Browsing by Author "Huynh, Inchuen"

Hawkes processes are a special class of inhomogenous Poisson processes used to model events exhibiting interdependencies. Initially introduced in Hawkes [1971], Hawkes processes have since found applications in various fields such as seismology, finance, and criminology. The defining feature of Hawkes processes lies in their ability to capture self-exciting behaviour, where the occurrence of an event increases the risk of experiencing subsequent events. This effect is quantified in their conditional intensity function which takes into account the history of the process in the kernel. This thesis focuses on the modeling of event histories using Hawkes processes. We define both the univariate and multivariate forms of Hawkes processes and discuss the selection of kernels, which determine whether the process is a jump or a non-jump process. In a jump Hawkes process, the conditional intensity spikes at an occurrence of an event and the risk of experiencing new events is the highest immediately after an event. For non-jump processes, the risk increases more gradually and can be more flexible. Additionally, we explore the choice of baseline intensity and the inclusion of covariates in the conditional intensity of the process. For parameter estimation, we derive the log-likelihood functions and discuss goodness of fit methods. We show that by employing the time-rescaling theorem to transform event times, assessing the fit of a Hawkes process reduces to that of an unit rate Poisson process. Finally, we illustrate the application of Hawkes processes by exploring whether an exponential Hawkes process can be used to model occurrences of diabetes-related comorbidities using data from the Diabetes Register of the Finnish Institute for Health and Welfare (THL). Based on our analysis, the process did not adequately describe our data, however, exploring alternative kernel functions and incorporating time-varying baseline intensities hold potential for improving the compatibility.