针对核极限学习机（Extreme Learning Machine with Kernel, KELM）在线应用过程中，核矩阵膨胀，导致算法复杂性不断上升，且难以跟踪系统时变特征的问题，以滑动时间窗为基本建模策略，提出了一种新的KELM在线稀疏学习算法.在前向与后向稀疏化过程中，基于提出的构造与修剪策略，通过在线最小化字典的积累一致性，可选择一组具有预定规模的关键节点.在增样学习与减样学习过程中，基于节点选择结果，利用矩阵的初等变换与分块矩阵求逆公式，模型参数能被在线递推更新.提出的算法被用于混沌时间序列预测与音频放大器状态预测.实验结果表明：相比于4种流形的在线序贯ELM算法，提出的方法在花费相似的测试时间的条件下，能够显著提升预测精度，且具有较好的稳定性.
It is difficult for extreme learning machine with kernel (KELM) to curb kernel matrix expansion and track the system dynamic changes effectively when it is applied to solve online learning tasks. So the sliding time window method is regarded as the basic modeling strategy, and a new online sparsification learning algorithm for KELM is proposed in this paper. In the process of forward sparsification and backward sparsification, a sparse dictionary with predefined size can be selected by online minimization of its cumulative coherence based on our proposed constructive and pruning strategy. In the process of incremental learning and decremental learning, the model parameters can be directly updated by elementary transformation of matrices and block matrix inversion formula based on the selected dictionary. The performance of the proposed algorithm is compared with several well-known online sequential ELM algorithms. The simulation results show that the proposed algorithm can achieve higher prediction accuracy and better stability, meanwhile, it costs the similar testing time.
［1］张弦, 王宏力.局域极端学习机及其在状态在线监测中的应用［J］.上海交通大学学报, 2011, 45(2): 236-240.
ZHANG Xian, WANG Hongli. Local extreme learning machine and its application to condition on-line monitoring［J］. Journal of Shanghai Jiao Tong University, 2011, 45(2): 236-240.
［2］TIAN Z, QIAN C, GU B, et al. Electric vehicle air conditioning system performance prediction based on artificial neural network［J］. Applied Thermal Engineering, 2015, 89: 101-114.
［3］ZHAO S L, CHEN B D, ZHU P P, et al. Fixed budget quantized kernel least-mean-square algorithm［J］. Signal Processing, 2013, 93（9）: 2759-2770.
［4］FAN H J, SONG Q, YANG X L, et al. Kernel online learning algorithm with state feedbacks［J］. Knowledge-Based System, 2015, 89(C): 173-180.
［5］HUANG G B, ZHOU H M, DING X J, et al. Extreme learning machine for regression and multiclass classification［J］. IEEE Transactions on Systems, Man and Cybernetics-Part B: Cybernetics, 2011, 42(2): 513-529.
［6］LIANG N Y, HUANG G B, SUNDARARAJAN N. A fast and accurate online sequential learning algorithm for feedforward networks［J］. IEEE Transactions on Neural Networks, 2006, 17(6): 1411-1423.
［7］HUYNH H T, WON Y. Regularized online sequential learning algorithm for single-hidden layer feedforward neural networks［J］. Pattern Recognition Letters, 2011, 32(14): 1930-1935.
［8］GUO L, HAO J H, LIU M. An incremental extreme learning machine for online sequential learning problems［J］. Neurocomputing, 2014, 128(27): 50-58.
［9］FAN H J, SONG Q. A sparse kernel algorithm for online time series data prediction［J］. Expert Systems with Applications, 2013, 40(6): 2174-2181.
［10］张英堂，马超，李志宁，等. 基于快速留一交叉验证的核极限学习机在线建模［J］. 上海交通大学学报， 2014, 48(5): 641-646.
ZHANG Yingtang, MA Chao, Li Zhining, et al. Online modeling of kernel extreme learning machine based on fast leave-one-out cross-validation［J］. Journal of Shanghai Jiao Tong University, 2014, 48(5): 641-646.
［11］LIN M, ZHANG L J, JIN R, et al. Online kernel learning with nearly constant support vectors［J］. Neurocomputing, 2016, 179(C): 26-36.
［12］ZHOU X R, LIU Z J, ZHU C X. Online regularized and kernelized extreme learning machines with forgetting mechanism ［J］. Mathematical Problems in Engineering, 2014, 2014(1): 1-11.
［13］ZHOU X R, WANG C S. Cholesky factorization based online regularized and kernelized extreme learning machines with forgetting mechanism［J］. Neurocomputing, 2016, 174(1): 1147-1155.
［14］SCARDAPANCE S, COMMINIELLO D, SCARPINITI M, et al. Online sequential extreme learning machine with kernel［J］. IEEE Transactions on Neural Networks and Learning Systems, 2015, 26(9): 2214-2220.
［15］RICHARD C, BERMUDEZ J C M, HONEINE P. Online prediction of time series data with kernels［J］. IEEE Transactions on Signal Processing, 2009, 57(3): 1058-1067.
［16］HUANG G B, ZHU Q Y, SIEW C K. Extreme learning machine: Theory and application［J］. Neurocomputing, 2006, 70(1/3): 489-501.
［17］FAN H J, SONG Q, SHRESTHA S B. Kernel online learning with adaptive kernel width［J］. Neurocomputing, 2016, 175(Part A): 233-242.
