Hi Christian and thanks for your message. 
 
>From reading "standard" mixture model books, I don't recall any of them talking directly about shape, orientation, and volume. So, in the finite mixture context I'm not exactly sure what these terms even mean as they relate to the covariance matrix. Maybe I'm missing something but it seems that the Fraley and Raftery framework is different than any mixture approach I've read about. 
 
I have a three dimensional model. Thus, the covariance matrix will be 3 by 3 for each of the G classes. I want the covariance matrix to be unrestricted (i.e., each of the 6 elements free to be estimated), yet for each of the G classes to share a common covariance matrix. That is, Sigma_1=Sigma_2=...=Sigma_G, where the elements are themselves unrestricted. 
 
My problem is translating the structure of the covariance matrix to the Fraley and Raftery framework. Reading their work I kept thanking they would have a table that translated the structure of the covariance matrix to their method of parameterization. You mention that it is the most intuitive framework, and I would agree if what was interested was the volume, shape, orientation, and distribution. My impression though is that people think in terms of the covariance structure. Where are these terms even defined? 
 
Anyway, thanks for your help. Any insight would be greatly appreciated. 
Ken


Christian Hennig <fm3a004@math.uni-hamburg.de> wrote:
Dear Ken,

in principle you have all relevant informations already in your mail.
As far as I know, the parameterization of Fraley and Raftery is the most 
intuitive one. I don't know for which kind of application you need 
direct parameterization,
but in my experience the parameters volume, shape and orientation are 
more interesting in most applications than the direct values of Sigma_k.

However, not all possible structures seem to be implemented. Your examples
are not, I suspect:

> 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. 

This would be VEE. If you assume det(Sigma_1)=1 (which is necessary for your
parameterization to be identified), then sigma^2 is lambda, i.e., 
the volume parameter, and Sigma_1 would be the remaining matrix product.
However, VEE is not implemented. You may mail to Chris Fraley and ask why...
You see that the problem is not the parameterization, but the fact that 
VEE is missing in mclust.

(It is somewhat confusing the you use I for the covariance matrix, because
emclust uses this letter for a covariance matrix, which is the identity 
matrix.)


> 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 do not really understand this. Do you want to assume that the elements of
Sigma_1 should be pairwise different? Why do you need such an assumption?
That's not a very favourable choice for estimation, I think, and it would 
be estimated by VEE as well (which would yield such a solution with
probability 1), if it would be implemented.

Best,
Christian

***********************************************************************
Christian Hennig
Fachbereich Mathematik-SPST/ZMS, Universitaet Hamburg
hennig@math.uni-hamburg.de, http://www.math.uni-hamburg.de/home/hennig/
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ich empfehle www.boag-online.de


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