Wishart: Wishart distribution

In mixAK: Multivariate Normal Mixture Models and Mixtures of Generalized Linear Mixed Models Including Model Based Clustering

Wishart distribution

Description

Wishart distribution

\mbox{Wishart}(\nu, \boldsymbol{S}),

where \nu are degrees of freedom of the Wishart distribution and \boldsymbol{S} is its scale matrix. The same parametrization as in Gelman (2004) is assumed, that is, if \boldsymbol{W}\sim\mbox{Wishart}(\nu,\,\boldsymbol{S}) then

\mbox{E}(\boldsymbol{W}) = \nu \boldsymbol{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.

Usage

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

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),

where p is number of rows and columns of the matrix \boldsymbol{W}.

In the univariate case, \mbox{Wishart}(\nu,\,S) is the same as \mbox{Gamma}(\nu/2, 1/(2S)).

Generation of random numbers is performed by the algorithm described in Ripley (1987, pp. 99).

Value

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 \boldsymbol{W}) in rows is returned.

Author(s)

Arnošt Komárek [e-mail]

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.

Examples

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]))

mixAK documentation built on Sept. 17, 2024, 1:06 a.m.