SOME THEORETICAL RESULTS ON 2ND-ORDER CALIBRATION METHODS FOR DATA WITH AND WITHOUT RANK OVERLAP

GRAM, a method for second-order calibration, has been introduced by Sanchez 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 profiles of (rank-one) analytes present in sample and mixture. This claim has not been proven rigorously. In the present paper, rigorous proofs are given for situations where the claims are valid indeed. In addition, 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 overlapping 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 framework is provided for a variety of second-order calibration problems and the extent to which quantitative and qualitative information can be expected is given.