Computing Fractal Correlation Dimension

Fractal correlation dimension serves as a crucial metric for characterizing the complexity of nonlinear systems, primarily used to quantify the distribution characteristics of point sets in phase space. The core concept involves revealing a system's dynamic properties by analyzing the clustering degree of phase space trajectories. Below is an analysis of the implementation approach for this MATLAB program:

The computation of correlation dimension typically consists of three key steps: phase space reconstruction, correlation integral calculation, and dimension estimation.

Phase Space Reconstruction The program first maps time series data to high-dimensional phase space using delay embedding method, which involves two critical parameters: delay time and embedding dimension. Appropriate parameter selection directly influences subsequent calculation accuracy. The program may employ autocorrelation function or mutual information method to automatically determine the optimal delay time through algorithm optimization.

Correlation Integral Calculation The core procedure involves calculating correlation integral C(r) by statistically determining the proportion of point pairs in phase space whose distances are smaller than a given radius r. The program iterates through a series of radius values, computing the number of point pairs satisfying distance conditions for each r. Logarithmic interval sampling is typically implemented to enhance computational efficiency, using vectorized operations to handle large distance matrices.

Dimension Estimation through Linear Fitting In the double logarithmic coordinate system, the slope of correlation integral C(r) versus radius r provides the correlation dimension estimate. The program identifies the linear region (scaling region) and applies least squares fitting to obtain the final dimension value. To improve robustness, the implementation may include data denoising procedures or multiple sampling average processing using statistical functions.

Typical application scenarios for this method include: - Complexity analysis of biological signals (e.g., EEG/ECG) - Fault detection in mechanical vibration systems - Nonlinear feature identification in meteorological/financial time series

Important considerations: Practical applications require verification of scaling region existence to avoid false fractal phenomena caused by noise. For short data sequences, improved algorithms such as GP algorithm might be necessary for calibration through additional algorithmic modifications.