Phase Space Reconstruction Function (PSR) for Chaotic Time Series Analysis

Phase Space Reconstruction (PSR) is a fundamental technique in chaotic time series analysis, primarily used to map one-dimensional time series data into a higher-dimensional phase space to reveal hidden dynamical system characteristics. This technique typically involves implementing delay coordinate embedding algorithms that reconstruct the system's underlying dynamics from observed data.

In chaos theory, many complex dynamical systems cannot be directly observed in their multi-dimensional states, but we can reconstruct their phase space through observations of single variables (such as time series). The core concept of phase space reconstruction involves selecting appropriate embedding dimensions and time delays to reconstruct the topological structure of the original system, thereby enabling analysis of its dynamical properties. Implementation typically requires calculating optimal parameters using methods like mutual information or false nearest neighbors.

PSR commonly employs the Delay Coordinate Embedding method, based on Takens' theorem, which states that under certain conditions, with proper time delay and embedding dimension selection, the reconstructed phase space can preserve the geometric and dynamical characteristics of the original system. The key algorithmic steps involve determining optimal parameters through methods such as autocorrelation function analysis for time delay and false nearest neighbors (FNN) method for embedding dimension selection. Code implementation often involves calculating mutual information minima or average displacement for parameter optimization.

Phase space reconstruction lays the foundation for subsequent chaos analysis techniques (such as Lyapunov exponent calculation, fractal dimension estimation, etc.) and finds widespread applications in meteorology, finance, biological signal processing, and other fields. This enables researchers to discover underlying patterns from seemingly disordered time series data through proper algorithmic implementation of reconstruction parameters and phase space visualization techniques.