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Multifractal detrended fluctuation analysis is an effective tool for dealing with the non-uniformity and singularity of nonstationary time series. For the serious issues of the trend extraction and the inefficient computation in the traditional polynomial fitting based multifractal detrended fluctuation analysis, based on the dual-tree complex wavelet transform, a novel multifractal analysis is proposed. To begin with, as the dual-tree complex wavelet transform has the anti-aliasing and nearly shift-invariance, it is first utilized to decompose the signal through the pyramid algorithm, and the scale-dependent trends and the fluctuations are extracted from the wavelet coefficients. Then, using the wavelet coefficients, the length of the non-overlapping segment on a corresponding time scale is computed through the Hilbert transform, and each of the extracted fluctuations is divided into a series of non-overlapping segments whose sizes are identical. Next, on each scale, the detrended fluctuation function for each segment is calculated, and the overall fluctuation function can be obtained by averaging all segments with different orders. Finally, the generalized Hurst index and scaling exponent spectrum are determined from the logarithmic relations between the overall detrended fluctuation function and the time scale and the standard partition function, respectively, and then the multifractal singularity spectrum is calculated with the help of Legendre transform. We assess the performance of the dual-tree-complex wavelet transform based multifractal detrended fluctuation analysis (MFDFA) procedure through the classic multiplicative cascading process and the fractional Brownian motions, which have the theoretical fractal measures. For the multiplicative cascading process, compared with the traditional polynomial fitting based MFDFA methods, the proposed multifractal approach defines the trends and the length of non-overlapping segments adaptively and obtains a more precise result, while for the traditional MFDFA method, for the negative orders, no matter the generalized Hurst index, scaling exponents spectrum, or the multifractal singularity spectrum, the acquired results each have a significant deviation from the theoretical one. For the time series with different sizes, the proposed method can also give a stable result. Compared with the other adaptive method such as maximal overlap discrete wavelet transform based MFDFA and the discrete wavelet transfrom based MFDFA, the proposed approach obtains a very accurate result and has a fast calculation speed. For another time series of fractional Brownian motions with different Hurst indexes of 0.4, 0.5 and 0.6, which represent the anticorrelated, uncorrelated, correlated process, respectively, the results of the proposed method are consistent with those analytical results, while the results of the polynomial fitting based MFDFA methods are most greatly affected by the order of the fitting polynomial. The method in this article provides a valuable reference for how to use the dual-tree complex wavelet transform to realize the multifractal detrended fluctuation analysis, and we can benefit from the signal self-adaptive trend extraction and the high computation efficiency.
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Keywords:
- nonstationary time series /
- multifractal /
- detrended fluctuation analysis /
- dual-tree complex wavelet transform
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[2] Muzy J F, Bacry E, Arneodo A 1993 Phys. Rev. E 47 875
[3] Wang D L, Yu Z G, Anh V 2012 Chin. Phys. B 21 080504
[4] Caraiani P 2012 Physica A 391 3629
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[7] Xiong J, Chen S K, Wei W, Liu S, Guan W 2014 Acta Phys. Sin. 63 200504 (in Chinese) [熊杰,陈绍宽,韦伟,刘爽,关伟 2014 63 200504]
[8] Liu N B, Guan J, Song J, Huang Y, He Y 2013 Sci. China: Inform. Sci. 43 768 (in Chinese) [刘宁波, 关键, 宋杰, 黄勇, 何友 2013 中国科学: 信息科学 43 768]
[9] Xing H Y, Zhang Q, Xu W 2015 Acta Phys. Sin. 64 110502 (in Chinese) [行鸿彦, 张强, 徐伟 2015 64 110502]
[10] Zhou Y, Leung Y 2010 J. Stat. Mech-Theory E. 2010 P12006
[11] Lin M, Yan S X, Zhao G, Wang G 2013 Commun. Theor. Phys. 59 1
[12] Telesca L, Matcharashvili T, Chelidze T, Zhukova N, Javakhishvili Z 2013 Nat. Hazards 77 117
[13] Loiseau P, Mdigue C, Gonalves P, Attia N, Seuret S, Cottin F, Chemla D, Sorine M, Barral J 2012 Physica A 391 5658
[14] Lafouti M, Ghoranneviss M 2015 Chin. Phys. Lett. 32 105201
[15] Xi C P, Zhang S N, Xiong G, Zhao H C 2015 Acta Phys. Sin. 64 136403 (in Chinese) [奚彩萍, 张淑宁, 熊刚, 赵惠昌 2015 64 136403]
[16] Qian X Y, Gu G F, Zhou W X 2011 Physica A 390 4388
[17] Zhou J, Manor B, Liu D, Hu K, Zhang J, Fang J 2013 Plos One 8 e62585
[18] Guo T, Lan J L, Huang W W, Zhang Z 2013 J. Commun. 34 38 (in Chinese) [郭通,兰巨龙,黄万伟,张震 2013 通信学报 34 38]
[19] Peng Z K, Tse P W, Chu F L 2005 Mech. Syst. Signal. Pr. 19 974
[20] Muzy J, Bacry E, Arneodo A 1991 Phys. Rev. Lett. 67 3515
[21] Manimaran P, Panigrahi P K, Parikh J C 2009 Physica A 388 2306
[22] Liang Z, Li D, Ouyang G, Wang Y, Voss L J, Sleigh J W, Li X 2012 Clin. Neurophysiol. 123 681
[23] Selesnick I W, Baraniuk R G, Kingsbury N G 2005 IEEE Signal Proc. Mag. 22 123
[24] Nelson J, Kingsbury N 2012 IET Signal Process. 6 484
[25] Nafornita C, Isar A, Nelson J D B 2014 Proceedings of the 2014 IEEE International Conference on Image Processing New York, USA, January 28, 2014 p2689
[26] Macek W M, Wawrzaszek A 2011 Nonlinear Proc. Geoph. 18 287
[27] Cheng Q 2012 Nonlinear Proc. Geoph. 19 57
[28] Sezer A 2012 Sci. Iran. 19 1456
[29] Cao G, Xu W 2016 Physica A 444 505
[30] Arshad S, Rizvi S A R 2015 Physica A 419 158
[31] Sun K, Jin G, Wang C Y, Ma C W, Qian W P, Gao M G 2015 J. Electr. Inform. Technol. 37 982 (in Chinese) [孙康, 金钢, 王超宇,马超伟,钱卫平,高梅国 2015 电子与信息学报 37 982]
[32] Lin P L, Huang P W, Lee C H, Wu M T 2013 Pattern Recogn. 46 3279
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[1] Ni H J, Zhou L P, Zeng P, Huang X L, Liu H X, Ning X B 2015 Chin. Phys. B 24 070502
[2] Muzy J F, Bacry E, Arneodo A 1993 Phys. Rev. E 47 875
[3] Wang D L, Yu Z G, Anh V 2012 Chin. Phys. B 21 080504
[4] Caraiani P 2012 Physica A 391 3629
[5] Lin J S, Chen Q 2013 Mech. Syst. Signal. Pr. 38 515
[6] Xiao H, L Y, Wang T 2015 J. Vib. Eng. 28 331 (in Chinese) [肖涵, 吕勇, 王涛 2015 振动工程学报 28 331]
[7] Xiong J, Chen S K, Wei W, Liu S, Guan W 2014 Acta Phys. Sin. 63 200504 (in Chinese) [熊杰,陈绍宽,韦伟,刘爽,关伟 2014 63 200504]
[8] Liu N B, Guan J, Song J, Huang Y, He Y 2013 Sci. China: Inform. Sci. 43 768 (in Chinese) [刘宁波, 关键, 宋杰, 黄勇, 何友 2013 中国科学: 信息科学 43 768]
[9] Xing H Y, Zhang Q, Xu W 2015 Acta Phys. Sin. 64 110502 (in Chinese) [行鸿彦, 张强, 徐伟 2015 64 110502]
[10] Zhou Y, Leung Y 2010 J. Stat. Mech-Theory E. 2010 P12006
[11] Lin M, Yan S X, Zhao G, Wang G 2013 Commun. Theor. Phys. 59 1
[12] Telesca L, Matcharashvili T, Chelidze T, Zhukova N, Javakhishvili Z 2013 Nat. Hazards 77 117
[13] Loiseau P, Mdigue C, Gonalves P, Attia N, Seuret S, Cottin F, Chemla D, Sorine M, Barral J 2012 Physica A 391 5658
[14] Lafouti M, Ghoranneviss M 2015 Chin. Phys. Lett. 32 105201
[15] Xi C P, Zhang S N, Xiong G, Zhao H C 2015 Acta Phys. Sin. 64 136403 (in Chinese) [奚彩萍, 张淑宁, 熊刚, 赵惠昌 2015 64 136403]
[16] Qian X Y, Gu G F, Zhou W X 2011 Physica A 390 4388
[17] Zhou J, Manor B, Liu D, Hu K, Zhang J, Fang J 2013 Plos One 8 e62585
[18] Guo T, Lan J L, Huang W W, Zhang Z 2013 J. Commun. 34 38 (in Chinese) [郭通,兰巨龙,黄万伟,张震 2013 通信学报 34 38]
[19] Peng Z K, Tse P W, Chu F L 2005 Mech. Syst. Signal. Pr. 19 974
[20] Muzy J, Bacry E, Arneodo A 1991 Phys. Rev. Lett. 67 3515
[21] Manimaran P, Panigrahi P K, Parikh J C 2009 Physica A 388 2306
[22] Liang Z, Li D, Ouyang G, Wang Y, Voss L J, Sleigh J W, Li X 2012 Clin. Neurophysiol. 123 681
[23] Selesnick I W, Baraniuk R G, Kingsbury N G 2005 IEEE Signal Proc. Mag. 22 123
[24] Nelson J, Kingsbury N 2012 IET Signal Process. 6 484
[25] Nafornita C, Isar A, Nelson J D B 2014 Proceedings of the 2014 IEEE International Conference on Image Processing New York, USA, January 28, 2014 p2689
[26] Macek W M, Wawrzaszek A 2011 Nonlinear Proc. Geoph. 18 287
[27] Cheng Q 2012 Nonlinear Proc. Geoph. 19 57
[28] Sezer A 2012 Sci. Iran. 19 1456
[29] Cao G, Xu W 2016 Physica A 444 505
[30] Arshad S, Rizvi S A R 2015 Physica A 419 158
[31] Sun K, Jin G, Wang C Y, Ma C W, Qian W P, Gao M G 2015 J. Electr. Inform. Technol. 37 982 (in Chinese) [孙康, 金钢, 王超宇,马超伟,钱卫平,高梅国 2015 电子与信息学报 37 982]
[32] Lin P L, Huang P W, Lee C H, Wu M T 2013 Pattern Recogn. 46 3279
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