C_C Algorithm for Determining Embedding Dimension and Time Delay in Phase Space Reconstruction

In nonlinear time series analysis, phase space reconstruction is a critical step for revealing system dynamics, where the selection of embedding dimension (m) and time delay (τ) directly impacts reconstruction quality. The C_C algorithm is a classical computational method whose core concept relies on correlation integrals and statistical properties of time series. Algorithm Principles The C_C algorithm jointly optimizes m and τ through the following steps: Statistical Bias Removal: Computes the correlation integral of the time series to eliminate bias caused by random noise. Delay Window τ_w: Introduces τ_w = (m-1)τ as a global parameter to avoid local optima resulting from independent optimization of m and τ. Objective Function: Defines an indicator S(m,τ) that measures statistical dependence, with optimal parameters determined through its extremum points. MATLAB Implementation Key Points Preprocessing requires normalizing the time series to eliminate dimensional effects. A loop iterates through candidate m and τ values to compute the corresponding S(m,τ) matrix. Result visualization typically includes two subplots: S versus τ curve (identifying the first minimum point) and stability interval analysis for τ_w. This algorithm's strength lies in simultaneously considering statistical dependence and geometric structure, making it suitable for chaotic systems and high-noise environments. Practical applications require careful setting of parameter search ranges to avoid computational complexity explosion. Code implementation typically involves using nested loops for parameter scanning, with the correlation integral calculation optimized through vectorization techniques. Key functions would include normalization routines, distance matrix computations, and statistical averaging operations for robust parameter estimation.