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摘要:
针对核超限学习机（KELM）在线状态预测过程中，核矩阵阶数不断增长且难以跟踪时变动态特征的问题，提出了一个具有遗忘因子的稀疏KELM在线状态预测方法。通过引入遗忘因子构建新的目标函数，使稀疏字典中各元素依据时间远近具有不同权重，保证了模型对动态变化的有效跟踪；通过最小化字典的快速留一交叉验证（FLOO-CV）误差，选择具有预定规模的关键节点构成字典；基于当前字典，通过矩阵初等变换和分块求逆，实现相关参数的递推更新。某型飞机发动机的状态预测结果表明，与目前已有的3种在线序贯KELM相比，所提方法在6个监测项目上的平均训练时间分别缩短了7.5%、62.0%和81.9%，平均预测精度分别提升了44.0%、19.9%和50.9%。
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关键词:
- 状态预测 /
- 在线序贯学习 /
- 快速留一交叉验证(FLOO-CV) /
- 超限学习机 /
- 核方法
Abstract:In order to curb kernel matrix expansion and track the time-varying dynamic characteristics when kernel extreme learning machine (KELM) is applied to online condition prediction, a sparse KELM online prediction algorithm with forgetting factor is proposed. By introducing forgetting factor, a new objective function is constructed, which makes every element in sparse dictionary has different weights related to timestamp and ensures the effective tracking of the dynamic changes. By minimizing the fast leave-one-out cross-validation (FLOO-CV) error, key nodes with predetermined size are selected to form a dictionary. At the same time, the online recursive updating of model parameters is realized based on the elementary transformation of matrix and the inverse formula of block matrix. The proposed algorithm is compared with the recently proposed three online sequential KELM algorithms. The experimental results of aero-engine condition prediction show that the average training time of the proposed algorithm on six monitoring items is reduced by 7.5%, 62.0% and 81.9% respectively, and the average prediction accuracy is improved by 44.0%, 19.9% and 50.9% respectively.
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图 2 对Mackey-Glass混沌时间序列学习的训练样本数
Figure 2. Number of learned samples for chaotic Mackey-Glass time series
图 5 某发动机排气温度与滑油压力变化曲线
Figure 5. Curves of exhaust gas temperature and oil pressure of an aero-engine
表 1 实验1参数设置
Table 1. Parameter setup in Experiment 1
方法 正则化参数c 核参数σ 其他参数 ReOS-ELM 2×103 L=50 KB-IELM 10 10 FF-KB-IELM 10 10 γ=0.999 FOKELM 2×104 10 z=50 FF-FOKELM 2×104 10 z=50, γ=0.999 ALD-KOS-ELM 1×103 10 b=0.000 01 NOS-KELM 2×104 10 m=50, δ=0.01, η=0.8 FF-OSKELM 2×104 10 m=50, γ=0.999 注：L为ReOS-ELM中隐层神经元个数；z为FOKELM中时间窗长度；ALD-KOS-ELM中的b表示ALD准则需要设置的阈值；NOS-KELM中的δ和η均是梯度下降法中运用动态学习率时需要设置的常数，详细请参见相关文献[17]。 
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表 2 实验1预测结果
Table 2. Prediction results of Experiment 1
方法 训练 测试 时间/s RMSE 时间/s RMSE MAPE MRPE ReOS-ELM 7.940 5 0.019 5 0.00 11 0.018 6 0.050 0 0.012 1 KB-IELM 8.665 9 0.017 5 0.010 7 0.015 3 0.042 0 0.010 8 FF-KB-IELM 8.632 2 0.015 3 0.010 1 0.013 2 0.037 5 0.009 4 FOKELM 0.092 2 0.026 8 4.446 5×10-4 0.012 2 0.031 7 0.008 3 FF-FOKELM 0.088 6 0.025 9 3.314 5×10-4 0.011 0 0.028 6 0.007 4 ALD-KOS-ELM 0.146 4 0.011 2 3.716 0×10-4 0.010 9 0.025 0 0.008 1 NOS-KELM 0.565 0 0.010 4 2.405 9×10-4 0.009 3 0.026 3 0.006 3 FF-OSKELM 0.045 4 0.003 4 3.512 2×10-4 0.003 3 0.011 0 0.002 3 注：ELM输入层初始权重的随机性导致ReOS-ELM的实验结果具有很大的随机性，表中只列出了ReOS-ELM一次实验的结果。
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表 3 实验2参数设置
Table 3. Parameter setup in Experiment 2
监测项目 正则化
参数c核参数
σm 阈值δ 发动机扭矩 2×104 5×104 30 2×10-5 发动机转速 2×104 1×109 30 2×10-8 排气温度 2×104 1×107 30 2×10-9 滑油温度 2×104 2×105 30 2×10-9 滑油压力 2×104 2×104 30 2×10-9 燃油瞬时流量 2×103 2×105 30 2×10-6 注：阈值δ为ALD-KOS-ELM的参数，m为其他3种方法的参数，表示字典规模或时间窗长度。
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表 4 实验2预测结果
Table 4. Prediction results of Experiment 2
监测项目 方法 训练 测试 时间/s RMSE RMSE MAPE MRPE 发动机扭矩 FOKELM 0.006 8 0.846 3 1.453 8 3.342 7 0.166 5 ALD-KOS-ELM 0.006 9 0.784 6 1.020 5 2.581 4 0.110 8 NOS-KELM 0.045 0 0.710 8 1.457 0 3.421 6 0.167 5 FF-OSKELM 0.005 7 0.653 0 1.304 1 3.062 2 0.150 7 发动机转速 FOKELM 0.003 1 333.634 4 241.756 6 627.155 1 0.007 2 ALD-KOS-ELM 0.011 4 91.332 5 167.500 5 451.946 1 0.005 2 NOS-KELM 0.053 4 156.557 5 276.815 6 684.522 2 0.007 9 FF-OSKELM 0.004 4 82.935 1 132.873 9 385.796 6 0.004 0 排气温度 FOKELM 0.005 6 4.859 3 7.474 8 19.777 0 0.012 5 ALD-KOS-ELM 0.031 0 4.192 1 5.402 7 13.672 6 0.009 3 NOS-KELM 0.038 8 5.386 8 7.648 3 20.021 0 0.012 8 FF-OSKELM 0.005 2 3.286 17 3.855 3 9.694 2 0.006 7 滑油温度 FOKELM 0.004 3 0.183 7 0.321 1 0.511 7 0.008 1 ALD-KOS-ELM 0.044 4 0.044 7 0.160 4 0.250 0 0.004 2 NOS-KELM 0.062 6 0.218 9 0.701 0 0.985 9 0.018 5 FF-OSKELM 0.004 7 0.031 7 0.108 5 0.180 9 0.002 8 滑油压力 FOKELM 0.004 6 0.111 5 0.105 4 0.140 0 0.030 9 ALD-KOS-ELM 0.035 0 0.061 4 0.040 7 0.056 3 0.012 5 NOS-KELM 0.039 1 0.077 5 0.050 4 0.075 7 0.015 2 FF-OSKELM 0.025 7 0.036 7 0.028 0 0.045 7 0.008 4 燃油瞬时流量 FOKELM 0.033 5 3.353 4 7.456 0 18.800 3 0.031 7 ALD-KOS-ELM 0.012 2 2.419 3 6.589 6 23.668 6 0.024 4 NOS-KELM 0.057 4 2.619 0 7.698 1 18.667 9 0.033 6 FF-OSKELM 0.007 8 2.163 2 6.508 9 23.396 4 0.024 1 注：因为方法的测试时间均过于短暂，已能满足绝大部分在线应用需求，此表不再将之作为对比项。
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