Noise-assisted estimation of attractor invariants

Area of Science:

• Dynamical systems theory
• Nonlinear time series analysis
• Statistical physics

Background:

• Estimating invariants of dynamical systems is crucial for understanding system behavior.
• Existing methods for correlation dimension and entropy estimation can be sensitive to noise.
• A unified framework for noise-assisted invariant estimation is needed.

Purpose of the Study:

• To propose the noise-assisted correlation integral (NCI) for estimating correlation dimension (D), correlation entropy (K2), and noise level (σ).
• To introduce the U-correlation integral as a specific case of NCI and derive coarse-grained estimators (DmU, KmU, σmU).
• To evaluate the performance of these estimators using time series from the Henon map and Mackey-Glass system.

Main Methods:

• Development of the noise-assisted correlation integral (NCI) algorithm.
• Modification of the correlation algorithm by incorporating random noise.
• Derivation of coarse-grained estimators (DmU, KmU, σmU) from the U-correlation integral.
• Analysis of estimator behavior using simulated time series (Henon map, Mackey-Glass) under varying noise levels and data lengths.

Main Results:

• The NCI framework unifies existing correlation integral methods (Grassberger et al., Diks' GCI).
• The derived estimators DmU and σmU show performance comparable to GCI-based methods.
• The KmU estimator demonstrates superior performance over GCI-based counterparts for estimating correlation entropy (K2).
• An automatic algorithm for estimating D, K2, and σ from time series was developed and validated.

Conclusions:

• The noise-assisted correlation integral (NCI) provides a flexible and effective framework for invariant estimation.
• The U-correlation integral and its derived estimators offer reliable tools for analyzing dynamical systems.
• The proposed automatic algorithm enables statistically reliable estimation of key dynamical invariants from time series data.