# Extracting features or gene from PCA after calculating PCA for downstream analysis

The quote below is from this paper:

We performed principal component analysis (PCA) of low-coverage sequencing data to identify genes explaining variation across cells. PCA separated the cells into groups corresponding to the source populations (Fig. 2c and Supplementary Figs. 3–5), and the genes distinguishing each group reflected the biological properties of the cell types (Supplementary Fig. 5 and Supplementary Table 3). PCA of low- and high-coverage sequencing data revealed a remarkably similar graphical distribution of analyzed cells, and the majority (78%) of the top 500 genes determined by PCA were shared between PCA performed on low- and high-coverage data (Supplementary Figs. 4 and 6 and Supplementary Table 4).

How are they performing PCA and then finding out the components which contribute to the strongest PC and then taking out the genes for downstream analysis? Let's say there are genes involved in PC1 and PC2 which define a certain cell type and that distinguish it. I think for a biologist, like me, I just want to get the genes or gene list that are getting involved in determining lineage or cell type. How can I do this?

As a to test what they might be doing i did this

test<- read.csv('NON_CODING.csv',header = T,row.names = 1)

# preform PCA
pca = prcomp(t(test), center=TRUE, scale=TRUE)


then after calculating PCA i do see this in my console when I type pca

> pca
Standard deviations (1, .., p=16):
[1] 3.826851e+01 2.080405e+01 1.568739e+01 1.349256e+01 1.119348e+01 9.980547e+00 8.365034e+00
[8] 8.098841e+00 7.519507e+00 6.184505e+00 5.880260e+00 5.139609e+00 4.851091e+00 4.335606e+00
[15] 3.870918e+00 3.173450e-14

Rotation (n x k) = (2899 x 16):
PC1           PC2           PC3           PC4
5S_rRNA                      -1.090574e-02 -2.412665e-03  1.637689e-02 -3.603865e-02
AB019441.29                  -1.928250e-02  1.821083e-03 -9.713724e-03 -1.978737e-03
ABBA01017803.1               -1.823266e-02 -3.727144e-03 -9.790131e-04 -3.937062e-02
ABC14-1080714F14.1            2.438024e-02  3.816019e-03  1.657784e-04 -7.141480e-03
ABC7-481722F1.1               2.467403e-02  2.873432e-03  3.781153e-03  3.790824e-03
AC000036.4                   -1.066777e-02  2.838305e-02  2.642673e-02 -7.998608e-03
AC000089.3                   -1.249026e-02 -7.870864e-04 -2.569602e-02 -3.165073e-03
AC000120.7                   -1.313696e-02  5.923245e-03  1.633255e-02  4.077602e-02
AC000123.4                    2.415996e-02  1.029732e-02  8.930792e-03 -1.118660e-02
AC000403.4                    8.692835e-03 -3.060170e-02  3.746839e-02 -6.968971e-03
AC002064.4                   -1.799590e-02  2.147160e-02 -2.303742e-02  7.647937e-03


Now how do i find which are the list or set of genes that gives me PC1 or PC2 so on

Any suggestion or help how to get the genes that make most difference between two major component that would be very helpful

The help page explains that there is a component which:

rotation

So using using the rotation one can use it to calculate the importance of each feature in the input:

C <- chol(S <- toeplitz(.9 ^ (0:31))) # Cov.matrix and its root
set.seed(17)
X <- matrix(rnorm(32000), 1000, 32)
Z <- X %*% C  ## ==>  cov(Z) ~=  C'C = S
pZ <- prcomp(Z)
genesImportance <- Z %*% pZ$rotation genesImportance2 <- pZ$x
all.equal(genesImportance, genesImportance2)
## [1] "Mean relative difference: 0.03680457"


This new matrix has the loading for each gene for each component. So you can now select them according to how much do they participate in each component or whatever you want. Or you can use it to predict the position of samples you didn't set in the initial data.

• Im using prcomp function would that work as you mentioned princomp ? – krushnach Chandra Dec 13 '17 at 4:44
• Z%*%pZ$rotation should be the same as pZ$x (that’s in fact how pZ$x is calculated by prcomp). – Konrad Rudolph Dec 13 '17 at 10:47 • Totally true, I didn't went further in the help page, but with rotation one can predict the position of new samples not included in the original data. I'll edit it. – llrs Dec 13 '17 at 10:50 • @KonradRudolph I'm still not getting where is this coming from italic pZ$x im little ignorant as I see examples and do it so i have to do a back referencing to in R methods and tutorials , is it in my data frame as well ?with context to my data frame are you suggesting this italic"pca\$x" and what is that Z actually – krushnach Chandra Dec 13 '17 at 11:41
• @krushnachChandra Z is the initial data for the PCA. pZ is the output of prcomp. See the edit, I might not explain it clearly enough – llrs Dec 13 '17 at 11:53