Estimating fractal dimensions: A comparative review and open source implementations.

Autor:	Datseris G; Department of Mathematics and Statistics, University of Exeter, EX4 4QF Exeter, United Kingdom., Kottlarz I; Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 Göttingen, Germany.; Department of Pharmacology and Toxicology, University Medical Center Göttingen (UMG), Robert-Koch-Str. 40, 37075 Göttingen, Germany.; Institute for the Dynamics of Complex Systems, University of Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany., Braun AP; Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 Göttingen, Germany.; Institute for the Dynamics of Complex Systems, University of Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany., Parlitz U; Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 Göttingen, Germany.; Institute for the Dynamics of Complex Systems, University of Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany.
Jazyk:	angličtina
Zdroj:	Chaos (Woodbury, N.Y.) [Chaos] 2023 Oct 01; Vol. 33 (10).
DOI:	10.1063/5.0160394
Abstrakt:	The fractal dimension is a central quantity in nonlinear dynamics and can be estimated via several different numerical techniques. In this review paper, we present a self-contained and comprehensive introduction to the fractal dimension. We collect and present various numerical estimators and focus on the three most promising ones: generalized entropy, correlation sum, and extreme value theory. We then perform an extensive quantitative evaluation of these estimators, comparing their performance and precision using different datasets and comparing the impact of features like length, noise, embedding dimension, and falsify-ability, among many others. Our analysis shows that for synthetic noiseless data, the correlation sum is the best estimator with extreme value theory following closely. For real experimental data, we found the correlation sum to be more strongly affected by noise vs the entropy and extreme value theory. The recent extreme value theory estimator seems powerful as it has some of the advantages of both alternative methods. However, using four different ways for checking for significance, we found that the method yielded "significant" low-dimensional results for inappropriate data like stock market timeseries. This fact, combined with some ambiguities we found in the literature of the method applications, has implications for both previous and future real-world applications using the extreme value theory approach, as, for example, the argument for small effective dimensionality in the data cannot come from the method itself. All algorithms discussed are implemented as performant and easy to use open source code via the DynamicalSystems.jl library. (© 2023 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).)
Databáze:	MEDLINE
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