GP Algorithm for Computing Optimal Embedding Dimension and Time Delay in Time Series Phase Space Reconstruction

Phase space reconstruction serves as a fundamental method for analyzing nonlinear time series, with its core challenge lying in determining two critical parameters: optimal embedding dimension (m) and time delay (τ). The Grassberger-Procaccia (GP) algorithm provides quantitative guidance for parameter selection through correlation integral calculations and correlation dimension estimation. Determination of Time Delay (τ) The GP algorithm typically combines autocorrelation functions or mutual information methods to obtain τ. The optimal delay time should ensure that adjacent coordinate components are neither completely correlated nor entirely independent. The algorithm calculates correlation integral curves of time series and observes their saturation behavior to validate the appropriateness of τ selection. In code implementation, this involves computing delayed coordinate vectors using tau*(0:m-1) time shifts and analyzing the decay pattern of correlation sums. Optimization of Embedding Dimension (m) The correlation dimension (D) converges gradually as m increases. The GP algorithm computes D-values across different m-values, where the m corresponding to stabilized D-values (no significant changes) represents the optimal embedding dimension. This process requires balancing between overfitting (excessive m) and information loss (insufficient m). Programmatically, this involves iterative dimension expansion using embedding matrices and monitoring dimension convergence through slope changes in log-log plots. Key Implementation Aspects Correlation Integral Calculation: Counts the proportion of point pairs in phase space with distances smaller than a given radius r. Code implementation typically uses vectorized distance calculations and histogram accumulation for efficiency. Double Logarithmic Linear Fitting: Estimates correlation dimension through the slope of ln(C) versus ln(r) relationship. This requires robust linear regression on the scaling region of the correlation integral curve. Parameter Sensitivity Analysis: Validates robustness against noise and sampling intervals through perturbation tests and Monte Carlo simulations. The algorithm finds wide applications in chaos system analysis and physiological signal processing, though practical considerations require balancing computational efficiency against data volume requirements. Implementation often involves optimizations like box-assisted methods for handling large-scale time series data.