Branching + admixture simulation: effect of initialization

Last updated: 2020-01-16

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Knit directory: drift-workflow/analysis/

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Rmd	dbc9609	Jason Willwerscheid	2020-01-16	wflow_publish(“analysis/admix_sim4.Rmd”)
html	eb82fe7	Jason Willwerscheid	2020-01-15	Build site.
Rmd	93ec986	Jason Willwerscheid	2020-01-15	wflow_publish(“analysis/admix_sim4.Rmd”)
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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
eb82fe7	Jason Willwerscheid	2020-01-15
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
eb82fe7	Jason Willwerscheid	2020-01-15
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$iterations, aes(x = iter, y = elbo)) + geom_line()

Version	Author	Date
eb82fe7	Jason Willwerscheid	2020-01-15
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


plot_loadings(drift.flg$EL, labs, paste("s2:", round(drift.flg$prior_s2, 2)))

Version	Author	Date
eb82fe7	Jason Willwerscheid	2020-01-15
fb2ed3f	Jason Willwerscheid	2020-01-14

plot_cov(drift.flg)

Version	Author	Date
eb82fe7	Jason Willwerscheid	2020-01-15

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
eb82fe7	Jason Willwerscheid	2020-01-15
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$iterations, aes(x = iter, y = elbo)) + geom_line()

Version	Author	Date
eb82fe7	Jason Willwerscheid	2020-01-15
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


plot_loadings(drift.flb$EL, labs, paste("s2:", round(drift.flb$prior_s2, 2)))

Version	Author	Date
eb82fe7	Jason Willwerscheid	2020-01-15
fb2ed3f	Jason Willwerscheid	2020-01-14

plot_cov(drift.flb)

Version	Author	Date
eb82fe7	Jason Willwerscheid	2020-01-15

Hclust initialization

Initial values

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

Version	Author	Date
eb82fe7	Jason Willwerscheid	2020-01-15
fb2ed3f	Jason Willwerscheid	2020-01-14

Drift results

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

Version	Author	Date
eb82fe7	Jason Willwerscheid	2020-01-15
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


plot_loadings(drift.hclust$EL, labs, paste("s2:", round(drift.hclust$prior_s2, 2)))

Version	Author	Date
eb82fe7	Jason Willwerscheid	2020-01-15
fb2ed3f	Jason Willwerscheid	2020-01-14

plot_cov(drift.hclust)

Version	Author	Date
eb82fe7	Jason Willwerscheid	2020-01-15

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
eb82fe7	Jason Willwerscheid	2020-01-15
fb2ed3f	Jason Willwerscheid	2020-01-14

Drift results

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

Version	Author	Date
eb82fe7	Jason Willwerscheid	2020-01-15
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


plot_loadings(drift.true$EL, labs, paste("s2:", round(drift.true$prior_s2, 2)))

Version	Author	Date
eb82fe7	Jason Willwerscheid	2020-01-15
fb2ed3f	Jason Willwerscheid	2020-01-14

plot_cov(drift.true)

Version	Author	Date
eb82fe7	Jason Willwerscheid	2020-01-15

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)

Version	Author	Date
eb82fe7	Jason Willwerscheid	2020-01-15

Drift results

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

Version	Author	Date
eb82fe7	Jason Willwerscheid	2020-01-15

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


plot_loadings(drift.3factor$EL, labs, paste("s2:", round(drift.3factor$prior_s2, 2)))

Version	Author	Date
eb82fe7	Jason Willwerscheid	2020-01-15

plot_cov(drift.3factor)

Version	Author	Date
eb82fe7	Jason Willwerscheid	2020-01-15

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$iterations, aes(x = iter, y = elbo)) + geom_line()

Version	Author	Date
eb82fe7	Jason Willwerscheid	2020-01-15
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


plot_loadings(drift.rand$EL, labs, paste("s2:", round(drift.rand$prior_s2, 2)))

Version	Author	Date
eb82fe7	Jason Willwerscheid	2020-01-15
fb2ed3f	Jason Willwerscheid	2020-01-14

plot_cov(drift.rand)

Version	Author	Date
eb82fe7	Jason Willwerscheid	2020-01-15

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$iterations, aes(x = iter, y = elbo)) + geom_line()

Version	Author	Date
eb82fe7	Jason Willwerscheid	2020-01-15
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


plot_loadings(drift.perpop$EL, labs, paste("s2:", round(drift.perpop$prior_s2, 2)))

Version	Author	Date
eb82fe7	Jason Willwerscheid	2020-01-15
fb2ed3f	Jason Willwerscheid	2020-01-14

plot_cov(drift.perpop)

Version	Author	Date
eb82fe7	Jason Willwerscheid	2020-01-15

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$iterations$elbo[1]),
  FinalELBO = sapply(all.drift, function(x) x$elbo),
  ELBOdiff = sapply(all.drift, function(x) x$elbo - x$iterations$elbo[1]),
  n_iter = sapply(all.drift, function(x) max(x$iterations$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       
#> [16] pillar_1.3.1      rlang_0.4.2       lazyeval_0.2.2   
#> [19] pscl_1.5.2        readxl_1.3.1      rstudioapi_0.10  
#> [22] ebnm_0.1-24       whisker_0.3-2     Matrix_1.2-15    
#> [25] rmarkdown_1.12    labeling_0.3      munsell_0.5.0    
#> [28] mixsqp_0.3-10     broom_0.5.1       compiler_3.5.3   
#> [31] modelr_0.1.5      xfun_0.6          pkgconfig_2.0.2  
#> [34] SQUAREM_2017.10-1 htmltools_0.3.6   tidyselect_0.2.5 
#> [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       
#> [61] colorspace_1.4-1  ashr_2.2-38       rvest_0.3.4      
#> [64] knitr_1.22        haven_2.1.1

Branching + admixture simulation: effect of initialization

Jason Willwerscheid

1/10/2020

Flash initialization (greedy)

Initial values

Drift results

Flash initialization (backfit)

Initial values

Drift results

Hclust initialization

Initial values

Drift results

Initialization from “true” solution

Initial values

Drift results

Initialization using three factors

Initial values

Drift results

Random initialization

Initial values

Drift results

One factor per population

Initial values

Drift results

Results summary