基于EM-EKF算法的RBF-AR模型参数估计

Yanhui Xi; Hui Peng; Hong Mo

doi:10.16383/j.aas.2017.e160216

基于EM-EKF算法的RBF-AR模型参数估计

doi: 10.16383/j.aas.2017.e160216

席燕辉^1,2, ,,
彭辉^3,,
莫红^1,

1.
长沙理工大学电气与信息工程学院长沙 410076
2.
国防科学技术大学信息系统与管理学院长沙 410073
3.
中南大学信息科学与工程学院长沙 410083

基金项目:

the National Natural Science Foundation of China 70921001

the National Natural Science Foundation of China 51425701

Hunan Province Science and Technology Program 2015NK3035

the National Natural Science Foundation of China 71271215

the National Natural Science Foundation of China 61233008

the National Natural Science Foundation of China 51577014

the National Natural Science Foundation of China 51507015

the Natural Science Foundation of Hunan Province 2015JJ3008

the National Natural Science Foundation of China 61773402

the Key Laboratory of Renewable Energy Electric-Technology of Hunan Province 2014ZNDL002

the National Natural Science Foundation of China 61540037

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出版历程
- 收稿日期: 2016-11-14
- 录用日期: 2016-12-20
- 刊出日期: 2017-09-20

Parameter Estimation of RBF-AR Model Based on the EM-EKF Algorithm

Yanhui Xi^{1,2
, ,},
Hui Peng^3
,,
Hong Mo^1
,

1.
School of Electrical and Information Engineering, Changsha University of Science and Technology, Changsha 410076, China
2.
School of Information Systems and Management, National University of Defense Technology, Changsha 410073, China
3.
School of Information Science and Engineering, Central South University, Changsha 410083, China

Funds:

the National Natural Science Foundation of China 70921001

the National Natural Science Foundation of China 51425701

Hunan Province Science and Technology Program 2015NK3035

the National Natural Science Foundation of China 71271215

the National Natural Science Foundation of China 61233008

the National Natural Science Foundation of China 51577014

the National Natural Science Foundation of China 51507015

the Natural Science Foundation of Hunan Province 2015JJ3008

the National Natural Science Foundation of China 61773402

the Key Laboratory of Renewable Energy Electric-Technology of Hunan Province 2014ZNDL002

the National Natural Science Foundation of China 61540037

More Information

Author Bio:
Hui Peng received the Ph.D. degree at the School of Mathematical and Physical Science, the Graduate University for Advanced Studies, Japan, in 2003. He is a Professor at the School of Information Science and Engineering, Central South University. His research interests include predictive control, adaptive control, system modeling and identification, modeling and dynamic asset allocation in financial markets. E-mail: huipeng@mail.csu.edu.cn

Hong Mo received the Ph.D. degree from Graduate University of Chinese Academy of Sciences in 2004. She is a Professor at the School of Electrical and Information Engineering, Changsha University of Sciences and Technology. Her research interests include linguistic dynamic systems, time-varying universe, and type-2 fuzzy sets. E-mail: mohong198@163.com

Corresponding author: Yanhui Xi received the Ph.D. degree in control science and engineering from Central South University, China, in 2013. She is an Associate Professor at Changsha University of Science and Technology. Her research interests include system modeling and identification, nonlinear optimization, and traveling identification. Corresponding author of this paper.E-mail: xiyanhui@126.com

摘要

摘要: 为了利用EKF（extended Kalman filter）算法对RBF-AR（radial basis function network-based autoregressive）模型进行参数估计，重构了RBF-AR模型的网络结构，将其变换成一种新型的广义径向基函数（radial basis function，RBF）神经网络.与典型三层RBF网络结构相比，该广义RBF网络增加了线性输出加权层.为了克服基于EKF神经网络学习算法由于噪声统计特性未知导致滤波发散或者滤波精度不高的问题，利用EM（expectation maximization）算法对RBF-AR模型噪声协方差矩阵进行估计.同时，通过EKF滤波实时估计RBF-AR模型参数（系统状态），EKF平滑过程得到了更加准确的期望估计.仿真结果显示，该方法用在此变形的RBF-AR模型结构中是有效的，特别在信噪比低的情况下，估计效果比SNPOM（structured nonlinear parameter optimization method）方法好，而且还能估计出噪声方差.F检验显示了两方法估计得到的标准偏差有显著性差异.
- EM算法 /
- 扩展卡尔曼滤波 /
- 平滑算法 /
- 径向基函数网络 /
- RBF-AR模型
Abstract: RBF-AR (radial basis function network-based autoregressive) model is reconstructed as a new type of general radial basis function (RBF) neural network, which has additional linear output weight layer in comparison with the traditional three-layer RBF network. The extended Kalman filter (EKF) algorithm for RBF training has low filtering accuracy and divergence because of unknown prior knowledge, such as noise covariance and initial states. To overcome the drawback, the expectation maximization (EM) algorithm is used to estimate the covariance matrices of noises and the initial states. The proposed method, called the EM-EKF (expectation-maximization extended Kalman filter) algorithm, which combines the expectation maximization, extended Kalman filtering and smoothing process, is developed to estimate the parameters of the RBF-AR model, the initial conditions and the noise variances simultaneously. It is shown by the simulation tests that the EM-EKF method for the reconstructed RBF-AR network provides better results than structured nonlinear parameter optimization method (SNPOM) and the EKF, especially in low SNR (signal noise ratio). Moreover, the EM-EKF method can accurately estimate the noise variance. F test indicates there is significant difference between results obtained by the SNPOM and the EM-EKF.
- Expectation maximization (EM) algorithm /
- extended Kalman filtering (EKF) and smoothing /
- #160; /
- radial basis function (RBF) neural network /
- RBF-AR model

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Fig. 1 The schematic of the RBF-AR model.

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Fig. 2 The original Mackey-Glass time series.

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Fig. 3 The convergence process of the Log-likelihood, the measurement and process noise variance.

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Fig. 4 The Mackey-Glass chaotic time series corrupted by additive white noise.

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Fig. 5 The convergence process of the Log-likelihood, the measurement and process noise variance.

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Fig. 6 The convergence process of the Log-likelihood, the measurement and process noise variance.

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Table Ⅰ COMPARISON RESULTS FOR MACKEY-GLASS TIME SERIES

Method	MSE (Training)	MSE (Testing)	$R$
SNPOM	1.0800E-7	1.2600E-7	unknown
EKF	1.9560E-7	2.1786E-7	0.002 (given)
EKF	1.2559E-7	1.9793E-7	0.0002 (given)
EM-EKF	7.1765E-8	1.2008E-7	1.8146E-7

下载: 导出CSV

Table Ⅱ COMPARISON RESULTS FOR MACKEY-GLASS TIME SERIES CORRUPTES BY ADDITIVE WHITE NOISE

Noise variance	Method	MSE (Training)	MSE (Testing)	R
0.25	SNPOM	0.26606	0.28120	unknown
	EKF	0.26577	0.28082	0.2 (given)
	EKF	0.27343	0.28825	0.1 (given)
	EM-EKF	0.25750	0.27825	0.25950
1	SNPOM	0.97452	1.1644	unknown
	EKF	0.97215	1.1512	0.8 (given)
	EKF	0.98262	1.1785	0.6 (given)
	EM-EKF	0.96590	1.13907	1.0589

下载: 导出CSV

Table Ⅲ THE RESULTS OF STATISTICAL F TEST AT LEVEL 0.05

Case	$F_{\alpha=0.05}$	$F$	Results
Case 1 (Training)	1.16	1.4642	Reject; Difference
Case 1 (Testing)	1.16	1.0105	No reject; No difference
Case 2 (Training)	1.16	14.1212	Reject; Significant difference
Case 2 (Testing)	1.16	9.2984	Reject; Significant difference
Case 3 (Training)	1.16	37.8297	Reject; Significant difference
Case 3 (Testing)	1.16	15.5751	Reject; Significant difference

下载: 导出CSV

Table Ⅳ THE COMPUTATION TIME OF DIFFERENT METHODS (S)

Method	Time
SNPOM	10.690
EKF	0.40572
EM-EKF	30.766

下载: 导出CSV

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