Screening Effect in Isotropic Gaussian Processes

The screening effect is the phenomenon that optimal linear prediction of a spatial process on an unobserved location mostly depends on nearby observations. That is, the optimal predictor based on just nearby observations yields a good approximation of which based on the whole large dataset.However, the approximation does not always perform well since the screening effect may not hold in all situations. To determine when the screening effect holds is an important issue in spatial statistics.This paper provides some sufficient conditions for ensuring an asymptotic screening effect in Rdbased on the spectral density of the underlying isotropic Gaussian process and the geometries of nearby observations. These results apply to isotropic processes with an arbitrary degree of differentiability.Assuming we are predicting at origin, the conditions are(1) the spectral density is nearly a constant in balls of finite radius far from the origin,(2) the positions of nearby observations do not fall on a curve with non-zero intercept whose degree is less or equal to the order of mean square differentiability of the process. These conditions are easy to verify in practice. Convergence rates of the asymptotic screening effect are also obtained. These rates depend on the rate of decrease of the spectral density.Simulation studies on the screening effect for finite samples are also provided.