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Recently, adaptive filters have been widely used to perform the prediction of chaotic time series. Generally, the Gaussian noise is considered for the system noise. However, many non-Gaussian noises, e.g., impulse noise and alpha noise, exist in real systems. Adaptive filters are therefore required to reduce such non-Gaussian noises for practical applications. For improving the robustness against non-Gaussian noise, the maximum correntropy criterion (MCC) is successfully used to derive various robust adaptive filters. In these robust adaptive filters, the steepest ascent method based on the first-order derivative is generally utilized to construct the weight update form. It is well known that the traditional derivative can be generalized by the fractional-order derivative effectively. Therefore, to further improve the performance of adaptive filters based on the MCC, the fractional-order derivative is applied to the MCC-based algorithm, generating a novel fractional-order maximum correntropy criterion (FMCC) algorithm. Under the non-Gaussian noises, the proposed FMCC algorithm can be applied to predicting the chaotic time series effectively. In the proposed FMCC algorithm, the weight update form is constructed by using a combination of the first-order derivative based term and the fractional-order derivative based term. The Riemann-Liouville definition is utilized for calculating the fractional-order derivative in the proposed FMCC algorithm. The order of the fractional-order derivative is a crucial parameter of the proposed FMCC algorithm. However, it is difficult to obtain the optimal fractional order for different nonlinear systems theoretically. Therefore, the influence of the fractional order on the prediction performance is determined by trials for different nonlinear systems. The appropriate fractional order corresponds to the optimum of prediction accuracy, and can be chosen in advance. Simulations in the context of prediction of Mackey-Glass time series and Lorenz time series demonstrate that in the case of non-Gaussian noises the proposed FMCC algorithm achieves better prediction accuracy and faster convergence rate than the least mean square (LMS) algorithm, the MCC algorithm, and the fractional-order least mean square (FLMS) algorithm. In addition, the computational complexity of different filters is compared with each other under the example of the prediction of Marckey-Glass time series by using mean consumed time. It can be found that the computational complexity of FMCC algorithm is higher than those of the MCC and the LMS algorithms, but only slightly higher than that of the FLMS algorithm. As a result, comparing with other filters, the FMCC algorithm can improve the prediction performances of chaotic time series at the cost of the increasing computational complexity.
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Keywords:
- prediction of chaotic time series /
- adaptive filter /
- fractional-order derivative /
- maximum correntropy criterion
[1] Tang Z J, Ren F, Peng T, Wang W B 2014 Acta Phys. Sin. 63 050505(in Chinese) [唐舟进, 任峰, 彭涛, 王文博 2014 63 050505]
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[7] Takens F 1981 Lecture Notes Math. 898 366
[8] Al-saggaf U M, Moinuddin M, Arif M, Zerguine A 2015 Signal Process. 111 50
[9] Gui G, Peng W, Adachi F 2014 Int. J. Commun. Syst. 27 2956
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[11] Van V S, Lazarogredilla M, Santamaria I 2012 IEEE Trans. Neural Netw. Learn. Syst. 23 1313
[12] Qiao B Q, Liu S M, Zeng H D, Li X, Dai B Z 2017 Sci. China: Phys. Mech. 60 040521
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[15] Chen B D, Xing L, Liang J L, Zheng N N, Principe J C 2014 Signal Process. Lett. 21 880
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[20] Zhou Y, Ionescu C, Machado J A T 2015 Nonlinear Dyn. 80 1661
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[22] Santamaria I, Pokharel P P, Principe J C 2006 IEEE Trans. Signal Process. 54 2187
[23] Liu W, Pokharel P P, Principe J C 2007 IEEE Trans. Signal Process. 55 5286
[24] Aronszajn A 1950 IEEE Trans. Am. Math. Soc. 68 337
[25] Duan J W, Ding X, Liu T 2017 Sci. China: Inf. Sci. 60 1
[26] Huang S, Zhang R, Chen D 2016 J. Computat. Nonlinear Dyn. 11 031007
[27] Shoaib B, Qureshi I M 2014 Chin. Phys.. 23 050503
[28] Mackey M C, Glass L 1977 Science 197 87
[29] Lorenz E N 1963 J. Atmos. Sci. 20 130
[30] Li B B, Ma H S, Liu M Q 2014 J. Electron. Inf. Technol. 36 868(in Chinese) [李兵兵, 马洪帅, 刘明骞 2014 电子与信息学报 36 868]
[31] Stewart I 2000 Nature 406 948
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[1] Tang Z J, Ren F, Peng T, Wang W B 2014 Acta Phys. Sin. 63 050505(in Chinese) [唐舟进, 任峰, 彭涛, 王文博 2014 63 050505]
[2] Song T, Li H 2012 Acta Phys. Sin. 61 080506(in Chinese) [宋彤, 李菡 2012 61 080506]
[3] Zhang J S, Xiao X C 2000 Chin. Phys. Lett. 17 88
[4] Farmer J D, Sidorowich J J 1987 Phys. Rev. Lett. 59 845
[5] Zheng Y F, Wang S Y, Feng J C, Tse C K 2016 Digit. Signal Process. 48 130
[6] Meng Q F, Zhang Q, Mou W Y 2006 Acta Phys. Sin. 55 1666(in Chinese) [孟庆芳, 张强, 牟文英 2006 55 1666]
[7] Takens F 1981 Lecture Notes Math. 898 366
[8] Al-saggaf U M, Moinuddin M, Arif M, Zerguine A 2015 Signal Process. 111 50
[9] Gui G, Peng W, Adachi F 2014 Int. J. Commun. Syst. 27 2956
[10] Ozeki K, Umeda T 1984 Electr. Commun. Jpn. 67 19
[11] Van V S, Lazarogredilla M, Santamaria I 2012 IEEE Trans. Neural Netw. Learn. Syst. 23 1313
[12] Qiao B Q, Liu S M, Zeng H D, Li X, Dai B Z 2017 Sci. China: Phys. Mech. 60 040521
[13] Erdogmus D, Principe J C 2002 IEEE Trans. Neural Netw. 13 1035
[14] Hu T, Wu Q, Zhou D X 2016 IEEE Trans. Signal Process 64 6571
[15] Chen B D, Xing L, Liang J L, Zheng N N, Principe J C 2014 Signal Process. Lett. 21 880
[16] Shi L M, Lin Y 2014 Signal Process. Lett. 21 1385
[17] Chen B D, Principe J C 2012 IEEE Trans. Process. Lett. 19 491
[18] Chen Y, Li S G, Liu H 2016 Acta Phys. Sin. 65 170501(in Chinese) [陈晔, 李生刚, 刘恒 2016 65 170501]
[19] Shah S M, Samar R, Khan N M, Raja M A Z 2016 Nonlinear Dyn. 88 839
[20] Zhou Y, Ionescu C, Machado J A T 2015 Nonlinear Dyn. 80 1661
[21] Shah S M, Samar R, Raja M A Z, Chambers J A 2014 Electron. Lett. 50 973
[22] Santamaria I, Pokharel P P, Principe J C 2006 IEEE Trans. Signal Process. 54 2187
[23] Liu W, Pokharel P P, Principe J C 2007 IEEE Trans. Signal Process. 55 5286
[24] Aronszajn A 1950 IEEE Trans. Am. Math. Soc. 68 337
[25] Duan J W, Ding X, Liu T 2017 Sci. China: Inf. Sci. 60 1
[26] Huang S, Zhang R, Chen D 2016 J. Computat. Nonlinear Dyn. 11 031007
[27] Shoaib B, Qureshi I M 2014 Chin. Phys.. 23 050503
[28] Mackey M C, Glass L 1977 Science 197 87
[29] Lorenz E N 1963 J. Atmos. Sci. 20 130
[30] Li B B, Ma H S, Liu M Q 2014 J. Electron. Inf. Technol. 36 868(in Chinese) [李兵兵, 马洪帅, 刘明骞 2014 电子与信息学报 36 868]
[31] Stewart I 2000 Nature 406 948
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