Efficient Computation of Gaussian Likelihoods for Stationary Markov Random Field Models

We introduce methods for efficiently computing the Gaussian likelihood for stationary Markov random field models, when the data locations fall on a possibly incomplete regular grid. The calculations rely on the availability of the covariances, which we show can be computed to any user-specified accuracy with fast Fourier transform algorithms. Several methods are presented, covering models with and without additive error terms and situations where either conserving memory or reducing computation time are favored, and some of the algorithms are easily parallelized. The examples presented highlight frequentist inference, but access to the likelihood allows for Bayesian inference as well. We demonstrate our results in simulation and timing studies and with an application to gridded satellite data, where we use the likelihood both for parameter estimation and likelihood ratio model comparison. In the data analysis, stochastic partial differential equation approximations are outperformed by an independent block approximation.