On the Computation of the Extremal Index for Time Series

• Published in: Journal of statistical physics Vol. 179; no. 5-6; pp. 1666 - 1697
• Main Authors: Caby, Th, Faranda, D., Vaienti, S., Yiou, P.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-06-2020; Springer; Springer Nature B.V; Springer Verlag
• Subjects: Dynamical Systems; Extreme value theory; Extreme values; Liapunov exponents; Mathematical and Computational Physics; Mathematics; Periodic variations; Physical Chemistry; Physics; Physics and Astronomy; Quantum Physics; Statistical Physics and Dynamical Systems; Theoretical; Extremal index; Extreme value theory; Random dynamical systems; Poisson statistics
• ISSN: 0022-4715; 1572-9613
• DOI: 10.1007/s10955-019-02423-z

The extremal index is a quantity introduced in extreme value theory to measure the presence of clusters of exceedances. In the dynamical systems framework, it provides important information about the dynamics of the underlying systems. In this paper we provide a review of the meaning of the extremal index in dynamical systems. Depending on the observables used, this quantity can inform on local properties of attractors such as periodicity, stability and persistence in phase space, or on global properties such as the Lyapunov exponents. We also introduce a new estimator of the extremal index and show its relation with those previously introduced in the statistical literature. We reserve a particular focus to the systems perturbed with noise as they are a good paradigm of many natural phenomena. Different kind of noises are investigated in the annealed and quenched situations. Applications to climate data are also presented.