Variance stablizing transformation

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Last updated: 2020-09-07

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Knit directory: smash-gen/

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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=seq(0,10,length.out = 500)
ans=c()
sr=c()
set.seed(12345)
for (i  in 1:500) {
  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))

Past versions of unnamed-chunk-1-1.png

Version	Author	Date
e24d0a7	Dongyue Xie	2018-10-16

Apprently, if the mean is small say 1 or 2 then the transformed variable has variance smaller than 0.25. One way to adjust for small counts is to take all possibilities into account. For example, if we observe x=0, then we can assign the following variance to its transformation, let \(y\sim Poisson(\lambda)\): \(\frac{\int p(x=0;\lambda)var(\sqrt{y+c};\lambda)d\lambda}{\int p(x=0;\lambda)d\lambda}\). In practice we can generate a grid of discrete values and evaluate the above quantity.

for(i in c(0,1,2)){
 p0 = dpois(i,mu)
 p0 = p0/sum(p0)
 p0 = sum(p0*ans)
 print(p0)
}

[1] 0.1382905
[1] 0.2018222
[1] 0.2293388

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=30,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,sigma = sig)
    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: (mean function (0.1,6))

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 = 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)})})

par(mfrow=c(2,2))

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

}

Past versions of unnamed-chunk-5-1.png

Version	Author	Date
7386626	Dongyue Xie	2018-10-23
d45f4a0	Dongyue Xie	2018-10-16
e24d0a7	Dongyue Xie	2018-10-16

boxplot(mses,names = c('square root','anscombe','smashgen'),main='nugget=0',ylab='MSE')

Past versions of unnamed-chunk-6-1.png

Version	Author	Date
7386626	Dongyue Xie	2018-10-23

########
result=simu_study(m,sig=1)


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

#unlist(lapply(mses, mean))

par(mfrow=c(2,2))
for (j in 1:4) {
  plot(m,type='l',main='nugget=1')
#lines(result$sr[1,],col=2)
lines(result$an[j,],col=3)
lines(result$ashp[j,],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))

}

Past versions of unnamed-chunk-7-1.png

Version	Author	Date
7386626	Dongyue Xie	2018-10-23

boxplot(mses,names = c('square root','anscombe','smashgen'),main='nugget=1',ylab='MSE')

Past versions of unnamed-chunk-8-1.png

Version	Author	Date
7386626	Dongyue Xie	2018-10-23

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

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)})})

#unlist(lapply(mses, mean))

par(mfrow=c(2,2))

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

}

Past versions of unnamed-chunk-9-1.png

Version	Author	Date
7386626	Dongyue Xie	2018-10-23

boxplot(mses,names = c('square root','anscombe','smashgen'),main='nugget=0',ylab='MSE')

Past versions of unnamed-chunk-10-1.png

Version	Author	Date
7386626	Dongyue Xie	2018-10-23

Step function

mean: (1,7)

m=c(rep(1,n/4),rep(5,n/4),rep(7,n/4),rep(1,n/4))

result=simu_study(m)

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

#unlist(lapply(mses, mean))

par(mfrow=c(2,2))

for (j in 1:4) {
  plot(m,type='l',main='nugget=1',ylim=c(-1,8))
#lines(result$sr[1,],col=2)
lines(result$an[j,],col=3)
lines(result$ashp[j,],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))

}

Past versions of unnamed-chunk-11-1.png

Version	Author	Date
7386626	Dongyue Xie	2018-10-23

boxplot(mses,names = c('square root','anscombe','smashgen'),main='nugget=0',ylab='MSE')

Past versions of unnamed-chunk-12-1.png

Version	Author	Date
7386626	Dongyue Xie	2018-10-23

Heavi Sine function

mean: (1,24)

m=wavethresh::DJ.EX(n)
m=m$heavi+15
range(m)

[1]  0.8749178 24.4167215

result=simu_study(m)

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

#unlist(lapply(mses, mean))

par(mfrow=c(2,2))

for (j in 1:4) {
  plot(m,type='l',main='nugget=1')
#lines(result$sr[1,],col=2)
lines(result$an[j,],col=3)
lines(result$ashp[j,],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))

}

Past versions of unnamed-chunk-13-1.png

Version	Author	Date
7386626	Dongyue Xie	2018-10-23

boxplot(mses,names = c('square root','anscombe','smashgen'),main='nugget=0',ylab='MSE')

Session information

sessionInfo()

R version 3.6.1 (2019-07-05)
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/3.6/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.6/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_0.1.2  wavethresh_4.6.8 MASS_7.3-51.4    caTools_1.17.1.2
[5] smashr_1.2-7     ashr_2.2-38     

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.2        compiler_3.6.1    later_1.0.0      
 [4] git2r_0.26.1      workflowr_1.5.0   bitops_1.0-6     
 [7] iterators_1.0.12  tools_3.6.1       digest_0.6.21    
[10] evaluate_0.14     lattice_0.20-38   rlang_0.4.5      
[13] Matrix_1.2-17     foreach_1.4.7     yaml_2.2.0       
[16] parallel_3.6.1    xfun_0.10         stringr_1.4.0    
[19] knitr_1.25        fs_1.3.1          rprojroot_1.3-2  
[22] grid_3.6.1        data.table_1.12.6 glue_1.3.1       
[25] R6_2.4.0          rmarkdown_1.16    mixsqp_0.1-97    
[28] magrittr_1.5      whisker_0.4       backports_1.1.5  
[31] promises_1.1.0    codetools_0.2-16  htmltools_0.4.0  
[34] httpuv_1.5.2      stringi_1.4.3     doParallel_1.0.15
[37] pscl_1.5.2        truncnorm_1.0-8   SQUAREM_2017.10-1