Partial least squares discriminant analysis for chemometrics and metabolomics: How scores, loadings, and weights differ according to two common algorithms

Two common algorithms for partial least squares discriminant analysis developed by Wold (NIPALS) and by Martens are compared. Although their classification performance is identical, their projections into variable space (often called scores) and into object space (loadings and weights) are quite different. Martens' algorithm can be visualised as a rotation in variable space, but Wold's results in quite complex distortions. Most software presents scores plots using the Wold algorithm but fails to appreciate that variable space is distorted, so scores from both algorithms are different. Weights, which can be obtained from both methods, are identical, although loadings (as commonly defined) from the Wold algorithm differ. The paper illustrates the two methods graphically to review the difference between these two methods.