On Hidden Rank Deficiency in MCR Problems

Abstract

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.