Different approaches to examine inhibiting Hawkes processes: likelihood-based and moment-based - Nguyen, Duong Quynh Chi (Nanyang Technological University, 2022)
Hawkes processes are widely used in different research fields to study the excitation effects among events, such as earthquakes, neurons membranes, etc. Using a neural network-based approach with a variation of likelihood function as the loss function, we perform intensity and occurrence time prediction on three different scenarios of Hawkes processes: self-exciting, inhibiting with ReLU transfer function, and inhibiting with softplus transfer function. Numerical and graphical results of experiments indicate that the neural network only fails to capture the inhibition effects if underlying intensities are passed through ReLU function. This suggests the practicality of the softplus function in defining inhibition effects in Hawkes processes, potentially owing to the invertibility characteristics of softplus function. Meanwhile, examining the cumulants of multivariate Hawkes processes, we apply method of moments to estimate relevant parameters of the Hawkes process. From our experimental results, we are required to utilize all 5 cumulants formulas of a two-dimensional Hawkes processes to obtain satisfactory estimates of 2 parameters. Thus, it is deduced that the highly complex and sensitive cumulants formulas require the usage of all cumulants functions, equivalently, the usage of all information available, to yield satisfactory estimates of the Hawkes processes parameters.