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基于灵敏度分析的模块化回声状态网络修剪算法

王磊 乔俊飞 杨翠丽 朱心新

王磊, 乔俊飞, 杨翠丽, 朱心新. 基于灵敏度分析的模块化回声状态网络修剪算法. 自动化学报, 2019, 45(6): 1136-1145. doi: 10.16383/j.aas.c180288
引用本文: 王磊, 乔俊飞, 杨翠丽, 朱心新. 基于灵敏度分析的模块化回声状态网络修剪算法. 自动化学报, 2019, 45(6): 1136-1145. doi: 10.16383/j.aas.c180288
WANG Lei, QIAO Jun-Fei, YANG Cui-Li, ZHU Xin-Xin. Pruning Algorithm for Modular Echo State Network Based on Sensitivity Analysis. ACTA AUTOMATICA SINICA, 2019, 45(6): 1136-1145. doi: 10.16383/j.aas.c180288
Citation: WANG Lei, QIAO Jun-Fei, YANG Cui-Li, ZHU Xin-Xin. Pruning Algorithm for Modular Echo State Network Based on Sensitivity Analysis. ACTA AUTOMATICA SINICA, 2019, 45(6): 1136-1145. doi: 10.16383/j.aas.c180288

基于灵敏度分析的模块化回声状态网络修剪算法

doi: 10.16383/j.aas.c180288
基金项目: 

北京市博士后工作经费资助项目 2017ZZ-028

北京市教委项目 KM201710005025

国家自然科学基金 61603012

国家自然科学基金 61533002

详细信息
    作者简介:

    乔俊飞  北京工业大学信息学部教授.主要研究方向为污水处理过程智能控制, 神经网络结构设计与优化.E-mail:junfeq@bjut.edu.cn

    杨翠丽  北京工业大学信息学部讲师.主要研究方向为神经网络和智能优化算法.E-mail:clyang5@bjut.edu.cn

    朱心新  北京工业大学硕士研究生.主要研究方向为神经网络结构设计与优化.E-mail:1205580412@emails.bjut.edu.cn

    通讯作者:

    王磊  北京工业大学信息学部博士研究生.主要研究方向为神经网络结构设计与优化.本文通信作者.E-mail:jade_wanglei@163.com

Pruning Algorithm for Modular Echo State Network Based on Sensitivity Analysis

Funds: 

Beijing Postdoctoral Research Foundation 2017ZZ-028

Beijing Municipal Education Commission Foundation KM201710005025

Supported by National Natural Science Foundation of China 61603012

Supported by National Natural Science Foundation of China 61533002

More Information
    Author Bio:

      Professor at the Faculty of Information Technology, Beijing University of Technology. His research interest covers intelligent control of wastewater treatment process, structure design and optimization of neural networks

     i Lecturer at the Faculty of Information Technology, Beijing University of Technology. Her research interest covers neural network and intelligent optimization algorithm

      Master student at the Faculty of Information Technology, Beijing University of Technology. Her research interest covers structure design and optimization of neural networks

    Corresponding author: WANG Lei   Ph. D. candidate at the Faculty of Information Technology, Beijing University of Technology. His research interest covers structure design and optimization of neural networks. Corresponding author of this paper
  • 摘要: 针对回声状态网络(Echo state network,ESN)的结构设计问题,提出基于灵敏度分析的模块化回声状态网络修剪算法(Pruning algorithm for modular echo state network,PMESN).该网络由相互独立的子储备池模块构成.首先利用矩阵的奇异值分解(Singular value decomposition,SVD)构造子储备池模块的权值矩阵,并利用分块对角阵原理生成储备池.然后利用子储备池模块输出和相应的输出层权值向量,定义学习残差对于子储备池模块的灵敏度以及网络规模适应度.利用灵敏度大小判断子储备池模块的贡献度,并根据网络规模适应度确定子储备池模块的个数,删除灵敏度低的子模块.在网络的修剪过程中,不需要缩放权值就可以保证网络的回声状态特性.实验结果说明,所提出的算法有效解决了ESN的网络结构设计问题,基本能够确定与样本数据相匹配的网络规模,具有较好的泛化能力和鲁棒性.
    1)  本文责任编委 鲁仁全
  • 图  1  无输出反馈的基本ESN结构

    Fig.  1  The basic architecture of the OESN without feedback

    图  2  无输出反馈的具有多个子储备池的模块化ESN结构

    Fig.  2  The architecture of MESN without feedback

    图  3  基于PMESN和OESN的含噪声的Lorenz时间序列预测结果

    Fig.  3  Prediction results based on PMESN and OESN for Lorenz time series with noise

    图  4  基于PMESN和OESN的含噪声的Lorenz时间序列的模型设计成功率

    Fig.  4  Successful design ratio based on PMESN and OESN for Lorenz time series with noise

    图  5  基于PMESN和OESN的含噪声的非线性系统辨识预测结果

    Fig.  5  Prediction results based on PMESN and OESN for nonlinear system identification with noise

    图  6  基于PMESN和OESN的含噪声的非线性系统辨识的模型设计成功率

    Fig.  6  Successful design ratio based on PMESN and OESN for nonlinear system identification with noise

    图  7  基于PMESN和OESN的出水NH4-N浓度预测结果

    Fig.  7  Prediction results based on PMESN and OESN for effluent NH4-N prediction

    图  8  基于PMESN和OESN的出水NH4-N浓度预测的模型设计成功率

    Fig.  8  Successful design ratio based on PMESN and OESN for effluent NH4-N prediction

    表  1  子储备池规模对PMESN性能的影响

    Table  1  Influence of sub-reservoir size on PMESN

    子储备池
    规模
    训练时间(s) 测试NRMSE
    平均值 标准差
    3 25.12 8.92×10-3 8.87×10-4
    5 27.23 4.56×10-3 4.82×10-4
    10 24.35 5.13×10-3 5.48×10-4
    15 23.68 6.35×10-3 6.69×10-4
    20 22.19 6.98×10-3 6.95×10-4
    下载: 导出CSV

    表  2  储备池初始规模对PMESN性能的影响

    Table  2  Influence of initial reservoir size on PMESN

    储备池初
    始规模
    训练时间(s) 测试NRMSE
    平均值 标准差
    100 22.15 8.92 × 10-3 8.78 × 10-4
    200 24.54 4.89 × 10-3 4.82 × 10-4
    300 26.36 4.72 × 10-3 4.89 × 10-4
    400 27.26 4.35 × 10-3 4.48 × 10-4
    500 28.39 3.99 × 10-3 4.06 × 10-4
    600 32.68 4.82 × 10-3 5.38 × 10-4
    下载: 导出CSV

    表  3  网络规模适应度阈值对PMESN性能的影响

    Table  3  Influence of fitness threshold of network size on PMESN

    网络规模适
    储备池初
    始规模
    测试NRMSE 储备池最
    终规模
    平均值 标准差
    1 500 4.12 × 10-3 4.23 × 10-4 432
    0.9 500 4.23 × 10-3 4.19 × 10-4 413
    0.8 500 4.08 × 10-3 4.29 × 10-4 395
    0.7 500 4.21 × 10-3 4.36 × 10-4 382
    0.6 500 4.19 × 10-3 4.06 × 10-4 365
    0.5 500 4.15 × 10-3 4.13 × 10-4 329
    0.4 500 4.02 × 10-3 4.09 × 10-4 298
    0.3 500 9.58 × 10-3 9.37 × 10-4 275
    0.2 500 5.58 × 10-3 5.62 × 10-4 246
    0.1 500 8.69 × 10-3 8.36 × 10-4 213
    下载: 导出CSV

    表  4  基于不同模型的含噪声的Lorenz时间序列预测的参数和仿真结果对比

    Table  4  Comparison of some parameters and simulation results of different models for Lorenz time series with noise

    网络模型 储备池
    初始规模
    储备池
    最终规模
    谱半径 稀疏度 网络规模适
    应度阈值
    训练时间(s) NRMSE
    平均值 标准差
    PMESN 500 285 0.8500 0.0100 0.4 28.85 4.01 × 10-3 3.64 × 10-4
    OESN[1] 500 500 0.8500 0.0500 - 25.32 8.38 × 10-3 6.38 × 10-4
    SCR[8] 500 500 0.8000 0.0020 - 22.15 8.28 × 10-3 8.16 × 10-4
    DESN[9] 500 500 0.8000 0.0238 - 27.35 9.12 × 10-3 9.43 × 10-4
    GESN[6] 50 400 0.9236 0.0200 - 81.35 3.96× 10-3 4.15 × 10-4
    SIPA-SCR[11] 500 463 0.8500 0.0020 - 41.39 5.65 × 10-3 5.68 × 10-4
    AEESN[13] 500 385 0.8500 0.0500 - 31.39 5.31 × 10-3 5.06 × 10-4
    “–”表示原文献中无此参数
    下载: 导出CSV

    表  5  基于不同模型的含噪声的非线性系统辨识的参数和仿真结果对比

    Table  5  Comparison of some parameters and simulation results of different models for nonlinear system identification with noise

    网络模型 储备池
    初始规模
    储备池
    最终规模
    谱半径 稀疏度 网络规模适
    应度阈值
    训练时间(s) NRMSE
    平均值 标准差
    PMESN 500 245 0.8500 0.0100 0.5 39.88 0.0359 0.0020
    OESN[1] 500 500 0.8500 0.0500 - 34.46 0.0723 0.0023
    SCR[8] 500 500 0.8000 0.0020 - 29.86 0.0692 0.0021
    DESN[9] 500 500 0.8000 0.0238 - 36.85 0.0812 0.0022
    GESN[6] 50 400 0.9236 0.0200 - 83.69 0.0436 0.0019
    SIPA-SCR[11] 500 445 0.8500 0.0020 - 45.66 0.0582 0.0024
    AEESN[13] 500 376 0.8500 0.0500 - 37.79 0.0519 0.0018
    “–”表示原文献中无此参数
    下载: 导出CSV

    表  6  基于不同模型的出水NH4-N浓度预测的参数和仿真结果对比

    Table  6  Comparison of some parameters and simulation results of different models for effluent NH4-N prediction

    网络模型 储备池
    初始规模
    储备池
    最终规模
    谱半径 稀疏度 网络规模适
    应度阈值
    训练时间(s) NRMSE
    平均值 标准差
    PMESN 500 255 0.8500 0.0100 0.4 38.83 0.2039 0.0198
    OESN[1] 500 500 0.8500 0.0500 - 32.19 0.3328 0.0232
    SCR[8] 500 500 0.8000 0.0020 - 29.86 0.2938 0.0286
    DESN[9] 500 500 0.8000 0.0238 - 35.92 0.3426 0.0312
    GESN[6] 50 400 0.9236 0.0200 - 91.08 0.2236 0.0022
    SIPA-SCR[11] 500 458 0.8500 0.0020 - 44.26 0.2935 0.0301
    AEESN[13] 500 365 0.8500 0.0500 - 39.33 0.2899 0.0268
    “–”表示原文献中无此参数
    下载: 导出CSV
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  • 收稿日期:  2018-05-08
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