Phase Space Reconstruction Parameter Selection: Delay Time and Embedding Dimension

When selecting parameters for phase space reconstruction, a fundamental technique for analyzing complex dynamical systems, two critical parameters require careful consideration: the delay time (τ) and the embedding dimension (m). The delay time represents the time interval between successive observations in the time series data, while the embedding dimension determines the number of dimensions required to properly unfold the system's dynamics in phase space. In practical implementations, the delay time is commonly estimated using mutual information methods or autocorrelation functions. For mutual information, developers typically calculate the first minimum of the mutual information function between the original time series and its delayed version. For autocorrelation, the delay time corresponds to the point where the autocorrelation function drops to 1/e of its initial value. The embedding dimension is frequently determined using the false nearest neighbors (FNN) method. This algorithm incrementally increases the embedding dimension and computes the percentage of false neighbors that disappear as the dimension becomes sufficient to unfold the attractor. Alternatively, the Cao's method provides a more robust approach by examining the behavior of nearest neighbors as the embedding dimension increases. Proper parameter selection can be implemented using MATLAB's Phase Space Reconstruction toolbox or Python's nolitsa package, which provide built-in functions for calculating optimal delay times and embedding dimensions. These parameters significantly impact the quality of reconstructed attractors and subsequent analysis such as Lyapunov exponent calculation and predictability assessment. Careful optimization of these parameters enables researchers to obtain accurate phase space reconstructions that faithfully represent the underlying system dynamics. By implementing appropriate algorithms for parameter selection, analysts can achieve optimal reconstructions that preserve the topological properties of the original dynamical system, leading to more reliable nonlinear time series analysis results.