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Area of Science:

  • Dynamical Systems and Chaos Theory
  • Statistical Physics
  • Data Analysis and Dimensionality

Background:

  • Accurate local dimension estimation is vital for analyzing multi-fractal dynamical systems and their degrees of freedom.
  • Traditional methods rely on pairwise distances, assuming constant local dimension for continuous random variables.

Purpose of the Study:

  • To derive and assess approximate analytical expressions for variations in estimated local dimensions of absolutely continuous random variables.
  • To investigate how probability density function, threshold, and phase-space dimension influence local dimension estimation accuracy.

Main Methods:

  • Application of extreme value theory to estimate local dimensions from pairwise distance distributions.
  • Derivation of approximate analytical expressions for local dimension variations.
  • Numerical simulations across dimensions 1 to 30 to validate analytical findings.

Main Results:

  • Local dimension estimates can diverge from theoretical values due to uneven data sampling.
  • Variations depend on the probability density function (especially its Laplacian) and the chosen threshold.
  • Deviations are more pronounced for probability density functions with low absolute values and high Laplacian values.

Conclusions:

  • The derived analytical expressions provide insights into local dimension estimation errors in multi-fractal systems.
  • These effects are significant for moderately high-dimensional systems and limited dataset sizes.
  • Recommends accounting for these local dimension variations in future empirical data studies, with implications for fields like weather regime analysis.