Cluster size distributions of discrete random fields

TL;DR

This paper analyzes the geometry of threshold-exceedance regions for discrete random fields by deriving exact cluster-size distributions for stationary fields and introducing a peak-based cluster-size distribution to handle nonstationarity. The authors introduce the root-at-origin density $w_k$ (and its inside variant $w_k^{\rm inside}$) to express $P(S_u=k)$, and develop Monte Carlo estimators to compute these quantities in higher dimensions. They extend the framework to ${\mathbb Z}^d$ with different neighborhood structures (nearest and Moore) and demonstrate that peak-based distributions converge to exact results at high thresholds, providing a practical tool for spatial extent inference in Gaussian and non-Gaussian fields. Numerical results across 1D and 2D Gaussian and chi-squared fields validate the theoretical formulas and show the utility of the peak-based approach in nonstationary settings, with clear implications for medical imaging, geoscience, and environmental monitoring where thresholded discrete fields arise.

Abstract

We study discrete random fields $\{X_t: t\in \mathbb{Z}^d\}$ parameterized on the $d$-dimensional integer lattice $\mathbb{Z}^d$. For a fixed threshold $u$, the excursion set $\{t \in \mathbb{Z}^d : X_t > u\}$ decomposes into connected components or clusters, whose size, defined as the number of lattice points they contain, are random. This paper investigates the probability distribution of these cluster sizes. For stationary random fields, we derive exact expressions for the cluster size distribution. To address nonstationary settings, we introduce a peak-based cluster size distribution, which characterizes the distribution of cluster sizes conditional on the presence of a local maximum above $u$. This formulation provides a tractable alternative when exact cluster size distributions are analytically inaccessible. The proposed framework applies broadly to Gaussian and non-Gaussian random fields, relying only on their joint dependence structure. Our results provide a theoretical foundation for quantifying spatial extent in discretely sampled data, with applications to medical imaging, geoscience, environmental monitoring, and other scientific areas where thresholded random fields naturally arise.

Figures (3)

• Figure 1: The left panel is the scatter plot of a simulated realization of a centered stationary Gaussian process with covariance $C(t,s)=e^{-(t-s)^2}$, where exceedances above the threshold $u=0.5$ are marked in red, and the corresponding clusters are marked in consecutive red dots along the index ${\mathbb Z}$. This realization exhibits 14 observed clusters: 7 of size one, 5 of size two, 1 of size three, and 1 of size four. The right panel displays the empirical cluster size distribution obtained from repeated simulations.
• Figure 2: Two neighbourhood systems in ${\mathbb Z}^2$. Left: nearest neighbor connectivity as in \ref{['eq:nearest']}, where each point has 4 neighbors. Right: Moore neighbor connectivity as in \ref{['eq:Moore']}, where each point has 8 neighbors (including diagonals).
• Figure 3: Simulation of clusters and empirical cluster size distributions for an isotropic Gaussian field on ${\mathbb Z}^2$ with covariance $C(t,s)=e^{-\|t-s\|^2}$ and threshold $u=0.5$. Panels (a) and (b) use the nearest neighbors, while panels (c) and (d) use Moore neighbors. In (a), distinct clusters above $u$ are highlighted in blue and red (note that diagonal sites are not connected under nearest neighbors); while in (c), they are marked in red. Panels (b) and (d) display the corresponding empirical cluster size distributions from repeated simulations.

Theorems & Definitions (7)

• Theorem 2.1
• proof
• Theorem 2.3
• proof
• Corollary 2.8
• Theorem 3.2
• proof