# Leibniz MMS Days 2019 - Abstract

**Neymeyr, Klaus**

*How nonnegative matrix factorizations can help to decompose spectral mixture data*

Nonnegative matrix factorization is a key tool for solving the blind source separation (BSS) problem. BSS has the aim to reconstruct the unknown nonnegative source terms from observable mixture data assuming a multilinear mixing model.

Here, we consider the bilinear Lambert-Beer “mixing” model in application to the spectral observation of a chemical reaction system. So-called multivariate curve resolution (MCR) methods serve to recover the pure component information from the spectral mixture data. However, MCR methods suffer from the intrinsic ambiguity of such pure component factorizations. Typically, there are no unique nonnegative solutions for the spectra and the concentration profiles of the chemical components.

In this talk we present the multivariate curve resolution problem for chemical spectral data. We discuss a systematic approach for representing the sets of feasible pure component factors in the form of the area of feasible solutions. Some applications are presented from an interdisciplinary cooperation with the Leibniz Institute for Catalysis (LIKAT), e.g., for the rhodium catalyzed hydroformylation process.

*References:*

[1] K. Neymeyr, M. Sawall, On the set of solutions of the nonnegative matrix factorization problem, SIAM J. Matrix Anal. Appl. 39, 1049-1069 (2018).

[2] H. Schröder, M. Sawall, C. Kubis, A. Jürß, D. Selent, A. Brächer, A. Börner, R. Franke, K. Neymeyr, Comparative multivariate curve resolution study in the area of feasible solutions, Chemom. Intell. Lab. Syst. 163, 55-63 (2017).

[3] M. Sawall, A. Jürß, H. Schröder, K. Neymeyr, On the analysis and computation of the area of feasible solutions for two-, three- and four-component systems. Book contribution in volume 30 of Data Handling in Science and Technology, “Resolving Spectral Mixture”, Chapter 5, pages 135-184, Ed. Cyril Ruckebusch, Elsevier 2016.

*Software:*http://www.math.uni-rostock.de/facpack/