Towards a complete characterization of indicator variograms and madograms

Indicator variograms and madograms are structural tools used in many disciplines of the natural sciences and engineering to describe random sets and random fields. To date, several necessary conditions are known for a function to be a valid indicator variogram but, except for intractable corner-positive inequalities, a complete characterization of indicator variograms is missing. Likewise, only partial characterizations of madograms are known. This paper provides novel necessary and sufficient conditions for a given function to be the variogram of an indicator random field with constant mean value or to be the madogram of a random field, and establishes under which conditions these two families of functions coincide. Our results apply to any set of points where the random field is defined and rely on distance geometry and Gaussian random field theory.

💡 Research Summary

This paper addresses a long‑standing gap in the theory of spatial statistics: a complete characterization of indicator variograms and madograms. Indicator variograms arise from the 0‑1 indicator field of a random closed set, while madograms are the first‑order variograms (½ E|Z(x)−Z(y)|) of real‑valued random fields. Although it is known that every indicator variogram is a madogram and every madogram is a variogram, the converse statements are not generally true, and existing necessary conditions (upper bound, triangle inequality, Matheron, Shepp, corner‑positive inequalities) are not sufficient.

The authors first review the classical result that variograms are exactly the conditionally negative semidefinite, symmetric functions vanishing on the diagonal. They then emphasize that indicator variograms form a strict subclass, requiring additional constraints such as the upper bound g≤½, triangle inequality, and a hierarchy of higher‑order inequalities.

The core contribution is a set of necessary and sufficient conditions expressed through integral mixture representations. Theorem 1 shows that a function g(x,y) is an indicator variogram of a first‑order stationary indicator random field if and only if it can be written as

 g(x,y)=½ π ∫₀^∞ arccos ρ_ω(x,y) F(dω),

where for each frequency ω, ρ_ω is a correlation function (positive semidefinite, unit diagonal) on the index set X, and F is a distribution function on (0,∞). This representation reveals that the class of indicator variograms is precisely the closed convex hull of the “median indicator variograms” of Gaussian random fields (Proposition 1, Corollary 1). An equivalent formulation using arcsin (Theorem 2) involves the variograms of standard Gaussian fields, highlighting the link to Hermitian random fields.

Proposition 4 establishes closure of the indicator‑variogram class under scalar multiplication, convex combinations, and the non‑linear operation g+g′−4gg′. Theorem 3 provides a comprehensive suite of inequalities—polygonal, odd‑clique, hypermetric, positive‑semidefinite, rounded‑positive‑semidefinite, gap, and corner‑positive—that any indicator variogram must satisfy. These include and strengthen the classical Matheron and Shepp conditions, offering a practically checkable set of sufficient constraints.

For madograms, Theorem 4 proves that any madogram γ¹ can be expressed as

 γ¹(x,y)=∫₀^∞ p γ_ω(x,y) F(dω),

with γ_ω a variogram and p a non‑negative weight. Consequently, every madogram is a mixture of variograms. Theorem 5 shows that when the madogram is bounded by ½ and the underlying field takes values in