  Discrete time spatial time series data arise routinely in meteorological and
environmental studies. Inference and prediction associated with them are mostly
carried out using any of the several variants of the linear state space model
that are collectively called linear dynamic spatio-temporal models (LDSTMs).
However, real world environmental processes are highly complex and are seldom
representable by models with such simple linear structure. Hence, nonlinear
dynamic spatio-temporal models (NLDSTMs) based on the idea of nonlinear
observational and evolutionary equation have been proposed as an alternative.
However, in that case, the caveat lies in selecting the specific form of
nonlinearity from a large class of potentially appropriate nonlinear functions.
Moreover, modeling by NLDSTMs requires precise knowledge about the dynamics
underlying the data. In this article, we address this problem by introducing
the Gaussian random functional dynamic spatio-temporal model (GRFDSTM). Unlike
the LDSTMs or NLDSTMs, in GRFDSTM both the functions governing the
observational and evolutionary equations are composed of Gaussian random
functions. We exhibit many interesting theoretical properties of the GRFDSTM
and demonstrate how model fitting and prediction can be carried out coherently
in a Bayesian framework. We also conduct an extensive simulation study and
apply our model to a real, SO2 pollution data over Europe. The results are
highly encouraging.
