Calculation of the Critical Hurst Exponent for Fractal Analysis

In fractal analysis, the computation of the Hurst exponent holds significant importance as it quantifies long-range dependencies within time series data. Our implementation utilizes rescaled range (R/S) analysis methodology, where the algorithm calculates the ratio of the range of cumulative deviations to the standard deviation across varying time intervals. The program includes two validated sample sequences with Hurst exponents of 0.7 and 0.8, serving as reference datasets for understanding the exponent's behavioral characteristics and practical applications in time series interpretation. The computational workflow involves: 1) partitioning the time series into multiple segments, 2) computing partial series and cumulative deviations, 3) determining range-to-standard-deviation ratios, and 4) deriving the Hurst exponent through logarithmic regression. When employing this analytical technique, practitioners may complement the analysis with related metrics such as fractal dimensions and Lyapunov exponents to achieve comprehensive characterization. Ultimately, fractal analysis serves as a powerful toolkit for revealing underlying patterns and persistence properties in temporal data structures.