Optimal Spatial Prediction for Non-negative Spatial Processes Using a Phi-divergence Loss Function

Abstract

A major component of inference in spatial statistics is that of spatial prediction of an unknown value from an underlying spatial process, based on noisy measurements of the process taken at various locations in a spatial domain. The most commonly used predictor is the conditional expectation of the unknown value given the data, and its calculation is obtained from assumptions about the probability distribution of the process and the measurements of that process. The conditional expectation is unbiased and minimises the mean-squared prediction error, which can be interpreted as the unconditional risk based on the squared-error loss function. Cressie [4, p. 108] generalised this approach to other loss functions, to obtain spatial predictors that are optimal (i.e., that minimise the unconditional risk) but not necessarily unbiased. This chapter is concerned with spatial prediction of processes that take non-negative values, for which there is a class of loss functions obtained by adapting the phi-divergence goodness-of-fit statistics [6]. The important sub-class of power-divergence loss functions is featured, from which a new class of spatial predictors can be defined by choosing the predictor that minimises the corresponding unconditional risk. An application is given to spatial prediction of zinc concentrations in soil on a floodplain of the Meuse River in the Netherlands.