The focus of this research project is on development of flexible Bayesian statistical approaches to modeling and inference for point processes. The research will develop methods within the widely expanding field of Bayesian nonparametrics to provide a general model-based inference framework for a large class of problems involving point pattern data. The project will study inferential methods for non-homogeneous Poisson processes in time, space, and space-time, including extensions to incorporate time- and space-varying covariates as well as marked point processes. It also will consider techniques for model checking as well as extensions to modeling for non-Poisson point processes. This general methodology will be developed around the area of extreme value analysis, which consists of the exploration of events that occur in the tails of probability distributions. Key applications arise in fields as diverse as finance, actuarial sciences and climatology. Point process modeling provides a general approach to addressing scientifically important questions in the study of extremes. The theory for this approach has been extensively developed, but the limited existing work on statistical methods relies on restrictive parametric assumptions. The Bayesian nonparametric methodology will provide a natural framework for more flexible inference and prediction with important practical implications in enhancing our ability to quantify the risks associated with the occurrence of relatively unlikely events. Of particular interest will be assessment of the extreme behavior of environmental variables that are likely to be affected by climate change.
The study of extremes (very large or very small values) of a physical process observed in time, space, or space-time is of critical importance in several fields, including econometrics, geosciences, and environmental policy making. A powerful approach to statistical modeling for extreme value analysis draws from the theory of point processes, which are stochastic models for random events over time and/or space. This research will formulate a general statistical framework for analysis of extremes through a novel synthesis of methods from point process modeling and Bayesian nonparametrics, a rapidly growing area of Bayesian statistics. Due to their generality, the statistical methods that will be developed under this research project have the potential of impacting many scientific fields where point processes are applied. In the context of extreme value analysis, the methodology will focus on appropriate quantification of uncertainty for rare but catastrophic events such as torrential rains, severe droughts, or stock index crashes. For these and related applications, improved prediction of the probability of occurrence of extreme events and understanding of associated factors can have an important impact on effective decision making.
The project advanced the field of statistical modeling and inference for point processes. These are stochastic processes that are used to describe events occurring at random times or locations. It is a class of probabilistic models that has wide applicability in social sciences, risk analysis, environmental sciences, and reliability, to mention a few areas. The theory of point processes is well developed. However, methods to learn the properties of such processes from data are comparatively less developed. This project has developed a range of methods to perform inference on point processes that are general and flexible as well as computationally efficient. The new methods for modeling and inference were built within Bayesian nonparametrics, a rapidly growing area of Bayesian statistics. To test and exemplify the methodology developed in this project, we have considered a number of applications to different fields. Among them are: study of the evolution of hurricane occurrences, and the corresponding wind speed and damage, along the U.S. Gulf and Atlantic coasts; analysis of neuronal intensity rates recorded under multiple experimental conditions; description of the intensity of crime occurrence in an urban environment; prediction of extreme rainfall occurrence over specific geographic regions and across time; and systemic risk assessment in correlated financial markets. The project had an important educational component. In particular, five different Ph.D. students in the Department of Applied Mathematics and Statistics at University of California, Santa Cruz were involved in different research topics related to the project. For two of them the development of the project was central to their Ph.D. dissertations. The project involved intense dissemination efforts that resulted in numerous presentations at national and international meetings and seminars, as well as several journal publications.
