基于改进稀疏KELM的在线非平稳动态系统状态预测方法

本文针对基于核的增量超限学习机(kernel based incremental extreme learning machine，KB-IELM)对非平稳动态系统的时变状态跟踪能力不足的问题，提出一种新型的状态预测方法。通过融合遗忘因子和自适应时变正则化因子构建新的目标函数。通过最小化字典的快速留一交叉验证(fast leave-one-out cross-validation, FLOO-CV)误差，选择具有预定规模的关键节点以构成字典。通过融合遗忘因子，为字典中各关键节点按时间顺序分配不同权重。基于FLOO-CV原则，使用天牛须搜索(beetle antennae search，BAS)算法为不同的非线性区域赋予不同的正则化参数。通过矩阵初等变换和分块求逆，实现核权重向量的在线递推更新。将模型应用于非平稳Mackey-Glass混沌时间序列预测和某型飞机发动机的状态预测。所提算法相比于最新的非平稳在线序列核超限学习机(non-stationary online sequential kernel extreme learning machine，NOS-KELM)和融合自适应正则化因子的在线稀疏核超限学习机(online sparse kernel extreme learning machine with adaptive regulation factor, OSKELM-ARF)两种方法，其训练精度分别提升了66.67%、50.72%、预测精度提升了67.02%、56.34%，最大预测误差减少了67.27%、51.09%，平均相对误差率分别减少了67.18%、59.62%。实验证明所提算法有效提升了在线预测的精度。

State prediction method of online non-stationary dynamic system based on improved sparse KELM

A new state prediction method for the problem is proposed that the kernel based incremental extreme learning machine (KB-IELM) has insufficient ability to track the time-varying state of non-stationary dynamic systems. A new objective function is constructed by integrating the forgetting factor and the adaptive time-varing regulation factor. The key nodes with the predetermined scale are selected to form the dictionary by minimizing the fast leave-one-out cross-validation (FLOO-CV) error of the dictionary. Different weights are assigned to the key nodes in the dictionary according to the sequential order via mixing the forgetting factor. Based on the FLOO-CV principle, the different regulation parameters are assigned to different nonlinear regions by utilizing the algorithm of the beetle antennae search (BAS). The online recursive update of weight vectors is realized by matrix elementary transformation and block inversion. The model is applied to the non-stationary Mackey-Glass chaotic time series prediction and the state prediction of an aircraft engine. Compared with the latest two methods of the non-stationary online sequential kernel extreme learning machine (NOS-KELM) and the online sparse kernel extreme learning machine with adaptive regulation factor (OSKELM-ARF), the training accuracy is improved by 66.67% and 50.72 %. The prediction accuracy is improved by 67.02% and 56.34%, and the maximum prediction error is reduced by 67.27% and 51.09%. Then the average relative error rate is reduced by 67.18% and 59.62% respectively. It is proved that the proposed algorithm effectively improve the accuracy of online prediction.