Likelihood approximations for big nonstationary spatial temporal lattice data

We propose a nonstationary Gaussian likelihood approximation for the class of evolutionary spectral models for data on a regular lattice. Lattice data include many important environmental data sources such as weather model output or gridded data products derived from satellite observations. The likelihood approximation is an extension of the Whittle likelihood and is computationally efficient to evaluate when the evolutionary transfer function can be expressed in a flexible low-dimensional form. The low-dimensional form for the evolutionary transfer function is an attractive modeling framework since it allows the practitioner to build nonstationary models in a sequential manner and choose the appropriate dimension based on changes in approximate loglikelihood. While the transfer functions are low-dimensional, the resulting covariance matrices are generally full rank, and thus no rank reduction is required for the computational efficiency of the methods. We study the covariance matrix implied by the likelihood approximation and give its asymptotic rate of approximation to the exact covariance matrix. We evaluate the likelihood approximation in a simulation study and show that it can produce asymptotically efficient parameter estimates when an operation similar to tapering is applied. We introduce an algorithm based on the Ising model to partition the domain into stationary subregions and show in a simulation that the methods can reliably recover an unknown partition. We apply our modeling and estimation framework to analyze spatial-temporal output from a regional weather model comprised of 151,200 wind speed values, and we demonstrate that the fitted covariances are consistent with local empirical variograms.