样本协方差矩阵和样本相关矩阵及其在样本主成分中的应用

1、 张应应，荣腾中 (R2. 1.77908842 * % cov Z = = 1.14726052 , 0.07365107 ( (R3. 0.628753 0.93516170 0.03380488 * * % % cov X= ,Z Q = 1.028615 0.66087530 0.04255799 , 1.308349 0.07016503 0.04970440 ( (R4. 1.0000000 0.1556371 0.4967672 % cor X = R = 0.1556371 1.0000000 0.6873084 , 0.4967672 0.6873084 1.0000000

2、( (R5. 1 % Z E = cor = 3×3 1 , 1 ( (R6. 0.4713911 0.8730833 0.1245632 1 * * 2 %,Z % X Q cor = = 0.7711769 0.6170047 0.1568164 , 0.9809002 0.0655073 0.1831493 ( ( (R7. % = 3= tr cov Z % . tr cov X ( ( ( ( 由数值结果可以看出，(S1-(S7和(R1-(R7均是正确的。 下面给一个 S 0 且 R 0 的例子，即 S 和 R 都有 1 个 0 特征值，此时(S5，(S6，(R5和(R6是

3、不成 立的，但其余性质是成立的。 例 2. X = (x ij n× p ,n = p = 3, xij N ( 0,1 . 为重复本文的结果，我们在 R 软件中使用 set.seed(1，此时 -0.6264538 1.5952808 0.4874291 X = 0.1836433 0.3295078 0.7383247 0.8356286 0.8204684 0.5757814 0.28982112 0.10867553 0.05840744 = S 0.10867553 1.46007844 0.05734883 , 0.05840744 0.05734883 0.016195

4、85 59 张应应，荣腾中 1.0000000 0.1670624 0.8525136 = R 0.1670624 1.0000000 0.3729362 , 0.8525136 0.3729362 1.0000000 1 = = 2 3 1.471929 0.2941663 , 7.139463e 17 0.08974467 0.97207413 0.21683584 Q= ( a1 , a2 , a3 = 0.99532799 0.07975163 0.05442315 . 0.03561033 0.22070697 0.97468987 = 0 ，故 S 0 。 由于 3 (S1. 0.

5、28982112 0.10867553 0.05840744 cov ( X = = 0.10867553 1.46007844 0.05734883 , S 0.05840744 0.05734883 0.01619585 (S2. 1.471929 cov ( Z = 0.2941663 , = 0 (S3. 0.13209779 0.28595150 = Q cov ( X , Z = 1.46505219 0.02346024 0.05241588 0.06492456 (S4. 0 0, 0 1.0000000 0.1670624 0.8525136 cor ( X = = 0.16

6、70624 1.0000000 0.3729362 , R 0.8525136 0.3729362 1.0000000 (S5. 1 0 0.7037615 1 = cor ( Z 0 1 0.4139885 = 1 E3×3 , 0.7037615 0.4139885 1 1 (S6. 60 张应应，荣腾中 0.2022493 0.97933406 0.2630977 cor ( X , Z = 0.9993591 0.03579711 0.7181300 0.3394826 0.94061233 0.6283174 0.2022493 0.97933406 NaN 0.99935

7、91 0.03579711 NaN 0.3394826 0.94061233 NaN 2Q 2 = D = 1 1 ( Diag ( S 1 2 2, Q 1 (S7. tr = = ( cov ( X tr ( cov ( Z 1.766095. 此时 0.3720764 1.01558921 0.8885750 * % X X = = 1.1327004 0.03194387 1.0828999 , 0.7606241 0.98364534 0.1943249 * 1 * * = = 2 * 3 1.879171 1.120829 , 0 0.6669312 0.3826931 0.639

8、3346 * * * * 1 , a 2 , a 3 = Q = a 0.1799815 0.9153613 0.3601670 . 0.7230556 0.1251382 0.6793607 ( * = 0 ，故 R 0 。 由于 3 (R1. 1.0000000 0.1670624 0.8525136 % cov X = R = 0.1670624 1.0000000 0.3729362 , 0.8525136 0.3729362 1.0000000 ( (R2. 1.879171 * %= cov Z = 1.120829 , 0 ( (R3. 1.253278 0.4289336 * * % % Q = cov X , Z = 0.338216 1.0259635 1.358745 0.14