On the Computation of the Extremal Index for Time Series
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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.
KeywordsExtreme value theory Extremal index Random dynamical systems Poisson statistics
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 .
- 12.Caby, T., Faranda, D., Mantica, G., Vaienti, S., Yiou, P.: Generalized dimensions, large deviations and the distribution of rare events. arXiv:1812.00036
- 16.Abadi, M., Freitas, A.C.M., Freitas, J.M.: Dynamical counterexamples regarding the Extremal Index and the mean of the limiting cluster size distribution. arXiv:1808.02970
- 19.Leadbetter, M.R.: On extreme values in stationary sequences. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 28, 289–303 (1973/1974)Google Scholar
- 27.Freitas, A.C.M., Freitas, J.M., Rodrigues, F.B., Soares, J.V.: Rare events for cantor target sets. arXiv:1903.07200 (2019)
- 31.Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.L., De Waal, D., (Contributions by), Ferro, C. (Contributions by): Statistics of Extremes: Theory and Applications Google Scholar
- 38.Arnold, L.: Random Dynamics. Springer Monographs in Mathematics. Springer, Berlin (1998)Google Scholar
- 39.Gonzalez-Tokman, C.: Multiplicative ergodic theorems for transfer operators: towards the identification and analysis of coherent structures in non-autonomous dynamical systems, Contributions of Mexican mathematicians abroad in pure and applied mathematics Guanajuato, Mexico American Mathematical Society. Contemp. Math. 709, 31–52 (2018)MathSciNetCrossRefGoogle Scholar
- 40.Freitas, A.C.M., Freitas, J.M., Magalhaes, M., Vaienti, S.: Point processes of non stationary sequences generated by sequential and random fynamical systems. arXiv:1904.05761 (2019)
- 45.Haydn, N., Vaienti, S.: Limiting entry times distribution for arbitrary sets. arXiv:1904.08733 (2019)
- 48.Gallo, S., Haydn, N., Vaienti, S.: in preparationGoogle Scholar