
Hello all (especially MCLUS users). 

I'm trying to make use of the MCLUST package by C. Fraley and A. Raftery. My problem is trying to figure out how the (model) identifier (e.g, EII, VII, VVI, etc.) relates to the covariance matrix. The parameterization of the covariance matrix makes use of the method of decomposition in Banfield and Rraftery (1993) and Fraley and Raftery (2002) where 

Sigma_k = lambda_k*D_k*A_k*D_k^' 

where Sigma_k is the covariance matrix for the kth (k=1,...,G), lambda_k is the kth groups constant of proportionality, D_k is the orthogonal matrix of eigenvectors for the kth group, and A_k is a diagonal matrix whose elements are proportional to the eigenvalues. The parameterization of the covariance matrix Sigma_k depends on the distribution (whether spherical, diagonal, or ellipsoidal), volume (equal or variable), shape (equal or variable), and orientation (coordinate axes, equal, or variable). The distribution, volume, shape and orientation are a function of  lambda_k, D_k, and A_k. Thus, depending on whether or not these values are constant across class defines Sigma_k. 

What I'm trying to figure out is how the distribution, volume, shape, and orientation relate to Sigma_k. As far as the parameterization of Sigma_k, what do "distribution," "volume," "shape," and "orientation" even mean. Does a table exist of how these values relate to the Sigma_k? I know a table exists in the MCLUST software manual on the MCLUST website, but this table doesn't relate the values of distribution, volume, shape, and orientation to Sigma_k directly, only to how Sigma_k would be parameterized (this isn’t helpful unless you know what distribution, volume, shape and orientation  mean in terms of the within class covariance matrix) So, just what do the distribution, volume, shape, and orientation mean in the context of Sigma_k?

 
What do the distribution, volume, shape, and orientation mean for a Sigma_k=sigma^2*I where I is a p by p covariance matrix, sigma^2 is the constant variance and Sigma_1=Sigma_2=…=Sigma_G. What about when a Sigma_k=sigma^2_k*I, or when Sigma_1=Sigma_2=…=Sigma_G in situations where each element of the (constant across class) covariance matrix is different?

I would say I have a pretty good understanding of finite mixture modeling, but nothing I've read (expect the works cited in the 2002 JASA paper) talks about parameterizing the Sigma_k matrix in such a way. It would be nice to specify a structure directly for Sigma_k (as most books talk about). Any help on this issue would be greatly appreciated. 

Thanks, 
Ken.

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