[R] MCLUST Covariance Parameterization.

[Christian Hennig]

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 at math.uni-hamburg.de, http://www.math.uni-hamburg.de/home/hennig/
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