The stats implementation of rWishart is in C and is very fast. It is often the case that we do not want a sample from the Wishart distribution, but rather from the inverse of it or from the Cholesky decomposition of a sample from the Wishart distribution. Or even from the inverse of the Cholesky decomposition of a draw from the Wishart distribution. Funnily enough (if you have a weird sense of humor), when you inspect the source code for the rWishart distribution, it generates the Cholesky decomposition and then multiplies it out. Meanwhile, drawing from the rWishart and then inverting or doing a Cholesky decomposition or whatever in R is just slow – comparatively.
This suggests some obvious efficiencies: perhaps, if we would rather have the Cholesky decomposition of the Wishart random matrix, we could tell the function to stop right there.
library('CholWishart')
set.seed(20180220)
A <- stats::rWishart(1,10,5*diag(4))[,,1]
set.seed(20180220)
B <- rInvWishart(1,10,.2*diag(4))[,,1]
set.seed(20180220)
C <- rCholWishart(1,10,5*diag(4))[,,1]
set.seed(20180220)
D <- rInvCholWishart(1,10,.2*diag(4))[,,1]How does rWishart(n, df, Sigma) work (supposing Sigma is a \(p \times p\) matrix)? First, it generates a sample from the Cholesky decomposition of a Wishart distribution with \(\Sigma = \mathbf{I}_p\). How this is done: on the \(i^{th}\) element of the main diagonal, draw from \(\sqrt{\chi_{p-i+1}^2}\). On the upper triangle of the matrix, draw from \(N(0,1)\) in each square. Then, this can be multiplied by the Cholesky decomposition of the provided Sigma to obtain the Cholesky factor of the desired sample from the Wishart random variable. The rWishart function multiplies it out. Therefore, if the Cholesky decomposition is desired, one only needs to stop there.
If \(X \sim \textrm{W}(\nu,\Sigma)\). Then we define the Inverse Wishart as \(X^{-1} = Y \sim \textrm{IW}(\nu , \Sigma^{-1})\). There are other parameterizations of this, mostly coming down to different ways of writing the \(\nu\) parameter - be aware of this when using any package drawing from the Inverse Wishart distribution. This comes up directly in Bayesian statistics. We are also interested in the Cholesky decomposition of this, as it is directly relevant to the generation of the matrix variate \(t\)-distribution. In this package it is done by taking the covariance matrix, inverting it, computing the Cholesky decomposition of the inverted covariance matrix, drawing the Cholesky factor of a Wishart matrix using that, and then inverting based on that (as finding \(\Psi^{-1}\) given the Cholesky factorization of \(\Psi\) is relatively fast). This can then be converted into the Cholesky factor of the Inverse Wishart if that is what is desired. This would be slow to do in R, but in C it is not so bad.
So here is what happens with the results of the above:
A %*% B
## [,1] [,2] [,3] [,4]
## [1,] 1.000000e+00 -2.775558e-17 -1.387779e-16 -5.551115e-17
## [2,] -4.718448e-16 1.000000e+00 1.387779e-17 0.000000e+00
## [3,] -1.249001e-16 1.387779e-17 1.000000e+00 -5.551115e-17
## [4,] 1.110223e-16 -1.110223e-16 0.000000e+00 1.000000e+00
crossprod(C) %*% crossprod(D) # note: we do not expect C = D^-1, we expect this!
## [,1] [,2] [,3] [,4]
## [1,] 1.000000e+00 -2.775558e-17 -2.081668e-16 -1.110223e-16
## [2,] -4.718448e-16 1.000000e+00 1.387779e-16 0.000000e+00
## [3,] -1.249001e-16 1.110223e-16 1.000000e+00 -1.110223e-16
## [4,] 1.110223e-16 -8.326673e-17 -8.326673e-17 1.000000e+00
crossprod(D) %*% A
## [,1] [,2] [,3] [,4]
## [1,] 1.000000e+00 -4.718448e-16 -1.249001e-16 1.110223e-16
## [2,] -2.775558e-17 1.000000e+00 1.110223e-16 -8.326673e-17
## [3,] -2.081668e-16 1.387779e-16 1.000000e+00 -8.326673e-17
## [4,] -1.110223e-16 0.000000e+00 -1.110223e-16 1.000000e+00
crossprod(C) %*% B
## [,1] [,2] [,3] [,4]
## [1,] 1.000000e+00 -2.775558e-17 -1.387779e-16 -5.551115e-17
## [2,] -4.718448e-16 1.000000e+00 1.387779e-17 0.000000e+00
## [3,] -1.249001e-16 1.387779e-17 1.000000e+00 -5.551115e-17
## [4,] 1.110223e-16 -1.110223e-16 0.000000e+00 1.000000e+00There is some roundoff error.