On rotational ambiguity in parallel factor analysis

Abstract

Although, in many cases parallel factor analysis (PARAFAC) resolves the trilinear data arrays to the true physical factors that form the data, i.e., unique solution can be found, the algorithm does not always converge to chemically meaningful solutions. Kiers and Smilde [J. Chemom. 1995; 9: 179–195] rigorously proved that unique decomposition does not hold in cases with ‘rank overlap’. They showed when PARAFAC is applied on a three-way data array which has rank overlap in one of its loading modes; the solution obtained is not unique and at best cannot be easily compared with the underlying physical factors owing to a rotational ambiguity. An aspect which is significantly less documented in the previous publications is the reliable detection of rotational ambiguities in multi-way methods. A few reported methods are based on bilinear models for calculating the feasible bands of three-way data. In this paper we propose a method to calculate feasible bands of resolved profiles of components in three-way methods and visualize the rotational ambiguity in three-way data in the results of the three-way methods. Most of discussion is in the PARAFAC algorithm. The principle behind the algorithm is described in detail and tested for simulated data set. Completely general and exhaustive results are presented for the two-component cases. In particular, the effect of the noise is investigated and a comparison is made between feasible solutions obtained from PARAFAC and matrix-augmented with trilinearity. It is shown that the results obtained from both methods are identical.