Special Issue Article
Visualization and establishment of partial uniqueness and uniqueness rules in parallel factor analysis
Article first published online: 2 AUG 2013
Copyright © 2013 John Wiley & Sons, Ltd.
Journal of Chemometrics
Special Issue: Chemometrics in Iran
Volume 27, Issue 10, pages 330–340, October 2013
How to Cite
Omidikia, N., Abdollahi, H. and Kompany-Zareh, M. (2013), Visualization and establishment of partial uniqueness and uniqueness rules in parallel factor analysis. J. Chemometrics, 27: 330–340. doi: 10.1002/cem.2520
- Issue published online: 17 OCT 2013
- Article first published online: 2 AUG 2013
- Manuscript Accepted: 17 JUN 2013
- Manuscript Revised: 10 JUN 2013
- Manuscript Received: 15 FEB 2013
- partial uniqueness;
- three-component parallel factor analysis;
- rotational ambiguity;
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.