A comparison of nine PLS1 algorithms

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 kerne...

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Vydáno v:Journal of chemometrics Ročník 23; číslo 10; s. 518 - 529
Hlavní autor: Andersson, Martin
Médium: Journal Article
Jazyk:angličtina
Vydáno: Chichester, UK John Wiley & Sons, Ltd 01.10.2009
Wiley
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Témata:
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
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Popis
Shrnutí: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.
Bibliografie:istex:0D728252638AA50BF7F8884215A9C9072A02021B
ark:/67375/WNG-RFVG5T0X-N
ArticleID:CEM1248
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ISSN:0886-9383
1099-128X
1099-128X
DOI:10.1002/cem.1248