n<<p n<<p p: # of features: ~ 104 (e.g., number of human genes) n:# of samples: ~10 (e.g., number of human patients) p: Number of features(genes) n:Number of samples(patients) Features (genes) selected
Conventional approach: Conventional approach: Apply statistical test to individual feature ↓ Attribute P-values to individual feature to reject null hypothesis (e.g., expression of the gene is identical between healthy control and patients) ↓ Correct P-values with considering large p (P-value as small as 1/p can occur accidentally) ↓ Select features associated with adjusted P-values less than threshold values (e.g. <0.05) P-value 1-(P-value) 0 1 D(1-P)
of feayures)→ P-values are heavily corrected (become larger, i.e., less significant) Adjusted P-value are hard to be less than threshold values Advanced approach? LASSO (add penalty term to restrict the number of features selected) ↓ Lack of stability (biologically, selected feature must be stable regardless to samples considered, if the same target (e.g., a specific disease) Why is feature selection with “large p small n” difficult?
distinction between healthy controls (HC) and patients (PA)) ↓ Select features with large enough contribution to the selected sample vectors (i.e., large enough to reject null hypothesis, e.g., projection obeys Gaussian Gaussian) Individual p p features (genes) can be represented as a cloud of points in the space spanned by n n sample vectors (e.g. gene expression of samples sample vector Projection to sample vectors Sample 1 (PA) Sample 2 (HC) Sample n (HC)
are often associated with multiple characters e.g: “Healthy controls vs Patients” (j) + “Tissue specificity” (k) that can be represented as tensor, x ijk , that represents ith gene expression HC vs PA Tissue Genes G u l1i u l2j u l3k x ijk =∑ l 1 ∑ l 2 ∑ l 3 G(l 1 l 2 l 3 )u l 1 i u l 2 j u l 3 k sample vector Projection to sample vectors
≃∑ 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 j u l 2 k u l 3 m u l 4 i u l1j : l 1 th cell lines dependence u l2k : l 2 th with and without SARS-CoV-2 infection u l3m : l 3 th dependence upon biological replicate u l4i : l 4 th gene dependence G: weight of individual terms
independent of cell lines and biological replicates （u l1j ,u l3m take constant regardless j,m） and dependent upon with or wothout SARS-CoV-2 infection（u l21 =-u l22 ） Heavy “large p small n” problem Number of variables(=p): 21797 ~ 104 Number of samples (=n): 5 ⨉2 ⨉3 =30 ~10 p/n ~ 103
Cell lines With and without SARS-CoV-2 infection biological replicate Independent of cell lines and biological replicate, but dependent upon SARS-CoV-2 infection.
but dependent upon SARS-CoV-2 infection is associated with u 5i (l 4 =5) P i =P χ2 [> (u 5i σ5 )2] Computed P-values are corrected with considering multiple comparison corrections by Benjamini-Hochberg method. 163 genes with corrected P-values <0.01 are selected among 21,797 genes.
we do not know how many genes should be selected, lasso and random forest is useless. Instead we employed SAM and limma, which are gene selection specific algorithm (adjusted P-values are used ). t test SAM limma P>0.01 P≦0.01 P>0.01 P≦0.01 P>0.01 P≦0.01 Calu3 21754 43 21797 0 335 3789 NHBE 21797 0 21797 0 342 3906 A549 MOI 0.2 21797 0 21797 0 319 4391 MOI 2.0 21472 325 21797 0 208 4169 ACE2 expressed 21796 1 21797 0 182 4245
Calu3 7278 16432 NHBE 23383 327 A549 MOI 0.2 7858 15852 MOI 2.0 16279 7431 ACE2 expressed 16201 7509 After the publication of our paper, we have found the paper[*] that originally studied this GEO data was published (when we have done this study, only GEO data set was provides and no papers were published). The paper includes DESeq2 results. It is similar to limma; it detected most of genes as DEGs wheras it identified limited number of DEGs for NHBE cell lines [*]Daniel Blanco-Melo et al, Cell, 2020; 181(5): 1036-1045.e9. doi: 10.1016/j.cell.2020.04.026. Imbalanced Host Response to SARS-CoV-2 Drives Development of COVID-19
l2j u l3k L1 L2 L3 j=1,..,n 1 k=1,…,n 2 i=1,…,p x ij’k’ j=1,..,n 1 k=1,…,n 2 i=1,…,p ⨉ x jkj ' k ' =∑ i x ijk x ij' k ' (Linear kernel) This can reduce the data size: ℝp n1 n2 ⨉ ⨉ → ℝn1 n2 n1 n2 ⨉ ⨉ ⨉ , n 1 ,n 2 <<p
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 j u l 2 k u l 3 j' u l 4 k ' x jkj’k’ ∈ ℝn1 n2 n1 ⨉ ⨉ n2 ⨉ G u l3j’ u l1j u l2k L3 L1 L2 u l4k’ L4 Kernel Trick x jkj’k’ → k(x ijk ,x ij’k’ ):non-negative definite
)=exp(−α∑i ( x ijk −x ij ' k ' )2) Radial base function kernel k (x ijk , x ij ' k ' )=(1+∑ i x ijk x ij ' k ' ) d Polynomial kernel k(x ijk ,x ij’k’ )→ tensor decomposition
u l1j , u l2k u l 1 i ∝∑ jk x ijk u l 1 j u l 2 k P i =P χ2 [> (u l 1 i σl 1 )2] Computed P-values are corrected with considering multiple comparison corrections by Benjamini-Hochberg method. Features with corrected P-values <0.01 are selected. TD
Recompute x jkj’k’ x jkj’k’ → u l1j ⨉ u l2k TD Estimate coincidence between u l1j , u l2k and classification of (k,j) Rank i i based upon the amount of decreased coincidence u l1j ⨉ u l2k k
=687582 p k =35829 p k =1588 p k =1588 Integrate four omics profiles that share same samples, n 1 , n 2 with distinct number distinct number of features, of features, p p k k p k ⨉n 1 ⨉n 2 p k ⨉n 1 ⨉n 2 p k ⨉n 1 ⨉n 2 p k ⨉n 1 ⨉n 2 n 1 =5, n 2 =15
feature extraction. It can reduce the required data size from p⨉n to n⨉n. Since p>>n, it is a great achievement. It can integrate multiomics data sets that have distinct number of features if they share samples.