Last updated: 2018-10-22

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Introduction

Data generation

Assume \(m\) has smooth structure, \(\mu_t=(m_t+\epsilon)^2\) where \(\epsilon\sim N(0,\sigma^2)\), and \(X_t\sim Pois(\mu_t)\). Then \(Y_t=\sqrt{X_t}\approx \sqrt{\mu_t}+N(0,1/4)=m+\epsilon+N(0,1/4)\) and \(\hat\mu_t=\hat m_t^2\).

Recover smooth structure

If we do square-root type variance stablizing transformation of poisson \(X\sim Poi(\mu)\), then \(Y=\sqrt{X}\) and \(E(Y)\approx \sqrt{\mu}\) and \(Var(Y)\approx 1/4\). Assume \(\mu=(m+\epsilon)^2\), then \(E(Y)=m+\epsilon\). So we can have a version of vst to deal with nugget effect. For example, if we observe \(X_t\sim Pois(\mu)\) then form \(y_t=\sqrt{x_t}\). Apply any smoothing method with variance \((\sigma^2+1/4)\) to \(y=(y_1,...,y_t)\) to get \(\hat{m}\) then \(\hat\mu_{smooth}=\hat{m}^2\).

Advancetage: no need to worry about 0; homoscedastic variance which is easier to estimate.

When the nugget effect is unknown, three neighboring points(Gasser et al, 1986) were used to estimate the variance ,\(\sigma^2+1/4\). The formula is \[\frac{2}{3(n-2)}\Sigma_{i=1}^{n-2} (\frac{1}{2}y_i-y_{i+1}+\frac{1}{2}y_{i+2})^2,\] where \(y=\sqrt{x}\). If the estimated variance is smaller than \(1/4\), I make it to be \(1/4\). Or, we can simply treat var as unkown and use smash.gaus(y)

Experiments

library(smashrgen)
vst_gen=function(x,sigma=NULL,c=3/8,nug_est=T){
  n=length(x)
  y=sqrt(x+c)
  x.var=rep(1/4,n)
  x.var[x==0]=0
  if(is.null(sigma)){
    if(nug_est){
      sigma=sqrt(homo_var(sqrt(x),1/4)-1/4)
      m=smashr::smash.gaus(y,sqrt(x.var+sigma^2))
    }else{
      m=smashr::smash.gaus(y)
    }
  }else{
    m=smashr::smash.gaus(y,sqrt(x.var+sigma^2))
  }
  return(m^2-c)
}

homo_var=function(x,minv=0){
  #second order method 
  n=length(x)
  ssq=0
  for (i in 1:(n-2)) {
    ssq=ssq+(0.5*x[i]-x[i+1]+0.5*x[i+2])^2
  }
  var.hat=ssq*2/(3*(n-2))
  return(ifelse(var.hat>=minv,var.hat,minv))
}
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

set.seed(12345)
sigma=0
mm=(sqrt(m)+rnorm(n,0,sigma))^2
x=rpois(n,mm)
mu=vst_gen(x,sigma)

plot(x,col='grey80',main='no nugget')
lines(m)
lines(mu,col=2)

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Version Author Date
a31c75e Dongyue Xie 2018-10-18 

set.seed(12345)
sigma=0.3
mm=(sqrt(m)+rnorm(n,0,sigma))^2
x=rpois(n,mm)
mu=vst_gen(x,sigma)
mu.s=vst_gen(x,NULL)
mu.su=vst_gen(x,NULL,nug_est = F)
plot(x,col='grey80',main='with nugget 0.3')
lines(m)
lines(mu,col=2)
lines(mu.s,col=3)
lines(mu.su,col=4)
legend('topright',c('smooth mean','known var','unkown var-est nugget','unkown var'),lty=c(1,1,1,1),col=c(1,2,3,4))

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Version Author Date
a31c75e Dongyue Xie 2018-10-18 

m=m*5+20
set.seed(12345)
sigma=0.3
x=rpois(n,(sqrt(m)+rnorm(n,0,sigma))^2)

mu=vst_gen(x,sigma)
mu.s=vst_gen(x,NULL)
mu.su=vst_gen(x,NULL,nug_est = F)
plot(x,col='grey80',main='with nugget 0.3')
lines(m)
lines(mu,col=2)
lines(mu.s,col=3)
lines(mu.su,col=4)
legend('topright',c('smooth mean','known var','unkown var-est nugget','unkown var'),lty=c(1,1,1,1),col=c(1,2,3,4))

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Version Author Date
a31c75e Dongyue Xie 2018-10-18 

sigma=0.5
x=rpois(n,(sqrt(m)+rnorm(n,0,sigma))^2)

mu=vst_gen(x,sigma)
mu.s=vst_gen(x,NULL)
mu.su=vst_gen(x,NULL,nug_est = F)
plot(x,col='grey80',main='with nugget 0.5')
lines(m)
lines(mu,col=2)
lines(mu.s,col=3)
lines(mu.su,col=4)
legend('topright',c('smooth mean','known var','unkown var-est nugget','unkown var'),lty=c(1,1,1,1),col=c(1,2,3,4))

Real data comparison

extract_counts_CTCF <- function(filename){
  bed_counts <- read.table(filename, header = F, stringsAsFactors = F)
  colnames(bed_counts) <- c("chr", "start", "end", "name", "width", "counts")

  counts <- strsplit(bed_counts$counts, split = ",")[[1]]
  counts[counts == "NA"] <- 0
  counts <- as.numeric(counts)

  return(counts.l = list(chr = bed_counts$chr, start = bed_counts$start, end = bed_counts$end, counts = counts))
}

chipexo1 <- extract_counts_CTCF("/Users/dongyue/Documents/smash-gen/data/chipexo_examples/example_CTCF_MACE_wgEncodeOpenChromChipHelas3CtcfAlnRep1_forward_counts.txt")

smash.out=smash.poiss(chipexo1$counts)
y=reflect(chipexo1$counts,'both',c(300,299))
smashgen.out=smash_gen_lite(y)
vst.out=vst_gen(y,NULL,nug_est = F)

plot(chipexo1$counts, col = "gray80", ylab = "rep1 forward", xlab = "", main = "EncodeOpenChromChipHelas - Rep 1")
lines(smash.out, col = 2)
lines(smashgen.out[301:725],col=4)
lines(vst.out[301:725],col=3)

legend("topright", # places a legend at the appropriate place
       c("truth","smash-poiss",'smashgen','vst-gen'), # puts text in the legend
       lty=c(0,1,1,1), # gives the legend appropriate symbols (lines)
       pch=c(1,NA,NA,NA),
       lwd=c(1,1,1,1),
       cex = 0.5,
       col=c("gray80","red", "blue",3))

Expand here to see past versions of unnamed-chunk-3-1.png:
Version Author Date
a31c75e Dongyue Xie 2018-10-18 

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

chipexo1 <- extract_counts_CTCF("/Users/dongyue/Documents/smash-gen/data/chipexo_examples/example_CTCF_MACE_wgEncodeBroadHistoneHelas3CtcfStdAlnRep1_forward_counts.txt")

smash.out=smash.poiss(chipexo1$counts)
y=reflect(chipexo1$counts,'both',c(300,299))
smashgen.out=smash_gen_lite(y)
vst.out=vst_gen(y,NULL,nug_est = F)

plot(chipexo1$counts, col = "gray80", ylab = "rep1 forward", xlab = "", main = "EncodeBroadHistoneHelas - Rep 1")
lines(smash.out, col = 2)
lines(smashgen.out[301:725],col=4)
lines(vst.out[301:725],col=3)

legend("topright", # places a legend at the appropriate place
       c("truth","smash-poiss",'smashgen','vst-gen'), # puts text in the legend
       lty=c(0,1,1,1), # gives the legend appropriate symbols (lines)
       pch=c(1,NA,NA,NA),
       lwd=c(1,1,1,1),
       cex = 0.5,
       col=c("gray80","red", "blue",3))

########
chipexo1 <- extract_counts_CTCF("/Users/dongyue/Documents/smash-gen/data/chipexo_examples/example_CTCF_MACE_wgEncodeOpenChromChipHelas3CtcfAlnRep1_forward_counts.txt")

smash.out=smash.poiss(chipexo1$counts)
y=reflect(chipexo1$counts,'both',c(300,299))
smashgen.out=smash_gen_lite(y)
vst.out=vst_gen(y,NULL,nug_est = F)

plot(chipexo1$counts, col = "gray80", ylab = "rep1 forward", xlab = "", main = "EncodeOpenChromChipHelas - Rep 1")
lines(smash.out, col = 2)
lines(smashgen.out[301:725],col=4)
lines(vst.out[301:725],col=3)

legend("topright", # places a legend at the appropriate place
       c("truth","smash-poiss",'smashgen','vst-gen'), # puts text in the legend
       lty=c(0,1,1,1), # gives the legend appropriate symbols (lines)
       pch=c(1,NA,NA,NA),
       lwd=c(1,1,1,1),
       cex = 0.5,
       col=c("gray80","red", "blue",3))

#########

chipexo1 <- extract_counts_CTCF("/Users/dongyue/Documents/smash-gen/data/chipexo_examples/example_CTCF_MACE_wgEncodeOpenChromChipHelas3CtcfAlnRep2_forward_counts.txt")

smash.out=smash.poiss(chipexo1$counts)
y=reflect(chipexo1$counts,'both',c(300,299))
smashgen.out=smash_gen_lite(y)
vst.out=vst_gen(y,NULL,nug_est = F)

plot(chipexo1$counts, col = "gray80", ylab = "rep2 forward", xlab = "", main = "EncodeOpenChromChipHelas - Rep 2
")
lines(smash.out, col = 2)
lines(smashgen.out[301:725],col=4)
lines(vst.out[301:725],col=3)

legend("topright", # places a legend at the appropriate place
       c("truth","smash-poiss",'smashgen','vst-gen'), # puts text in the legend
       lty=c(0,1,1,1), # gives the legend appropriate symbols (lines)
       pch=c(1,NA,NA,NA),
       lwd=c(1,1,1,1),
       cex = 0.5,
       col=c("gray80","red", "blue",3))

Reference

Gasser, T., Sroka, L. and Jennen-Steinmetz, C. (1986). Residual variance and residual pattern in nonlinear regression. Biometrika 73 625–633.

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] ashr_2.2-7       smashr_1.2-0    

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

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