Conditionally specified space-time models for multivariate processes

We propose a class of conditionally specified models for the analysis of multivariate space-time processes. Such models are useful in situations where there is sparse spatial coverage of one of the processes and much more dense coverage of the other process(es). The dependence structure across processes and over space, and time is completely specified through a neighborhood structure. These models are applicable to both point and block sources; for example, multiple pollutant monitors (point sources) or several county level exposures (block sources). We introduce several computational tricks which are integral for model fitting, give some simple sufficient and necessary conditions for the space-time covariance matrix to be positive definite, and implement a Gibbs sampler, using Hybrid MC steps, to sample from the posterior distribution of the parameters. Model fit is assessed via the DIC. Predictive accuracy, both over time and space, is assessed both relatively and absolutely via mean squared prediction error and coverage probabilities. As an illustration of these models, we fit them to particulate matter and ozone data collected in the LA area in 1995 over a 3 month period. In this data, the spatial coverage of particulate matter was sparse relative to that of ozone.