G-P Algorithm in Chaotic Time Series Analysis

In chaotic time series analysis, the G-P (Grassberger-Procaccia) method is a widely used technique for calculating the correlation dimension and embedding dimension necessary for phase space reconstruction. The algorithm implementation typically involves three key computational steps: First, compute the first-order difference of the original time series to generate a new sequence, often implemented using numpy's diff() function or similar vectorized operations. Second, sort the new sequence to obtain an ordered series, which can be efficiently achieved through quicksort or mergesort algorithms. Finally, perform statistical analysis on the sorted sequence to determine appropriate embedding and correlation dimensions through techniques like correlation integral calculation and log-log plot analysis. The correlation dimension is derived from the slope of the log-log plot of correlation sums versus distance thresholds, while the embedding dimension is identified when the dimension estimate stabilizes. Overall, the G-P method serves as an effective computational tool that enables researchers to better understand the characteristics and patterns of chaotic time series through robust dimension estimation, with common implementations involving distance matrix computations and scaling region identification in programming languages like Python or MATLAB.