A Multifractal Detrended Fluctuation Analysis approach using generalized functions

Detrended Fluctuation Analysis (DFA) and its generalization for multifractal signals, Multifractal Detrend Fluctuation Analysis (MFDFA), are widely used techniques to investigate the fractal properties of time series by estimating their Hurst exponent (H). These methods involve calculating the fluctuation functions, which represent the square root of the mean square deviation from the detrended cumulative curve of a given time series. However, in the multifractal variant of the method, a particular case arises when the multifractal index vanishes. Consequently, it becomes necessary to define the fluctuation function differently for this specific case. In this paper, we propose an approach that eliminates the need for a piecewise definition of the fluctuation function, thereby enabling a unified formulation and interpretation in both DFA and MFDFA methodologies. Our formulation provides a more compact algorithm applicable to mono and multifractal time series. To achieve this, we express the fluctuation functions as generalized means, using the generalized logarithm and exponential functions from the context of the non-extensive statistical mechanics. We identified that the generalized formulation is the Box–Cox transformation of the dataset; hence we established a relationship between statistics parametrization and multifractality. Furthermore, this equivalence is related to the entropic index of the generalized functions and the multifractal index of the MFDFA method. To validate our formulation, we assess the efficacy of our method in estimating the (generalized) Hurst exponents H using commonly used signals such as the fractional Ornstein–Uhlenbeck (fOU) process, the symmetric Lévy distribution, pink, white, and Brownian noises. In addition, we apply our proposed method to a real-world dataset, further demonstrating its effectiveness in estimating the exponents H and uncovering the fractal nature of the data.