DelayTime_MutualInformation Method for Computing Delay Time in Chaotic Time Series with MATLAB Implementation

This article presents the implementation of the DelayTime_MutualInformation method for computing the optimal delay time in chaotic time series analysis using MATLAB. This method provides a robust approach for understanding chaotic phenomena through computational analysis. First, we define chaotic time series - these are time series that appear random but are actually generated by complex dynamical systems. The inherent uncertainty and pseudo-randomness of chaotic time series make them crucial in various fields including meteorology, finance, and biological systems analysis. The core algorithm employs mutual information theory, which quantifies the statistical dependence between two random variables. In MATLAB implementation, the key function calculates the mutual information between the original time series and its time-delayed version. The optimal delay time corresponds to the first local minimum of the mutual information function, indicating sufficient independence between delayed coordinates while preserving system dynamics. The MATLAB code typically involves these computational steps: loading time series data, creating delayed versions of the series, computing probability distributions using histogram methods, calculating joint and marginal probabilities, and finally determining mutual information values across different delay parameters. The implementation utilizes MATLAB's built-in functions for statistical calculations and vector operations to ensure computational efficiency. We demonstrate how to apply these computational results to practical problems, including phase space reconstruction and chaotic system characterization. The method provides a systematic approach for determining appropriate embedding parameters in nonlinear time series analysis.