Kriging and Cross-validation for Massive Spatial Data

Spatial prediction such as kriging involves the inversion of a covariance matrix. When the number of locations is very large as in many studies, inversion of the covariance matrix may not be practical. Covariance tapering, predictive process models and low rank kriging are some methods for overcoming the large matrix problem, all of which can be regarded as approximations to the underlying stationary process. Therefore, efficient cross-validation is very helpful for spatial prediction with large data to assess how well an approximation works. This work studies the calculation of drop-one prediction and various prediction scores. Estimators are constructed to minimize some prediction scores. One advantage of this approach is that it integrates estimation and cross-validation and does not treat them as two separate procedures. Further simplification of calculation is studied that is based on the infill asymptotic theory. The methods are illustrated through the analysis of a US precipitation dataset.