Generate n random matrices, distributed according
to the Cholesky factorization of a Wishart distribution with
parameters Sigma and df, \(W_p(Sigma, df)\)
(known as the Bartlett decomposition
in the context of Wishart random matrices).
rCholWishart(n, df, Sigma)
| n | integer sample size. |
|---|---|
| df | numeric parameter, "degrees of freedom". |
| Sigma | positive definite \(p \times p\) "scale" matrix, the matrix parameter of the distribution. |
a numeric array, say R, of dimension
\(p \times p \times n\),
where each R[,,i] is a Cholesky decomposition of a sample
from the Wishart distribution \(W_p(Sigma, df)\). Based on a
modification of the existing code for the rWishart function.
Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis (3rd ed.). Hoboken, N. J.: Wiley Interscience.
Mardia, K. V., J. T. Kent, and J. M. Bibby (1979) Multivariate Analysis, London: Academic Press.
A. K. Gupta and D. K. Nagar 1999. Matrix variate distributions. Chapman and Hall.
# How it is parameterized: set.seed(20180211) A <- rCholWishart(1L, 10, 3 * diag(5L))[, , 1] A #> [,1] [,2] [,3] [,4] [,5] #> [1,] 6.732026 1.653646 1.53888250 1.840051 0.9673595 #> [2,] 0.000000 3.605802 -0.02454492 1.376210 3.3807539 #> [3,] 0.000000 0.000000 3.88257652 1.754274 -0.6427580 #> [4,] 0.000000 0.000000 0.00000000 4.634089 3.5618443 #> [5,] 0.000000 0.000000 0.00000000 0.000000 5.4805713 set.seed(20180211) B <- rInvCholWishart(1L, 10, 1 / 3 * diag(5L))[, , 1] B #> [,1] [,2] [,3] [,4] [,5] #> [1,] 0.17569 -0.1133604 -0.06754304 -0.03553973 0.02019099 #> [2,] 0.00000 0.2909057 -0.03454705 -0.01907355 -0.06546147 #> [3,] 0.00000 0.0000000 0.28079294 -0.13256269 0.05760802 #> [4,] 0.00000 0.0000000 0.00000000 0.21687541 -0.08522708 #> [5,] 0.00000 0.0000000 0.00000000 0.00000000 0.13422902 crossprod(A) %*% crossprod(B) #> [,1] [,2] [,3] [,4] [,5] #> [1,] 1.000000e+00 8.326673e-17 -1.665335e-16 -1.110223e-16 2.775558e-17 #> [2,] 6.245005e-17 1.000000e+00 -1.110223e-16 -1.110223e-16 5.551115e-17 #> [3,] 7.719519e-17 -2.775558e-17 1.000000e+00 3.816392e-17 -1.387779e-17 #> [4,] 8.326673e-17 -5.551115e-17 -3.330669e-16 1.000000e+00 1.110223e-16 #> [5,] 1.110223e-16 2.220446e-16 -1.110223e-16 2.220446e-16 1.000000e+00 set.seed(20180211) C <- chol(stats::rWishart(1L, 10, 3 * diag(5L))[, , 1]) C #> [,1] [,2] [,3] [,4] [,5] #> [1,] 6.732026 1.653646 1.53888250 1.840051 0.9673595 #> [2,] 0.000000 3.605802 -0.02454492 1.376210 3.3807539 #> [3,] 0.000000 0.000000 3.88257652 1.754274 -0.6427580 #> [4,] 0.000000 0.000000 0.00000000 4.634089 3.5618443 #> [5,] 0.000000 0.000000 0.00000000 0.000000 5.4805713