The estimation of the five parameters (λ0, K 0, α, c , p ) of the ETAS model (equation (3)) is carried out by the maximum likelihood method. The log likelihood with respect to the occurrence times of the earthquakes t i is given by

where t s and t e define the starting and the ending time of the activity [Ogata , 1993].

[23] Model selection, particularly the determination of the number of parameters, is carried out using the Akaike information criterion (AIC) [Akaike , 1974; Parzen et al. , 1998]. The statistic AIC = −2 MLL + 2k is computed for each of the models fit to the same data, where MLL is the maximum log likelihood value with respect to the parameters, and k is the total number of fitted parameters. In comparing models with different numbers of parameters, addition of the quantity 2k roughly compensates for the additional flexibility which the extra parameters provide. The model with the lower AIC value is taken as giving the better choice for forward prediction purposes. Insofar as it depends on the likelihood ratio, the AIC can also be used as a rough guide for testing the model. In testing a model with k + d parameters against a null hypothesis with just k parameters, we take a difference of 2 in AIC values as a rough estimate of significance at the 5% level. An alternative to the Akaike information criterion is the modified version of Schwarz's information criterion, BIC = MLL − 0.5k ln(N /2π), where N is the number of events [Main et al. , 1999]. In this case, the best model has the largest BIC value. The BIC criterion has been shown to be superior in the case of larger data sets [Koehler and Murphree , 1988].

--Please refer to "Detecting fluid signals in seismicity data through statistical earthquake modeling"