Partial least squares discrimination with heterogeneous covariance structures

Barker and Rayens 1 argued that partial least squares (PLS) is to be preferred over principal components analysis (PCA) when linear discrimination is the goal and dimension reduction is required as a first step. In particular, it is now known that when PLS is used as the dimension reduction tool, information involving Fisher's among-groups sums-of-squares and cross-products matrix is embedded in the structures extracted to achieve that reduction. Liu and Rayens 2 followed up with a formal proof for the superiority of PLS over PCA in the two-group case (with respect to formal misclassification probabilities) and pointed to a subclass of oriented PLS (OrPLS) 3 techniques that would always produce a lower misclassification rate than would PLS, for the two-group problem with like covariance matrices. This paper addresses the situation where these within-groups covariances are potentially not the same, and hence a single reduction step followed by linear discrimination may not be appropriate. A variety of techniques are compared by way of a large-scale simulation study, and a version of the ridged subclass studied by Liu and Rayens 2 emerges as the recommended approach from among those studied. An application aimed at distinguishing asymptomatic women at risk for Alzheimer's disease from women not at risk, based on functional magnetic resonance imaging (fMRI) scans, is also presented.