A hierarchical approach to multivariate spatial modeling and prediction

We propose a hierarchical model for multivariate spatial modeling and prediction under which one specifies a joint distribution for a multivariate spatial process indirectly through specification of simpler conditional models. This approach is similar to standard methods known as cokriging and kriging with external drift,' but avoids some of the inherent difficulties in these two approaches including specification of valid joint covariance models and restriction to exhaustively sampled covariates. Moreover, both existing approaches can be formulated in this hierarchical framework. The hierarchical approach is ideally suited for, but not restricted for use in, situations in which known cause/effect' relationships exist. Because the hierarchical approach models dependence between variables in conditional means, as opposed to cross-covariances, very complicated relationships are more easily parameterized. We suggest an iterative estimation procedure that combines generalized least squares with imputation of missing values using the best linear unbiased predictor. An example is given that involves prediction of a daily ozone summary from maximum daily temperature in the Midwest.