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The results by soft modeling multivariate curve resolution methods often are not unique and are questionable because of the rotational ambiguity. It means a range of feasible solutions equally fits experimental data and fulfills the constraints. Regarding to chemometric literature, the reduction of the rotational ambiguity in multivariate curve resolution problems is a major challenge in order to construct effective chemometric methods. It is worth to study the effects of applying constraints on the reduction of the rotational ambiguity, since it can help us to choose the useful constraints for multivariate curve resolution methods for analyzing data sets. The aim of this work is to demonstrate the impact of monotonicity and unimodality constraints on the full set of all feasible, nonnegative solutions. We compared the results of two constraints in different two- and three-component systems. To reach this goal, two simulated kinetic and equilibrium data sets are used. Moreover, an experimental data set related to a mixture of two uniprotic acid solutions at different pH values as a model for equilibrium systems is used to extend the discussions to real cases. It is shown in this work that monotonicity is a meaningful chemistry-based constraint which in some cases is more effective than unimodality. • Reduction of both AFS-sets by the monotonicity and unimodality. constraints in SMCR. • Computation of monotone-unimodal areas in outer polygon for U-Space. • The results of two constraints, monotonicity and unimodality, in different two- and three-component systems were compared. [ABSTRACT FROM AUTHOR]

The boundaries of the set of feasible bilinear solutions for three-component systems have been estimated in the presence of noise, as a sequel of a previous report on two-component data sets. Simulations involving a significant degree of overlap among component profiles in both matrix modes have been analyzed using FACPACK and the recently described sensor-wise N-BANDS model. The latter computes the upper and lower envelopes of the set of profiles showing maximum and minimum intensity at each sensor. For three and more components, the graphical representation of the extreme profiles corresponding to maximum and minimum signal contribution function provided by methods such as MCR-BANDS and N-BANDS do not enclose, in general, the full set of feasible profiles. The present results were obtained under non-negativity as the only constraint on the profiles, and were successfully compared with those obtained using a noise addition procedure. A three-component experimental system was also analyzed regarding the combined effect of rotational ambiguity and instrumental noise. • The combined effect of rotational ambiguity and noise is studied. • Sensor wise N-BANDS and FACPACK algorithms are applied. • Results for a simulated three-component system are compared. • An experimental system is analyzed using both models. [ABSTRACT FROM AUTHOR]

Multivariate curve resolution methods suffer from the non-uniqueness of the solutions of the nonnegative matrix factorization 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. • The strongly varying restrictiveness of the equality constraint in MCR computations is analyzed by two estimators. • The estimator values are visualized in the area of feasible solutions (AFS) by means of an additional color coordinate. • The additional AFS color coordinate improves the information content and can be generalized to other indicator variables. [ABSTRACT FROM AUTHOR]