A fast polygon inflation algorithm to compute the area of feasible solutions for three‐component systems. I: concepts and applications

Abstract

The multicomponent factorization of multivariate data often results in nonunique solutions. The so‐called rotational ambiguity paraphrases the existence of multiple solutions that can be represented by the area of feasible solutions (AFS). The AFS is a bounded set that may consist of isolated subsets. The numerical computation of the AFS is well understood for two‐component systems and is an expensive numerical process for three‐component systems. In this paper, a new fast and accurate algorithm is suggested that is based on the inflation of polygons. Starting with an initial triangle located in a topologically connected subset of the AFS, an automatic extrusion algorithm is used to form a sequence of growing polygons that approximate the AFS from the interior. The polygon inflation algorithm can be generalized to systems with more than three components. The efficiency of this algorithm is demonstrated for a model problem including noise and a multicomponent chemical reaction system. Further, the method is compared with the recent triangle‐boundary‐enclosing scheme of Golshan, Abdollahi, and Maeder (Anal. Chem. 2011, 83, 836–841). Copyright © 2013 John Wiley & Sons, Ltd.

On generalized Borgen plots. I: From convex to affine combinations and applications to spectral dataSpectra

“…Similarly to the identity matrix T -1 T, a permutation matrix and its inverse (transposed matrix) can be inserted in the truncated SVD. This shows that the set of all feasible spectra is completely determined by the set of all possible first rows to T [8]. Additionally, these s degrees of freedom in the first row of T can be further reduced by one degree of freedom by the row sum scaling condition for A [4,6].…”

“…Later, the RS-scaling is used (e.g., in [6]). This scaling does not require a singular value decomposition of D. In contrast to this, the numerical algorithms to compute the AFS work within the basis of left and right singular vectors of D [8]. With respect to this basis, the FSV-scaling can be used.…”

“…(The idea of considering slightly negative components within the factorization problem has already been used in [11] to handle noisy data.) (3) The gap between the geometric constructive Borgen plots [4,6,15] and the numerical AFS algorithms in [7][8][9] with their different factor scalings is bridged. An explicit transformation between row sum scaling and first singular vector scaling is derived.…”

“…The genuinely numerical procedures to compute the AFS in [7,8,[19][20][21] Similarly, a regular matrix T 2 R s s is used in order to define the factors…”

“…First, the triangle enclosure algorithm has been presented in 2011 [7]. Second, the polygon inflation algorithm has been suggested in 2013 [8,9]. These two numerical algorithms are based on a different scaling compared with the geometric construction in [6] so that the result can look quite different.…”