Last updated: 2020-01-15

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Rmd 93ec986 Jason Willwerscheid 2020-01-15 wflow_publish(“analysis/admix_sim4.Rmd”)
html fb2ed3f Jason Willwerscheid 2020-01-14 Build site.
Rmd 5657aa2 Jason Willwerscheid 2020-01-14 wflow_publish(“analysis/admix_sim4.Rmd”) 
suppressMessages({
  library(flashier)
  library(drift.alpha)
  library(ggplot2)
  library(reshape2)
  library(tidyverse)
})

In this admixture simulation, I introduce a branching event. Namely, I allow two populations to drift until time \(t\), at which point the first population diverges into Populations 1 and 2. The three populations then each experience independent drift until time \(2t\), at which point Populations 2 and 4 admix in equal proportions:

Version Author Date
465356c Jason Willwerscheid 2020-01-14
898bc42 Jason Willwerscheid 2020-01-10

The covariance matrix appears as follows:

set.seed(666)
simple.admix <- admix_graph_sim(n_per_pop = 20, p = 1000, 
                                c1 = 2, c2 = 1, c3 = 0, c4 = 0,
                                c5 = 1, c6 = 1, c7 = 0,
                                w = 0.5, sigma_e = sqrt(0.25))
plot_cov(simple.admix$covmat, as.is = TRUE)

Version Author Date
fb2ed3f Jason Willwerscheid 2020-01-14

Flash initialization (greedy)

Initial values

fl <- flash(simple.admix$Y, prior.family = c(prior.bimodal(), prior.normal()))
#> Adding factor 1 to flash object...
#> Adding factor 2 to flash object...
#> Adding factor 3 to flash object...
#> Adding factor 4 to flash object...
#> Adding factor 5 to flash object...
#> Adding factor 6 to flash object...
#> Factor doesn't significantly increase objective and won't be added.
#> Wrapping up...
#> Done.
#> Nullchecking 5 factors...
#> Done.
labs <- rep(c("A", "B", "C", "D"), each = 20)
plot_loadings(fl$flash.fit$EF[[1]], labs)

Version Author Date
fb2ed3f Jason Willwerscheid 2020-01-14

Drift results

drift.flg <- drift(init_from_flash(fl), miniter = 2, maxiter = 500, tol = 0.01, verbose = FALSE)
ggplot(drift.flg$elbo.df, aes(x = iter, y = elbo)) + geom_line()

Version Author Date
fb2ed3f Jason Willwerscheid 2020-01-14 
drift.flg[c("elbo", "KL_l", "KL_f")]
#> $elbo
#> [1] -65978.01
#> 
#> $KL_l
#> [1]   -7.09820  -45.12987 -138.11891 -135.45788    0.00000
#> 
#> $KL_f
#> [1] -7733.409

lblr <- paste("s2:", round(drift.flg$prior_s2, 2))
names(lblr) <- 1:length(drift.flg$prior_s2)
lblr <- as_labeller(lblr)
plot_loadings(drift.flg$EL, labs, lblr)

Version Author Date
fb2ed3f Jason Willwerscheid 2020-01-14 
plot_cov(drift.flg$EL * rep(sqrt(drift.flg$prior_s2), each = 80))

Flash initialization (backfit)

Initial values

fl <- fl %>% flash.backfit() %>% flash.nullcheck(remove = TRUE)
#> Backfitting 5 factors (tolerance: 1.19e-03)...
#>   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...
#> Wrapping up...
#> Done.
#> Nullchecking 5 factors...
#> Wrapping up...
#> Done.
plot_loadings(fl$flash.fit$EF[[1]], labs)

Version Author Date
fb2ed3f Jason Willwerscheid 2020-01-14

Drift results

drift.flb <- drift(init_from_flash(fl), miniter = 2, maxiter = 500, tol = 0.01, verbose = FALSE)
ggplot(drift.flb$elbo.df, aes(x = iter, y = elbo)) + geom_line()

Version Author Date
fb2ed3f Jason Willwerscheid 2020-01-14 
drift.flb[c("elbo", "KL_l", "KL_f")]
#> $elbo
#> [1] -66052.64
#> 
#> $KL_l
#> [1] -188.64520  -45.12987 -144.26448
#> 
#> $KL_f
#> [1] -7753.763

lblr <- paste("s2:", round(drift.flb$prior_s2, 2))
names(lblr) <- 1:length(drift.flb$prior_s2)
lblr <- as_labeller(lblr)
plot_loadings(drift.flb$EL, labs, lblr)

Version Author Date
fb2ed3f Jason Willwerscheid 2020-01-14 
plot_cov(drift.flb$EL * rep(sqrt(drift.flb$prior_s2), each = 80))

Hclust initialization

Initial values

init <- init_using_hclust(simple.admix$Y, k = 4)
plot_loadings(init$EL, labs)

Version Author Date
fb2ed3f Jason Willwerscheid 2020-01-14

Drift results

drift.hclust <- drift(init, miniter = 2, maxiter = 500, tol = 0.01, verbose = FALSE)
ggplot(drift.hclust$elbo.df, aes(x = iter, y = elbo)) + geom_line()

Version Author Date
fb2ed3f Jason Willwerscheid 2020-01-14 
drift.hclust[c("elbo", "KL_l", "KL_f")]
#> $elbo
#> [1] -67207.06
#> 
#> $KL_l
#> [1]  1.307399e-12 -5.586292e+01 -5.586292e+01 -4.512987e+01 -4.512987e+01
#> [6] -7.105427e-15 -4.512987e+01
#> 
#> $KL_f
#> [1] -9026.252

lblr <- paste("s2:", round(drift.hclust$prior_s2, 2))
names(lblr) <- 1:length(drift.hclust$prior_s2)
lblr <- as_labeller(lblr)
plot_loadings(drift.hclust$EL, labs, lblr)

Version Author Date
fb2ed3f Jason Willwerscheid 2020-01-14 
plot_cov(drift.hclust$EL * rep(sqrt(drift.hclust$prior_s2), each = 80))

Initialization from “true” solution

Initial values

# I can't give init_from_EL a singular matrix, so I need to fudge the loadings a bit.
EL <- cbind(c(rep(1, 40), rep(0.25, 20), rep(0, 20)),
            c(rep(1, 20), rep(0, 60)),
            c(rep(0, 20), rep(1, 20), rep(0.5, 20), rep(0, 20)),
            c(rep(0, 40), rep(0.5, 20), rep(1, 20)))
init <- init_from_EL(simple.admix$Y, EL)
plot_loadings(init$EL, labs)

Version Author Date
fb2ed3f Jason Willwerscheid 2020-01-14

Drift results

drift.true <- drift(init, miniter = 2, maxiter = 500, tol = 0.01, verbose = FALSE)
ggplot(drift.true$elbo.df, aes(x = iter, y = elbo)) + geom_line()

Version Author Date
fb2ed3f Jason Willwerscheid 2020-01-14 
drift.true[c("elbo", "KL_l", "KL_f")]
#> $elbo
#> [1] -66036.96
#> 
#> $KL_l
#> [1] -134.57291  -45.12987 -129.44262 -143.92071
#> 
#> $KL_f
#> [1] -7671.43

lblr <- paste("s2:", round(drift.true$prior_s2, 2))
names(lblr) <- 1:length(drift.true$prior_s2)
lblr <- as_labeller(lblr)
plot_loadings(drift.true$EL, labs, lblr)

Version Author Date
fb2ed3f Jason Willwerscheid 2020-01-14 
plot_cov(drift.true$EL * rep(sqrt(drift.true$prior_s2), each = 80))

Initialization using three factors

Initial values

EL <- cbind(c(rep(1, 20), rep(0, 60)),
            c(rep(0, 20), rep(1, 20), rep(0.5, 20), rep(0, 20)),
            c(rep(0, 40), rep(0.5, 20), rep(1, 20)))
init <- init_from_EL(simple.admix$Y, EL)
plot_loadings(init$EL, labs)

Drift results

drift.3factor <- drift(init, miniter = 2, maxiter = 500, tol = 0.01, verbose = FALSE)
ggplot(drift.3factor$elbo.df, aes(x = iter, y = elbo)) + geom_line()

drift.3factor[c("elbo", "KL_l", "KL_f")]
#> $elbo
#> [1] -66023.31
#> 
#> $KL_l
#> [1]  -45.12987 -139.63585 -143.91220
#> 
#> $KL_f
#> [1] -7772.31

lblr <- paste("s2:", round(drift.3factor$prior_s2, 2))
names(lblr) <- 1:length(drift.3factor$prior_s2)
lblr <- as_labeller(lblr)
plot_loadings(drift.3factor$EL, labs, lblr)

plot_cov(drift.3factor$EL * rep(sqrt(drift.3factor$prior_s2), each = 80))

Random initialization

Initial values

init <- init_using_hclust(simple.admix$Y, k = 4, as_tree = FALSE)
init$EL <- matrix(rnorm(80 * 4), ncol = 4)
init$EL2 <- init$EL^2
plot_loadings(init$EL, labs)

Version Author Date
fb2ed3f Jason Willwerscheid 2020-01-14

Drift results

drift.rand <- drift(init, miniter = 2, maxiter = 500, tol = 0.01, verbose = FALSE)
ggplot(drift.rand$elbo.df, aes(x = iter, y = elbo)) + geom_line()

Version Author Date
fb2ed3f Jason Willwerscheid 2020-01-14 
drift.rand[c("elbo", "KL_l", "KL_f")]
#> $elbo
#> [1] -66456
#> 
#> $KL_l
#> [1] -2.108788e+02 -1.810760e+02 -1.566267e+02 -1.421085e-14
#> 
#> $KL_f
#> [1] -7983.783

lblr <- paste("s2:", round(drift.rand$prior_s2, 2))
names(lblr) <- 1:length(drift.rand$prior_s2)
lblr <- as_labeller(lblr)
plot_loadings(drift.rand$EL, labs, lblr)

Version Author Date
fb2ed3f Jason Willwerscheid 2020-01-14 
plot_cov(drift.rand$EL * rep(sqrt(drift.rand$prior_s2), each = 80))

One factor per population

Initial values

init <- init_using_hclust(simple.admix$Y, k = 4, as_tree = FALSE)
plot_loadings(init$EL, labs)

Version Author Date
fb2ed3f Jason Willwerscheid 2020-01-14

Drift results

drift.perpop <- drift(init, miniter = 2, maxiter = 500, tol = 0.01, verbose = FALSE)
ggplot(drift.perpop$elbo.df, aes(x = iter, y = elbo)) + geom_line()

Version Author Date
fb2ed3f Jason Willwerscheid 2020-01-14 
drift.perpop[c("elbo", "KL_l", "KL_f")]
#> $elbo
#> [1] -67865.59
#> 
#> $KL_l
#> [1] -45.12987 -45.12987 -66.30345 -45.12987
#> 
#> $KL_f
#> [1] -9726.515

lblr <- paste("s2:", round(drift.perpop$prior_s2, 2))
names(lblr) <- 1:length(drift.perpop$prior_s2)
lblr <- as_labeller(lblr)
plot_loadings(drift.perpop$EL, labs, lblr)

Version Author Date
fb2ed3f Jason Willwerscheid 2020-01-14 
plot_cov(drift.perpop$EL * rep(sqrt(drift.perpop$prior_s2), each = 80))

Results summary

all.drift <- list(drift.flg, drift.flb, drift.hclust, drift.true, drift.3factor, drift.rand, drift.perpop)

res.df <- data.frame(
  Name = c("flash.greedy", "flash.backfit", "hclust", "true.4factor", "three.factors", "random", "one.factor.per.pop"),
  InitialELBO = sapply(all.drift, function(x) x$elbo.df$elbo[1]),
  FinalELBO = sapply(all.drift, function(x) x$elbo),
  ELBOdiff = sapply(all.drift, function(x) x$elbo - x$elbo.df$elbo[1]),
  n_iter = sapply(all.drift, function(x) max(x$elbo.df$iter)),
  KL_l = sapply(all.drift, function(x) sum(x$KL_l)),
  KL_f = sapply(all.drift, function(x) x$KL_f),
  ResidS2 = sapply(all.drift, function(x) x$resid_s2)
)

knitr::kable(res.df, digits = 3)
Name InitialELBO FinalELBO ELBOdiff n_iter KL_l KL_f ResidS2
flash.greedy -66879.59 -65978.01 901.575 98 -325.805 -7733.409 0.249
flash.backfit -66054.71 -66052.64 2.070 16 -378.040 -7753.763 0.249
hclust -67704.69 -67207.06 497.625 194 -247.115 -9026.252 0.249
true.4factor -66206.07 -66036.96 169.116 28 -453.066 -7671.430 0.249
three.factors -66030.96 -66023.31 7.653 7 -328.678 -7772.310 0.249
random -133446.73 -66456.00 66990.736 156 -548.582 -7983.783 0.249
one.factor.per.pop -67889.72 -67865.59 24.138 50 -201.693 -9726.515 0.249 


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_0.0.6 flashier_0.2.2   
#> 
#> loaded via a namespace (and not attached):
#>  [1] Rcpp_1.0.1        lubridate_1.7.4   lattice_0.20-38  
#>  [4] assertthat_0.2.1  rprojroot_1.3-2   digest_0.6.18    
#>  [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  
#> [13] evaluate_0.13     highr_0.8         httr_1.4.0       
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#> [22] ebnm_0.1-24       whisker_0.3-2     Matrix_1.2-15    
#> [25] rmarkdown_1.12    labeling_0.3      munsell_0.5.0    
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#> [37] workflowr_1.2.0   codetools_0.2-16  crayon_1.3.4     
#> [40] withr_2.1.2       MASS_7.3-51.1     grid_3.5.3       
#> [43] nlme_3.1-137      jsonlite_1.6      gtable_0.3.0     
#> [46] git2r_0.25.2      magrittr_1.5      scales_1.0.0     
#> [49] cli_1.1.0         stringi_1.4.3     fs_1.2.7         
#> [52] doParallel_1.0.14 xml2_1.2.0        generics_0.0.2   
#> [55] iterators_1.0.10  tools_3.5.3       glue_1.3.1       
#> [58] hms_0.4.2         parallel_3.5.3    yaml_2.2.0       
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#> [64] knitr_1.22        haven_2.1.1