A fast polygon inflation algorithm to compute the area of feasible solutions for three‐component systems. I: concepts and applications. (19th April 2013)

• Record Type: Journal Article
• Authors: Sawall, Mathias; Kubis, Christoph; Selent, Detlef; Börner, Armin; Neymeyr, Klaus
• Is Part Of: Journal of chemometrics. Volume 27:Number 5(2013:May)
• Page Start: 106
• Page End: 116
• Publication Date: 2013-04-19
• Subjects: factor analysis -- pure component decomposition -- nonnegative matrix factorization -- spectral recovery -- band boundaries of feasible solutions -- polygon inflation; Chemistry -- Mathematics -- Periodicals; Chemistry -- Statistical methods -- Periodicals; 542.85
• DOI: 10.1002/cem.2498
• Languages: English
• ISSNs: 0886-9383

Abstract

Abstract : The multicomponent factorization of multivariate data often results in nonunique solutions. The so‐called rotational ambiguity paraphrases the existence of multiple solutions that can be represented by the area of feasible solutions (AFS). The AFS is a bounded set that may consist of isolated subsets. The numerical computation of the AFS is well understood for two‐component systems and is an expensive numerical process for three‐component systems. In this paper, a new fast and accurate algorithm is suggested that is based on the inflation of polygons. Starting with an initial triangle located in a topologically connected subset of the AFS, an automatic extrusion algorithm is used to form a sequence of growing polygons that approximate the AFS from the interior. The polygon inflation algorithm can be generalized to systems with more than three components. The efficiency of this algorithm is demonstrated for a model problem including noise and a multicomponent chemical reaction system. Further, the method is compared with the recent triangle‐boundary‐enclosing scheme of Golshan, Abdollahi, and Maeder (Anal. Chem. 2011, 83, 836–841). Copyright © 2013 John Wiley & Sons, Ltd. Abstract : The rotational ambiguity of a model‐free MCR technique can be represented by the area of feasible solutions (AFS). In this paper, a new algorithm is presented for a fast numerical computation of the AFS for three‐component systems. The algorithm is based on an adaptive inflation of polygons.