How to reverse PCA in prcomp to get original data

Question

I want to reverse the PCA calculated from prcomp to get back to my original data.

I thought something like the following would work:

pca$x %*% t(pca$rotation)

but it doesn't.

The following link shows how to get back the original data from PCs, but explains it only for PCA using eigen on the covariance matrix http://www.di.fc.ul.pt/~jpn/r/pca/pca.html

prcomp doesn't calcluate PCs that way.

"The calculation is done by a singular value decomposition of the (centered and possibly scaled) data matrix, not by using eigen on the covariance matrix." -prcomp

Answer 1

I hope this will help as well.

rm(list = ls())

# ---- 
# create a dataset feature of class 1, 100 samples
f1 <- rnorm(n = 100, mean = 5, sd = 1)

# ---- 
# still in the same feature, create class 2, also 100 samples
f1 <- c(f1,rnorm(n = 100, mean = 10, sd = 1))

# ----
# create another feature, of course it has 200 samples
f2 <- (f1 * 1.25) + rnorm(n = 200, mean = 7, sd = 0.75)

# ----
# put them together in one container i.e dataset
# feature #1 could better represent the separation of the two class
# since it spread from about 4 to 11, while feature #2 spread from about
# 6 to 8 (without addition 1.5 of feature #1)
mydataset <- cbind(f1,f2)

# ----
# create coloring label
class.color <- c(rep(2,100),rep(3,100))

# ----
# plot the dataset
plot(mydataset, col = class.color, main = 'the original formation')

# ----
# transform it...!!!!
pca.result <- prcomp(mydataset,scale. = TRUE, center = TRUE, retx = TRUE)

# ----
# plot the samples on their new axis
# recall that when a line was drawn at the zero value of PC 1, it could separate the red and green class
# but not when it was drawn at the zero value of PC 2
# the line at the zero of PC 1 put red on its left and green on its right (or vice versa)
# the line at the zero of PC 2 put BOTH red AND green on its upper part, and ALSO BOTH red AND green on its
# lower part... i.e. PC 2 could not separate the red and green class
plot(pca.result$x, col = class.color, main = 'samples on their new axis')

# ----
# calculate the variance explained by the PCs in percent
# PC 1 could explain approximately 98% while PC 2 only 2%
variance.total <- sum(pca.result$sdev^2)
variance.explained <- pca.result$sdev^2 / variance.total * 100
print(variance.explained)

# ----
# drop PC 2 ---> samples drawn at PC 1's axis ---> this is the desired new representation of dataset
plot(x = pca.result$x[,1], y = rep(0,200), col = class.color, 
     main = 'over PC 1', ylab = '', xlab = 'PC 1')

# ----
# drop PC 1 ---> samples drawn at PC 2's axis ---> this is the UNdesired new representation of dataset
plot(x = pca.result$x[,2], y = rep(0,200), col = class.color, 
     main = 'over PC 2', ylab = '', xlab = 'PC 2')

# ----
# now choose only PC 1 and get it back to the original dataset, let's see what it's like
# take all PC 1 value, put it on first column of the new dataset, and zero pad the second column
new.dataset <- cbind(
  pca.result$x[,1],
  rep(0,200)
)

# ----
# take alook at a glance the new dataset
# remember, although the choosen one was only PC 1, doesn't mean that there would be only one column
# the second column (and all column for a larger feature) must also exist
# but now they are all set to zero 
(new.dataset)

# ----
# transform it back
new.dataset <- new.dataset %*% solve(pca.result$rotation)

# ----
# plot the new dataset that is constructed with only one PC 
# (a little clumsy though, for we already have a new better axis system, why would we use the old one?)
plot(new.dataset,col = class.color,
     main = 'centered and scaled\nnew dataset with only one pc ---> PC 1', xlab = 'f1', ylab = 'f2')
# ----
# remember, the dots are stil in scale and center position
# must be stretched and dragged first
scalling.matrix <- matrix(rep(pca.result$scale,200),ncol = 2, byrow = TRUE)
centering.matrix <- matrix(rep(pca.result$center,200),ncol = 2, byrow = TRUE)

# ----
# obtain original values
new.dataset <- (new.dataset * scalling.matrix) + centering.matrix

# ----
# compare the result before and after centering
# all dots reside the same position, but with different values
plot(new.dataset,col = class.color,
     main = 'stretched and dragged\nnew dataset with only one pc ---> PC 1', xlab = 'f1', ylab = 'f2')

# ----
# what if all PCs were all used in construction the data?
# they'll be forming back (but OF COURSE that's not the principal component analysis here on earth for)
new.dataset <- cbind(
  pca.result$x[,1],
  pca.result$x[,2]
)
new.dataset <- new.dataset %*% solve(pca.result$rotation)
new.dataset <- (new.dataset * scalling.matrix) + centering.matrix
plot(new.dataset,col = class.color,
     main = 'new dataset with\nboth pc included ---> PC 1 & 2 present', xlab = 'f1', ylab = 'f2')

# ----
# compare the inverted dots with those from the original formation, they're all the same

Answer 2

prcomp will center the variables so you need to add the subtracted means back

t(t(pca$x %*% t(pca$rotation)) + pca$center)

If pca$scale is TRUE you will also need to re-scale

t(t(pca$x %*% t(pca$rotation)) * pca$scale + pca$center)