On the Computation of the Extremal Index for Time Series

Abstract

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.

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Notes

  1. 1.

    Abadi et al. [16] built a dynamically generated stochastic processes with an extremal index for which that equality does not hold. They considered observable functions maximised at least two points of the phase space, where one of them is an indifferent periodic point and another one is either a repelling periodic point or a non periodic point. We will not consider these kind of observables in this paper.

  2. 2.

    This condition holds for instance when the invariant measure \(\mu \) is mixing with decay of correlation fast enough; sometimes a rate of decay as \(n^{-2}\) is sufficient.

  3. 3.

    Notice that this is equivalent to Eq. (22) by defining the map \(\omega \rightarrow f_{(\omega )'}\), where \((\omega )'\) is the \(\omega _1\) coordinate of \(\omega \).

  4. 4.

    These conditions essentially ensure that the transfer operator associated with the map T has a spectral gap and that the density h has finite oscillation in the neighborhood of the diagonal.

  5. 5.

    This time scale is relevant for the underlying dynamics of the atmospheric circulation.

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Acknowledgements

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].

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Appendix

Appendix

As mentioned in Sect. 3, results proving the existence of extreme value laws for non-stationary sequential systems are lacking. Nevertheless, the methods of estimation described in the paper can still be used to evaluate a quantity \(\theta \) that would correspond to the extremal index in the eventuality that an extreme value law holds for this kind of systems, which by the way was the case in [35]. We could also alternatively define the extremal index by computing the statistics of the number of visit and evaluating the expectation in Eq. (46). We choose this second option and consider the motion given by the concatenation

$$\begin{aligned} f_{\bar{\xi }}^n(x)=f_{\xi _n} \circ \cdots \circ f_{\xi _1} (x), \end{aligned}$$

where the probability law of the \(\xi _i\) changes over time. In particular, we consider the 10 maps

$$\begin{aligned} f_i(x)=2x+b_i \mod \ 1, \end{aligned}$$

where \(b_i\) is the ith component of a vector \({\bar{b}}\) of size 10, with entries equally spaced between 0 and 1 / 2. We consider sequences of time intervals \([\tau k+1,(k+1)\tau ]\), with \(\tau =10\), for \(k=0,1,2,\dots \) in which the weights associated to \(\xi _i\) are equal to \(p_i^k\). For every \(\tau \) iterations, the weight associated to each \(\xi _i\) changes randomly, with the only constraint that they sum to 1. \({\hat{\theta }}_5\) is computed considering a trajectory of \(5\times 10^7\) points, and a threshold u corresponding to the 0.995-quantile of the observable distribution. Using the same trajectory, we computed the empirical distribution of the number of visits in a ball centered at the origin and of radius \(r=e^{-u}\) in intervals of time of length \(\lfloor 2tr \rfloor \), with \(t=50\). This is indeed t times the Lebesgue measure (which is the invariant measure associated to all the maps \(f_i\)) of a ball of radius r centered in the origin. Choosing the same trajectory for the computation of \(\theta \) and for the statistics of visits is of crucial importance here, because different probability laws imply large variations of \(\theta \) depending on the trajectory considered. In Fig. 12, we observe again a perfect agreement between the empirical distribution of the number of visits and the Polyà-Aeppli distribution of parameters \(t=50\) and \({\hat{\theta }}_5\), which is equal to 0.92 for the trajectory presented in the Fig. 12. Although different trajectories give variations for \(\theta \), this agreement is stable against different trajectories and different values of t and \(\tau \), suggesting that a geometric Poisson distribution for the number of visits given by \(\theta \) is a universal feature, even for non-stationary scenarii.

Fig. 12
figure12

Comparison between the empirical distribution \(\mu (N_n(50)=k)\) and the Polyà-Aeppli distribution of parameters \(t=50\) and \({\hat{\theta }}_5\approx 0.92\), computed with a trajectory of length \(5\times 10^7\) of the sequential system described in the text. The procedure used to compute the empirical distribution is described in the text

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Caby, T., Faranda, D., Vaienti, S. et al. On the Computation of the Extremal Index for Time Series. J Stat Phys 179, 1666–1697 (2020). https://doi.org/10.1007/s10955-019-02423-z

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Keywords

  • Extreme value theory
  • Extremal index
  • Random dynamical systems
  • Poisson statistics