Two Critical Parameters in Phase Space Reconstruction

Phase space reconstruction is a fundamental method for analyzing nonlinear time series, which maps one-dimensional temporal data to higher-dimensional space to reveal underlying dynamical characteristics of systems. Two crucial parameters must be determined: the delay time (τ) and the correlation dimension (d).

Delay Time (τ) The selection of delay time significantly impacts the quality of reconstructed phase space. Common methodologies include autocorrelation function and mutual information approaches. The autocorrelation method computes self-similarity coefficients of the time series, where the delay time corresponds to the first instance when autocorrelation drops to 1/e of its initial value. The mutual information method accounts for nonlinear dependencies by identifying the first minimum point of the mutual information function. Code Implementation: In MATLAB, use `autocorr()` for linear correlation analysis or implement mutual information calculation through probability distribution estimation of binned data.

Correlation Dimension (d) This parameter quantifies system complexity, typically computed using Grassberger-Procaccia (G-P) algorithm. The method derives correlation integrals from point-pair distances in embedded space, with correlation dimension estimated as the slope of double logarithmic curves. A stable slope value with increasing embedding dimensions indicates valid correlation dimension. Algorithm Implementation: The G-P algorithm involves calculating distance matrices between embedded vectors, followed by logarithmic regression analysis. MATLAB implementation can leverage vectorized operations for efficient distance computation using `pdist2()` function.

MATLAB applications can utilize toolboxes like TSTOOL or custom scripts. For instance, delay time calculation may combine built-in statistical functions with nonlinear dependence estimators, while G-P algorithm implementation requires careful threshold selection for correlation sum calculation and linear fitting procedures.

Proper parameter selection enables accurate reconstruction of phase space topology, forming the foundation for subsequent chaos analysis including Lyapunov exponent computation and predictive modeling.