Last updated: 2018-05-21

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    File Version Author Date Message
    Rmd 5b61f90 stephens999 2018-05-21 workflowr::wflow_publish(“eb_vs_soft.Rmd”)
    Rmd fcfdba2 stephens999 2018-04-09 add comparison of eb vs soft thresholding

Simulate some data

Here we simulate a pretty sparse situation: 90% sparsity.

set.seed(1)
theta = c(rep(0,900),rnorm(100,0,4))
x = theta + rnorm(1000)
hist(theta,breaks = seq(-10,10,length=20),xlim=c(-10,10))

Now do inference under the EBNM model. Notice how the small values are shrunk hard, and the big values are hardly shrunk at all.

library("ashr")
library("ebnm")
x.pn = ebnm_point_normal(x,1)
plot(x,get_pm(x.pn),main="observation (x) vs posterior mean", ylab="posterior mean")
abline(a=0,b=1,col="red")

What about soft thresholding?

soft_thresh = function(x,lambda){
  ifelse(abs(x)>lambda, sign(x)*(abs(x)-lambda), 0)
}
plot(x,soft_thresh(x,1.5),main="observation (x) vs soft-thresholding estimate (lambda=1.5)",ylab="soft-thresholded estimate")
abline(a=0,b=1,col="red")

Now look at Mean Squared Error. Notice that both do much better than no shrinkage!!

mse = rep(0,100)
l = seq(0,5,length=100)
for(i in 1:100){
  mse[i] = mean((theta-soft_thresh(x,l[i]))^2)
}
plot(l,mse,ylim=c(0,1.2), main="black = no shrinkage; red=Empirical Bayes; points= soft-thresholding", xlab="lambda")
abline(h=mean((theta-get_pm(x.pn))^2),col="red")
abline(h=mean((theta-x)^2),col="blue")

Session information

sessionInfo()
R version 3.3.2 (2016-10-31)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: OS X El Capitan 10.11.6

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

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

other attached packages:
[1] ebnm_0.1-10 ashr_2.2-7 

loaded via a namespace (and not attached):
 [1] Rcpp_0.12.16      knitr_1.20        whisker_0.3-2    
 [4] magrittr_1.5      workflowr_1.0.1   MASS_7.3-49      
 [7] pscl_1.5.2        doParallel_1.0.11 SQUAREM_2017.10-1
[10] lattice_0.20-35   foreach_1.4.4     stringr_1.3.0    
[13] tools_3.3.2       parallel_3.3.2    grid_3.3.2       
[16] R.oo_1.22.0       git2r_0.21.0      htmltools_0.3.6  
[19] iterators_1.0.9   yaml_2.1.18       rprojroot_1.3-2  
[22] digest_0.6.15     Matrix_1.2-14     R.utils_2.6.0    
[25] codetools_0.2-15  evaluate_0.10.1   rmarkdown_1.9    
[28] stringi_1.1.7     backports_1.1.2   R.methodsS3_1.7.1
[31] truncnorm_1.0-7

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