A Time-Scaled ETAS (Epidemic Type Aftershock Sequence)) Model based on the works of Y. Ogata and J. Zhuang post time-scaling as per the works of J. F. Lawless, T. Duchesne

The time-scaled ETAS model is a statistical model that describes the temporal and spatial patterns of aftershocks following a main shock, altering the time-scale such that the time between events are such that it is easier to distinguish between whether the event is a background event or triggered event. The model is based on the Epidemic Type Aftershock Sequence (ETAS) model and it takes into account the time-dependent behaviour of aftershocks. The model is defined by a set of parameters, including the background rate of earthquakes, the intensity of the main shock, and the rate at which aftershocks decay with time. These parameters are estimated from the data using a maximum likelihood estimation technique. The time-scaled ETAS model also includes a spatiotemporal component, which accounts for the spatial distribution of earthquakes and the temporal clustering of aftershocks, and is defined by a set of spatial and temporal kernels, which are estimated via maximum likelihood estimation. We have obtained results for model performance of the ETAS model via two approaches. The first approach leads to assuming the ETAS model as a Marked Point Process and maximizing the resulting log-likelihood (read: minimizing negative log-likelihood) iteratively using the optimization procedure proposed by Nelder – Mead (1965). By this approach, we have, for each type of time-scaled data, considered the ETAS model as a Ground Intensity Function for two models where the underlying probability distribution of event magnitudes is either exponential or gamma, and estimates of the parameters have been thus compared. The second approach entails a similar path, except that for likelihood maximization (or negative log likelihood minimization), we opt for the Davidon – Fletcher – Powell (DFP) algorithm for parameter optimization, similar to the model proposed by Ogata (1998) & Zhuang (2006), coined Iterative Stochastic De-clustering Method (ISDM) by A. Jalilian (2019).