Last updated: 2019-05-03

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File	Version	Author	Date	Message
Rmd	280cc1f	jhmarcus	2019-05-03	added printing of noise in simpler
html	280cc1f	jhmarcus	2019-05-03	added printing of noise in simpler
Rmd	912c5ff	jhmarcus	2019-05-03	added simpler tree sim
html	912c5ff	jhmarcus	2019-05-03	added simpler tree sim

In this analysis I simulate data from the same tree as described in Simple Tree Simulation (also see below) but parameterize the simulation as a factor analysis model i.e. simulating under the model we are fitting. I also removed the additional binomial sampling from the allele frequencies at the tips and just directly modeled Gaussian data.

Import

Here I import the some required packages:

library(ggplot2)
library(dplyr)
library(tidyr)
library(flashier)
source("../code/viz.R")

Functions

#' @title Simpler Tree Simulation
#'
#' @description Simulates genotypes under a simple population 
#'              tree as described in Pickrell and Pritchard 2012 via 
#'              a factor analysis model:
#'
#'              https://journals.plos.org/plosgenetics/article?id=10.1371/journal.pgen.1002967
#'
#' @param n_per_pop number of individuals per population
#' @param p number of SNPs
#' @param sigma_e std. dev of noise
simpler_tree_simulation = function(n_per_pop, p, sigma_e){
  
  n = n_per_pop * 4
  L = matrix(0, nrow=4*n_per_pop, ncol=6)
  L[1:n_per_pop, 2] = L[1:n_per_pop, 6] = 1
  L[(n_per_pop + 1):(2*n_per_pop), 2] = L[(n_per_pop + 1):(2*n_per_pop), 5] = 1
  L[(2*n_per_pop + 1):(3*n_per_pop), 1] = L[(2*n_per_pop + 1):(3*n_per_pop), 3] = 1
  L[(3*n_per_pop + 1):(4*n_per_pop), 1] = L[(3*n_per_pop + 1):(4*n_per_pop), 4] = 1

  Z = matrix(rnorm(p*6, 0, 1), nrow=p, ncol=6)
  E = matrix(rnorm(n*p, 0, sigma_e), nrow=n, ncol=p)
  Y = L %*% t(Z) + E

  res = list(Y=Y, L=L, Z=Z)
  
  return(res)
  
}

plot_flash_loadings = function(flash_fit, n_per_pop){

  l_df = as.data.frame(flash_fit$loadings$normalized.loadings[[1]])
  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),
               rep("Pop3", n_per_pop), rep("Pop4", n_per_pop))
  
  gath_l_df = l_df %>% gather(K, value, -ID, -pop) 

  p1 = ggplot(gath_l_df, aes(x=ID, y=value, color=pop)) + 
       geom_point() +
       facet_wrap(K~., scale="free") +
       theme_bw() 
  
  p2 = structure_plot(gath_l_df, 
                      colset="Set3", 
                      facet_grp="pop", 
                      facet_levels=paste0("Pop", 1:4),
                      keep_leg=TRUE,
                      fact_type="nonnegative") 
  
  return(list(p1=p1, p2=p2))
  
}

Low Noise

I display the true loadings matrix and population covariance matrix for a simulation of 20 individuals per population with 10000 independent SNPs.

set.seed(1234)

n_per_pop = 20
sigma_e = .01
p = 10000

sim_res = simpler_tree_simulation(n_per_pop, p, sigma_e)
Y = sim_res$Y
L = sim_res$L
LLt = L %*% t(L)
plot_covmat(LLt)

Version	Author	Date
912c5ff	jhmarcus	2019-05-03

l_df = data.frame(L)
colnames(l_df) = paste0("K", 1:6)
l_df$iid = 1:nrow(L)
l_df$pop = c(rep("Pop1", n_per_pop), rep("Pop2", n_per_pop),
             rep("Pop3", n_per_pop), rep("Pop4", n_per_pop))

gath_l_df = l_df %>% gather(K, value, -iid, -pop)
p = ggplot(gath_l_df, aes(x=iid, y=value, color=pop)) + 
    geom_point() +
    facet_wrap(K~., scale="free") +
    theme_bw()
p

Version	Author	Date
912c5ff	jhmarcus	2019-05-03

Here we can see the block-like structure to the covariance matrix.

Greedy

I fit greedy flash:

flash_fit = flashier::flashier(Y, 
                               greedy.Kmax=10,
                               prior.type=c("nonnegative", "point.normal"),
                               ebnm.param=list(fixg=TRUE, g=list(pi0=0, a=1, mu=0)),
                               var.type=0,
                               backfit="none")

Initializing flash object...
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...
Adding factor 7 to flash object...
Factor doesn't increase objective and won't be added.
Nullchecking 6 factors...
Wrapping up...
Done.

Lhat = flash_fit$loadings$normalized.loadings[[1]]
p_res = plot_flash_loadings(flash_fit, n_per_pop)
print(p_res$p1)

Version	Author	Date
912c5ff	jhmarcus	2019-05-03

print(p_res$p2)

Version	Author	Date
912c5ff	jhmarcus	2019-05-03

print(paste0("objective=", flash_fit$objective))

[1] "objective=72473.5307279732"

print(paste0("est_sd=", sqrt(1 / flash_fit$fit$tau)))

[1] "est_sd=0.188101535612845"

print(plot_covmat(Lhat %*% t(Lhat)))

We can see the loadings matrix is recovered pretty well.

Backfit

I add a final backfitting step.

flash_fit = flashier::flashier(Y, 
                               flash.init = flash_fit,
                               prior.type=c("nonnegative", "point.normal"),
                               ebnm.param=list(fixg=TRUE, g=list(pi0=0, a=1, mu=0)),
                               var.type=0,
                               backfit="final",
                               backfit.order="dropout",
                               backfit.reltol=10)

Initializing flash object...
Adding factor 7 to flash object...
Factor doesn't increase objective and won't be added.
Backfitting 6 factors...
Nullchecking 6 factors...
Wrapping up...
Done.

Lhat = flash_fit$loadings$normalized.loadings[[1]]
p_res = plot_flash_loadings(flash_fit, n_per_pop)
print(p_res$p1)

Version	Author	Date
912c5ff	jhmarcus	2019-05-03

print(p_res$p2)

Version	Author	Date
912c5ff	jhmarcus	2019-05-03

print(paste0("objective=", flash_fit$objective))

[1] "objective=2169849.35240762"

print(paste0("est_sd=", sqrt(1 / flash_fit$fit$tau)))

[1] "est_sd=0.0101417353225806"

print(plot_covmat(Lhat %*% t(Lhat)))

The loadings matrix is recovered pretty well and also looks pretty similar to the greedy run

Medium Noise

Now I simulate data with a standard deviation of the errors set to be 5 times higher than the last simulation:

n_per_pop = 20
sigma_e = .5
p = 10000
sim_res = simpler_tree_simulation(n_per_pop, 10000, sigma_e)
Y = sim_res$Y

Greedy

flash_fit = flashier::flashier(Y, 
                               greedy.Kmax=10,
                               prior.type=c("nonnegative", "point.normal"),
                               ebnm.param=list(fixg=TRUE, g=list(pi0=0, a=1, mu=0)),
                               var.type=0,
                               backfit="none")

Initializing flash object...
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...
Adding factor 7 to flash object...
Factor doesn't increase objective and won't be added.
Nullchecking 6 factors...
Wrapping up...
Done.

Lhat = flash_fit$loadings$normalized.loadings[[1]]
p_res = plot_flash_loadings(flash_fit, n_per_pop)
print(p_res$p1)

Version	Author	Date
912c5ff	jhmarcus	2019-05-03

print(p_res$p2)

Version	Author	Date
912c5ff	jhmarcus	2019-05-03

print(paste0("objective=", flash_fit$objective))

[1] "objective=-745056.156106639"

print(paste0("est_sd=", sqrt(1 / flash_fit$fit$tau)))

[1] "est_sd=0.538844063230669"

print(plot_covmat(Lhat %*% t(Lhat)))

The greedy solution recovers the loadings pretty well.

Backfit

flash_fit = flashier::flashier(Y, 
                               flash.init = flash_fit,
                               prior.type=c("nonnegative", "point.normal"),
                               ebnm.param=list(fixg=TRUE, g=list(pi0=0, a=1, mu=0)),
                               var.type=0,
                               backfit="final",
                               backfit.order="dropout",
                               backfit.reltol=10)

Initializing flash object...
Adding factor 7 to flash object...
Factor doesn't increase objective and won't be added.
Backfitting 6 factors...
Nullchecking 6 factors...
Wrapping up...
Done.

Lhat = flash_fit$loadings$normalized.loadings[[1]]
p_res = plot_flash_loadings(flash_fit, n_per_pop)
print(p_res$p1)

Version	Author	Date
912c5ff	jhmarcus	2019-05-03

print(p_res$p2)

Version	Author	Date
912c5ff	jhmarcus	2019-05-03

print(paste0("objective=", flash_fit$objective))

[1] "objective=-710595.815816299"

print(paste0("est_sd=", sqrt(1 / flash_fit$fit$tau)))

[1] "est_sd=0.501086281220637"

print(plot_covmat(Lhat %*% t(Lhat)))

The backfitting begins to get funky / noisy.

High Noise

Now the standard deviation of the errors is 10 times higher than the original simulation

n_per_pop = 20
sigma_e = 1.0
p = 10000
sim_res = simpler_tree_simulation(n_per_pop, p, sigma_e)
Y = sim_res$Y

Greedy

flash_fit = flashier::flashier(Y, 
                               greedy.Kmax=10,
                               prior.type=c("nonnegative", "point.normal"),
                               ebnm.param=list(fixg=TRUE, g=list(pi0=0, a=1, mu=0)),
                               var.type=0,
                               backfit="none")

Initializing flash object...
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...
Adding factor 7 to flash object...
Factor doesn't increase objective and won't be added.
Nullchecking 6 factors...
Wrapping up...
Done.

Lhat = flash_fit$loadings$normalized.loadings[[1]]
p_res = plot_flash_loadings(flash_fit, n_per_pop)
print(p_res$p1)

Version	Author	Date
912c5ff	jhmarcus	2019-05-03

print(p_res$p2)

Version	Author	Date
912c5ff	jhmarcus	2019-05-03

print(paste0("objective=", flash_fit$objective))

[1] "objective=-1238486.22414427"

print(paste0("est_sd=", sqrt(1 / flash_fit$fit$tau)))

[1] "est_sd=1.0288417142139"

print(plot_covmat(Lhat %*% t(Lhat)))

The greedy solution still recovers the loadings well!

Backfit

flash_fit = flashier::flashier(Y, 
                               flash.init = flash_fit,
                               prior.type=c("nonnegative", "point.normal"),
                               ebnm.param=list(fixg=TRUE, g=list(pi0=0, a=1, mu=0)),
                               var.type=0,
                               backfit="final",
                               backfit.order="dropout",
                               backfit.reltol=10)

Initializing flash object...
Adding factor 7 to flash object...
Factor doesn't increase objective and won't be added.
Backfitting 6 factors...
Nullchecking 6 factors...
Factor 5 removed, increasing objective by 1.234e-01.
Adding factor 7 to flash object...
Factor doesn't increase objective and won't be added.
Backfitting 6 factors...
Nullchecking 6 factors...
Factor 4 removed, increasing objective by 8.795e-02.
Adding factor 7 to flash object...
Factor doesn't increase objective and won't be added.
Backfitting 6 factors...
Nullchecking 6 factors...
Wrapping up...
Done.

Lhat = flash_fit$loadings$normalized.loadings[[1]]
p_res = plot_flash_loadings(flash_fit, n_per_pop)
print(p_res$p1)

Version	Author	Date
912c5ff	jhmarcus	2019-05-03

print(p_res$p2)

Version	Author	Date
912c5ff	jhmarcus	2019-05-03

print(paste0("objective=", flash_fit$objective))

[1] "objective=-1208183.23583764"

print(paste0("est_sd=", sqrt(1 / flash_fit$fit$tau)))

[1] "est_sd=0.999720375541533"

print(plot_covmat(Lhat %*% t(Lhat)))

Now the backfitting is looking like the results of the previous simulation I ran with Binomial noise etc in Simple Tree Simulation. So perhaps backfitting is having a hard time in the face of “high” noise scenarios and somehow greedy is robust to this noise? Either way backfitting seems to givea fitted covariance matrix that look very close to the truth accross all the simulations so it is indeed fitting the data well.

sessionInfo()

R version 3.5.1 (2018-07-02)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: macOS  10.14.2

Matrix products: default
BLAS/LAPACK: /Users/jhmarcus/miniconda3/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] RColorBrewer_1.1-2 flashier_0.1.1     tidyr_0.8.2       
[4] dplyr_0.8.0.1      ggplot2_3.1.0     

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.0        compiler_3.5.1    pillar_1.3.1     
 [4] git2r_0.23.0      plyr_1.8.4        workflowr_1.2.0  
 [7] iterators_1.0.10  viridis_0.5.1     tools_3.5.1      
[10] digest_0.6.18     lattice_0.20-38   viridisLite_0.3.0
[13] evaluate_0.12     tibble_2.0.1      gtable_0.2.0     
[16] pkgconfig_2.0.2   rlang_0.3.1       foreach_1.4.4    
[19] Matrix_1.2-15     parallel_3.5.1    yaml_2.2.0       
[22] ebnm_0.1-17       xfun_0.4          gridExtra_2.3    
[25] withr_2.1.2       stringr_1.4.0     knitr_1.21       
[28] fs_1.2.6          rprojroot_1.3-2   grid_3.5.1       
[31] tidyselect_0.2.5  glue_1.3.0        R6_2.4.0         
[34] rmarkdown_1.11    mixsqp_0.1-115    ashr_2.2-37      
[37] purrr_0.3.0       reshape2_1.4.3    magrittr_1.5     
[40] whisker_0.3-2     MASS_7.3-51.1     codetools_0.2-16 
[43] backports_1.1.3   scales_1.0.0      htmltools_0.3.6  
[46] assertthat_0.2.0  colorspace_1.4-0  labeling_0.3     
[49] stringi_1.2.4     pscl_1.5.2        doParallel_1.0.14
[52] lazyeval_0.2.1    munsell_0.5.0     truncnorm_1.0-8  
[55] SQUAREM_2017.10-1 crayon_1.3.4

Simpler tree simulation

jhmarcus

2019-05-02

Import

Functions

Low Noise

Greedy

Backfit

Medium Noise

Greedy

Backfit

High Noise

Greedy

Backfit