A Comparison of Random Field Models Beyond Bivariate Distributions

In order to determine to what extent a spatial random field can be characterized by its low-order distributions, we consider four models (specifically, random spatial tessellations) with exactly the same univariate and bivariate distributions and we compare the statistics associated with various multiple-point configurations and the responses to specific transfer functions. The three- and four-point statistics are found to be the same or experimentally hardly distinguishable because of ergodic fluctuations, whereas change of support and flow simulation produce very different outcomes. This example indicates that low-order distributions may not discriminate between contending random field models, that simulation algorithms based on such distributions may not reproduce the spatial properties of a given model or training image, and that the inference of high-order distribution may require very large training images.