Obtaining spatial predictions by kriging is a common approach in geostatistics. This is usually accomplished by assuming a Gaussian random field (GRF), estimating covariance parameters by maximum likelihood estimation, and using the kriging equation to obtain predictions. For massive data sets, kriging becomes computationally intensive, both in terms of CPU time and memory, and this burden is even more restrictive for multivariate data. Cressie and Johannesson (2008) proposed fixed rank kriging as a solution, with maximum likelihood estimation of the covariance parameters later addressed by Katzfuss and Cressie (2011b). The disadvantage to this method is that accuracy in prediction is bounded by the predetermined fixed components of the model. We propose two methods that utilize the Spatial Random Effects (SRE) model of Cressie and Johannesson (2008), but allow for estimation of the fixed components. In the first method called Reduced Basis Kriging, we use restricted maximum likelihood estimation and sparse matrix methodology to obtain additional gains in computational efficiency without loss of accuracy in prediction. Reduced Basis Kriging does require additional model assumptions, therefore the alternating expectation conditional maximization (AECM) algorithm is suggested as a second method which maintains a very flexible covariance structure and provides estimation of the fixed components. These methods are then extended to handle multivariate data for either a large sample size or a large number of response variables. Unlike previous methods of efficient cokriging, this methodology does not require that observations are recorded at the same locations. Experiments show that our methodology can provide a consistent improvement in accuracy while minimizing the additional computational burden of extra parameter estimation. The methodology is extended to climate data archived by the National Climate Data Center.
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