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Fig 1.

Boxplots of sample singular value vectors x_ℓ3,j (a) when TD was applied to the type I tensor and (b), (c), 1 ≤ ℓ₃ ≤ 5, when TD was applied to the type II tensor, generated from mRNA and miRNA expression profiles of multi-omics datasets. (d) Sample singular value vectors when HO GSVD was applied to multi-omics datasets. P-values computed by categorical regression attributed to (a) to (d) were below the figures.

https://doi.org/10.1371/journal.pone.0200451.g001

There are errors in the second paragraph of the subsection titled, “Definition and terminology of TD” in the Materials and Methods section. All instances of the following, “G(n₁, n₂, …, n_m)” should instead read as, . The correct paragraph should be: TD is the expansion of tensor , n_k = 1, …, N_k, 1 ≤ k ≤ m in the form where , 1 ≤ k ≤ m, are orthogonal matrices. Since is as large as , this formula is clearly overcomplete and does not give unique expansion. In this study, in order to decide , , 1 ≤ k ≤ m uniquely, I employ the higher order singular value decomposition (HOSVD) algorithm [23], which has successfully used to analyse microarrays [24] previously. is a core matrix. , 1 ≤ k ≤ m, are singular value matrices and their column vectors are singular value vectors. , having larger absolute values, has more contribution to . Since the combination of , 1 ≤ k ≤ m, associated with to which larger absolute values were attributed contributes more collectively to , they are more likely to be associated with one another.