Bayesian spatial prediction of random space-time fields with application to mapping PM2.5 exposure

This article presents a multivariate spatial prediction methodology in a Bayesian framework, The method is especially suited for use in environmetrics, where vector-valued responses are observed at a small set of ambient monitoring stations ''(gauged sites)'' at successive time points. However, the stations may have varying start-up times so that the data have a ''staircase'' pattern (''monotone'' pattern in the terminology of Rubin and Shaffer). The lowest step corresponds to the newest station in the monitoring network. We base our approach on a hierarchical Bayes prior involving a Gaussian generalized inverted Wishart model. For given hyperparameters, we derive the predictive distribution for currently gauged sites at times before their start-up when no measurements were taken. The resulting, predictive distribution is a matric t distribution with appropriate covariance parameters and degrees of freedom. We estimate die hyperparameters using the method of moments (MOM) as an easy-to-implement alternative to the more complex EM algorithm. The MOM in particular gives exact parameter estimates and involves less cumbersome calculations than the EM algorithm. Finally, we obtain the predictive distribution for unmeasured responses at ''ungauged'' sites. The results obtained here allow us to pool the data from different sites that measure different pollutants, and also to treat cases where the observed data monitoring stations have a monotonic ''staircase'' Meture. We demonstrate the use of this methodology by mapping PM., fields for Philadelphia during the period of May 1992 to September 1993. Large amounts of data missing by design make this application particularly challenging. We give empirical evidence that the method performs well.