Towards a complete characterization of indicator variograms and madograms

TL;DR

The paper develops necessary and sufficient conditions (NSCs) for the variograms of indicator random fields and the madograms of general random fields on arbitrary point sets. It provides a unifying representation: indicator variograms are expressible as $g(x,y)=\frac{1}{2\pi}\int_{0}^{+\infty} \arccos\rho_{\omega}(x,y)\,F(d\omega)$ with matrix-valued or scalar correlations in Hermitian constructions, and madograms admit $\gamma^{(1)}(x,y)=\int_{0}^{+\infty}\sqrt{\gamma_{\omega}(x,y)}\,F(d\omega)$ representations, together with a suite of negative-type, polygonal, hypermetric, and corner-positive inequalities. The authors prove equivalences and sufficiency results, derive abundant examples across Euclidean spaces, spheres, graphs, and abstract sets, and discuss implications for model design and simulation algorithms. They also highlight the special role of median indicators and the limitations of variogram-based realizations, showing how NSCs can ensure internal consistency and guide practical generation of indicator fields. Overall, the work provides a comprehensive, geometry- and Gaussian-theory-based framework for characterizing and constructing indicator variograms and madograms with broad applicability. $g$ and $\gamma^{(1)}$ representations, together with the related inequalities, offer robust tools for consistent modeling of random sets and boundary structure in high dimensions and on non-Euclidean domains.

Abstract

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.

Figures (3)

• Figure 1: Elementary models that are admissible as indicator variograms in Euclidean spaces: hyperbolic tangent model 1 with $\lambda = \frac{\pi}{2}$ and $\varpi=1$, and I-Bessel model with $\lambda = 10$ and $\varpi=1$.
• Figure 2: Top: Experimental variograms (green lines) and average experimental variogram (blue line) of $100$ realizations of an indicator with mean value $0.5$ simulated on a regular grid with $120 \times 80$ points in the 2D plane. Experimental variograms are calculated along the first axis. The theoretical variogram inputted in the sequential simulation algorithm is indicated in black and consists of an isotropic cubic (a), exponential (b) or spherical (c) model. Bottom: Examples of indicator realizations for the cubic (d), exponential (e) and spherical (f) models.
• Figure 3: Realizations of indicator random fields on the $2$-sphere with mean value $0.5$ and exponential variogram $g(x,y)=\frac{1}{4}(1-\exp(-t\, d_\text{GC}(x,y))$. From left to right: $t=10$, $t=30$ and $t=50$.

Theorems & Definitions (76)

• Proposition 1: McMillan McMillan1955
• Proposition 2: Matheron Matheron1989
• Definition 1
• Proposition 3
• Theorem 1
• Corollary 1
• Remark 1
• Remark 2
• Corollary 2: Existence theorem for indicator variograms
• Proposition 4