Modelling for Poisson process intensities over irregular spatial domains

TL;DR

This paper introduces nonparametric Bayesian models for Poisson process intensities over irregular spatial domains, enabling flexible inference with efficient computation and addressing basis selection and prior specification.

Contribution

It presents two novel classes of models for irregular domains, with methods for basis number estimation, prior setting, and posterior inference, supported by synthetic and real data examples.

Findings

Models effectively capture spatial Poisson intensities.

Methodology supports inference on point process functionals.

Models are computationally efficient and flexible.

Abstract

We develop nonparametric Bayesian modelling approaches for Poisson processes, using weighted combinations of structured beta densities to represent the point process intensity function. For a regular spatial domain, such as the unit square, the model construction implies a Bernstein-Dirichlet prior for the Poisson process density, which supports general inference for point process functionals. The key contribution of the methodology is two classes of flexible and computationally efficient models for spatial Poisson process intensities over irregular domains. We address the choice or estimation of the number of beta basis densities, and develop methods for prior specification and posterior simulation for full inference about functionals of the point process. The methodology is illustrated with both synthetic and real data sets.