Density-Induced Variations of Local Dimension Estimates for Absolutely Continuous Random Variables.

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.