Covariance-Based Rational Approximations of Fractional SPDEs for Computationally Efficient Bayesian Inference

The stochastic partial differential equation (SPDE) approach is widely used for modeling large spatial datasets. It is based on representing a Gaussian random field u on R-d as the solution of an elliptic SPDE L(beta)u = W where L is a second- order differential operator, 2 beta is an element of N is a positive parameter that controls the smoothness of u and W is Gaussian white noise. A few approaches have been suggested in the literature to extend the approach to allow for any smoothness parameter satisfying beta > d/4. Even though those approaches work well for simulating SPDEs with general smoothness, they are less suitable for Bayesian inference since they do not provide approximations which are Gaussian Markov random fields (GMRFs) as in the original SPDE approach. We address this issue by proposing a new method based on approximating the covariance operator L-2 beta of the Gaussian field u by a finite element method combined with a rational approximation of the fractional power. This results in a numerically stable GMRF approximation which can be combined with the integrated nested Laplace approximation (INLA) method for fast Bayesian inference. A rigorous convergence analysis of themethod is performed and the accuracy of themethod is investigated with simulated data. Finally, we illustrate the approach and corresponding implementation in the R package rSPDE via an application to precipitation data which is analyzed by combining the rSPDE package with the R-INLA software for full Bayesian inference. Supplementary materials for this article are available online.

Automated summary

The stochastic partial differential equation (SPDE) approach is widely used for modeling large spatial datasets. A new method is proposed for Bayesian inference using a stable Gaussian Markov random fields (GMRF) approximation, which approximates the covariance operator of the Gaussian field by a finite element method combined with a rational approximation of the fractional power. The method is rigorously analyzed for convergence and the accuracy is investigated with simulated data.