Wishart: Wishart distribution

Description Usage Arguments Details Value Value for dWISHART Value for rWISHART Author(s) References See Also Examples

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.

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

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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)

Arno<c5><a1>t Kom<c3><a1>rek 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.

See Also

rWishart.

Examples

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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 May 29, 2017, 9:20 p.m.

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