When are Extreme Events the better predictable, the larger they are?

TL;DR

This paper explores when large extreme events are more predictable than smaller ones in time series, revealing that predictability depends on the underlying probability distribution, with Gaussian distributions favoring larger event predictability.

Contribution

It analytically and numerically demonstrates how the predictability of large events varies with the probability distribution of the underlying process, using a simple prediction algorithm and ROC analysis.

Findings

Large increments are more predictable in Gaussian processes.

Large increments are harder to predict in power law tail distributions.

No significant dependence on event size for exponential distributions.

Abstract

We investigate the predictability of extreme events in time series. The focus of this work is to understand under which circumstances large events are better predictable than smaller events. Therefore we use a simple prediction algorithm based on precursory structures which are identified using the maximum likelihood principle. Using the receiver operator characteristic curve as a measure for the quality of predictions we find that the dependence on the event magnitude is closely linked to the probability distribution function of the underlying stochastic process. We evaluate this dependence on the probability distribution function analytically and numerically. If we assume that the optimal precursory structures are used to make the predictions, we find that large increments are better predictable if the underlying stochastic process has a Gaussian probability distribution function, whereas larger increments are harder to predict if the underlying probability distribution function has a power law tail. In the case of an exponential distribution function we find no significant dependence on the event magnitude. Furthermore we compare these results with predictions of increments in correlated data, namely, velocity increments of a free jet flow. The velocity increments in the free jet flow are in dependence on the time scale either asymptotically Gaussian or asymptotically exponential distributed. The numerical results for predictions within free jet data are in good agreement with the previous analytical considerations for random numbers.