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I have fitted an ETAS model (unmarked) with an independent exponential magnitude distribution to a set of earthquake data. However, I do not know how to calculate the probability of the earthquake occurrence (or the probability of no events) during a certain time interval.
Is there an explicit formula to compute the probability, like the one in the Poisson model?
An epidemic-type aftershock sequence (ETAS) model is a marked point process model.
Typically, a temporal model has the form (Ogata, 1988):
$$ \lambda^*(t) = \mu + \sum k10^{\alpha(M_i-M_c)} (t_i+c)^{-p} $$
Where $\lambda^*(t)$ is a conditional intensity function. If you are using no marks (e.g. earthquake magnitudes) you may be dealing with a Hawkse process.
Then the likelihood of an event can be written as:
$$f(t|\mathcal{H}_{t_n}) = \lambda^*(t)e^{-\int^t_{t_n}{\lambda^*(s) ds}} $$
This is in fact what is optimized in the maximum likelihood estimation, typically using the log likelihood of all the events:
$$ logL = \sum log\lambda^*(t_i) -\int_0^T \lambda^*(s)ds $$
If I understand the question correctly. The probabibily that an event does not occur in between the time of an event $t_i$ and some time $t$ can be written as $1-F(t)$ where $F(t)$ is the cumulative distribution of the event likelihood such that:
$$1-F(t) = \exp\left(-\int_{t_n}^t \lambda^*(s)ds\right)$$
I do not know the exact form of the ETAS model you are using, but it should be reasonably straightforward to implement from here on end.
Refer to this excellent note set for more: https://arxiv.org/pdf/1806.00221.pdf
