Last updated: 2018-10-07

workflowr checks: (Click a bullet for more information)

✔ R Markdown file: up-to-date

Great! Since the R Markdown file has been committed to the Git repository, you know the exact version of the code that produced these results.
✔ Environment: empty

Great job! The global environment was empty. Objects defined in the global environment can affect the analysis in your R Markdown file in unknown ways. For reproduciblity it’s best to always run the code in an empty environment.
✔ Seed: set.seed(20180501)

The command set.seed(20180501) was run prior to running the code in the R Markdown file. Setting a seed ensures that any results that rely on randomness, e.g. subsampling or permutations, are reproducible.
✔ Session information: recorded

Great job! Recording the operating system, R version, and package versions is critical for reproducibility.
✔ Repository version: cb9be5d
Great! You are using Git for version control. Tracking code development and connecting the code version to the results is critical for reproducibility. The version displayed above was the version of the Git repository at the time these results were generated.

Note that you need to be careful to ensure that all relevant files for the analysis have been committed to Git prior to generating the results (you can use wflow_publish or wflow_git_commit). workflowr only checks the R Markdown file, but you know if there are other scripts or data files that it depends on. Below is the status of the Git repository when the results were generated:
```
Ignored files:
    Ignored:    .DS_Store
    Ignored:    .Rhistory
    Ignored:    .Rproj.user/
    Ignored:    data/.DS_Store

Untracked files:
    Untracked:  analysis/chipexoeg.Rmd
    Untracked:  data/chipexo_examples/
    Untracked:  data/chipseq_examples/

Unstaged changes:
    Modified:   analysis/sigma.Rmd
```
Note that any generated files, e.g. HTML, png, CSS, etc., are not included in this status report because it is ok for generated content to have uncommitted changes.

Expand here to see past versions:

File	Version	Author	Date	Message
Rmd	cb9be5d	Dongyue Xie	2018-10-07	add

The problem

Model: \(X\sim Poi(\mu)\) and define \(y=\log m+\frac{x-m}{m}\)

Previously, we used \(m=\)ash posterior mean of x. The problem is that shrinkage effect is too strong that for large observations \(x\), the approximated normal data points are too large. Hence, after taking exponential of estimated normal mean, the estimation ‘blow up’.

Now we try to do Taylor series expansion around \(\log x\)(MLE) for non-zero \(x\) and around ash posterior mean around zero \(x\). In other words, now \(m=\)ash posterior mean of zero \(x\)s and \(m=x\) for non-zero \(x\)s.

Spike mean function

library(ashr)
library(smashrgen)
spike.f = function(x) (0.75 * exp(-500 * (x - 0.23)^2) + 1.5 * exp(-2000 * (x - 0.33)^2) + 3 * exp(-8000 * (x - 0.47)^2) + 2.25 * exp(-16000 * 
    (x - 0.69)^2) + 0.5 * exp(-32000 * (x - 0.83)^2))
n = 256
t = 1:n/n
m = spike.f(t)

m=m*2+0.1
range(m)

[1] 0.100000 6.025467

sig=0

set.seed(12345)
lambda=exp(log(m)+rnorm(n,0,sig))
x=rpois(n,lambda)

x.ash=ash(rep(0,n),1,lik=lik_pois(x))$result$PosteriorMean
m.hat=x.ash
m.hat[which(x!=0)]=(x[which(x!=0)])
y=log(m.hat)+(x-m.hat)/m.hat
m.tilde=exp(smash.gaus(y,sigma = sqrt(sig^2+1/m.hat)))
m.tilde2=exp(smash.gaus(y))

par(mfrow=c(2,2))

plot(x,col='grey80',ylab='',xlab='',main='n=256,nugget=0')
lines(m,col='grey60')
lines(m.tilde,col=2)
lines(m.tilde2,col=4)

legend('topleft',c('data','true mean','smashgen-known var','smashgen-unknown var'),lty=c(0,1,1,1),pch=c(1,NA,NA,NA),col=c('grey80','grey60',2,4))
#################

sig=0.1

set.seed(12345)
lambda=exp(log(m)+rnorm(n,0,sig))
x=rpois(n,lambda)


x.ash=ash(rep(0,n),1,lik=lik_pois(x))$result$PosteriorMean
m.hat=x.ash
m.hat[which(x!=0)]=(x[which(x!=0)])
y=log(m.hat)+(x-m.hat)/m.hat
m.tilde=exp(smash.gaus(y,sigma = sqrt(sig^2+1/m.hat)))
m.tilde2=exp(smash.gaus(y))

plot(x,col='grey80',ylab='',xlab='',main='n=256,nugget=0.1')
lines(m,col='grey60')
lines(m.tilde,col=2)
lines(m.tilde2,col=4)

legend('topleft',c('data','true mean','smashgen-known var','smashgen-unknown var'),lty=c(0,1,1,1),pch=c(1,NA,NA,NA),col=c('grey80','grey60',2,4))

#################

sig=1

set.seed(12345)
lambda=exp(log(m)+rnorm(n,0,sig))
x=rpois(n,lambda)

x.ash=ash(rep(0,n),1,lik=lik_pois(x))$result$PosteriorMean
m.hat=x.ash
m.hat[which(x!=0)]=(x[which(x!=0)])
y=log(m.hat)+(x-m.hat)/m.hat
m.tilde=exp(smash.gaus(y,sigma = sqrt(sig^2+1/m.hat)))
m.tilde2=exp(smash.gaus(y))

plot(x,col='grey80',ylab='',xlab='',main='n=256,nugget=1')
lines(m,col='grey60')
lines(m.tilde,col=2)
lines(m.tilde2,col=4)

legend('topleft',c('data','true mean','smashgen-known var','smashgen-unknown var'),lty=c(0,1,1,1),pch=c(1,NA,NA,NA),col=c('grey80','grey60',2,4))

plot(x,col='grey80',ylab='',xlab='',main='Previous verison using ash posterior mean, nugget=1')
lines(m,col='grey60')
lines(smash_gen_lite(x))
legend('topleft',c('data','true mean','fit'),lty=c(0,1,1),pch=c(1,NA,NA),col=c('grey80','grey60',1))

n = 512
t = 1:n/n
m = spike.f(t)

m=m*2+0.1
range(m)

[1] 0.100000 6.076316

sig=0

set.seed(12345)
lambda=exp(log(m)+rnorm(n,0,sig))
x=rpois(n,lambda)



x.ash=ash(rep(0,n),1,lik=lik_pois(x))$result$PosteriorMean
m.hat=x.ash
m.hat[which(x!=0)]=(x[which(x!=0)])
y=log(m.hat)+(x-m.hat)/m.hat
m.tilde=exp(smash.gaus(y,sigma = sqrt(sig^2+1/m.hat)))
m.tilde2=exp(smash.gaus(y))

par(mfrow=c(2,2))

plot(x,col='grey80',ylab='',xlab='',main='n=512,nugget=0')
lines(m,col='grey60')
lines(m.tilde,col=2)
lines(m.tilde2,col=4)

legend('topleft',c('data','true mean','smashgen-known var','smashgen-unknown var'),lty=c(0,1,1,1),pch=c(1,NA,NA,NA),col=c('grey80','grey60',2,4))
#################

sig=0.1

set.seed(12345)
lambda=exp(log(m)+rnorm(n,0,sig))
x=rpois(n,lambda)



x.ash=ash(rep(0,n),1,lik=lik_pois(x))$result$PosteriorMean
m.hat=x.ash
m.hat[which(x!=0)]=(x[which(x!=0)])
y=log(m.hat)+(x-m.hat)/m.hat
m.tilde=exp(smash.gaus(y,sigma = sqrt(sig^2+1/m.hat)))
m.tilde2=exp(smash.gaus(y))

plot(x,col='grey80',ylab='',xlab='',main='n=512,nugget=0.1')
lines(m,col='grey60')
lines(m.tilde,col=2)
lines(m.tilde2,col=4)

legend('topleft',c('data','true mean','smashgen-known var','smashgen-unknown var'),lty=c(0,1,1,1),pch=c(1,NA,NA,NA),col=c('grey80','grey60',2,4))

#################

sig=1

set.seed(12345)
lambda=exp(log(m)+rnorm(n,0,sig))
x=rpois(n,lambda)



x.ash=ash(rep(0,n),1,lik=lik_pois(x))$result$PosteriorMean
m.hat=x.ash
m.hat[which(x!=0)]=(x[which(x!=0)])
y=log(m.hat)+(x-m.hat)/m.hat
m.tilde=exp(smash.gaus(y,sigma = sqrt(sig^2+1/m.hat)))
m.tilde2=exp(smash.gaus(y))

plot(x,col='grey80',ylab='',xlab='',main='n=512,nugget=1')
lines(m,col='grey60')
lines(m.tilde,col=2)
lines(m.tilde2,col=4)

legend('topleft',c('data','true mean','smashgen-known var','smashgen-unknown var'),lty=c(0,1,1,1),pch=c(1,NA,NA,NA),col=c('grey80','grey60',2,4))

plot(x,col='grey80',ylab='',xlab='',main='Previous verison using ash posterior mean, nugget=1')
lines(m,col='grey60')
lines(smash_gen_lite(x))
legend('topleft',c('data','true mean','fit'),lty=c(0,1,1),pch=c(1,NA,NA),col=c('grey80','grey60',1))

Session information

sessionInfo()

R version 3.5.1 (2018-07-02)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.6

Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRlapack.dylib

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] smashrgen_0.1.0  wavethresh_4.6.8 MASS_7.3-50      caTools_1.17.1.1
[5] smashr_1.2-0     ashr_2.2-7      

loaded via a namespace (and not attached):
 [1] Rcpp_0.12.18      compiler_3.5.1    git2r_0.23.0     
 [4] workflowr_1.1.1   R.methodsS3_1.7.1 R.utils_2.7.0    
 [7] bitops_1.0-6      iterators_1.0.10  tools_3.5.1      
[10] digest_0.6.17     evaluate_0.11     lattice_0.20-35  
[13] Matrix_1.2-14     foreach_1.4.4     yaml_2.2.0       
[16] parallel_3.5.1    stringr_1.3.1     knitr_1.20       
[19] REBayes_1.3       rprojroot_1.3-2   grid_3.5.1       
[22] data.table_1.11.6 rmarkdown_1.10    magrittr_1.5     
[25] whisker_0.3-2     backports_1.1.2   codetools_0.2-15 
[28] htmltools_0.3.6   assertthat_0.2.0  stringi_1.2.4    
[31] Rmosek_8.0.69     doParallel_1.0.14 pscl_1.5.2       
[34] truncnorm_1.0-8   SQUAREM_2017.10-1 R.oo_1.22.0

This reproducible R Markdown analysis was created with workflowr 1.1.1

Poisson spike issues

Dongyue Xie

2018-10-07

The problem

Spike mean function

Session information