Multiclass partial least squares discriminant analysis: Taking the right way—A critical tutorial

Vityaz at the Crossroads by Viktor Mikhailovich Vasnetsov, 1882. Decision making is an important part of our lives, and chemometrics contributes to making choices more purposeful. Sometimes choosing the wrong way can lead to serious trouble. As written on the stone in the illustration, “If you ride to the left, you will lose your horse, if you ride to the right, you will lose your head.” Partial least squares discriminant analysis (PLS‐DA) is an enormously popular method in various scientific areas: genomics, proteomics, metabolomics, and in food and pharmaceutical sciences. In this paper, we provide a detailed PLS‐DA theory that could be used as an itinerary in this perilous world. A distinct feature of this theory is that it does not utilize the PLS scores but is entirely based on the predicted dummy responses. We demonstrate that the results of the direct multi‐class PLS‐DA can be presented in a straightforward way by projecting the response matrix on the “super‐score” space by means of Principal Component Analysis. Two approaches to discrimination are considered: the hard and the soft way of allocation. Correspondingly, two versions of PLS‐DA are presented: the conventional hard PLS‐DA and the newly introduced soft PLS‐DA that seems to be a novel approach in chemometrics. The quality of classification is assessed using figures of merit (sensitivity, specificity, and efficiency). It is shown how these characteristics are used for the selection of the model complexity. A number of practical problems are scrutinized, such as unbalanced sizes of classes, comparison of discriminant and class‐modeling approaches, and authentication by the “one against all” strategy. The paper is illustrated by real‐world and simulated examples.