Overdispersed splitting method dealing with NB data

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Last updated: 2022-11-30

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Knit directory: gsmash/

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Introduction

We simulate mean parameter \(\lambda\) from \(\pi_0\delta_0 + \pi_1Exp(0.1)\).

Then generate data using a NB distribution \(NB(r,p)\). Then \(r(1-p)/p = \lambda\) so \(p = r/(r+\lambda)\). The variance is \(r(1-p)/p^2 = \lambda + \lambda^2/r\).

What’s the corresponding \(\sigma^2\) in \(Poisson(\exp(\mu+\sigma^2))\)?

Since \(\exp(\mu+\sigma2/2)=\lambda\), we have \(\mu = \log\lambda - \sigma^2/2\). Then by matching the variance of NB and the Poisson model, we solve \((\exp(\sigma^2)-1)\exp(2\mu+\sigma^2) = \lambda^2/r\) and we have \(\sigma^2 = \log(1+1/r)\). The smaller the \(r\), the larger oversidpersion.

library(vebpm)
library(ashr)

simu_func = function(n_simu=10,n,r,prior_rate=0.1,w=0.8,seed = 12345,n_plot = 3){
  set.seed(seed)
  mse_mean = c()
  mse_non0_mean = c()
  sigma2 = log(1+1/r)
  
  for(i in 1:n_simu){
    lambda = c(rep(0,round(n*w)) , rexp(round(n*(1-w)),prior_rate))
    non0_idx = which(lambda!=0)
    y = rnbinom(n,r,mu  = lambda)
    fit_ash = ash_pois(y)
    fit_split_ash_init = pois_mean_split(y,sigma2=sigma2,est_sigma2 = FALSE,mu_pm_init = fit_ash$result$PosteriorMean,tol=1e-3)
    #fit_split_logx = pois_mean_split(y,sigma2=sigma2,est_sigma2 = FALSE,mu_pm_init = NULL)
    # mse_b = rbind(mse_b,c(mse(log(fit_ash$result$PosteriorMean),(b)),
    #                       mse(fit_split_ash_init$posterior$mean_b,b),
    #                       mse(fit_split_logx$posterior$mean_b,b)))
    mse_mean = rbind(mse_mean,c(mse(fit_ash$result$PosteriorMean,lambda),
                                  mse(fit_split_ash_init$posterior$mean_exp_b,lambda),
                                 mse(exp(fit_split_ash_init$posterior$mean_b),lambda)))
    mse_non0_mean = rbind(mse_non0_mean,c(mse(fit_ash$result$PosteriorMean[non0_idx],lambda[non0_idx]),
                              mse(fit_split_ash_init$posterior$mean_exp_b[non0_idx],lambda[non0_idx]),
                              mse(exp(fit_split_ash_init$posterior$mean_b[non0_idx]),lambda[non0_idx])))

    colnames(mse_mean) = c('ash','split','split exp(b_pm)')
    colnames(mse_non0_mean) = c('ash','split','split exp(b_pm)')
    if(i<=n_plot){
      par(mfrow=c(4,1))
      ylim = range(c(y,lambda,fit_split_ash_init$posterior$mean_exp_b,exp(fit_split_ash_init$posterior$mean_b)))
      plot(y,col='grey80',ylab='y',main='true mean',ylim=ylim)
      lines(lambda,col='grey20')
      plot(y,col='grey80',ylab='y',main='ash',ylim=ylim)
      lines(fit_ash$result$PosteriorMean)
      plot(y,col='grey80',ylab='y',main='splitting',ylim=ylim)
      lines(fit_split_ash_init$posterior$mean_exp_b)
      plot(y,col='grey80',ylab='y',main='splitting, exp(b_pm)',ylim=ylim)
      lines(exp(fit_split_ash_init$posterior$mean_b))
    }
    
  }
  return(list(mse_mean = mse_mean,mse_non0_mean = mse_non0_mean,sigma2=sigma2))
  
}

r = 10

res = simu_func(n_simu=10,n=1000,r = 10,n_plot = 10)

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colMeans(res$mse_mean)

            ash           split split exp(b_pm) 
       5.519197        6.371801        5.822060 

colMeans(res$mse_non0_mean)

            ash           split split exp(b_pm) 
       27.59361        31.83897        29.09030 

r = 5

res = simu_func(n_simu=10,n=1000,r = 5,n_plot = 10)

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colMeans(res$mse_mean)

            ash           split split exp(b_pm) 
       9.715968       11.669435        9.867543 

colMeans(res$mse_non0_mean)

            ash           split split exp(b_pm) 
       48.57728        58.31219        49.30285 

r = 1

res = simu_func(n_simu=10,n=1000,r = 1,n_plot = 10)

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colMeans(res$mse_mean)

            ash           split split exp(b_pm) 
       30.42936        40.33983        24.16137 

colMeans(res$mse_non0_mean)

            ash           split split exp(b_pm) 
       152.1421        201.6679        120.7762 

r = 50

res = simu_func(n_simu=10,n=1000,r = 50,n_plot = 10)

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colMeans(res$mse_mean)

            ash           split split exp(b_pm) 
       2.723380        2.905060        2.842274 

colMeans(res$mse_non0_mean)

            ash           split split exp(b_pm) 
       13.61450        14.50211        14.18819 

Session information

sessionInfo()

R version 4.2.1 (2022-06-23)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Ubuntu 20.04.5 LTS

Matrix products: default
BLAS:   /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.9.0
LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.9.0

locale:
 [1] LC_CTYPE=C.UTF-8       LC_NUMERIC=C           LC_TIME=C.UTF-8       
 [4] LC_COLLATE=C.UTF-8     LC_MONETARY=C.UTF-8    LC_MESSAGES=C.UTF-8   
 [7] LC_PAPER=C.UTF-8       LC_NAME=C              LC_ADDRESS=C          
[10] LC_TELEPHONE=C         LC_MEASUREMENT=C.UTF-8 LC_IDENTIFICATION=C   

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] ashr_2.2-54     vebpm_0.3.1     workflowr_1.7.0

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.9         horseshoe_0.2.0    invgamma_1.1       lattice_0.20-45   
 [5] nleqslv_3.3.3      getPass_0.2-2      ps_1.7.1           assertthat_0.2.1  
 [9] rprojroot_2.0.3    digest_0.6.29      utf8_1.2.2         truncnorm_1.0-8   
[13] R6_2.5.1           rootSolve_1.8.2.3  evaluate_0.17      highr_0.9         
[17] httr_1.4.4         ggplot2_3.3.6      pillar_1.8.1       rlang_1.0.6       
[21] rstudioapi_0.14    ebnm_1.0-9         irlba_2.3.5.1      nloptr_2.0.3      
[25] whisker_0.4        callr_3.7.2        jquerylib_0.1.4    Matrix_1.5-1      
[29] rmarkdown_2.17     splines_4.2.1      stringr_1.4.1      munsell_0.5.0     
[33] mixsqp_0.3-48      compiler_4.2.1     httpuv_1.6.6       xfun_0.33         
[37] pkgconfig_2.0.3    SQUAREM_2021.1     htmltools_0.5.3    tidyselect_1.2.0  
[41] tibble_3.1.8       matrixStats_0.62.0 fansi_1.0.3        dplyr_1.0.10      
[45] later_1.3.0        grid_4.2.1         jsonlite_1.8.2     gtable_0.3.1      
[49] lifecycle_1.0.3    DBI_1.1.3          git2r_0.30.1       magrittr_2.0.3    
[53] scales_1.2.1       ebpm_0.0.1.3       cli_3.4.1          stringi_1.7.8     
[57] cachem_1.0.6       fs_1.5.2           promises_1.2.0.1   bslib_0.4.0       
[61] generics_0.1.3     vctrs_0.4.2        trust_0.1-8        tools_4.2.1       
[65] glue_1.6.2         parallel_4.2.1     processx_3.7.0     fastmap_1.1.0     
[69] yaml_2.3.5         colorspace_2.0-3   deconvolveR_1.2-1  knitr_1.40        
[73] sass_0.4.2