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摘要: 提出一种基于压缩感知(Compressive sensing, CS)和多分辨分析(Multi-resolution analysis, MRA)的多尺度最小二乘支持向量机(Least squares support vector machine, LS-SVM). 首先将多尺度小波函数作为支持向量核, 推导出多尺度最小二乘支持向量机模型, 然后基于压缩感知理论, 利用最小二乘匹配追踪(Least squares orthogonal matching pursuit, LS-OMP)算法对多尺度最小二乘支持向量机的支持向量进行稀疏化, 最后用稀疏的支持向量实现函数回归. 实验结果表明, 本文方法利用不同尺度小波核逼近信号的不同细节, 而且以比较少的支持向量能达到很好的泛化性能, 大大降低了运算成本, 相比普通最小二乘支持向量机, 具有更优越的表现力.Abstract: A multi-scale least squares support vector machine (LS-SVM) based on compressive sensing (CS) and multi-resolution analysis (MRA) is proposed. First, a multi-scale LS-SVM model is conducted, in which a support vector kernel with the multi-resolution wavelet function is employed; then inspired by CS theory, sparse support vectors of multi-scale LS-SVM are constructed via least squares orthogonal matching pursuit (LS-OMP); finally, sparse support vectors are applied to function approximation. Simulation experiments demonstrate that the proposed method can estimate diverse details of signal by means of wavelet kernel with different scales. What is more, it can achieve good generalization performance with fewer support vectors, reducing the operation cost greatly, performing more superiorly compared to ordinary LS-SVM.
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图 2 两尺度径向基小波核LS-SVM的实验结果
Fig. 2 Experimental results of two-scale RBF wavelet kernel LS-SVM
图 3 标准LS-SVM、稀疏LS-SVM、两尺度LS-SVM、稀 疏两尺度LS-SVM 四算法在不同稀疏度下NMSE 比较
Fig. 3 NMSE comparison of standard LS-SVM, sparse LS-SVM, two-scale LS-SVM, sparse two-scale LS-SVM algorithms under di®erent sparse degrees
图 4 标准LS-SVM和两尺度LS-SVM的回归结果比较
Fig. 4 Regression results comparison of standard LS-SVM and two-scale LS-SVM
图 6 90% 稀疏度两尺度LS-SVM 逼近效果
Fig. 6 Approximation result of 90% sparse degree two-scale LS-SVM
图 7 90% 稀疏度标准LS-SVM 逼近效果
Fig. 7 Approximation result of 90% sparse degree standard LS-SV
图 8 标准LS-SVM 和两尺度LS-SVM 在50% 稀疏度下拟合结果比较
Fig. 8 Approximation results comparison of standard LS-SVM and two-scale LS-SVM under 50% sparse degree
图 9 标准LS-SVM 和两尺度LS-SVM 在不同稀疏度下 NMSE 比较
Fig. 9 NMSE comparison of standard LS-SVM and two-scale LS-SVM under di®erent sparse degrees
图 11 标准LS-SVM、稀疏LS-SVM、两尺度LS-SVM、稀 疏两尺度LS-SVM 四算法在不同稀疏度下NMSE 比较
Fig. 11 NMSE comparison of standard LS-SVM, sparse LS-SVM, two-scale LS-SVM, sparse two-scale LS-SVM algorithms under di®erent sparse degrees
图 12 标准LS-SVM 和两尺度LS-SVM 在50% 稀疏度下的拟合结果比较
Fig. 12 Approximation results comparison of standard LS-SVM and two-scale LS-SVM under 50% sparse degree
表 1 不同小波核函数的两尺度LS-SVM NMSE 比较
Table 1 NMSE comparison of two-scale LS-SVM with di®erent wavelet kernel functions
核函数 参数选择 准确率(NMSE) RBF 小波核 $\gamma_1=50, \gamma_2=100, \sigma_1^2=0.5, \sigma_2^2=3.5$ -47.1780 Morlet 小波核 $\gamma_1=80, \gamma_2=150, \sigma_1^2=0.8, \sigma_2^2=4$ -46.4707 Mexican hat 小波核 $\gamma_1=100, \gamma_2=200, \sigma_1^2=0.35, \sigma_2^2=6.25$ -46.6829 
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表 2 不同核函数的两尺度LS-SVM NMSE 比较
Table 2 NMSE comparison of two-scale LS-SVM with di®erent kernel functions
核函数 参数选择 准确率(NMSE) RBF 小波核 $\gamma_1=50, \gamma_2=100, \sigma_1^2=0.5, \sigma_2^2=3.5$ -47.1780 RBF 核 $\gamma_1=100, \gamma_2=200, \sigma_1^2=0.3, \sigma_2^2=3$ -44.7617 Sinc 小波核 $\gamma_1=60, \gamma_2=220, \sigma_1^2=0.5, \sigma_2^2=5$ -44.1170
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表 4 稀疏两尺度LS-SVM 和稀疏标准LS-SVM 在不同稀疏度下NMSE 比较
Table 4 NMSE comparison of sparse two-scale LS-SVM and sparse standard LS-SVM algorithms under di®erent sparse degrees
稀疏度(%) 90 80 70 60 50 40 30 20 10 NMSE 两尺度 -52.8800 -59.7540 -69.7735 -72.7535 -72.9664 -72.9800 -72.9805 -72.9805 -72.9805 标准 -49.9373 -54.9217 -58.7773 -63.1308 -64.8218 -65.3600 -65.4205 -65.4312 -65.4312
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