(Debiased) Complex Autocorrelation Method for Calculating Optimal Delay Time in Chaotic Time Series

This article presents a MATLAB implementation of the (debiased) complex autocorrelation method for calculating the optimal delay time in chaotic time series analysis. We begin by explaining the relationship between chaotic phenomena and time series data, highlighting the importance of determining appropriate delay parameters for phase space reconstruction. The methodology section details the complex autocorrelation algorithm, which extends traditional autocorrelation by incorporating imaginary components to better capture nonlinear dynamics. Key implementation aspects include signal preprocessing, complex autocorrelation computation using FFT-based convolution, and bias correction techniques to improve statistical accuracy. The provided MATLAB code demonstrates practical implementation with commented sections covering data normalization, correlation function calculation, and delay threshold detection based on predefined criteria. Finally, we offer a tested MATLAB prototype that includes visualization functions to plot autocorrelation results and automatically identify the optimal delay point, enabling researchers to effectively apply this method to their chaotic systems analysis.