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Chin. Phys. B, 2011, Vol. 20(9): 090507    DOI: 10.1088/1674-1056/20/9/090507
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Relationships of exponents in multifractal detrended fluctuation analysis and conventional multifractal analysis

Zhou Yu(周煜)a), Leung Yee(梁怡)a)b)c)†, and Yu Zu-Guo(喻祖国) d)e)
a Department of Geography and Resource Management, The Chinese University of Hong Kong, Hong Kong, China; b Center for Environmental Policy and Resource Management, The Chinese University of Hong Kong, Hong Kong, Chinac Institute of Space and Earth Information Science, The Chinese University of Hong Kong, Hong Kong, China; d Discipline of Mathematical Sciences, Faculty of Science and Technology, Queensland University of Technology, GPO Box 2434, Brisbane, Q 4001, Australiae School of Mathematics and Computational Science, Xiangtan University, Hunan 411105, China
Abstract  Multifractal detrended fluctuation analysis (MF-DFA) is a relatively new method of multifractal analysis. It is extended from detrended fluctuation analysis (DFA), which was developed for detecting the long-range correlation and the fractal properties in stationary and non-stationary time series. Although MF-DFA has become a widely used method, some relationships among the exponents established in the original paper seem to be incorrect under the general situation. In this paper, we theoretically and experimentally demonstrate the invalidity of the expression τ(q)=qh(q)-1 stipulating the relationship between the multifractal exponent τ(q) and the generalized Hurst exponent h(q). As a replacement, a general relationship is established on the basis of the universal multifractal formalism for the stationary series as τ(q)=qh(q)-qH'-1, where H' is the nonconservation parameter in the universal multifractal formalism. The singular spectra, α and f(α), are also derived according to this new relationship.
Keywords:  fractals      Hurst exponent      multifractal detrended fluctuation analysis      time series analysis  
Received:  23 January 2011      Revised:  12 May 2011      Accepted manuscript online: 
PACS:  05.45.Df (Fractals)  
  47.53.+n (Fractals in fluid dynamics)  
  05.45.Tp (Time series analysis)  

Cite this article: 

Zhou Yu(周煜), Leung Yee(梁怡), and Yu Zu-Guo(喻祖国) Relationships of exponents in multifractal detrended fluctuation analysis and conventional multifractal analysis 2011 Chin. Phys. B 20 090507

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