title: Analytical solution and meaning of feasible regions in two-component three-way arrays.
abstract: Although many efforts have been directed to the development of approximation methods for determining the extent of feasible regions in two- and three-way data sets; analytical determination (i.e. using only finite-step direct calculation(s) instead of the less exact numerical ones) of feasible regions in three-way arrays has remained unexplored. In this contribution, an analytical solution of trilinear decomposition is introduced which can be considered as a new direct method for the resolution of three-way two-component systems. The proposed analytical calculation method is applied to the full rank three-way data array and arrays with rank overlap (a type of rank deficiency) loadings in a mode. Close inspections of the analytically calculated feasible regions of rank deficient cases help us to make clearer the information gathered from multi-way problems frequently emerged in physics, chemistry, biology, agricultural, environmental and clinical sciences, etc. These examinations can also help to answer, e.g., the following practical question: "Is two-component three-way data with proportional loading in a mode actually a three-way data array?" By the aid of the additional information resulted from the investigated feasible regions of two-component three-way data arrays with proportional profile in a mode, reasons for the inadequacy of the seemingly trilinear data treatment methods published in the literature (e.g., U-PLS/RBL-LD that was used for extraction of quantitative and qualitative information reported by Olivieri et al. (Anal. Chem. 82 (2010) 4510-4519)) could be completely understood.
authors.affiliation: Institute of Process Engineering, Faculty of Engineering, University of Szeged, Moszkvai krt. 5-7, H-6725 Szeged, Hungary; Department of Chemistry, Institute for Advanced Studies in Basic Sciences, P.O. Box 45195-1159, Zanjan, Iran.
authors.forename: Nematollah
authors.initials: N
authors.lastname: Omidikia
authors.affiliation: Department of Chemistry, Institute for Advanced Studies in Basic Sciences, P.O. Box 45195-1159, Zanjan, Iran.
authors.forename: Hamid
authors.initials: H
authors.lastname: Abdollahi
authors.affiliation: Department of Chemistry, Institute for Advanced Studies in Basic Sciences, P.O. Box 45195-1159, Zanjan, Iran.
authors.forename: Mohsen
authors.initials: M
authors.lastname: Kompany-Zareh
authors.affiliation: Institute of Process Engineering, Faculty of Engineering, University of Szeged, Moszkvai krt. 5-7, H-6725 Szeged, Hungary. Electronic address: rajko@sol.cc.u-szeged.hu.
authors.forename: Róbert
authors.initials: R
authors.lastname: Rajkó
mesh_terms: D002623:Chemistry Techniques, Analytical; D005240:Feasibility Studies; D016014:Linear Models; D008962:Models, Theoretical; D013223:Statistics as Topic / Q000379:methods*
keywords: Direct and iterative trilinear decompositions; Feasible regions and profiles; Lawton and Sylvestre self-modelling curve resolution method; Trilinear and non-trilinear two-component three-way data arrays; Trilinearity constraint