Computing return times or return periods with rare event algorithms

Abstract

The average time between two occurrences of the same event, referred to as its return time (or return period), is a useful statistical concept for practical applications. For instance insurances or public agency may be interested by the return time of a 10 m flood of the Seine river in Paris. However, due to their scarcity, reliably estimating return times for rare events is very difficult using either observational data or direct numerical simulations. For rare events, an estimator for return times can be built from the extrema of the observable on trajectory blocks. Here, we show that this estimator can be improved to remain accurate for return times of the order of the block size. More importantly, we show that this approach can be generalised to estimate return times from numerical algorithms specifically designed to sample rare events. So far those algorithms often compute probabilities, rather than return times. The approach we propose provides a computationally extremely efficient way to estimate numerically the return times of rare events for a dynamical system, gaining several orders of magnitude of computational costs. We illustrate the method on two kinds of observables, instantaneous and time-averaged, using two different rare event algorithms, for a simple stochastic process, the Ornstein-Uhlenbeck process. As an example of realistic applications to complex systems, we finally discuss extreme values of the drag on an object in a turbulent flow.

Computation of Extreme Values of Time Averaged Observables in Climate Models with Large Deviation Techniques

“…If on the contrary the resampling time is much larger than the autocorrelation time, the trajectories will fall back to the typical states, and the importance sampling efficiency will be lost. Tests with simpler systems [22] have shown that the precise value of τ does not affect the results, as long as it is of the order of the autocorrelation time. The trajectories are perturbed after cloning by adding a small random field to the surface pressure, as described in [26].…”

“…Thanks to the rare event algorithm, we can extend the estimate of the return time curve to much rarer events. The return time can be computed from the output of the rare event algorithm as described in details in [26] and [22]. The red line in figure 7b) has been obtained by computing return time functions from the experiments with the algorithm (the case k * = 20 and k * = 40 repeated twice with different sets of initial conditions to improve the statistics) and averaging the results in the areas of overlap.…”

“…Using (24) instead of (23) does not change the results of the estimate or the convergence region, but gives smaller statistical errors where they can be computed. In the following we keep the simpler notation (22) for ease of presentation. As discussed in the main text, in a practical application one is constrained by the fixed length T of the time-series, and the choice of τ b and N b has to be considered carefully.…”

Adaptive multilevel splitting: Historical perspective and recent results

“…Finally, because AMS is used with a deterministic final time, it is worth noted that the best importance function should depend on time (see [37] section IIIC, see also [7]). Also note that using all these trajectories with intermediate levels , it is easy to generate an approximation of the curve → r( ).…”

Machine Learning of committor functions for predicting high impact climate events

“…A committor function is the probability that an event will occur or not in the future, as a function of the current state of the system. For the El-Niño case, this committor function will be the probability that an observable O of the system reaches a given threshold within a time T [18]. The definition and mathematical properties of committor functions are introduced in section III.…”