MATLAB Code Implementation for Calculating Hurst Exponent

The Hurst exponent is a crucial metric for analyzing long-term memory in time series, widely applied in finance, hydrology, and signal processing. MATLAB implementations of Hurst exponent calculations typically involve the following classical methods, each suited for different data characteristics: Rescaled Range Analysis (R/S Method) This traditional approach calculates the ratio of cumulative deviation range to standard deviation in time series, observing its variation with subsample lengths. Ideal for data with distinct fractal characteristics but may yield biases for non-stationary series. Code implementation involves partitioning data into segments, calculating R/S statistics for each segment, and performing linear regression on log-log plots to estimate H. Detrended Fluctuation Analysis (DFA) This method computes scaling behavior of fluctuation functions after removing local trends. DFA demonstrates greater robustness for non-stationary series, commonly used in physiological signals or stock price fluctuation analysis. Implementation requires integrating profiles, detrending through polynomial fitting, and calculating root-mean-square fluctuations across multiple scales. Variance Method Estimates H-value based on the rate of variance change after temporal aggregation. Offers high computational efficiency but shows sensitivity to periodic data. The algorithm involves aggregating time series at different scales and analyzing variance scaling through linear regression. Wavelet Transform Method Utilizes multiscale characteristics of wavelet coefficients to decompose time series, capturing both high-frequency and low-frequency components simultaneously. Suitable for high-frequency financial data or complex oscillatory signals. Implementation employs discrete wavelet transform to compute wavelet variances across scales. Key Implementation Considerations Data Preprocessing: Verify input series stationarity, applying differencing or logarithmic transformations when necessary. Subinterval Selection: For R/S analysis, appropriately set subinterval length ranges, typically using powers of 2 (2^n) to ensure statistical significance. Scaling Range Determination: During linear fitting in log-log plots, manually or algorithmically identify valid scaling ranges. For financial time series analysis, prioritizing combined R/S and DFA methods is recommended, while wavelet transform methods are preferable for engineering vibration signals. MATLAB's matrix operation advantages enable concise implementation of these algorithms, though sensitivity to data length and noise varies across methods. (Note: Practical code implementation typically involves sliding window processing, linear regression fitting, and other steps, with specific code structure descriptions omitted here)