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Abstract:
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Spatio-temporal data sets can be very large. Prediction (Kriging) relies on the inversion of an n by n matrix, which grows computationally at a cubic rate as the sample size, n, increases. Additionally, covariance estimation methods which rely on iterative inversion of covariance matrices, such as Restricted Maximum Likelihood (REML), are infeasible if the covariance matrix is too large. We describe the product-sum spatio-temporal model in terms of linear mixed models with spatial and temporal random effects. This model is in general nonseparable, but contains a separable model as a special case. We show how the additive structure of the product-sum model allows for recursive use of the Sherman-Morrison-Woodbury identity to greatly increase the computational speed associated with inverting the covariance matrix. We also show how results from partitioned matrices can be used to accommodate some missing values. We compare the product-sum model to other spatio-temporal methods with simulated data and give computational benchmarks. We then apply the methods to a temperature data set to estimate fixed effects and predict in space and time.
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