Predictive spatio-temporal models for spatially sparse environmental data

We present a family of spatio-temporal models which are geared to provide time-forward predictions in environmental applications where data is spatially sparse but temporally dense. That is measurements are made at few spatial locations (stations), but at many regular time intervals. When predictions in the time direction is the purpose of the analysis, then spatial-stationarity assumptions which are commonly used in spatial modeling, are not necessary. The family of models proposed does not make such assumptions and consists in a vector autoregressive (VAR) specification, where there are as many time series as stations. However, by taking into account the spatial dependence structure, a model building strategy is introduced, which borrows its simplicity from the Box-Jenkins strategy for univariate autoregressive (AR) models for time series. As for AR models, model building may be performed either by displaying sample partial correlation functions, or by minimizing an information criterion. Two environmental data sets are studied. In particular, we find evidence that a parametric modeling of the spatio-temporal correlation function is not appropriate because it rests on too strong assumptions. Moreover, we propose to compare model selection strategies with an out-of-sample validation method based on recursive prediction errors.