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Introduction

In this example, we test out symEBcovMF on unbalanced, star-structured data.

Example

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))

Version Author Date
4ad25a4 Annie Xie 2025-04-30

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)

Version Author Date
4ad25a4 Annie Xie 2025-04-30

symEBcovMF

symebcovmf_unbal_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))

Progression of ELBO

symebcovmf_unbal_full_elbo_vec <- symebcovmf_unbal_fit$vec_elbo_full[!(symebcovmf_unbal_fit$vec_elbo_full %in% c(1:length(symebcovmf_unbal_fit$vec_elbo_K)))]
ggplot() + geom_line(data = NULL, aes(x = 1:length(symebcovmf_unbal_full_elbo_vec), y = symebcovmf_unbal_full_elbo_vec)) + xlab('Iter') + ylab('ELBO')

Version Author Date
4ad25a4 Annie Xie 2025-04-30

Visualization of Estimate

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

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

Version Author Date
4ad25a4 Annie Xie 2025-04-30

This is the objective function value attained:

symebcovmf_unbal_fit$elbo
[1] 1087.677

Visualization of Fit

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

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

Version Author Date
4ad25a4 Annie Xie 2025-04-30

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_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')

Version Author Date
4ad25a4 Annie Xie 2025-04-30

Observations

symEBcovMF does a relatively good job at recovering the four population effects. Interestingly, while factor 3 primarily recovers the third group effect, it has a small (but still notable) loading on the first group effect. I explore this more in another analysis. I did also try symEBcovMF with generalized binary prior, and in that case, the factors primarily captured one group effect each.

symEBcovMF with refit step

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)

Progression of ELBO

symebcovmf_unbal_refit_full_elbo_vec <- symebcovmf_unbal_refit_fit$vec_elbo_full[!(symebcovmf_unbal_refit_fit$vec_elbo_full %in% c(1:length(symebcovmf_unbal_refit_fit$vec_elbo_K)))]
ggplot() + geom_line(data = NULL, aes(x = 1:length(symebcovmf_unbal_refit_full_elbo_vec), y = symebcovmf_unbal_refit_full_elbo_vec)) + xlab('Iter') + ylab('ELBO')

Version Author Date
4ad25a4 Annie Xie 2025-04-30

A note: I don’t think I save the ELBO value after the refitting step in vec_elbo_full. But the refitting does change this vector since it changes the residual matrix that is used when you add a new vector.

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)

Version Author Date
4ad25a4 Annie Xie 2025-04-30

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))

Version Author Date
4ad25a4 Annie Xie 2025-04-30

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')

Version Author Date
4ad25a4 Annie Xie 2025-04-30

Comparison of Estimates

Comparison of the factors:

for (i in 1:4){
  print(ggplot(data = NULL, aes(x = symebcovmf_unbal_fit$L_pm[,i], y = symebcovmf_unbal_refit_fit$L_pm[,i])) + geom_point() + geom_abline(slope = 1, intercept = 0, color = 'red') + ylab('symEBcovMF with refitting') + xlab('Regular symEBcovMF') + labs(title=paste('Factor',i)))
}

Version Author Date
4ad25a4 Annie Xie 2025-04-30

Version Author Date
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Version Author Date
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Version Author Date
4ad25a4 Annie Xie 2025-04-30

Observations

The estimate from symEBcovMF with refitting qualitatively looks the same as the estimate from regular symEBcovMF. Plotting the values of the two estimates show that they are nearly the same.


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