On the Computation of the Extremal Index for Time Series

  • Th. Caby
  • D. Faranda
  • S. VaientiEmail author
  • P. Yiou


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.


Extreme 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 [14].


  1. 1.
    Faranda, D., Moreira Freitas, A.C., Milhazes Freitas, J., Holland, M., Kuna, T., Lucarini, V., Nicol, M., Todd, M., Vaienti, S.: Extremes and Recurrence in Dynamical Systems. Wiley, New York (2016)zbMATHGoogle Scholar
  2. 2.
    Faranda, D., Lucarini, V., Manneville, P., Wouters, J.: On using extreme values to detect global stability thresholds in multi-stable systems: the case of transitional plane Couette flow. Chaos Solit. Fractals 64, 26–35 (2014)ADSCrossRefGoogle Scholar
  3. 3.
    Lucarini, V., Faranda, D., Wouters, J., Kuna, T.: Towards a general theory of extremes for observables of chaotic dynamical systems. J. Stat. Phys. 154(3), 723–750 (2014)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Lucarini, V., Blender, R., Herbert, C., Ragone, F., Pascale, S., Wouters, J.: Mathematical and physical ideas for climate science. Rev. Geophys. 52(4), 809–859 (2014)ADSCrossRefGoogle Scholar
  5. 5.
    Moloney, N., Faranda, D., Sato, Y.: An overview of the extremal index. J. Nonlinear Sci. 29, 022101 (2018)MathSciNetGoogle Scholar
  6. 6.
    Faranda, D., Messori, G., Yiou, P.: Dynamical proxies of North Atlantic predictability and extremes. Sci. Rep. 7, 41278 (2017)ADSCrossRefGoogle Scholar
  7. 7.
    Caballero, R., Faranda, D., Messori, G.: A dynamical systems approach to studying midlatitude weather extremes. Geophys. Res. Lett. 44(7), 3346–3354 (2017)ADSCrossRefGoogle Scholar
  8. 8.
    Faranda, D., Sato, Y., Messori, G., Moloney, N.R., Yiou, P.: Minimal dynamical systems model of the northern hemisphere jet stream via embedding of climate data. Earth Syst. Dyn. 10, 555–567 (2019). ADSCrossRefGoogle Scholar
  9. 9.
    Faranda, D., Alvarez-Castro, M.C., Messori, G., Rodrigues, D., Yiou, P.: The hammam effect or how a warm ocean enhances large scale atmospheric predictability. Nat. Commun. 10(1), 1316 (2019)ADSCrossRefGoogle Scholar
  10. 10.
    Faranda, D., Vaienti, S.: Correlation dimension and phase space contraction via extreme value theory. Chaos 28, 041103 (2018)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Freitas, A.C.M., Freitas, J.M., Todd, M.: The extremal index, hitting time statistics and periodicity. Adv. Math. 231(5), 2626–2665 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Caby, T., Faranda, D., Mantica, G., Vaienti, S., Yiou, P.: Generalized dimensions, large deviations and the distribution of rare events. arXiv:1812.00036
  13. 13.
    Keller, G., Liverani, C.: Rare events, escape rates and quasistationarity: some exact formulae. J. Stat. Phys. 135, 519–534 (2009)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Azevedo, D., Freitas, A.-C.M., Freitas, J.-M., Rodrigues, F.B.: Extreme value laws for dynamical systems with countable extremal sets. J. Stat. Phys. 167(5), 12441261 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Keller, G.: Rare events, exponential hitting times and extremal indices via spectral perturbation. Dyn. Syst. 27, 11–27 (2012)MathSciNetCrossRefGoogle Scholar
  16. 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
  17. 17.
    Haydn, N., Vaienti, S.: The compound Poisson distribution and return times in dynamical systems. Probab. Theory Related Fields 144(3–4), 517–542 (2009)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Aytac, H., Freitas, J.M., Vaienti, S.: Laws of rare events for deterministic and random dynamical systems. Trans. Am. Math. Soc. 367(11), 8229–8278 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Leadbetter, M.R.: On extreme values in stationary sequences. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 28, 289–303 (1973/1974)Google Scholar
  20. 20.
    O’Brien, G.L.: Extreme values for stationary and Markov sequences. Ann. Probab. 15, 281–291 (1987)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Bradley, R.C.: Basic properties of strong mixing conditions. A survey and some open questions. Probab. Surv. 2, 107–144 (2005)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Chernick, M.R., Hsing, T., McCormick, W.P.: Calculating the extremal index for a class of stationary sequences. Adv. Appl. Probab. 23(4), 835–850 (1981)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Carvalho, M., Freitas, A.C.M., Milhazes Freitas, J., Holland, M., Nicol, M.: Extremal dichotomy for uniformly hyperbolic systems. Dyn. Syst. 30(4), 383–403 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Holland, M., Nicol, M., Torok, A.: Extreme value theory for non-uniformly expanding dynamical systems. Trans. Am. Math. Soc. 364(2), 661–688 (2012)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Azevedo, D., Freitas, A.C.M., Freitas, J.M., Rodrigues, F.B.: Clustering of extreme events created by multiple correlated maxima. Phys. D 315, 3348 (2016)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Mantica, G., Perotti, L.: Extreme value laws for fractal intensity functions in dynamical systems: Minkowski analysis. J. Phys. A 49(37), 374001 (2016)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Freitas, A.C.M., Freitas, J.M., Rodrigues, F.B., Soares, J.V.: Rare events for cantor target sets. arXiv:1903.07200 (2019)
  28. 28.
    Chazottes, J.-R., Coelho, Z., Collet, P.: Poisson processes for subsystems of nite type in symbolic dynamics. Stoch. Dyn. 9(3), 393422 (2009)CrossRefGoogle Scholar
  29. 29.
    Keller, G., Liverani, C.: Rare events, escape rates and quasistationarity: some exact formulae. J. Stat. Phys. 135(3), 519–534 (2009)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Faranda, D., Ghoudi, H., Guiraud, P., Vaienti, S.: Extreme value theory for synchronization of coupled map lattices. Nonlinearity 31(7), 332–3358 (2018)MathSciNetCrossRefGoogle Scholar
  31. 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
  32. 32.
    Süveges, M.: Likelihood estimation of the extremal index. Extremes 10, 41–55 (2017)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Ferreira, M.: Heuristic tools for the estimation of the extremal index: a comparison of methods. REVSTAT Stat. J. 16, 115–136 (2018)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Arnold, V.I., Avez, A.: Ergodic Problems of Classical Mechanics. The Mathematical Physics Monograph Series. W. A. Benjamin Inc., New York (1968)zbMATHGoogle Scholar
  35. 35.
    Freitas, A.C.M., Freitas, J.M., Vaienti, S.: Extreme value laws for nonstationary processes generated by sequential and random dynamical systems. Annales de l’Institut Henri Poincaré 53, 1341–1370 (2017)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Conze, J.P., Raugi, A.: Limit theorems for sequential expanding dynamical systems on \([0, 1]\). Ergod. Theory Relat. Fields. Contemp. Math. 430, 89–121 (2007)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Haydn, N., Nicol, M., Torok, A., Vaienti, S.: Almost sure invariance principle for sequential and non-stationary dynamical systems. Trans. Am. Math. Soc. 36, 5293–5316 (2017)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Arnold, L.: Random Dynamics. Springer Monographs in Mathematics. Springer, Berlin (1998)Google Scholar
  39. 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. 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)
  41. 41.
    Faranda, D., Freitas, J.-M., Lucarini, V., Turchetti, G., Vaienti, S.: Extreme value statistics for dynamical systems with noise. Nonlinearity 26, 2597–2622 (2013)ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    Faranda, D., Ghoudi, H., Guiraud, P., Vaienti, S.: Extreme value theory for synchronization of coupled map lattices. Nonlinearity 31, 3326–3358 (2018)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Faranda, D., Vaienti, S.: Extreme value laws for dynamical systems under observational noise. Phys. D 280, 86–94 (2014)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Moreira Freitas, A.C., Milhazes Freitas, J., Todd, M.: The compound Poisson limit ruling periodic extreme behaviour of non-uniformly hyperbolic dynamics. Commun. Math. Phys. 321(2), 483527 (2013)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Haydn, N., Vaienti, S.: Limiting entry times distribution for arbitrary sets. arXiv:1904.08733 (2019)
  46. 46.
    Coelho, Z., Collet, P.: Asymptotic limit law for the close approach of two trajectories in expanding maps of the circle. Prob. Theory Relat. Fields 99, 237–250 (1994)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Gora, P., Boyarsky, A.: Laws of Chaos. Birkhäuser, Basel (1997)zbMATHGoogle Scholar
  48. 48.
    Gallo, S., Haydn, N., Vaienti, S.: in preparationGoogle Scholar

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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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