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 equal to the number of absorbing species, S. However, in many practical reaction networks, such an assumption does not hold. This paper examines various scenarios where ‘rank deficiency’ can occur. The most important 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 rotation in the reaction-spectra space of dimension R. Another possibility is to augment the rank of the data matrix to S for which two experimental methods are developed. Rank augmentation is performed by (i) multiple process runs and (ii) addition of reactants or products during the reaction. The composite data matrices are of rank S and, hence, suited to factor analysis. The number of necessary runs or additions can be detected by determining the rank of both the original and column-mean-centered data matrices. With rank augmentation, it is possible to determine both the number of independent reactions and the number of absorbing species. Furthermore, the influence on the rank of data pretreatment such as mean centering, normalization, auto-scaling, and differentiation with respect to time or wavelength is examined.