Last updated: 2022-11-21

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Rmd 03048ee DongyueXie 2022-11-21 wflow_publish("analysis/overdispersed_splitting_nb.Rmd")

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)

r = 10

first run

set.seed(12345)
n = 3000
lambda = c(rep(0,n*0.8),rexp(n*0.2,0.1))
r = 10
y = rnbinom(n,r,mu  = lambda)
sigma2 = log(1+1/r)
sigma2
[1] 0.09531018

fix \(\sigma2\)

fit = pois_mean_split2(y,sigma2=sigma2,est_sigma2 = FALSE,mu_pm_init = log(y+1))
fit$elbo
[1] -7394.922
plot(lambda,col='grey80')
lines(fit$posterior$mean)

fit$fitted_g
$sigma2
[1] 0.09531018

$g_b
$pi
[1] 0.8402475 0.1597525

$mean
[1] -2.315816 -2.315816

$scale
[1] 0.000000 4.057725

attr(,"class")
[1] "laplacemix"
attr(,"row.names")
[1] 1 2

second run

lambda = c(rep(0,n*0.8),rexp(n*0.2,0.1))
y = rnbinom(n,r,mu  = lambda)

fix \(\sigma2\)

fit = pois_mean_split2(y,sigma2=sigma2,est_sigma2 = FALSE,mu_pm_init = log(y+1))
fit$elbo
[1] -7463.391
plot(lambda,col='grey80')
lines(fit$posterior$mean)

fit$fitted_g
$sigma2
[1] 0.09531018

$g_b
$pi
[1] 0.8344718 0.1655282

$mean
[1] -2.403966 -2.403966

$scale
[1] 0.000000 4.176129

attr(,"class")
[1] "laplacemix"
attr(,"row.names")
[1] 1 2

third run

lambda = c(rep(0,n*0.8),rexp(n*0.2,0.1))
y = rnbinom(n,r,mu  = lambda)

fix \(\sigma2\)

fit = pois_mean_split2(y,sigma2=sigma2,est_sigma2 = FALSE,mu_pm_init = log(y+1))
fit$elbo
[1] -7394.452
plot(lambda,col='grey80')
lines(fit$posterior$mean)

fit$fitted_g
$sigma2
[1] 0.09531018

$g_b
$pi
[1] 0.8412745 0.1587255

$mean
[1] -2.332336 -2.332336

$scale
[1] 0.000000 4.137975

attr(,"class")
[1] "laplacemix"
attr(,"row.names")
[1] 1 2

r = 5

first run

set.seed(12345)
n = 3000
lambda = c(rep(0,n*0.8),rexp(n*0.2,0.1))
r = 5
y = rnbinom(n,r,mu  = lambda)
sigma2 = log(1+1/r)
sigma2
[1] 0.1823216

fix \(\sigma2\)

fit = pois_mean_split2(y,sigma2=sigma2,est_sigma2 = FALSE,mu_pm_init = log(y+1))
fit$elbo
[1] -7338.985
plot(lambda,col='grey80')
lines(fit$posterior$mean)

fit$fitted_g
$sigma2
[1] 0.1823216

$g_b
$pi
[1] 0.8542314 0.1457686

$mean
[1] -1.950497 -1.950497

$scale
[1] 0.000000 3.709187

attr(,"class")
[1] "laplacemix"
attr(,"row.names")
[1] 1 2

second run

lambda = c(rep(0,n*0.8),rexp(n*0.2,0.1))
y = rnbinom(n,r,mu  = lambda)

fix \(\sigma2\)

fit = pois_mean_split2(y,sigma2=sigma2,est_sigma2 = FALSE,mu_pm_init = log(y+1))
fit$elbo
[1] -7372.572
plot(lambda,col='grey80')
lines(fit$posterior$mean)

fit$fitted_g
$sigma2
[1] 0.1823216

$g_b
$pi
[1] 0.8516417 0.1483583

$mean
[1] -1.957995 -1.957995

$scale
[1] 0.000000 3.720176

attr(,"class")
[1] "laplacemix"
attr(,"row.names")
[1] 1 2

third run

lambda = c(rep(0,n*0.8),rexp(n*0.2,0.1))

y = rnbinom(n,r,mu  = lambda)

fix \(\sigma2\)

fit = pois_mean_split2(y,sigma2=sigma2,est_sigma2 = FALSE,mu_pm_init = log(y+1))
fit$elbo
[1] -7357.373
plot(lambda,col='grey80')
lines(fit$posterior$mean)

fit$fitted_g
$sigma2
[1] 0.1823216

$g_b
$pi
[1] 0.8561683 0.1438317

$mean
[1] -1.888776 -1.888776

$scale
[1] 0.000000 3.652401

attr(,"class")
[1] "laplacemix"
attr(,"row.names")
[1] 1 2

r = 50

first run

set.seed(12345)
n = 3000
lambda = c(rep(0,n*0.8),rexp(n*0.2,0.1))
r = 50
y = rnbinom(n,r,mu  = lambda)
sigma2 = log(1+1/r)
sigma2
[1] 0.01980263

fix \(\sigma2\)

fit = pois_mean_split2(y,sigma2=sigma2,est_sigma2 = FALSE,mu_pm_init = log(y+1))
Warning in pois_mean_split2(y, sigma2 = sigma2, est_sigma2 = FALSE, mu_pm_init =
log(y + : An iteration decreases ELBO. This is likely due to numerical issues.
fit$elbo
[1] -11559.26
plot(lambda,col='grey80')
lines(fit$posterior$mean)

fit$fitted_g
$sigma2
[1] 0.01980263

$g_b
$pi
[1] 0.0004275021 0.9995724979

$mean
[1] -1.222233 -1.222233

$scale
[1] 0.0000000 0.6113245

attr(,"class")
[1] "laplacemix"
attr(,"row.names")
[1] 1 2

second run

lambda = c(rep(0,n*0.8),rexp(n*0.2,0.1))

y = rnbinom(n,r,mu  = lambda)

fix \(\sigma2\)

fit = pois_mean_split2(y,sigma2=sigma2,est_sigma2 = FALSE,mu_pm_init = log(y+1))
Warning in pois_mean_split2(y, sigma2 = sigma2, est_sigma2 = FALSE, mu_pm_init =
log(y + : An iteration decreases ELBO. This is likely due to numerical issues.
fit$elbo
[1] -8852.339
plot(lambda,col='grey80')
lines(fit$posterior$mean)

fit$fitted_g
$sigma2
[1] 0.01980263

$g_b
$pi
[1] 0.6725263 0.3274737

$mean
[1] -1.216211 -1.216211

$scale
[1] 0.0000000 0.8433485

attr(,"class")
[1] "laplacemix"
attr(,"row.names")
[1] 1 2

third run

lambda = c(rep(0,n*0.8),rexp(n*0.2,0.1))

y = rnbinom(n,r,mu  = lambda)

fix \(\sigma2\)

fit = pois_mean_split2(y,sigma2=sigma2,est_sigma2 = FALSE,mu_pm_init = log(y+1))
fit$elbo
[1] -7723.969
plot(lambda,col='grey80')
lines(fit$posterior$mean)

fit$fitted_g
$sigma2
[1] 0.01980263

$g_b
$pi
[1] 0.831282 0.168718

$mean
[1] -2.427278 -2.427278

$scale
[1] 0.000000 4.317392

attr(,"class")
[1] "laplacemix"
attr(,"row.names")
[1] 1 2

r = 500

first run

set.seed(12345)
n = 3000
lambda = c(rep(0,n*0.8),rexp(n*0.2,0.1))
r = 500
y = rnbinom(n,r,mu  = lambda)
sigma2 = log(1+1/r)
sigma2
[1] 0.001998003

fix \(\sigma2\)

fit = pois_mean_split2(y,sigma2=sigma2,est_sigma2 = FALSE,mu_pm_init = log(y+1))
Warning in pois_mean_split2(y, sigma2 = sigma2, est_sigma2 = FALSE, mu_pm_init =
log(y + : An iteration decreases ELBO. This is likely due to numerical issues.
fit$elbo
[1] -14715.5
plot(lambda,col='grey80')
lines(fit$posterior$mean)

fit$fitted_g
$sigma2
[1] 0.001998003

$g_b
$pi
[1] 1.234136e-11 1.000000e+00

$mean
[1] -0.694044 -0.694044

$scale
[1] 0.0000000 0.4826454

attr(,"class")
[1] "laplacemix"
attr(,"row.names")
[1] 1 2

second run

lambda = c(rep(0,n*0.8),rexp(n*0.2,0.1))

y = rnbinom(n,r,mu  = lambda)

fix \(\sigma2\)

fit = pois_mean_split2(y,sigma2=sigma2,est_sigma2 = FALSE,mu_pm_init = log(y+1))
Warning in pois_mean_split2(y, sigma2 = sigma2, est_sigma2 = FALSE, mu_pm_init =
log(y + : An iteration decreases ELBO. This is likely due to numerical issues.
fit$elbo
[1] -14951.34
plot(lambda,col='grey80')
lines(fit$posterior$mean)

fit$fitted_g
$sigma2
[1] 0.001998003

$g_b
$pi
[1] 3.637649e-08 1.000000e+00

$mean
[1] -0.3579676 -0.3579676

$scale
[1] 0.0000000 0.4422295

attr(,"class")
[1] "laplacemix"
attr(,"row.names")
[1] 1 2

third run

lambda = c(rep(0,n*0.8),rexp(n*0.2,0.1))

y = rnbinom(n,r,mu  = lambda)

fix \(\sigma2\)

fit = pois_mean_split2(y,sigma2=sigma2,est_sigma2 = FALSE,mu_pm_init = log(y+1))
Warning in pois_mean_split2(y, sigma2 = sigma2, est_sigma2 = FALSE, mu_pm_init =
log(y + : An iteration decreases ELBO. This is likely due to numerical issues.
fit$elbo
[1] -14954.5
plot(lambda,col='grey80')
lines(fit$posterior$mean)

fit$fitted_g
$sigma2
[1] 0.001998003

$g_b
$pi
[1] 7.026071e-08 9.999999e-01

$mean
[1] -0.2497244 -0.2497244

$scale
[1] 0.0000000 0.4171243

attr(,"class")
[1] "laplacemix"
attr(,"row.names")
[1] 1 2

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] vebpm_0.2.6     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_0.3-43      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   ashr_2.2-54        deconvolveR_1.2-1 
[73] knitr_1.40         sass_0.4.2