Predictive spatio-temporal models for spatially sparse environmental data

We present a family of spatio-temporal models which are geared to pro- vide time-forward predictions in environmental applications where data is spatially sparse but temporally rich. That is measurements are made at few spatial locations (stations), but at many regular time intervals. When predictions in the time di- rection 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 of 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 univari- ate 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. A simulation study illustrates the gain re- sulting from our modeling strategy. Two environmental data sets are studied. In particular, we find evidence that a parametric modeling of the spatio-temporal cor- relation 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. In this article we present a model building strategy designed to work within a family of vector autoregressive models for time series being recorded at specific spatial locations. The methodology has been developed with environmental ap- plications in mind where measurements on a variable are made at regular time intervals and at several stations located within a specific area. More specifically, we focus on situations were measurements are available at a few stations −the spatio-temporal data is sparse in space but rich in time. We put the discussion into concrete form with two examples treated previously in the literature. The first data set we consider consists of average daily wind speeds mea- sured at 11 synoptic meteorological stations located in Ireland during the period 1961-78, 6,570 observations per location. Gneiting (2002) used this data set to

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