Nonparametric inference for nonstationary spatial point processes

Abstract

Point pattern data often exhibit features such as abrupt changes, hotspots and spatially varying dependence in local intensity. Under a Poisson process framework, these correspond to discontinuities and nonstationarity in the underlying intensity function. These features are difficult to capture with standard modeling approaches. This paper proposes a spatial Cox process model in which nonstationarity is induced through a random partition of the spatial domain, with conditionally independent Gaussian process priors specified across the resulting regions. This construction allows for heterogeneous spatial behavior, including sharp transitions in intensity. A discretization-free MCMC algorithm is developed to target the infinite-dimensional posterior distribution without approximation, thus ensuring exact inference. The random partition framework via Voronoi tessellation also reduces the computational burden associated with Gaussian process models. Spatial covariates can be incorporated to account for structured variation in intensity. The proposed methodology is evaluated through synthetic examples and real-world applications, demonstrating its ability to flexibly capture complex spatial structures. The model performs competitively, outperforming stationary and nonstationary alternatives in a variety of scenarios. Recent computational methods are used, enabling scalability to large datasets while preserving exactness. The paper concludes with a discussion of potential extensions and directions for future work.