An overview of the extremal index - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Chaos: An Interdisciplinary Journal of Nonlinear Science Année : 2019

An overview of the extremal index

Résumé

For a wide class of stationary time series, extreme value theory provides limiting distributions for rare events. The theory describes not only the size of extremes, but also how often they occur. In practice, it is often observed that extremes cluster in time. Such short-range clustering is also accommodated by extreme value theory via the so-called extremal index. This review provides an introduction to the extremal index by working through a number of its intuitive interpretations. Thus, depending on the context, the extremal index may represent (i) the loss of iid degrees of freedom, (ii) the multiplicity of a compound Poisson point process, (iii) the inverse mean duration of extreme clusters. More recently, the extremal index has also been used to quantify (iv) recurrences around unstable fixed points in dynamical systems. Whether extreme events occur in isolation or in clusters is an important question for their prediction and mitigation. Extreme value theory can accommodate clustering via the extremal index which, heuristically, measures the size of the cluster. Mathematically, clustering is most conveniently understood within the framework of point processes. Without clustering, extreme events occur in the manner of a Poisson process. With clustering, extreme events are bunched together in a compound Poisson process. In order to develop an intuition for the extremal index, we survey some simple examples from stochastic processes , real-world time series and dynamical systems .
Fichier principal
Vignette du fichier
extremal_index.pdf (677.9 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-02334271 , version 1 (28-10-2019)

Identifiants

Citer

Nicholas Moloney, Davide Faranda, Yuzuru Sato. An overview of the extremal index. Chaos: An Interdisciplinary Journal of Nonlinear Science, 2019, 29 (2), pp.022101. ⟨10.1063/1.5079656⟩. ⟨hal-02334271⟩
83 Consultations
1324 Téléchargements

Altmetric

Partager

Gmail Mastodon Facebook X LinkedIn More