二维变量的主成分分析和回归分析的关系研究

对二维变量进行主成分分析相当于对坐标轴按某一角度进行旋转,而对其进行回归分析则可以求出与原坐标轴有夹角的回归直线,探讨了这两个角度之间的关系.结果表明,当二维变量间完全线性相关时,两个角度相等;若两者均在0～π/2范围内,则前者大于等于后者,若两者均在π/2～π范围内,则前者小于等于后者;两者不可能一个在0～π/2范围内,而另一个在π/2～π范围内.

A Study between the Principal Component Analysis and The Regression Analysis for a Two-Dimension Variable

The principal component analysis for a two-dimensional variable is equivalent to rotating the coordinate axes with certain angle.However,through regression analysis,we could derive regression line which is of certain included angle to the original coordinate axes.This paper discussed the relation between the above mentioned two angles.The result indicates that:the two angles are equal when the two-dimensional variables show linear correlation completely;the former angle is bigger than the latter when the two angles are at0～π/2;when the two angles are atπ/2～π,the latter angle is bigger than the former;it is impossible that one angle is at0～π/2 and the other is atπ/2～π.