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For multivariate chaotic time series, a method of conditional entropy extending dimension(CEED) in the reconstructed phase space is proposed. First, the delay time of any variable time series is selected by mutual information method, and then the embedding dimension of phase space is extended by the conditional entropy. This method can ensure the independence of reconstructed coordinates from low space to high space and eliminate the redundancy of phase space, because the largest condition entropy is choosen. The effective input vector for the prediction of multivariate time series is given. Simulations of the Lorenz system and Henon system show that the neural network predictions of multivariate time series are much better than the prediction of univariate and existing multivariate. Therefore, CEED is effective for multivariate chaotic systems.
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Keywords:
- multivariate chaotic time series /
- phase space reconstruction /
- conditional entropy /
- neural network prediction
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[3] Ma Q L, Zheng Q L, Peng H, Zhong T W, Qin J W 2008 Chin. Phys. B 17 536
[4] Zhang S Q, Jia J, Gao M, Han X 2010 Acta Phys. Sin. 59 1576 (in Chinese) [张淑清、贾 健、高 敏、韩 叙 2010 59 1576]
[5] Fraser A M, Swinney H L 1986 Phys. Rev. A 33 11340
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[7] Yan H,Wei P,Xiao X C 2009 Chin. Phys. B 18 3287
[8] Zhang C T, Ma Q L, Peng H 2010 Acta Phys. Sin. 7623 (in Chinese) [张春涛、马千里、彭 宏 2010 7623]
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[10] Cao L Y, Mees A, Judd K 1998 Physica D 121 75
[11] Rombowts S A R B, Keunen R WM, Stom C J 1995 Phys. Lett. A 202 352
[12] Lu S,Wang H Y 2006 Acta Phys. Sin. 55 572 (in Chinese) [卢 山、王海燕 2006 55 572]
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[14] Wang H Y, Sheng Z H, Zhang J 2003 J. Southeast Univ. (Natural Science Edition) 33 115 (in Chinese) [王海燕、盛昭瀚、张 进 2003 东南大学学报(自然科学版) 33 115 ]
[15] Broomhead D S, King G P 1986 Physica D 20 217
[16] Fan C J 2009 J. University of Shanghai for Science and Technology 31 273 (in Chinese) [樊重俊 2009 上海理工大学学报 31 273]
[17] Han M, Fan M M, Xi J H 2005 Lecture Notes in Computer Science 3497 618
[18] Han M, Wang Y 2009 Expert Systems with Applications 36 1280
[19] Tian Y C, Fu X T, Lu Y Z 1993 Control and Decision 8 345 (in Chinese) [田玉楚、符雪桐、吕勇哉 1993 控制与决策 8 345]
[20] Elman J L 1990 Cognitive Science 14 179
[21] Ye M Y, Wang X D, Zhang H R 2005 Acta Phys. Sin. 54 2568 (in Chinese) [叶美盈、汪晓东、张浩然 2005 54 2568]
[22] Ren R, Xu J, Zhu S H 2006 Acta Phys. Sin. 55 555 (in Chinese) [任 韧、徐 进、朱世华 2006 55 555]
[23] Mao J Q, Yao J, Ding H S 2009 Acta Phys. Sin.58 2220 (in Chinese) [毛剑琴、姚 健、丁海山 2009 58 2220]
[24] Chiu S 1994 J. Intell. Fuzzy Syst. 2 267
[25] Jang J S R 1993 IEEE Trans. Syst. Man 23 665
[26] Zhang J S, Li H C, Xiao X C 2005 Chin. Phys. 14 49
[27] Jan S, Guatavo D 1997 Physics D 108 335
[28] Zhang J F, Hu S S 2008 Acta Phys. Sin. 57 2708 (in Chinese) [张军峰、胡寿松 2008 57 2708]
[29] Gholpour A, Araabi B N, Lucas C 2006 Neural Proc. Lett. 24 217
[30] Han M,Wei R 2008 14th International Conference of Neural Information Processing Kitakyushu, Japan, November 13—16, 2007 p415
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[1] Peng S G, Yu S M 2009 Chin. Phys. B 18 3758
[2] Ma Q L, Zheng Q L, Peng H, Qin J W 2009 Acta Phys. Sin. 58 1410 (in Chinese) [马千里、郑启伦、彭 宏、覃姜维 2009 58 1410]
[3] Ma Q L, Zheng Q L, Peng H, Zhong T W, Qin J W 2008 Chin. Phys. B 17 536
[4] Zhang S Q, Jia J, Gao M, Han X 2010 Acta Phys. Sin. 59 1576 (in Chinese) [张淑清、贾 健、高 敏、韩 叙 2010 59 1576]
[5] Fraser A M, Swinney H L 1986 Phys. Rev. A 33 11340
[6] Xiao F H,Yan G R, Han Y H 2005 Acta Phys. Sin. 54 550 (in Chinese) [肖方红、阎桂荣、韩宇航 2005 54 550]
[7] Yan H,Wei P,Xiao X C 2009 Chin. Phys. B 18 3287
[8] Zhang C T, Ma Q L, Peng H 2010 Acta Phys. Sin. 7623 (in Chinese) [张春涛、马千里、彭 宏 2010 7623]
[9] Kantz H, Schreiber T 1997 Nonlinear Time Series Analysis(Cambridge: Cambridge University Press) p7
[10] Cao L Y, Mees A, Judd K 1998 Physica D 121 75
[11] Rombowts S A R B, Keunen R WM, Stom C J 1995 Phys. Lett. A 202 352
[12] Lu S,Wang H Y 2006 Acta Phys. Sin. 55 572 (in Chinese) [卢 山、王海燕 2006 55 572]
[13] Zhang Y, Guan W 2009 Acta Phys. Sin. 58 756 (in Chinese) [张 勇、关 伟 2009 58 756]
[14] Wang H Y, Sheng Z H, Zhang J 2003 J. Southeast Univ. (Natural Science Edition) 33 115 (in Chinese) [王海燕、盛昭瀚、张 进 2003 东南大学学报(自然科学版) 33 115 ]
[15] Broomhead D S, King G P 1986 Physica D 20 217
[16] Fan C J 2009 J. University of Shanghai for Science and Technology 31 273 (in Chinese) [樊重俊 2009 上海理工大学学报 31 273]
[17] Han M, Fan M M, Xi J H 2005 Lecture Notes in Computer Science 3497 618
[18] Han M, Wang Y 2009 Expert Systems with Applications 36 1280
[19] Tian Y C, Fu X T, Lu Y Z 1993 Control and Decision 8 345 (in Chinese) [田玉楚、符雪桐、吕勇哉 1993 控制与决策 8 345]
[20] Elman J L 1990 Cognitive Science 14 179
[21] Ye M Y, Wang X D, Zhang H R 2005 Acta Phys. Sin. 54 2568 (in Chinese) [叶美盈、汪晓东、张浩然 2005 54 2568]
[22] Ren R, Xu J, Zhu S H 2006 Acta Phys. Sin. 55 555 (in Chinese) [任 韧、徐 进、朱世华 2006 55 555]
[23] Mao J Q, Yao J, Ding H S 2009 Acta Phys. Sin.58 2220 (in Chinese) [毛剑琴、姚 健、丁海山 2009 58 2220]
[24] Chiu S 1994 J. Intell. Fuzzy Syst. 2 267
[25] Jang J S R 1993 IEEE Trans. Syst. Man 23 665
[26] Zhang J S, Li H C, Xiao X C 2005 Chin. Phys. 14 49
[27] Jan S, Guatavo D 1997 Physics D 108 335
[28] Zhang J F, Hu S S 2008 Acta Phys. Sin. 57 2708 (in Chinese) [张军峰、胡寿松 2008 57 2708]
[29] Gholpour A, Araabi B N, Lucas C 2006 Neural Proc. Lett. 24 217
[30] Han M,Wei R 2008 14th International Conference of Neural Information Processing Kitakyushu, Japan, November 13—16, 2007 p415
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