  We describe spatio-temporal random processes using linear mixed models. We
show how many commonly used models can be viewed as special cases of this
general framework and pay close attention to models with separable or
product-sum covariances. The proposed linear mixed model formulation
facilitates the implementation of a novel algorithm using Stegle
eigendecompositions, a recursive application of the Sherman-Morrison-Woodbury
formula, and Helmert-Wolf blocking to efficiently invert separable and
product-sum covariance matrices, even when every spatial location is not
observed at every time point. We show our algorithm provides noticeable
improvements over the standard Cholesky decomposition approach. Via
simulations, we assess the performance of the separable and product-sum
covariances and identify scenarios where separable covariances are noticeably
inferior to product-sum covariances. We also compare likelihood-based and
semivariogram-based estimation and discuss benefits and drawbacks of both. We
use the proposed approach to analyze daily maximum temperature data in Oregon,
USA, during the 2019 summer. We end by offering guidelines for choosing among
these covariances and estimation methods based on properties of observed data.
