Building the connection between cosine similarity and correlation in R

Question

According to some articles (e.g. here) correlation is just a centered version of cosine similarity. I use the following code to calculate the cosine similarity matrix of the column vectors of a matrix X (slightly modified from here):

cos.sim <- function(ix) 
{
  A = X[,ix[1]]
  B = X[,ix[2]]
  return(t(A)%*%B/sqrt(sum(A^2)*sum(B^2)))
}   
n <- ncol(X) 
cmb <- expand.grid(i=1:n, j=1:n) 
C <- matrix(apply(cmb,1,cos.sim),n,n)

My question
Which modifications of the code above are needed to get the correlation matrix cor(X) instead of the cosine similarity matrix. I guess the changes are minimal but I can't see them at the moment.

Answer 1

The answer is really right there in your linked articles. From the first, here are the formulae for cosine and correlation (lightly edited for brevity and clarity):

\begin{align} {\rm CosSim}(x,y) &= \frac{\sum_i x_i y_i}{ \sqrt{ \sum_i x_i^2} \sqrt{ \sum_i y_i^2 } } \\ \ \\ \ \\ {\rm Corr}(x,y) &= \frac{ \sum_i (x_i-\bar{x}) (y_i-\bar{y}) }{ \sqrt{\sum (x_i-\bar{x})^2} \sqrt{ \sum (y_i-\bar{y})^2 } } \\ \ \\ {\rm Corr}(x,y) &= {\rm CosSim}(x-\bar{x},\ y-\bar{y}) \end{align}

So the simplest adaptation is just to subtract the means from your input vectors:

library(MASS)  # we need this package to generate correlated data below
set.seed(2641) # this makes the example exactly reproducible
  # now I generate correlated data:
X <- mvrnorm(1000, mu=c(100, 150), Sigma=rbind(c(30, 17),
                                               c(17, 50) ) )
  # I adapted the function somewhat, as the original was keyed to its context
cos.sim <- function(X, corr=FALSE){
  if(corr){ 
    A = X[,1] - mean(X[,1])
    B = X[,2] - mean(X[,2])
  } else {
    A = X[,1]
    B = X[,2]
  }
  return( t(A)%*%B / sqrt(sum(A^2)*sum(B^2)) )
} 
cos.sim(X)
#           [,1]
# [1,] 0.9985756
cos.sim(X, corr=TRUE)
#           [,1]
# [1,] 0.4604822
cor(X)
#           [,1]      [,2]
# [1,] 1.0000000 0.4604822
# [2,] 0.4604822 1.0000000

---

Here is a matrix version:

set.seed(6616)
X3 <- mvrnorm(1000, mu=c(100, 150, 175), Sigma=rbind(c(30, 17, 12),
                                                     c(17, 50, 29),
                                                     c(12, 29, 46) ))
cos.sim.mat <- function(X, corr=FALSE){
  if(corr){ X = apply(X, 2, function(x){ x-mean(x) }) }
  denom = solve(diag(sqrt(diag(t(X)%*%X))))
  return( denom%*%(t(X)%*%X)%*%denom )
} 
cos.sim.mat(X3)
#           [,1]      [,2]      [,3]
# [1,] 1.0000000 0.9984552 0.9983700
# [2,] 0.9984552 1.0000000 0.9992154
# [3,] 0.9983700 0.9992154 1.0000000
cos.sim.mat(X3, corr=TRUE)
#           [,1]      [,2]      [,3]
# [1,] 1.0000000 0.3990872 0.2584569
# [2,] 0.3990872 1.0000000 0.5900067
# [3,] 0.2584569 0.5900067 1.0000000
cor(X3)
#           [,1]      [,2]      [,3]
# [1,] 1.0000000 0.3990872 0.2584569
# [2,] 0.3990872 1.0000000 0.5900067
# [3,] 0.2584569 0.5900067 1.0000000

Answer 2

The only line that has to be added is X <- apply(X, 2, function(x){ x-mean(x) }) which just subtracts the mean column-wise:

X <- apply(X, 2, function(x){ x-mean(x) })     
cos.sim <- function(ix) 
{
  A = X[,ix[1]]
  B = X[,ix[2]]
  return(t(A)%*%B/sqrt(sum(A^2)*sum(B^2)))
}   
n <- ncol(X) 
cmb <- expand.grid(i=1:n, j=1:n) 
C <- matrix(apply(cmb,1,cos.sim),n,n)