A parameterization of anisotropic Gaussian fields with penalized complexity priors

TL;DR

This paper introduces a new parameterization and penalized complexity priors for anisotropic Gaussian fields, enhancing Bayesian spatial modeling by better controlling covariance structures.

Contribution

It develops a smooth, invertible parameterization of anisotropic Gaussian fields and constructs weakly informative PC priors for these parameters.

Findings

The parameterization allows flexible modeling of anisotropic GFs.

The PC priors effectively penalize complexity, favoring simpler models.

The approach improves covariance estimation in Bayesian spatial analysis.

Abstract

Gaussian random fields (GFs) are fundamental tools in spatial modeling and can be represented flexibly and efficiently as solutions to stochastic partial differential equations (SPDEs). The SPDEs depend on specific parameters, which enforce various field behaviors and can be estimated using Bayesian inference. However, even under in-fill asymptotics, the likelihood only provides limited insights into the covariance structure. In response, it is essential to leverage priors to achieve appropriate, meaningful covariance structures in the posterior. This study introduces a smooth, invertible parameterization of the correlation length and diffusion matrix of an anisotropic GF and constructs penalized complexity (PC) priors for the model when the parameters are constant in space. The formulated prior is weakly informative, effectively penalizing complexity by pushing the correlation range toward infinity and the anisotropy to zero.