  Spatial generalized linear mixed models (SGLMMs) are popular and flexible
models for non-Gaussian spatial data. They are useful for spatial
interpolations as well as for fitting regression models that account for
spatial dependence, and are commonly used in many disciplines such as
epidemiology, atmospheric science, and sociology. Inference for SGLMMs is
typically carried out under the Bayesian framework at least in part because
computational issues make maximum likelihood estimation challenging, especially
when high-dimensional spatial data are involved. Here we provide a
computationally efficient projection-based maximum likelihood approach and two
computationally efficient algorithms for routinely fitting SGLMMs. The two
algorithms proposed are both variants of expectation maximization algorithm,
using either Markov chain Monte Carlo or a Laplace approximation for the
conditional expectation. Our methodology is general and applies to both
discrete-domain (Gaussian Markov random field) as well as continuous-domain
(Gaussian process) spatial models. We show, via simulation and real data
applications, that our methods perform well both in terms of parameter
estimation as well as prediction. Crucially, our methodology is computationally
efficient and scales well with the size of the data and is applicable to
problems where maximum likelihood estimation was previously infeasible.
