Introduction

Variance stablizing transformation.

\(E(X)=\mu\) and \(Var(X)=g(\mu)\), want to find \(f(\cdot)\) s.t \(Var(f(X))\) has constant variance. Consider the Taylor series expansion of \(f(X)\) around \(\mu\): \(f(X)\approx f(\mu)+(Y-\mu)f'(\mu)\) so we have \([f(X)-f(\mu)]^2\approx (X-\mu)^2(f'(\mu))^2 \Rightarrow Var(f(X))\approx Var(X)(f'(\mu))^2\).

For poisson distribution, \((f'(\mu))^2\propto \mu^{-1}\) so if we take \(Y=\sqrt{X}\) then \(Var(Y)\approx \frac{1}{4}\). This was original proposed by Bartlett in 1936.

For Binomial data, \((f'(\mu))^2\propto 1/(np(1-p))\) so if we take \(Y=sin^{-1}(\sqrt{X/n})\) then \(Var(Y)\approx \frac{1}{2}\).

Anscombe(1948) shows that for \(Y=\sqrt{X+c}\), \(Var(Y)\approx \frac{1}{4}[1+\frac{3-8c}{8\mu}+\frac{32c^2-52c+17}{2\mu^2}]]\). If take \(c=3/8\) and for large \(\mu\), \(Var(Y)\approx 1/4\). Also clearly, \(\lim_{\mu\to 0}Var(\sqrt{X+c})=0\).

mu=c(seq(0,1,length.out = 50),seq(1,10,length.out = 50))
ans=c()
sr=c()
set.seed(12345)
for (i  in 1:100) {
  x=rpois(1e6,mu[i])
  ans[i]=var(sqrt(x+3/8))
  sr[i]=var(sqrt(x))
}
plot(mu,ans,type='l',ylim=c(0,0.5),ylab='')
lines(mu,sr,col=4)
abline(a=0.25,b=0,lty=2)
legend('bottomright',c('anscombe','square root'),lty=c(1,1),col=c(1,4))

Compare log and anscombe transformation

Poisson variance stablizing trasformations: square root and Anscombe transformation.

For vst, if we observe \(x=0\), then I use \(var(\sqrt{X+3/8})=0\) instead of \(1/4\).

vst_smooth=function(x,method,ep=1e-5){
  n=length(x)
  if(method=='sr'){
    x.t=sqrt(x)
    x.var=rep(1/4,n)
    x.var[x==0]=0
    mu.hat=(smashr::smash.gaus(x.t,sigma=sqrt(x.var)))^2
    
  }
  if(method=='anscombe'){
    x.t=sqrt(x+3/8)
    x.var=rep(1/4,n)
    x.var[x==0]=0
    mu.hat=(smashr::smash.gaus(x.t,sigma=sqrt(x.var)))^2-3/8
  }
  if(method=='log'){
    x.t=x
    x.t[x==0]=ep
    x.var=1/x.t
    x.t=log(x.t)
    mu.hat=exp(smashr::smash.gaus(x.t,sigma=sqrt(x.var)))
  }
  return(mu.hat)
}

simu_study=function(m,sig=0,nsimu=5,seed=12345){
  set.seed(seed)
  sr=c()
  an=c()
  ashp=c()
  for (i in 1:nsimu) {
    lambda=exp(log(m)+rnorm(n,0,sig))
    x=rpois(n,lambda)
    mu.sr=vst_smooth(x,'sr')
    mu.an=vst_smooth(x,'anscombe')
    mu.ash=smash_gen_lite(x)
    sr=rbind(sr,mu.sr)
    an=rbind(an,mu.an)
    ashp=rbind(ashp,mu.ash)
  }
  return(list(sr=sr,an=an,ashp=ashp))
}

When there are a number of \(0s\) in the observation:

library(ashr)
library(smashrgen)

Loading required package: smashr

Loading required package: caTools

Loading required package: MASS

Loading required package: wavethresh

WaveThresh: R wavelet software, release 4.6.8, installed

Copyright Guy Nason and others 1993-2016

Note: nlevels has been renamed to nlevelsWT

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

m=m*2+0.1
range(m)

[1] 0.100000 6.076316

result=simu_study(m)


mses=lapply(result, function(x){apply(x, 1, function(y){mean((y-m)^2)})})

boxplot(mses,names = c('square root','anscombe','smashgen'))

plot(m,type='l',main='nugget=0')
lines(result$sr[1,],col=2)
lines(result$an[1,],col=3)
lines(result$ashp[1,],col=4)
legend('topright',c('mean','square root','anscombe','smashgen'),lty=c(1,1,1,1),col=c(1,2,3,4))

########
result=simu_study(m,sig=0.5)


mses=lapply(result, function(x){apply(x, 1, function(y){mean((y-m)^2)})})

boxplot(mses,names = c('square root','anscombe','smashgen'))

plot(m,type='l',main='nugget=0.5')
#lines(result$sr[1,],col=2)
lines(result$an[1,],col=3)
lines(result$ashp[1,],col=4)
legend('topright',c('mean','anscombe','smashgen'),lty=c(1,1,1),col=c(1,3,4))

#legend('topright',c('mean','square root','anscombe','smashgen'),lty=c(1,1,1,1),col=c(1,2,3,4))

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

Increase range of mean function:

m=m*20+30
range(m)

[1]  32.0000 151.5263

result=simu_study(m)

mses=lapply(result, function(x){apply(x, 1, function(y){mean((y-m)^2)})})

boxplot(mses,names = c('square root','anscombe','smashgen'))

plot(m,type='l',main='nugget=0')
lines(result$sr[1,],col=2)
lines(result$an[1,],col=3)
lines(result$ashp[1,],col=4)
legend('topright',c('mean','square root','anscombe','smashgen'),lty=c(1,1,1,1),col=c(1,2,3,4))

result=simu_study(m,sig=0.5)

mses=lapply(result, function(x){apply(x, 1, function(y){mean((y-m)^2)})})

boxplot(mses,names = c('square root','anscombe','smashgen'))

plot(m,type='l',main='nugget=0.5')
lines(result$sr[1,],col=2)
lines(result$an[1,],col=3)
lines(result$ashp[1,],col=4)
legend('topright',c('mean','square root','anscombe','smashgen'),lty=c(1,1,1,1),col=c(1,2,3,4))

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