综合得分的表达式

特点

第一类

$F_{11}^c={\displaystyle \sum _{j=1}^m{F}_j}$

$F_{11}^c$ 为各因子得分之和

$F_{12}^c={\displaystyle \sum _{j=1}^m\frac{1}{m}{F}_j}=\frac{1}{m}{\displaystyle \sum _{j=1}^m{F}_j}\propto {\displaystyle \sum _{j=1}^m{F}_j}$

权重 $w_j=\frac{1}{m},j=1,2,\cdots ,m,{\displaystyle \sum _{j=1}^m{w}_j=1}$

$F_{13}^c=\frac{1}{\sqrt{m}}{\displaystyle \sum _{j=1}^m{F}_j}\propto {\displaystyle \sum _{j=1}^m{F}_j}$

$Var\left(F_{13}^c\right)=1$

第二类

$F_{21}^c={\displaystyle \sum _{j=1}^m{\alpha }_j{F}_j}={\displaystyle \sum _{j=1}^m\frac{g_j^2}{p}{F}_j}=\frac{1}{p}{\displaystyle \sum _{j=1}^m{g}_j^2{F}_j}\propto {\displaystyle \sum _{j=1}^m{g}_j^2{F}_j}$

权重 ${\alpha }_j=\frac{g_j^2}{p}$ 为公共因子 $F_j$ 对 $X_1,X_2,\cdots ,X_p$ 的 总方差贡献率

$F_{22}^c={\displaystyle \sum _{j=1}^m\frac{g_j^2}{{\displaystyle \sum _{i=1}^m{g}_i^2}}{F}_j}=\frac{1}{{\displaystyle \sum _{i=1}^m{g}_i^2}}{\displaystyle \sum _{j=1}^m{g}_j^2{F}_j}\propto {\displaystyle \sum _{j=1}^m{g}_j^2{F}_j}$

权重 $w_j=\frac{g_j^2}{{\displaystyle \sum _{i=1}^m{g}_i^2}}\propto g_j^2,j=1,2,\cdots ,m,{\displaystyle \sum _{j=1}^m{w}_j=1}$

$F_{23}^c={\displaystyle \sum _{j=1}^m{g}_j^2{F}_j}$

权重 $g_j^2$ 为公共因子 $F_j$ 对 $X_1,X_2,\cdots ,X_p$ 的 总方差贡献

$F_{24}^c=\frac{1}{\sqrt{{\displaystyle \sum _{i=1}^m{g}_i^4}}}{\displaystyle \sum _{j=1}^m{g}_j^2{F}_j}\propto {\displaystyle \sum _{j=1}^m{g}_j^2{F}_j}$

$Var\left(F_{24}^c\right)=1$