| % File src/library/stats/man/rWishart.Rd |
| % Part of the R package, https://www.R-project.org |
| % Copyright 2012-2014 R Core Team |
| % Distributed under GPL 2 or later |
| |
| \name{rWishart} |
| \alias{rWishart} |
| \title{Random Wishart Distributed Matrices} |
| \description{ |
| Generate \code{n} random matrices, distributed according to the |
| Wishart distribution with parameters \code{Sigma} and \code{df}, |
| \eqn{W_p(\Sigma, m),\ m=\code{df},\ \Sigma=\code{Sigma}}{W_p(Sigma, df)}. |
| } |
| \usage{ |
| rWishart(n, df, Sigma) |
| } |
| \arguments{ |
| \item{n}{integer sample size.} |
| \item{df}{numeric parameter, \dQuote{degrees of freedom}.} |
| \item{Sigma}{positive definite (\eqn{p\times p}{p * p}) \dQuote{scale} |
| matrix, the matrix parameter of the distribution.} |
| } |
| \details{ |
| If \eqn{X_1,\dots, X_m, \ X_i\in\mathbf{R}^p}{X1,...,Xm, Xi in R^p} is |
| a sample of \eqn{m} independent multivariate Gaussians with mean (vector) 0, and |
| covariance matrix \eqn{\Sigma}, the distribution of |
| \eqn{M = X'X} is \eqn{W_p(\Sigma, m)}. |
| |
| Consequently, the expectation of \eqn{M} is |
| \deqn{E[M] = m\times\Sigma.}{E[M] = m * Sigma.} |
| Further, if \code{Sigma} is scalar (\eqn{p = 1}), the Wishart |
| distribution is a scaled chi-squared (\eqn{\chi^2}{chi^2}) |
| distribution with \code{df} degrees of freedom, |
| \eqn{W_1(\sigma^2, m) = \sigma^2 \chi^2_m}{W_1(sigma^2, m) = sigma^2 chi[m]^2}. |
| |
| The component wise variance is |
| \deqn{\mathrm{Var}(M_{ij}) = m(\Sigma_{ij}^2 + \Sigma_{ii} \Sigma_{jj}).}{% |
| Var(M[i,j]) = m*(S[i,j]^2 + S[i,i] * S[j,j]), where S=Sigma.} |
| } |
| \value{ |
| a numeric \code{\link{array}}, say \code{R}, of dimension |
| \eqn{p \times p \times n}{p * p * n}, where each \code{R[,,i]} is a |
| positive definite matrix, a realization of the Wishart distribution |
| \eqn{W_p(\Sigma, m),\ \ m=\code{df},\ \Sigma=\code{Sigma}}{W_p(Sigma, df)}. |
| } |
| \references{ |
| Mardia, K. V., J. T. Kent, and J. M. Bibby (1979) |
| \emph{Multivariate Analysis}, London: Academic Press. |
| } |
| \author{Douglas Bates} |
| \seealso{ |
| \code{\link{cov}}, \code{\link{rnorm}}, \code{\link{rchisq}}. |
| } |
| \examples{ |
| ## Artificial |
| S <- toeplitz((10:1)/10) |
| set.seed(11) |
| R <- rWishart(1000, 20, S) |
| dim(R) # 10 10 1000 |
| mR <- apply(R, 1:2, mean) # ~= E[ Wish(S, 20) ] = 20 * S |
| stopifnot(all.equal(mR, 20*S, tolerance = .009)) |
| |
| ## See Details, the variance is |
| Va <- 20*(S^2 + tcrossprod(diag(S))) |
| vR <- apply(R, 1:2, var) |
| stopifnot(all.equal(vR, Va, tolerance = 1/16)) |
| } |
| \keyword{multivariate} |