UCSC-SOE-09-31: Dirichlet Process Mixture Modeling for Marked Poisson Processes
Matthew Taddy and Athanasios Kottas
We propose a general modeling framework for marked non-homogeneous Poisson processes observed over time or space. The modeling approach exploits the connection of the non-homogeneous 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. A key feature of such models is that they yield general specifications for the corresponding conditional mark distribution resulting in flexible inference for different types of multivariate marks. 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.
We propose a general modeling framework for marked non-homogeneous Poisson processes observed over time or space. The modeling approach exploits the connection of the non-homogeneous 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. A key feature of such models is that they yield general specifications for the corresponding conditional mark distribution resulting in flexible inference for different types of multivariate marks. 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.