Bayesian Nonparametric Modeling and Inference Methods for Point Processes

University Of California-Santa Cruz, Santa Cruz CA

Investigators

Abstract

This research project will develop flexible statistical models and corresponding inference methods for different classes of point processes. The theory of point processes was developed to study the distribution of events that occur at random times and/or spatial locations. This project will advance statistical methodology for a range of problems involving data that arise from point processes. A key objective will be to increase the scope of several classes of point processes by developing statistical models that relax the restrictive assumptions of state-of-the-art methods. Point process methods are important for the field of statistics and have applications in a variety of areas, including climatology, criminology, finance, and seismology. The new models will be applied to important societal problems including the study of clustering patterns for different types of crime and the analysis of earthquake occurrences. To facilitate use of the methods by practitioners and researchers from other fields, publicly available software will be developed for implementing several of the statistical models. The project also will create educational opportunities for graduate students and seek to increase the participation of women and underrepresented groups in the research. This research project will develop a general model-based framework for point processes over time or space, including Poisson processes and Hawkes processes. The modeling framework involves structured mixture representations for point process intensities that achieve a balance between model flexibility and computational efficiency in implementation of statistical inference and prediction. The project will develop tractable inference methods for spatial point processes observed over irregular domains; for instance, city, state, or country boundaries. It also will increase the inferential scope for marked Hawkes processes, a versatile class of stochastic point process models that have been applied in diverse areas, including earthquake modeling, finance, criminology, and analysis of social networks. Even though the theory for Hawkes processes is well studied, statistical inference methods are relatively less developed, especially under general settings that allow for non-standard data features and for full uncertainty quantification. The research project has a substantial analytic component with regards to methodological development for the various point process models, as well as a significant computational component with regards to achieving efficient model fitting that can be scaled to large amounts of data. The practical utility of the new methods will be investigated with several simulation studies and through substantive applications involving analysis of earthquake and crime data. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.