Wishart {mixAK}R Documentation

Wishart distribution

Description

Wishart distribution

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

E(W) = nu*S.

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(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).

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 W) in rows is returned.

Author(s)

Arnošt Komárek arnost.komarek[AT]mff.cuni.cz

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.

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]))

[Package mixAK version 0.4 Index]