ON THE RANK DEFICIENCY AND RANK AUGMENTATION OF THE SPECTRAL MEASUREMENT MATRIX

The analysis of spectral measurements using standard factor-analytical (FA) techniques requires the rank of the absorbance matrix to be equa l to the number of absorbing species, S. However, in many practical re action networks, such an assumption does not hold. This paper examines various scenarios where 'rank deficiency' can occur. The most importa nt case is when the number of independent reactions, R, is less than S . In such a case, rank analysis can only reveal R and, hence, standard FA techniques will fail. Hence, one possibility is to perform rotatio n in the reaction-spectra space of dimension R. Another possibility is to augment the rank of the data matrix to S for which two experimenta l methods are developed. Rank augmentation is performed by (i) multipl e process runs and (ii) addition of reactants or products during the r eaction. The composite data matrices are of rank S and, hence, suited to factor analysis. The number of necessary runs or additions can be d etected by determining the rank of both the original and column-mean-c entered data matrices. With rank aug mentation, it is possible to dete rmine both the number of independent reactions and the number of absor bing species. Furthermore, the influence on the rank of data pretreatm ent such as mean centering, normalization, auto-scaling, and different iation with respect to time or wavelength is examined.