Rational approximation and intrinsic Gaussian processes

TL;DR

This paper introduces a systematic framework for intrinsic Gaussian processes, leveraging rational approximation to improve modeling, inference, and computation, thus enhancing robustness and interpretability in spatial data analysis.

Contribution

It develops practical algorithms and dependence models for intrinsic GPs, bridging the gap with traditional approaches and enabling more efficient spatial modeling.

Findings

New algorithms for intrinsic GPs improve computational efficiency.

Rational approximation clarifies the structure of intrinsic GP models.

Numerical examples demonstrate robustness and interpretability benefits.

Abstract

Gaussian processes (GPs) defined through intrinsic random fields provide a flexible framework for modeling spatial phenomena, and have been advocated in a variety of applications over the past several decades. Nevertheless, their adoption has lagged behind traditional, covariance-based approaches, in part because the intrinsic formulation has lacked an accompanying toolkit of computational methods and dependence specifications that facilitate fitting and prediction. We develop here a systematic framework for modeling intrinsic GPs and introduce practical algorithms and dependence/variogram models for modeling, inference and computation that parallel those of traditional, stationary GPs. We explore a close connection between intrinsic GP models and rational approximation, which clarifies the underlying problem structure. Numerical examples illustrate how the new tools can be deployed in practice, highlighting the advantages of intrinsic-field modeling in terms of robustness, interpretability, and computational efficiency.