BAYESIAN MULTIVARIATE SPATIAL INTERPOLATION WITH DATA MISSING BY DESIGN

In a network of s(g) sites, responses like levels of airborne pollutan t concentrations may be monitored over time. The sites need not all me asure the same set of response items and unmeasured items are consider ed as data missing by design. We propose a hierarchical Bayesian appro ach to interpolate the levels of, say, k responses at s(u) other locat ions called ungauged sites and also the unmeasured levels of the k res ponses at the gauged sites. Our method involves two steps. First, when all hyperparameters are assumed to be known, a predictive distributio n is derived. In turn, an interpolator, its variance and a simultaneou s interpolation region are obtained. in step two, we propose the use o f an empirical Bayesian approach to estimate the hyperparameters throu gh an EM algorithm. We base our theory on a linear Gaussian model and the relationship between a multivariate normal and matrix T-distributi on. Our theory allows us to pool data from several existing networks t hat measure different subsets of response items for interpolation.