The SPDE approach for Gaussian and non-Gaussian fields: 10 years and still running

TL;DR

This paper reviews a decade of the SPDE approach for modeling Gaussian and non-Gaussian spatial and spatio-temporal fields, highlighting its connections, extensions, and practical applications.

Contribution

It provides a comprehensive overview of the SPDE methodology, including recent developments, theoretical insights, and computational strategies for diverse types of random fields.

Findings

Unified framework connecting SPDEs with various dependence structures

Extensions to non-Gaussian, non-stationary, and manifold-based fields

Practical computational methods for large-scale spatial modeling

Abstract

Gaussian processes and random fields have a long history, covering multiple approaches to representing spatial and spatio-temporal dependence structures, such as covariance functions, spectral representations, reproducing kernel Hilbert spaces, and graph based models. This article describes how the stochastic partial differential equation approach to generalising Mat\'ern covariance models via Hilbert space projections connects with several of these approaches, with each connection being useful in different situations. In addition to an overview of the main ideas, some important extensions, theory, applications, and other recent developments are discussed. The methods include both Markovian and non-Markovian models, non-Gaussian random fields, non-stationary fields and space-time fields on arbitrary manifolds, and practical computational considerations.