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Introduction

In this example, we test out symEBcovMF on balanced, 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 <- rep(40, 4)
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 <- c(rep('A', 40), rep('B', 40), rep('C', 40), rep('D', 40))
plot_loadings(sim_data$LL, pop_vec)

symEBcovMF

symebcovmf_bal_fit <- sym_ebcovmf_fit(S = sim_data$YYt, ebnm_fn = ebnm_point_exponential, K = 5, maxiter = 100, rank_one_tol = 10^(-8), tol = 10^(-8))

Progression of ELBO

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

Visualization of Estimate

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

bal_pops <- c(rep('A', 40), rep('B', 40), rep('C', 40), rep('D', 40))
plot_loadings(symebcovmf_bal_fit$L_pm %*% diag(sqrt(symebcovmf_bal_fit$lambda)), bal_pops)

This is the objective function value attained:

symebcovmf_bal_fit$elbo
[1] -17646.82

Visualization of Fit

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

symebcovmf_bal_fitted_vals <- tcrossprod(symebcovmf_bal_fit$L_pm %*% diag(sqrt(symebcovmf_bal_fit$lambda)))
plot_heatmap(symebcovmf_bal_fitted_vals, brks = seq(0, max(symebcovmf_bal_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_bal_fitted_vals)[off_diag_idx])) + geom_point() + ylim(-1, 15) + xlim(-1,15) + xlab('Observed Values') + ylab('Fitted Values') + geom_abline(slope = 1, intercept = 0, color = 'red')

Observations

Similar to EBMFcov, symEBcovMF adds an intercept-like factor as the first factor. Therefore, when Kmax = 4, the method finds one intercept like factor and three population-effect factors. When Kmax = 5, the method recovers the last population effect factor.

symEBcovMF with refit step

symebcovmf_bal_refit_fit <- sym_ebcovmf_fit(S = sim_data$YYt, ebnm_fn = ebnm_point_exponential, K = 5, maxiter = 100, rank_one_tol = 10^(-8), tol = 10^(-8), refit_lam = TRUE)

Progression of ELBO

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

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:

bal_pops <- c(rep('A', 40), rep('B', 40), rep('C', 40), rep('D', 40))
plot_loadings(symebcovmf_bal_refit_fit$L_pm %*% diag(sqrt(symebcovmf_bal_refit_fit$lambda)), bal_pops)

This is the objective function value attained:

symebcovmf_bal_refit_fit$elbo
[1] 1755.434

Visualization of Fit

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

symebcovmf_bal_refit_fitted_vals <- tcrossprod(symebcovmf_bal_refit_fit$L_pm %*% diag(sqrt(symebcovmf_bal_refit_fit$lambda)))
plot_heatmap(symebcovmf_bal_refit_fitted_vals, brks = seq(0, max(symebcovmf_bal_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_bal_refit_fitted_vals)[off_diag_idx])) + geom_point() + ylim(-1, 15) + xlim(-1,15) + xlab('Observed Values') + ylab('Fitted Values') + geom_abline(slope = 1, intercept = 0, color = 'red')

Observations

We see that symEBcovMF with the refitting step improves upon the fit. Now, the first factor, which is similar to an intercept factor, has lower weight. Would it be possible for lambda to become zero without changing the vector? Perhaps if the residual matrix had multiple negative eigenvalues?


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