Basis-Function Models in Spatial Statistics

TL;DR

This paper reviews the use of basis-function models in spatial statistics, highlighting their flexibility and computational efficiency for modeling complex spatial dependencies across various applications.

Contribution

It provides a comprehensive overview of basis-function models, illustrating their application in diverse spatial data contexts and discussing available software tools.

Findings

Basis-function models effectively handle non-stationary spatial processes.

They are widely applicable in geophysics and other fields.

Software implementations facilitate practical adoption.

Abstract

Spatial statistics is concerned with the analysis of data that have spatial locations associated with them, and those locations are used to model statistical dependence between the data. The spatial data are treated as a single realisation from a probability model that encodes the dependence through both fixed effects and random effects, where randomness is manifest in the underlying spatial process and in the noisy, incomplete, measurement process. The focus of this review article is on the use of basis functions to provide an extremely flexible and computationally efficient way to model spatial processes that are possibly highly non-stationary. Several examples of basis-function models are provided to illustrate how they are used in Gaussian, non-Gaussian, multivariate, and spatio-temporal settings, with applications in geophysics. Our aim is to emphasise the versatility of these spatial statistical models and to demonstrate that they are now centre-stage in a number of application domains. The review concludes with a discussion and illustration of software currently available to fit spatial-basis-function models and implement spatial-statistical prediction.