  A method for dimension reduction with clustering, classification, or
discriminant analysis is introduced. This mixture model-based approach is based
on fitting generalized hyperbolic mixtures on a reduced subspace within the
paradigm of model-based clustering, classification, or discriminant analysis. A
reduced subspace of the data is derived by considering the extent to which
group means and group covariances vary. The members of the subspace arise
through linear combinations of the original data, and are ordered by importance
via the associated eigenvalues. The observations can be projected onto the
subspace, resulting in a set of variables that captures most of the clustering
information available. The use of generalized hyperbolic mixtures gives a
robust framework capable of dealing with skewed clusters. Although dimension
reduction is increasingly in demand across many application areas, the authors
are most familiar with biological applications and so two of the five real data
examples are within that sphere. Simulated data are also used for illustration.
The approach introduced herein can be considered the most general such approach
available, and so we compare results to three special and limiting cases.
Comparisons with several well established techniques illustrate its promising
performance.
