Hi Ken, it seems that you want equal covariance matrices, which means equal, but free volume, orientation and shape. That's "EEE", and it *is* implemented.
I still do not really understand your "translation" problem. All information is in the formula which appeared in your first posting: Sigma_k = lambda_k*D_k*A_k*D_k^' That is, if you have lambda (volume>0), D (orientation, orthogonal) and A (shape, diagonal>0), you can compute Sigma and if you have Sigma, you can compute lambda, A, and D by spectral decomposition. Hope this helps, Christian On Mon, 7 Jun 2004, [EMAIL PROTECTED] wrote: > 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 <[EMAIL PROTECTED]> 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 > [EMAIL PROTECTED], http://www.math.uni-hamburg.de/home/hennig/ > ####################################################################### > ich empfehle www.boag-online.de > > > __________________________________________________ > > > > [[alternative HTML version deleted]] > > ______________________________________________ > [EMAIL PROTECTED] mailing list > https://www.stat.math.ethz.ch/mailman/listinfo/r-help > PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html > *********************************************************************** Christian Hennig Fachbereich Mathematik-SPST/ZMS, Universitaet Hamburg [EMAIL PROTECTED], http://www.math.uni-hamburg.de/home/hennig/ ####################################################################### ich empfehle www.boag-online.de ______________________________________________ [EMAIL PROTECTED] mailing list https://www.stat.math.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html
