Puntuaciones de PC a partir de las matrices de correlación y covarianza mediante cálculos matriciales y prcomp

Question

Quiero obtener las puntuaciones de los PC mediante un enfoque matricial. Mis puntuaciones de PCs calculadas para la matriz de correlación coinciden con prcomp pero las puntuaciones de PC para la matriz de covarianza no coinciden con los resultados de prcomp . ¿Podría indicarme qué es lo que me falta? Gracias

PCA en la matriz de correlación

# 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 en la matriz de covarianza

# 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

No conozco a R pero puedo ver su error. Cuando lo haces con correlaciones, multiplicas correctamente estandarizado datos por los vectores propios (supongo que es scale(X) que estandariza) para obtener las puntuaciones del PC.

Cuando se analizan las covarianzas, hay que multiplicar centrado datos por los vectores propios. Pero en lugar de eso, estás multiplicando crudo datos por los vectores propios. De ahí que se obtengan puntuaciones incorrectas.

Answer 2

Traduciendo la respuesta de @ttnphns en R .

PCA en la matriz de correlación

# 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 en la matriz de covarianza

# 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