UCLA

Self-exciting point processes describe random sequences of events where the occurrence of an event increases the likelihood that subsequent events occur nearby in time and space. Models for self-exciting point processes have many important applications to diverse topics such as earthquake and crime forecasting, epidemiology, invasive species, and social networks.

The first part of this dissertation discusses a new application of self-exciting point processes to modeling the times when e-mails are sent by individuals in a social network. The proposed models are fit to datasets from West Point Military Academy and the Enron Corporation, and the resulting parameter estimates characterize communication behaviors and leadership roles for users in each network. We argue that the self-exciting models adequately capture major temporal clustering features in the data and perform better than traditional stationary Poisson models.

The second part of this dissertation discusses the nonparametric method of Marsan and Lengline (2008) for estimating space-time Hawkes point process models of earthquake occurrences. Their method provides an estimate of a stationary background rate for mainshocks, and a histogram estimate of the triggering function for the rate of aftershocks following an earthquake. At each step of the procedure the model estimates rely on computing the probability each earthquake is a mainshock or aftershock of a previous event. We focus on improving Marsan and Lengline's method by proposing novel ways to incorporate a non-stationary background rate, and adding error bars to the histogram estimates which capture the sampling variability and bias in the estimation of the underlying seismic process. A simulation study is designed to validate and assess new methodology. An application to earthquake data from the Tohoku District in Japan is also discussed, and the results are compared to a well established parametric model of seismicity for this region.