Global, local and unique decompositions in OnPLS for multiblock data analysis.

OnPLS is an extension of O2PLS that decomposes a set of matrices, in either multiblock or path model analysis, such that each matrix consists of two parts: a globally joint part containing variation shared with all other connected matrices, and a part that contains locally joint and unique variation, i.e. variation that is shared with some, but not all, other connected matrices or that is unique in a single matrix. A further extension of OnPLS suggested here decomposes the part that is not globally joint into locally joint and unique parts. To achieve this it uses the OnPLS method to first find and extract a globally joint model, and then applies OnPLS recursively to subsets of matrices that contain the locally joint and unique variation remaining after the globally joint variation has been extracted. This results in a set of locally joint models. The variation that is left after the globally joint and locally joint variation has been extracted is (by construction) not related to the other matrices and thus represents the strictly unique variation in each matrix. The method's utility is demonstrated by its application to both a simulated data set and a real data set acquired from metabolomic, proteomic and transcriptomic profiling of three genotypes of hybrid aspen. The results show that OnPLS can successfully decompose each matrix into global, local and unique models, resulting in lower numbers of globally joint components and higher intercorrelations of scores. OnPLS also increases the interpretability of models of connected matrices, because of the locally joint and unique models it generates.