Second-order (matrix-valued) calibration by multivariate curve resolution–alternating least squares (MCR-ALS) can quantify an analyte in the presence of uncalibrated interferents, yet the underlying bilinear decomposition is generally not unique: infinitely many profile pairs reproduce the data equally well. The source work (Alcaraz, Culzoni, Ibañez, Lozano and Olivieri, 2020) mapped, for highly overlapped systems, which component profiles become unique and which remain rotationally ambiguous, and concluded that further theoretical work is needed. Later literature (2020–2026) has delivered tractable algorithms that bound the feasible profile range for many components, and a parallel single-matrix uniqueness theory (Manne-type conditions and the general rule of uniqueness / data-based uniqueness theorems, GRU/DBU), but no general predictive theory for the augmented multi-sample calibration geometry, nor general ambiguity-removal strategies. The open problem is to build that theory: conditions, provable where possible, that determine which row-/column-space sub-profiles (analyte vs interferent) of an augmented second-order calibration decomposition are unique under realistic overlap and constraint sets.

Background

In second-order analytical calibration, each sample yields a data matrix, e.g. luminescence intensity as a function of emission wavelength and time decay, or chromatography- or capillary-electrophoresis spectra recorded across elution time. Under a bilinear model, such a matrix is written as a sum of contributions, each the outer product of a concentration-related profile in one instrumental mode and a pure-signal profile (spectrum, decay curve) in the other. Multivariate curve resolution–alternating least squares (MCR-ALS) decomposes such data by alternately refitting the two profile matrices subject to chemical constraints (non-negativity, correspondence of species across samples, etc.). To calibrate, the matrices of pure-analyte standards and of test samples are stacked into a single augmented matrix, usually by appending samples along the concentration (row) direction so that the pure-signal profiles are shared by all samples.

Two facts make this setting chemically central. First, stacked second-order data permit the second-order advantage: estimation of a calibrated analyte in samples containing unmodeled, unexpected constituents. Second, the bilinear decomposition is generically subject to rotational ambiguity: if D=CSTD = C S^{\mathrm{T}} with CC the concentration-type profiles and SS the pure-signal profiles, then for many invertible transforms TT, D=(CT)(T−1ST)D = (C T)(T^{-1} S^{\mathrm{T}}) holds exactly, and only the applied constraints prune the set of feasible solutions. This ambiguity propagates to predicted analyte concentrations. Tools to size it include MCR-BANDS (nonlinear optimizations returning extreme feasible profiles per component) and the area of feasible solutions (AFS), the set of all constraint-compatible profiles, computable exactly only for small systems (grid-search sum-of-squared-residuals mapping, or geometric packages such as FACPACK). Sensor-wise variants of MCR-BANDS (N-BANDS) instead bound one measured variable at a time, returning a lower and an upper feasible bound for each component profile at that variable, which keeps the computation tractable for more components.

For single matrices, classical uniqueness results go back to Manne's conditions: roughly, a species' profiles are uniquely recoverable when other species leave selective (interference-free) windows in both data modes. A modern line — the general rule of uniqueness (GRU) and data-based uniqueness (DBU) theorem of Karimvand, Akbari Lakeh, Tavakkoli, Ghaffari, Omidikia, Khalili Ali Abad, Rajkó and Abdollahi (2020), operationalized by Akbari Lakeh, Abdollahi and Rajkó (2022) — predicts full, partial or fractional uniqueness of individual non-negative profiles at no extra computational cost. A 2026 re-tuning by Rajkó and Razaq still works on unifying and correcting interpretations of these conditions (including a new counterexample), showing the theory itself remains in flux.

The source work studied the highly overlapped regime, where selective windows are absent and one might expect total ambiguity. Surprisingly, in simulated and experimental systems (ciprofloxacin in the presence of salicylate by decay-luminescence matrices; ciprofloxacin among other fluoroquinolones by capillary electrophoresis with ultraviolet detection, CE-UV), some component profiles came out unique while others remained ambiguous — the pattern depending on the data mode (row vs column space), on whether a species was the calibrated analyte or an uncalibrated interferent, and on the initialization of the least-squares iterations from "purest variables" (approximate pure-species regions used to seed MCR-ALS). The authors mapped these instances by grid-search feasible-region mapping and noted that existing tools do not scale (MCR-BANDS-type optimizers do not in general cover the full feasible range; exhaustive grids are limited to a few components), closing with the remark that further theoretical work is still needed to explain the phenomenon.

Problem Statement

Establish a general, predictive theory of uniqueness in multivariate curve resolution of augmented second-order calibration data with highly overlapped profiles. Specifically: (1) give checkable conditions — provable wherever possible, in the spirit of Manne's selective-window conditions or the GRU/DBU framework but extended to multi-sample augmented matrices — under which a given component sub-profile (row-space, i.e. concentration-type, vs column-space, i.e. pure-signal; calibrated analyte vs uncalibrated interferent) of the bilinear decomposition is unique, partially unique, or rotationally ambiguous, as a function of the overlap pattern between species profiles, the augmentation direction, the number of components, the applied constraints (non-negativity, correspondence of species across samples, trilinearity — the assumption that each component keeps the same profile shape in every sample in both instrumental modes — when it holds), and the noise level; (2) explain within this theory why analyte quantitation can remain accurate while profiles are ambiguous, i.e. the structural origin of the second-order advantage (quantifying a calibrated analyte despite uncalibrated interferents) in the highly overlapped regime, and when it fails; (3) explain, as a consequence of the theory, when initialization of the least-squares iterations from purest variables leads to accurate prediction of the analyte of interest despite rotational ambiguity. Parts (1)–(3) are all required components of the theory: (2) and (3) are not optional elaborations but claims the theory must deliver. The factor list in (1) delimits the scope of dependence over which the uniqueness conditions are claimed — their dependence on these factors is to be stated where it is scientifically meaningful, not as a checklist of separately mandatory variables (in particular the continuous noise level).

References

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