When transforming vectors to their Principal Components, are their relations preserved?

eduardofv Leave a comment



pca test






Question: When transforming vectors to their Principal Components, are their relations preserved?

We have a set of vectors. After performing Principal Component Analysis (PCA) we now use the “rotated” vectors to perform analysis. Can we be confident that the original relations (cosine similarity between vectors) are preserved on the new vector space?

knitr::opts_chunk$set(echo = TRUE)
set.seed(1234)
#Cosine Similarity
cos.sim <- function(A,B) 
{
  return( sum(A*B)/sqrt(sum(A^2)*sum(B^2)) )
}

1 – Generate a test set y=x+err

#"noisy" x=y
x = runif(n = 1000,min=-100,max=100) # x has a uniform distribution
y = x + rnorm(n=1000,mean=0,sd=20) # y has a normal distribution

d <- data.frame(x=x,y=y)
#ver primeros puntos
head(d)
##           x            y
## 1 -77.25932 -57.56371837
## 2  24.45988  -0.03487656
## 3  21.85495  36.04947094
## 4  24.67589  22.49148848
## 5  72.18308 107.83523462
## 6  28.06212  23.19322747
plot(d)

2 – Perform PCA

pca <- prcomp(d,center = F,scale. = F)
summary(pca)
## Importance of components:
##                           PC1      PC2
## Standard deviation     83.040 13.30478
## Proportion of Variance  0.975  0.02503
## Cumulative Proportion   0.975  1.00000
pca$rotation
##        PC1       PC2
## x 0.691987 -0.721910
## y 0.721910  0.691987
dt <- pca$x #d transformed
head(dt)
##            PC1        PC2
## [1,] -95.01826  15.940928
## [2,]  16.90074 -17.681966
## [3,]  41.14781   9.168461
## [4,]  33.31222  -2.249953
## [5,] 127.79708  22.510896
## [6,]  36.16204  -4.208914
plot(dt)