Classical GP Algorithm for Computing Correlation Dimension

The classical GP algorithm (Grassberger-Procaccia algorithm) serves as a fundamental tool for analyzing correlation dimensions in chaotic time series, widely applied in nonlinear dynamics and complex system studies. This method reveals fractal characteristics of systems by computing correlation integrals of point pairs in phase space, providing mathematical basis for quantifying time series complexity. Implementation logic can be divided into four key steps: Phase Space Reconstruction Using time-delay embedding method to map one-dimensional time series into higher-dimensional phase space, requiring determination of embedding dimension m and time delay τ. Takens' theorem guarantees that reconstructed trajectories can preserve the topological properties of the original system with appropriate parameters. Code implementation typically involves creating a Hankel matrix or using specialized embedding functions. Correlation Integral Calculation Statistical analysis of distance distribution among all point pairs in phase space. By setting radius r, the algorithm calculates the proportion of point pairs with distances smaller than r, defining the correlation integral C(r). Practical implementations often use logarithmically uniform sampling of r values to optimize computational efficiency. This can be achieved through vectorized distance calculations or efficient pairwise computation methods. Scale-Free Interval Identification In double logarithmic coordinates, the linear region between C(r) and r represents the scale-free interval. The correlation dimension estimate is obtained by performing linear regression on the slope within this interval. Boundary determination commonly employs manual inspection or adaptive algorithms like the maximum slope method. Dimension Convergence Verification Gradually increasing embedding dimension m until the correlation dimension estimate stabilizes without further growth with increasing m. This stable value represents the true correlation dimension, a phenomenon known as dimension saturation. Implementation requires iterative computation across multiple embedding dimensions. Optimization techniques include: Utilizing KD-tree structures for accelerated nearest neighbor searches Parallelizing distance matrix computations using multiprocessing or GPU acceleration Implementing Theiler windows to exclude temporally correlated point pairs The algorithm finds significant applications in EEG signal analysis, turbulence studies, and other domains, though it requires attention to noise sensitivity and data quantity requirements. Practical applications should incorporate surrogate data tests to ensure result reliability, often implemented through statistical significance testing routines.