A self-exciting point process with cyclic component, trend component, triggering function, and response function

A point process is a collection of random points located in a particular area. A point process such that one event can trigger other events at a particular time is called a self-exciting point process. In a point process, an intensity function plays a crucial role in stating the probability of the number of events per time unit conditional on past events. The conditional intensity function can be estimated by considering the cyclic component, trend component, internal factor, and external factor impacting the occurrence of earthquakes. This research aims to derive the conditional intensity function of self-exciting point processes and to estimate the parameters of the conditional intensity function for earthquake data in Java Island and Sumatra Island. Conditional intensity function of the process can be presented as a function of time by considering cyclic component as a Fourier series and trend component as a sum of polynomials, while each of triggering function and response function as a sum of exponential functions. Data analysis showed that earthquakes that occurred in Sumatra have the same probability of triggering aftershocks as earthquakes that occurred in Java.