Analytical solution and meaning of feasible regions in two-component three-way arrays

Nematollah Omidikia, Hamid Abdollahi, Mohsen Kompany-Zareh, R. Rajkó

Research output: Contribution to journalArticle

4 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)42-53
Number of pages12
JournalAnalytica Chimica Acta
Publication statusPublished - Oct 5 2016



  • 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

ASJC Scopus subject areas

  • Analytical Chemistry
  • Environmental Chemistry
  • Biochemistry
  • Spectroscopy

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