mutual information map for unraveling the independence of solutions in the area offeasible solutions for three-component systems

Investigating rotational ambiguity in decomposition-based methods is crucial in chemometrics, as it posessignificant challenges [1]. the presence of rotational ambiguity gives rise to an area known as the area offeasible solution (afs) [1]. hence, this study focuses on exploring the relationship between afs and theconcept of independence, which is the main constraint in independent component analysis (ica)-basedtechniques [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 ourunderstanding 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). anexperimental chromatographic dataset was also used for validation (x6). the range of allowable responseswas computed using the facpack algorithm, and multiple responses were obtained for each dataset, withmi calculated for each response. examination of the findings revealed that the different solutions withinthe 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 profilesdemonstrate higher mi values, suggesting greater dependence. we examined four well-known icaalgorithms (mean-field ica (mf-ica), mutual information-based least dependent component analysis(milca) and joint approximate diagonalization of eigenmatrices (jade)) as well as multivariate curveresolution-alternating least squares (mcr-als) to understand their solutions and mi values for thesimulated and experimental datasets. the mi maps demonstrated that the solutions from mf-ica andmcr-als fell within the afs, aligning with the expectation that ica algorithms aim to minimize mivalues for signals. in contrast, the solutions from milca and jade were outside the afs, as they solelyfocused on maximizing independence. furthermore, we assessed the alignment of mi values by obtainingmi values with 10 repetitions for the noisy data, confirming the consistency of mi ratios across theserepetitions. additionally, the concept of independence was re-evaluated for triangles with equal areas inafs using mi calculations. this investigation contributes to a better understanding of permissibleresponses in three-component systems and sheds light on the implications of using different ica algorithmsin chemometrics. the results support the validity and applicability of our approach and provide insights forfurther analysis and research.