Fig. 1

Decomposing data tensors using independent component analysis. Tensorial ICA (tICA) works by decomposing a data tensor, here depicted as an order-3 tensor with three dimensions representing features (CpGs/genes), samples and tissue or data type, into a source tensor S and two mixing matrices defined over tissue/data type and samples, respectively. The key property of tICA is that the independent components in S are as statistically independent from each other as possible. Statistical independence is a stronger criterion than linear decorrelation and allows improved inference of sparse sources of data variation. Positive kurtosis can be used to rank independent components to select the most sparse factors. The largest absolute weights within each independent component can be used for feature selection, while the corresponding component in the mixing matrices informs about the pattern of variation of this component across tissue/data types and samples, respectively. In the latter case, the weights can be correlated to sample phenotypes, such as normal/cancer status or genotype. For the first mixing matrix, the weights inform us about the relation between data types (e.g. if the copy-number change is positively correlated with gene expression), or for a multi-cell EWAS, whether mQTLs are cell type independent or not. + ve positive, −ve negative, CNV copy-number variation, DNAm DNA methylation, EWAS epigenome-wide association study, mQTL methylation quantitative trait locus, mRNA messenger RNA