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Rmd fce089f Annie Xie 2025-05-05 Add exploration of unbalanced nonoverlapping example

Introduction

In this analysis, I explore the unbalanced non-overlapping setting.

Packages and Functions

library(ebnm)
library(pheatmap)
library(ggplot2)
source('code/symebcovmf_functions.R')
source('code/visualization_functions.R')

Data Generation

# adapted from Jason's code
# args is a list containing pop_sizes, branch_sds, indiv_sd, n_genes, and seed
sim_star_data <- function(args) {
  set.seed(args$seed)
  
  n <- sum(args$pop_sizes)
  p <- args$n_genes
  K <- length(args$pop_sizes)
  
  FF <- matrix(rnorm(K * p, sd = rep(args$branch_sds, each = p)), ncol = K)
  
  LL <- matrix(0, nrow = n, ncol = K)
  for (k in 1:K) {
    vec <- rep(0, K)
    vec[k] <- 1
    LL[, k] <- rep(vec, times = args$pop_sizes)
  }
  
  E <- matrix(rnorm(n * p, sd = args$indiv_sd), nrow = n)
  Y <- LL %*% t(FF) + E
  YYt <- (1/p)*tcrossprod(Y)
  
  return(list(Y = Y, YYt = YYt, LL = LL, FF = FF, K = ncol(LL)))
}
pop_sizes <- c(20,50,30,60)
n_genes <- 1000
branch_sds <- rep(2,4)
indiv_sd <- 1
seed <- 1
sim_args = list(pop_sizes = pop_sizes, branch_sds = branch_sds, indiv_sd = indiv_sd, n_genes = n_genes, seed = seed)
sim_data <- sim_star_data(sim_args)

This is a heatmap of the scaled Gram matrix:

plot_heatmap(sim_data$YYt, colors_range = c('blue','gray96','red'), brks = seq(-max(abs(sim_data$YYt)), max(abs(sim_data$YYt)), length.out = 50))

This is a scatter plot of the true loadings matrix:

pop_vec <- rep(c('A','B','C','D'), times = pop_sizes)
plot_loadings(sim_data$LL, pop_vec)

symEBcovMF with refitting

symebcovmf_unbal_refit_fit <- sym_ebcovmf_fit(S = sim_data$YYt, ebnm_fn = ebnm_point_exponential, K = 4, maxiter = 100, rank_one_tol = 10^(-8), tol = 10^(-8), refit_lam = TRUE)

Visualization of Estimate

This is a scatter plot of \(\hat{L}_{refit}\), the estimate from symEBcovMF:

plot_loadings(symebcovmf_unbal_refit_fit$L_pm %*% diag(sqrt(symebcovmf_unbal_refit_fit$lambda)), pop_vec)

This is the objective function value attained:

symebcovmf_unbal_refit_fit$elbo
[1] 1097.095

Visualization of Fit

This is a heatmap of \(\hat{L}_{refit}\hat{\Lambda}_{refit}\hat{L}_{refit}'\):

symebcovmf_unbal_refit_fitted_vals <- tcrossprod(symebcovmf_unbal_refit_fit$L_pm %*% diag(sqrt(symebcovmf_unbal_refit_fit$lambda)))
plot_heatmap(symebcovmf_unbal_refit_fitted_vals, brks = seq(0, max(symebcovmf_unbal_refit_fitted_vals), length.out = 50))

This is a scatter plot of fitted values vs. observed values for the off-diagonal entries:

diag_idx <- seq(1, prod(dim(sim_data$YYt)), length.out = ncol(sim_data$YYt))
off_diag_idx <- setdiff(c(1:prod(dim(sim_data$YYt))), diag_idx) 

ggplot(data = NULL, aes(x = c(sim_data$YYt)[off_diag_idx], y = c(symebcovmf_unbal_refit_fitted_vals)[off_diag_idx])) + geom_point() + ylim(-1, 5) + xlim(-1,5) + xlab('Observed Values') + ylab('Fitted Values') + geom_abline(slope = 1, intercept = 0, color = 'red')

Observations

As noted in a different analysis, symEBcovMF does a relatively good job at recovering the four population effects. However, one undesired aspect of the estimate is that factor 3 contains the effects of two different groups. I would like to figure out why this is happening. Furthermore, the group 1 effect is much smaller than the group 4 effect, so I’m wondering if it’s possible to shrink the group 1 effect down to zero so that the factor only captures one group.

Trying a smaller convergence tolerance

In this section, we explore whether a smaller convergence tolerance will yield a factor which captures only one group. Perhaps the group 1 effect is in the process of being (slowly) shrunk down to zero, and a smaller tolerance will allow for more shrinkage.

symebcovmf_unbal_refit_smaller_tol_fit <- sym_ebcovmf_fit(S = sim_data$YYt, ebnm_fn = ebnm_point_exponential, K = 4, maxiter = 100, rank_one_tol = 10^(-15), tol = 10^(-8), refit_lam = TRUE)
[1] "elbo decreased by 7.27595761418343e-12"
[1] "elbo decreased by 1.45519152283669e-11"
[1] "elbo decreased by 1.2732925824821e-11"
[1] "elbo decreased by 1.90993887372315e-11"

Visualization of Estimate

This is a scatter plot of \(\hat{L}_{refit}\), the estimate from symEBcovMF:

plot_loadings(symebcovmf_unbal_refit_smaller_tol_fit$L_pm %*% diag(sqrt(symebcovmf_unbal_refit_smaller_tol_fit$lambda)), pop_vec)

This is the objective function value attained:

symebcovmf_unbal_refit_smaller_tol_fit$elbo
[1] 1097.121

Comparison of factor 3 estimates:

ggplot(data = NULL, aes(x = symebcovmf_unbal_refit_fit$L_pm[,3], y = symebcovmf_unbal_refit_smaller_tol_fit$L_pm[,3])) + geom_point() + geom_abline(slope = 1, intercept = 0, color = 'red')

Observations

Decreasing the convergence tolerance does not lead to more shrinkage of the group 1 effect in the factor 3 estimate. The optimization procedure for factor 3 does stop because the ELBO slightly decreases (on the order of \(10^{-11}\)). My guess is the decrease is caused by a numerical issue. So it’s possible that we would see more shrinkage if we let the optimization run for more iterations.

Try initializing from true factor

To check if this is a convergence issue, I try fitting the third factor initialized from the true single population effect factor.

First, we fit the first two factors.

symebcovmf_unbal_refit_rank2_fit <- sym_ebcovmf_fit(S = sim_data$YYt, ebnm_fn = ebnm_point_exponential, K = 2, maxiter = 100, rank_one_tol = 10^(-8), tol = 10^(-8), refit_lam = TRUE)

This is a heatmap of the residual matrix, \(S - \sum_{k=1}^{2} \hat{\lambda}_k \hat{\ell}_k \hat{\ell}_k'\):

R <- sim_data$YYt - tcrossprod(symebcovmf_unbal_refit_rank2_fit$L_pm %*% diag(sqrt(symebcovmf_unbal_refit_rank2_fit$lambda)))
plot_heatmap(R, colors_range = c('blue','gray96','red'), brks = seq(-max(abs(R)), max(abs(R)), length.out = 50))

Now, we fit the third factor, initialized with the true population effect factor.

symebcovmf_unbal_true_init_fac3_fit <- sym_ebcovmf_r1_fit(S = sim_data$YYt, symebcovmf_unbal_refit_rank2_fit, ebnm_fn = ebnm::ebnm_point_exponential, maxiter = 100, tol = 10^(-8), v_init = rep(c(0,0,1,0), times = pop_sizes))

This is a plot of the estimate of the third factor.

plot(symebcovmf_unbal_true_init_fac3_fit$L_pm[,3], ylab = 'Third Factor')

This is a plot of the ELBO during the fit of the third factor.

fac3.idx <- which(symebcovmf_unbal_true_init_fac3_fit$vec_elbo_full == 3)
plot(symebcovmf_unbal_true_init_fac3_fit$vec_elbo_full[(fac3.idx+1):length(symebcovmf_unbal_true_init_fac3_fit$vec_elbo_full)], ylab = 'ELBO')

true_fac3 <- rep(c(0,0,1,0), times = pop_sizes)
true_fac3 <- true_fac3/sqrt(sum(true_fac3^2))
estimates_list <- list(true_fac3)
for (i in 1:11){
  estimates_list[[(i+1)]] <- sym_ebcovmf_r1_fit(sim_data$YYt, symebcovmf_unbal_refit_rank2_fit, ebnm_fn = ebnm::ebnm_point_exponential, maxiter = i, tol = 10^(-8), v_init = rep(c(0,0,1,0), times = pop_sizes))$L_pm[,3]
}

This is a plot of the progression of the estimate.

par(mfrow = c(6,2), mar = c(2, 2, 1, 1) + 0.1)
max_y <- max(sapply(estimates_list, max))
min_y <- min(sapply(estimates_list, min))
for (i in 1:12){
  plot(estimates_list[[i]], main = paste('Iter', (i-1)), ylab = 'L', ylim = c(min_y, max_y))
}

par(mfrow = c(1,1))

Observations

When initialized with the true factor, the method still yields an estimate that has non-zero loading on the first group. This suggests that the method does prefer this estimate. Perhaps an easier way of getting a representation comprised of four single group effect factors is to backfit. Question I have: why does this method have this issue but EBMFcov-greedy does not?


sessionInfo()
R version 4.3.2 (2023-10-31)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS Sonoma 14.4.1

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRblas.0.dylib 
LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.11.0

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

time zone: America/Chicago
tzcode source: internal

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] ggplot2_3.5.1   pheatmap_1.0.12 ebnm_1.1-34     workflowr_1.7.1

loaded via a namespace (and not attached):
 [1] gtable_0.3.5       xfun_0.48          bslib_0.8.0        processx_3.8.4    
 [5] lattice_0.22-6     callr_3.7.6        vctrs_0.6.5        tools_4.3.2       
 [9] ps_1.7.7           generics_0.1.3     tibble_3.2.1       fansi_1.0.6       
[13] highr_0.11         pkgconfig_2.0.3    Matrix_1.6-5       SQUAREM_2021.1    
[17] RColorBrewer_1.1-3 lifecycle_1.0.4    truncnorm_1.0-9    farver_2.1.2      
[21] compiler_4.3.2     stringr_1.5.1      git2r_0.33.0       munsell_0.5.1     
[25] getPass_0.2-4      httpuv_1.6.15      htmltools_0.5.8.1  sass_0.4.9        
[29] yaml_2.3.10        later_1.3.2        pillar_1.9.0       jquerylib_0.1.4   
[33] whisker_0.4.1      cachem_1.1.0       trust_0.1-8        RSpectra_0.16-2   
[37] tidyselect_1.2.1   digest_0.6.37      stringi_1.8.4      dplyr_1.1.4       
[41] ashr_2.2-66        labeling_0.4.3     splines_4.3.2      rprojroot_2.0.4   
[45] fastmap_1.2.0      grid_4.3.2         colorspace_2.1-1   cli_3.6.3         
[49] invgamma_1.1       magrittr_2.0.3     utf8_1.2.4         withr_3.0.1       
[53] scales_1.3.0       promises_1.3.0     horseshoe_0.2.0    rmarkdown_2.28    
[57] httr_1.4.7         deconvolveR_1.2-1  evaluate_1.0.0     knitr_1.48        
[61] irlba_2.3.5.1      rlang_1.1.4        Rcpp_1.0.13        mixsqp_0.3-54     
[65] glue_1.8.0         rstudioapi_0.16.0  jsonlite_1.8.9     R6_2.5.1          
[69] fs_1.6.4