Implementation and Simulation of Chaotic Time Series Prediction Methods

Chaotic time series prediction is a forecasting approach based on nonlinear systems theory, applicable to dynamic systems that appear random but actually follow deterministic patterns. The core concept involves reconstructing the phase space to extract key characteristic parameters (such as time delay, correlation dimension, and Lyapunov exponents), thereby building a prediction model. In code implementation, this typically requires creating functions for phase space reconstruction using time-delay embedding techniques.

Time delay is crucial for phase space reconstruction, where optimal intervals are determined through autocorrelation functions or mutual information methods. This ensures the reconstructed trajectory preserves the topological properties of the original system. The correlation dimension reflects the fractal characteristics of system complexity, calculated using the Grassberger-Procaccia algorithm - implementable through functions that compute correlation integrals from embedded data points. The maximum Lyapunov exponent quantifies the system's sensitivity to initial conditions, with positive values indicating chaotic behavior. Code implementation often involves Rosenstein's algorithm for practical Lyapunov exponent calculation from time series data.

Simulation implementation typically follows these steps: First, preprocessing raw time series through denoising and normalization functions; Second, reconstructing phase space using delay embedding methods with optimized parameters; Third, computing correlation dimensions and Lyapunov exponents to verify chaotic properties; Finally, employing prediction techniques like local linear approximation, neural networks, or support vector machines. The method finds wide applications in meteorology, finance, and biological signal processing, but requires careful attention to noise interference and parameter sensitivity during algorithm implementation.