Calculating Delay Time for Chaotic Time Series Analysis

In chaotic time series analysis, calculating the delay time is a critical step that directly affects the quality of phase space reconstruction. The selection of delay time must satisfy the fundamental requirements of Takens' embedding theorem: it should be sufficiently large to ensure independence between coordinates, yet sufficiently small to preserve the system's dynamical information.

There are two mainstream classical approaches for calculating delay time:

The autocorrelation function method is the most intuitive linear approach. It calculates the autocorrelation function of the time series and typically selects the delay where the function first crosses zero or drops to 1/e of its initial value. This method is computationally simple but insensitive to nonlinear characteristics. In code implementation, one would use functions like numpy.correlate() or MATLAB's xcorr() to compute the autocorrelation, then find the first zero-crossing or 1/e threshold point.

The mutual information method is a superior nonlinear approach based on information theory, which calculates the information correlation between delayed sequences. Typically, the delay corresponding to the first local minimum of the mutual information function is selected, ensuring that reconstructed coordinates contain sufficient new information while maintaining appropriate correlation. Implementation involves calculating joint probability distributions using histogram methods or kernel density estimation, followed by entropy computations to derive mutual information values.

Several key factors must be considered in practical applications: sample length affects the reliability of statistical measures; noise levels can interfere with correlation function calculations; multivariate systems may require multidimensional mutual information methods. The optimal delay time selection should also be combined with embedding dimension determination to jointly ensure the topological equivalence of the reconstructed phase space. Algorithm implementations often incorporate validation techniques like false nearest neighbors to coordinate these parameters effectively.