Using curvature to select the time lag for delay reconstruction

Area of Science:

• Dynamical Systems Theory
• Nonlinear Dynamics
• Time Series Analysis

Background:

• Choosing an appropriate time-delay parameter (τ) is crucial for accurate delay reconstructions of dynamical systems.
• Suboptimal time delays can lead to geometric distortions, such as flattening or overfolding, in the reconstructed phase space.
• Existing methods like average mutual information can be unreliable with noisy or filtered data.

Purpose of the Study:

• To propose and validate a new curvature-based heuristic for selecting the optimal time-delay parameter (τ).
• To demonstrate the effectiveness of this method in identifying appropriate delays for delay reconstructions.
• To show the robustness of the proposed method, especially in scenarios where conventional techniques fail.

Main Methods:

• Calculating the mean Menger curvature of trajectory segments in two-dimensional delay reconstructions.
• Analyzing the curvature as a function of the time-delay parameter (τ).
• Identifying the optimal time delay by finding the minimum value of the mean Menger curvature.

Main Results:

• The minimum of the mean Menger curvature provides an effective heuristic for choosing the time-delay parameter (τ).
• This curvature-based approach successfully identifies delays that avoid geometric distortions like flattening and overfolding.
• The method demonstrates superior performance compared to average mutual information, particularly with noisy or filtered time-series data.

Conclusions:

• A curvature-based approach offers a reliable and effective method for selecting the time-delay parameter in delay reconstructions.
• This technique provides a robust alternative to traditional methods, especially in challenging data conditions.
• The proposed heuristic enhances the accuracy and reliability of dynamical system reconstructions from time-series data.