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Rmd | 91313d4 | Matthew Stephens | 2020-07-20 | workflowr::wflow_publish(“tree_pca.Rmd”) |
I want to take a look at the simulated data from here where there is a tree structure plus admixture event, to see how simple things like PCA behave.
Here is the code used there to simulate the data.
set.seed(666)
n_per_pop <- 60
p <- 10000
a <- rnorm(p)
b <- rnorm(p)
c <- rnorm(p)
d <- rnorm(p, sd = 0.5)
e <- rnorm(p, sd = 0.5)
f <- rnorm(p, sd = 0.5)
g <- rnorm(p, sd = 0.5)
popA <- c(rep(1, n_per_pop), rep(0, 4 * n_per_pop))
popB <- c(rep(0, n_per_pop), rep(1, n_per_pop), rep(0, 3 * n_per_pop))
popC <- c(rep(0, 2 * n_per_pop), rep(1, n_per_pop), rep(0, 2 * n_per_pop))
popD <- c(rep(0, 3 * n_per_pop), rep(1, n_per_pop), rep(0, n_per_pop))
popE <- c(rep(0, 4 * n_per_pop), rep(1, n_per_pop))
E.factor <- (a + b + e) / 2 + (a + c + f) / 3 + (a + c + g) / 6
Y <- cbind(popA, popB, popC, popD, popE) %*%
rbind(a + b + d, a + b + e, a + c + f, a + c + g, E.factor)
Y <- Y + rnorm(5 * n_per_pop * p, sd = 0.1)
rownames(Y) <- c(rep("A",n_per_pop),rep("B",n_per_pop),rep("C",n_per_pop),rep("D",n_per_pop),
rep("E",n_per_pop))
Here I do an SVD of Y
and plot the first two left singular vectors (which I will call the PCs for now):
Y.svd = svd(Y)
plot(Y.svd$u[,1], main="PC1")
plot(Y.svd$u[,2], main="PC2")
So we can see that the second PC captures the deepest split of the tree (A,B vs C,D). This should be expected, at least in hindsight, as the first PC captures the mean term.
Check this by looking at the right singular vectors: PC2 should correspond to the drift event c-b
:
plot(Y.svd$v[,2],c-b, main="PC2 vs c-b")
And PC1 should be the mean
plot(Y.svd$v[,1],colMeans(Y), main="PC1 vs mean")
Here I repeat that process hierarchically to see how it goes…
First I split on the second PC
split =Y.svd$u[,2]>0
Y.0 = Y[!split,]
Y.1 = Y[split,]
Now apply svd to left and right splits, and plot.
The left split contains only one admixed individual (E), so the second PC nicely splits A vs B:
Y.0.svd = svd(Y.0)
plot(Y.0.svd$u[,2],type="n")
text(Y.0.svd$u[,2],rownames(Y.0))
However, the right split contains most of the admixed individuals and these throw off the PCA from splitting on C vs D (maybe not suprisingly):
Y.1.svd = svd(Y.1)
plot(Y.1.svd$u[,2],type="n")
text(Y.1.svd$u[,2],rownames(Y.1))
In fact here PC3 is closer to the split we want:
plot(Y.1.svd$u[,3],type="n")
text(Y.1.svd$u[,3],rownames(Y.1))
sessionInfo()
R version 3.6.0 (2019-04-26)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Mojave 10.14.6
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRlapack.dylib
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets methods base
loaded via a namespace (and not attached):
[1] workflowr_1.6.1 Rcpp_1.0.4.6 rprojroot_1.3-2 digest_0.6.25
[5] later_1.0.0 R6_2.4.1 backports_1.1.5 git2r_0.26.1
[9] magrittr_1.5 evaluate_0.14 stringi_1.4.6 rlang_0.4.5
[13] fs_1.3.2 promises_1.1.0 whisker_0.4 rmarkdown_2.1
[17] tools_3.6.0 stringr_1.4.0 glue_1.4.0 httpuv_1.5.2
[21] xfun_0.12 yaml_2.2.1 compiler_3.6.0 htmltools_0.4.0
[25] knitr_1.28