Relationships of exponents in multifractal detrended fluctuation analysis and conventional multifractal analysis

doi:10.1088/1674-1056/20/9/090507

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

[1]	Mandelbrot B B 1983 The Fractal Geometry of Nature (New York: W.H. Freeman & Co Ltd)
[2]	Hurst H E 1951 Trans. Amer. Soc. Civ. Eng. 116 770
[3]	Halsey T C, Jensen M H, Kadanoff L P, Procaccia I and Shraiman B I 1986 Phys. Rev. A 33 1141
[4]	Yu Z G, Anh V V, Gong Z M and Long S C 2002 Chin. Phys. 11 1313
[5]	Yu Z G, Xiao Q J, Shi L, Yu J W and Anh V 2010 Chin. Phys. B 19 068701
[6]	Zhu S M, Yu Z G and Anh V 2011 Chin. Phys. B 20 010505
[7]	Han J J and Fu W J 2010 Chin. Phys. B 19 010205
[8]	Peng C K, Buldyrev S V, Havlin S, Simons M, Stanley H E and Goldberger A L 1994 Phys. Rev. E 49 1685
[9]	Taqqu M S, Teverovsky V and Willinger W 1995 Fractals 3 785
[10]	Kantelhardt J W, Zschiegner S A, Koscielny-Bunde E, Havlin S, Bunde A and Stanley H E 2002 Physica A 316 87
[11]	Movahed M S, Jafari G R, Ghasemi F, Rahvar S and Tabar M R R 2006 J. Stat. Mech. P02003
[12]	Movahed M S and Hermanis E 2008 Physica A 387 915
[13]	Telesca L, Lapenna V and Macchiato M 2004 Chaos Solition. Fract. 19 1
[14]	Telesca L, Lapenna V and Macchiato M 2005 New J. Phys. 7 214
[15]	Kimiagar S, Movahed M S, Khorram S, Sobhanian S and Tabar M R R 2009 J. Stat. Mech. P03020
[16]	Luo S H and Zeng J S 2009 Acta Phys. Sin. 58 150 (in Chinese)
[17]	Chen Z, Ivanov P Ch, Hu K and Stanley H E 2002 Phys. Rev. E 65 041107
[18]	Hu K, Ivanov P Ch, Chen Z, Carpena P and Stanley H E 2001 Phys. Rev. E 64 011114
[19]	Ma Q D Y, Bartsch R P, Bernaola-Galvan P, Yoneyama M and Ivanov P Ch 2010 Phys. Rev. E 81 031101
[20]	Heneghan C and McDarby G 2000 Phys. Rev. E 62 6103
[21]	Xu L, Ivanov P Ch, Hu K, Chen Z, Carbone A and Stanley H E 2005 Phys. Rev. E 71 051101
[22]	Bashan A, Bartsch R, Kantelhardt J W and Havlin S 2008 Physica A 387 5080
[23]	Anh V, Yu Z G and Wanliss J 2007 Nonlinear. Proc. Geophys. 14 701
[24]	Anh V V, Yong J M and Yu Z G 2008 J. Geophys. Res. 113 A10215
[25]	Yu Z G, Anh V and Eastes R 2009 J. Geophys. Res. 114 A05214
[26]	Zhang Q, Xu C Y, Chen Y D and Yu Z 2008 Hydrol. Proc. 22 4997
[27]	Zhang Q, Xu C Y, Yu Z, Liu C and Chen Y D 2009 Physica A 388 927
[28]	Talkner P and Weber R O 2000 Phys. Rev. E 62 150
[29]	Varotsos P A, Sarlis N V and Skordas E S 2002 Phys. Rev. E 66 011902
[30]	Varotsos P A, Sarlis N V and Skordas E S 2003 Phys. Rev. E 67 021109
[31]	Telesca L, Colangelo G, Lapenna V and Macchiato M 2004 Phys. Lett. A 332 398
[32]	Muzy J F, Bacry E and Arneodo A 1994 Int. J. Bifur. Chaos 4 245
[33]	Oswiecimka P, Kwapien J and Drozdz S 2006 Phys. Rev. E 74 016103
[34]	Mandelbrot B B and Van Ness J W 1968 SIAM Rev. 10 422
[35]	Barton R J and Poor H V 1988 IEEE Trans. Inf. Theory 34 943
[36]	Flandrin P 1992 IEEE Trans. Inf. Theory 38 910
[37]	Lovejoy S, Schertzer D and Allaire V C 2008 Atmos. Res. 90 10
[38]	Schertzer D and Lovejoy S 1987 J. Geophys. Res. 92 9693
[39]	Lavallee D, Lovejoy S, Schertzer D and Ladoy P 1993 Fractals in Geography (Englewood Cliffs: PTR Prentice Hall) p. 158
[40]	Yu Z G, Leung Y, Chen Y D, Zhang Q and Anh V 2011 Multifractal Analyses of Daily Rainfall in the Pearl River Basin of China (submitted to J. Hydrol.)
[41]	Seuront L, Schmitt F, Lagadeuc Y, Schertzer D and Lovejoy S 1999 J. Plankton Res. 21 877
[42]	Davis A, Marshak A, Wiscombe W and Cahalan R 1994 J. Geophys. Res. 99 8055
[43]	Koscielny-Bunde E, Kantelhardt J W, Braun P, Bunde A and Havlin S 2006 J. Hydrol. 322 120
[44]	Tessier Y, Lovejoy S, Hubert P, Schertzer D and Pecknold S 1996 J. Geophys. Res. 101 26427

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Relationships of exponents in multifractal detrended fluctuation analysis and conventional multifractal analysis

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