Generalized rank annihilation factor analysis GRAFA

1 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. 

In the next step, the rank is calculated of the difference matrix X = X - kX. For any value of k, the rank of X is equal to 1 + n, except for the case where k is exactly equal to the contribution of the analyte to the signal. In this case the rank of X is / - 1. Thus the concentration of the analyte in the unknown sample can be found by determining the k-value for which the rank of Xj is equal to / - 1. The amount of the analyte in the sample is then equal to kc where is the concentration of the analyte in the standard solution. In order to find this k-value Ho et al. proposed an iterative procedure which plots the eigenvalues of the least significant PC of X as a function of k. This eigenvalue becomes minimal when k exactly compensates the signal of the analyte in the sample. For other k-values the signal is under- or 

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). 

More than one analyte can be quantified simultaneously in the presence of interfering compounds. The required measurements are identical to RAFA a data matrix X of the unknown sample and a calibration matrix with the analytes X.  

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