A linear mixed model formulation for spatio-temporal random processes with computational advances for the product, sum, and product–sum covariance functions

To properly characterize a spatio-temporal random process, it is necessary to understand the process’ dependence structure. It common describe this using single error having complicated covariance. Instead of approach, we processes linear mixed models several errors; each describes specific quality This model formulation general, intuitive, and contains many commonly used covariance functions as special cases. We focus on express three functions: product (separable), sum (linear), product–sum. discuss benefits drawbacks function propose novel algorithms Stegle eigendecompositions, recursive application Sherman–Morrison–Woodbury formula, Helmert–Wolf blocking efficiently invert their matrices, even when every spatial location not observed at time point. Via simulation study an analysis temperature data in Oregon, USA, assess performance computational efficiency these estimated restricted maximum likelihood (likelihood-based) Cressie’s weighted least squares (semivariogram-based). end by offering guidelines for choosing among combinations estimation methods based properties desired balance between efficiency.