Prediction of nonlinear spatial functionals
Spatial statistical methodology can be useful in the arena of environmental regulation. Some regulatory questions may be addressed by predicting linear functionals of the underlying signal, but other questions may require the prediction of nonlinear functionals of the signal. For example, in order to be in regulatory compliance, air-pollution levels have to fall within specified limits over some geographic region; whether or not they are in compliance and where they are out of compliance are nonlinear functionals. We propose a spatial empirical Bayes model for environmental data collected over a region.
Further, we propose a predictor, based on the kriging methodology with extra constraints, that implies useful unbiasedness properties in predicting nonlinear spatial functionals. This predictor, called covariance-matching constrained kriging, is an optimal linear predictor that matches not only first moments but second moments (including specified covariances) as well.