  Advances in Geographical Information Systems (GIS) have led to the enormous
recent burgeoning of spatial-temporal databases and associated statistical
modeling. Here we depart from the rather rich literature in space-time modeling
by considering the setting where space is discrete (e.g., aggregated data over
regions), but time is continuous. Our major objective in this application is to
carry out inference on gradients of a temporal process in our data set of
monthly county level asthma hospitalization rates in the state of California,
while at the same time accounting for spatial similarities of the temporal
process across neighboring counties. Use of continuous time models here allows
inference at a finer resolution than at which the data are sampled. Rather than
use parametric forms to model time, we opt for a more flexible stochastic
process embedded within a dynamic Markov random field framework. Through the
matrix-valued covariance function we can ensure that the temporal process
realizations are mean square differentiable, and may thus carry out inference
on temporal gradients in a posterior predictive fashion. We use this approach
to evaluate temporal gradients where we are concerned with temporal changes in
the residual and fitted rate curves after accounting for seasonality,
spatiotemporal ozone levels and several spatially-resolved important
sociodemographic covariates.
