On the restrictiveness of equality constraints in multivariate curve resolution Mathias Sawall a , Somaye Vali Zade b , Christoph Kubis c , Henning Schr¨ oder a,c , Denise Meinhardt a,c , Alexander Br¨ acher d , Robert Franke d,e , Armin B¨ orner c , Hamid Abdollahi b , Klaus Neymeyr a,c a Universit¨ at Rostock, Institut f¨ ur Mathematik, Ulmenstraße 69, 18057 Rostock, Germany b Faculty of Chemistry, Institute for Advanced Studies in Basic Sciences, 45195-1159 Zanjan, Iran c Leibniz-Institut f¨ ur Katalyse, Albert-Einstein-Straße 29a, 18059 Rostock d Evonik Performance Materials GmbH, Paul-Baumann Straße 1, 45772 Marl, Germany e Lehrstuhl f¨ ur Theoretische Chemie, Ruhr-Universit¨ at Bochum, 44780 Bochum, Germany Abstract Multivariate curve resolution methods suffer from the non-uniqueness of the solutions of the nonnegative matrix fac- torization problem. The solution ambiguity can be considerably reduced by equality constraints in the form of known spectra or concentration profiles. Two measures are suggested that indicate the impact of the equality constraints. The representation of these measures in the area of feasible solutions show strong variations in the restrictiveness of equality constraints. The measures are tested for a three-component model problem and experimental data sets from the hydroformylation process and a catalyst cluster formation. Key words: multivariate curve resolution, rotational ambiguity, area of feasible solutions, Borgen plot, equality constraint. 1. Introduction Multivariate curve resolution (MCR) methods aim at the extraction of pure component information from spectral mixtures [13, 3, 15, 14, 19]. Typically, the spectral data is given on a time × frequency-grid in form of an absorption matrix D ∈ R k×n . Therein, k is the number of spectra and n is the number of frequency channels. The Lambert-Beer law predicts that D can, at least approximately, be represented as the product of a concentration matrix C ∈ R k×s and a matrix of pure component spectra S ∈ R n×s , see [15, 11, 14]. Therein s is the number of chemical components. The resulting pure component factors C and S are nonnegative matrices. Typically, many nonnegative matrix factorizations D ≈ CS T exist. The chemically correct factors are among all these nonnegative matrix factorizations. These facts are well known under the keyword of the rotational ambiguity of the factorization [30, 1]. The so-called Area of Feasible Solutions (AFS) is a low-dimensional representation of this ambiguity [6, 25, 5, 21]. Two AFS sets exist for a data matrix D. The first one represents the possible concentration profiles and is denoted by M C . The second one contains representations of the possible pure component spectra and is denoted by M S . The rotational ambiguity can be considerably reduced by known spectra or concentration profiles. For instance, a pure component spectrum of a reactant or of a reaction product might be known or a concentration profile of a certain component can sometimes be measured. Such additional knowledge on the reaction system can be fed into the factorization process in form of so-called equality constraints. The effect of equality constraints on the reduction of the rotational ambiguity can be very different. The relative position of the known component in the AFS related to the certain polygons, namely the inner polygon and the outer polygon, explains the behavior. The aim of this paper is to suggest two measures of ambiguity reduction. To each point of the AFS we assign a characteristic number that estimates the size of the AFS after addition of the equality constraints. As the size of the AFS correlates to some extent to the rotational ambiguity we consider these measures as ambiguity estimators. These characteristic numbers can also be considered as restrictiveness measures. A relatively large reduced AFS shows that the equality constraint cannot significantly reduce the rotational ambiguity. The profile that is locked by the equality constraint is compatible to many other solutions. In contrast to this a small reduced AFS means that the choice of the equality constraint is very restrictive for the factorization problem. We summarize these notions as follows: An equality constraint strongly reduces the AFS if it has a high restrictiveness. Similarly, a weak AFS reduction corresponds to a low restrictiveness. Then the fixed profile is compatible to many other spectral or concentration profiles. December 20, 2019