Generalization of Higuchi’s fractal dimension for multifractal analysis of time series with limited length

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.