Robust extremes in chaotic deterministic systems

Vitolo, Renato; Holland, Martin; Ferro, Christopher A. T.

doi:10.1063/1.3270389

Cited by 12 publications

(13 citation statements)

References 45 publications

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“…Obviously, it is of crucial relevance, for both mathematical reasons and for devising a working framework to be used in applications, to understand under which circumstances the time series of observables of deterministic dynamical systems can be treated using the EVT. Empirical studies show that in some cases, the extremes of dynamical observables of chaotic systems obey up to a high degree of precision the GEV statistics, even if it is apparent that the asymptotic convergence is highly dependent on the considered observables (Felici et al, 2007;Vannitsem 2007;and Vitolo et al, 2009a;2009b). Instead, Balakrishnan et al (1995) and, more recently, Nicolis et al (2006) showed that when considering a dynamical system featuring a regular (periodic or quasiperiodic) motion, the extremes of a generic dynamical observable do not obey any statistics compatible with those of GEV distributions.…”

Section: B Extreme Value Theory For Dynamical Systemsmentioning

confidence: 98%

Extreme value theory for singular measures

Lucarini

Faranda

Turchetti

et al. 2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

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In this paper, we perform an analytical and numerical study of the extreme values of specific observables of dynamical systems possessing an invariant singular measure. Such observables are expressed as functions of the distance of the orbit of initial conditions with respect to a given point of the attractor. Using the block maxima approach, we show that the extremes are distributed according to the generalised extreme value distribution, where the parameters can be written as functions of the information dimension of the attractor. The numerical analysis is performed on a few low dimensional maps. For the Cantor ternary set and the Sierpinskij triangle, which can be constructed as iterated function systems, the inferred parameters show a very good agreement with the theoretical values. For strange attractors like those corresponding to the Lozi and Hènon maps, a slower convergence to the generalised extreme value distribution is observed. Nevertheless, the results are in good statistical agreement with the theoretical estimates. It is apparent that the analysis of extremes allows for capturing fundamental information of the geometrical structure of the attractor of the underlying dynamical system, the basic reason being that the chosen observables act as magnifying glass in the neighborhood of the point from which the distance is computed.

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Section: B Extreme Value Theory For Dynamical Systemsmentioning

confidence: 98%

Extreme value theory for singular measures

Lucarini

Faranda

Turchetti

et al. 2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…Until now, rigorous results have been obtained assuming the existence of an invariant measure for the dynamical systems and the fulfillment of independence requirement on the series of maxima achieved by imposing D ′ and D 2 mixing conditions, or, alternatively, assuming an exponential hitting time statistics [22][23][24][25]. The parameters of the GEV distribution obtained choosing as observables the function g i , i = 1, 2, 3 defined above depend on the local dimension of the attractor D and numerical algorithms to perform statistical inference can be set up for mixing systems having both absolutely continuous and singular invariant measures [27,28,39,40]. Instead, when considering systems with regular dynamics, the statistics of the block maxima of any observable does not converge to the GEV family [18,19].…”

Section: Discussionmentioning

confidence: 99%

Universal Behaviour of Extreme Value Statistics for Selected Observables of Dynamical Systems

2012

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The main results of the extreme value theory developed for the investigation of the observables of dynamical systems rely, up to now, on the Gnedenko approach. In this framework, extremes are basically identified with the block maxima of the time series of the chosen observable, in the limit of infinitely long blocks. It has been proved that, assuming suitable mixing conditions for the underlying dynamical systems, the extremes of a specific class of observables are distributed according to the so called Generalized Extreme Value (GEV) distribution. Direct calculations show that in the case of quasi-periodic dynamics the block maxima are not distributed according to the GEV distribution. In this paper we show that, in order to obtain a universal behaviour of the extremes, the requirement of a mixing dynamics can be relaxed if the Pareto approach is used, based upon considering the exceedances over a given threshold. Requiring that the invariant measure locally scales with a well defined exponent -the local dimension -, we show that the limiting distribution for the exceedances of the observables previously studied with the Gnedenko approach is a Generalized Pareto distribution where the parameters depends only on the local dimensions and the value of the threshold. This result allows to extend the extreme value theory for dynamical systems to the case of regular motions. We also provide connections with the results obtained with the Gnedenko approach. In order to provide further support to our findings, we present the results of numerical experiments carried out considering the well-known Chirikov standard map.

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“…Maximum likelihood is a common estimation method [10]: in this case, standard asymptotic theory also provides confidence intervals (uncertainties) for the point estimates. The estimated GEV parameters and associated uncertainties can then be used to derive other quantities of interest, such as return periods for given return levels of the variable of interest, see the above references and [16,17,52,53] for examples.…”

Section: Background On Extreme Value Theorymentioning

confidence: 99%

Extreme value laws in dynamical systems under physical observables

Holland

Vitolo

Rabassa

et al. 2012

Physica D: Nonlinear Phenomena

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Extreme value theory for chaotic dynamical systems is a rapidly expanding area of research. Given a system and a real function (observable) defined on its phase space, extreme value theory studies the limit probabilistic laws obeyed by large values attained by the observable along orbits of the system. Based on this theory, the so-called block maximum method is often used in applications for statistical prediction of large value occurrences. In this method, one performs inference for the parameters of the Generalised Extreme Value (GEV) distribution, using maxima over blocks of regularly sampled observations along an orbit of the system. The observables studied so far in the theory are expressed as functions of the distance with respect to a point, which is assumed to be a density point of the system's invariant measure. However, this is not the structure of the observables typically encountered in physical applications, such as windspeed or vorticity in atmospheric models. In this paper we consider extreme value limit laws for observables which are not functions of the distance from a density point of the dynamical system. In such cases, the limit laws are no longer determined by the functional form of the observable and the dimension of the invariant measure: they also depend on the specific geometry of the underlying attractor and of the observable's level sets. We present a collection of analytical and numerical results, starting with a toral hyperbolic automorphism as a simple template to illustrate the main ideas. We then formulate our main results for a uniformly hyperbolic system, the solenoid map. We also discuss non-uniformly hyperbolic examples of maps (H\'enon and Lozi maps) and of flows (the Lorenz63 and Lorenz84 models). Our purpose is to outline the main ideas and to highlight several serious problems found in the numerical estimation of the limit laws

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Robust extremes in chaotic deterministic systems

Cited by 12 publications

References 45 publications

Extreme value theory for singular measures

Extreme value theory for singular measures

Universal Behaviour of Extreme Value Statistics for Selected Observables of Dynamical Systems

Extreme value laws in dynamical systems under physical observables

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