Estimation of space-time branching process models in seismology using an EM-type algorithm

Maximum likelihood estimation of branching point process models via numerical optimization procedures can be unstable and computationally intensive. We explore an alternative estimation method based on the expectation-maximization algorithm. The method involves viewing the estimation of such branching processes as analogous to incomplete data problems. Using an application from seismology, we show how the epidemic-type aftershock sequence (ETAS) model can, in fact, be estimated this way, and we propose a computationally efficient procedure to maximize the expected complete data log-likelihood function. Using a space-time ETAS model, we demonstrate that this method is extremely robust and accurate and use it to estimate declustered background seismicity rates of geologically distinct regions in Southern California. All regions show similar declustered background intensity estimates except for the one covering the southern section of the San Andreas fault system to the east of San Diego in which a substantially higher intensity is observed.