How To Apply the Complementarity and Coupling Theorems in MCR Methods: Practical Implementation and Application to the Rhodium-Catalyzed Hydroformylation

Abstract

Multivariate curve resolution techniques can be used in order to extract from spectroscopic data of chemical mixtures the contributions from the pure components, namely, their concentration profiles and their spectra. The curve resolution problem is by nature a matrix factorization problem, which suffers from the difficulty that the pure component factors are not unique. In chemometrics the so-called rotational ambiguity paraphrases the existence of numerous, feasible solutions. However, most of these solutions are not chemically meaningful. The rotational ambiguity can be reduced by adding additional information on the pure factors such as known pure component spectra or measured concentration profiles of the components. The complementarity and coupling theory (as developed in J. Chemometrics 2013 27, 106–116) provides a theoretical basis for exploiting such adscititious information in order to reduce the ambiguity. In this work the practical application of the complementarity and coupling theory is explained, a user-friendly MATLAB implementation is presented, and the techniques are applied to spectral data from the rhodium-catalyzed hydroformylation process.

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

“…The computational costs are low. Then in a second step, the duality theory, see, eg, literature, can be applied to compute the dual inner polygons and . These polygons are consistent to the weakened nonnegativity constraint.…”

On an SVD‐free approach to the complementarity and coupling theory A note on the elimination of unknowns in sums of dyadic products

“…with the column space operator col as defined in (6). Similarly, if s -1 concentration profiles are given by C 1 , then the spectrum of the complementary component is given by Figure 1.…”

“…In [6], Algorithm 1, the MATLAB code is provided for an implementation of the complementarity theory for the special case of s 0 = s -1. With the simplified form (10) or (11), the implementation is possible without referring to the SVD of D. In line one of Algorithm 1, the matrix C whose columns are all multiples of the desired complementary concentration profile c is constructed by a single command.…”

“…In fact, setting˛= -2 p 2 results in a = (2, 0) T . This is the first column of C in (6). Alternatively,˛= -p 2 gives a = (3, 1) T , which is the first column of Q C in the second factorization in (1).…”

“…The complementarity and coupling theory has already successfully been applied to model and experimental FT-IR data from the hydroformylation process in [6,13] and to UV-VIS data from the formation of hafnacyclopentene in [1,12]. Aspects of its implementation in its SVD-bound form are presented in [6]. Further, implications on the area of feasible solutions are treated in [13].…”