Mutual Information Map for Unraveling the Independence of Solutions in the Area of Feasible Solutions for Three-Component Systems

Investigating rotational ambiguity in decomposition-based methods is crucial in chemometrics, as it poses significant challenges [1]. The presence of rotational ambiguity gives rise to an area known as the Area of Feasible Solution (AFS) [1]. Hence, this study focuses on exploring the relationship between AFS and the concept of independence, which is the main constraint in Independent Component Analysis (ICA)-based techniques [2]. The investigation involves calculating the independence using mutual information (MI) [3, 4] for feasible solutions. Additionally, a novel concept called the MI map is introduced to enhance our understanding of AFS in this context. To achieve this, we simulated various chromatographic datasets, ranging from simple to complex, for three-component systems, taking noise into account (X1 to X5). An experimental chromatographic dataset was also used for validation (X6). The range of allowable responses was computed using the Facpack algorithm, and multiple responses were obtained for each dataset, with MI calculated for each response. Examination of the findings revealed that the different solutions within the feasible bands are associated with distinct different MI values. As anticipated by the duality concept, spectral profiles exhibit lower MI values, indicating higher independence, while concentration profiles demonstrate higher MI values, suggesting greater dependence. We examined four well-known ICA algorithms (mean-field ICA (MF-ICA), mutual information-based least dependent component analysis (MILCA) and joint approximate diagonalization of eigenmatrices (JADE)) as well as multivariate curve resolution-alternating least squares (MCR-ALS) to understand their solutions and MI values for the simulated and experimental datasets. The MI maps demonstrated that the solutions from MF-ICA and MCR-ALS fell within the AFS, aligning with the expectation that ICA algorithms aim to minimize MI values for signals. In contrast, the solutions from MILCA and JADE were outside the AFS, as they solely focused on maximizing independence. Furthermore, we assessed the alignment of MI values by obtaining MI values with 10 repetitions for the noisy data, confirming the consistency of MI ratios across these repetitions. Additionally, the concept of independence was re-evaluated for triangles with equal areas in AFS using MI calculations. This investigation contributes to a better understanding of permissible responses in three-component systems and sheds light on the implications of using different ICA algorithms in chemometrics. The results support the validity and applicability of our approach and provide insights for further analysis and research.