Visualization and establishment of partial uniqueness and uniqueness rules in parallel factor analysis

Abstract

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.

Definition and detection of data-based uniqueness in evaluating bilinear (two-way) chemical measurements

“…(1) The concentration and absorption profiles of the first component are determined uniquely. This phenomenon is the so-called partial uniqueness [31] that refers to systems in which one factor is recovered uniquely. (2) Because of the unique resolution of the first component profiles, the potential locations of the feasible solutions are restricted to line segments for the complementary components and simplified, exactly, to Lawton-Sylvestre (LS) plot [26].…”

“…Although, considerable efforts have been devoted to investigate the uniqueness conditions in three-way models [30][31], the exploration of the uniqueness has been disregarded in two-way data sets. A new term, fractional uniqueness, is introduced to refer to systems in which a single profile of a component is recovered uniquely while others in the corresponding modes are not.…”

Usefulness of PARAFAC for the Quantification, Identification, and Description of Analytical Data

“…[15], there is a list of six principles (or rules) that can guide researchers on full or partial uniqueness in their data. With the idea of evaluating partial uniqueness for specific data with three PARAFAC factors, the "bands" that include the effect of the lack of uniqueness on the estimation of the profiles can be visualized [16], an application of this strategy to analyze to what extent the uniqueness of the PARAFAC solution degrades as the profiles of two factors overlap is in Ref. [17].…”

Non‐rotational Tucker3 core simplification

“…Closure is because of at least one closed system such as an equilibrium (e.g. titration of an acid with a base ) or a second order reaction (e.g. U + Y➔Z) in which the summation of two or more concentration profiles is constant during all stages of the process.…”

“…Although there is RO in all modes in case (5), 222(RO13, RO23, RO13), the pseudo array rank is smaller than the chemical rank (array rank deficient). This is because RO is between the same profiles in two modes (RO13 is in first and third modes) . Therefore, the pseudo array rank is reduced to two.…”