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Sampling Hyperspheres via Extreme Value Theory: Implications for Measuring Attractor Dimensions

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Abstract

The attractor Hausdorff dimension is an important quantity bridging information theory and dynamical systems, as it is related to the number of effective degrees of freedom of the underlying dynamical system. By using the link between extreme value theory and Poincaré recurrences, it is possible to estimate this quantity from time series of high-dimensional systems without embedding the data. In general \(d \le n\), where n is the dimension of the full phase-space, as the dynamics freezes some of the available degrees of freedom. This is equivalent to constraining trajectories on a compact object in phase space, namely the attractor. Information theory shows that the equality \(d=n\) holds for random systems. However, applying extreme value theory, we show that this result cannot be recovered and that \(d<n\). We attribute this effect to the curse of dimensionality, and in particular to the phenomenon of concentration of the norm observed in high-dimensional systems. We derive a theoretical expression for d(n) for Gaussian random vectors, and we show numerically that similar curse of dimensionality effects are found for random systems characterized by non-Gaussian distributions. Finally, we show that the effect of the curse of dimensionality can be quantified using the extreme value theory, thus enabling to retrieve the degree of non-randomness of a system. We provide examples issued from real-world climate and financial datasets.

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Notes

  1. For a definition of the concept of curse of dimensionality, we quote Cabestany et al. [9], who define it as: “the expression of all phenomena that appear with high-dimensional data, and that have most often unfortunate consequences on the behavior and performances of learning algorithms”

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Acknowledgements

We thank P. Yiou, B. Dubrulle and S. Vaienti for useful discussion. D. F., M. C. A. C., G.M. acknowledge the support of the ERC Grant A2C2 n No. 338965. MCAC was further supported by the Swedish Research Council Vetenskapsrådet Grant No. C0629701 and GM was further supported by a grant from the Department of Meteorology of Stockholm University and by the Swedish Research Council Vetenskapsrådet Grant No. 2016-03724.

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Authors and Affiliations

  1. LSCE-IPSL, CEA Saclay l’Orme des Merisiers, CNRS UMR 8212 CEA-CNRS-UVSQ, Université Paris-Saclay, 91191, Gif-sur-Yvette, France

    Flavio Maria Emanuele Pons, Gabriele Messori, M. Carmen Alvarez-Castro & Davide Faranda

  2. Department of Earth Sciences, Uppsala University, 752 36, Uppsala, Sweden

    Gabriele Messori

  3. Department of Meteorology and Bolin Centre for Climate Research, Stockholm University, 106 91, Stockholm, Sweden

    Gabriele Messori

  4. Climate Simulation and Prediction Division, Centro Euro-Mediterraneo sui Cambiamenti Climatici, Bologna, 40127, Italy

    M. Carmen Alvarez-Castro

  5. London Mathematical Laboratory, 14 Buckingham Street, London, WC2N 6DF, UK

    Davide Faranda

  6. Department of Statistics, University of Bologna, via delle Belle Arti 41, 40126, Bologna, Italy

    Flavio Maria Emanuele Pons

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  1. Flavio Maria Emanuele Pons
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Correspondence to Flavio Maria Emanuele Pons.

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Communicated by Valerio Lucarini.

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Pons, F.M.E., Messori, G., Alvarez-Castro, M.C. et al. Sampling Hyperspheres via Extreme Value Theory: Implications for Measuring Attractor Dimensions. J Stat Phys 179, 1698–1717 (2020). https://doi.org/10.1007/s10955-020-02573-5

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