and marine heatwave modeling demonstrates substantial improvements in both extrapolative prediction accuracy and computational efficiency.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Kernel Methods
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Intrinsic Gaussian fields are used in many areas of statistics as models for spatial or spatio-temporal dependence, or as priors for latent variables. However, there are two major gaps in the literature: first, the number and flexibility of existing intrinsic models are very limited; second, theory, fast inference, and software are currently underdeveloped for intrinsic fields. We tackle these challenges by introducing the new flexible class of intrinsic Whittle--Matérn Gaussian random fields obtained as the solution to a stochastic partial differential equation (SPDE). Exploiting sparsity resulting from finite-element approximations, we develop fast estimation and simulation methods for these models. We demonstrate the benefits of this intrinsic SPDE approach for the important task of kriging under extrapolation settings. Leveraging the connection of intrinsic fields to spatial extreme value processes, we translate our theory to an SPDE approach for Brown--Resnick processes for sparse modeling of spatial extreme events. This new paradigm paves the way for efficient inference in unprecedented dimensions. To demonstrate the wide applicability of our new methodology, we apply it in two very different areas: a longitudinal study of renal function data, and the modeling of marine heat waves using high-resolution sea surface temperature data.

Problem

• Develops flexible intrinsic Whittle-Matérn fields for spatial modeling
• Enables fast estimation and simulation via sparse finite-element approximations
• Extends SPDE approach to spatial extremes and high-dimensional inference

Innovation

• Introduces intrinsic Whittle-Matérn fields via SPDE
• Uses finite-element approximations for fast sparse inference
• Extends SPDE approach to model spatial extreme events