A number of reviews of mixture models, such as Fraley and Raftery (2002) describe three common models, in terms of their geometric interpretation:
- All mixture components are spherical and of the same size
- Equal variance
- Unconstrained variance
Helpfully, though for some beginners like me confusingly, MCLUST in R provides a wider range of model names that include the three common models above, according to the MCLUST documentation:
multivariate mixture "EII" = spherical, equal volume "VII" = spherical, unequal volume "EEI" = diagonal, equal volume and shape "VEI" = diagonal, varying volume, equal shape "EVI" = diagonal, equal volume, varying shape "VVI" = diagonal, varying volume and shape "EEE" = ellipsoidal, equal volume, shape, and orientation "EVE" = ellipsoidal, equal volume and orientation "VEE" = ellipsoidal, equal shape and orientation "VVE" = ellipsoidal, equal orientation "EEV" = ellipsoidal, equal volume and equal shape "VEV" = ellipsoidal, equal shape "EVV" = ellipsoidal, equal volume "VVV" = ellipsoidal, varying volume, shape, and orientation
Which of the MCLUST model names do the three common models described by Fraley and Raftery correspond to?
My educated guesses, assuming that varying volume and shape (and orientation) are simply less-constrained parameterizations of the covariance matrix, and therefore equal volume and shape (and orientation) are the same for equal variance, are:
- All mixture components are spherical and the same size: EII
- Equal variance across mixture components: EEE
- Unconstrained variance across mixture components: VVV
I ask because in my area of research / field, Latent Profile Analysis (LPA) (or Latent Class Analysis [LCA]) are commonly used to do mixture modeling as part of a latent variable model approach.
In this approach, analysts commonly specify models in which the measured variables' residual variances and covariances are constrained to be the same across profiles (or classes) or to be freely-estimated across classes.
More generally than about this specific question, I'm searching for advice about how to interpret these geometric model descriptions to the way models are specified in an LPA / LCA approach.