Probability distribution of forecasts based on the ETAS model

Earthquake probability forecasts based on a point process model, which is defined with a conditional intensity function (e.g. ETAS), are generally made by using the history of the process to the current point in time, and by then simulating over the future time interval over which a forecast is required. By repeating the simulation multiple times, an estimate of the mean number of events together with the empirical probability distribution of event counts can be derived. This can involve a considerable amount of computation. Here we derive a recursive procedure to approximate the expected number of events when forecasts are based on the ETAS model. To assess the associated uncertainty of this expected number, we then derive the probability generating function of the distribution of the forecasted number of events. This theoretically derived distribution is very complex; hence we show how it can be approximated using the negative binomial distribution.