Point processes models describe random sequences of events. One key model is the self-exciting point process model where the covariance between events is positive. Models of this type have many applications including seismology, epidemiology, and crime. While model estimation is a primary focus, quantification and assessment of model performance are also useful at identifying departures of model fit from the data. This dissertation discusses two applications of self-exciting point process models. The proposed models are fit to data sets from California seismology and plague data. Performance is verified using simulation studies. We also introduce various model evaluation techniques and conduct detailed model evaluations.
This dissertation is organized as follows: Chapter 1 of this dissertation provides preliminary background on point process models. Chapter 2 introduces a nonparametric technique for estimating self-exciting point process models and proposes an extension which seeks to capture anisotropy in the spatial distribution of aftershocks. A forecasting approach is developed and model performance is evaluated retrospectively in comparison to Helmstetter et al. 2007. Chapter 3 describes a mathematical curiosity that allows one to compute exact maximum likelihood estimates of the triggering function in a direct and extremely rapid manner when the number p of intervals on which the nonparametric estimate is sought equals the number n of observed points. Chapter 4 uses Voronoi residuals, super-thinning, and some other residual analysis methods to evaluate a selection of earthquake forecast models in the Collaboratory for the Study of Earthquake Predictability (CSEP). Chapter 5 introduces NonParametricHawkes, an R package for estimation, forecast, and evaluation of spatial-temporal models, created to increase the accessibility of the methods introduced here.
