Example for HW2
Rick Farouni
September 12, 2016
PCA
First, lets read the data, extract the variables we need, then center and optionaly scale the data.
strength_df <- read.table("~/Documents/repos/PSYCH7821/assets/assignments/HW2/strength.csv", header = TRUE)
X <- as.matrix(scale(strength_df[ , 6:11],
center = TRUE,
scale = TRUE)
)
X## dm1 nd1 dm2 nd2 dm3 nd3
## [1,] -0.9588 -1.1144 -0.864 -0.99887 -0.9076 -0.9320
## [2,] -0.3131 -0.0662 -0.267 0.00762 -0.3316 -0.1434
## [3,] -0.1517 0.1959 -0.118 0.17537 -0.2596 0.0319
## [4,] -0.4745 0.1959 -0.341 0.00762 -0.6196 -0.0558
## [5,] -1.1203 -1.2891 -1.237 -1.33437 -1.3395 -1.3701
## [6,] -0.1517 -0.7650 -0.491 -0.66337 -0.2596 -0.4939
## [7,] -0.3131 -0.7650 -0.491 -0.66337 -0.4756 -0.7568
## [8,] -0.7974 -0.9397 -1.088 -0.99887 -1.0515 -1.1949
## [9,] -0.0709 0.0212 -0.043 -0.16012 -0.3316 -0.1434
## [10,] -0.4745 -0.2409 -0.416 -0.16012 -0.6196 -0.6691
## [11,] -0.4745 0.4579 0.405 1.26574 0.9643 1.7844
## [12,] -1.1203 -0.7650 -1.088 -0.83112 -1.1235 -1.1073
## [13,] -0.9588 -0.9397 -0.938 -0.66337 -0.8356 -0.4063
## [14,] -1.1203 -0.9397 -1.088 -0.83112 -1.1955 -1.0196
## [15,] -1.2817 -1.8132 -1.237 -1.75374 -1.4835 -1.6330
## [16,] -1.4431 -1.4638 -1.386 -1.33437 -1.1955 -1.3701
## [17,] 1.1398 1.5061 0.927 1.18187 0.9643 0.7329
## [18,] 1.3013 1.5061 1.225 1.34962 1.2523 1.6967
## [19,] -0.3131 0.2832 -0.192 0.67862 0.2443 0.7329
## [20,] 0.8170 1.3314 1.300 1.51737 1.3962 0.9957
## [21,] 0.7362 0.6326 0.629 0.34312 0.6043 0.3824
## [22,] 1.9470 1.3314 1.449 0.93024 1.6842 0.9957
## [23,] 2.5120 2.0302 2.942 2.10449 2.3322 1.7844
## [24,] 0.2519 0.4579 0.405 0.17537 0.2443 -0.2310
## [25,] 1.4627 0.9820 0.852 1.51737 0.9643 1.7844
## [26,] 0.3327 -0.0662 -0.192 -0.32787 0.2443 -0.4939
## [27,] 0.8170 -0.1535 0.852 -0.32787 0.2443 0.0319
## [28,] -0.6360 -1.2891 -0.640 -1.41824 -0.1876 -0.7568
## [29,] 0.9784 0.8073 1.002 0.67862 0.9643 0.5576
## [30,] 0.1712 0.5453 -0.043 0.17537 -0.0436 0.4700
## [31,] 0.8170 1.3314 1.151 1.34962 1.3962 1.4339
## [32,] 0.1712 -0.2409 0.106 -0.16012 0.1004 0.0319
## [33,] -1.2817 -0.7650 -1.088 -0.83112 -1.3395 -0.6691
## attr(,"scaled:center")
## dm1 nd1 dm2 nd2 dm3 nd3
## 28.9 25.8 31.6 28.9 35.6 32.6
## attr(,"scaled:scale")
## dm1 nd1 dm2 nd2 dm3 nd3
## 12.4 11.4 13.4 11.9 13.9 11.4You can fit the model with a single call to the function prcomp like this
pca_fit <- prcomp(X)The proportion of variance explained can be obtained using the summary function
summary(pca_fit)## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6
## Standard deviation 2.358 0.5189 0.301 0.20207 0.16276 0.11612
## Proportion of Variance 0.927 0.0449 0.015 0.00681 0.00442 0.00225
## Cumulative Proportion 0.927 0.9715 0.987 0.99334 0.99775 1.00000Now pca_fit is a list with three objects (rotation, sdev, X). You can access the objects with the $ operator, so for example:
Q is the loading matrix (eigenvectors)
Q <- pca_fit$rotation
Q## PC1 PC2 PC3 PC4 PC5 PC6
## dm1 0.399 -0.602 0.1065 -0.6222 0.1288 0.25248
## nd1 0.413 0.149 0.6804 0.0749 0.2489 -0.52646
## dm2 0.413 -0.348 -0.0746 0.3689 -0.7345 -0.16596
## nd2 0.410 0.444 0.2423 0.1548 -0.0936 0.73744
## dm3 0.414 -0.165 -0.4804 0.4673 0.5934 0.00422
## nd3 0.401 0.520 -0.4803 -0.4784 -0.1456 -0.29616sigma is the vector of standard deviations
sigma <- pca_fit$sdev
sigma ## [1] 2.358 0.519 0.301 0.202 0.163 0.116Z is the matrix of sample PC scores
Z <- X %*% Q
Z## PC1 PC2 PC3 PC4 PC5 PC6
## [1,] -2.35773 -0.0666 -0.1542 0.06154 -0.0759 0.023514
## [2,] -0.45402 0.2549 0.1716 0.00622 -0.0375 0.046756
## [3,] -0.05106 0.2985 0.2778 -0.04375 -0.0595 -0.003102
## [4,] -0.52517 0.5101 0.4345 -0.07771 -0.1219 -0.146745
## [5,] -3.14030 -0.1711 0.0742 -0.03294 -0.0273 0.017165
## [6,] -1.15607 -0.3603 -0.2988 -0.13172 0.1304 0.101842
## [7,] -1.41540 -0.3642 -0.0860 -0.00646 0.0197 0.138025
## [8,] -2.47874 -0.1728 0.1940 -0.04993 0.1056 0.086696
## [9,] -0.29775 -0.0302 0.1994 -0.08128 -0.1332 -0.098943
## [10,] -1.05103 0.0776 0.3969 0.12948 -0.0708 0.153522
## [11,] 1.80065 1.5436 -0.7828 0.27184 -0.0504 -0.019040
## [12,] -2.46135 0.1793 0.3116 0.11446 0.0363 0.010655
## [13,] -1.93819 0.3958 -0.2356 -0.11881 -0.0428 0.035931
## [14,] -2.52809 0.2108 0.1852 0.02581 -0.0626 0.076372
## [15,] -3.75787 -0.4510 -0.2057 0.02177 -0.1865 0.020306
## [16,] -3.34310 0.0254 -0.1370 0.16708 0.0827 0.052990
## [17,] 2.63624 -0.0378 0.5480 0.02865 0.1957 -0.000391
## [18,] 3.39855 0.2895 -0.0177 -0.26223 0.0122 -0.169706
## [19,] 0.58599 0.9394 -0.1313 0.01378 0.1461 0.088165
## [20,] 3.01116 0.2150 0.1145 0.48211 0.0233 0.119526
## [21,] 1.35833 -0.3162 0.0711 -0.02619 0.0614 -0.109136
## [22,] 3.40216 -0.8250 -0.0569 -0.12225 0.2851 -0.051673
## [23,] 5.59751 -0.7559 -0.0382 0.23639 -0.4044 0.110549
## [24,] 0.53678 -0.3069 0.3443 0.27871 0.0114 -0.045860
## [25,] 3.07735 0.4117 -0.1924 -0.69011 -0.0228 0.305422
## [26,] -0.20525 -0.5858 0.0452 0.01685 0.4151 0.056237
## [27,] 0.59388 -0.9803 -0.2932 -0.15716 -0.3880 -0.104562
## [28,] -2.01200 -0.5784 -0.7870 0.11783 0.1988 -0.198270
## [29,] 2.03780 -0.3851 0.0121 0.11018 0.0188 -0.004857
## [30,] 0.51793 0.3225 0.2301 -0.29958 0.0786 -0.246754
## [31,] 3.05667 0.4204 -0.1255 0.19152 0.0848 -0.109170
## [32,] 0.00149 -0.2469 -0.2559 -0.07851 -0.0460 0.025309
## [33,] -2.43938 0.5400 0.1877 -0.09560 -0.1764 -0.160772F (pattern matrix) is the loadings multiplied by a scaling factor
F <- Q %*% diag(sigma)
F## [,1] [,2] [,3] [,4] [,5] [,6]
## dm1 0.940 -0.3123 0.0320 -0.1257 0.0210 0.02932
## nd1 0.973 0.0772 0.2045 0.0151 0.0405 -0.06113
## dm2 0.973 -0.1806 -0.0224 0.0745 -0.1195 -0.01927
## nd2 0.966 0.2303 0.0728 0.0313 -0.0152 0.08563
## dm3 0.977 -0.0856 -0.1444 0.0944 0.0966 0.00049
## nd3 0.946 0.2699 -0.1444 -0.0967 -0.0237 -0.03439To extract the first factor, use subsetting with []. For example, the first factor is
f1 <- F[ , 1]
as.matrix(f1)## [,1]
## dm1 0.940
## nd1 0.973
## dm2 0.973
## nd2 0.966
## dm3 0.977
## nd3 0.946sample PC scores scaled
Zs <- Z %*% diag(1/sigma)
Zs## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] -0.999936 -0.1283 -0.5131 0.3046 -0.4662 0.20251
## [2,] -0.192554 0.4912 0.5708 0.0308 -0.2301 0.40266
## [3,] -0.021655 0.5753 0.9243 -0.2165 -0.3658 -0.02671
## [4,] -0.222728 0.9829 1.4456 -0.3845 -0.7488 -1.26376
## [5,] -1.331833 -0.3298 0.2470 -0.1630 -0.1678 0.14782
## [6,] -0.490303 -0.6942 -0.9942 -0.6519 0.8009 0.87706
## [7,] -0.600284 -0.7019 -0.2860 -0.0320 0.1208 1.18866
## [8,] -1.051259 -0.3330 0.6454 -0.2471 0.6488 0.74663
## [9,] -0.126281 -0.0581 0.6636 -0.4022 -0.8186 -0.85209
## [10,] -0.445753 0.1495 1.3206 0.6408 -0.4349 1.32212
## [11,] 0.763673 2.9746 -2.6048 1.3453 -0.3094 -0.16397
## [12,] -1.043885 0.3456 1.0368 0.5664 0.2232 0.09176
## [13,] -0.822007 0.7628 -0.7840 -0.5880 -0.2630 0.30944
## [14,] -1.072191 0.4062 0.6163 0.1277 -0.3847 0.65771
## [15,] -1.593753 -0.8692 -0.6845 0.1077 -1.1457 0.17487
## [16,] -1.417844 0.0489 -0.4560 0.8269 0.5079 0.45635
## [17,] 1.118056 -0.0729 1.8236 0.1418 1.2024 -0.00337
## [18,] 1.441361 0.5578 -0.0589 -1.2977 0.0749 -1.46150
## [19,] 0.248525 1.8104 -0.4369 0.0682 0.8979 0.75927
## [20,] 1.277063 0.4144 0.3810 2.3858 0.1433 1.02935
## [21,] 0.576084 -0.6093 0.2366 -0.1296 0.3775 -0.93988
## [22,] 1.442893 -1.5897 -0.1893 -0.6050 1.7515 -0.44501
## [23,] 2.373963 -1.4566 -0.1270 1.1698 -2.4847 0.95204
## [24,] 0.227656 -0.5915 1.1455 1.3792 0.0698 -0.39494
## [25,] 1.305137 0.7933 -0.6401 -3.4151 -0.1399 2.63028
## [26,] -0.087050 -1.1289 0.1502 0.0834 2.5502 0.48431
## [27,] 0.251869 -1.8891 -0.9754 -0.7777 -2.3839 -0.90049
## [28,] -0.853309 -1.1147 -2.6188 0.5831 1.2212 -1.70749
## [29,] 0.864251 -0.7420 0.0403 0.5453 0.1156 -0.04183
## [30,] 0.219661 0.6215 0.7657 -1.4825 0.4829 -2.12503
## [31,] 1.296363 0.8101 -0.4175 0.9478 0.5213 -0.94016
## [32,] 0.000634 -0.4758 -0.8514 -0.3885 -0.2829 0.21796
## [33,] -1.034565 1.0406 0.6245 -0.4731 -1.0838 -1.38456The first standardized principle component pc1
z1 <- Zs[ , 1]
as.matrix(z1)## [,1]
## [1,] -0.999936
## [2,] -0.192554
## [3,] -0.021655
## [4,] -0.222728
## [5,] -1.331833
## [6,] -0.490303
## [7,] -0.600284
## [8,] -1.051259
## [9,] -0.126281
## [10,] -0.445753
## [11,] 0.763673
## [12,] -1.043885
## [13,] -0.822007
## [14,] -1.072191
## [15,] -1.593753
## [16,] -1.417844
## [17,] 1.118056
## [18,] 1.441361
## [19,] 0.248525
## [20,] 1.277063
## [21,] 0.576084
## [22,] 1.442893
## [23,] 2.373963
## [24,] 0.227656
## [25,] 1.305137
## [26,] -0.087050
## [27,] 0.251869
## [28,] -0.853309
## [29,] 0.864251
## [30,] 0.219661
## [31,] 1.296363
## [32,] 0.000634
## [33,] -1.034565plots
biplot(pca_fit, scale = 0)
plot(pca_fit)
comparison with SVD (Advanced)
svd_fit <- svd(X)
# singular values
Sigma_svd <- diag(svd_fit$d)
Sigma_svd ## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 13.3 0.00 0.0 0.00 0.000 0.000
## [2,] 0.0 2.94 0.0 0.00 0.000 0.000
## [3,] 0.0 0.00 1.7 0.00 0.000 0.000
## [4,] 0.0 0.00 0.0 1.14 0.000 0.000
## [5,] 0.0 0.00 0.0 0.00 0.921 0.000
## [6,] 0.0 0.00 0.0 0.00 0.000 0.657#the left singular vectors
U <- svd_fit$u
U## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] -0.176765 -0.02268 -0.09071 0.05384 -0.0824 0.035798
## [2,] -0.034039 0.08683 0.10091 0.00544 -0.0407 0.071181
## [3,] -0.003828 0.10170 0.16340 -0.03828 -0.0647 -0.004722
## [4,] -0.039373 0.17376 0.25556 -0.06798 -0.1324 -0.223403
## [5,] -0.235437 -0.05829 0.04366 -0.02882 -0.0297 0.026132
## [6,] -0.086674 -0.12272 -0.17575 -0.11523 0.1416 0.155043
## [7,] -0.106116 -0.12407 -0.05056 -0.00565 0.0214 0.210128
## [8,] -0.185838 -0.05888 0.11410 -0.04368 0.1147 0.131986
## [9,] -0.022323 -0.01027 0.11731 -0.07111 -0.1447 -0.150630
## [10,] -0.078799 0.02643 0.23345 0.11327 -0.0769 0.233720
## [11,] 0.135000 0.52583 -0.46046 0.23781 -0.0547 -0.028986
## [12,] -0.184535 0.06109 0.18328 0.10013 0.0395 0.016221
## [13,] -0.145312 0.13484 -0.13860 -0.10394 -0.0465 0.054701
## [14,] -0.189538 0.07181 0.10895 0.02258 -0.0680 0.116268
## [15,] -0.281738 -0.15365 -0.12100 0.01905 -0.2025 0.030914
## [16,] -0.250642 0.00864 -0.08061 0.14617 0.0898 0.080672
## [17,] 0.197646 -0.01288 0.32237 0.02506 0.2126 -0.000596
## [18,] 0.254799 0.09861 -0.01042 -0.22940 0.0132 -0.258359
## [19,] 0.043933 0.32003 -0.07723 0.01205 0.1587 0.134222
## [20,] 0.225755 0.07325 0.06734 0.42176 0.0253 0.181965
## [21,] 0.101838 -0.10771 0.04182 -0.02291 0.0667 -0.166148
## [22,] 0.255070 -0.28103 -0.03347 -0.10694 0.3096 -0.078667
## [23,] 0.419661 -0.25750 -0.02244 0.20680 -0.4392 0.168299
## [24,] 0.040244 -0.10456 0.20250 0.24382 0.0123 -0.069817
## [25,] 0.230718 0.14024 -0.11315 -0.60372 -0.0247 0.464972
## [26,] -0.015388 -0.19957 0.02656 0.01474 0.4508 0.085614
## [27,] 0.044525 -0.33395 -0.17244 -0.13748 -0.4214 -0.159185
## [28,] -0.150845 -0.19705 -0.46294 0.10308 0.2159 -0.301844
## [29,] 0.152779 -0.13118 0.00712 0.09639 0.0204 -0.007394
## [30,] 0.038831 0.10987 0.13536 -0.26208 0.0854 -0.375656
## [31,] 0.229167 0.14321 -0.07381 0.16754 0.0922 -0.166199
## [32,] 0.000112 -0.08410 -0.15051 -0.06868 -0.0500 0.038530
## [33,] -0.182887 0.18395 0.11041 -0.08363 -0.1916 -0.244758#the right singular vectors
Q <- svd_fit$v
Q## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0.399 -0.602 0.1065 -0.6222 0.1288 0.25248
## [2,] 0.413 0.149 0.6804 0.0749 0.2489 -0.52646
## [3,] 0.413 -0.348 -0.0746 0.3689 -0.7345 -0.16596
## [4,] 0.410 0.444 0.2423 0.1548 -0.0936 0.73744
## [5,] 0.414 -0.165 -0.4804 0.4673 0.5934 0.00422
## [6,] 0.401 0.520 -0.4803 -0.4784 -0.1456 -0.29616#reconstuction error residual
res <- X - U %*% Sigma_svd %*% t(Q)
res ## dm1 nd1 dm2 nd2 dm3 nd3
## [1,] 0.00e+00 -3.77e-15 -1.11e-16 -3.77e-15 2.22e-16 -1.11e-16
## [2,] 2.78e-16 2.08e-16 2.22e-16 -1.72e-16 2.22e-16 3.05e-16
## [3,] 2.78e-16 -2.50e-16 1.11e-16 3.05e-16 3.33e-16 -2.08e-17
## [4,] 4.44e-16 -2.78e-16 1.67e-16 1.02e-16 4.44e-16 -1.18e-16
## [5,] -2.22e-16 -4.44e-16 6.66e-16 0.00e+00 4.44e-16 0.00e+00
## [6,] -1.94e-16 -1.11e-16 3.89e-16 0.00e+00 1.67e-16 -2.22e-16
## [7,] -2.22e-16 0.00e+00 1.67e-16 0.00e+00 1.11e-16 1.11e-16
## [8,] 0.00e+00 -4.44e-16 4.44e-16 0.00e+00 2.22e-16 0.00e+00
## [9,] -2.78e-17 -3.82e-17 2.08e-17 0.00e+00 2.22e-16 -1.11e-16
## [10,] 3.33e-16 -2.50e-16 2.78e-16 5.55e-17 1.11e-16 2.22e-16
## [11,] 1.05e-15 1.67e-16 -3.33e-16 -2.22e-16 -5.55e-16 4.44e-16
## [12,] 2.22e-16 -3.33e-16 0.00e+00 0.00e+00 6.66e-16 4.44e-16
## [13,] 3.33e-16 -2.22e-16 5.55e-16 -2.22e-16 3.33e-16 1.67e-16
## [14,] 2.22e-16 -4.44e-16 6.66e-16 0.00e+00 4.44e-16 2.22e-16
## [15,] 0.00e+00 0.00e+00 4.44e-16 0.00e+00 4.44e-16 4.44e-16
## [16,] 2.22e-16 -2.22e-16 4.44e-16 -2.22e-16 4.44e-16 6.66e-16
## [17,] 2.22e-16 0.00e+00 -2.22e-16 4.44e-16 -6.66e-16 -5.55e-16
## [18,] 0.00e+00 2.22e-16 -6.66e-16 -2.22e-16 -6.66e-16 -4.44e-16
## [19,] 5.55e-16 0.00e+00 1.94e-16 0.00e+00 -2.78e-17 3.33e-16
## [20,] 1.11e-16 2.22e-16 -6.66e-16 0.00e+00 -1.33e-15 -2.22e-16
## [21,] -1.11e-16 1.11e-16 -2.22e-16 2.22e-16 -1.11e-16 -3.33e-16
## [22,] -8.88e-16 2.22e-16 -6.66e-16 1.11e-16 -8.88e-16 -9.99e-16
## [23,] -4.44e-16 8.88e-16 -1.78e-15 4.44e-16 -1.78e-15 -4.44e-16
## [24,] -5.55e-17 1.11e-16 -3.33e-16 3.05e-16 -1.67e-16 2.22e-16
## [25,] -2.22e-16 -3.33e-16 0.00e+00 0.00e+00 -7.77e-16 -1.11e-15
## [26,] -3.89e-16 1.25e-16 2.78e-17 2.78e-16 -1.39e-16 -3.33e-16
## [27,] -9.99e-16 2.78e-16 -3.33e-16 -1.11e-16 -4.44e-16 -4.72e-16
## [28,] -7.77e-16 2.22e-16 -3.33e-16 -4.44e-16 -3.61e-16 -1.11e-16
## [29,] -6.66e-16 2.22e-16 -6.66e-16 3.33e-16 -6.66e-16 -1.11e-16
## [30,] 8.33e-17 -1.11e-16 2.29e-16 1.11e-16 4.02e-16 -2.22e-16
## [31,] 3.33e-16 4.44e-16 -2.22e-16 4.44e-16 -6.66e-16 0.00e+00
## [32,] -2.22e-16 8.33e-17 -6.94e-17 -1.67e-16 -8.33e-17 -1.25e-16
## [33,] 4.44e-16 -4.44e-16 2.22e-16 -1.11e-16 6.66e-16 3.33e-16
## attr(,"scaled:center")
## dm1 nd1 dm2 nd2 dm3 nd3
## 28.9 25.8 31.6 28.9 35.6 32.6
## attr(,"scaled:scale")
## dm1 nd1 dm2 nd2 dm3 nd3
## 12.4 11.4 13.4 11.9 13.9 11.4Z_svd1 <- U %*% Sigma_svd
Z_svd1## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] -2.35773 -0.0666 -0.1542 0.06154 -0.0759 0.023514
## [2,] -0.45402 0.2549 0.1716 0.00622 -0.0375 0.046756
## [3,] -0.05106 0.2985 0.2778 -0.04375 -0.0595 -0.003102
## [4,] -0.52517 0.5101 0.4345 -0.07771 -0.1219 -0.146745
## [5,] -3.14030 -0.1711 0.0742 -0.03294 -0.0273 0.017165
## [6,] -1.15607 -0.3603 -0.2988 -0.13172 0.1304 0.101842
## [7,] -1.41540 -0.3642 -0.0860 -0.00646 0.0197 0.138025
## [8,] -2.47874 -0.1728 0.1940 -0.04993 0.1056 0.086696
## [9,] -0.29775 -0.0302 0.1994 -0.08128 -0.1332 -0.098943
## [10,] -1.05103 0.0776 0.3969 0.12948 -0.0708 0.153522
## [11,] 1.80065 1.5436 -0.7828 0.27184 -0.0504 -0.019040
## [12,] -2.46135 0.1793 0.3116 0.11446 0.0363 0.010655
## [13,] -1.93819 0.3958 -0.2356 -0.11881 -0.0428 0.035931
## [14,] -2.52809 0.2108 0.1852 0.02581 -0.0626 0.076372
## [15,] -3.75787 -0.4510 -0.2057 0.02177 -0.1865 0.020306
## [16,] -3.34310 0.0254 -0.1370 0.16708 0.0827 0.052990
## [17,] 2.63624 -0.0378 0.5480 0.02865 0.1957 -0.000391
## [18,] 3.39855 0.2895 -0.0177 -0.26223 0.0122 -0.169706
## [19,] 0.58599 0.9394 -0.1313 0.01378 0.1461 0.088165
## [20,] 3.01116 0.2150 0.1145 0.48211 0.0233 0.119526
## [21,] 1.35833 -0.3162 0.0711 -0.02619 0.0614 -0.109136
## [22,] 3.40216 -0.8250 -0.0569 -0.12225 0.2851 -0.051673
## [23,] 5.59751 -0.7559 -0.0382 0.23639 -0.4044 0.110549
## [24,] 0.53678 -0.3069 0.3443 0.27871 0.0114 -0.045860
## [25,] 3.07735 0.4117 -0.1924 -0.69011 -0.0228 0.305422
## [26,] -0.20525 -0.5858 0.0452 0.01685 0.4151 0.056237
## [27,] 0.59388 -0.9803 -0.2932 -0.15716 -0.3880 -0.104562
## [28,] -2.01200 -0.5784 -0.7870 0.11783 0.1988 -0.198270
## [29,] 2.03780 -0.3851 0.0121 0.11018 0.0188 -0.004857
## [30,] 0.51793 0.3225 0.2301 -0.29958 0.0786 -0.246754
## [31,] 3.05667 0.4204 -0.1255 0.19152 0.0848 -0.109170
## [32,] 0.00149 -0.2469 -0.2559 -0.07851 -0.0460 0.025309
## [33,] -2.43938 0.5400 0.1877 -0.09560 -0.1764 -0.160772Z_svd2 <- X %*% Q
Z_svd2## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] -2.35773 -0.0666 -0.1542 0.06154 -0.0759 0.023514
## [2,] -0.45402 0.2549 0.1716 0.00622 -0.0375 0.046756
## [3,] -0.05106 0.2985 0.2778 -0.04375 -0.0595 -0.003102
## [4,] -0.52517 0.5101 0.4345 -0.07771 -0.1219 -0.146745
## [5,] -3.14030 -0.1711 0.0742 -0.03294 -0.0273 0.017165
## [6,] -1.15607 -0.3603 -0.2988 -0.13172 0.1304 0.101842
## [7,] -1.41540 -0.3642 -0.0860 -0.00646 0.0197 0.138025
## [8,] -2.47874 -0.1728 0.1940 -0.04993 0.1056 0.086696
## [9,] -0.29775 -0.0302 0.1994 -0.08128 -0.1332 -0.098943
## [10,] -1.05103 0.0776 0.3969 0.12948 -0.0708 0.153522
## [11,] 1.80065 1.5436 -0.7828 0.27184 -0.0504 -0.019040
## [12,] -2.46135 0.1793 0.3116 0.11446 0.0363 0.010655
## [13,] -1.93819 0.3958 -0.2356 -0.11881 -0.0428 0.035931
## [14,] -2.52809 0.2108 0.1852 0.02581 -0.0626 0.076372
## [15,] -3.75787 -0.4510 -0.2057 0.02177 -0.1865 0.020306
## [16,] -3.34310 0.0254 -0.1370 0.16708 0.0827 0.052990
## [17,] 2.63624 -0.0378 0.5480 0.02865 0.1957 -0.000391
## [18,] 3.39855 0.2895 -0.0177 -0.26223 0.0122 -0.169706
## [19,] 0.58599 0.9394 -0.1313 0.01378 0.1461 0.088165
## [20,] 3.01116 0.2150 0.1145 0.48211 0.0233 0.119526
## [21,] 1.35833 -0.3162 0.0711 -0.02619 0.0614 -0.109136
## [22,] 3.40216 -0.8250 -0.0569 -0.12225 0.2851 -0.051673
## [23,] 5.59751 -0.7559 -0.0382 0.23639 -0.4044 0.110549
## [24,] 0.53678 -0.3069 0.3443 0.27871 0.0114 -0.045860
## [25,] 3.07735 0.4117 -0.1924 -0.69011 -0.0228 0.305422
## [26,] -0.20525 -0.5858 0.0452 0.01685 0.4151 0.056237
## [27,] 0.59388 -0.9803 -0.2932 -0.15716 -0.3880 -0.104562
## [28,] -2.01200 -0.5784 -0.7870 0.11783 0.1988 -0.198270
## [29,] 2.03780 -0.3851 0.0121 0.11018 0.0188 -0.004857
## [30,] 0.51793 0.3225 0.2301 -0.29958 0.0786 -0.246754
## [31,] 3.05667 0.4204 -0.1255 0.19152 0.0848 -0.109170
## [32,] 0.00149 -0.2469 -0.2559 -0.07851 -0.0460 0.025309
## [33,] -2.43938 0.5400 0.1877 -0.09560 -0.1764 -0.160772