The Spatial Kernel Predictor based on Huge Observation Sets

Abstract

Spatial prediction in an arbitrary location, based on a spatial set of observations, is usually performed by Kriging, being the best linear unbiased predictor (BLUP) in a least-square sense. In order to predict a continuous surface over a spatial domain a grid representation is most often used. Kriging predictions and prediction variances are computed in the nodes of a grid covering the spatial domain, and the continuous surface is assessed from this grid representation. A precise representation usually requires the number of grid nodes to be considerably larger than the number of observations. For a Gaussian random field model the Kriging predictor coinsides with the conditional expectation of the spatial variable given the observation set. An alternative expression for this conditional expectation provides a spatial predictor on functional form which does not rely on a spatial grid discretization. This functional predictor, called the Kernel predictor, is identical to the asymptotic grid infill limit of the Kriging-based grid representation, and the computational demand is primarily dependent on the number of observations - not the dimension of the spatial reference domain nor any grid discretization. We explore the potential of this Kernel predictor with associated prediction variances. The predictor is valid for Gaussian random fields with any eligible spatial correlation function, and large computational savings can be obtained by using a finite-range spatial correlation function. For studies with a huge set of observations, localized predictors must be used, and the computational advantage relative to Kriging predictors can be very large. Moreover, model parameter inference based on a huge observation set can be efficiently made. The methodology is demonstrated in a couple of examples.