On the Computation of the Extremal Index for Time Series

Many insights have recently emerged from applications of extreme value theory to the study of time series generated by dynamical systems. In particular the extremal index offers a powerful statistical indicator to discriminate among different qualitative types of dynamical behaviours. In a recent paper, LML External Fellow Davide Faranda, working with colleagues Théophile Caby, Sandro Vaienti and Pascal Yiou, give an overview of some rigorous mathematical results which show that the extremal index is very sensitive to stochastic perturbations affecting the otherwise deterministic evolution of a dynamical system.
The main aim of their work is to illustrate precisely how the extremal index is affected by noise. The authors start from a general formula for the index obtained by Keller and Liverani in the context of stationary dynamical systems, which enjoy a so-called “spectral gap” property. This formula encompasses all other rigorous formulae previously obtained for the computation of the extremal index. They then show that when this spectral formula holds, it implies a generalized version of another formula – the O’Brien formula – which has inspired several important numerical algorithms and statistical estimators.
As the researchers emphasize, this method can be used to compute the extremal index to any desired level of precision. The formula is particularly helpful in delicate computations for dynamical systems near unstable fixed points or periodic points. In the climate system, for example, unstable fixed points correspond to interesting weather regimes and the extremal index is related to their persistence. This technique, the authors believe, will be of use in computing precise extremal indices to improve the estimates of the typical lengths of cold or hot waves.
The paper is available at https://doi.org/10.1007/s10955-019-02423-z

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