Determining the number of principal factors by eigenvector comparison of the original bi-linear data matrix and the one reconstructed from key variables

• A method was proposed to determine the number of principal factors of a data matrix.
• It is based on eigenvector comparison of the original and a reconstructed data matrix.
• The reconstruction of a data matrix is based on key variables of the original matrix.
• The novel method is mathematically rigorous, and the determination was clear.
• The novel method yielded accurate results for both simulated and experimental data.

It is an essential step in analyzing hyphenated chromatographic data of complex chemical systems to determine the number of principal factors of the bi-linear matrix. The determination is difficult due to the co-existence of non-chemical factors, such as background, noise, etc. A new method was proposed for the determination based on comparing eigenvectors of the original data matrix and the one reconstructed from key spectral variables that are selected with orthogonal projection approach (OPA). The proposed method is mathematically rigorous and the determination is clear. In comparison with other four indices, i.e., NPFPCA (noise perturbation in functional principal component analysis), RESO (the ratio of eigenvalues calculated by smoothed principal component analysis and those calculated by ordinary principal component analysis), DRAUG (determination of rank by augmentation) and DRMAD (determination of rank by median absolute deviation), this proposed method was proven to have good performance in both simulated GC-IR and experimental HPLC-DAD data.