Bayesian Multivariate Spatial Interpolation with Data Missing by Design

In a network of s _g sites, responses like levels of airborne pollutant concentrations may be monitored over time. The sites need not all measure the same set of response items and unmeasured items are considered as data missing by design . We propose a hierarchical Bayesian approach to interpolate the levels of, say, k responses at s _u other locations called ungauged sites and also the unmeasured levels of the k responses at the gauged sites. Our method involves two steps. First, when all hyperparameters are assumed to be known, a predictive distribution is derived. In turn, an interpolator, its variance and a simultaneous interpolation region are obtained. In step two, we propose the use of an empirical Bayesian approach to estimate the hyperparameters through an EM algorithm. We base our theory on a linear Gaussian model and the relationship between a multivariate normal and matrix T -distribution. Our theory allows us to pool data from several existing networks that measure different subsets of response items for interpolation.