A comparison of nine PLS1 algorithms

• Published in: Journal of chemometrics Vol. 23; no. 10; pp. 518 - 529
• Main Author: Andersson, Martin
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 01.10.2009; Wiley; Wiley Subscription Services, Inc
• Subjects: Algorithms; Analytical chemistry; Chemistry; Comparative studies; comparison; Exact sciences and technology; Foods; General and physical chemistry; General. Nomenclature, chemical documentation, computer chemistry; Least squares method; Mathematical analysis; Mathematical models; numerical; Numerical analysis; Numerical stability; PLS; Regression; regression vector; Spectrometers; speed; stability; Statistical methods; Theory of reactions, general kinetics. Catalysis. Nomenclature, chemical documentation, computer chemistry; Vectors (mathematics); Speed; algorithms; Stability; comparison; regression vector; Spectrometer; Algorithm; PLS regression; numerical; PLS; Calculation; Models; Chemometrics; Numerical stability; Food
• ISSN: 0886-9383; 1099-128X; 1099-128X
• DOI: 10.1002/cem.1248

Nine PLS1 algorithms were evaluated, primarily in terms of their numerical stability, and secondarily their speed. There were six existing algorithms: (a) NIPALS by Wold; (b) the non‐orthogonalized scores algorithm by Martens; (c) Bidiag2 by Golub and Kahan; (d) SIMPLS by de Jong; (e) improved kernel PLS by Dayal; and (f) PLSF by Manne. Three new algorithms were created: (g) direct‐scores PLS1 based on a new recurrent formula for the calculation of basis vectors yielding scores directly from X and y; (h) Krylov PLS1 with its regression vector defined explicitly, using only the original X and y; (i) PLSPLS1 with its regression vector recursively defined from X and the regression vectors of its previous recursions. Data from IR and NIR spectrometers applied to food, agricultural, and pharmaceutical products were used to demonstrate the numerical stability. It was found that three methods (c, f, h) create regression vectors that do not well resemble the corresponding precise PLS1 regression vectors. Because of this, their loading and score vectors were also concluded to be deviating, and their models of X and the corresponding residuals could be shown to be numerically suboptimal in a least squares sense. Methods (a, b, e, g) were the most stable. Two of them (e, g) were not only numerically stable but also much faster than methods (a, b). The fast method (d) and the moderately fast method (i) showed a tendency to become unstable at high numbers of PLS factors. Copyright © 2009 John Wiley & Sons, Ltd. Nine PLS1 algorithms were evaluated in terms of their numerical stability and their speed. It was found that the models of Bidiag2, PLSF and the new Krylov PLS1 algorithm were deviating from the precise PLS solution. They were numerically unstable and suboptimal in a least‐squares sense. The most stable were: NIPALS, the non‐orthogonalized PLS1 algorithm, the improved kernel PLS algorithm, and the new direct‐scores PLS1 algorithm. The last two were not only numerically stable but also 2–4 times faster.