UC Santa Cruz

Model-based inferential methods for point processes have received less attention than the corresponding theory of point processes and is more scarcely developed than other areas of statistical inference.

Classical inferential methods for point processes include likelihood-based and nonparametric methods. Bayesian analysis provides simulation-based estimation of several statistics of interest for point processes. However, a challenge of Bayesian modeling, specifically for point processes, is selecting an appropriate parametric form for the intensity function. Bayesian nonparametric methods aim to avoid the narrow focus of parametric assumptions by imposing priors that can support the entire space of distributions and functions. It is naturally a more flexible and adaptable approach than those based on parametric models.

In this dissertation, we focus on developing methodology for some classes of temporal point processes modeling and inference in the context of Bayesian nonparametric methods, mainly with applications in environmental science.

Firstly, we are motivated to study seasonal marked point process by an application of

hurricanes occurrences. We develop nonparametric Bayesian methodology to study the dynamic

evolution of a seasonal marked point process intensity under the assumption that the point process is a non-homogeneous Poisson process. The dynamic model for time-varying intensities provides both the intra-seasonal and inter-seasonal variability of occurrences of events. Considering marks, we provide a full probabilistic model for the point process over the joint marks-points space which allows for different types of inferences, including full inference for dynamically evolving conditional mark densities given a time point, a particular time period, and even a subset of marks.

We apply this method to study the evolution of the intensity of the process of hurricane landfall occurrences, and the respective maximum wind speed and associated damages. We show several novel inferences which are explored for the first time in the analysis of hurricane occurrences.

Then we look beyond Poisson processes and propose a flexible approach to modeling and inference for homogeneous renewal processes.

This modeling method is based on a structured mixture of Erlang densities with common scale parameter for the renewal process inter-arrival density. The mixture weights are defined through an underlying distribution function modeled nonparametrically with a Dirichlet process prior. This model specification enables flexible shapes for the inter-arrival time density, including heavy tailed and multimodal densities. Moreover, the choice of the Dirichlet process centering distribution controls clustering or declustering patterns for the point process.

Finally we extend our model to accommodate point processes with time-varying inter-arrivals, which are referred to as modulated renewal processes in the literature. We introduce time dependence in the scale parameter of the Erlang mixture by replacing it with a latent stochastic process. A number of synthetic data sets and real data sets are used to illustrate the modeling approaches.

The main contribution of this thesis is to provide Bayesian nonparametric modeling and inference methods for some classes of point processes, which are more flexible than existing methods.

Moreover, the key complication for Bayesian inference is that the likelihood of a generic point process involves a normalizing constant which is, most of the times, analytically intractable. Discretization is very often used in existing methods to get likelihood approximations that facilitate computations, especially for models based on Gaussian process priors. Superior to these methods, our work uses the exact likelihood without approximation in all of our developed models.