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提出了KLD系数和归一化KLD系数来刻画多维序列的相关结构, 以解决KLD维密度固有的局限性. 利用完全相关和完全不相关的多维序列, 导出KLD维密度的上界和下界函数, 进而导出KLD系数的上界和下界, 在此基础上提出归一化KLD系数. 解析分析和数值仿真都证明, 多维序列相关结构的变化会引起归一化KLD系数线性的变化. 数值仿真还证明, 即使多维序列中仅有其中的两个时间序列的相关结构发生改变, 归一化KLD系数仍能灵敏地检测到. 不仅如此, 归一化KLD系数还可用于非平稳时间序列的分析. 耦合映象格子的数值仿真结果表明, 归一化KLD系数还能够分析非线性系统的相关结构.The KLD coefficient and the normalized KLD coefficient are proposed to characterize the correlation of multivariable series in order to overcome the intrinsic limitations of the KLD dimension density. Using the uncorrelated or perfectly correlated multivariable series, the upper and the lower bound functions of the KLD dimension density, and furthermore the upper and the lower bounds of the KLD coefficient are analytically deduced. Then, the normalized KLD coefficient is proposed in the paper. The analyses and numerical simulations prove that the changes of correlation of multivariable series can lead to linear variation of the normalized KLD coefficient. The simulations also prove that the normalized KLD coefficient can detect the changes of correlation sensitively, even if these are induced by only two channels of multivariable series. Furthermore, the normalized KLD coefficient can be used to analyze the nonstationary time series. The simulation results of coupled map lattice prove that the normalized KLD coefficient can also be used for the nonlinear system analysis.
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Keywords:
- multivariable series analysis /
- correlation /
- detection /
- KLD
[1] Müller M, Baier G 2005 Phys. Rev. E 71 046116
[2] He G G, Zhu P, Chen H P, Xie X P 2010 Acta Phys. Sin. 59 5307 (in Chinese) [何国光, 朱萍, 陈宏平, 谢小平 2010 59 5307]
[3] Quiroga R Q, Kraskov A, Kreuz T, Grassberger P 2002 Phys. Rev. E 65 041903
[4] Pereda E, Quiroga R Q, Bhattacharya J 2005 Progress in Neurobiology 77 1
[5] Stam C J 2005 Clinical Neurophysiology 116 2266
[6] Bloomfield P 2000 Fourier Analysis of Time Series (New York: A Wiley-Interscience Publication, John Wiley & Sons, INC.) p1
[7] Xie X P, Cao Z T, Weng X C 2008 NeuroImage 40 1672
[8] Zoldi S M, Greenside H S 1997 Phys. Rev. Lett. 78 1687
[9] Mallat S 1989 IEEE Pattern Anal. and Machine Intell. 11 674
[10] Raab C, Kurths J 2001 Phys. Rev. E 64 016216
[11] Bauer M, Heng H, Martienssen W 1993 Phys. Rev. Lett. 71 521
[12] Bünner M J, Hegger R 1999 Physics Letters A 258 25
[13] Zeng X, Eykholt R, Pielke R A 1991 Phys. Rev. Lett. 66 3229
[14] Sano M, Sawada Y 1985 Phys. Rev. Lett. 55 1082
[15] Kramer M A 1991 Amer. Inst. Chem. Engin. Journal 37 233
[16] Roweis S T, Saul L K 2000 Science 290 2323
[17] Common P 1991 Signal Processing 24 1
[18] Meixner M, Zoldi S M, Bose S, Schöll E 2000 Phys. Rev. E 61 1382
[19] Xie X P, Zhao X H, Fang Y T, Cao Z T, He G G 2011 Physics Letters A 375 1789
[20] Varela H, Beta C, Bonnefont A, Krischer K 2005 Phys. Rev. Lett. 94 174104
[21] Plerou V, Gopikrishnan P, Rosenow B, Amaral L A N, Guhr T, Stanley H 2002 Phys. Rev. E 65 066126
[22] Bazzani A, Castellani G C 2010 Phys. Rev. E 81 051917
[23] Yang W M 1994 Spatiotemporal Chaos and Coupled Map Lattices (Shanghai: Shanghai Scientific and Technological Education Publishing House) p12 (in Chinese) [杨维明 1994 时空混沌和耦合映象格子(上海: 上海科技教育出版社) 第12页]
[24] Wolf A, Swift J B, Swinney H L, Vastano J A 1985 Physica D 16 285
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[1] Müller M, Baier G 2005 Phys. Rev. E 71 046116
[2] He G G, Zhu P, Chen H P, Xie X P 2010 Acta Phys. Sin. 59 5307 (in Chinese) [何国光, 朱萍, 陈宏平, 谢小平 2010 59 5307]
[3] Quiroga R Q, Kraskov A, Kreuz T, Grassberger P 2002 Phys. Rev. E 65 041903
[4] Pereda E, Quiroga R Q, Bhattacharya J 2005 Progress in Neurobiology 77 1
[5] Stam C J 2005 Clinical Neurophysiology 116 2266
[6] Bloomfield P 2000 Fourier Analysis of Time Series (New York: A Wiley-Interscience Publication, John Wiley & Sons, INC.) p1
[7] Xie X P, Cao Z T, Weng X C 2008 NeuroImage 40 1672
[8] Zoldi S M, Greenside H S 1997 Phys. Rev. Lett. 78 1687
[9] Mallat S 1989 IEEE Pattern Anal. and Machine Intell. 11 674
[10] Raab C, Kurths J 2001 Phys. Rev. E 64 016216
[11] Bauer M, Heng H, Martienssen W 1993 Phys. Rev. Lett. 71 521
[12] Bünner M J, Hegger R 1999 Physics Letters A 258 25
[13] Zeng X, Eykholt R, Pielke R A 1991 Phys. Rev. Lett. 66 3229
[14] Sano M, Sawada Y 1985 Phys. Rev. Lett. 55 1082
[15] Kramer M A 1991 Amer. Inst. Chem. Engin. Journal 37 233
[16] Roweis S T, Saul L K 2000 Science 290 2323
[17] Common P 1991 Signal Processing 24 1
[18] Meixner M, Zoldi S M, Bose S, Schöll E 2000 Phys. Rev. E 61 1382
[19] Xie X P, Zhao X H, Fang Y T, Cao Z T, He G G 2011 Physics Letters A 375 1789
[20] Varela H, Beta C, Bonnefont A, Krischer K 2005 Phys. Rev. Lett. 94 174104
[21] Plerou V, Gopikrishnan P, Rosenow B, Amaral L A N, Guhr T, Stanley H 2002 Phys. Rev. E 65 066126
[22] Bazzani A, Castellani G C 2010 Phys. Rev. E 81 051917
[23] Yang W M 1994 Spatiotemporal Chaos and Coupled Map Lattices (Shanghai: Shanghai Scientific and Technological Education Publishing House) p12 (in Chinese) [杨维明 1994 时空混沌和耦合映象格子(上海: 上海科技教育出版社) 第12页]
[24] Wolf A, Swift J B, Swinney H L, Vastano J A 1985 Physica D 16 285
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