提出一种基于最优样本子集的在线模糊最小二乘支持向量机(least squares support vector machine,LSSVM) 混沌时间序列预测方法. 算法选择与预测样本时间上以及欧氏距离最近的样本点构成最优样本子集，并采用" 不敏感函数对其进行模糊化处理，通过模糊LSSVM 训练获得预测模型. 随着时间窗口的滑动，最优样本子
集和预测模型实时更新，模型更新采用分块矩阵方法降低运算复杂度. 实验中对时变Ikeda 序列进行预测，表明所提出的方法与离线和在线LSSVM 相比，训练速度更快，预测精度更高.
An optimal training subset online fuzzy least squares support vector machine (OTSOF-LSSVM) is proposed for chaotic time series prediction. Samples nearest to the prediction sample in both time and space are chosen to form the optimal training subset. An "-insensitive function is introduced to formulate the fuzzy membership. Thus a prediction model is established by fuzzy LSSVM. The subset and model are updated with the moving time window. Computational complexity is reduced by matrix partitioning. Experiment of predicting the time-variant chaotic time series Ikeda shows that the proposed method has better accuracy and high training speed as compared to offline and online LSSVM.
[1] KHATIBI R, Bellie SIVAKUMAR B, Mohammad Ali GHORBANI M A, Ozgur KISI O, Kasim KOÇAK K, Davod Farsadi ZADEH D F. Investigating chaos in river stage and discharge time series [J]. Journal of Hydrology, 2012, 414-415: 108-117.
[2] LI Huaqing, LIAO Xiaofeng, LI Chuandong, LI Chaojie. Chaos control and synchronization via a novel chatter free sliding mode control strategy [J]. Neurocomputing, 2011, 74: 3212-3222.
[3] DU Jianguo, HUANG Tingwen, SHENG Zhaohan, ZHANG Haibin. A new method to control chaos in an economic system [J]. Applied Mathematics and Computation, 2010, 217: 2370-2380.
[4] De PAULA A S, SAVI M A. Comparative analysis of chaos control methods: a mechanical system case study [J]. International Journal of Non-Linear Mechanics 2011, 46(8): 1076-1089.
[5] SMITH R K, GRABOWSKI M, CAMLEY R E. Period doubling toward chaos in a driven magnetic macrospin [J]. Journal of Magnetism and Magnetic Materials, 2010, 322: 2127-2134.
[6] LI Hengchao, ZHANG Jiashu, XIAO Xianci. Neural Volterra(是人名吗？) filter for chaotic time series prediction [J]. Chinese physics, 2005, 14(11): 2181-2188.（是人名，滤波方法）
[7] Alan Wolf, Jack B Swift, Harry L Swinney, John A. Vastano. Determining Lyapunov exponents from a time series [J]. Physics D, 1985, 16(2): 285-317.
[8] 张勇，关伟. 基于最大李亚普诺夫指数的改进混沌时间序列预测 [J]. 信息与控制，2009, 38(3): 360-364.
Zhang Yong, Guan Wei. An improved method for forecasting chaotic time series based on maximum Lyapunov exponent [J]. Information and Control, 2009, 38(3): 360-364. (in Chinese)
[9] Han M, Xi J, Xu S, Yin F L. Prediction of chaotic time series based on the recurrent predictor neural network [J]. IEEE Transaction on Signal Processing, 2004, 52(12): 3409-3416.
[10] MA Qianli, ZHENG Qilun, PENG Hong, ZHONG Tanwei, QIN Jiangwei. Multi-step-prediction of chaotic time series based on co-evolutionary recurrent neural network [J]. Chinese Physics B, 2008, 17(2): 536-542.
[11] DU H, ZHANG N. Time series prediction using evolving radial basis function networks with encoding scheme [J]. Neurocomputing, 2008, 71: 1388-1400.
[12] VAPNIK V. Statistical learning theory [M]. New York: Wiley, 1998.
[13] 崔万照，朱长纯，保文星. 混沌时间序列的支持向量机预测 [J]. 物理学报，2004, 53(10): 3303-3310.
CUI Wanzhao, ZHU Changchun, BAO Wenxing. Prediction of the chaotic time series using support vector machines [J]. Acta Physica Sinica, 2004, 53(10): 3303-3310. (in Chinese)
[14] CUI Wanzhao, ZHU Changchun, BAO Wenxing, LIU Junhua. Chaotic time series prediction using mean-field theory for support vector machine [J]. Chinese physics, 2005, 14(4): 922-927.
[15] MA J, THEILER J, PERKINS S. Accurate online support vector regression [J]. Neural Computation, 2003, 15(11): 2683-2703.
[16] SUYKENS J A K, VANDEWALLE J. Least squares support vector machine classifiers [J]. Neural Processing Letters, 1999, 9(3): 293-300.
[17] 叶美盈，汪晓东，张浩然. 基于在线最小二乘支持向量机回归的混沌时间序列预测 [J]. 物理学报，2005, 54(6): 2568-2573.
YE Meiying, WANG Xiaodong, ZHANG Haoran. Chaotic time series forecasting using online least squares support vector machine regression [J]. Acta Physica Sinica, 2005, 54(6): 2568-2573. (in Chinese)
[18] 肖支才，王杰，王永生. 基于在线LS-SVM算法的变参数混沌时间序列预测 [J]. 航空计算技术，2010, 40(3): 29-33.
XIAO Zhicai, WANG Jie, WANG Yongsheng. Predict the time series of the parameter-varying chaotic system based on recursive lease square support vector machine (RLS-SVM) [J]. Aeronautical Computing Technique, 2010, 40(3): 29-33. (in Chinese)
