Integration of Random Fields

Abstract

This chapter studies the prediction of integrals of random fields based on observations on a lattice. The goal here is not to give a full exposition of the topic (see Ritter (1995) for a more detailed treatment) but to make two specific points about properties of systematic designs. The first is that simple averages over observations from systematic designs can be very poor predictors of integrals of random fields, especially in higher dimensions. The second is that, at least for random fields that are not too anisotropic, the problem with this predictor is the simple average aspect of it, not the systematic design. These two points are of interest on their own, but they are also critical to understanding a serious flaw in an argument of Matheron (1971) purporting to demonstrate that statistical inference is “impossible” for differentiable random fields (see 6.3).