Cao Method for Determining Optimal Embedding Dimension in Chaotic Time Series Phase Space Reconstruction

In chaotic time series analysis, phase space reconstruction is a crucial step for revealing system dynamics. Determining the optimal embedding dimension is essential for reconstruction quality. The Cao method is a practical and reliable technique that automatically identifies the minimum sufficient embedding dimension by calculating the distance change rate between neighboring points.

The core concept of the Cao method involves reconstructing phase space using time-delay coordinates and examining how neighbor point behavior changes as embedding dimension increases. In implementation, the algorithm computes two key indicators: E1 and E2. E1 determines whether the embedding dimension is sufficient - when its value stops changing significantly, the corresponding dimension represents the optimal solution. E2 distinguishes random sequences from deterministic chaotic sequences.

In MATLAB implementation, the original time series is typically normalized first, followed by phase space construction under different embedding dimensions. By calculating the distance change rate between all data points and their nearest neighbors as dimension increases, an E1 curve is plotted. The saturation point observed in this curve indicates the optimal embedding dimension. This method eliminates subjective judgment and provides objective quantification standards for nonlinear time series analysis.

The algorithm has significant application value in prediction and fault diagnosis domains, effectively improving the accuracy of chaotic time series analysis. Compared with traditional false nearest neighbor methods, the Cao method requires less data length, offers higher computational efficiency, and is particularly suitable for practical engineering applications.