Statistical Framework for Bivariate Point Processes: Conditional Intensity and Parameter Estimation Techniques

The point process is a model that is suitable to describe the number of events that occur randomly in a given interval through its intensity function. The conditional intensity of the bivariate point process can be seen more specifically in two separate groups. The purpose of this study is to construct the conditional intensity for the homogeneous bivariate point process and to estimate the parameters using the maximum likelihood method. The conditional intensity construction is obtained from the ratio of the event-time probability density function to the event time survival function. Estimation of the conditional intensity parameter is carried out by constructing a likelihood function of the probability of one event in a small interval multiplied by the probability of no event in the remaining observation time. The results show that the conditional intensity of a bivariate point process depends on the number of events <img src=image/13440812_01.gif> and the observation time interval <img src=image/13440812_02.gif>. The pattern application was performed on two datasets, namely dataset the number of active cases of Covid-19 in Indonesia and dataset the number of earthquake in Sulawesi Island. The application to Dataset Covid-19 or Dataset earthquake reveals that the conditional intensity of the type-1 and type-2 in both datasets exhibits a directly proportional to the average frequency of events and inversely proportional to the observation time. Application to Dataset A or Dataset B shows that the conditional intensity of type A and type B events for the two Datasets is inversely proportional to the observation time.