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Multifractal detrended fluctuation analysis (MFDFA) and multifractal detrended moving average (MFDMA) algorithm have been established as two important methods to estimate the multifractal spectrum of the one-dimensional random fractal signals. They have been generalized to deal with two-dimensional and higher-dimensional fractal signals. This paper gives a brief introduction of the two algorithms, and a detail description of the numerical experiments on the one-dimensional time series by using the two methods. By applying the two methods to the series generated from the binomial multiplicative cascades (BMC), we systematically carry out comparative analysis to get the advantages, disadvantages and the applicability of the two algorithms, for the first time so far as we know, from six aspects: the similarities and differences of the algorithm models, the statistical accuracy, the sensitivities of the sample size, the selection of scaling range, the choice of the q-orders, and the calculation amount. For one class of signals, the larger the sample size, the more accurate the estimated multifractal spectrum. Selection of appropriate scaling range affects the statistical accuracy in comparison of the two methods for almost all examples. The presence of scale invariance should be checked by first running the two methods over a large scaling range (e.g., from 10 to (N+1)/11 in this paper) and then plot log10 (Fq (scale)) against log10 (scale). In the MFDFA-m (m is the polynomial order, and in this paper m=1) method, the scaling range can be selected from {m + 2, 10} to N/10, N is the sample size of the time series. In the MFDMA algorithm, the scaling range should be from 10 to (N+1)/11. It is favorable to have an equal spacing between scales and the number of the scales should be larger than 10 and usually be selected from 20 to 40. The q-orders should consist of both positive and negative q's. When |q| = 5, the calculated results will not be sensitive with the increase of Δq from 0.05 to 1. If Δq = 0.1, the calculation error will be relatively small when 0 q|≤ 10. With the increase of |q|, the width of the multifractal spectrum will obviously become wider when 0 q|≤10 and the change will be smaller when |q|≥20. If |q| continues to increase, the local fluctuations will approach zero when the scale is small. The critical steps exist in the calculation of local trends for the MFDFA-m and the running moving average for the MFDMA. If the sample size N is fixed and the scale is relatively small, the runtime of the critical steps of MFDFA-1 will be longer than that of MFDMA. When the scale increases from 4 to N/4, it will be shorter than that of MFDMA. Results provide a valuable reference on how to choose the algorithm between MFDFA and MFDMA, and how to make the schemes of the parameter setting of the two algorithms when dealing with specific signals in practical applications.
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Keywords:
- multifractal detrended fluctuation analysis /
- multifractal detrended moving average algorithm /
- random fractal signal /
- comparative analysis
[1] Serrano E, Figliola A 2009 Physica A 388 2793
[2] Ge E J, Leung Y 2013 J. Geogr. Syst. 15 115
[3] Kantelhardt J W, Zschiegner S A, Koscielny-Bunde E, Bunde A, Havlin S, Stanley H E 2002 Physica A 316 87
[4] Kantelhardt J W 2008 arXiv:0804.0747v1[physics. data-an]
[5] Espen A F, Ihlen 2012 Front. Physiol. 3 141
[6] Espen A F, Ihlen 2013 Behav. Res. Methods 45 928
[7] Chen Z, Ivanov P C, Hu K, Stanley H E 2002 Phys. Rev. E 65 041107
[8] Rybski D, Bunde A, Havlin S, Kantelhardt J W, Koscielny-Bunde E 2011 In Extremis (Berlin Heidelberg: Springer-Verlag) pp216-248
[9] Zhou Y, Leung Y, Yu Z G 2011 Chin. Phys. B 20 090507
[10] Bashan A, Bartsch R, Kantelhardt J W, Havlin S 2008 Physica A 387 5080
[11] Vandewalle N, Ausloos M 1998 Phys. Rev. E 58 6832
[12] Gu G F, Zhou W X 2010 Phys. Rev. E 82 011136
[13] Arianos S, Carbone A 2007 Physica A 382 9
[14] Shao Y H, Gu G F, Jiang Z Q, Zhou W X, Sornette D 2012 Sci. Rep. 2 835
[15] Gu G F, Zhou W X 2006 Phys. Rev. E 74 061104
[16] Xiong G, Zhang S N, Zhao H C 2014 Acta Phys. Sin. 63 150503 (in Chinese) [熊刚, 张淑宁, 赵惠昌 2014 63 150503]
[17] Xiong G, Zhang S N, Zhao H C 2014 Chaos Soliton. Fract. 65 97
[18] Xu L M, Ivanov P C, Hu K, Chen Z, Carbone A, Stanley H E 2005 Phys. Rev. E 71 051101
[19] Alessio E, Carbone A, Castelli G, Frappietro V 2002 Eur. Phys. J. B 27 197
[20] Carbone A, Castelli G, Stanley H E 2004 Physica A 344 267
[21] Jiang Z Q, Zhou W X 2011 Phys. Rev. E 84 016106
[22] Guan J, Liu N B, Huang Y 2011 Radar Target Detection and Application of Fractal Theory (Beijing: Publishing House Of Electronics Industry) p68 (in Chinese) [关键, 刘宁波, 黄勇 2011 雷达目标检测的分形理论及应用(北京:电子工业出版社) 第68页]
[23] Zhang B 2013 Ph. D. Dissertation (Xian: Xidian University) (in Chinese) [张波 2013 博士学位论文(西安: 西安电子科技大学)]
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[1] Serrano E, Figliola A 2009 Physica A 388 2793
[2] Ge E J, Leung Y 2013 J. Geogr. Syst. 15 115
[3] Kantelhardt J W, Zschiegner S A, Koscielny-Bunde E, Bunde A, Havlin S, Stanley H E 2002 Physica A 316 87
[4] Kantelhardt J W 2008 arXiv:0804.0747v1[physics. data-an]
[5] Espen A F, Ihlen 2012 Front. Physiol. 3 141
[6] Espen A F, Ihlen 2013 Behav. Res. Methods 45 928
[7] Chen Z, Ivanov P C, Hu K, Stanley H E 2002 Phys. Rev. E 65 041107
[8] Rybski D, Bunde A, Havlin S, Kantelhardt J W, Koscielny-Bunde E 2011 In Extremis (Berlin Heidelberg: Springer-Verlag) pp216-248
[9] Zhou Y, Leung Y, Yu Z G 2011 Chin. Phys. B 20 090507
[10] Bashan A, Bartsch R, Kantelhardt J W, Havlin S 2008 Physica A 387 5080
[11] Vandewalle N, Ausloos M 1998 Phys. Rev. E 58 6832
[12] Gu G F, Zhou W X 2010 Phys. Rev. E 82 011136
[13] Arianos S, Carbone A 2007 Physica A 382 9
[14] Shao Y H, Gu G F, Jiang Z Q, Zhou W X, Sornette D 2012 Sci. Rep. 2 835
[15] Gu G F, Zhou W X 2006 Phys. Rev. E 74 061104
[16] Xiong G, Zhang S N, Zhao H C 2014 Acta Phys. Sin. 63 150503 (in Chinese) [熊刚, 张淑宁, 赵惠昌 2014 63 150503]
[17] Xiong G, Zhang S N, Zhao H C 2014 Chaos Soliton. Fract. 65 97
[18] Xu L M, Ivanov P C, Hu K, Chen Z, Carbone A, Stanley H E 2005 Phys. Rev. E 71 051101
[19] Alessio E, Carbone A, Castelli G, Frappietro V 2002 Eur. Phys. J. B 27 197
[20] Carbone A, Castelli G, Stanley H E 2004 Physica A 344 267
[21] Jiang Z Q, Zhou W X 2011 Phys. Rev. E 84 016106
[22] Guan J, Liu N B, Huang Y 2011 Radar Target Detection and Application of Fractal Theory (Beijing: Publishing House Of Electronics Industry) p68 (in Chinese) [关键, 刘宁波, 黄勇 2011 雷达目标检测的分形理论及应用(北京:电子工业出版社) 第68页]
[23] Zhang B 2013 Ph. D. Dissertation (Xian: Xidian University) (in Chinese) [张波 2013 博士学位论文(西安: 西安电子科技大学)]
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