################
## PCA based on SVD: prcomp
################
load("datV5video.rda")
dat2 ## big numbers indicate good performance
datC <- scale(dat2, center = TRUE, scale = FALSE)

## Using eigenvalue decomposition (using correlation matrix)
?princomp
heptathlon_pca <- princomp(datC, cor = TRUE)
summary(heptathlon_pca, loadings = TRUE)
head(heptathlon_pca$scores[,1:3]) ## coordinates on first 3 PCs

## Using singular value decomposition (using correlation matrix)
?prcomp
prRes <- prcomp(dat2, center = TRUE, scale = TRUE, retx = TRUE)
summary(prRes)
prRes$rotation ## loadings
head(prRes$x[,1:3]) ## coordinates on first 3 PCs

## In high-dimensional settings (p>n), only the SVD-approach works
set.seed(32)
datHD <- matrix(rnorm(50*100), 50, 100)
dim(datHD) ## more variables than samples
EVapproach <- princomp(datHD) ## does not work
SVDapproach <- prcomp(datHD) ## works
SVDapproach$x[,1:3] ## coordinates on first 3 PCs