Adaptive step-size modified fractional least mean square algorithm for chaotic time series prediction

doi:10.1088/1674-1056/23/5/050503

Abstract This paper presents an adaptive step-size modified fractional least mean square (AMFLMS) algorithm to deal with a nonlinear time series prediction. Here we incorporate adaptive gain parameters in the weight adaptation equation of the original MFLMS algorithm and also introduce a mechanism to adjust the order of the fractional derivative adaptively through a gradient-based approach. This approach permits an interesting achievement towards the performance of the filter in terms of handling nonlinear problems and it achieves less computational burden by avoiding the manual selection of adjustable parameters. We call this new algorithm the AMFLMS algorithm. The predictive performance for the nonlinear chaotic Mackey Glass and Lorenz time series was observed and evaluated using the classical LMS, Kernel LMS, MFLMS, and the AMFLMS filters. The simulation results for the Mackey glass time series, both without and with noise, confirm an improvement in terms of mean square error for the proposed algorithm. Its performance is also validated through the prediction of complex Lorenz series.

Keywords: fractional LMS kernel LMS Reimann-Lioville derivative Mackey glass and Lorenz time series

Received: 08 October 2013
Revised: 21 November 2013
Accepted manuscript online:

PACS:	05.45.Df	(Fractals)
	05.30.Pr	(Fractional statistics systems)
	05.45.Tp	(Time series analysis)
	92.60.Wc	(Weather analysis and prediction)

Fund: Project supported by the Higher Education Commission of Pakistan.

Corresponding Authors: Bilal Shoaib
E-mail: bilal.phdee42@iiu.edu.pk

About author: 05.45.Df; 05.30.Pr; 05.45.Tp; 92.60.Wc

Cite this article:

Bilal Shoaib, Ijaz Mansoor Qureshi, Shafqatullah, Ihsanulhaq Adaptive step-size modified fractional least mean square algorithm for chaotic time series prediction 2014 Chin. Phys. B 23 050503

[1]	Miller K S and Ross B 1993 An Introduction to the Fractional Calculus and Fractional Derivative (New York: John Willey & Sons)
[2]	Samko S, Killbas A and Marichev O 1993 Fractional Integrals and Derivatives: Theory and Applications (Verdon: Breach Science Publishers)
[3]	Anatoly K A, Srivastav H M and Trijillo J J 2006 Theory and Applications of Fractional Differential Equations (North Holland: Mathematics Studiesn)
[4]	Ross B 1992 Int. J. Math. Statist. Sci. 1 21
[5]	Ross B 1997 Historia Math. 4 75
[6]	Ross B 1975 International Conference at the University of New Heaven, June 15-16, Berlin, Germany
[7]	Machado J T, Kiryakova V and Mianardi F 2011 Communication in Nonlinear Science and Numerical Simulation 16 1140
[8]	Methews J V and Xie Z 1993 IEEE Trans. Signal Processing 41 2075
[9]	Tseng, Cheng C and Lee S L 2012 Signal Processing 92 553
[10]	Raja M A and Qureshi I M 2009 Eur. J. Sci. Res. 35 14
[11]	Akhtar P and Yasin M 2012 2nd International Multi-topic Conference IMTIC, March 28-30, Jamshoro, Pakistan
[12]	Geravanchizadeh, Masood and Osgouei S G 2011 19th Iranian Conference on Electrical Engineering, May 17-19, Tehran, Iran
[13]	Li X T and Yin M H 2012 Chin. Phys. B 21 050507
[14]	Harris R W, Chabries D M and Bishop F A 1985 IEEE Trans. Accoustic Speech and Signal Processing 33 222
[15]	Kwong C P 1992 IEEE Trans. Sign. Process. 40 2613
[16]	Mikheal W B, et al. 1986 IEEE Trans. Circ. Syst. 33 677
[17]	Wei R, Lu J G, Li J and Wang Z Q 2002 Acta Electron. Sin. 30 73
[18]	Bernard W and Streams S D 1985 Advance Signal Processing (New Jersy: Prentice Hall)
[19]	Liu W, Pockarel P and Principe J C 2008 IEEE Trans. Sign. Process. 56 543
[20]	Ali H S 2003 Fundamental of Adaptive Filtering (New York: John Willey & Sons)
[21]	Pockarel P, Liu W and Principe J C 2009 Sign. Process. 89 257
[22]	Liu W, Principe J C and Haykin S 2010 Kernel Adaptive Filtering (New York: John Willey & Sons)
[23]	Shoaib B, Qureshi I M, Ihsanulhaq and Shafqatullah 2014 Chin. Phys. B 23 030502
[24]	Cheng E S, Chen S and Mulgrew B 1996 IEEE Trans. Neural Networks 7 190

[1]	Multifractal analysis of the software evolution in software networks Meili Liu(刘美丽), Xiaogang Qi(齐小刚), and Hao Pan(潘浩). Chin. Phys. B, 2022, 31(3): 030501.
[2]	Fractal sorting vector-based least significant bit chaotic permutation for image encryption Yong-Jin Xian(咸永锦), Xing-Yuan Wang(王兴元), Ying-Qian Zhang(张盈谦), Xiao-Yu Wang(王晓雨), and Xiao-Hui Du(杜晓慧). Chin. Phys. B, 2021, 30(6): 060508.
[3]	Analysis and implementation of new fractional-order multi-scroll hidden attractors Li Cui(崔力), Wen-Hui Luo(雒文辉), and Qing-Li Ou(欧青立). Chin. Phys. B, 2021, 30(2): 020501.
[4]	Analysis of secondary electron emission using the fractal method Chun-Jiang Bai(白春江), Tian-Cun Hu(胡天存), Yun He(何鋆), Guang-Hui Miao(苗光辉), Rui Wang(王瑞), Na Zhang(张娜), and Wan-Zhao Cui(崔万照). Chin. Phys. B, 2021, 30(1): 017901.
[5]	Image encryption technique based on new two-dimensional fractional-order discrete chaotic map and Menezes-Vanstone elliptic curve cryptosystem Zeyu Liu(刘泽宇), Tiecheng Xia(夏铁成), Jinbo Wang(王金波). Chin. Phys. B, 2018, 27(3): 030502.
[6]	Detection of meso-micro scale surface features based on microcanonical multifractal formalism Yuanyuan Yang(杨媛媛), Wei Chen(陈伟), Tao Xie(谢涛), William Perrie. Chin. Phys. B, 2018, 27(1): 010502.
[7]	Multifractal modeling of the production of concentrated sugar syrup crystal Sheng Bi(闭胜), Jianbo Gao(高剑波). Chin. Phys. B, 2016, 25(7): 070502.
[8]	Exploring the relationship between fractal features and bacterial essential genes Yong-Ming Yu(余永明), Li-Cai Yang(杨立才), Qian Zhou(周茜), Lu-Lu Zhao(赵璐璐), Zhi-Ping Liu(刘治平). Chin. Phys. B, 2016, 25(6): 060503.
[9]	Multifractal analysis of white matter structural changes on 3D magnetic resonance imaging between normal aging and early Alzheimer's disease Ni Huang-Jing (倪黄晶), Zhou Lu-Ping (周泸萍), Zeng Peng (曾彭), Huang Xiao-Lin (黄晓林), Liu Hong-Xing (刘红星), Ning Xin-Bao (宁新宝), the Alzheimer's Disease Neuroimaging Initiative. Chin. Phys. B, 2015, 24(7): 070502.
[10]	Directional region control of the thermalfractal diffusion of a space body Qiao Wei (乔威), Sun Jie (孙洁), Liu Shu-Tang (刘树堂). Chin. Phys. B, 2015, 24(5): 050504.
[11]	Space–time fractional KdV–Burgers equation for dust acoustic shock waves in dusty plasma with non-thermal ions Emad K. El-Shewy, Abeer A. Mahmoud, Ashraf M. Tawfik, Essam M. Abulwafa, Ahmed Elgarayhi. Chin. Phys. B, 2014, 23(7): 070505.
[12]	Row-column visibility graph approach to two-dimensional landscapes Xiao Qin (肖琴), Pan Xue (潘雪), Li Xin-Li (李信利), Mutua Stephen, Yang Hui-Jie (杨会杰), Jiang Yan (蒋艳), Wang Jian-Yong (王建勇), Zhang Qing-Jun (张庆军). Chin. Phys. B, 2014, 23(7): 078904.
[13]	A fractal approach to low velocity non-Darcy flow in a low permeability porous medium Cai Jian-Chao (蔡建超). Chin. Phys. B, 2014, 23(4): 044701.
[14]	A modified fractional least mean square algorithm for chaotic and nonstationary time series prediction Bilal Shoaib, Ijaz Mansoor Qureshi, Ihsanulhaq, Shafqatullah. Chin. Phys. B, 2014, 23(3): 030502.
[15]	Subcooled pool boiling heat transfer in fractal nanofluids：A novel analytical model Xiao Bo-Qi (肖波齐), Yang Yi (杨毅), Xu Xiao-Fu (许晓赋). Chin. Phys. B, 2014, 23(2): 026601.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Adaptive step-size modified fractional least mean square algorithm for chaotic time series prediction

Cite this article:

Online attention