In 1995, Manne published his profile-based theorems giving the (later shown to be incomplete)
necessary and sufficient conditions of the unique resolution:
Theorem 1. If all interfering compounds that appear inside the concentration window of a given
analyte also appear outside this window, it is possible to calculate the concentration profile of
the analyte.
Theorem 2. If for every interferent the concentration window of the analyte has a subwindow
where the interferent is absent, then it is possible to calculate the spectrum of the analyte.
Theorem 3. For a resolution based only upon rank information in the chromatographic direction
the conditions of Theorems 1 and 2 are not only sufficient but also necessary
In 2015, a new term, i.e., data-based uniqueness was defined and investigated in detail, and a general procedure was suggested for detection of uniquely recovered profile(s) on the basis of data set structure in both abstract subspaces. Differences between the data-based uniqueness and profile-based uniqueness were described, and it was shown that Manne’s theorems are not sufficient in general. It means that the “unique” solution obtained by using local rank information and Manne’s theorems may be incorrect, and this fact needs further investigation. It seems that the data-based uniqueness theorem is the correct unification and combination of Manne’s Theorem 1 and 2.
In 2020, a general rule for uniqueness (GRU) was proposed to unify all the different information that led to a unique solution in one framework. The GRU states that any information, able to fix the subspace of all complementary components in one space, results in a unique solution for the analyte in the other space, which is a powerful use of the duality concept. In 2009, using Borgen plot, a hitherto unknown discrepancy of MCR-ALS (Multivariate Curve Resolution – Alternating Least Squares) algorithms was revealed, i.e. the sub- and even the final solutions can be outside of the range of the data matrix R.
In 2010, Rolf Manne asked the data used for the previous paper from the first author, and a fruitful discussion started. Based on the data and Rolf’s basic ideas, we could develop and implement a new MCR algorithm which was coined as Moving Nodes. Unfortunately, the discussion did not progress further, the results were not disclosed, and Rolf passed away in 2024.
Very recently, we have decided to revitalize the development/implementation and we demonstrate Manne’s Moving Nodes algorithm in memory of Manne's great work in chemometrics.
In this lecture, we use LC-MS proteomic data for illustrations.