Matrix Methods for Multiple Measures

Abstract

We return now to the context of Chap. 3 where rotational rescaling was introduced and then exploited in the manner of principal components via the princomp() function facility of R. We generalize that work to encompass linear transformation of multiple measures to generate a secondary set of what we will call virtual variates. We reserve the term virtual variates for situations in which the transformation is reversible so that the original multiple measures can be reproduced from the secondary set through a suitable transformation. In such circumstances, the transformation effectively provides a particular perspective from which to view the data without loss of information. With principal components, the perspective was one of independent (uncorrelated) dimensions (Raykov 2008; Timm 2002; Hair et al. 2010). In the earlier venture, we did not undertake to reverse the principal component transformation, but will do so in the course of current consideration. Virtual variates from principal components are also not the only interesting perspectives of this nature to be explored (Johnson and Wichern 2007; Sengupta 2003; Mukhopadhyay 2009), and linear transformations will entail several additional operators in R.