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

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A major problem of multivariate curve resolution methods is the underlying non-uniqueness of the pure component decompositions. This raises the question how a chemical experiment should be designed so that the solution ambiguity is as small as possible. Changes of the reaction conditions belong to the possible variations whereas for a fixed chemical reaction system, the pure component spectra appear to be unchangeable. The paper investigates and discusses the possibility to design a chemical experiment in a way that minimizes the ambiguity of the factorization. The analysis identifies regions of the spectra that are responsible for a small ambiguity. Certain sources are identified that are responsible for an increased ambiguity by means of an a posteriori analysis. This results in recommendations how to construct spectral measurements incorporating a reduced factorization ambiguity. Furthermore, lower bounds on an unavoidable base level of ambiguity are specified under the constraint of fixed reactants. The problem analysis is accompanied by investigations of several experimental data sets. KEYWORDS ambiguity, design of experiments, multivariate curve resolutionJournal of Chemometrics. 2020;34:e3159.wileyonlinelibrary.com/journal/cem

T = (\begin{array}{llll} 1 & x_{1} & ⋯ & x_{s - 1} \\ 1 \\ ⋮ & normalW \\ 1 \end{array}) .

The AFS for the factor S reads with x =( x 1 ,…, x s −1 ) T and W

M_{S} = false{x \in ℝ^{s - 1} : exists W \in ℝ^{false(s - 1 false) \times false(s - 1 false)} such that rank false(T false) = s and C, S \geq 0 false} .

Its pendant for C is denoted

M_{C}

F_{S}

and

F_{C}

(the name derives from first polygon ) and the polyhedra INNPOL

I_{S}

and

I_{C}

(the name derives from inner polygon as FIRPOL includes INNPOL).…”

Section: Borgen Plots and Rotational Ambiguity In Afs Constructionsmentioning

confidence: 99%

“…Instead, the non‐trivial non‐uniqueness, the so‐called rotational ambiguity, is a major problem . The AFS, see Section for details, is a low‐dimensional representation of the possible columns of either C or S and has extensively been studied in recent years …”

Section: Introductionmentioning

confidence: 99%

Multivariate curve resolution methods and the design of experiments

Sawall

Kubis

Schröder

et al. 2019

Self Cite

“…Later, Rajko and Istvan recast the famous analytical generalization of the self‐modeling curve resolution (SMCR) method made by Borgen et al for three‐component systems. Borgen plot as a geometric construction was specially designed to analyze non‐negative data sets, but recently Jürß et al proposed the concept of generalized Borgen plots for spectral data which contains small negative entries …”

Section: Introductionmentioning

confidence: 99%

Trilinear self‐modeling curve resolution using Borgen‐Rajkó plot

Omidikia

Abdollahi

Kompany‐Zareh

et al. 2019

Modern analytical instruments provide measurement data arrays with full of hidden and redundant information. Multivariate curve resolution (MCR) techniques decompose data set to physic‐chemically meaningful abstract profiles. On the other hand, for such data matrices, Borgen‐Rajkó self‐modeling curve resolution (SMCR) techniques reveal all possible solutions analytically under the minimal assumption. Although Lawton‐Sylvestre (LS) and Borgen methods have been proposed for the non‐negative curve resolution of two‐component and three‐component systems, there is still a great deal of interest to include further restrictions on the Borgen‐Rajkó SMCR. As modern hyphenated analytical instruments produce multiway (eg, three‐way) arrays, multiway analysis (eg, trilinear decomposition) was received much more popularity by chemists. This contribution is decicated to the extension of the Borgen algorithm to the trilinear data sets. The Borgen method incorporating trilinearity constraint is approached in an analogy to the trilinear Lawton‐Sylvester method. The proposed analytical triadic decomposition is applied to the simulated three‐way arrays with full rank and rank overlap (a type of rank deficiency) loadings. Finally, the proposed algorithm is further exemplified with a three‐component spectrofluorimetric study of acid‐base equilibrium of the pyridoxine. Investigating feasible regions of the simulated and experimental three‐component arrays reveal interesting additional information.

“…The topic of this paper is the simultaneous construction of pairs of dual Borgen plots. Borgen plots represent in a low‐dimensional form the possible nonnegative factors of a spectral data matrix whose rows contain a series of spectra recorded from a chemical reaction system. For a detailed introduction to the underlying multivariate curve resolution (MCR) problem, we refer to the first part of this paper and the literature cited therein.…”

Section: Introductionmentioning

confidence: 99%

“…For a detailed introduction to the underlying multivariate curve resolution (MCR) problem, we refer to the first part of this paper and the literature cited therein. See also previous works for an introduction and definition of the so‐called Area of Feasible Solutions (AFS). Its geometric construction is developed by Borgen and Kowalski in their landmark paper from 1985 …”

Section: Introductionmentioning

confidence: 99%

Simultaneous construction of dual Borgen plots. II: Algorithmic enhancement for applications to noisy spectral data

Sawall

Moog

Kubis

et al. 2018

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Borgen plots are low‐dimensional representations of the set of all nonnegative factorizations of spectral data matrices. Classical Borgen plots are limited to nonnegative data and can be constructed for the spectral factor or for the concentration profiles. In the first part of this paper, a simultaneous construction of the two dual Borgen plots is presented, which intensively exploits the underlying duality principles. The second part introduces algorithmic enhancements which make the simultaneous Borgen plot construction possible for noisy experimental data matrices which can contain small negative matrix entries. The new method is tested for FT‐IR spectral data from the Rhodium catalyzed hydroformylation process. The results are compared to those by the FACPACK‐implementation of the polygon inflation method.