Last updated: 2025-04-21

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Knit directory: symmetric_covariance_decomposition/

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Rmd d31c16e Annie Xie 2025-04-21 Add symebmfcov null setting analysis

Introduction

In this example, we test out symEBcovMF on a null dataset – the data matrix \(X\) is generated from normal noise and then \(S = XX'\) is formed.

Example

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

Data Generation

sim_null_data <- function(args) {
  set.seed(args$seed)
  
  Y <- matrix(rnorm(args$n * args$p, sd = args$indiv_sd), nrow = n)
  YYt <- (1/args$p)*tcrossprod(Y)
  
  return(list(Y = Y, YYt = YYt))
}
n <- 200
p <- 1000
indiv_sd <- 1
seed <- 1
sim_args = list(n = n, p = p, indiv_sd = indiv_sd, seed = seed)
sim_data <- sim_null_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))

symEBcovMF

symebcovmf_fit <- sym_ebcovmf_fit(S = sim_data$YYt, ebnm_fn = ebnm_point_exponential, K = 1, maxiter = 100, rank_one_tol = 10^(-8), tol = 10^(-8), refit_lam = FALSE)

Progression of ELBO

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

Visualization of Estimate

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

pop_vec <- rep('A', n)
plot_loadings(symebcovmf_fit$L_pm %*% diag(sqrt(symebcovmf_fit$lambda), ncol = length(symebcovmf_fit$lambda)), pop_vec, legendYN = FALSE)

This is the objective function value attained:

symebcovmf_fit$elbo
[1] 22796.37

Visualization of Fit

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

symebcovmf_fitted_vals <- tcrossprod(symebcovmf_fit$L_pm %*% diag(sqrt(symebcovmf_fit$lambda), ncol = length(symebcovmf_fit$lambda)))
plot_heatmap(symebcovmf_fitted_vals, brks = seq(0, max(symebcovmf_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_fitted_vals)[off_diag_idx])) + geom_point() + ylim(-0.5, 0.5) + xlim(-0.5,0.5) + xlab('Observed Values') + ylab('Fitted Values') + geom_abline(slope = 1, intercept = 0, color = 'red')

Observations

In this null setting, we would hope that the method would not recover any factors. However, the method does find factors to add. I tested Kmax = 1, 10, 50, 100, and 200, and in all settings, the method utilizes the full factor allowance (i.e. for Kmax = 100, the method adds 100 factors). Flash also has this behavior in the covariance space. Some questions of interest: 1) Why does this happen? Is it because of model misspecification? 2) How can we remedy this? Matthew thought that the estimation of \(\sigma^2\) could help.


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