# Wishart distribution Wishart distribution [REMOVE_ME]$$ \mbox{Wishart}(\nu, \boldsymbol{S}),Wishart(nu, S), [REMOVE_ME_2]$$ where $nu$ are degrees of freedom of the Wishart distribution and $S$ is its scale matrix. The same parametrization as in Gelman (2004) is assumed, that is, if $W~Wishart(nu,S)$ then [REMOVE_ME]$$ \mbox{E}(\boldsymbol{W}) = \nu \boldsymbol{S}.E(W) = nu*S. [REMOVE_ME_2]$$ Prior to version 3.4-1 of this package, functions `dWISHART` and `rWISHART` were called as `dWishart` and `rWishart`, respectively. The names were changed in order to avoid conflicts with `rWishart` from a standard package `stats`. ## Description Wishart distribution $$ \mbox{Wishart}(\nu, \boldsymbol{S}),Wishart(nu, S), $$ where $nu$ are degrees of freedom of the Wishart distribution and $S$ is its scale matrix. The same parametrization as in Gelman (2004) is assumed, that is, if $W~Wishart(nu,S)$ then $$ \mbox{E}(\boldsymbol{W}) = \nu \boldsymbol{S}.E(W) = nu*S. $$ Prior to version 3.4-1 of this package, functions `dWISHART` and `rWISHART` were called as `dWishart` and `rWishart`, respectively. The names were changed in order to avoid conflicts with `rWishart` from a standard package `stats`. ## Details The density of the Wishart distribution is the following $$ f(\boldsymbol{W}) = \Bigl(2^{\nu\,p/2}\,\pi^{p(p-1)/4}\,\prod_{i=1}^p\Gamma\bigl(\frac{\nu + 1 - i}{2}\bigr)\Bigr)^{-1}\,|\boldsymbol{S}|^{-\nu/2}\,|\boldsymbol{W}|^{(\nu - p - 1)/2}\,\exp\Bigl(-\frac{1}{2}\mbox{tr}(\boldsymbol{S}^{-1}\boldsymbol{W})\Bigr),%f(W) = (2^{nu*p/2} * pi^{p*(p-1)/4} * prod[i=1]^p Gamma((nu + 1 -i)/2))^{-1}* |S|^{-nu/2} * |W|^{(nu - p - 1)/2} * exp(-0.5*tr(S^{-1}*W)), $$ where $p$ is number of rows and columns of the matrix $W$. In the univariate case, $Wishart(nu,S)$ is the same as $Gamma(nu/2, 1/(2*S)).$ Generation of random numbers is performed by the algorithm described in Ripley (1987, pp. 99). ```r dWISHART(W, df, S, log=FALSE) rWISHART(n, df, S) ``` ## Arguments - `W`: Either a matrix with the same number of rows and columns as `S` (1 point sampled from the Wishart distribution) or a matrix with `ncol` equal to `ncol`*`(ncol+1)`/2 and `n` rows (`n` points sampled from the Wishart distribution for which only lower triangles are given in rows of the matrix `W`). - `n`: number of observations to be sampled. - `df`: degrees of freedom of the Wishart distribution. - `S`: scale matrix of the Wishart distribution. - `log`: logical; if `TRUE`, log-density is computed ## Returns Some objects. ## Value for dWISHART A numeric vector with evaluated (log-)density. ## Value for rWISHART If `n` equals 1 then a sampled symmetric matrix W is returned. If `n` > 1 then a matrix with sampled points (lower triangles of $W$) in rows is returned. ## References Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (2004). **Bayesian Data Analysis, Second edition**. Boca Raton: Chapman and Hall/CRC. Ripley, B. D. (1987). **Stochastic Simulation**. New York: John Wiley and Sons. ## See Also `rWishart`. ## Author(s) Arnošt Komárek arnost.komarek@mff.cuni.cz ## Examples ```r set.seed(1977) ### The same as gamma(shape=df/2, rate=1/(2*S)) df <- 1 S <- 3 w <- rWISHART(n=1000, df=df, S=S) mean(w) ## should be close to df*S var(w) ## should be close to 2*df*S^2 dWISHART(w[1], df=df, S=S) dWISHART(w[1], df=df, S=S, log=TRUE) dens.w <- dWISHART(w, df=df, S=S) dens.wG <- dgamma(w, shape=df/2, rate=1/(2*S)) rbind(dens.w[1:10], dens.wG[1:10]) ldens.w <- dWISHART(w, df=df, S=S, log=TRUE) ldens.wG <- dgamma(w, shape=df/2, rate=1/(2*S), log=TRUE) rbind(ldens.w[1:10], ldens.wG[1:10]) ### Bivariate Wishart df <- 2 S <- matrix(c(1,3,3,13), nrow=2) print(w2a <- rWISHART(n=1, df=df, S=S)) dWISHART(w2a, df=df, S=S) w2 <- rWISHART(n=1000, df=df, S=S) print(w2[1:10,]) apply(w2, 2, mean) ## should be close to df*S (df*S)[lower.tri(S, diag=TRUE)] dens.w2 <- dWISHART(w2, df=df, S=S) ldens.w2 <- dWISHART(w2, df=df, S=S, log=TRUE) cbind(w2[1:10,], data.frame(Density=dens.w2[1:10], Log.Density=ldens.w2[1:10])) ### Trivariate Wishart df <- 3.5 S <- matrix(c(1,2,3,2,20,26,3,26,70), nrow=3) print(w3a <- rWISHART(n=1, df=df, S=S)) dWISHART(w3a, df=df, S=S) w3 <- rWISHART(n=1000, df=df, S=S) print(w3[1:10,]) apply(w3, 2, mean) ## should be close to df*S (df*S)[lower.tri(S, diag=TRUE)] dens.w3 <- dWISHART(w3, df=df, S=S) ldens.w3 <- dWISHART(w3, df=df, S=S, log=TRUE) cbind(w3[1:10,], data.frame(Density=dens.w3[1:10], Log.Density=ldens.w3[1:10])) ```

Wishart function