Admixture simulation 1: Simple admixture
Jason Willwerscheid
1/10/2020
Last updated: 2020-01-10
Checks: 6 0
Knit directory: drift-workflow/analysis/
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| File | Version | Author | Date | Message |
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| Rmd | 6138c6e | Jason Willwerscheid | 2020-01-10 | correct cov calculation |
| html | fe3a797 | Jason Willwerscheid | 2020-01-10 | Build site. |
| Rmd | 8f061f6 | Jason Willwerscheid | 2020-01-10 | workflowr::wflow_publish(“analysis/admix_sim1.Rmd”) |
| html | 65769eb | Jason Willwerscheid | 2020-01-10 | Build site. |
| Rmd | e32e21c | Jason Willwerscheid | 2020-01-10 | workflowr::wflow_publish(“analysis/admix_sim1.Rmd”) |
suppressMessages({
library(flashier)
library(drift.alpha)
library(ggplot2)
library(reshape2)
library(tidyverse)
})I want to do a series of simulations to see whether driftr can handle very simple admixture events. I start with the simplest event I could imagine: allow two populations to drift until time \(t\), admix them in equal proportions (still at time \(t\)), and then terminate immediately. Means for Populations 1 and 3 (the non-admixed populations) will be independent, while means for Population 2 will simply be averaged from the means for Populations 1 and 3:
| Version | Author | Date |
|---|---|---|
| be4c90a | Jason Willwerscheid | 2020-01-10 |
The covariance matrix appears as follows:
set.seed(666)
simple.admix <- admix_graph_sim(n_per_pop = 20, p = 500)
plot_cov(simple.admix$CovMat, as.is = TRUE)
| Version | Author | Date |
|---|---|---|
| 65769eb | Jason Willwerscheid | 2020-01-10 |
Flash: greedy
Even in this simple case, greedy flash does not do well. It appears more as if Populations 1 and 3 split off from a main trunk and less as if a subsequent admixture event generated Population 2.
Y <- simple.admix$Y
fl.greed <- flash.init(Y) %>%
flash.add.greedy(Kmax = 20, prior.family = c(prior.bimodal(), prior.normal()), tol = 1)#> Adding factor 1 to flash object...
#> Adding factor 2 to flash object...
#> Adding factor 3 to flash object...
#> Adding factor 4 to flash object...
#> Factor doesn't significantly increase objective and won't be added.
#> Wrapping up...
#> Done.
labs <- rep(c("A", "B", "C"), each = 20)
plot_loadings(fl.greed$flash.fit$EF[[1]], labs)
| Version | Author | Date |
|---|---|---|
| 65769eb | Jason Willwerscheid | 2020-01-10 |
plot_cov(fl.greed$flash.fit$EF[[1]] * rep(sapply(fl.greed$fitted.g[[2]], `[[`, "sd"), each = 60))
Flash: backfit
Backfitting doesn’t solve the problem:
fl.bf <- fl.greed %>% flash.backfit(maxiter = 30)#> Backfitting 3 factors (tolerance: 4.47e-04)...
#> Difference between iterations is within 1.0e+03...
#> Difference between iterations is within 1.0e+02...
#> Difference between iterations is within 1.0e+01...
#> Difference between iterations is within 1.0e+00...
#> Difference between iterations is within 1.0e-01...
#> Difference between iterations is within 1.0e-02...
#> Difference between iterations is within 1.0e-03...
#> --Maximum number of iterations reached!
#> Wrapping up...
#> Done.plot_loadings(fl.bf$flash.fit$EF[[1]], labs)
| Version | Author | Date |
|---|---|---|
| 65769eb | Jason Willwerscheid | 2020-01-10 |
plot_cov(fl.bf$flash.fit$EF[[1]] * rep(sapply(fl.greed$fitted.g[[2]], `[[`, "sd"), each = 60))
Drift: \(k\) correctly specified
When it’s initialized to a tree with two leaves, driftr almost exactly recovers the correct covariance matrix. Further, the loadings are easily interpretable as the result of an admixture event:
drift.res <- init_using_hclust(simple.admix$Y, k = 2) %>%
drift(miniter = 2, maxiter = 30)#> 1 : -1062.472
#> 2 : 14785.771
#> 3 : 21948.179
#> 4 : 22189.537
#> 5 : 22225.170
#> 6 : 22255.595
#> 7 : 22280.569
#> 8 : 22297.774
#> 9 : 22308.707
#> 10 : 22315.724
#> 11 : 22320.603
#> 12 : 22324.315
#> 13 : 22327.361
#> 14 : 22330.136
#> 15 : 22333.208
#> 16 : 22337.342
#> 17 : 22341.704
#> 18 : 22345.266
#> 19 : 22348.987
#> 20 : 22353.588
#> 21 : 22358.211
#> 22 : 22360.427
#> 23 : 22361.833
#> 24 : 22362.771
#> 25 : 22363.462
#> 26 : 22364.382
#> 27 : 22365.220
#> 28 : 22366.047
#> 29 : 22366.649
#> 30 : 22367.367plot_loadings(drift.res$EL, labs)
| Version | Author | Date |
|---|---|---|
| 65769eb | Jason Willwerscheid | 2020-01-10 |
plot_cov(drift.res$EL * rep(sqrt(drift.res$prior_s2), each = 60))
Drift: \(k\) over-specified
Finally, I over-specify \(k\) by initializing to a three-leaf tree. The loadings incorrectly suggest that Population 3 split from 1 and 2 first, and then Populations 1 and 2 diverged. Note, however, that the ELBO is quite a bit lower than the ELBO of the driftr solution initialized using the correct \(k\):
drift.res <- init_using_hclust(simple.admix$Y, k = 3) %>%
drift(miniter = 2, maxiter = 30)#> 1 : 20844.551
#> 2 : 21010.666
#> 3 : 21091.488
#> 4 : 21134.886
#> 5 : 21161.693
#> 6 : 21179.963
#> 7 : 21193.468
#> 8 : 21204.050
#> 9 : 21212.566
#> 10 : 21219.477
#> 11 : 21225.115
#> 12 : 21229.752
#> 13 : 21233.606
#> 14 : 21236.846
#> 15 : 21239.600
#> 16 : 21241.964
#> 17 : 21244.014
#> 18 : 21245.804
#> 19 : 21247.381
#> 20 : 21248.779
#> 21 : 21250.027
#> 22 : 21251.146
#> 23 : 21252.155
#> 24 : 21253.069
#> 25 : 21253.902
#> 26 : 21254.662
#> 27 : 21255.359
#> 28 : 21256.001
#> 29 : 21256.594
#> 30 : 21257.143plot_loadings(drift.res$EL, labs)
| Version | Author | Date |
|---|---|---|
| 65769eb | Jason Willwerscheid | 2020-01-10 |
plot_cov(drift.res$EL * rep(sqrt(drift.res$prior_s2), each = 60))
sessionInfo()#> R version 3.5.3 (2019-03-11)
#> 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/3.5/Resources/lib/libRblas.0.dylib
#> LAPACK: /Library/Frameworks/R.framework/Versions/3.5/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
#>
#> other attached packages:
#> [1] forcats_0.4.0 stringr_1.4.0 dplyr_0.8.0.1
#> [4] purrr_0.3.2 readr_1.3.1 tidyr_0.8.3
#> [7] tibble_2.1.1 tidyverse_1.2.1 reshape2_1.4.3
#> [10] ggplot2_3.2.0 drift.alpha_0.0.5 flashier_0.2.2
#>
#> loaded via a namespace (and not attached):
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#> [7] foreach_1.4.4 truncnorm_1.0-8 R6_2.4.0
#> [10] cellranger_1.1.0 plyr_1.8.4 backports_1.1.3
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#> [31] xfun_0.6 pkgconfig_2.0.2 SQUAREM_2017.10-1
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