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

TL;DR

This paper presents a generalized Higuchi-based method for multifractal analysis of short univariate time series, effectively distinguishing between mono- and multifractal dynamics with practical advantages over existing techniques.

Contribution

It introduces MF-HDA, a novel multifractal Higuchi dimension analysis method that estimates the multifractal spectrum using a generalized approach suitable for limited data.

Findings

MF-HDA reliably estimates the multifractal spectrum for moderate time series lengths.

The method effectively distinguishes mono- from multifractal dynamics.

Practical advantages include applicability to short time series.

Abstract

We introduce a generalization of Higuchi's estimator of the fractal dimension as a new way to characterize the multifractal spectrum of univariate time series. The resulting 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 as well as 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 multi-fractal dynamics based on relatively short time series.