Extreme value distributions of observation recurrences

Over the past few years, extreme value theory (EVT) has increasingly found use in the analysis of dynamical systems, especially to quantify the probabilities of rare events, defined as visits to particular small subsets of the phase space. This approach allows the description of the asymptotic statistics of hitting times and the number of small-set visits. It has been usefully applied, in particular, to understanding the recurrence properties of chaotic systems in climate science. This includes applications to the recurrence properties of the North-Atlantic sea-level pressure fields, as well as a number of instantaneous metrics tracking the rarity, predictability and persistence of atmospheric jet states and circulation patterns. The same metrics have also been used to classify and evaluate the dynamical consistency of state-of-the art climate models in representing the atmospheric dynamics, revealing significant shifts in the local dimensions between 1850 and 2100 in various datasets.

In a new paper, LML External Fellow Davide Faranda and colleagues offer an in-depth review of the fundamental basis of the EVT approach, as well as practical techniques useful in studying the recurrence statistics of smooth observables of chaotic dynamical systems. They formulate general results which can be applied to a wide range of situations. The dynamical systems they consider all have strong mixing properties, exhibiting exponential decay of correlations on suitable spaces of observables. In this context, the researchers examine how an efficient perturbative theory can be used to compute the extremal index in a broad variety of situations.

Faranda and colleagues expect that probabilistic analyses should make it possible to extend these results to a larger class of systems, including non-uniformly hyperbolic systems or those exhibiting intermittency. The perspective offered in this paper also considers how more complex systems and physical time series can be tested and interpreted in the framework of the EVT approach.

The paper is Th Caby, D Faranda, S Vaienti & P Yiou, 2021 Nonlinearity 34 118 and is available at https://hal.archives-ouvertes.fr/hal-02904585

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