The geometry of PLS1 explained properly: 10 key notes on mathematical properties of and some alternative algorithmic approaches to PLS1 modelling

Journal of Chemometrics

Volumn 28, Issue 3, 2014, Pages 168-180

  Indahl, Ulf G ^a  

^a NORWEGIAN UNIVERSITY OF LIFE SCIENCES   (Norway)

Author keywords

Bidiagonalization; Change of coordinates and bases; Orthogonal and non orthogonal weights; PLS1 algorithms; QR factorization; Reorthogonalization; Scores and projections; Singular value decomposition; Truncation

Indexed keywords

FACTORIZATION; ITERATIVE METHODS; LEAST SQUARES APPROXIMATIONS; MATLAB;

BIDIAGONALIZATION; CHANGE OF COORDINATE AND BASE; NON-ORTHOGONAL; ORTHOGONAL AND NON-ORTHOGONAL WEIGHT; ORTHONORMAL; PLS1 ALGORITHM; QR FACTORIZATIONS; REORTHOGONALIZATIONS; SCORE AND PROJECTION; TRUNCATION;

SINGULAR VALUE DECOMPOSITION;

EID: 84897618381     PISSN: 08869383     EISSN: 1099128X     Source Type: Journal    
DOI: 10.1002/cem.2589     Document Type: Article

References (31)

1. Wold S, Martens H, Wold H. The multivariate calibration method in chemistry solved by the PLS method. In Proceedings of the Conference of Matrix Pencils, Ruhe A, Kågström B (eds), Lecture Notes in Mathematics. Springer Verlag: Heidelberg, 1983; 286-293.
2. Wold S, Ruhe A, Wold H, Dunn WJ, III. The collinearity problem in linear regression. The partial least squares (PLS) approach to generalized inverses. SIAM J. Scientific Comput 1984; 5: 735-743.
3. Martens H, Naes T. Multivariate Calibration (2nd edn). Wiley: Chichester, UK, 1989.
4. Wold S, Sjöström M, Eriksson L. PLS-regression: a basic tool of chemometrics. Chemom. Intell. Lab. Syst 2001; 58: 109-130.
5. Manne R. Analysis of two partial-least-squares algorithms for multivariate calibration. Chemom. Intell. Lab. Syst. 1987; 2: 187-197.
6. Höskuldsson A. PLS regression methods. J. Chemom 1988; 2: 211-228.
7. Helland IS. On the structure of partial least squares regression. Commun. Statist.(Simul. Comput.) 1988; 17: 581-607.
8. Frank IE, Friedman JH. A statistical view of some chemometrics regression tools. Technometrics 1993; 35(2): 109-148.
9. Garthwaite PH. An interpretation of partial least squares. J. Am. Stat. Assoc. 1994; 89(425): 122-127.
10. ter Braak CJF, de Jong S. The objective function of partial least squares regression. J. Chemom. 1998; 12: 41-54.
11. Helland IS. Some theoretical aspects of partial least squares regression. Chemom. Intell. Lab. Syst. 2001; 58: 97-107.
12. Ergon R. PLS score-loading correspondence a bi-orthogonal factorization. J. Chemom. 2002; 16: 368-373.
13. Eldén L. Partial least-squares vs. Lanczos bidiagonalization-I:analysis of a projection method for multiple regression. Comput. Stat. Data Anal. 2004; 46: 11-31.
14. Boulesteix AL, Strimmer K. Partial least squares: a versatile tool for the analysis of high-dimensional genomic data. Brief. Bioinform. 2006; 8(1): 32-44.
15. Pell RJ, Ramos LS, Manne R. The model space in partial least squares regression. J. Chemom. 2007; 21: 165-172.
16. Wold S, Høy M, Martens H, Trygg J, Westad F, MacGregor J, Wise BM. The PLS model space revisited. J. Chemom. 2009; 23: 67-68.
17. Bro R, Eldén L. PLS works. J. Chemom. 2009; 23: 69-71.
18. Ergon R. Re-interpretation of NIPALS results solves PLSR inconsistency problem. J. Chemom. 2009; 23: 72-75.
19. Manne R, Pell RJ, Ramos LS. The PLS model space: the inconsistency persists. J. Chemom. 2009; 23: 76-77.
20. Andersson M. A comparison of nine PLS1 algorithms. J. Chemom. 2009; 23: 518-529.
21. Ergon R. Finding Y-relevant part of X by use of PCR and PLSR model reduction methods. J. Chemom. 2007; 21: 537-546.
22. de Jong S. SIMPLS: an alternative approach to partial least squares regression. Chemom. Intell. Lab. Syst. 1993; 18: 251-263.
23. Golub GH, Kahan W. Calculating the singular values and pseudoinverse of a matrix. SIAM J. Numer. Anal. 1965; 2: 205-224.
24. Phatak A, de Jong S. The geometry of partial least squares. J. Chemom. 1997; 11: 311-338.
25. Higham NJ. Efficient algorithms for computing the condition number of a tridiagonal matrix. SIAM J. Sci. Stat. Comput. 1986; 7(1): 150-165.
26. Paige C, Saunders M. LSQR: an algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Softw. 1982; 8: 43-71.
27. Simon HD. The Lanczos algorithm with partial reorthogonalization. Math. Comp. 1984; 42(165): 115-142.
28. Björck Å. Numerical Methods for Least Squares Problems. SIAM: Philadelphia, 1996.
29. Larsen RM. Lanczos bidiagonalization with partial reorthogonalization. Technical Report, DAIMI PB-357, Department of Computer Science, Aarhus University, September 1998.
30. Faber NM, Ferré J. On the numerical stability of two widely used PLS algorithms. J. Chemom. 2002; 22: 101-105.
31. Björck Å. Available from: http://www.mai.liu.se/~akbjo/ (accessed Jan 7th, 2014).