Quantitative features of multifractal subtleties in time series

¹  Institute of Nuclear Physics, Polish Academy of Sciences - ul. Radzikowskiego 152, PL-31-342 Kraków, Poland, EU
²  Faculty of Mathematics and Natural Sciences, University of Rzeszów - PL-35-310 Rzeszów, Poland, EU

Corresponding author: jaroslaw.kwapien@ifj.edu.pl

Received: 14 July 2009
Accepted: 30 November 2009

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 its practical significance this fact may reflect another manifestation of a conjectured q-generalized Central-Limit Theorem.