Accelerating the estimation of renewal Hawkes self-exciting point processes

The renewal Hawkes process is a nascent point process model that generalizes the Hawkes process. Although it has shown strong application potential, fitting the renewal Hawkes process to data remains a challenging task, especially on larger datasets. This article tackles this challenge by providing two approaches that significantly reduce the time required to fit renewal Hawkes processes. Since derivative-based methods for optimization, in general, converge faster than derivative-free methods, our first approach is to derive algorithms for evaluating the gradient and Hessian of the log-likelihood function and then use a derivative-based method, such as the Newton-Raphson method, in maximizing the likelihood, instead of the derivative-free method currently being used. Our second approach is to seek linear time algorithms that produce accurate approximations to the likelihood function, and then directly optimize the approximation to the log-likelihood function. Our simulation experiments show that the Newton-Raphson method reduces the computational time by about 30%. Furthermore, the approximate likelihood methods produce equally accurate estimates compared to the methods based on the exact likelihood and are about 20-40 times faster on datasets with about 10,000 events. We conclude with an analysis of price changes of several currencies relative to the US Dollar.