Angle Distribution of Loading Subspace (ADLS) for estimating chemical rank in multivariate analysis: Applications in...

Multivariate analyses are increasingly popular to explore the underlying structure of multivariate datasets, which are more and more prevalent in analytical chemistry. However, difficulties can be associated with estimating the number of components for the data with considerable coherence and noise. The method of Angle Distribution of Loading Subspace (ADLS) has been proposed to estimate the number of components for Principal Component Analysis (PCA) and PARAllel FACtor analysis (PARAFAC), which showed some advantages, in particular in the case of datasets with high coherence, over the commonly used methods (scree plot and cross-validation in PCA, and core consistency diagnostics (CORCONDIA) in PARAFAC). In this paper, we systematically improved and applied ADLS to estimate the number of components in different multivariate methods including, Multivariate Curve Resolution (MCR), PARAFAC and four-way PARAFAC. Firstly, we showed that ADLS performed better when estimating the chemical rank for MCR analysis, compared with scree plots. As well as this, we improved ADLS in multi-way analysis (three- and four-way PARAFAC) by calculating the loading subspace in advance using the Khatri-Rao product. The improved ADLS in multi-way analysis provided the correct result for the simulated three-way fluorescence datasets with unevenly distributed coherence at different dimensions, while the previous version of ADLS showed biased results and CORCONDIA / split-half analysis provided relatively unstable results. Moreover, ADLS was used to estimate the chemical rank for a four-way real-life fluorescence dataset analyzed by four-way PARAFAC. In this case the result of chemical rank results from ADLS was more precise and informative compared with CORCONDIA /split-half analysis in four-way analysis.