EM Algorithm for the Estimation of the RETAS Model

The Renewal Epidemic-Type Aftershock Sequence (RETAS) model is a recently proposed point process model that can fit event sequences such as earthquakes better than preexisting models. Evaluating the log-likelihood function and directly maximizing it has been shown to be a viable approach to obtain the maximum likelihood estimator (MLE) of the RETAS model. However, the direct likelihood maximization suffers from numerical issues such as premature termination of parameter searching and sensitivity to the initial value. In this work, we propose to use the Expectation-Maximization (EM) algorithm as a numerically more stable alternative to obtain the MLE of the RETAS model. We propose two choices of the latent variables, leading to two variants of the EM algorithm. As well as deriving the conditional distribution of the latent variables given the observed data required in the E-step of each EM-cycle, we propose an approximation approach to speed up the E-step. The resulting approximate EM algorithms can obtain the MLE much faster without compromising on the accuracy of the solution. These newly developed EM algorithms are shown to perform well in simulation studies and are applied to an Italian earthquake catalog. Supplementary materials for this article are available online.