On generalized Borgen plots. I: From convex to affine combinations and applications to spectral dataSpectra

Abstract

In 1985, Borgen and Kowalski (Anal. Chim. Acta 1985; 174: 1‐26) published their landmark paper on the geometric construction of feasible regions for nonnegative factorizations of spectral data matrices for three‐component systems. These geometric constructions are called Borgen plots. Borgen plots are principally restricted to nonnegative data and are sometimes considered as analytical tool. Major contributions to this theory have been given by Rajkó. In contrast to these geometric constructions, numerical methods to compute the so‐called area of feasible solutions (AFS) have been studied by Golshan et al. (Anal. Chem. 2011; 83 (3): 836‐841) and by Sawall et al. (J. Chemom. 2013; 27: 106‐116). These numerical methods can even treat spectral data, which include slightly negative components.In this work, the concept of generalized Borgen plots is introduced for spectral data, which are polluted by small negative entries. The analysis is not restricted to three‐component systems but can be applied to general s‐component systems. Generalized Borgen plots are identical to the classical Borgen plots for nonnegative data. The analysis in this work also bridges the gap between the different scalings (Borgen norms) used for AFS computations.The algorithmic procedure of generalized Borgen plots for three‐component systems and its implementation in the FAC‐PACK software are described in the second part of this paper. Copyright © 2015 John Wiley & Sons, Ltd.