# ETAS Space-Time Modeling of Chile Triggered Seismicity Using Covariates: Some Preliminary Results
## 3 Branching Point Processes and ETAS Model  
Branching processes are used to model reproduction phenomena. These models have been recently considered for the description of different application fields: biology [27], demography [28], epidemiology [29; 30]. Any analytic spatio-temporal point process is uniquely characterized by its associated _conditional intensity function_ (CIF) [31] that represents the instantaneous rate or hazard for events at time \(t\) and location \(\mathbf{s}\) given all the observations up to time \(t\), conditioning on the random past history of the process \(\mathcal{H}_{t}\). The conditional intensity function of the branching model is defined as the sum of a term describing the large time- scale variation (spontaneous activity or background) and one relative to the small time-scale variation due to the interaction with the events in the past (triggered activity or offspring):  
\[\lambda_{\boldsymbol{\theta}}(t,\mathbf{s}|\mathcal{H}_{t})=\mu f(\mathbf{s} )+\tau_{\boldsymbol{\phi}}(t,\mathbf{s}) \tag{1}\]  
with \(\boldsymbol{\theta}=(\boldsymbol{\phi},\mu)^{\prime}\), the vector of parameters of the triggered intensity (\(\boldsymbol{\phi}\)) together with the parameter of the background general intensity (\(\mu\)), \(f(\mathbf{s})\) the space density, and \(\tau_{\boldsymbol{\phi}}(t,\mathbf{s})\) the triggered intensity, given by:  
\[\tau_{\boldsymbol{\phi}}(t,\mathbf{s})=\sum_{t_{j}<t}\
u_{\boldsymbol{\phi}} (t-t_{j},\mathbf{s}-\mathbf{s}_{j}).\]  
where \(\
u_{\boldsymbol{\phi}}(t-t_{j},\mathbf{s}-\mathbf{s}_{j})\) is the space-time intensity at \((t,\mathbf{s})\) triggered by a previous event at time \(t_{j}\). The self-exciting component of the model basically provides a description of the intensity at a space-time location \((t,\mathbf{s})\) caused by each previous event. As introduced above, a branching process for earthquake description, widely used in seismological context is the Epidemic Type Aftershocks-Sequences (ETAS) model [9]. The ETAS conditional intensity function can be written, starting from model (1), as follows:  
\[\lambda_{\boldsymbol{\theta}}(t,\mathbf{s}|H_{t})=\mu f(\mathbf{s})+\sum_{t_{ j}<t}\frac{\kappa_{0}\,\exp\{\alpha(m_{j}-m_{0})\}}{(t-t_{j}+c)^{p}}\Big{\{}( \mathbf{s}-\mathbf{s}_{j})^{2}+d\Big{\}}^{-q} \tag{2}\]  
where \(f(\mathbf{s})\) is the space density of the background/long term component, stationary in time. The aftershock (triggered) component is the product of the density of aftershocks in time, i.e., the Omori law representing the occurrence rate of aftershocks at time \(t\), following the earthquake of time \(t_{j}\) and magnitude \(m_{j}\), and the density of aftershocks in space. In particular, \(m_{j}\) is the magnitude of the \(j\)-th event and \(m_{0}\) the threshold magnitude, \(k\) is a normalizing constant, \(c\) and \(p\) characteristic parameters of the seismic activity of the given region; \(p\) is useful for characterizing the pattern of seismicity, indicating the decay rate of aftershocks in time; \(d\) and \(q\) are two parameters related to the spatial influence of the mainshock. For the spatial triggered distribution, conditioned to magnitude of the generating event, the distribution is:  
\[\ell(\mathbf{s}-\mathbf{s}_{j}|m_{j})=\left\{\frac{(\mathbf{s}-\mathbf{s}_{j })^{2}}{e^{\gamma\,\,(m_{j}-m_{0})}}+d\right\}^{-q} \tag{3}\]  
relating the occurrence rate of aftershocks to the mainshock magnitude \(m_{j}\), through the parameters \(\alpha,\gamma\) that measure the influence on the relative weight of each sequence. The FLP approach, which is a nonparametric estimation procedure based on the subsequent increments of log-likelihood obtained by adding an observation one at a time, to account for the information of the observations until \(t_{k}\) on the next one, has been developed for estimating the background component [11; 32]. The package etasFLP[11; 33; 34] provides tools to implement this mixed approach for a wide class of ETAS models for the description of seismic events, developed in an R environment [20].