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Algorithm to estimate the Hurst exponent of high-dimensional fractals
1Physics Department, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|February 1, 2008
Summary
We developed a new algorithm to estimate the Hurst exponent for complex, high-dimensional fractals. This method accurately analyzes fractal surfaces, offering insights into their roughness and scaling properties.
Area of Science:
- Fractal Geometry
- Data Analysis
Background:
- Estimating fractal dimensions is crucial for characterizing complex systems.
- Existing methods struggle with high-dimensional data and computational efficiency.
Purpose of the Study:
- To propose a novel algorithm for Hurst exponent estimation in high-dimensional fractals.
- To validate the algorithm's accuracy and efficiency using synthetic fractal surfaces.
Main Methods:
- Developed a generalized high-dimensional variance approach.
- Utilized a moving average low-pass filter for analysis.
- Generated surrogate fractal surfaces using random midpoint displacement and Cholesky-Levinson factorization.
Main Results:
- The algorithm accurately estimated Hurst exponents from 0.1 to 0.9.
- Demonstrated effectiveness on surfaces of varying sizes and complexities.
- Evaluated computational efficiency and accuracy.
Conclusions:
- The proposed algorithm provides a robust method for Hurst exponent estimation.
- It is suitable for analyzing high-dimensional fractal surfaces generated by various methods.
- Offers a computationally efficient and accurate tool for fractal analysis.
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