Last updated: 2020-06-25

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Rmd af4af59 Jason Willwerscheid 2020-06-25 workflowr::wflow_publish(“analysis/extrapolate2.Rmd”) 
suppressMessages({
  library(flashier)
  library(drift.alpha)
  library(tidyverse)
})

I test extrapolation on a four-population tree with an additional admixed population. The latter has admixture proportions of \(1/2\) Population B, \(1/3\) Population C, and \(1/6\) Population D.

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)

plot_dr <- function(dr) {
  sd <- sqrt(dr$prior_s2)
  L <- dr$EL
  LDsqrt <- L %*% diag(sd)
  K <- ncol(LDsqrt)
  plot_loadings(LDsqrt[,1:K], rep(c("A", "B", "C", "D", "E"), each = n_per_pop)) +
    scale_color_brewer(palette="Set2")
}

Extrapolation again finds a much better solution within 2000 iterations:

greedy <- init_from_data(Y)

dr_slow <- drift(greedy, verbose = FALSE, extrapolate = FALSE, maxiter = 2000, tol = 1e-4)

options(extrapolate.control = list(beta.max = 1))
dr_xtrap <- drift(greedy, verbose = FALSE, extrapolate = TRUE, maxiter = 2000, tol = 1e-4)

df <- dr_slow$iterations %>%
  mutate(extrapolate = "FALSE") %>%
  bind_rows(dr_xtrap$iterations %>% mutate(extrapolate = "TRUE"))
ggplot(df, aes(x = iter, y = elbo, col = extrapolate)) + geom_line()

Without extrapolation, the solution is very difficult to interpret:

plot_dr(dr_slow)

The extrapolated solution isn’t perfect, but it’s pretty good. It captures the underlying tree fairly well, and it correctly represents population E as an admixture (even though it’s missing the contribution from Population C).

plot_dr(dr_xtrap)


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      ggplot2_3.2.0     tidyverse_1.2.1  
#> [10] drift.alpha_0.0.9 flashier_0.2.4   
#> 
#> loaded via a namespace (and not attached):
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#>  [4] lattice_0.20-38    assertthat_0.2.1   rprojroot_1.3-2   
#>  [7] digest_0.6.18      truncnorm_1.0-8    R6_2.4.0          
#> [10] cellranger_1.1.0   plyr_1.8.4         backports_1.1.3   
#> [13] evaluate_0.13      httr_1.4.0         pillar_1.3.1      
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#> [37] withr_2.1.2        crayon_1.3.4       grid_3.5.3        
#> [40] nlme_3.1-137       jsonlite_1.6       gtable_0.3.0      
#> [43] git2r_0.25.2       magrittr_1.5       scales_1.0.0      
#> [46] cli_1.1.0          stringi_1.4.3      reshape2_1.4.3    
#> [49] fs_1.2.7           xml2_1.2.0         generics_0.0.2    
#> [52] RColorBrewer_1.1-2 tools_3.5.3        glue_1.3.1        
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