Natural duality in minimal constrained self modeling curve resolution

Self modeling curve resolution (SMCR) was introduced by Lawton and Sylvestre (LS) [Technometrics 1971; 13: 617–633] to decompose raw spectroscopic data of two component systems into product of two physically interpretable profile matrices provided that both concentrations and absorbances are non‐negative, accepting both as minimal constraints. Later Borgen et al. in 1985–86 [Anal. Chim. Acta 1985; 174: 1–26; Microchim. Acta 1986; 11: 63–73] generalized LS method for three‐component systems with the same minimal constraints. The concepts of Borgen were rather difficult to understand and to implement, that is why several chemometricians turned to developing approximation methods. Very recently, Rajkó and István [J. Chemom. 2005; 19: 448–463] have revisited Borgen's method and they have given clearer interpretation and used computational geometry tools to find inner and outer polygons. In the meantime Henry [Chemom. Intell. Lab. Syst. 2005; 77: 59–63] has introduced the duality relationship, but he has described it only for multivariate receptor modeling of compositional data of airborne pollution. Generalization of his duality principle will be given in this paper for universally using it in SMCR which is based on singular value decomposition (SVD) or principal component analysis (PCA). Copyright © 2007 John Wiley & Sons, Ltd.

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