# ETAS Space-Time Modeling of Chile Triggered Seismicity Using Covariates: Some Preliminary Results
## 3 Branching Point Processes and ETAS Model
ETAS Model with CovariatesThe ETAS model is widely used in the framework of statistical modeling of earthquake occurrence; however, in some applications, it may not adequately describe the triggered seismicity. In the classical ETAS model formulation, as in Equation (2), the average number of events triggered by an event at time \(t_{j}\) depends only on its magnitude \(m_{j}\). Ref. [15] proposed an ETAS model with covariates in the framework of the survival analysis. This model relates the average number of events triggered by an event at time \(t_{j}\) with the value of some covariates, including magnitude, corresponding to the \(j-\)th event. From a statistical point of view, the inclusion of new explanatory variables can significantly improve the fitting of the model to real observed data. These variables may also vary continuously in space, and their effect is incorporated in the triggering part of the model. The background component of the model is estimated using the Forward Likelihood for Prediction (FLP) approach [10; 11; 12]. As proposed by [35] in a context of infection occurrences, in [15] we incorporate the space-time phenomenological laws of the triggering part of the ETAS model with the effects of covariates. In particular, we model the covariates of the ETAS model as in a GLM framework, such that \(\eta_{j}\) is a classical linear predictor given by \(\eta_{j}=\mathbf{\beta}^{\prime}\mathbf{Z}_{j}\), where \(\mathbf{Z}_{j}\) is the vector of covariates observed for the \(j\)-th event and \(\mathbf{\beta}\) is a vector of unknown parameters. Therefore, the triggering function is factorized into separate effects of marks, time, and relative location:  
\[\lambda_{\tilde{\mathbf{\theta}}}(\mathbf{t},\mathbf{s}|H_{t})=\mu f(\mathbf{s})+\sum_{t_{j}<t }\frac{\kappa_{0}\;exp(\eta_{j})}{(t-t_{j}+c)^{p}}\Big{\{}(\mathbf{s}-\mathbf{s}_{j})^ {2}+d\Big{\}}^{-q} \tag{4}\]  
where \((t_{j},\mathbf{s}_{j})\) is the time and location of individual occurrence \(j\), \(\eta_{j}=\mathbf{\beta}^{\prime}\mathbf{Z}_{j}\) is a linear predictor, with \(\mathbf{Z}_{j}\) the external known covariate vector, including the magnitude (usually coinciding with the first covariate), acting in a multiplicative fashion on the base risk and \(\tilde{\theta}=(\mu,\kappa_{0},c,p,d,q,\mathbf{\beta})^{\prime}\), with \(\mathbf{\beta}\) a \(k\) component vector, to be estimated. More in details, in the usual ETAS model [9; 36], \(k=1\), \(\mathbf{Z}_{j1}=m_{j}-m_{0}\), and \(\beta_{1}=\alpha-\gamma\). In this model formulation, for an easier correspondence with the ETAS parametrization, in the \(\mathbf{\beta}\) vector a constant term is not included, since the presence of the parameter \(\kappa_{0}\) in the model. In the seismic context, that would provide a more general formalism for the earthquake occurrence in space and time. Indeed, the main idea is that the effect on the future activity depends not only on the closeness of the previous events but also on other characteristics of the main event, such as magnitude, as usual, and the distance from the faults, or other geological sources. An extended version of the package etasFLP for generalized offspring component is available at the CRAN [21].