Generalized rank annihilation

In this section we focus on methods for the quantitation of a compound in the presence of an unknown interference without the requirement that this interference should be identified first or its spectrum should be estimated. Hyphenated methods are the main application domain. The methods we discuss are generalized rank annihilation method (GRAM) and residual bilinearization (RBL). 

Generalized rank annihilation factor analysis (GRAFA)... 

When several analytes have to be determined, this procedure needs to be repeated for each analyte. Because this algorithm requires that a PCA is calculated for each considered value of k, RAFA is computationally intensive. Sanchez and Kowalski [34] introduced generalized rank annihilation factor analysis (GRAFA). 

M.J.P. Gerritsen, H. Tanis, B.G.M. Vandeginste and G. Kateman, Generalized rank annihilation factor analysis, iterative target transformation factor analysis and residual bilinearization for the quantitative analysis of data from liquid-chromatography with photodiode array detection. Anal. Chem., 64 (1992) 2042-2056. 

E. Sanchez and B.R. Kowalski, Generalized rank annihilation factor analysis. Anal. Chem., 58... 

Chemometric methods can greatly increase the number of analyzable peaks in MDLC in particular, the generalized rank annihilation method (GRAM) can quantify overlapping peaks by deconvoluting the combined signal to those of each dimension. Standards with precise retention time are required, and there must be some resolution in both dimensions [60,61]. 

Windig, W. and Antalek, B., Direct exponential curve resolution algorithm (Decra) — a novel application of the generalized rank annihilation method for a single spectral mixture data set with exponentially decaying contribution profiles, Chemom. Intell. Lab. Syst., 1997, 37, 241-254. 

Three-way calibrations methods, such as the generalized rank annihilation method (GRAM) and parallel factor analysis (PARAFAC), are becoming increasingly prevalent tools to solve analytical challenges. The main advantage of three-way calibration is estimation of analyte concentrations in the presence of unknown, uncalibrated... 

An alternative method to solving Equation 12.3 is to reduce both Rj and R2 to square, nonsingular, nonidentity matrices by projecting each matrix independently onto the space formed jointly by the two matrices. This permits calculation of A and F, via the QZ algorithm [23] and, by extension, relative concentration estimates, Z, and estimates of the true underlying factors in the X- and Y-ways. This is known as the generalized rank annihilation method (GRAM) [24, 25],... 

Li, S. and Gemperline, P.J., Generalized rank annihilation method using similarity transformations, Anal. Chem., 64, 599-607, 1992. 

Generalized Rank Annihilation Method as per Wilson, Sanchez, and Kowalski. %INPUT... 

The basic theory behind the generalized rank annihilation method is that the rank reduction can be re-expressed and automated. A scalar y (relative concentration of the analyte in the unknown sample) is sought such that the matrix pencil... 

A practical numerical issue in the use of the generalized rank annihilation method is that, at certain times, complex-valued solutions may arise. Means have been provided, though, for eliminating this problem by simple similarity transformations [Faber 1997, Li et al. 1992],... 

Given an array X of size I x J x K, two slices in, say, the third mode are needed in order to be able to use the generalized rank annihilation method. These may be formed as weighted averages of all the slices. A sensible way to define two such samples is to determine two slices that preserve the total variation in X maximally in a least squares sense. Additionally these two slices must be within an I -dimensional subspace (R is the number of components in the PARAFAC model) in the first and second mode in order to maximize directly the appropriateness of the span of the data matrices. Thus, two slices Gi and G2 of size R x R are sought such that these are representative of the variation in X. This may be accomplished in a least squares sense by fitting a Tucker3 model with dimensionality R x R x 2,... 

The basic principle of the generalized rank annihilation method has been (re-)invented several times, e.g., in signal processing under the name ESPRIT [Roy Kailath 1989] and... 

The first article of Sanchez and Kowalski [1986] on generalized rank annihilation factor analysis is purely based on equations, but later articles of the same group on spectral curve resolution contain line plots of the estimated spectra after the rank annihilation analysis [Sanchez et al. 1987, Ramos et al. 1987], Sanchez [et al. 1987] is mainly about simulated examples. See Figures 8.6 and 8.7. 

Figure 8.6. The line plot shows chromatograms resulting from a rank annihilation analysis of LC-UV data of a mixture of aromatic chemicals as used in [Ramos et al. 1987]. Reprinted from Journal of Chromatography, 385, Ramos S, Sanchez E, Kowalski B, Generalized rank annihilation method. II. Analysis of bimodal chromatographic data, 165-180, Copyright (1987), with permission from...

A set of standards was prepared containing five different concentrations of Pb2+ (see first column of Table 10.3). Also a set of mixtures containing a fixed amount of interferents (all interferents Co2+, Mn2+, Ni2+ and Zn2+ present at 0.5 /xM each), and varying amounts of Pb2+ (again at five concentrations) was available. Each standard was used to calibrate each mixture, resulting in 25 calibration models with the structure of Equation (10.5). Generalized rank annihilation was used to fit the PARAFAC model to obtain the concentrations of the analyte Pb2+ in the mixtures. The results are reported in Table 10.3. 

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