Multi-way principal components-and PLS-analysis

The Lohmöller–Wold decomposition of multi‐way (three‐way, four‐way, etc.) data arrays is combined with the non‐linear partial least squares (NIPALS) algorithms to provide multi‐way solutions of principal components analysis (PCA) and partial least squares modelling in latent variables (PLS). The decomposition of a multi‐way array is developed as the product of a score vector and a loading array, where the score vectors have the same properties as those of ordinary two‐way PCA and PLS. In image analysis, the array would instead be decomposed as the product of a loading vector and an image score matrix. The resulting methods are equivalent to the method of unfolding a multi‐way array to a two‐way matrix followed by ordinary PCA or PLS analysis. This automatically proves the eigenvector and least squares properties of the multi‐way PCA and PLS methods. The methodology is presented; the algorithms are outlined and illustrated with a small chemical example.