Hal. Kiers et Ak. Smilde, SOME THEORETICAL RESULTS ON 2ND-ORDER CALIBRATION METHODS FOR DATA WITH AND WITHOUT RANK OVERLAP, Journal of chemometrics, 9(3), 1995, pp. 179-195
GRAM, a method for second-order calibration, has been introduced by Sa
nchez and Kowalski and later modified by Wilson, Sanchez and Kowalski.
The methods are based on the claim that, in cases without measurement
error they yield correct estimates for the concentration ratios and p
rofiles of (rank-one) analytes present in sample and mixture. This cla
im has not been proven rigorously. In the present paper, rigorous proo
fs are given for situations where the claims are valid indeed. In addi
tion, it is shown that PARAFAC, an alternative method for second-order
calibration, can be used to obtain the same results. Next it is shown
that the claims do not hold in cases with 'rank overlap' (partly over
lapping profiles) and it is proven that a procedure by Wang et al. can
still be used to assess some of the concentration ratios. A general f
ramework is provided for a variety of second-order calibration problem
s and the extent to which quantitative and qualitative information can
be expected is given.