PCA在克服变量多重相关性中的局限作用

Limitations of PCA in solving the multiple correlations between variables

针对主成分分析(PCA)在非线性特征的观测变量中应用的局限作用,对PCA进行了理论研究。基于欧氏空间和统计方法,讨论了PCA的数学本质,以变量高度多重相关为例,分析了非线性系统结构,提出并证明了PCA在克服变量多重相关性和多指标系统评估中存在局限性的必然原因。针对一些具体的非线性问题,提出了若干改进的PCA方法,以及消除其局限性的方法和建议。

In view of the limitations of PCA＇s application to variables observed with nonlinear features, Principal Component Analysis （PCA） was studied theoretically. Based on the coordinates of Euclidean-space and statistical method, the study discussed the mathematical essence of PCA, and analyzed the structure of nonlinear system, and then proposed and proved the causes of the limitations of PCA in （a） solving the multiple correlations between variables efficiently and （b） using in multiple indexes system evaluation under the condition of the multiple correlations by an example of multiple correlations between variables in high degree. For some nonlinear problems, the improvement on PCA and solutions to the limitations were presented accordingly.