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It is shown that the sum of squares of the standardised scores of all non-zero principal components (PCs) equals the squared Mahalanobis distance. A new distance measure, the reduced Mahalanobis distance, is explored in which the number of PCs retained is less than the full rank model. It is illustrated by both one-class and two-class classifiers. Linear discriminant analysis can be employed as a soft model, and principal component analysis using the pooled variance-covariance matrix is introduced as an intermediate view between conjoint and disjoint models allowing linear discriminant analysis to be used on these reduced rank models. By choosing the most discriminatory PCs, it can be shown that the reduced Mahalanobis distance has superior performance over the full rank model for discriminating via soft models. Copyright © 2016 John Wiley & Sons, Ltd.

Partial least squares discriminant analysis (PLS-DA) has been available for nearly 20 years yet is poorly understood by most users. By simple examples, it is shown graphically and algebraically that for two equal class sizes, PLS-DA using one partial least squares (PLS) component provides equivalent classification results to Euclidean distance to centroids, and by using all nonzero components to linear discriminant analysis. Extensions where there are unequal class sizes and more than two classes are discussed including common pitfalls and dilemmas. Finally, the problems of overfitting and PLS scores plots are discussed. It is concluded that for classification purposes, PLS-DA has no significant advantages over traditional procedures and is an algorithm full of dangers. It should not be viewed as a single integrated method but as step in a full classification procedure. However, despite these limitations, PLS-DA can provide good insight into the causes of discrimination via weights and loadings, which gives it a unique role in exploratory data analysis, for example in metabolomics via visualisation of significant variables such as metabolites or spectroscopic peaks. Copyright © 2014 John Wiley & Sons, Ltd.