Investigating error structure effect on possible solutions in the three-way models

The main assumption behind most of chemometrics tool is identical independent distribution (i.i.d) for noises structure [1]. Alternating lest-squares and principal component analysis provides maximum likelihood (ML) estimation in the i.i.d condition [1]. Unfortunately this assumption will be violated in most of experimental cases. Hence, maximum likelihood paves the way to a modification on the basic chemometrics tools [2]. MLPCA and MLPARAFAC are state of art chemometrics tool for the analysis of fallible two-way and three-way data sets [3]. Error structures can be categorized in six different cases and these cases encompass all of the possible structures that will be dealt with. Error structure of a data set can be a combination of these cases [4]. Different MLPARAFAC algorithms were developed in order to a trilinear decomposition of three-way data sets [3]. The aim of this contribution is two-folded. 1) The effect of noise structure on the possible solution of multi-way models will be highlighted. In other words calculation of feasible regions will extended to the MLPARAFAC models. Data sets with different known error structure were simulated and feasible regions were calculated. It was shown that the error effect is non-uniform and complex on the possible solutions. 2) In the same analogy with MLPCA-MCR-ALS, MLPCA-MA-MCR-ALS will be used for handling error structure of three-way data sets. It should be highlighted that MLPCA-MCR-ALS provides reliable estimation of profiles such as MCR-WLAS in noisy measurements [4,5]. Finally, the results confirmed that the resolved profiles obtained by MLPCA-MA-MCR-ALS are practically identical to those obtained by ML-PARAFAC and that they can differ from those resolved by ordinary PARAFAC-ALS, especially in the case of high noise. In MLPCA-MA-MCR-ALS with trilinearity constraint, MLPCA is only used as a preliminary data pretreatment before MA-MCR analysis and this is the possible advantage over MLPARAFAC and it does not require changing the traditional PARAFAC algorithm.