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Several curve resolution techniques have been developed to analyze higher-order data structures, among which one can name parallel profiles with linear dependencies (PARALIND). The PARALIND is a variant of the Parallel Factor Analysis (PARAFAC) to cope with linear dependencies. Applying a PARAFAC model with different random initializations to a noisy three-way data with linearly dependent loadings always converges to a best-fitted noisy solution. Since PARAFAC maximally fits the noise part of the data, rank-deficiency is broken, and it causes a noise-induced unique decomposition. The uniqueness due to the presence of noise in rank-deficient cases is an artificial phenomenon called 'surface uniqueness.' The PARALIND model causes non-uniqueness to be revealed, ie, it leads to a different set of profiles in each run. However, it might not give us all the possible solutions. Therefore, there is still a demand for determining all and every possible solution. The aim of this paper is 3-fold: (1) A method is developed for determining feasible regions using the PARALIND model. (2) The surface uniqueness phenomenon is clarified. (3) The effect of interaction matrices in PARALIND as an active constraint is emphasized and discussed in detail using simulated noisy and noise-free two- and three-component data arrays. [ABSTRACT FROM AUTHOR]

Parallel factor analysis (PARAFAC) is believed to be one of the most widely used multiway curve resolution techniques. The superiority of PARAFAC is due to its uniqueness property that is provided under mild assumptions. Unambiguously recovered chemically and physically meaningful profiles by PARAFAC are reliable for quantitative and qualitative application. The essential uniqueness condition of PARAFAC breaks totally or partially in data sets with linearly dependent loadings. PARAFAC uniqueness levels include full uniqueness, partial uniqueness, uni-mode uniqueness, and full non-uniqueness. Although PARAFAC uniqueness conditions are more or less known, they cannot cover the whole of existing uniqueness problems, especially complex rank-deficient systems. The questions investigated here are as follows: Under which conditions is PARAFAC still unique and what is the extent of deterioration of PARAFAC uniqueness by linearly dependent loadings? These are among the most important points in dealing with three-way data arrays with PARAFAC models. Direct visualization of feasible bands using an explicit, well-defined, and trustworthy method is a key feature of our approach to the investigation of uniqueness and recasting partial uniqueness rules in this study. Different simulated examples with different uniqueness levels were designed, and rotational ambiguity was calculated for each case. In this way, the full implication of partial uniqueness was investigated. Direct visualization of feasible bands is beneficial not only for confirmation of previous rules of uniqueness and partial uniqueness but also for development of new theorems. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]