Fast likelihood computation of epidemic type aftershock‐sequence model

The Epidemic Type Aftershock‐Sequences (ETAS) model is a point process representing the temporal activity of earthquakes in a certain geophysical region. A method for fast and accurate computation of the likelihood is suggested for the ETAS model. Double sums of variables in the logarithm of the likelihood function with respect to the earthquake occurrence data is reduce to a single sum by using the recursive relation of the integrands in the Gamma transformation of the modified Omori functions. Then the integral is computed in unusually high accuracy by using the Double Exponential transformation followed by an application of the trapezoidal rule with an equal mesh size. The computing run time is greatly improved to be proportional to the number of recorded earthquakes N compared to the order of N ² for the naive computation method. The present numerical procedures put the parameter estimation of the ETAS model to practical use in large number of occurrence records.