PCs scores from Correlation and Covariance matrices through matrix computations and prcomp

Question

I'm want to get PCs scores through matrix approach. My calculated PCs scores for correlation matrix matches with prcomp results but the PCs scores for covariance matrix do not match with the results of prcomp. Could you point out what am I missing? Thanks

PCA on Correlation matrix

# PCA on Correlation matrix
X <- USArrests
Cor <- cor(X)

EigenCor <- eigen(Cor)
ECor <- EigenCor$vectors
head(t(t(ECor) %*% t(scale(X))))


                 [,1]       [,2]        [,3]         [,4]
Alabama    -0.9756604  1.1220012 -0.43980366  0.154696581
Alaska     -1.9305379  1.0624269  2.01950027 -0.434175454
Arizona    -1.7454429 -0.7384595  0.05423025 -0.826264240
Arkansas    0.1399989  1.1085423  0.11342217 -0.180973554
California -2.4986128 -1.5274267  0.59254100 -0.338559240
Colorado   -1.4993407 -0.9776297  1.08400162  0.001450164

PCACor <- prcomp(x = X, retx = TRUE, center = TRUE, scale. = TRUE)
summary(PCACor)


Importance of components:
                          PC1    PC2     PC3     PC4
Standard deviation     1.5749 0.9949 0.59713 0.41645
Proportion of Variance 0.6201 0.2474 0.08914 0.04336
Cumulative Proportion  0.6201 0.8675 0.95664 1.00000

head(PCACor$x)
                  PC1        PC2         PC3          PC4
Alabama    -0.9756604  1.1220012 -0.43980366  0.154696581
Alaska     -1.9305379  1.0624269  2.01950027 -0.434175454
Arizona    -1.7454429 -0.7384595  0.05423025 -0.826264240
Arkansas    0.1399989  1.1085423  0.11342217 -0.180973554
California -2.4986128 -1.5274267  0.59254100 -0.338559240
Colorado   -1.4993407 -0.9776297  1.08400162  0.001450164

PCA on Covariance matrix

# PCA on Covariance matrix
Cov <- var(X)
EigenCov <- eigen(Cov)
ECov <- EigenCov$vectors
head(t(t(ECov) %*% t(X)))
                [,1]      [,2]      [,3]       [,4]
Alabama    -239.7035 -46.45394 -5.873077  5.7840485
Alaska     -267.7288 -39.91901 16.748431 -0.7178995
Arizona    -298.9695 -66.73235 -5.065592 -0.9775376
Arkansas   -193.2414 -41.19804 -3.167955  2.8551540
California -282.3243 -80.42202  3.367729  0.5643217
Colorado   -209.8773 -71.62154  8.901219  1.6546839



PCACov <- prcomp(x = X, retx = TRUE, center = TRUE, scale. = FALSE)
summary(PCACov)



Importance of components:
                           PC1      PC2    PC3     PC4
Standard deviation     83.7324 14.21240 6.4894 2.48279
Proportion of Variance  0.9655  0.02782 0.0058 0.00085
Cumulative Proportion   0.9655  0.99335 0.9991 1.00000

head(PCACov$x)
                 PC1        PC2        PC3        PC4
Alabama     64.80216 -11.448007 -2.4949328 -2.4079009
Alaska      92.82745 -17.982943 20.1265749  4.0940470
Arizona    124.06822   8.830403 -1.6874484  4.3536852
Arkansas    18.34004 -16.703911  0.2101894  0.5209936
California 107.42295  22.520070  6.7458730  2.8118259
Colorado    34.97599  13.719584 12.2793628  1.7214637

Answer 1

I don't know R but can see your mistake. When you do it with correlations, you correctly multiply standardized data by the eigenvectors (I guess it's scale(X) which standardizes) to get the PC scores.

When you analyse covariances, you must multiply centered data by the eigenvectors. But instead, you are multiplying raw data by the eigenvectors. Hence you get incorrect scores.

Answer 2

Translating the answer of @ttnphns into R.

PCA on Correlation matrix

# PCA on Correlation matrix
X <- USArrests
Cor <- cor(X)

EigenCor <- eigen(Cor)
ECor <- EigenCor$vectors
head(t(t(ECor) %*% t(scale(X))))


                 [,1]       [,2]        [,3]         [,4]
Alabama    -0.9756604  1.1220012 -0.43980366  0.154696581
Alaska     -1.9305379  1.0624269  2.01950027 -0.434175454
Arizona    -1.7454429 -0.7384595  0.05423025 -0.826264240
Arkansas    0.1399989  1.1085423  0.11342217 -0.180973554
California -2.4986128 -1.5274267  0.59254100 -0.338559240
Colorado   -1.4993407 -0.9776297  1.08400162  0.001450164

PCACor <- prcomp(x = X, retx = TRUE, center = TRUE, scale. = TRUE)
summary(PCACor)


Importance of components:
                          PC1    PC2     PC3     PC4
Standard deviation     1.5749 0.9949 0.59713 0.41645
Proportion of Variance 0.6201 0.2474 0.08914 0.04336
Cumulative Proportion  0.6201 0.8675 0.95664 1.00000

head(PCACor$x)
                  PC1        PC2         PC3          PC4
Alabama    -0.9756604  1.1220012 -0.43980366  0.154696581
Alaska     -1.9305379  1.0624269  2.01950027 -0.434175454
Arizona    -1.7454429 -0.7384595  0.05423025 -0.826264240
Arkansas    0.1399989  1.1085423  0.11342217 -0.180973554
California -2.4986128 -1.5274267  0.59254100 -0.338559240
Colorado   -1.4993407 -0.9776297  1.08400162  0.001450164

PCA on Covariance matrix

# PCA on Covariance matrix
Cov <- var(X)
EigenCov <- eigen(Cov)
ECov <- EigenCov$vectors
head(t(t(ECov) %*% t(scale(x = X, center = TRUE, scale = FALSE))))

                 [,1]       [,2]       [,3]       [,4]
Alabama     -64.80216  11.448007 -2.4949328  2.4079009
Alaska      -92.82745  17.982943 20.1265749 -4.0940470
Arizona    -124.06822  -8.830403 -1.6874484 -4.3536852
Arkansas    -18.34004  16.703911  0.2101894 -0.5209936
California -107.42295 -22.520070  6.7458730 -2.8118259
Colorado    -34.97599 -13.719584 12.2793628 -1.7214637

# Here is the change
PCACov <- prcomp(x = X, retx = TRUE, center = TRUE, scale. = FALSE)
summary(PCACov)



Importance of components:
                           PC1      PC2    PC3     PC4
Standard deviation     83.7324 14.21240 6.4894 2.48279
Proportion of Variance  0.9655  0.02782 0.0058 0.00085
Cumulative Proportion   0.9655  0.99335 0.9991 1.00000

head(PCACov$x)
                 PC1        PC2        PC3        PC4
Alabama     64.80216 -11.448007 -2.4949328 -2.4079009
Alaska      92.82745 -17.982943 20.1265749  4.0940470
Arizona    124.06822   8.830403 -1.6874484  4.3536852
Arkansas    18.34004 -16.703911  0.2101894  0.5209936
California 107.42295  22.520070  6.7458730  2.8118259
Colorado    34.97599  13.719584 12.2793628  1.7214637