Last updated: 2025-06-05

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Rmd 1e1f127 Annie Xie 2025-06-05 Add analysis of backfit in overlapping setting

Introduction

In this example, we test out symEBcovMF with backfit on overlapping (but not necessarily hierarchical)-structured data.

Motivation

I am interested in testing whether backfitting helps symEBcovMF in the overlapping setting (particularly in the setting where we set Kmax to be the true number of factors). Our high level goal is to develop a method that does well in both the tree setting and the overlapping setting.

Packages and Functions

library(ebnm)
library(pheatmap)
library(ggplot2)
library(lpSolve)
source('code/symebcovmf_functions.R')
source('code/visualization_functions.R')
compute_crossprod_similarity <- function(est, truth){
  K_est <- ncol(est)
  K_truth <- ncol(truth)
  n <- nrow(est)
  
  #if estimates don't have same number of columns, try padding the estimate with zeros and make cosine similarity zero
  if (K_est < K_truth){
    est <- cbind(est, matrix(rep(0, n*(K_truth-K_est)), nrow = n))
  }
  
  if (K_est > K_truth){
    truth <- cbind(truth, matrix(rep(0, n*(K_est - K_truth)), nrow = n))
  }
  
  #normalize est and truth
  norms_est <- apply(est, 2, function(x){sqrt(sum(x^2))})
  norms_est[norms_est == 0] <- Inf
  
  norms_truth <- apply(truth, 2, function(x){sqrt(sum(x^2))})
  norms_truth[norms_truth == 0] <- Inf
  
  est_normalized <- t(t(est)/norms_est)
  truth_normalized <- t(t(truth)/norms_truth)
  
  #compute matrix of cosine similarities
  cosine_sim_matrix <- abs(crossprod(est_normalized, truth_normalized))
  
  assignment_problem <- lp.assign(t(cosine_sim_matrix), direction = "max")
  return((1/K_truth)*assignment_problem$objval)
}
permute_L <- function(est, truth){
  K_est <- ncol(est)
  K_truth <- ncol(truth)
  n <- nrow(est)

  #if estimates don't have same number of columns, try padding the estimate with zeros and make cosine similarity zero
  if (K_est < K_truth){
    est <- cbind(est, matrix(rep(0, n*(K_truth-K_est)), nrow = n))
  }

  if (K_est > K_truth){
    truth <- cbind(truth, matrix(rep(0, n*(K_est - K_truth)), nrow = n))
  }

  #normalize est and truth
  norms_est <- apply(est, 2, function(x){sqrt(sum(x^2))})
  norms_est[norms_est == 0] <- Inf

  norms_truth <- apply(truth, 2, function(x){sqrt(sum(x^2))})
  norms_truth[norms_truth == 0] <- Inf

  est_normalized <- t(t(est)/norms_est)
  truth_normalized <- t(t(truth)/norms_truth)

  #compute matrix of cosine similarities
  cosine_sim_matrix <- abs(crossprod(est_normalized, truth_normalized))
  assignment_problem <- lp.assign(t(cosine_sim_matrix), direction = "max")

  perm <- apply(assignment_problem$solution, 1, which.max)
  return(est[,perm])
}

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

# args is a list containing n, p, k, indiv_sd, pi1, and seed
sim_binary_loadings_data <- function(args) {
  set.seed(args$seed)
  
  FF <- matrix(rnorm(args$k * args$p, sd = args$group_sd), ncol = args$k)
  if (args$constrain_F) {
    FF_svd <- svd(FF)
    FF <- FF_svd$u
    FF <- t(t(FF) * rep(args$group_sd, args$k) * sqrt(p))
  }
  LL <- matrix(rbinom(args$n*args$k, 1, args$pi1), nrow = args$n, ncol = args$k)
  E <- matrix(rnorm(args$n * args$p, sd = args$indiv_sd), nrow = args$n)
  
  Y <- LL %*% t(FF) + E
  YYt <- (1/args$p)*tcrossprod(Y)
  
  return(list(Y = Y, YYt = YYt, LL = LL, FF = FF, K = ncol(LL)))
}
n <- 100
p <- 1000
k <- 10
pi1 <- 0.1
indiv_sd <- 1
group_sd <- 1
seed <- 1
sim_args = list(n = n, p = p, k = k, pi1 = pi1, indiv_sd = indiv_sd, group_sd = group_sd, seed = seed, constrain_F = FALSE)
sim_data <- sim_binary_loadings_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('A', n)
plot_loadings(sim_data$LL, pop_vec, legendYN = FALSE)

This is a heatmap of the true loadings matrix:

plot_heatmap(sim_data$LL)

Regular symEBcovMF

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

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

plot_loadings(symebcovmf_overlap_refit_fit$L_pm %*% diag(sqrt(symebcovmf_overlap_refit_fit$lambda)), pop_vec, legendYN = FALSE)

This is a heatmap of the true loadings matrix:

plot_heatmap(sim_data$LL)

This is a heatmap of \(\hat{L}_{refit}\). The columns have been permuted to match the true loadings matrix:

symebcovmf_overlap_refit_fit_L_permuted <- permute_L(symebcovmf_overlap_refit_fit$L_pm, sim_data$LL)
plot_heatmap(symebcovmf_overlap_refit_fit_L_permuted, brks = seq(0, max(symebcovmf_overlap_refit_fit_L_permuted), length.out = 50))

This is a heatmap of \(\hat{L}_{refit}\) where the columns have not been permuted. The order corresponds to the order in which the factors were added.

plot_heatmap(symebcovmf_overlap_refit_fit$L_pm %*% diag(sqrt(symebcovmf_overlap_refit_fit$lambda)), brks = seq(0, max(symebcovmf_overlap_refit_fit$L_pm %*% diag(sqrt(symebcovmf_overlap_refit_fit$lambda))), length.out = 50))

This is the objective function value attained:

symebcovmf_overlap_refit_fit$elbo
[1] 1346.566

This is the crossproduct similarity of \(\hat{L}_{refit}\):

compute_crossprod_similarity(symebcovmf_overlap_refit_fit$L_pm, sim_data$LL)
[1] 0.8462461

Observations

Greedy symEBcovMF does an okay job at recovering the overlapping structure. Some of estimates match the corresponding true factors very well, e.g. factors 1, 2, and 4. However, some estimates are more dense than their corresponding true factors, e.g. factors 5, 6, 9, and 10. There are also a couple of estimates which differ from their corresponding true factors by a couple of samples, e.g. factors 3, 7, and 8.

Looking at the heatmap where the columns are ordered by the order they were added, we see that the earlier factors are more dense, while the later factors are more sparse. The factors added later are the factors which more closely match their corresponding true factors.

For comparison, EBCD

For comparison, I try running greedy EBCD on the data.

library(ebcd)
ebcd_init_obj <- ebcd_init(X = t(sim_data$Y))
greedy_ebcd_fit <- ebcd_greedy(ebcd_init_obj, Kmax = 10, ebnm_fn = ebnm::ebnm_point_exponential)

This is a heatmap of the estimate from greedy EBCD, \(\hat{L}_{ebcd-greedy}\):

plot_heatmap(greedy_ebcd_fit$EL, brks = seq(0, max(greedy_ebcd_fit$EL), length.out = 50))

This is the crossproduct similarity of \(\hat{L}_{ebcd-greedy}\):

compute_crossprod_similarity(greedy_ebcd_fit$EL, sim_data$LL)
[1] 0.852768

Now, we run EBCD’s backfit method.

ebcd_backfit_fit <- ebcd_backfit(greedy_ebcd_fit)

This is a heatmap of the estimate from EBCD with backfit, \(\hat{L}_{ebcd-backfit}\):

plot_heatmap(ebcd_backfit_fit$EL, brks = seq(0, max(ebcd_backfit_fit$EL), length.out = 50))

This is a heatmap of the true loadings matrix:

plot_heatmap(sim_data$LL)

This is a heatmap of \(\hat{L}_{ebcd-backfit}\). The columns have been permuted to best match the true loadings matrix.

ebcd_backfit_fit_L_permuted <- permute_L(ebcd_backfit_fit$EL, sim_data$LL)
plot_heatmap(ebcd_backfit_fit_L_permuted, brks = seq(0, max(ebcd_backfit_fit_L_permuted), length.out = 50))

This is the crossproduct similarity of \(\hat{L}_{ebcd-backfit}\):

compute_crossprod_similarity(ebcd_backfit_fit$EL, sim_data$LL)
[1] 0.9990405

We see that the greedy EBCD method performs comparably to greedy symEBcovMF. Therefore, part of the reason EBCD performs so well in this setting is its backfitting. So I suspect that backfitting will help us obtain a better estimate.

symEBcovMF with Backfit

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

symebcovmf_fit_backfit <- sym_ebcovmf_backfit(sim_data$YYt, symebcovmf_overlap_refit_fit, ebnm_fn = ebnm::ebnm_point_exponential, backfit_maxiter = 100)
[1] "elbo decreased by 7.27595761418343e-12"
[1] "elbo decreased by 1.09139364212751e-11"
[1] "elbo decreased by 8.18545231595635e-12"
[1] "elbo decreased by 1.2732925824821e-11"
[1] "elbo decreased by 1.81898940354586e-12"
[1] "elbo decreased by 1.04591890703887e-11"
[1] "elbo decreased by 6.3664629124105e-12"
[1] "elbo decreased by 1.36424205265939e-11"
[1] "elbo decreased by 2.27373675443232e-12"
[1] "elbo decreased by 2.72848410531878e-12"
[1] "elbo decreased by 4.54747350886464e-12"
[1] "elbo decreased by 1.40971678774804e-11"
[1] "elbo decreased by 1.36424205265939e-12"
[1] "elbo decreased by 4.09272615797818e-12"
[1] "elbo decreased by 5.45696821063757e-12"
[1] "elbo decreased by 1.31876731757075e-11"
[1] "elbo decreased by 3.18323145620525e-12"
[1] "elbo decreased by 8.18545231595635e-12"
[1] "elbo decreased by 3.18323145620525e-12"
[1] "elbo decreased by 8.18545231595635e-12"
[1] "elbo decreased by 4.54747350886464e-13"
[1] "elbo decreased by 1.40971678774804e-11"
[1] "elbo decreased by 2.00088834390044e-11"
[1] "elbo decreased by 9.09494701772928e-13"
[1] "elbo decreased by 3.63797880709171e-12"
[1] "elbo decreased by 9.09494701772928e-13"
[1] "elbo decreased by 5.0022208597511e-12"
[1] "elbo decreased by 9.09494701772928e-13"
[1] "elbo decreased by 4.54747350886464e-13"
[1] "elbo decreased by 8.18545231595635e-12"
[1] "elbo decreased by 1.81898940354586e-12"
[1] "elbo decreased by 9.09494701772928e-13"
[1] "elbo decreased by 1.45519152283669e-11"
[1] "elbo decreased by 1.68256519827992e-11"
[1] "elbo decreased by 1.50066625792533e-11"
[1] "elbo decreased by 5.0022208597511e-12"
[1] "elbo decreased by 1.81898940354586e-12"
[1] "elbo decreased by 1.36424205265939e-12"
[1] "elbo decreased by 1.00044417195022e-11"
[1] "elbo decreased by 1.13686837721616e-11"
[1] "elbo decreased by 5.45696821063757e-12"
[1] "elbo decreased by 3.63797880709171e-12"
[1] "elbo decreased by 4.54747350886464e-13"
[1] "elbo decreased by 2.27373675443232e-12"
[1] "elbo decreased by 1.36424205265939e-11"
[1] "elbo decreased by 5.91171556152403e-12"
[1] "elbo decreased by 6.3664629124105e-12"
[1] "elbo decreased by 2.00088834390044e-11"
[1] "elbo decreased by 4.09272615797818e-12"
[1] "elbo decreased by 1.18234311230481e-11"
[1] "elbo decreased by 1.54614099301398e-11"
[1] "elbo decreased by 1.81898940354586e-12"
[1] "elbo decreased by 2.72848410531878e-12"
[1] "elbo decreased by 4.54747350886464e-13"
[1] "elbo decreased by 4.54747350886464e-12"
[1] "elbo decreased by 7.27595761418343e-12"
[1] "elbo decreased by 1.36424205265939e-12"
[1] "elbo decreased by 2.72848410531878e-12"
[1] "elbo decreased by 2.27373675443232e-12"
[1] "elbo decreased by 2.27373675443232e-12"
[1] "elbo decreased by 4.54747350886464e-13"
[1] "elbo decreased by 5.45696821063757e-12"

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

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

This is a heatmap of the true loadings matrix:

plot_heatmap(sim_data$LL)

This is a heatmap of \(\hat{L}_{backfit}\). The columns have been permuted to match the true loadings matrix:

symebcovmf_fit_backfit_L_permuted <- permute_L(symebcovmf_fit_backfit$L_pm, sim_data$LL)
plot_heatmap(symebcovmf_fit_backfit_L_permuted, brks = seq(0, max(symebcovmf_fit_backfit_L_permuted), length.out = 50))

This is the objective function value attained:

symebcovmf_fit_backfit$elbo
[1] 3019.79

This is the crossproduct similarity of \(\hat{L}_{backfit}\):

compute_crossprod_similarity(symebcovmf_fit_backfit$L_pm, sim_data$LL)
[1] 0.8984952

Observations

Visually, the estimate after the backfitting appears to better match the true loadings matrix. This is corroborated by the increased cross-product similarity. For many of the factors, the estimates are very close to the true values. However, the estimate for factor 5 is noticeably off. After further inspection, we see that the best estimate for factor 3 captures the shared effects from the true factor 5 along with the shared effects from the true factor 3.

Here, I look at the residual matrix minus the estimate for factor 5 (the two-individual factor):

R <- sim_data$YYt - tcrossprod(symebcovmf_fit_backfit$L_pm[,c(2:10)] %*% diag(sqrt(symebcovmf_fit_backfit$lambda[2:10])))
plot_heatmap(R, colors_range = c('blue','gray96','red'), brks = seq(-max(abs(R)), max(abs(R)), length.out = 50))

This is the component generated from the estimate of factor 5:

component_fac5 <- tcrossprod(sqrt(symebcovmf_fit_backfit$lambda[1])*symebcovmf_fit_backfit$L_pm[,1])
plot_heatmap(component_fac5, colors_range = c('blue','gray96','red'), brks = seq(-max(abs(component_fac5)), max(abs(component_fac5)), length.out = 50))

It doesn’t make sense to me why this component would be chosen given the residual matrix.


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] ebcd_0.0.0.9000 lpSolve_5.6.20  ggplot2_3.5.1   pheatmap_1.0.12
[5] 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