On uniqueness and selectivity in three-component parallel factor analysis

• Calculation of rotational ambiguity in PARAFAC extends to three components systems.
• Selective window in profiles causes uniqueness in PARAFAC on rank deficient systems.
• An independent factor in two modes with selective window recovered uniquely by PARAFAC.
• Kruskal condition in sufficient, not necessary for R = 2 and R = 3.
• Effect of selective windows on uniqueness in PARAFAC is investigated systematically.

Unambiguous recovery of profiles is a distinguishable advantage of Parallel Factor Analysis (PARAFAC) as a trilinear model and has made it a promising exploratory tool for data analysis. Linear dependency in profiles destroys trilinearity and will increase ambiguity in the curve resolution of three-way data sets. PARAFAC uniqueness deteriorates totally or partially in data sets with linearly dependent loadings. Exploiting a reliable method for determination and direct visualization of feasible bands in the PARAFAC model can be helpful not only in full characterization of uniqueness conditions but also in the investigation of the effects of constraints on the PARAFAC feasible solutions. The purpose of this paper is twofold. First, the calculation of rotational ambiguity in the PARAFAC model extends to three components system. The principle behind the algorithm is described in detail and tested for simulated and real data sets. Completely general and thoroughly investigated results are presented for the three component cases. Secondly, the effects of selective regions in the profiles on the resolution of systems that suffered from the rank deficiency problem, due to rank overlap, are emphasized. In the case of two-way data sets the effect of selectivity constraint on the unique recovery of profiles was investigated and applied. However, to our knowledge, in this report, for the first time, the effect of the presence of selective windows in the profiles, on the unique resolution of three-way data sets has been systematically investigated.

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