Estimation of chemical rank of a three-way array using a two-mode subspace comparison approach

When two matrices are formed by unfolding a three-way array along two full-rank modes and when two subspaces with the same size are constructed by taking the first principal component vectors of the same mode space in these two unfolded matrices, the principal components corresponding to the chemical species should behave identically in both subspaces, while the components corresponding to the noise contribution should behave differently in these two subspaces. Based on the difference of the behavior of the chemical signal and noise components, the two-mode subspace comparison (TMSC) approach has been proposed to estimate the chemical rank of a three-way array with two full-rank modes. Two outstanding features of the proposed method have been demonstrated. It is robust to a very high degree of collinearity between the spectra or chromatograms involved, or to a very high level of noise contained in a three-way array. The method has been shown to be useful for the treatment of three-way analytical data sets obtained by high-performance liquid chromatography-diode array detector (HPLC)-DAD and excitation–emission fluorescence spectroscopy.