On the area of feasible solutions for rank‐deficient problems: I. Introduction of a generalized concept

Sawall, Mathias; Neymeyr, Klaus

doi:10.1002/cem.3316

Cited by 4 publications

(2 citation statements)

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“…Rank deficiency can also be well-understood by a geometric representation in the abstract U-and V -space, 19,20 which are spanned by the bases of left and right singular vectors of D. The letters U and V refer to the SVD D ¼ UΣV T . The columns of U contain the left singular vectors and the columns of V the right singular vectors.…”

Section: Geometric Analysis In the U-and V -Spacementioning

confidence: 99%

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A note on rank deficiency and numerical modeling

Neymeyr,

Sawall,

Andersons

2024

Journal of Chemometrics

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Linearly dependent concentration profiles of a chemical reaction can result in a spectral data matrix with a chemical rank smaller than the number of absorbing chemical species. Such a rank deficiency is problematic for a factor analysis as some information on the pure component spectra cannot be recovered from the mixture data. Matrix augmentation can break rank deficiencies and enable successful pure component recovery. In contrast to this, an artificial breakdown of a rank deficiency can be caused by a numerical finite precision simulation of the underlying kinetic model and can fake a successful MCR analysis. This work discusses the problem and points out some remedies.

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Section: Geometric Analysis In the U-and V -Spacementioning

confidence: 99%

“…Rank deficiency can also be well‐understood by a geometric representation in the abstract

U

‐ and

V

‐space, 19,20 which are spanned by the bases of left and right singular vectors of

D

. The letters

U

and

V

refer to the SVD

D = U normalΣ V^{T}

.…”

Section: Geometric Analysis In the U‐ And V‐spacementioning

confidence: 99%

A note on rank deficiency and numerical modeling

Neymeyr,

Sawall,

Andersons

2024

Journal of Chemometrics

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On Hidden Rank Deficiency in MCR Problems

Andersons,

Sawall,

Beese

et al. 2024

Journal of Chemometrics

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Pure component decomposition problems in chemometrics can be classified into rank‐regular and rank‐deficient problems. Rank‐deficient problems are characterized by a spectral data matrix that has a lower rank than the number of chemical species. However, it is possible that there exists rank‐regular factorization of the spectral data matrix, but none of these solutions can be interpreted chemically, and only a solution of the MCR problem with rank deficiency is chemically meaningful. Then we say that the underlying problem suffers from a hidden rank deficiency. In this paper, MCR problems with hidden rank deficiency are introduced and analyzed with several examples for problems of rank 2 and rank 3. The area of feasible solutions is determined with the help of additional constraints on the solution.

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