第36卷第12期 北京科技大学学报 Vol.36 No.12 2014年12月 Journal of University of Science and Technology Beijing Dec.2014 多分辨率小波极限学习机 全丽萍，李晓理2，王巧智》 1)北京科技大学自动化学院，北京1000832)上海交通大学机械系统与振动国家重点实验室，上海200240 3)北京科技大学机械工程学院，北京100083 ☒通信作者，E-mail:chuanziyiwei@163.com 摘要针对一类具有空间不均匀性的辨识和回归问题，提出了基于小波分析的极限学习机方法.从多分辨率分析的思想出 发，构造一簇紧支撑正交小波作为隐层激活函数，并利用改进的误差最小化极限学习机训练输出层权重，避免了新加入高分 辨率子网络后的重新训练.同时，由一维多分辨分析的张量积构造了二维多分辨小波极限学习机.进而通过脊波变换将小波 学习机扩展到高维空间，对脊波函数的伸缩、方向和位置参数进行优化计算.对具有奇异性的函数仿真结果证明，与标准极限 学习机相比，小波极限学习机由于其聚微性能在极短的训练时间内更好地逼近目标.一些实际基准回归问题上的测试验证了 脊波极限学习机在其中大部分问题上达到更高的训练和泛化精度. 关键词学习算法：极限学习机：小波分析：多分辨分析：正交 分类号TP183 Multiresolution wavelet extreme learning machine QUAN Li-ping”,LI Xiao-i.2,WANG Qiao--hi》 1)School of Automation and Electrical Engineering,University of Science and Technology Beijing,Beijing 100083,China 2)State Key Laboratory of Mechanical System and Vibration,Shanghai Jiao Tong University,Shanghai 200240,China 3)School of Mechanical and Engineering,University of Science and Technology Beijing,Beijing 100083,China Corresponding author,E-mail:chuanziyiwei@163.com ABSTRACT An extrme learning machine (ELM)algorithm based on wavelet transform was designed for a class of indentification and regression problem with inhomogeneity in a space.From the standpoint of multiresolution analysis,a set of compactly supported or- thogonal wavelets was constructed as the hidden layer activation function,and the output layer weight of the network was trained by an error minimized extreme learning machine.This method avoided retraining the output layer parameter as adding a subnetwork with high- er resolution.The wavelet ELM was then extended into a two-dimensional space using the tensor product of a scaling function.To hur- dle high-dimensionality issues,ridgelet transform based on ELM was obtained,whose scaling,direction,and position parameters were determined by optimization methods.Simulation results on functions with singularity confirm that the wavelet ELM can approch the tar- get better.When being tested on some real benchmark problems,the ridgelet ELM demonstrates better training and testing accuracy on most cases. KEY WORDS learning algorithms;extreme learning machine:wavelet analysis:multiresolution analysis:orthogonal 极限学习机近年来一直是神经网络领域非常活 领域使得逼近回归理论产生了极大的飞跃.极限学 跃的研究方向，具有学习速率高、能达到全局最优、 习机的隐层激励函数通常采用任意分段连续的非线 结构简单、泛化性能好等多重优点.将其引入预测 性函数，如Sigmoid、Sin和Hardlim一类支撑集为无 收稿日期：2014-0909 基金项目：新世纪优秀人才支持计划资助项目(NCET-11O578):中央高校基本科研业务费专项资金资助项目(FRF-TP-12OO5B):高等学校 博士学科点专项科研基金资助项目(20130006110008)：机械系统与振动国家重点实验室开放课题(MSV-201409) DOI:10.13374/j.issn1001-053x.2014.12.019;http://journals.ustb.edu.cn第 36 卷 第 12 期 2014 年 12 月 北京科技大学学报 Journal of University of Science and Technology Beijing Vol． 36 No． 12 Dec． 2014 多分辨率小波极限学习机 全丽萍1) ，李晓理1，2) ，王巧智3) 1) 北京科技大学自动化学院，北京 100083 2) 上海交通大学机械系统与振动国家重点实验室，上海 200240 3) 北京科技大学机械工程学院，北京 100083  通信作者，E-mail: chuanziyiwei@ 163． com 摘 要 针对一类具有空间不均匀性的辨识和回归问题，提出了基于小波分析的极限学习机方法． 从多分辨率分析的思想出 发，构造一簇紧支撑正交小波作为隐层激活函数，并利用改进的误差最小化极限学习机训练输出层权重，避免了新加入高分 辨率子网络后的重新训练． 同时，由一维多分辨分析的张量积构造了二维多分辨小波极限学习机． 进而通过脊波变换将小波 学习机扩展到高维空间，对脊波函数的伸缩、方向和位置参数进行优化计算． 对具有奇异性的函数仿真结果证明，与标准极限 学习机相比，小波极限学习机由于其聚微性能在极短的训练时间内更好地逼近目标． 一些实际基准回归问题上的测试验证了 脊波极限学习机在其中大部分问题上达到更高的训练和泛化精度． 关键词 学习算法; 极限学习机; 小波分析; 多分辨分析; 正交 分类号 TP 183 Multiresolution wavelet extreme learning machine QUAN Li-ping1)  ，LI Xiao-li1，2) ，WANG Qiao-zhi3) 1) School of Automation and Electrical Engineering，University of Science and Technology Beijing，Beijing 100083，China 2) State Key Laboratory of Mechanical System and Vibration，Shanghai Jiao Tong University，Shanghai 200240，China 3) School of Mechanical and Engineering，University of Science and Technology Beijing，Beijing 100083，China  Corresponding author，E-mail: chuanziyiwei@ 163． com ABSTＲACT An extrme learning machine ( ELM) algorithm based on wavelet transform was designed for a class of indentification and regression problem with inhomogeneity in a space． From the standpoint of multiresolution analysis，a set of compactly supported or￾thogonal wavelets was constructed as the hidden layer activation function，and the output layer weight of the network was trained by an error minimized extreme learning machine． This method avoided retraining the output layer parameter as adding a subnetwork with high￾er resolution． The wavelet ELM was then extended into a two-dimensional space using the tensor product of a scaling function． To hur￾dle high-dimensionality issues，ridgelet transform based on ELM was obtained，whose scaling，direction，and position parameters were determined by optimization methods． Simulation results on functions with singularity confirm that the wavelet ELM can approch the tar￾get better． When being tested on some real benchmark problems，the ridgelet ELM demonstrates better training and testing accuracy on most cases． KEY WOＲDS learning algorithms; extreme learning machine; wavelet analysis; multiresolution analysis; orthogonal 收稿日期: 2014--09--09 基金项目: 新世纪优秀人才支持计划资助项目( NCET--11--0578) ; 中央高校基本科研业务费专项资金资助项目( FＲF--TP--12--005B) ; 高等学校 博士学科点专项科研基金资助项目( 20130006110008) ; 机械系统与振动国家重点实验室开放课题( MSV--2014--09) DOI: 10． 13374 /j． issn1001--053x． 2014． 12． 019; http: / /journals． ustb． edu． cn 极限学习机近年来一直是神经网络领域非常活 跃的研究方向，具有学习速率高、能达到全局最优、 结构简单、泛化性能好等多重优点． 将其引入预测 领域使得逼近回归理论产生了极大的飞跃． 极限学 习机的隐层激励函数通常采用任意分段连续的非线 性函数，如 Sigmoid、Sin 和 Hardlim 一类支撑集为无