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Modifications of the Takens-Ellner algorithm for medium- and high-dimensional signals
1Department of Biophysics, Poznań University of Medical Sciences, ul. Fredry 10, Poznań, PL-61-701, Poland. kmichalak@ump.edu.pl
Summary
This study refines the Takens-Ellner algorithm to accurately estimate dimensional complexity (d) in complex signals. Modifications improve accuracy, especially for noisy or high-dimensional data, by optimizing parameter selection and excluding close point pairs.
Area of Science:
- Dynamical Systems Analysis
- Nonlinear Time Series Analysis
- Signal Processing
Background:
- Estimating dimensional complexity (d) is crucial for characterizing dynamical systems.
- The classic Takens-Ellner (TE) algorithm has limitations with medium- and high-dimensional signals.
- Accurate estimation of d requires careful selection of embedding parameters.
Purpose of the Study:
- To modify the Takens-Ellner algorithm for accurate dimensional complexity estimation.
- To address limitations of the classic TE algorithm in handling complex signals.
- To improve the robustness of dimensional complexity analysis for noisy and high-dimensional data.
Main Methods:
- Fitting a fourth-degree polynomial to the d=fn(W) relation to find the minimum slope.
- Excluding attractor pairs closer than the autocorrelation time to prevent underestimation of d.
- Modifying embedding parameter selection: calculating lag (L) based on embedding dimension (m) and window width (W) using L=W/(m-1).
- Employing cubic interpolation for non-integer L values.
Main Results:
- The modified algorithm precisely estimates dimensional complexity for signals with d≈4, d≈6, and d≈8.
- Polynomial fitting effectively identifies optimal window width (W) ranges.
- Exclusion of close attractor pairs mitigates the tendency for d underestimation.
- Cubic interpolation and error estimation methods are presented.
Conclusions:
- The enhanced Takens-Ellner algorithm provides accurate dimensional complexity estimation for complex signals.
- The modifications successfully address limitations of the classic TE algorithm.
- The study validates the algorithm's performance on synthetic and noisy signals.
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