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Area of Science:

  • Nonlinear Dynamics
  • Complex Systems Analysis
  • Data Science

Background:

  • The fractal dimension quantifies complexity in dynamical systems.
  • Various numerical techniques exist for estimating fractal dimension.
  • A comprehensive review and evaluation of these estimators are needed.

Purpose of the Study:

  • To provide a self-contained introduction to fractal dimension estimation.
  • To evaluate and compare the performance of key numerical estimators.
  • To assess the impact of data characteristics on estimator precision.

Main Methods:

  • Review of numerical techniques for fractal dimension estimation.
  • Focus on generalized entropy, correlation sum, and extreme value theory.
  • Quantitative evaluation using synthetic and real experimental datasets.

Main Results:

  • Correlation sum is optimal for synthetic, noiseless data.
  • Entropy and extreme value theory are more noise-resilient for experimental data.
  • Extreme value theory may yield spurious low-dimensional results for inappropriate data.

Conclusions:

  • The choice of fractal dimension estimator depends on data characteristics (e.g., noise).
  • Extreme value theory shows promise but requires careful validation for real-world applications.
  • Open-source implementations are available for discussed algorithms.