Evaluating different sparsity measures for resolving LC/GC-MS data in the context of multivariate curve resolution

Abstract

Since mass spectra are sparse in nature, the “sparsity” has been recently proposed as a constraint in multivariate curve resolution-alternating least squares (MCR-ALS) methods for analyzing LC/GC-MS data. There are different ways for implementation of sparsity constraint, and the majority of such methods rely on imposing a penalty term on the norms of recovered mass spectra. However, the main question is which penalty method is more appropriate for implementation of sparsity constraint in MCR methods. In order to address this question, two- and three-component LC/GC-MS data were simulated and used for the case study, in the work. The areas of feasible solutions (AFS) were calculated using the grid search and Nelder-Mead simplex algorithms. Moreover, different measures of sparsity such as L0-norm, L1-norm, and L2/L1 for all mass spectra in the AFSs were calculated and visualized as contour plots. The results revealed that all mentioned measures of sparsity find the sparsest solution in AFS. However, from the optimization point of view, L1-norm and L2/L1 are easier to implement than L0-norm. Approximation methods for solving L0-norm problem can take advantage of both unique solution and fast optimization. The results of L0-norm, L1-norm, and L2/L1 criteria were compared with other sparsity measures such as Shannon entropy, Hoyer, Kurtosis, and Gini indices. The results revealed that L1-norm sparsity measure coincides with L0-norm, Hoyer, Kurtosis, and Gini indices from the accuracy point of view. Finally, the feasibility of least absolute shrinkage and selection operator (Lasso) was assessed for implementation of L1-norm penalty and finding the sparsest solution in MCR. It was found that for small values of λ parameter, Lasso is able to find the sparsest solution with the minimum sum of squares of errors. Thorough optimization of λ is necessary for obtaining accurate results. The lasso-MCR-ALS algorithm was tested with the real GC-MS datasets related to the analysis of Iranian red jujube oil.