物理学报2018，Vol.67Issue(1)：276-284,9.DOI:10.7498/aps.67.20171803
基于分数阶最大相关熵算法的混沌时间序列预测
Prediction of chaotic time series based on the fractional-order maximum correntropy criterion algorithm
摘要
Abstract
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.关键词
混沌时间序列预测/自适应滤波器/分数阶微分/最大相关熵准则Key words
prediction of chaotic time series/adaptive filter/fractional-order derivative/maximum correntropy criterion引用本文复制引用
王世元,史春芬,钱国兵,王万里..基于分数阶最大相关熵算法的混沌时间序列预测[J].物理学报,2018,67(1):276-284,9.基金项目
国家自然科学基金(批准号:61671389)、中国博士后科学基金(批准号:2017M610583)、重庆市博士后科研项目特别资助(批准号:Xm2017107)和中央高校基本科研业务费(批准号:XDJK2017D177, XDJK2017D178)资助的课题.Project supported by the National Natural Science Foundation of China(Grant No.61671389), China Postdoctoral Science Foundation Funded Project(Grant No.2017M610583), Chongqing Postdoctoral Science Foundation Special Funded Project, China(Grant No.Xm2017107), and the Fundamental Research Funds for the Central Universities, China(Grant Nos.XDJK2017D177, XDJK2017D178). （批准号:61671389）
