Nonparametric prediction of nonstationary spatio-temporal processes

In spatial statistics often the response variable at a given location and time is observed together with some covariates which are known to influence the response. In several applications the relationship between the response and covariates may be unknown, and to prevent misspecification of the model, a nonparametric approach could be appropriate. In this paper prediction and forecasting of the response variable, for spatially nonstationary, spatio-temporal processes, within a nonparametric framework is developed. The linear prediction of the response, which involves estimation of the covariance structure, and also the more general optimal predictor are investigated. The asymptotic sampling properties of the predictors are studied. It is shown that in order to avoid the curse of dimensionality the covariance estimator should be defined in terms of the dependence structure of the spatio-temporal process. Furthermore the rate of convergence of the prediction estimators depend on the temporal dependence of the covariates and the mixing rates of the spatio-temporal process.

The model defined and the estimation methodology has many possible applications. We consider a specific application and illustrate our methodology by modelling house prices in the Stockport area, United Kingdom, using the deprivation index and district as the covariates.