Rank annihilation factor analysis

Generalized rank annihilation factor analysis (GRAFA)... 

In 1978, Ho et al. [33] published an algorithm for rank annihilation factor analysis. The procedure requires two bilinear data sets, a calibration standard set Xj and a sample set X . The calibration set is obtained by measuring a standard mixture which contains known amounts of the analytes of interest. The sample set contains the measurements of the sample in which the analytes have to be quantified. Let us assume that we are only interested in one analyte. By a PCA we obtain the rank R of the data matrix X which is theoretically equal to 1 + n, where rt is the number of interfering compounds. Because the calibration set contains only one compound, its rank R is equal to one. 

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... 

J. Ohman, P. Geladi and S. Wold, Residual bilinearization, part 2 Application to HPLC-diode array data and comparison with rank annihilation factor analysis. J. Chemom., 4 (1990) 135-146. 

The processing of EEM-data, obtained for mixtures, consists of the determination of the number of components, their concentrations and pure spectra. A very powerful method in that respect is rank annihilation factor analysis (RAFA). Based on RAF A, Ho et al. derived a quantitative method for the determination of one or more... 

McCue, M. and Malinowski, E.R., Rank annihilation factor-analysis of unresolved LC peaks, J. Chromatog. Sci., 21, 229-234, 1983. 

Seven simulated LC-UV/Vis DAS data matrices were constructed in MATLAB 5.0 (MathWorks Inc., Natick, MA). Each sample forms a 25 x 50 matrix. The simulated LC and spectral profiles are shown in Figure 12.3a and Figure 12.3b, respectively. Spectral and chromatographic profiles are constructed to have a complete overlap of the analyte profile by the interferents. Three of the samples represent pure standards of unit, twice-unit, and thrice-unit concentration. These standards are designated SI, S2, and S3, respectively. Three three-component mixtures of relative concentrations of interferent 1 analyte interferent 2 are 1 1.5 0.5, 2 0.5 2, and 2 2.5 1, and these are employed for all examples. These mixtures are designated Ml, M2, and M3, respectively. An additional two-component mixture, 2 2.5 0, is employed as an example for rank annihilation factor analysis. This sample is designated M4. In most applications, normally distributed random errors are added to each digitized channel of every matrix. These errors are chosen to have a mean of zero and a standard deviation of 0.14, which corresponds to 10% of the mean response of the middle standard. In the rank annihilation factor analysis (RAFA) examples, errors were chosen to simulate noise levels of 2.5 and 5% of the mean response of S2. In some PARAFAC examples, the noise level was chosen to be 30% of the mean response of the second standard. 

Rank annihilation methods employ eigenvalue-eigenvector analyses for direct determination of analyte concentration with or without intrinsic profile determination. With the exception of rank annihilation factor analysis, these methods obtain a direct, noniterative solution by solving various reconstructions of the generalized eigenvalue-eigenvector problem. 

Lorber, A., Features of quantifying chemical composition from two-dimensional data arrays by the rank annihilation factor analysis method, Anal. Chem., 57, 2395-2397, 1985. 

Booksh, K.S. and Kowalski, B.R., Comments on the DATa ANalysis (DATAN) algorithm and rank annihilation factor analysis for the analysis of correlated spectral data, J. Chemom., 8, 287-292, 1994. 

The situation changes dramatically when two-way bilinear data are adopted to attack this kind of problem. Ho et al. [31] introduced rank annihilation factor analysis (RAFA) method to quantify the desired analytes in the presence of unknown interferents. The iterative procedure of RAFA was modified by Lorber [32], a direct approach with the generalized... 

Rank annihilation factor analysis treats the special situation in which there are only two slices, where each slice contains the second-order measurements of one sample. One sample is usually a pure standard containing only the analyte of interest in a known concentration and the other is an unknown sample containing unknown interferents as well as the analyte of interest. The purpose in rank annihilation factor analysis is to estimate the concentration of the analyte of interest in the unknown sample. It is assumed that the signal from the analyte of interest corresponds to a rank-one component with the intensity proportional to the concentration. Thus the standard sample, say Xi, can be described... 

Figure 6.8. Example of the use of rank annihilation factor analysis for determining the concentration of tryptophane using fluorescence excitation-emission spectroscopy. In the top left plot the unknown sample is shown. It contains three different analytes. The standard sample (only tryptophane) is shown in the top right plot. In the lower right plot, it is shown that the smallest significant (third from top) singular value of the analyte-corrected unknown sample matrix reaches a clear minimum at the value 0.6. In the lower left plot the unknown sample is shown with 0.6 times the standard sample subtracted. It is evident that the contribution from the analyte is practically absent.

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. 

Nprgaard L, Ridder C, Rank annihilation factor analysis applied to flow injection analysis with photodiode-array detection, Chemometrics and Intelligent Laboratory Systems, 1994a, 23, 107-114. 

In the mid-1980s a new technique called rank annihilation factor analysis (RAFA and GRAM) was developed which performed this process non-iteratively [42, 43]. Now an exact solution can be estimated directly on the sample matrix if the pure component response matrix of the analyte of interest is known. The third technique uses a range of concentrations yielding several pure component data matrices. This approach, called trilinear decomposition (TLD), stabilizes the prediction of the unknown sample due to the range of known response matrices... 

Sanchez, E. Kowalski, B.R. (1986). Generalized Rank Annihilation Factor Analysis, Analytical Chemistry, Vol.58, pp. 496-499, ISSN 0003-2700 Szabadai Z. (2005). Bazele fizico-chimice ale metodelor de control analitic al medicamentelor, Editura Mirton, Vol. II., pp. 4.38-4.96, ISBN 973-661-677-0, Timisoara, Romania... 

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