从相关矩阵生成协方差矩阵

Question

I have a correlation matrix:

a <- matrix(c(1, .8, .8, .8, 1, .8, .8, .8, 1), 3)

##      [,1] [,2] [,3]
## [1,]  1.0  0.8  0.8
## [2,]  0.8  1.0  0.8
## [3,]  0.8  0.8  1.0

I would now like to create a covariance matrix from the correlation matrix. How can this be done in R?

I tried:

e1.sd <- 3
e2.sd <- 10
e3.sd <- 3
e.cov <- a * as.matrix(c, e1.sd, e2.sd, e3.sd) %*% t(as.matrix(c(e1.sd, e2.sd, e3.sd)))

But I get the error:

Error in a * as.matrix(c, e1.sd, e2.sd, e3.sd) %*% t(as.matrix(c(e1.sd,  : 
  non-conformable arrays

What am I doing wrong?

Answer 1

If you know the standard deviations of your individual variables, you can:

stdevs <- c(e1.sd, e2.sd, e3.sd)
#stdevs is the vector that contains the standard deviations of your variables
b <- stdevs %*% t(stdevs)  
# b is an n*n matrix whose generic term is stdev[i]*stdev[j] (n is your number of variables)
a_covariance <- b * a  #your covariance matrix

On the other hand, if you don't know the standard deviations, it's impossible.

Answer 2

require(MBESS)
a <- matrix(c(1,.8,.8,.8,1,.8,.8,.8,1),3)
> cor2cov(a,c(3,10,3))
    [,1] [,2] [,3]
[1,]  9.0   24  7.2
[2,] 24.0  100 24.0
[3,]  7.2   24  9.0

Answer 3

Building on S4M's answer, in base R, I would write this function:

cor2cov <- function(V, sd) {
  V * tcrossprod(sd)
}

tcrossprod will calculate the product of each combination of elements of the sd vector (equivalent to x %*% t(x) ), which we then (scalar) multiply by the variance-covariance matrix

Here's a quick check that the function is correct using the built in mtcars data set:

all.equal(
  cor2cov(cor(mtcars), sapply(mtcars, sd)), 
  cov(mtcars)
)

Answer 4

The answer marked as correct is wrong.

The correct solution seems to be the one provided by MBESS package, so see the post from dayne.

> a
     [,1] [,2] [,3]
[1,]  1.0  0.8  0.8
[2,]  0.8  1.0  0.8
[3,]  0.8  0.8  1.0
> b <- c(3,10,3)
> b %*% t(b)
     [,1] [,2] [,3]
[1,]    9   30    9
[2,]   30  100   30
[3,]    9   30    9
> c <- b %*% t(b)
> c %*% a
      [,1]  [,2]  [,3]
[1,]  40.2  44.4  40.2
[2,] 134.0 148.0 134.0
[3,]  40.2  44.4  40.2
> cor2cov(cor.mat=a, b )
     [,1] [,2] [,3]
[1,]  9.0   24  7.2
[2,] 24.0  100 24.0
[3,]  7.2   24  9.0
> a %*% c
     [,1] [,2] [,3]
[1,] 40.2  134 40.2
[2,] 44.4  148 44.4
[3,] 40.2  134 40.2
>