Sampling hyperspheres via extreme value theory: implications for measuring attractor dimensions
Résumé
The attractor dimension is an important quantity in information theory, 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 compute this quantity from time series of high-dimensional systems without embedding the data. In general d < 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 quanti ed 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 fi nancial datasets.
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