Last updated: 2019-01-06

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    Rmd 9b7cd1f Dongyue Xie 2019-01-06 wflow_publish(c(“analysis/index.Rmd”, “analysis/binomial2.Rmd”))

Introduction

Three methods to smooth binomial sequence: 1. likelihood expansion 2. vst 3. Poisson approx

# a function to estimate nugget effect: sigma^2

normaleqn=function(nug,y,mu,st){
  return(sum((y-mu)^2/(nug+st^2)^2)-sum(1/(nug+st^2)))
}

#a func output either nugget (sigma) or est mean 
NuggetEst=function(y,st,mean.out=F){
  #initialize nugget effect sigma^2
  n=length(y)
  x.m=c(y[n],y,y[1])
  st.m=c(st[n],st,st[1])
  nug.init=mean(((x.m[2:n]-x.m[3:(n+1)])^2+(x.m[2:n]-x.m[1:(n-1)])^2-2*st.m[2:n]^2-st.m[1:(n-1)]^2-st.m[3:(n+1)]^2)/4)
  nug.init=max(c(0,nug.init))
  #given st and nug to estimate mean
  mean.est=smashr::smash.gaus(y,sigma=sqrt(st^2+nug.init))
  #given mean estimate nugget effect
  nug.est=uniroot(normaleqn,c(-1e6,1e6),y=y,mu=mean.est,st=st)$root
  
  #if wanna mean estimation output, then estiamte mean again
  if(mean.out){return(smash.gaus(y,sigma=sqrt(st^2+nug.est)))}else{return(sqrt(nug.est))}
}

likelihood expansion

define \(\hat p_t=x_t/n_t\) if \(x_t\neq 0\) and \(x_t\neq n_t\); otherwise, \(\hat p_t\)=ash posterior mean using binomial likelihood(identity link)

pseudo-data: \(y_t=logit(\hat p_t)\); pseudo-data known variance \(s_t^2=\frac{1}{n_t\hat p_t(1-\hat p_t)}\); pseudo-data nugget effect: \(\sigma\), known/unknown.

library(ashr)
library(smashr)
logit=function(x){log(x/(1-x))}
sigmoid=function(x){exp(x)/(1+exp(x))}

binomial.smoothing.lik=function(x,nt,nugget,fil.num=1,family='DaubExPhase'){
  n=length(x)
  p.ash=ash(rep(0,n),1,lik=lik_binom(x,nt))$result$PosteriorMean
  p.hat=x/nt
  #subsititute 0/full obs by ash posterior mean
  p.hat[x==0]=p.ash[x==0]
  p.hat[x==nt]=p.ash[x==nt]
  y=logit(p.hat)
  #known sd
  st=sqrt(1/(nt*p.hat*(1-p.hat)))
  if(missing(nugget)){
    mu=NuggetEst(y,st,T)
  }else{
    mu=smash.gaus(y,sigma=sqrt(st^2+nugget^2),filter.number = fil.num,family = family)
  }
  
  return(sigmoid(mu))
}

Poisson approxiamtion

pseudo-data: \(\log(x_t)-\log(n_t)\). If \(x_t=0\), \(x_t\)=ash posterior mean. pseudo-data known var: \(s_t^2=1/x_t\)

binomial.smoothing.poi=function(x,nt,nugget,fil.num=1,family='DaubExPhase'){
  n=length(x)
  x.ash=ash(rep(0,n),1,lik=lik_pois(x))$result$PosteriorMean
  x[x==0]=x.ash[x==0]
  p.hat=x/nt
  y=log(p.hat)
  #known sd
  st=sqrt(1/x)
  if(missing(nugget)){
    mu=NuggetEst(y,st,T)
  }else{
    mu=smash.gaus(y,sigma=sqrt(st^2+nugget^2),filter.number = fil.num,family = family)
  }
  return(exp(mu))
}

# n=512
# p=sigmoid(c(rep(-2,128), rep(0, 128), rep(2, 128), rep(-2, 128)))
# set.seed(1234)
# nt=rpois(512,30)
# nugget=0.5
# ng=rnorm(n,0,nugget)
# q=sigmoid(logit(p)+ng)
# x=rbinom(n,nt,q)
# plot(x/nt,col='grey80')
# lines(p,col='grey80')
# lines(binomial.smoothing.poi(x,nt,nugget))

Variance stablizing transformation

VST on binomial data:

\(x\sim Binomial(n,p)\), let \(y=2\sqrt{n}\arcsin(\sqrt{\frac{x}{n}})\) then \(var(y)\approx1\).

pseudo-data: \(y_t=2\sqrt{n_t}\arcsin(\sqrt{\frac{x_t}{n_t}})\), pseudo data known variance: 1, pseudo data nugget: \(\sigma^2\), known/unkown.

binomial.smoothing.vst=function(x,nt,nugget,fil.num=1,family='DaubExPhase'){
  n=length(x)
  y=asin(sqrt((x)/(nt)))
  #known sd
  st=sqrt(1/(4*(nt)))
  if(missing(nugget)){
    mu=NuggetEst(y,st,T)
  }else{
    mu=smash.gaus(y,sigma=sqrt(st^2+nugget^2),filter.number = fil.num,family = family)
  }
  #inverse anscombe
  mu.inv=sin(mu)^2
  return(mu.inv)
}

# n=512
# p=sigmoid(c(rep(-2,128), rep(0, 128), rep(2, 128), rep(-2, 128)))
# set.seed(1234)
# nt=rpois(512,30)
# nugget=0.5
# ng=rnorm(n,0,nugget)
# mu=2*sqrt(nt)*asin(sqrt((nt*p+p/2-1/4)/(nt)))
# q=(sin((mu+ng)/sqrt(4*(nt)))^2*(nt)-1/4)/(nt)
# x=rbinom(n,nt,q)
# plot(x/nt,col='grey80')
# lines(p,col='grey80')
# lines(binomial.smoothing.vst(x,nt,nugget))

Experiment

nugget is chosen to be 0.5. It’s estiamted using mle.

ntri small = 3

set.seed(12345)
n=512
p=sigmoid(c(rep(-2,128), rep(0, 128), rep(2, 128), rep(-2, 128)))
nt=rep(3,512)
nugget=0.5
ng=rnorm(n,0,nugget)
q=sigmoid(logit(p)+ng)
x=rbinom(n,nt,q)
plot(x/nt,col='grey80',ylab = 'p')
lines(p,col='grey80')
lines(binomial.smoothing.lik(x,nt),col=1)
lines(binomial.smoothing.poi(x,nt),col=2)
lines(binomial.smoothing.vst(x,nt),col=3)
legend('topleft',c('mean','lik_binom','poi_approx','vst'),col=c('grey80',1,2,3),lty=c(1,1,1,1))

ntri small = 5

set.seed(12345)
n=512
p=sigmoid(c(rep(-2,128), rep(0, 128), rep(2, 128), rep(-2, 128)))
nt=rep(5,512)
nugget=0.5
ng=rnorm(n,0,nugget)
q=sigmoid(logit(p)+ng)
x=rbinom(n,nt,q)
plot(x/nt,col='grey80',ylab = 'p')
lines(p,col='grey80')
lines(binomial.smoothing.lik(x,nt),col=1)
lines(binomial.smoothing.poi(x,nt),col=2)
lines(binomial.smoothing.vst(x,nt),col=3)
legend('topleft',c('mean','lik_binom','poi_approx','vst'),col=c('grey80',1,2,3),lty=c(1,1,1,1))

ntri small = 10

set.seed(12345)
n=512
p=sigmoid(c(rep(-2,128), rep(0, 128), rep(2, 128), rep(-2, 128)))
nt=rep(10,512)
nugget=0.5
ng=rnorm(n,0,nugget)
q=sigmoid(logit(p)+ng)
x=rbinom(n,nt,q)
plot(x/nt,col='grey80',ylab = 'p')
lines(p,col='grey80')
lines(binomial.smoothing.lik(x,nt),col=1)
lines(binomial.smoothing.poi(x,nt),col=2)
lines(binomial.smoothing.vst(x,nt),col=3)
legend('topleft',c('mean','lik_binom','poi_approx','vst'),col=c('grey80',1,2,3),lty=c(1,1,1,1))

ntri small = 15

set.seed(12345)
n=512
p=sigmoid(c(rep(-2,128), rep(0, 128), rep(2, 128), rep(-2, 128)))
nt=rep(15,512)
nugget=0.5
ng=rnorm(n,0,nugget)
q=sigmoid(logit(p)+ng)
x=rbinom(n,nt,q)
plot(x/nt,col='grey80',ylab = 'p')
lines(p,col='grey80')
lines(binomial.smoothing.lik(x,nt),col=1)
lines(binomial.smoothing.poi(x,nt),col=2)
lines(binomial.smoothing.vst(x,nt),col=3)
legend('topleft',c('mean','lik_binom','poi_approx','vst'),col=c('grey80',1,2,3),lty=c(1,1,1,1))

ntri = 30

set.seed(12345)
n=512
p=sigmoid(c(rep(-2,128), rep(0, 128), rep(2, 128), rep(-2, 128)))
nt=rep(30,512)
nugget=0.5
ng=rnorm(n,0,nugget)
q=sigmoid(logit(p)+ng)
x=rbinom(n,nt,q)
plot(x/nt,col='grey80',ylab = 'p')
lines(p,col='grey80')
lines(binomial.smoothing.lik(x,nt),col=1)
lines(binomial.smoothing.poi(x,nt),col=2)
lines(binomial.smoothing.vst(x,nt),col=3)
legend('topleft',c('mean','lik_binom','poi_approx','vst'),col=c('grey80',1,2,3),lty=c(1,1,1,1))

ntri = 100

set.seed(12345)
n=512
p=sigmoid(c(rep(-2,128), rep(0, 128), rep(2, 128), rep(-2, 128)))
nt=rep(100,512)
nugget=0.5
ng=rnorm(n,0,nugget)
q=sigmoid(logit(p)+ng)
x=rbinom(n,nt,q)
plot(x/nt,col='grey80',ylab = 'p')
lines(p,col='grey80')
lines(binomial.smoothing.lik(x,nt),col=1)
lines(binomial.smoothing.poi(x,nt),col=2)
lines(binomial.smoothing.vst(x,nt),col=3)
legend('topleft',c('mean','lik_binom','poi_approx','vst'),col=c('grey80',1,2,3),lty=c(1,1,1,1))

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] smashr_1.2-0 ashr_2.2-7  

loaded via a namespace (and not attached):
 [1] Rcpp_0.12.18      knitr_1.20        whisker_0.3-2    
 [4] magrittr_1.5      workflowr_1.1.1   REBayes_1.3      
 [7] MASS_7.3-50       pscl_1.5.2        doParallel_1.0.14
[10] SQUAREM_2017.10-1 lattice_0.20-35   foreach_1.4.4    
[13] stringr_1.3.1     caTools_1.17.1.1  tools_3.5.1      
[16] parallel_3.5.1    grid_3.5.1        data.table_1.11.6
[19] R.oo_1.22.0       git2r_0.23.0      htmltools_0.3.6  
[22] iterators_1.0.10  assertthat_0.2.0  yaml_2.2.0       
[25] rprojroot_1.3-2   digest_0.6.17     Matrix_1.2-14    
[28] bitops_1.0-6      codetools_0.2-15  R.utils_2.7.0    
[31] evaluate_0.11     rmarkdown_1.10    wavethresh_4.6.8 
[34] stringi_1.2.4     compiler_3.5.1    Rmosek_8.0.69    
[37] backports_1.1.2   R.methodsS3_1.7.1 truncnorm_1.0-8

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