L1 L2 L3 HOSVD (Higher Order Singular Value Decomposition) Extension to tensor….. N M K x ijk ≃∑ l 1 =1 L 1 ∑ l 2 =1 L 2 ∑ l 3 =1 L 3 G(l 1 l 2 l 3 )u l 1 i u l 2 j u l 3 k N: number of genes (i) M: number of samples (j) K: number of tissues (k) xijk: gene expression Example
expression of gene i of sample j xkj: methylaion of region k of sample j x xijk ijk ≡ ≡ x xij ij ⨉ ⨉ x xkj kj G u l1i u l2j u l3k L1 L2 L3 x ijk N M K x ijk ≃∑ l 1 =1 L 1 ∑ l 2 =1 L 2 ∑ l 3 =1 L 3 G(l 1 l 2 l 3 )u l 1 i u l 2 j u l 3 k
Using Tensor Decomposition Based Unsupervised Feature Extraction –Comparison with DIABLO–” Y-h. Taguchi in De-Shuang Huang Vitoantonio Bevilacqua Prashan Premaratne (Eds.), Intelligent Computing Theories and Application, 15th International Conference, ICIC 2019 Nanchang, China, August 3–6, 2019 Proceedings, Part I, pp.565-574 https://doi.org/10.1007/978-3-030-26763-6_54 Preprint: https://doi.org/10.1101/591867
x kj :expression of kth miRNA of jth sample x pj :expression of pth protein of jth sample tensor：x ikpj =x ij・x kj・x pj Apply tensor decomposition (tensor version of singular vallue decomposition) x ikpj ≃∑ l 1 =1 L 1 ∑ l 2 =1 L 2 ∑ l 3 =1 L 3 ∑ l 4 =1 L 4 G (l 1 l 2 l 3 l 4 )u l 1 i u l 2 k u l 3 p u l 4 j ul1i: mRNA, ul2k: miRNA ul3p: proteome, ul4j: sample
Pros: Fast (because of no optimization) Robust (independent of label information) Unsupervised (no need to construct model in advance) Cons: Cons: No ways if it does not work Need more memories: 150 ⨉ (200+184+142) vs 150 ⨉ 200 ⨉ 184 ⨉ 142
set: GSE76381 scRNA-seq of human and mouse mid brain developments i:Genes j,k:cells Purpose of the analysis: Selection of genes associated with mid brain development commonly between human and mouse
Tensor is generated from product of cells using 13,384 common from product of cells using 13,384 common genes between human and mouse genes between human and mouse xijk = xij × xik ∈ ℝ13384×1977×1907 i:Genes j,k:Cells Size reduction needed because of too huge tensors xjk: decomposed by singular value decomposition vlj: lth human cell singular value vectors vlk: lth mouse cell singular value vectors x jk =∑ i x ijk
v lk =a l ' +∑ t b lt ' δkt δjt,δkt： 1 when cells j,k is measured at t 0 otherwise vlj and vlk with any kind of time dependence are selected with categorical regression(ANOVA)
( j)=∑ j v lj x ij u li (k)=∑ k v lk x ik lth human gene singular value vectors lth mouse gene singular value vectors P-values are attributed to gene singular value vectors by χ2 distribution, corrected by BH criterion, genes associated with adjusted P- values less than 0.01 are selected.
methods using TD for multi-omics data analysis as well as RNA-seq data analysis. I have published a monograph from Springer. I am happy if you can but it, although it is extremely expensive.