Mixture Modeling for Marked Poisson Processes

TL;DR

This paper introduces a flexible nonparametric Bayesian framework for modeling marked Poisson processes, enabling inference on mark distributions and intensities without complex dependence assumptions.

Contribution

It develops a joint nonparametric mixture model for the intensity and mark distribution, with guidelines for implementation and model checking.

Findings

Flexible inference for multivariate marks achieved

Method demonstrated on simulated data

Applied successfully to real data sets

Abstract

We propose a general modeling framework for marked Poisson processes observed over time or space. The modeling approach exploits the connection of the nonhomogeneous Poisson process intensity with a density function. Nonparametric Dirichlet process mixtures for this density, combined with nonparametric or semiparametric modeling for the mark distribution, yield flexible prior models for the marked Poisson process. In particular, we focus on fully nonparametric model formulations that build the mark density and intensity function from a joint nonparametric mixture, and provide guidelines for straightforward application of these techniques. A key feature of such models is that they can yield flexible inference about the conditional distribution for multivariate marks without requiring specification of a complicated dependence scheme. We address issues relating to choice of the Dirichlet process mixture kernels, and develop methods for prior specification and posterior simulation for full inference about functionals of the marked Poisson process. Moreover, we discuss a method for model checking that can be used to assess and compare goodness of fit of different model specifications under the proposed framework. The methodology is illustrated with simulated and real data sets.