主成分综合评价法的误区识别及其改进

大部分基于主成分分析的综合评价忽略了该方法排名结果不唯一的问题。本文通过理论证明了用多个主成分加权综合得到的指标进行排序一定是不可取的，进一步给出了用第一主成分进行综合评价的限制条件，指出了该评价法的存在误区的本质原因，使其应用更加规范和合理。在此基础上提出一种基于非线性投影的对数主成分评价法，改进了传统主成分分析的“线性化”约束，并从理论基础、几何意义和适用范围等方面阐述了本算法，还对2012年联合国人类发展指数的各评价方法进行分析和对比，验证了该方法的合理性和有效性。

Erroneous Zone Identification and Improvement of Synthesis Evaluation Based on Principal Component Analysis

The ranking order is often not unique when the principal component analysis (PCA) is used for the synthesis evaluation. In this paper, it is proved that it is not desirable to rank order by using comprehensive index from multiple principal components in the theory. Further, we give the limiting conditions for the synthesis evaluation of the first principal component, and point out the essential reason of the error of the evaluation method, which makes the application more standardized and reasonable. On this basis, we present a new idea of logarithmic principal component evaluation, improving the linear constraints of traditional principal component analysis. In addition to describing the theoretical basis, geometric meaning and scope of application of the algorithm, human development index of United Nations of 2012 are analyzed and compared using various evaluation methods and the results verify that this new method is reasonable and superior.