Generalized rank annihilation method. I: Derivation of eigenvalue problems

Abstract

SUMMARYRank 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.

The price paid for the second‐order advantage when using the generalized rank annihilation method (GRAM)

“…Equations (6) and (7) are consistent with the parametrization favoured by Faber et al [11]. This parametrization can be made unique by imposing constraints on X and Y.…”

“…Lorber's definition of multivariate sensitivity has been generalized by Faber et al [10] to multilinear data. The last step in (11) has been proved by Bauer et al [13].…”

Generalized rank annihilation method: standard errors in the estimated eigenvalues if the instrumental errors are heteroscedastic and correlated

“…14,16 The only difference lies in the way the left and right singular vectors are estimated. In the current method this is done by decomposing the column and row augmented matrices which can be recast into equation (12) using the reconstruction equations for the pure profiles.…”

“…* Recently a reformulation of the eigenvalue problem has led to a simplification of the error propagation. 16 Irrespective of the differences in approach taken in these theoretical contributions, they all have in common that the measurement error is assumed to be uncorrelated and homoscedastic. The advantage of this assumption is that it leads to a relatively simple expression for the standard error in the estimated eigenvalue.…”

An optimization approach for fitting canonical tensor decompositions.

“…Faber, Bro, and Hopke [15] compared ALS with a number of competing algorithms: direct trilinear decomposition (DTLD) [16,17,14,18,29,31,42], alternating trilinear decomposition (ATLD) [50], self-weighted alternating trilinear decomposition (SWATLD) [10,11], pseudo alternating least squares (PALS) [9], alternating coupled vectors resolution (ACOVER) [23], alternating slice-wise diagonalization (ASD) [22], and alternating coupled matrices resolution (ACOMAR) [30]. It is shown that while none of the algorithms is better than ALS in terms of the quality of solution, ASD may be an alternative to ALS when the computation time is a priority.…”