Multivariate spatiotemporal CDFs with random effects and measurement error

Spatial cumulative distributions (SCDFs) are useful in environmental applications - for example, by helping assess the fraction of a region exposed to harmful pollutants. Data sets containing the requisite spatial information often contain temporal data as well. We therefore extend the notion of an SCDF to a spatiotemporal cumulative distribution function (STCDF), with the goal of increasing precision by making use of repeated measurements. Ours is a hierarchical Bayesian approach, with estimation carried out by Markov chain Monte Carlo (MCMC) methods. We develop linear algebra results and corresponding computational techniques to handle the difficulties in evaluating the likelihood wrought by the large data sets (due to the added temporal component), the inclusion of spatial and temporal random effects, the need to account for measurement error, and the handling of missing data. We illustrate the concepts in a univariate setting with an Atlanta ozone data set, and in a bivariate (two pollutant) setting with a California NO/NO ₂ data set.