Perfect simulation of spatial point processes using dominated coupling from the past with application to a multiscale area-interaction point process

We consider perfect simulation algorithms for locally stable point processes based on dominated coupling from the past. A version of the algorithm is developed which is feasible for processes which are neither purely attractive nor purely repulsive. Such processes include multiscale area-interaction processes, which are capable of modelling point patterns whose clustering structure varies across scales. We prove correctness of the algorithm and existence of these processes. An application to the redwood seedlings data is discussed.

💡 Research Summary

The paper addresses the longstanding difficulty of obtaining exact samples from spatial point processes when using Markov chain Monte Carlo (MCMC) methods, where convergence diagnostics are often unavailable. Building on the Coupling From The Past (CFTP) framework, the authors develop a dominated CFTP algorithm that is computationally feasible for locally stable point processes that are neither purely attractive nor purely repulsive. The key innovation is a replacement of the original step that requires evaluating the Papangelou conditional intensity for every possible subset of the current configuration. In the new step (denoted “Step 5′”), the algorithm computes only a constant number of intensity bounds derived from the monotone components of the model’s density, compares these bounds with uniformly generated marks, and decides whether to accept a proposed birth. This reduces the per‑iteration computational cost from a quantity that grows with the number of points to a constant, while preserving the essential properties of domination, sandwiching, and eventual coalescence required for exact sampling. Theoretical results (Lemma 1, Theorem 1) prove that the modified algorithm remains valid and that the number of intensity evaluations is independent of the current configuration size.

To demonstrate the practical utility of the method, the authors introduce a multiscale area‑interaction point process. The standard area‑interaction model captures either clustering or regularity through a single interaction radius and a single attraction/repulsion parameter γ. The multiscale extension incorporates two interaction radii (G₁ and G₂) and two parameters (γ₁ > 1 for large‑scale attraction, γ₂ < 1 for small‑scale repulsion). Its density is the product of three monotone factors, fitting precisely into the framework required for Step 5′. The paper details how to simulate this model using the new algorithm, how to compute the Papangelou intensity, and how to estimate the parameters (λ, γ₁, γ₂) via maximum pseudo‑likelihood, with Monte‑Carlo approximations for the necessary integrals.

An empirical application to the classic Redwood seedlings data illustrates the approach. Compared with the single‑scale model, the multiscale model better reproduces the observed pattern of seedlings that exhibit regular spacing at short distances but form larger clusters at broader scales. Visual inspections and summary statistics (e.g., K‑function, pair‑correlation) confirm the improved fit.

The authors conclude that the dominated CFTP with the simplified intensity evaluation is a versatile tool for exact simulation of complex spatial point processes. They suggest extensions to other Gibbs‑type models, parallel implementations, and adaptive schemes for handling high‑dimensional or high‑intensity settings, pointing to a broad avenue for future research.