Abstract
There exist several methodologies for the multifractal characterization of nonstationary time series. However, when applied to sequences of limited length, these methods often tend to overestimate the actual multifractal properties. To address this aspect, we introduce here a generalization of Higuchi’s estimator of the fractal dimension as a new way to characterize the multifractal spectrum of univariate time series or sequences of relatively short length. This multifractal Higuchi dimension analysis (MF-HDA) method considers the order-q moments of the partition function provided by the length of the time series graph at different levels of subsampling. The results obtained for different types of stochastic processes, a classical multifractal model, and various real-world examples of word length series from fictional texts demonstrate that MF-HDA provides a reliable estimate of the multifractal spectrum already for moderate time series lengths. Practical advantages as well as disadvantages of the new approach as compared to other state-of-the-art methods of multifractal analysis are discussed, highlighting the particular potentials of MF-HDA to distinguish mono- from multifractal dynamics based on relatively short sequences.
Highlights
Since Benoit Mandelbrot applied the concept of fractality for the first time to real-world time series, different approaches have been proposed to detect fractal and multifractal scaling properties in diverse signals from complex systems [1,2,3,4,5,6]
A conceptually related approach based on time scale decomposition by means of a more data-adaptive technique has been developed more recently [25,26]. Another state of the art approach of multifractal analysis has been introduced by Kantelhardt et al, who proposed a generalization of detrended fluctuation analysis [27] to obtain an estimate of the multifractal spectrum of nonstationary time series
This multifractal detrended fluctuation analysis (MF-DFA), together with various algorithmic variants thereof differing in the way of how local time series detrending is performed, provides a robust way to determine the multifractal scaling characteristics in terms of generalized Hurst exponents h(q), where q represents the order of the statistical moment under study
Summary
Since Benoit Mandelbrot applied the concept of fractality for the first time to real-world time series, different approaches have been proposed to detect fractal and multifractal scaling properties in diverse signals from complex systems [1,2,3,4,5,6] (for recent reviews, see [7,8]). A conceptually related approach based on time scale decomposition by means of a more data-adaptive technique (the empirical mode decomposition) has been developed more recently [25,26] Another state of the art approach of multifractal analysis has been introduced by Kantelhardt et al, who proposed a generalization of detrended fluctuation analysis [27] to obtain an estimate of the multifractal spectrum of nonstationary time series. This multifractal detrended fluctuation analysis (MF-DFA), together with various algorithmic variants thereof differing in the way of how local time series detrending is performed, provides a robust way to determine the multifractal scaling characteristics in terms of generalized Hurst exponents h(q), where q represents the order of the statistical moment under study. A discussion and some conclusions and general remarks are provided in Sections 7 and 8, respectively
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