Last updated: 2025-05-12

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File	Version	Author	Date	Message
Rmd	58d7eb8	Annie Xie	2025-05-12	Add analysis of gb backfit initialized with point exp

Introduction

In this analysis, I want to test out initializing symEBcovMF with a binary prior with a point-exponential symEBcovMF fit.

Motivation

In the tree setting, I found that symEBcovMF with the point-exponential prior did a better job at recovering tree structure than symEBcovMF with the generalized binary prior. Matthew and I hypothesized that the point-exponential prior is better at finding sparse solutions. So we want to investigate a two-step procedure which finds a fit with point-exponential prior in the first step, and then refines the fit with the generalized binary (or other binary) prior in the second step. In this analysis, I investigate a backfitting procedure – first, I generate a fit with a point-exponential prior, and then use that as an initialization for a backfit with a binary prior.

Packages and Functions

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

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

To test this procedure, I will apply it to the tree-structured dataset. When testing out symEBcovMF, I found that the estimates from the two priors in the tree setting had the largest difference.

sim_4pops <- function(args) {
  set.seed(args$seed)
  
  n <- sum(args$pop_sizes)
  p <- args$n_genes
  
  FF <- matrix(rnorm(7 * p, sd = rep(args$branch_sds, each = p)), ncol = 7)
  # if (args$constrain_F) {
  #   FF_svd <- svd(FF)
  #   FF <- FF_svd$u
  #   FF <- t(t(FF) * branch_sds * sqrt(p))
  # }
  
  LL <- matrix(0, nrow = n, ncol = 7)
  LL[, 1] <- 1
  LL[, 2] <- rep(c(1, 1, 0, 0), times = args$pop_sizes)
  LL[, 3] <- rep(c(0, 0, 1, 1), times = args$pop_sizes)
  LL[, 4] <- rep(c(1, 0, 0, 0), times = args$pop_sizes)
  LL[, 5] <- rep(c(0, 1, 0, 0), times = args$pop_sizes)
  LL[, 6] <- rep(c(0, 0, 1, 0), times = args$pop_sizes)
  LL[, 7] <- rep(c(0, 0, 0, 1), times = args$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)))
}

sim_args = list(pop_sizes = rep(40, 4), n_genes = 1000, branch_sds = rep(2,7), indiv_sd = 1, seed = 1)
sim_data <- sim_4pops(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)

This is a plot of the eigenvalues of the Gram matrix:

S_eigen <- eigen(sim_data$YYt)
plot(S_eigen$values) + abline(a = 0, b = 0, col = 'red')

integer(0)

This is the minimum eigenvalue:

min(S_eigen$values)

[1] 0.3724341

Generalized binary

Generalized binary symEBcovMF

For comparison, we first try running greedy symEBcovMF with generalized binary prior.

symebcovmf_fit <- sym_ebcovmf_fit(S = sim_data$YYt, ebnm_fn = ebnm::ebnm_generalized_binary, K = 7, maxiter = 500, rank_one_tol = 10^(-8), tol = 10^(-8), refit_lam = TRUE)

[1] "elbo decreased by 0.185693113584421"
[1] "elbo decreased by 0.0114876429142896"
[1] "elbo decreased by 1.5825207810849e-10"

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_fit$L_pm %*% diag(sqrt(symebcovmf_fit$lambda)), bal_pops)

This is the objective function value attained:

symebcovmf_fit$elbo

[1] -14979.17

Generalized binary symEBcovMF backfit initialized with point-exponential fit

For this procedure, I start by running symEBcovMF with a point-exponential prior with Kmax set to the inputted Kmax value. From this, I get an estimate for \(L\), which I call \(\hat{L}_{exp}\). Then, I backfit with a generalized-binary prior.

symebcovmf_pexp_fit <- sym_ebcovmf_fit(S = sim_data$YYt, ebnm_fn = ebnm::ebnm_point_exponential, K = 7, maxiter = 500, rank_one_tol = 10^(-8), tol = 10^(-8), refit_lam = TRUE)

This is a scatter plot of \(\hat{L}_{exp}\), the estimate from point-exponential symEBcovMF. This is the initialization for the backfit:

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

Now, we run the backfit.

symebcovmf_gb_fit_backfit <- sym_ebcovmf_backfit(sim_data$YYt, symebcovmf_pexp_fit, ebnm_fn = ebnm_generalized_binary, backfit_maxiter = 100)

[1] "elbo decreased by 3.54702933691442e-11"
[1] "elbo decreased by 2.91038304567337e-11"
[1] "elbo decreased by 5.09771780343726e-09"
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[1] "elbo decreased by 2.65572452917695e-10"
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[1] "elbo decreased by 2.58205545833334e-08"
[1] "elbo decreased by 3.23780113831162e-09"
[1] "elbo decreased by 2.12821760214865e-10"
[1] "elbo decreased by 8.25821189209819e-09"
[1] "elbo decreased by 1.37333699967712e-09"
[1] "elbo decreased by 9.877112461254e-10"
[1] "elbo decreased by 4.58385329693556e-10"
[1] "elbo decreased by 1.08229869510978e-09"
[1] "elbo decreased by 4.05634636990726e-10"
[1] "elbo decreased by 9.64064383879304e-11"
[1] "elbo decreased by 7.2868715506047e-09"
[1] "elbo decreased by 4.4783519115299e-09"
[1] "elbo decreased by 4.72937244921923e-11"
[1] "elbo decreased by 3.0921000870876e-08"
[1] "elbo decreased by 6.12999428994954e-09"
[1] "elbo decreased by 3.00133251585066e-09"
[1] "elbo decreased by 6.87577994540334e-10"
[1] "elbo decreased by 1.01863406598568e-09"
[1] "elbo decreased by 2.5545887183398e-08"
[1] "elbo decreased by 4.0381564758718e-10"
[1] "elbo decreased by 8.91122908797115e-09"
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[1] "elbo decreased by 5.77347236685455e-09"
[1] "elbo decreased by 4.00177668780088e-10"
[1] "elbo decreased by 1.16124283522367e-08"
[1] "elbo decreased by 2.88855517283082e-09"
[1] "elbo decreased by 7.40328687243164e-10"
[1] "elbo decreased by 2.88491719402373e-09"
[1] "elbo decreased by 7.89441401138902e-10"
[1] "elbo decreased by 7.00310920365155e-10"
[1] "elbo decreased by 4.72937244921923e-11"
[1] "elbo decreased by 2.88491719402373e-09"
[1] "elbo decreased by 3.16504156216979e-10"
[1] "elbo decreased by 2.88491719402373e-09"
[1] "elbo decreased by 3.31110641127452e-08"
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[1] "elbo decreased by 1.43700162880123e-10"
[1] "elbo decreased by 5.10954123456031e-09"
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[1] "elbo decreased by 2.88491719402373e-09"
[1] "elbo decreased by 6.76664058119059e-10"
[1] "elbo decreased by 5.03678165841848e-09"
[1] "elbo decreased by 9.36779542826116e-10"
[1] "elbo decreased by 2.88491719402373e-09"
[1] "elbo decreased by 4.72937244921923e-11"
[1] "elbo decreased by 2.27373675443232e-10"
[1] "elbo decreased by 1.28056854009628e-09"
[1] "elbo decreased by 5.6206772569567e-10"
[1] "elbo decreased by 1.78260961547494e-10"
[1] "elbo decreased by 7.08678271621466e-09"
[1] "elbo decreased by 1.30785338114947e-09"
[1] "elbo decreased by 2.63753463514149e-10"
[1] "elbo decreased by 2.83762346953154e-10"
[1] "elbo decreased by 4.72937244921923e-11"
[1] "elbo decreased by 2.88127921521664e-09"
[1] "elbo decreased by 1.94813765119761e-08"
[1] "elbo decreased by 2.25190888158977e-09"
[1] "elbo decreased by 3.13229975290596e-09"
[1] "elbo decreased by 3.41970007866621e-10"
[1] "elbo decreased by 4.11091605201364e-10"
[1] "elbo decreased by 1.66437530424446e-08"
[1] "elbo decreased by 1.80625647772104e-09"
[1] "elbo decreased by 6.91215973347425e-11"
[1] "elbo decreased by 1.47338141687214e-10"
[1] "elbo decreased by 2.17551132664084e-09"
[1] "elbo decreased by 2.88855517283082e-09"
[1] "elbo decreased by 5.14774001203477e-10"
[1] "elbo decreased by 2.69210431724787e-10"
[1] "elbo decreased by 2.88855517283082e-09"
[1] "elbo decreased by 1.11103872768581e-08"
[1] "elbo decreased by 5.65705704502761e-10"
[1] "elbo decreased by 2.88491719402373e-09"
[1] "elbo decreased by 8.66202753968537e-09"
[1] "elbo decreased by 2.88855517283082e-09"
[1] "elbo decreased by 2.80761014437303e-08"
[1] "elbo decreased by 1.40062184073031e-10"
[1] "elbo decreased by 2.88855517283082e-09"
[1] "elbo decreased by 1.63709046319127e-10"
[1] "elbo decreased by 1.96814653463662e-08"
[1] "elbo decreased by 1.95723259821534e-09"
[1] "elbo decreased by 2.02453520614654e-09"
[1] "elbo decreased by 2.88855517283082e-09"
[1] "elbo decreased by 5.77347236685455e-09"
[1] "elbo decreased by 3.37058736477047e-09"
[1] "elbo decreased by 6.89760781824589e-09"
[1] "elbo decreased by 5.18411980010569e-10"
[1] "elbo decreased by 1.20053300634027e-10"
[1] "elbo decreased by 5.76983438804746e-09"
[1] "elbo decreased by 5.84077497478575e-09"
[1] "elbo decreased by 1.12049747258425e-09"
[1] "elbo decreased by 3.09937604470178e-08"
[1] "elbo decreased by 2.18278728425503e-09"
[1] "elbo decreased by 6.00266503170133e-11"
[1] "elbo decreased by 8.42192093841732e-10"
[1] "elbo decreased by 2.83762346953154e-10"
[1] "elbo decreased by 7.73252395447344e-09"
[1] "elbo decreased by 2.57932697422802e-09"
[1] "elbo decreased by 2.18278728425503e-10"
[1] "elbo decreased by 1.38243194669485e-10"
[1] "elbo decreased by 5.77347236685455e-09"
[1] "elbo decreased by 4.84942574985325e-09"
[1] "elbo decreased by 5.23868948221207e-10"
[1] "elbo decreased by 2.88491719402373e-09"
[1] "elbo decreased by 5.77347236685455e-09"
[1] "elbo decreased by 1.54159351950511e-08"
[1] "elbo decreased by 1.52795109897852e-09"
[1] "elbo decreased by 2.88491719402373e-09"
[1] "elbo decreased by 2.5465851649642e-11"
[1] "elbo decreased by 2.88491719402373e-09"
[1] "elbo decreased by 9.24046617001295e-10"
[1] "elbo decreased by 2.18278728425503e-10"
[1] "elbo decreased by 2.88127921521664e-09"
[1] "elbo decreased by 2.47036950895563e-08"
[1] "elbo decreased by 2.88491719402373e-09"
[1] "elbo decreased by 8.65838956087828e-09"
[1] "elbo decreased by 1.25510268844664e-10"
[1] "elbo decreased by 1.40062184073031e-09"
[1] "elbo decreased by 5.67524693906307e-10"
[1] "elbo decreased by 5.40785549674183e-09"
[1] "elbo decreased by 2.88491719402373e-09"
[1] "elbo decreased by 4.89308149553835e-10"
[1] "elbo decreased by 2.88491719402373e-09"
[1] "elbo decreased by 2.15186446439475e-09"
[1] "elbo decreased by 6.45741238258779e-10"
[1] "elbo decreased by 7.10133463144302e-09"
[1] "elbo decreased by 3.56521923094988e-09"
[1] "elbo decreased by 2.88491719402373e-09"
[1] "elbo decreased by 4.77484718430787e-09"
[1] "elbo decreased by 1.73895386978984e-09"
[1] "elbo decreased by 6.00266503170133e-11"
[1] "elbo decreased by 2.58296495303512e-10"
[1] "elbo decreased by 3.52883944287896e-10"
[1] "elbo decreased by 1.50066625792533e-08"
[1] "elbo decreased by 6.28278939984739e-09"
[1] "elbo decreased by 2.88491719402373e-09"
[1] "elbo decreased by 3.32875060848892e-10"
[1] "elbo decreased by 2.88855517283082e-09"
[1] "elbo decreased by 1.96250766748562e-08"
[1] "elbo decreased by 1.15323928184807e-09"
[1] "elbo decreased by 1.38243194669485e-10"
[1] "elbo decreased by 5.77347236685455e-09"
[1] "elbo decreased by 5.76983438804746e-09"
[1] "elbo decreased by 1.14978320198134e-08"
[1] "elbo decreased by 2.51020537689328e-09"
[1] "elbo decreased by 2.88491719402373e-09"
[1] "elbo decreased by 7.03039404470474e-09"
[1] "elbo decreased by 4.07271727453917e-09"
[1] "elbo decreased by 5.77347236685455e-09"
[1] "elbo decreased by 1.38243194669485e-10"
[1] "elbo decreased by 2.88855517283082e-09"
[1] "elbo decreased by 2.97586666420102e-09"
[1] "elbo decreased by 2.88491719402373e-09"
[1] "elbo decreased by 2.11002770811319e-10"
[1] "elbo decreased by 1.2732925824821e-10"
[1] "elbo decreased by 1.91394065041095e-08"
[1] "elbo decreased by 1.82626536116004e-09"
[1] "elbo decreased by 1.83717929758132e-10"
[1] "elbo decreased by 2.88491719402373e-09"
[1] "elbo decreased by 5.94809534959495e-10"
[1] "elbo decreased by 4.38194547314197e-09"
[1] "elbo decreased by 5.72981662116945e-10"
[1] "elbo decreased by 4.60204319097102e-10"
[1] "elbo decreased by 2.66118149738759e-09"
[1] "elbo decreased by 1.67347025126219e-10"
[1] "elbo decreased by 8.65838956087828e-09"
[1] "elbo decreased by 2.88491719402373e-09"
[1] "elbo decreased by 1.71185092767701e-08"
[1] "elbo decreased by 3.56340024154633e-09"
[1] "elbo decreased by 1.23873178381473e-09"
[1] "elbo decreased by 5.76983438804746e-09"
[1] "elbo decreased by 2.88491719402373e-09"
[1] "elbo decreased by 2.11857695830986e-08"
[1] "elbo decreased by 5.11136022396386e-10"
[1] "elbo decreased by 5.63886715099216e-11"
[1] "elbo decreased by 5.77347236685455e-09"
[1] "elbo decreased by 2.88491719402373e-09"
[1] "elbo decreased by 1.24073267215863e-08"
[1] "elbo decreased by 6.35736796539277e-09"
[1] "elbo decreased by 1.47883838508278e-09"
[1] "elbo decreased by 1.19689502753317e-09"
[1] "elbo decreased by 5.76983438804746e-09"
[1] "elbo decreased by 2.91602191282436e-08"
[1] "elbo decreased by 9.6588337328285e-10"
[1] "elbo decreased by 4.34374669566751e-09"
[1] "elbo decreased by 5.94809534959495e-10"
[1] "elbo decreased by 2.52839527092874e-10"
[1] "elbo decreased by 2.6358975446783e-08"
[1] "elbo decreased by 2.88855517283082e-09"
[1] "elbo decreased by 1.27693056128919e-09"
[1] "elbo decreased by 2.2846506908536e-09"
[1] "elbo decreased by 6.49379217065871e-10"
[1] "elbo decreased by 2.88491719402373e-09"
[1] "elbo decreased by 2.88127921521664e-09"
[1] "elbo decreased by 1.96923792827874e-08"
[1] "elbo decreased by 1.86628312803805e-09"
[1] "elbo decreased by 1.56433088704944e-10"
[1] "elbo decreased by 2.88491719402373e-09"
[1] "elbo decreased by 3.52883944287896e-10"
[1] "elbo decreased by 2.89219315163791e-09"
[1] "elbo decreased by 2.03726813197136e-08"
[1] "elbo decreased by 1.4715624274686e-09"
[1] "elbo decreased by 2.88491719402373e-09"
[1] "elbo decreased by 2.88491719402373e-09"
[1] "elbo decreased by 8.58562998473644e-09"
[1] "elbo decreased by 9.35324351303279e-09"
[1] "elbo decreased by 2.01089278561994e-08"
[1] "elbo decreased by 1.38606992550194e-09"
[1] "elbo decreased by 8.66202753968537e-09"
[1] "elbo decreased by 1.89174897968769e-10"
[1] "elbo decreased by 1.56232999870554e-08"
[1] "elbo decreased by 4.33647073805332e-09"
[1] "elbo decreased by 7.47604644857347e-10"
[1] "elbo decreased by 5.36601874046028e-10"
[1] "elbo decreased by 1.55214365804568e-08"
[1] "elbo decreased by 7.6397554948926e-11"
[1] "elbo decreased by 2.88491719402373e-09"
[1] "elbo decreased by 2.88491719402373e-09"
[1] "elbo decreased by 5.38420863449574e-10"
[1] "elbo decreased by 3.33220668835565e-08"
[1] "elbo decreased by 6.46832631900907e-09"
[1] "elbo decreased by 8.76752892509103e-10"
[1] "elbo decreased by 2.88127921521664e-09"
[1] "elbo decreased by 9.1313268058002e-10"
[1] "elbo decreased by 7.27595761418343e-10"
[1] "elbo decreased by 2.88491719402373e-09"
[1] "elbo decreased by 6.53017195872962e-10"
[1] "elbo decreased by 5.28962118551135e-09"
[1] "elbo decreased by 3.52338247466832e-09"
[1] "elbo decreased by 2.88855517283082e-09"
[1] "elbo decreased by 2.8321665013209e-09"
[1] "elbo decreased by 1.34605215862393e-10"
[1] "elbo decreased by 3.0740920919925e-10"
[1] "elbo decreased by 5.76983438804746e-09"
[1] "elbo decreased by 6.25914253760129e-09"
[1] "elbo decreased by 1.23691279441118e-09"
[1] "elbo decreased by 2.7648638933897e-10"
[1] "elbo decreased by 5.76619640924037e-09"
[1] "elbo decreased by 2.88855517283082e-09"
[1] "elbo decreased by 2.04199750442058e-08"
[1] "elbo decreased by 3.34694050252438e-10"
[1] "elbo decreased by 8.65475158207119e-09"

This is a scatter plot of \(\hat{L}_{gb-backfit}\), the estimate from the generalized binary backfit:

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

This is the objective function value attained:

symebcovmf_gb_fit_backfit$elbo

[1] 9092.392

Observations

The estimate from the backfit procedure looks closer to the tree-structured loadings matrix than the estimate from greedy generalized binary symEBcovMF. Furthermore, the backfit procedure attains a much higher objective function value. This suggests that the method does prefer the tree-structured loadings matrix, and the greedy generalized binary symEBcovMF method has a convergence issue. The second factor corresponding to a subtype effect could be more binary. This could potentially be fixed by using a more strict binary prior. Similarly, the first factor corresponding to the intercept could be more binary. Another observation is the single group effect factors all have a couple of small, non-zero loading values. I’m not exactly sure why those appear.

Binormal

In this section, I try applying the same method with the binormal prior instead of the generalized binary prior.

dbinormal = function (x,s,s0,lambda,log=TRUE){
  pi0 = 0.5
  pi1 = 0.5
  s2 = s^2
  s02 = s0^2
  l0 = dnorm(x,0,sqrt(lambda^2 * s02 + s2),log=TRUE)
  l1 = dnorm(x,lambda,sqrt(lambda^2 * s02 + s2),log=TRUE)
  logsum = log(pi0*exp(l0) + pi1*exp(l1))
 
  m = pmax(l0,l1)
  logsum = m + log(pi0*exp(l0-m) + pi1*exp(l1-m))
  if (log) return(sum(logsum))
  else return(exp(sum(logsum)))
}

ebnm_binormal = function(x,s, g_init = NULL, fix_g = FALSE, output = ebnm_output_default(), control = NULL){
  # Add g_init to make the method run
  if(is.null(dim(x)) == FALSE){
    x <- c(x)
  }
  s0 = 0.01
  lambda = optimize(function(lambda){-dbinormal(x,s,s0,lambda,log=TRUE)},
              lower = 0, upper = max(x))$minimum
  g = ashr::normalmix(pi=c(0.5,0.5), mean=c(0,lambda), sd=c(lambda * s0,lambda * s0))
  postmean = ashr::postmean(g,ashr::set_data(x,s))
  postsd = ashr::postsd(g,ashr::set_data(x,s))
  log_likelihood <- ashr::calc_loglik(g, ashr::set_data(x,s))
  return(list(fitted_g = g, posterior = data.frame(mean=postmean,sd=postsd), log_likelihood = log_likelihood))
}

Binormal symEBcovMF

For comparison, I try running regular symEBcovMF with the binormal prior.

symebcovmf_binormal_fit <- sym_ebcovmf_fit(S = sim_data$YYt, ebnm_fn = ebnm_binormal, K = 7, maxiter = 500, rank_one_tol = 10^(-8), tol = 10^(-8), refit_lam = TRUE)

[1] "elbo decreased by 15.6794903684367"

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

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

This is the objective function value attained:

symebcovmf_binormal_fit$elbo

[1] -14313.79

Binormal symEBcovMF backfit with point-exponential initialization

Now we try the backfit procedure with the binormal prior.

symebcovmf_binormal_fit_backfit <- sym_ebcovmf_backfit(sim_data$YYt, symebcovmf_pexp_fit, ebnm_fn = ebnm_binormal, backfit_maxiter = 100)

[1] "elbo decreased by 2.63753463514149e-11"
[1] "elbo decreased by 1.10958353616297e-10"
[1] "elbo decreased by 7.73070496506989e-12"
[1] "elbo decreased by 4.85852069687098e-09"
[1] "elbo decreased by 5.95900928601623e-09"
[1] "elbo decreased by 9.64064383879304e-11"
[1] "elbo decreased by 1.63709046319127e-11"
[1] "elbo decreased by 1.73131411429495e-08"
[1] "elbo decreased by 2.90310708805919e-09"
[1] "elbo decreased by 4.23096935264766e-09"
[1] "elbo decreased by 2.64117261394858e-08"
[1] "elbo decreased by 2.98132363241166e-09"
[1] "elbo decreased by 1.62071955855936e-09"
[1] "elbo decreased by 2.49146978603676e-08"
[1] "elbo decreased by 9.3496055342257e-10"
[1] "elbo decreased by 1.25619408208877e-08"
[1] "elbo decreased by 1.57597241923213e-08"
[1] "elbo decreased by 3.33238858729601e-09"
[1] "elbo decreased by 6.70115696266294e-09"
[1] "elbo decreased by 2.9987859306857e-08"
[1] "elbo decreased by 9.82254277914762e-10"
[1] "elbo decreased by 3.17704689223319e-08"
[1] "elbo decreased by 2.3919710656628e-09"
[1] "elbo decreased by 5.39330358151346e-09"
[1] "elbo decreased by 1.31149135995656e-09"
[1] "elbo decreased by 2.50020093517378e-08"
[1] "elbo decreased by 7.9689925769344e-09"
[1] "elbo decreased by 3.16358637064695e-08"
[1] "elbo decreased by 2.69574229605496e-09"
[1] "elbo decreased by 3.94720700569451e-10"
[1] "elbo decreased by 2.72848410531878e-10"
[1] "elbo decreased by 5.63886715099216e-10"
[1] "elbo decreased by 9.27684595808387e-11"
[1] "elbo decreased by 2.29192664846778e-09"
[1] "elbo decreased by 2.38469510804862e-08"
[1] "elbo decreased by 1.31512933876365e-09"
[1] "elbo decreased by 2.11002770811319e-10"
[1] "elbo decreased by 6.22094376012683e-10"
[1] "elbo decreased by 1.30057742353529e-09"
[1] "elbo decreased by 1.29148247651756e-10"
[1] "elbo decreased by 1.00044417195022e-09"
[1] "elbo decreased by 3.65616870112717e-10"
[1] "elbo decreased by 2.67809809884056e-08"
[1] "elbo decreased by 1.95559550775215e-08"
[1] "elbo decreased by 1.10958353616297e-10"
[1] "elbo decreased by 5.56974555365741e-09"
[1] "elbo decreased by 1.14596332423389e-10"
[1] "elbo decreased by 3.27054294757545e-09"
[1] "elbo decreased by 1.61890056915581e-10"
[1] "elbo decreased by 7.24867277313024e-09"
[1] "elbo decreased by 1.01681507658213e-09"
[1] "elbo decreased by 1.25510268844664e-10"
[1] "elbo decreased by 1.17379386210814e-08"
[1] "elbo decreased by 8.68021743372083e-09"
[1] "elbo decreased by 1.13504938781261e-09"
[1] "elbo decreased by 8.62200977280736e-10"
[1] "elbo decreased by 1.86082615982741e-09"
[1] "elbo decreased by 6.21366780251265e-09"
[1] "elbo decreased by 1.5788828022778e-09"
[1] "elbo decreased by 3.1574018066749e-08"
[1] "elbo decreased by 1.09139364212751e-09"
[1] "elbo decreased by 1.09139364212751e-09"
[1] "elbo decreased by 6.30825525149703e-09"
[1] "elbo decreased by 3.69254848919809e-10"
[1] "elbo decreased by 8.65838956087828e-10"
[1] "elbo decreased by 3.48718458553776e-08"
[1] "elbo decreased by 6.74663169775158e-09"
[1] "elbo decreased by 7.87986209616065e-09"
[1] "elbo decreased by 7.6397554948926e-10"
[1] "elbo decreased by 2.03908712137491e-09"
[1] "elbo decreased by 8.67657945491374e-10"
[1] "elbo decreased by 4.59658622276038e-09"
[1] "elbo decreased by 3.25599103234708e-10"
[1] "elbo decreased by 2.86054273601621e-08"
[1] "elbo decreased by 5.6206772569567e-09"
[1] "elbo decreased by 1.04046193882823e-09"
[1] "elbo decreased by 4.44742909166962e-09"
[1] "elbo decreased by 7.29414750821888e-10"
[1] "elbo decreased by 8.09450284577906e-10"
[1] "elbo decreased by 8.96761775948107e-10"
[1] "elbo decreased by 8.76934791449457e-09"
[1] "elbo decreased by 4.07453626394272e-09"
[1] "elbo decreased by 6.62657839711756e-09"
[1] "elbo decreased by 6.54836185276508e-11"
[1] "elbo decreased by 1.98269844986498e-09"
[1] "elbo decreased by 6.31189323030412e-10"
[1] "elbo decreased by 3.53611540049314e-09"
[1] "elbo decreased by 7.33780325390399e-09"
[1] "elbo decreased by 9.94987203739583e-10"
[1] "elbo decreased by 1.30967237055302e-09"
[1] "elbo decreased by 3.77331161871552e-08"
[1] "elbo decreased by 1.18961906991899e-09"
[1] "elbo decreased by 2.40106601268053e-10"
[1] "elbo decreased by 4.26916813012213e-09"
[1] "elbo decreased by 1.40062184073031e-09"
[1] "elbo decreased by 1.89174897968769e-09"
[1] "elbo decreased by 3.8198777474463e-10"
[1] "elbo decreased by 1.04228092823178e-09"
[1] "elbo decreased by 1.9990693544969e-09"
[1] "elbo decreased by 5.91571733821183e-08"
[1] "elbo decreased by 1.60616764333099e-09"
[1] "elbo decreased by 9.05856722965837e-10"
[1] "elbo decreased by 1.30785338114947e-09"
[1] "elbo decreased by 7.78527464717627e-10"
[1] "elbo decreased by 2.22371454583481e-08"
[1] "elbo decreased by 2.00270733330399e-09"
[1] "elbo decreased by 2.49019649345428e-09"
[1] "elbo decreased by 1.22927303891629e-08"
[1] "elbo decreased by 2.86672729998827e-09"
[1] "elbo decreased by 1.41881173476577e-09"
[1] "elbo decreased by 3.14685166813433e-10"
[1] "elbo decreased by 4.92946128360927e-10"
[1] "elbo decreased by 2.5465851649642e-10"
[1] "elbo decreased by 3.89954948332161e-08"
[1] "elbo decreased by 1.38970790430903e-09"
[1] "elbo decreased by 3.99086275137961e-09"
[1] "elbo decreased by 2.31339072342962e-08"
[1] "elbo decreased by 4.39467839896679e-09"
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[1] "elbo decreased by 3.47608875017613e-09"
[1] "elbo decreased by 7.88713805377483e-09"
[1] "elbo decreased by 2.78305378742516e-09"
[1] "elbo decreased by 6.25732354819775e-10"
[1] "elbo decreased by 2.40106601268053e-09"
[1] "elbo decreased by 3.09228198602796e-10"
[1] "elbo decreased by 1.23145582620054e-08"
[1] "elbo decreased by 4.37830749433488e-09"
[1] "elbo decreased by 5.63886715099216e-11"
[1] "elbo decreased by 1.36606104206294e-09"
[1] "elbo decreased by 1.54414010467008e-08"
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[1] "elbo decreased by 1.32786226458848e-10"
[1] "elbo decreased by 3.83624865207821e-09"
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[1] "elbo decreased by 1.16160663310438e-08"
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[1] "elbo decreased by 4.05634636990726e-10"
[1] "elbo decreased by 5.25687937624753e-10"
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[1] "elbo decreased by 1.16415321826935e-09"
[1] "elbo decreased by 1.43700162880123e-10"
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[1] "elbo decreased by 6.33008312433958e-10"
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[1] "elbo decreased by 4.53837856184691e-09"
[1] "elbo decreased by 2.8521753847599e-09"
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[1] "elbo decreased by 8.64019966684282e-10"
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[1] "elbo decreased by 7.45785655453801e-11"
[1] "elbo decreased by 1.28893589135259e-08"
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[1] "elbo decreased by 2.87218426819891e-09"
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[1] "elbo decreased by 2.72848410531878e-10"
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[1] "elbo decreased by 2.60843080468476e-09"
[1] "elbo decreased by 3.96539689972997e-10"
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[1] "elbo decreased by 4.56948328064755e-08"
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[1] "elbo decreased by 3.99995769839734e-09"
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[1] "elbo decreased by 9.27684595808387e-11"
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[1] "elbo decreased by 1.86792021850124e-08"
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[1] "elbo decreased by 3.51337803294882e-08"
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[1] "elbo decreased by 3.01588443107903e-09"
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[1] "elbo decreased by 3.93101800000295e-08"
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[1] "elbo decreased by 5.61522028874606e-09"
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[1] "elbo decreased by 1.52485881699249e-08"
[1] "elbo decreased by 1.65164237841964e-09"
[1] "elbo decreased by 6.22094376012683e-09"
[1] "elbo decreased by 3.96539689972997e-10"
[1] "elbo decreased by 6.584741640836e-10"
[1] "elbo decreased by 8.24002199806273e-10"
[1] "elbo decreased by 2.05964170163497e-08"
[1] "elbo decreased by 4.01996658183634e-10"
[1] "elbo decreased by 2.76813807431608e-08"
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[1] "elbo decreased by 9.62245394475758e-10"
[1] "elbo decreased by 4.94765117764473e-10"
[1] "elbo decreased by 7.22138793207705e-10"
[1] "elbo decreased by 5.07025106344372e-08"
[1] "elbo decreased by 2.20461515709758e-09"
[1] "elbo decreased by 2.83762346953154e-09"
[1] "elbo decreased by 1.18780008051544e-09"
[1] "elbo decreased by 5.89352566748857e-10"
[1] "elbo decreased by 1.40244083013386e-09"
[1] "elbo decreased by 1.58706825459376e-08"
[1] "elbo decreased by 8.21819412522018e-09"
[1] "elbo decreased by 7.67613528296351e-10"

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

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

This is the objective function attained:

symebcovmf_binormal_fit_backfit$elbo

[1] 9763.532

Observations

Similar to generalized binary symEBcovMF, the binormal symEBcovMF method fails to find all four single group effect factors. Instead, three of the factors combine two groups. The backfit procedure does a better job at recovering the tree structure in the data. Furthermore, the estimate from the backfit procedure attains a higher objective function value than the estimate from the greedy procedure. Comparing the binormal prior with the generalized binary prior, we see that the factors generated from the latter look more binary. Though this could be due to the settings of the priors. I’m guessing it would be possible to get a more binary estimate from the generalized binary prior by changing the scale parameter.

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

symebcovmf_gb_point_exp_backfit

Annie Xie

2025-05-07

Introduction

Motivation

Packages and Functions

Backfit Function

Data Generation

Generalized binary

Generalized binary symEBcovMF

Generalized binary symEBcovMF backfit initialized with point-exponential fit

Observations

Binormal

Binormal symEBcovMF

Binormal symEBcovMF backfit with point-exponential initialization

Observations