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On the Computation of the Extremal Index for Time Series

  • Th. Caby
  • D. Faranda
  • S. VaientiEmail author
  • P. Yiou
Article
  • 12 Downloads

Abstract

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.

Keywords

Extreme value theory Extremal index Random dynamical systems Poisson statistics 

Notes

Acknowledgements

We would like to thank Jorge Freitas, Michele Gianfelice, Nicolai Haydn and Giorgio Mantica for several discussions related to different parts of this work. PY was supported by ERC Grant No. 338965-A2C2. The authors thank the anonymous referees who suggested to formulate Proposition 1 in a better form and indicated us the appropriate reference [14].

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Aix Marseille Université, Université de Toulon, CNRS, CPTMarseilleFrance
  2. 2.Center for Nonlinear and Complex Systems, Dipartimento di Scienza ed Alta TecnologiaUniversità degli Studi dell’InsubriaComoItaly
  3. 3.Laboratoire des Sciences du Climat et de l’Environnement, UMR 8212 CEA-CNRS-UVSQIPSL and Université Paris-SaclayGif-sur-YvetteFrance
  4. 4.London Mathematical LaboratoryLondonUK

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