Estimation of Space-Time Branching Process Models in Seismology Using an EM-Type Algorithm

• Published in: Journal of the American Statistical Association Vol. 103; no. 482; pp. 614 - 624
• Main Authors: Veen, Alejandro, Schoenberg, Frederic P
• Format: Journal Article
• Language: English
• Published: Alexandria, VA Taylor & Francis 01.06.2008; American Statistical Association; Taylor & Francis Ltd
• Subjects: Algorithms; Applications; Applications and Case Studies; Branching process models; Disease models; Earthquakes; Epidemic-type aftershock sequence model; Epidemiology; Estimate reliability; Estimating techniques; Estimation; Estimation bias; Exact sciences and technology; Expectation-maximization algorithm; General topics; Geology; Markov processes; Mathematics; Maximum likelihood; Maximum likelihood estimation; Maximum likelihood method; Modeling; Optimization; Parametric inference; Probability and statistics; Probability theory and stochastic processes; Regression analysis; Robustness (mathematics); Sciences and techniques of general use; Seismicity; Seismology; Simulations; Space-time point process models; Spacetime; Statistical methods; Statistics; Stochastic processes; United States > US; Southern California; Expectation-maximization algorithm; Epidemic-type aftershock sequence model; Covering problem; Optimization method; Optimization; Hypothesis test; Statistical test; Branching process models; Earthquakes; Log likelihood; Incomplete information; Likelihood function; Mathematical programming; Epidemic; Maximization; Numerical model; Space time; Statistical method; Space-time point process models; Branching process; Numerical analysis; Point estimation; Point process; Maximum likelihood; Expectation; EM algorithm; Application
• ISSN: 0162-1459; 1537-274X
• DOI: 10.1198/016214508000000148

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.