Correlated Components

Principal components analysis is a much used and practical technique for analysing multivariate data, finding a particular set of linear compounds of the variables under consideration, such that covariances between all pairs are 0. An alternative view is that when the variables are considered as axes in a Cartesian coordinate system, then principal components analysis is the particular orthogonal rotation of the axes that makes all the pairwise covariances equal to 0. It is this view that is taken here, but instead of finding the rotation that makes all covariances equal to 0, an orthogonal rotation is found that maximizes the sum of the covariances. The rotation is not unique, except for the two or three component case, and so another criterion can be used alongside so that it too can also be optimized. The motivation is that two highly correlated components will tend to measure the same latent variable but with interesting differences because of the orthogonality between them. Theory is given for identifying the correlated components as well as algorithms for finding them. Two illustrative examples are provided, one involving gene expression data and the other consumer questionnaire data. Copyright (C) 2016 John Wiley & Sons, Ltd.