Nonparametric Inference for Nonstationary Spatial Point Processes

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 seminar 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 talk concludes with a discussion of potential extensions
and directions for future work.

Keywords: intractable likelihood, exact inference, data augmentation, MCMC, Voronoi
tessellation.