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Generalized rank annihilation method. I: Derivation of eigenvalue problems
Author(s) -
Faber N. M.,
Buydens L. M. C.,
Kateman G.
Publication year - 1994
Publication title -
journal of chemometrics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.47
H-Index - 92
eISSN - 1099-128X
pISSN - 0886-9383
DOI - 10.1002/cem.1180080206
Subject(s) - eigenvalues and eigenvectors , rank (graph theory) , degeneracy (biology) , principal component analysis , degenerate energy levels , mathematics , matrix (chemical analysis) , annihilation , calibration , transformation (genetics) , eigendecomposition of a matrix , physics , combinatorics , statistics , quantum mechanics , gene , biology , composite material , bioinformatics , biochemistry , materials science , chemistry
Rank annihilation factor analysis (RAFA) is a method for multicomponent calibration using two data matrices simultaneously, one for the unknown and one for the calibration sample. In its most general form, the generalized rank annihilation method (GRAM), an eigenvalue problem has to be solved. In this first paper different formulations of GRAM are compared and a slightly different eigenvalue problem will be derived. The eigenvectors of this specific eigenvalue problem constitute the transformation matrix that rotates the abstract factors from principal component analysis (PCA) into their physical counterparts. This reformulation of GRAM facilitates a comparison with other PCA‐based methods for curve resolution and calibration. Furthermore, we will discuss two characteristics common to all formulations of GRAM, i.e. the distinct possibility of a complex and degenerate solution. It will be shown that a complex solution‐contrary to degeneracy‐should not arise for components present in both samples for model data.