Erlang mixture modeling for Poisson process intensities

TL;DR

This paper introduces a Bayesian nonparametric Erlang mixture model for estimating Poisson process intensities, offering flexible shape modeling and efficient inference, with applications to spatial processes and comparisons to existing methods.

Contribution

It extends Erlang mixture models to intensity estimation for Poisson processes using a gamma process prior, enabling flexible shape modeling and efficient computation.

Findings

The model effectively captures diverse intensity shapes.

It outperforms Gaussian process-based models in certain scenarios.

Demonstrated successful application to synthetic and real data.

Abstract

We develop a prior probability model for temporal Poisson process intensities through structured mixtures of Erlang densities with common scale parameter, mixing on the integer shape parameters. The mixture weights are constructed through increments of a cumulative intensity function which is modeled nonparametrically with a gamma process prior. Such model specification provides a novel extension of Erlang mixtures for density estimation to the intensity estimation setting. The prior model structure supports general shapes for the point process intensity function, and it also enables effective handling of the Poisson process likelihood normalizing term resulting in efficient posterior simulation. The Erlang mixture modeling approach is further elaborated to develop an inference method for spatial Poisson processes. The methodology is examined relative to existing Bayesian nonparametric modeling approaches, including empirical comparison with Gaussian process prior based models, and is illustrated with synthetic and real data examples.