Rather than talking about the probability of occurrence of an event, it is more meaningful in many applications to talk about its return time: it is the average time between two occurrences of the event. Reliably estimating return times for rare events from observational data is very difficult, precisely because our sample is too small for such events. This problem made the headlines in the general press during the summer of 2017, when several tropical cyclones hit inhabited areas in the Carribean Islands and the continental US, wreaking havoc. For instance, in the wake of Hurricane Harvey, this article in the Washington Post discussed the concept of return time and touched upon the difficulty of estimating them.
With Thibault Lestang (ENS de Lyon), Francesco Ragone (University of Milan), Charles-Edouard Bréhier (Université Claude Bernard, Lyon) and Freddy Bouchet (ENS de Lyon), we recently published a preprint showing how to compute more efficiently return times using rare event algorithms. For a given dynamical system, these algorithms generate an ensemble of trajectories, just like classical Monte-Carlo methods. But unlike classical Monte-Carlo methods, these algorithms bias the ensemble statistics in a controlled way, so that our sample contains more rare events, defined in an appropriate manner. In that preprint, we show how these algorithms, which typically compute probabilities, can be modified to compute return times, and we suggest a variant of the adaptive multilevel splitting algorithm which is particularly simple to implement to carry out such computations for complex systems.
Computing return times or return periods with rare event algorithms
Thibault Lestang, Francesco Ragone, Charles-Edouard Bréhier, Corentin Herbert, Freddy Bouchet