autovariogram

Autovariogram is a function studied in stochastic geometry and spatial statistics that describes the second-order structure of a stationary random set by measuring how much of the set overlaps with its translate. For a stationary random closed set X in Euclidean space R^d, the autovariogram is defined as g(h) = E[Vol(X ∩ (X + h))], where Vol denotes Lebesgue measure and h ∈ R^d is a translation vector. Equivalently, g(h) can be written as the expected volume of the intersection between X and its shift by h. In the special case where X is a deterministic set A, this reduces to the covariogram g_A(h) = Vol(A ∩ (A + h)).

The autovariogram is typically nonnegative, finite, and often symmetric with respect to h → −h (g(h) = g(−h)). It attains its maximum at h = 0, with g(0) = E[Vol(X)], the mean volume (or area, etc.) of the random set. For bounded X, g(h) tends to zero as |h| grows large, and its smoothness depends on the regularity of the boundary of X.

In random-set theory, the autovariogram is related to the covariogram and to two-point coverage probabilities through g(h) = ∫ P(x ∈ X and x+h ∈ X) dx. It provides a concise summary of second-order spatial dependence and is used to characterize microstructures in materials, analyze binary images, and guide statistical inference for random sets.

Estimation of the autovariogram from data typically involves computing empirical averages of overlaps between the observed set and its translates over a observation window. Variations exist, including radial forms when X is isotropic.