Date of Award
12-2024
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Electrical Engineering and Computer Science
First Advisor
Georgios C. Anagnostopoulos
Second Advisor
Gnana Bhaskar. Tenali
Third Advisor
Adrian M. Peter
Fourth Advisor
Anthony O. Smith
Abstract
Hawkes self-exciting point processes have been widely used in fields such as seismology, finance and social media analysis. Despite decades of research focusing on these processes, several key aspects remain under-explored. These include broadly applicable methods for model evaluation and selection, intuitive nonparametric inference approaches akin to kernel density estimation for probability density functions, and strategies to enhance the performance of generative point process models in predictive tasks. In this dissertation, first, we extend the time-rescaling theorem, which is traditionally limited to non-terminating processes with complete observations, to accommodate terminating processes and incomplete observations as well. This extension allows for more robust model validation across diverse contexts, especially for Hawkes processes intended to model real-world, event dynamics. Second, we propose a nonparametric, kernel-based intensity estimation algorithm tailored to Hawkes processes. The algorithm iteratively updates declustering probabilities and estimates the intensity function of such processes with rigorously derived kernel weights and established asymptotic properties. Its simplicity and flexibility make it a valuable tool for many real-world scenarios. Third, we study the first- and second-order moments of counting processes associated to Hawkes processes and provide a closed-form expression for conditional mean counts. This facilitates a predictive learning of future event counts that yields models that are superior, when compared to predictions based on Hawkes processes fitted via maximum likelihood estimation. Through these contributions, this dissertation advances the theoretical understanding of Hawkes processes and expands applicability of such processes to a broader range of real-world problems.
Recommended Citation
Zhang, Xi, "Theoretical Advancements in Hawkes Processes and Their Practical Applications" (2024). Theses and Dissertations. 1512.
https://repository.fit.edu/etd/1512