M. Amrhein et al., ON THE RANK DEFICIENCY AND RANK AUGMENTATION OF THE SPECTRAL MEASUREMENT MATRIX, Chemometrics and intelligent laboratory systems, 33(1), 1996, pp. 17-33
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.