参考资料《统计建模与R软件》

典型相关的数学模型

设 X=(X1,X2,…,Xp)T,Y=(Y1,Y2,…,Yq)T 为两条随机向量, 我们希望找到向量 a,b 使得 U=aTX,V=aTY,ρ(U,V)达到最大 ，由于这样的向量有多组，我们加一个约束 Var(U)=Var(V)=1 这就变成了一个规划问题，我们可以求解(证明方法见参考资料P546) 类似的对于 ak,bk,Uk=aTkX,Vk=bkY 使得 maxρ(Uk,Vk),Var(Uk)=Var(VK)=1 成为第 k 个典型相关系数.

计算

令 M1=Σ−111Σ12Σ−122Σ21M2=Σ−122Σ21Σ−111Σ12

相关系数分别为 ρ21≥ρ22≥…,≥ρ2min(p,q) 分别为第 k 个相关系数，他们是 M1,M2 的特征值， ak,bk 为相应的特征向量.

估计

对于样本的相关阵往往不好知道确切值，我们可以计算 Σ 的最大似然估计去计算 M

R语言实现

> test<-data.frame( X1=c(191, 193, 189, 211, 176, 169, 154, 193, 176, 156, 189, 162, 182, 167, 154, 166, 247, 202, 157, 138), X2=c(36, 38, 35, 38, 31, 34, 34, 36, 37, 33, 37, 35, 36, 34, 33, 33, 46, 37, 32, 33), X3=c(50, 58, 46, 56, 74, 50, 64, 46, 54, 54, 52, 62, 56, 60, 56, 52, 50, 62, 52, 68), Y1=c( 5, 12, 13, 8, 15, 17, 14, 6, 4, 15, 2, 12, 4, 6, 17, 13, 1, 12, 11, 2), Y2=c(162, 101, 155, 101, 200, 120, 215, 70, 60, 225, 110, 105, 101, 125, 251, 210, 50, 210, 230, 110), Y3=c(60, 101, 58, 38, 40, 38, 105, 31, 25, 73, 60, 37, 42, 40, 250, 115, 50, 120, 80, 43) ) # 相关分析 > ca <- cancor(scale(test)[,1:3],scale(test)[,4:6]) > ca $cor #相关系数 [1] 0.79560815 0.20055604 0.07257029 #x的系数 $xcoef [,1] [,2] [,3] X1 -0.17788841 -0.43230348 -0.04381432 X2 0.36232695 0.27085764 0.11608883 X3 -0.01356309 -0.05301954 0.24106633 #y的系数 $ycoef [,1] [,2] [,3] Y1 -0.08018009 -0.08615561 -0.29745900 Y2 -0.24180670 0.02833066 0.28373986 Y3 0.16435956 0.24367781 -0.09608099 $xcenter X1 X2 X3 2.289835e-16 4.315992e-16 -1.778959e-16 $ycenter Y1 Y2 Y3 1.471046e-16 -1.776357e-16 4.996004e-17

相关分析变量及系数

> U <- as.matrix(scal_test)[,1:3] %*% ca$xcoef > V <- as.matrix(scal_test)[,4:6] %*% ca$ycoef > U [,1] [,2] [,3] [1,] -0.009969788 -0.121501078 -0.20419401 [2,] 0.186887139 -0.046163013 0.13223387 [3,] -0.101193522 -0.141661215 -0.37063341 [4,] 0.060964112 -0.346616669 0.03342558 [5,] -0.512831098 -0.458299483 0.44354554 [6,] -0.077780541 0.094512914 -0.23766491 [7,] 0.003955674 0.254201102 0.25701898 [8,] -0.016855040 -0.127105942 -0.34147617 [9,] 0.203734347 0.196310283 -0.00758741 [10,] -0.104800666 0.208124774 -0.11711820 [11,] 0.113834968 -0.016598895 -0.09752299 [12,] 0.063237343 0.213427257 0.21221151 [13,] 0.043586465 -0.008040409 0.01237648 [14,] -0.082181602 0.055998387 0.10021686 [15,] -0.094153311 0.228436101 -0.04670258 [16,] -0.173085857 0.047742282 -0.20173015 [17,] 0.718139369 -0.256090676 0.05898572 [18,] 0.001362964 -0.317746855 0.21374067 [19,] -0.221400693 0.120731486 -0.22201469 [20,] -0.001450263 0.420339649 0.38288931 > V [,1] [,2] [,3] [1,] -0.02909460 0.031027608 0.344302062 [2,] 0.23190170 0.084158321 -0.403047146 [3,] -0.12979237 -0.112030106 -0.133855684 [4,] 0.09063830 -0.150034732 -0.059921010 [5,] -0.39173848 -0.209788233 -0.008592976 [6,] -0.11930102 -0.288113119 -0.480186091 [7,] -0.22619839 0.122191086 -0.006091381 [8,] 0.21834490 -0.164740837 -0.074849953 [9,] 0.26809619 -0.165185828 0.003582504 [10,] -0.38258341 -0.041647344 0.042948559 [11,] 0.21737727 0.056375494 0.277291769 [12,] 0.01130349 -0.218167527 -0.264987331 [13,] 0.16412985 -0.065834278 0.157664098 [14,] 0.03462910 -0.097067107 0.157711712 [15,] 0.05393456 0.778658842 -0.283334468 [16,] -0.15965387 0.183746435 0.008766141 [17,] 0.43237930 -0.002016448 0.080198839 [18,] -0.12845980 0.223805116 0.055667431 [19,] -0.31879992 0.059073577 0.277587462 [20,] 0.16288721 -0.024410919 0.309145462

画图

可以看出 U[,1]与V[,1] 几乎成线性