Last updated: 2025-05-12

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File	Version	Author	Date	Message
Rmd	755d361	Annie Xie	2025-05-12	Add analysis of backfit in unbal nonoverlap setting

Introduction

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

Motivation

When applying greedy symEBcovMF with point-exponential prior to the unbalanced non-overlapping setting, the method did a relatively good job. However, the third factor it found was not a single group effect factor. For the third factor, the most prominent effect was the third group effect. But there was also a small non-zero loading on the first group effect. In my exploration, I found that the greedy method does prefer this solution (this solution yields a higher objective function). Therefore, I want to test if backfitting will change this factor to a single group effect factor. I hypothesize that it will since the first group effect is already captured in the fourth factor.

Packages and Functions

library(ebnm)
library(pheatmap)
library(ggplot2)

source('code/symebcovmf_functions.R')
source('code/visualization_functions.R')

Backfit Function

optimize_factor <- function(R, ebnm_fn, maxiter, tol, v_init, lambda_k, R2k, n, KL){
  R2 <- R2k - lambda_k^2
  resid_s2 <- estimate_resid_s2(n = n, R2 = R2)
  rank_one_KL <- 0
  curr_elbo <- -Inf
  obj_diff <- Inf
  fitted_g_k <- NULL
  iter <- 1
  vec_elbo_full <- NULL
  v <- v_init
  
  while((iter <= maxiter) && (obj_diff > tol)){
    # update l; power iteration step
    v.old <- v
    x <- R %*% v
    e <- ebnm_fn(x = x, s = sqrt(resid_s2), g_init = fitted_g_k)
    scaling_factor <- sqrt(sum(e$posterior$mean^2) + sum(e$posterior$sd^2))
    if (scaling_factor == 0){ # check if scaling factor is zero
      scaling_factor <- Inf
      v <- e$posterior$mean/scaling_factor
      print('Warning: scaling factor is zero')
      break
    }
    v <- e$posterior$mean/scaling_factor
    
    # update lambda and R2
    lambda_k.old <- lambda_k
    lambda_k <- max(as.numeric(t(v) %*% R %*% v), 0)
    R2 <- R2k - lambda_k^2
    
    #store estimate for g
    fitted_g_k.old <- fitted_g_k
    fitted_g_k <- e$fitted_g
    
    # store KL
    rank_one_KL.old <- rank_one_KL
    rank_one_KL <- as.numeric(e$log_likelihood) +
      - normal_means_loglik(x, sqrt(resid_s2), e$posterior$mean, e$posterior$mean^2 + e$posterior$sd^2)
    
    # update resid_s2
    resid_s2.old <- resid_s2
    resid_s2 <- estimate_resid_s2(n = n, R2 = R2) # this goes negative?????
    
    # check convergence - maybe change to rank-one obj function
    curr_elbo.old <- curr_elbo
    curr_elbo <- compute_elbo(resid_s2 = resid_s2,
                              n = n,
                              KL = c(KL, rank_one_KL),
                              R2 = R2)
    if (iter > 1){
      obj_diff <- curr_elbo - curr_elbo.old
    }
    if (obj_diff < 0){ # check if convergence_val < 0
      v <- v.old
      resid_s2 <- resid_s2.old
      rank_one_KL <- rank_one_KL.old
      lambda_k <- lambda_k.old
      curr_elbo <- curr_elbo.old
      fitted_g_k <- fitted_g_k.old
      print(paste('elbo decreased by', abs(obj_diff)))
      break
    }
    vec_elbo_full <- c(vec_elbo_full, curr_elbo)
    iter <- iter + 1
  }
  return(list(v = v, lambda_k = lambda_k, resid_s2 = resid_s2, curr_elbo = curr_elbo, vec_elbo_full = vec_elbo_full, fitted_g_k = fitted_g_k, rank_one_KL = rank_one_KL))
}

#nullcheck function
nullcheck_factors <- function(sym_ebcovmf_obj, L2_tol = 10^(-8)){
  null_lambda_idx <- which(sym_ebcovmf_obj$lambda == 0)
  factor_L2_norms <- apply(sym_ebcovmf_obj$L_pm, 2, function(v){sqrt(sum(v^2))})
  null_factor_idx <- which(factor_L2_norms < L2_tol)
  null_idx <- unique(c(null_lambda_idx, null_factor_idx))
  
  keep_idx <- setdiff(c(1:length(sym_ebcovmf_obj$lambda)), null_idx)
  
  if (length(keep_idx) < length(sym_ebcovmf_obj$lambda)){
    #remove factors
    sym_ebcovmf_obj$L_pm <- sym_ebcovmf_obj$L_pm[,keep_idx]
    sym_ebcovmf_obj$lambda <- sym_ebcovmf_obj$lambda[keep_idx]
    sym_ebcovmf_obj$KL <- sym_ebcovmf_obj$KL[keep_idx]
    sym_ebcovmf_obj$fitted_gs <- sym_ebcovmf_obj$fitted_gs[keep_idx]
  }
  
  #shouldn't need to recompute objective function or other things
  return(sym_ebcovmf_obj)
}

sym_ebcovmf_backfit <- function(S, sym_ebcovmf_obj, ebnm_fn, backfit_maxiter = 100, backfit_tol = 10^(-8), optim_maxiter= 500, optim_tol = 10^(-8)){
  K <- length(sym_ebcovmf_obj$lambda)
  iter <- 1
  obj_diff <- Inf
  sym_ebcovmf_obj$backfit_vec_elbo_full <- NULL
  
  # refit lambda
  sym_ebcovmf_obj <- refit_lambda(S, sym_ebcovmf_obj, maxiter = 25)
  
  while((iter <= backfit_maxiter) && (obj_diff > backfit_tol)){
    #print(iter)
    obj_old <- sym_ebcovmf_obj$elbo
    # loop through each factor
    for (k in 1:K){
      #print(k)
      # compute residual matrix
      R <- S - tcrossprod(sym_ebcovmf_obj$L_pm[,-k] %*% diag(sqrt(sym_ebcovmf_obj$lambda[-k]), ncol = (K-1)))
      R2k <- compute_R2(S, sym_ebcovmf_obj$L_pm[,-k], sym_ebcovmf_obj$lambda[-k], (K-1)) #this is right but I have one instance where the values don't match what I expect
      
      # optimize factor
      factor_proposed <- optimize_factor(R, ebnm_fn, optim_maxiter, optim_tol, sym_ebcovmf_obj$L_pm[,k], sym_ebcovmf_obj$lambda[k], R2k, sym_ebcovmf_obj$n, sym_ebcovmf_obj$KL[-k])
      
      # update object
      sym_ebcovmf_obj$L_pm[,k] <- factor_proposed$v
      sym_ebcovmf_obj$KL[k] <- factor_proposed$rank_one_KL
      sym_ebcovmf_obj$lambda[k] <- factor_proposed$lambda_k
      sym_ebcovmf_obj$resid_s2 <- factor_proposed$resid_s2
      sym_ebcovmf_obj$fitted_gs[[k]] <- factor_proposed$fitted_g_k
      sym_ebcovmf_obj$elbo <- factor_proposed$curr_elbo
      sym_ebcovmf_obj$backfit_vec_elbo_full <- c(sym_ebcovmf_obj$backfit_vec_elbo_full, factor_proposed$vec_elbo_full)
      
      #print(sym_ebcovmf_obj$elbo)
      sym_ebcovmf_obj <- refit_lambda(S, sym_ebcovmf_obj) # add refitting step?
      #print(sym_ebcovmf_obj$elbo)
    }
    
    iter <- iter + 1
    obj_diff <- abs(sym_ebcovmf_obj$elbo - obj_old)
    # need to add check if it is negative?
  }
  # nullcheck
  sym_ebcovmf_obj <- nullcheck_factors(sym_ebcovmf_obj)
  return(sym_ebcovmf_obj)
}

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)

Regular symEBcovMF

First, we apply greedy symEBcovMF with point-exponential prior to the Gram matrix:

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)

This is a scatter plot of \(\hat{L}\), 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

symEBcovMF with backfit

Now, we try backfitting (also with a point-exponential prior):

symebcovmf_fit_backfit <- sym_ebcovmf_backfit(sim_data$YYt, symebcovmf_unbal_refit_fit, ebnm_fn = ebnm::ebnm_point_exponential, backfit_maxiter = 100)

[1] "elbo decreased by 2.45563569478691e-10"
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[1] "elbo decreased by 8.47649062052369e-10"
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[1] "elbo decreased by 8.51287040859461e-10"
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[1] "elbo decreased by 8.47649062052369e-10"
[1] "elbo decreased by 8.54925019666553e-10"
[1] "elbo decreased by 8.51287040859461e-10"
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[1] "elbo decreased by 8.54925019666553e-10"
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[1] "elbo decreased by 8.54925019666553e-10"
[1] "elbo decreased by 8.18545231595635e-11"
[1] "elbo decreased by 1.01863406598568e-10"

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

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

This is the objective function value attained:

symebcovmf_fit_backfit$elbo

[1] 10300.7

Observations

We see that after backfitting, the third factor looks closer to a single group effect factor. The first group effect that was originally present has shrunk closer to zero. Furthermore, the objective function value after the backfit was quite a bit higher. One interesting observation is that some of the values that were previously zero are now small, non-zero values. Some of the other methods that work in the covariance space also have this characteristic, e.g. GBCD, EBMFcov. So perhaps this is related to working in the covariance space?

sessionInfo()

R version 4.3.2 (2023-10-31)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS 15.4.1

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

unbal_nonoverlap_backfit

Annie Xie

2025-05-09