Quantitative features of multifractal subtleties in time series

TL;DR

This paper investigates the origins of multifractality in time series using MFDFA and WTMM methods, revealing how nonlinear correlations influence multifractal properties and demonstrating rapid convergence of multifractal measures.

Contribution

It introduces a detailed analysis of multifractality origins in time series, highlighting the role of nonlinear correlations and the behavior at the border of Gaussian and Levy distributions.

Findings

Phase-like transition between monofractal and bifractal at q=5/3.

Multifractal measures converge rapidly within well-developed power-law correlations.

Results suggest a link to a q-generalized Central Limit Theorem.

Abstract

Based on the Multifractal Detrended Fluctuation Analysis (MFDFA) and on the Wavelet Transform Modulus Maxima (WTMM) methods we investigate the origin of multifractality in the time series. Series fluctuating according to a qGaussian distribution, both uncorrelated and correlated in time, are used. For the uncorrelated series at the border (q=5/3) between the Gaussian and the Levy basins of attraction asymptotically we find a phase-like transition between monofractal and bifractal characteristics. This indicates that these may solely be the specific nonlinear temporal correlations that organize the series into a genuine multifractal hierarchy. For analyzing various features of multifractality due to such correlations, we use the model series generated from the binomial cascade as well as empirical series. Then, within the temporal ranges of well developed power-law correlations we find a fast convergence in all multifractal measures. Besides of its practical significance this fact may reflect another manifestation of a conjectured q-generalized Central Limit Theorem.