Sampling from an isotropic Gaussian process

Abstract

Isotropic Gaussian processes, of which we shall give a formal definition presently, arise in various practical problems. The present inquiry arose from the consideration of the variability found in the yields of plots in agricultural field experiments. Samples of such patterns of variability can be obtained from uniformity trials, whereby a piece of land is treated uniformly throughout and the crop is harvested in small units or plots. The results from such trials have been widely used to determine optimum plot sizes for future experiments with the crop concerned, but it has never been clear how valid is the generalization from a uniformity trial to future experiments on other sites. One difficulty arises from the lack of a suitable model to express the variability; another difficulty arises from the formidable analytical problems besetting any attempts to apply deductive reasoning to even simple models. We may mention two approaches that have been made to the uniformity trial problem: Fairfield Smith (3) has supplied an empirical law connecting the variance of contiguous groups of plots with the size of the group, and Quenouille (6) and Whittle (8) have considered the fitting of two-dimensional isotropic Gaussian processes to uniformity trial data. Whittle has pointed out that a satisfactory model may have to incorporate an additional random element at each point, and has outlined the difficulties of estimation which arise when such an element is introduced. The connexion between Fairfield Smith's law and the approach via two-dimensional stochastic processes, if any, seems to be quite unknown.