PCA与SVD信号处理效果相似性与机理分析

Similarity of signal processing effect between PCA and SVD and its mechanism analysis

将主成分分析(Principal Component Analysis,PCA)用于信号处理,并与奇异值分解(Singular Value Decomposition,SVD)方法比较。分析总结PCA及SVD信号处理原理,提出基于PCA的特征值差分谱理论用于信号消噪。结果表明,PCA与SVD的处理效果较相似,相似性原因为原始矩阵右奇异向量即为协方差矩阵特征向量。SVD较PCA的重构误差小,因SVD无需计算协方差矩阵,可避免舍入误差产生。

The principal component analysis （PCA） was applied to signal processing and its effect was compared with that of the singular value decomposition （SVD）. The signal processing principles of PCA and SVD were analyzed and summarized, and the theory of eigenvalues difference spectrum based on PCA for signal denoising was introduced. It is pointed out that PCA has a very similar signal processing effect to that of SVD when applied to signal de-noising. The reason for this similarity was analyzed theoretically, and it is found that this is because the right singular vectors of the original matrix are just the eigenvectors of its covariance matrix, and leads to the similarity between PCA and SVD in signal processing. It is also pointed out that the reconstruction error of SVD is smaller than that of PCA, and the reason is that SVD does not need to compute the covariance matrix, so the rounding errors are avoided.