Visualization of local rank information using microscopic structure of bilinear chemical data

Multivariate Curve Resolution (MCR) is a methodology for analyzing chemical data used in many different application fields. However, MCR results are often complicated by the well-known rotational ambiguity problem. The reliability of the MCR results is highly dependent to the constraints that are used to apply the prior knowledge from the system under study. Local rank is one of the most powerful and widely used constraints in chemometrics [1]. Different methods in chemometrics rely on this constraint for building initial estimates [2], finding selective and/or zero concentration regions [3], and resolving concentration profile or spectra of analyte [4,5]. Furthermore, the prominent resolution theorems that define the uniqueness conditions in MCR are based on local rank information [6]. However, it has been shown very recently that local rank information may lead to incorrect solutions due to local rank deficiency problem [7]. More theoretical descriptions and generalizations are reported in this work to clarify the problem. Also, it is shown that the information related to the local rank can be extracted from the structure of data set by means of computational and geometry tools in addition to rank exploratory methods. Lawton-Sylvestre approach [8] in the case of two-componet and Borgen plot [9] in the case of three-component systems were used for this purpose.