UC Santa Cruz

We propose a Bayesian nonparametric modeling and inference framework for Hawkes processes. The objective is to increase the inferential scope for this practically important class of point processes by exploring flexible models for its conditional intensity function.

As a building block for conditional intensity models, we develop a prior probability model for temporal Poisson process intensities through structured mixtures of Erlang densities with common a scale parameter, mixing on the integer shape parameters. The mixture weights are constructed through increments of a cumulative intensity function modeled nonparametrically with a gamma process prior. This model specification provides a novel extension of Erlang mixtures for density estimation to the intensity estimation setting.

Turning to the main dissertation component, we develop different types of nonparametric prior models for the Hawkes process immigrant intensity and for the excitation function (or its normalized version, the offspring density), the two functions that define the point process conditional intensity. The prior models are carefully constructed such that, along with the Hawkes process branching structure, they enable efficient handling of the complex likelihood normalizing terms in implementation of inference. The methodology is further elaborated to construct a flexible and computationally efficient model for marked Hawkes processes. The motivating application involves earthquake data modeling, where the mark is given by the earthquake magnitude. The proposed model builds from a prior for the excitation function that allows flexible shapes for mark-dependent offspring densities. In the context of our motivating application, the modeling approach enables estimation of aftershock densities that can vary with the magnitude of the main shock, unlike existing marked Hawkes process models for earthquake occurrences.

For all proposed models, we develop approaches to prior specification and design posterior simulation algorithms to obtain inference for different point process functionals. The modeling approaches are studied empirically using several synthetic and real data examples, including data on earthquake occurrences from Japan and from the southwestern US.