On rotational ambiguity in parallel profiles with linear dependencies

Abstract

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 noisefree two-and three-component data arrays.