Chapter 5 - On the Analysis and Computation of the Area of Feasible Solutions for Two-, Three-, and Four-Component Systems

Abstract

Abstract The area of feasible solutions (AFS) is a low-dimensional representation of all possible concentration factors or spectral factors in nonnegative factorizations of a given spectral data matrix. The AFS analysis is a powerful methodology for the exploration of the rotational ambiguity inherent to the multivariate curve resolution problem. Up to now the AFS has been studied for two-, three-, and four-component systems: 1. The AFS for two-component systems was introduced by Lawton and Sylvestre in 1971. For these two-dimensional problems the AFS can be constructed analytically. 2. For three-component systems the AFS can either be constructed geometrically (classical approach by Borgen and Kowalski from 1985) or it can be computed by numerical algorithms. Various computational techniques have been suggested by different groups in the recent past. 3. For four-component systems a first numerical method for its computation has been published recently. A new polyhedron inflation algorithm is under development. In this chapter we explain the underlying concepts of the AFS theory and its contribution to a deepened understanding of the multivariate curve resolution problem. A survey is given on various methods for the computation of the AFS for two-, three-, and four-component systems. The focus is on methods which approximate the boundary of the AFS for three-component systems by inflating polygons and for four-component systems by inflating polyhedrons. Several numerical examples are discussed and the M at L ab -toolbox FACPACK for these AFS-computations is presented.