传统主元分析(Principal component analysis, PCA)方法因忽视量纲对系统的影响, 从而使选取的主元难以具有代表性; 而在进行量纲标准化后, 又因得到的特征值常常是近似相等的而无法进行有效的主元提取. 针对这一主要问题, 本文通过引入相对化变换(Relative transform, RT)、相对主元(Relative principal components, RPCs) 和分布"均匀"等概念, 建立起一种相对主元分析(Relative principal component analysis, RPCA)的新方法. 该方法首先对系统各分量进行量纲标准化; 其次再根据系统的先验信息分析和确定各分量的重要程度; 然后在系统能量守恒的准则下, 赋以系统各分量相应的权值; 最后利用已建立起的相对主元模型, 对系统实施RPCA. 同时运用数值例子, 开展了RPCA在数据压缩和系统故障诊断中的应用研究. 理论分析和仿真实验均表明, 采用RPCA方法选取出的主元更具代表性和显著几何意义, 加之选取主元的灵活性, 将使新方法具有更广泛的应用前景.
In traditional principal component analysis (PCA), because of the neglect of the influence of dimensions on the system, the selected principal components (PCs) often fail to be representative. For this problem, an improved algorithm, called relative principal component analysis (RPCA), is proposed by introducing some new concepts, such as relative transform (RT), relative principal components (RPCs), "rotundity" scatter and so on. Firstly, the algorithm standardizes every variable's dimension in this system. Secondly, according to priori information, it analyzes and determines the different important levels of different variables. And then it allocates weights for each variable under the criterion of conservation of system energy. Finally, the algorithm utilizes the relative-principal-component model established to analyze system. Meanwhile, its functions are illustrated by some numerical examples such as data compression and system fault diagnosis. Both theoretic analysis and computer simulation have shown that these selected RPCs are representative and their significance of geometry is notable. So we can say that the new method may have extensive applications, together with the flexibility of PCs selection.