Revisiting prior families for trees: Part VI

workflowr

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    Modified:   drift-workflow.Rproj

suppressMessages({
  library(flashier)
  library(drift.alpha)
  library(tidyverse)
})

I again simulate a tree with 8 populations, but I remove the admixed populations. I use identical branch lengths to illustrate the problems that can arise as a result.

set.seed(666)

p <- 10000
resid_sd <- 0.1

# Define tree by mean branch length at each depth:
branch_means <- rep(1, 4)
branch_sds <- rep(0, 4)
depth <- length(branch_means)
npop_pure <- 2^(depth - 1)

# Define admixtures by admixture proportions:
admix_pops <- matrix(nrow = 0, ncol = 0)
npop_admix <- ncol(admix_pops)

npop <- npop_pure + npop_admix

n <- sample(50, npop, replace = TRUE)
K <- 2^depth - 1

FF <- matrix(nrow = p, ncol = K)
k <- 1
for (d in 1:depth) {
  for (i in 1:(2^(d - 1))) {
    FF[, k] <- rnorm(p, sd = branch_means[d] + rnorm(1, sd = branch_sds[d]))
    k <- k + 1
  }
}

tree_mat <- matrix(0, nrow = npop_pure, ncol = K)
k <- 1
for (d in 1:depth) {
  size <- 2^(depth - d)
  for (i in 1:(2^(d - 1))) {
    tree_mat[((i - 1) * size + 1):(i * size), k] <- 1
    k <- k + 1
  }
}

pop_means <- FF %*% t(tree_mat)
if (npop_admix > 0) {
  pop_means <- cbind(pop_means, pop_means %*% admix_pops)
}

Y <- NULL
for (i in 1:npop) {
  Y <- rbind(Y, matrix(pop_means[, i], nrow = n[i], ncol = p, byrow = TRUE))
}
Y <- Y + rnorm(n * p, sd = resid_sd)

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(letters[1:npop], n)) +
    scale_color_brewer(palette="Set3")
}

tree.fn = function(x, s, g_init, fix_g, output, ...) {
  if (is.null(g_init)) {
    g_init <- ashr::unimix(rep(1/3, 3), c(-1, 0, 1), c(-1, 0, 1))
  }

  return(flashier:::ebnm.nowarn(x = x,
                                s = s,
                                g_init = g_init,
                                fix_g = fix_g,
                                output = output,
                                prior_family = "ash",
                                prior = c(10, 1, 10),
                                ...))
}

prior.tree = function(...) {
  return(as.prior(sign = 0, ebnm.fn = function(x, s, g_init, fix_g, output) {
    tree.fn(x, s, g_init, fix_g, output, ...)
  }))
}

flextree.fn = function(a = 0, b = 1) {
  return(function(x, s, g_init, fix_g, output, ...) {
  if (is.null(g_init)) {
    g_init <- ashr::unimix(rep(1/5, 5), c(-1, 0, 1, -b, a), c(-1, 0, 1, -a, b))
  }

  return(flashier:::ebnm.nowarn(x = x,
                                s = s,
                                g_init = g_init,
                                fix_g = fix_g,
                                output = output,
                                prior_family = "ash",
                                prior = c(10, 10, 10, 1, 1),
                                ...))
  })
}

prior.flextree = function(...) {
  return(as.prior(sign = 0, ebnm.fn = function(x, s, g_init, fix_g, output) {
    flextree.fn(x, s, g_init, fix_g, output, ...)
  }))
}

init.mean.factor <- function(resids, zero.idx) {
  u <- matrix(1, nrow = nrow(resids), ncol = 1)
  u[zero.idx, 1] <- 0
  v <- t(solve(crossprod(u), crossprod(u, resids)))
  return(list(u, v))
}

init.split.factor <- function(resids, zero.idx) {
  svd.res <- svd(resids, nu = 1, nv = 1)
  u <- svd.res$u
  u[zero.idx] <- 0
  u <- matrix(sign(u), ncol = 1)
  v <- t(solve(crossprod(u), crossprod(u, resids)))
  return(list(u, v))
}

I only show the initialization here (with no relaxation). Factor 2 looks good, but subsequent factors are jumbled.

fl <- flash.init(Y) %>%
  flash.set.verbose(0) %>%
  flash.init.factors(EF = init.mean.factor(Y, NULL), 
                     prior.family = c(prior.tree(), prior.normal())) %>%
  flash.fix.loadings(kset = 1, mode = 1L) %>%
  flash.backfit() %>%
  flash.add.greedy(Kmax = npop_pure - 1, 
                   prior.family = c(prior.tree(), prior.normal())) %>%
  flash.backfit()

fl2 <- fl

# Partial relaxation.
for (k in 1:fl2$n.factors) {
  fl2$flash.fit$ebnm.fn[[k]][[1]] <- flextree.fn(a = 0, b = 0.9)
}
fl2 <- fl2 %>% flash.backfit(warmstart = FALSE) %>% flash.backfit()

plot_dr(init_from_flash(fl))

Past versions of fl-1.png

One way to handle this problem is to impose a “hard” tree constraint on the initial fit. By doing so, each factor can be interpreted as a sub-partition of a previous partition, so that a binary tree can be easily reconstructed:

add_split <- function(fl, k) {
  set1 <- (fl$flash.fit$EF[[1]][, k] < -0.9)
  set2 <- (fl$flash.fit$EF[[1]][, k] > 0.9)
  set3 <- !set1 & !set2 # admixed individuals
  
  n.factors <- fl$n.factors
  
  fl2 <- fl %>%
    flash.init.factors(EF = init.mean.factor(Y - fitted(fl), set2 | set3),
                       prior.family = c(prior.tree(), prior.normal())) %>%
    flash.fix.loadings(kset = n.factors + 1, mode = 1L) %>%
    flash.backfit(n.factors + 1)
  
  fl2 <- fl2 %>%
    flash.init.factors(EF = init.split.factor(Y - fitted(fl2), set2 | set3),
                       prior.family = c(prior.tree(), prior.normal())) %>%
    flash.fix.loadings(kset = n.factors + 2, mode = 1L, is.fixed = set2 | set3) %>%
    flash.backfit(n.factors + 2)
  
  fl2 <- fl2 %>%
    flash.init.factors(EF = init.mean.factor(Y - fitted(fl2), set1 | set3),
                       prior.family = c(prior.tree(), prior.normal())) %>%
    flash.fix.loadings(kset = n.factors + 3, mode = 1L) %>%
    flash.backfit(n.factors + 3)
  
  fl2 <- fl2 %>%
    flash.init.factors(EF = init.split.factor(Y - fitted(fl2), set1 | set3),
                       prior.family = c(prior.tree(), prior.normal())) %>%
    flash.fix.loadings(kset = n.factors + 4, mode = 1L, is.fixed = set1 | set3) %>%
    flash.backfit(n.factors + 4)
  
  fl2 <- fl2 %>%
    flash.remove.factors(kset = c(n.factors + 1, n.factors + 3))
  
  return(fl2)
}

fl3 <- flash.init(Y) %>%
  flash.set.verbose(0) %>%
  flash.init.factors(EF = init.mean.factor(Y, NULL), 
                     prior.family = c(prior.tree(), prior.normal())) %>%
  flash.fix.loadings(kset = 1, mode = 1L) %>%
  flash.backfit() %>%
  flash.add.greedy(prior.family = c(prior.tree(), prior.normal())) %>%
  flash.backfit()

fl3 <- fl3 %>% add_split(k = 2)

fl3 <- fl3 %>% add_split(k = 3) %>% add_split(k = 4)

plot_dr(init_from_flash(fl3))

Past versions of fl3-1.png

Now I “relax” this fit. I relax the priors and then I unfix all loadings (with the exception of the first factor k = 1). It’s not perfect; in particular, smaller populations have loadings that are not as large as they should be. Still, it’s easy to spot the tree here:

fl4 <- fl3

for (k in 1:fl4$n.factors) {
  fl4$flash.fit$ebnm.fn[[k]][[1]] <- flextree.fn(a = 0, b = 0.9)
}
fl4 <- fl4 %>% flash.backfit(warmstart = FALSE) %>% flash.backfit()

fl5 <- fl4 %>%
  flash.fix.loadings(kset = 3:fl4$n.factors, mode = 1, is.fixed = FALSE) %>%
  flash.backfit()

plot_dr(init_from_flash(fl5))

Past versions of fl5-1.png

The second fit has a slightly lower ELBO, which can be explained by a lower KL-divergence for the loadings:

cat("ELBO (without tree constraint):", fl2$elbo,
    "\nELBO (with tree constraint):   ", fl5$elbo,
    "\nKL-div (loadings, no tree):  ", sum(fl2$flash.fit$KL[[1]]),
    "\nKL-div (loadings, with tree):", sum(fl5$flash.fit$KL[[1]]))

#> ELBO (without tree constraint): 2726712 
#> ELBO (with tree constraint):    2726650 
#> KL-div (loadings, no tree):   -1605.171 
#> KL-div (loadings, with tree): -2091.872

Session information

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_0.2.7   
#> 
#> loaded via a namespace (and not attached):
#>  [1] Rcpp_1.0.4.6       lubridate_1.7.4    invgamma_1.1      
#>  [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      
#> [16] rlang_0.4.2        lazyeval_0.2.2     readxl_1.3.1      
#> [19] rstudioapi_0.10    ebnm_0.1-21        irlba_2.3.3       
#> [22] whisker_0.3-2      Matrix_1.2-15      rmarkdown_1.12    
#> [25] labeling_0.3       munsell_0.5.0      mixsqp_0.3-40     
#> [28] broom_0.5.1        compiler_3.5.3     modelr_0.1.5      
#> [31] xfun_0.6           pkgconfig_2.0.2    SQUAREM_2017.10-1 
#> [34] htmltools_0.3.6    tidyselect_0.2.5   workflowr_1.2.0   
#> [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        
#> [55] hms_0.4.2          parallel_3.5.3     yaml_2.2.0        
#> [58] colorspace_1.4-1   ashr_2.2-51        rvest_0.3.4       
#> [61] knitr_1.22         haven_2.1.1