Calculating Multifractal Generalized Hurst Exponents and Multifractal Spectra Using the MFDMA Method

The MFDMA (Multifractal Detrended Moving Average) method is a computational approach for multifractal analysis that enables calculation of generalized Hurst exponents and multifractal spectra. In implementation, this method typically involves partitioning data into multiple scales through a sliding window approach, where each scale corresponds to different window sizes. For each scale, the algorithm performs detrending by subtracting the moving average from the original data series, then calculates fluctuation functions using root-mean-square deviations. The core algorithm involves computing q-th order fluctuation functions Fq(s) across various scales s, where q represents different statistical moments. Through logarithmic transformation and linear regression analysis between ln[Fq(s)] and ln(s), the method derives the generalized Hurst exponent h(q). The multifractal spectrum f(α) is then obtained via Legendre transformation, where α represents the singularity strength. Key implementation considerations include optimal window size selection, handling of boundary effects, and numerical stability for negative q-values. The method effectively captures relationships between detailed fluctuations and overall data characteristics, facilitating deeper understanding of data structures and improving predictive accuracy. By revealing scale-invariant properties and multifractal features, MFDMA serves as a robust tool for analyzing complex datasets across various domains including financial time series, physiological signals, and geophysical data.