Incorporating information on variables

All projection Procrustes methods may be treated as a generalized orthogonal rotation, part of which comprises the projection. It follows that after these types of Procrustes analyses, the embedded reference systems as well as the configurations themselves, coexist in the same P-dimensional space. Then, the reference system can be approximated in any sub-space from which biplot representations may be derived for all the orthogonal and projection Procrustes methods. In two-set problems, the relevant sub-space will be that which contains X. In K-sets problems the exhibited space is the centre of interest. Within this basic framework, there are many possibilities, depending less on the particular Procrustes method used, than on the multivariate method of deriving X from Y and on the types of variable — quantitative, nominal categorical, ordered categorical. It would be an impossible task to describe all the variants in detail but, fortunately, the same basic principles are valid in all cases. This chapter presents an overview of the more important special cases and illustrates methodology through an example.