rwc (version 1.12)

dGenWish: Density of the (singular) Generalized Wishart distribution

Description

Computes the density of the (possibly singular) Generalized Wishart distribution with null-space equal to the space spanned by the "one" vector. This corresponds to the case considered by McCullagh (2009) and Hanks and Hooten (2013).

Usage

dGenWish(Dobs, Sigma, df,log=FALSE)

Value

A numeric likelihood or log-likelihood

Arguments

Dobs

An observed squared-distance matrix.

Sigma

The covariance parameter of the Generalized Wishart.

df

An integer specifying the degrees of freedom.

log

Logical. If True, then the log-likelihood is computed.

Author

Ephraim M. Hanks

Details

Following McCullagh (2009), the likelihood can be computed by considering any contrast matrix L of full rank, and with n-1 rows and n columns, where n is the number of columns of 'Dobs'. If

Dobs ~ GenWish(Sigma,df,1)

is distributed as a generalized Wishart distribution with kernel (null space) equal to the one vector, and df degrees of freedom, then the likelihood can be computed by computing the likelihood of

L(-Dobs)L' ~ Wishart(L(2*Sigma)L',df)

Additionally, following Srivastava (2003), this likelihood holds (up to a proportionality constant) in the singular case where df<n.

Following this formulation, the log-likelihood computed here (up to an additive constant) is

-df/2*log|L(2*Sigma)L'| -1/2*tr( (L(2*Sigma)L')^-1 L(-D)L' )

References

McCullagh 2009. Marginal likelihood for distance matrices. Statistica Sinica 19: 631-649.

Srivastava 2003. Singular Wishart and multivariate beta distributions. The Annals of Statistics. 31(5), 1537-1560.

Hanks and Hooten 2013. Circuit theory and model-based inference for landscape connectivity. Journal of the American Statistical Association. 108(501), 22-33.

Examples

ras=raster(nrow=30,ncol=30)
extent(ras) <- c(0,30,0,30)
values(ras) <- 1
int=ras
cov.ras=ras

## get precision matrix of entire graph
B.int=get.TL(int)
Q.int=get.Q(B.int,1)

## get precision at a few nodes
Phi=get.Phi(Q.int,obs.idx=1:20)
S=ginv(as.matrix(Phi))

## simulate distance matrix
Dsim=rGenWish(df=20,Sigma=S)
image(Dsim)

## calculate log-likelihood
ll=dGenWish(Dsim,S,df=20,log=TRUE)
ll