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).
An observed squared-distance matrix.
The covariance parameter of the Generalized Wishart.
An integer specifying the degrees of freedom.
Logical. If True, then the log-likelihood is computed.
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'
A numeric likelihood or log-likelihood
Ephraim M. Hanks
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.
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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
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