Tree simulation (2pop)
jhmarcus
2019-11-20
Last updated: 2019-12-01
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Knit directory: drift-workflow/analysis/
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Here I simulate Gaussian data under a simple 2 population tree i.e. a split and explore the interpretability of flashier fits to the data. The data generated under this tree can be represented as a probabilistic matrix factorization model with 1 shared factor and 2 population specific factors … this is what we’d like to recover.
Import
Import the required packages and load helper scripts for this analysis:
library(ggplot2)
library(dplyr)
library(tidyr)
library(flashier)
source("../code/viz.R")
source("../code/sim.R")
source("../code/ebnm_functions.R")Functions
Load functions for this analysis:
plot_2pop_loadings = function(L){
l_df = as.data.frame(L)
colnames(l_df) = 1:ncol(l_df)
l_df$ID = 1:nrow(l_df)
l_df$pop = c(rep("Pop1", n_per_pop), rep("Pop2", n_per_pop))
gath_l_df = l_df %>% gather(K, value, -ID, -pop)
p = ggplot(gath_l_df, aes(x=ID, y=value, color=pop)) +
geom_point() +
facet_wrap(K~.) +
theme_bw()
return(p)
}Simulate
Here I simulate the data generated from a simple 2 population tree. I set the residual std. dev. to .5 and prior variances to 1. There are 50 individuals per population and 10000 simulated SNPs:
##### sim #####
set.seed(235)
n_per_pop = 50
sigma_e = .5
sigma_b = rep(1, 3)
p = 10000
sim_res = tree_simulation_2pop(n_per_pop, p, sigma_e, sigma_b)
K = 3
Y = sim_res$Y
##### viz #####
plot_2pop_loadings(sim_res$L)
Greedy
Run the greedy algorithm which seems to recover the tree:
##### fit #####
flash_res = flash.init(Y, var.type=0) %>%
flash.add.greedy(Kmax=K,
prior.family=list(c(prior.bimodal(),
prior.normal()),
c(prior.bimodal(),
prior.normal()),
c(prior.bimodal(),
prior.normal()),
c(prior.bimodal(),
prior.normal()),
c(prior.bimodal(),
prior.normal()),
c(prior.bimodal(),
prior.normal())))Adding factor 1 to flash object...
Adding factor 2 to flash object...
Adding factor 3 to flash object...
Wrapping up...
Done.##### viz #####
pm = flash_res$loadings.pm[[1]]
plot_2pop_loadings(pm)
Initialize from greedy
Initialize the backfitting algorithm with the greedy solution which recovers a sparser representation of the tree i.e. it zeros out the shared factor:
##### fit #####
flash_res = flash.init(Y, var.type=0) %>%
flash.add.greedy(Kmax=K,
prior.family=list(c(prior.bimodal(),
prior.normal()),
c(prior.bimodal(),
prior.normal()),
c(prior.bimodal(),
prior.normal()),
c(prior.bimodal(),
prior.normal()),
c(prior.bimodal(),
prior.normal()),
c(prior.bimodal(),
prior.normal()))) %>%
flash.backfit()Adding factor 1 to flash object...
Adding factor 2 to flash object...
Adding factor 3 to flash object...
Wrapping up...
Done.
Backfitting 3 factors (tolerance: 1.49e-02)...
Difference between iterations is within 1.0e+03...
Difference between iterations is within 1.0e+02...
Difference between iterations is within 1.0e+01...
Difference between iterations is within 1.0e+00...
Difference between iterations is within 1.0e-01...
Wrapping up...
Done.##### viz #####
pm = flash_res$loadings.pm[[1]]
elbo_gr_bf = flash_res$elbo
plot_2pop_loadings(flash_res$loadings.pm[[1]])
Initialize from truth and fix the priors
Initialize from the truth and fix the prior which as expected recovers the tree representation we want:
##### fit #####
flash_res = flash.init(Y, var.type=0) %>%
flash.init.factors(EF=list(u=sim_res$L, d=rep(1, K), v=sim_res$Z),
prior.family=list(c(prior.twopm(pi = c(0.0, 1.0)),
prior.fixnormal()),
c(prior.twopm(pi = c(.5, .5)),
prior.fixnormal()),
c(prior.twopm(pi = c(.5, .5)),
prior.fixnormal()))) %>%
flash.backfit()Backfitting 3 factors (tolerance: 1.49e-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...
Wrapping up...
Done.##### viz #####
pm = flash_res$loadings.pm[[1]]
elbo_init_true = flash_res$elbo
plot_2pop_loadings(flash_res$loadings.pm[[1]])
Compare the final elbos between the solution where we initialize from the truth and fix the prior to the solution where we initialize from the greedy fit and don’t fix the prior:
print(elbo_init_true)[1] -809767.7print(elbo_gr_bf)[1] -786235.7print(elbo_init_true>elbo_gr_bf)[1] FALSEThe solution where we initialize with greedy fit has a higher ELBO then when we initialize from the truth and fix the prior.
sessionInfo()R version 3.6.1 (2019-07-05)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: macOS Mojave 10.14.2
Matrix products: default
BLAS/LAPACK: /Users/jhmarcus/miniconda3/envs/flash_e/lib/R/lib/libRblas.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] ebnm_0.1-24 ashr_2.2-38 flashier_0.2.4 tidyr_0.8.3
[5] dplyr_0.8.0.1 ggplot2_3.1.1
loaded via a namespace (and not attached):
[1] Rcpp_1.0.2 compiler_3.6.1 pillar_1.3.1
[4] git2r_0.26.1 plyr_1.8.4 workflowr_1.4.0
[7] iterators_1.0.12 tools_3.6.1 digest_0.6.18
[10] lattice_0.20-38 evaluate_0.13 tibble_2.1.1
[13] gtable_0.3.0 pkgconfig_2.0.2 rlang_0.4.0
[16] foreach_1.4.7 Matrix_1.2-17 parallel_3.6.1
[19] yaml_2.2.0 xfun_0.6 withr_2.1.2
[22] stringr_1.4.0 knitr_1.22 fs_1.2.7
[25] rprojroot_1.3-2 grid_3.6.1 tidyselect_0.2.5
[28] glue_1.3.1 R6_2.4.0 rmarkdown_1.12
[31] mixsqp_0.2-4 purrr_0.3.2 magrittr_1.5
[34] MASS_7.3-51.4 codetools_0.2-16 backports_1.1.4
[37] scales_1.0.0 htmltools_0.3.6 assertthat_0.2.1
[40] colorspace_1.4-1 labeling_0.3 stringi_1.4.3
[43] pscl_1.5.2 doParallel_1.0.15 lazyeval_0.2.2
[46] munsell_0.5.0 truncnorm_1.0-8 SQUAREM_2017.10-1
[49] crayon_1.3.4