Last updated: 2018-05-17

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: 3c0ccfd

    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:    .Rhistory
    Ignored:    .Rproj.user/
    Ignored:    log/

Untracked files:
    Untracked:  analysis/binom.Rmd
    Untracked:  analysis/overdis.Rmd
    Untracked:  analysis/smashtutorial.Rmd
    Untracked:  docs/figure/smashtutorial.Rmd/

Unstaged changes:
    Modified:   analysis/ashpmean.Rmd
    Modified:   analysis/nugget.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 3c0ccfd Dongyue 2018-05-17 edit
    Rmd b7d1baa Dongyue 2018-05-17 edit
    Rmd 760a2fc Dongyue 2018-05-17 edit

Algorithm

Let \(X_t\) be a Binomial observation with number of trials \(n_t\), \(t=1,2,\dots,T\).

  1. Apply ash to binomial data \((X_t, n_t)\) and obtain the posterior mean \(\hat p=E(p|X,n)\), \(\hat p\in R^T\).
  2. Let \(s_t=\frac{1}{n_t\hat p_t(1-\hat p_t)}\) and \(Y_t=\log\frac{\hat p_t}{1-\hat p_t}+\frac{X_t/n_t-\hat p_t}{n_t\hat p_t(1-\hat p_t)}\).
  3. Apply smash.gaus to \(Y_t\) with standard deviation \(\sqrt(\sigma^2+s_t^2)\) if \(\sigma^2\) is known. If it’s unknown, then either \(\sigma^2\) or \(\sigma^2+s_t^2\) should be estimated first. The output of smash.gaus is denoted as \(\mu_t\).
  4. Export \(\frac{\exp(\mu_t)}{1+\exp(\mu_t)}\).

Methods for estimating \(\sigma^2\) or \(\sigma^2+s_t^2\) are discussed here.

library(smashrgen)
library(ggplot2)
simu_study=function(p,sigma,ntri,nsimu=100,seed=12345,
                    niter=1,family='DaubExPhase',ashp=TRUE,verbose=FALSE,robust=FALSE,
                    tol=1e-2){
  set.seed(seed)
  #smash.err=c()
  smashgen.err=c()
  smashgen.smashu.err=c()
  #smashgen.mle.err=c()
  n=length(p)
  true.p=exp(p)/(1+exp(p))
  for(k in 1:nsimu){
    ng=rnorm(n,0,sigma)
    m=exp(p+ng)
    q=m/(1+m)
    x=rbinom(n,ntri,q)
    #fit data
    #smash.out=smash.poiss(x)
    smashgen.out=smash_gen(x,dist_family = 'binomial',sigma = sigma,ntri=ntri)
    smashu.out=smash_gen(x,dist_family = 'binomial',y_var_est = 'smashu',ntri=ntri)
    #mle.out=smash_gen(x,dist_family = 'poisson',y_var_est = 'mle')
    #smash.err[k]=mse(exp(m),smash.out)
    smashgen.err[k]=mse(true.p,smashgen.out)
    smashgen.smashu.err[k]=mse(true.p,smashu.out)
    #smashgen.mle.err[k]=mse(exp(m),mle.out)
  }
  return(list(est=list(smashgen.out=smashgen.out,smashu.out=smashu.out,x=x),err=data.frame(smashgen=smashgen.err, smashgen.smashu=smashgen.smashu.err)))
}

Constant trend

\(\sigma=0.1\)

p=rep(0,128)
set.seed(111)
ntri=rpois(128,30)
result=simu_study(p,0.1,ntri)
par(mfrow=c(1,2))
plot(result$est$x/ntri,col='gray80',ylab='',main='smashgen: known variance')
lines(exp(p)/(1+exp(p)),col=1)
lines(result$est$smashgen.out,col=4)
legend("topright", # places a legend at the appropriate place
       c("truth","smashgen"), # puts text in the legend
       lty=c(1,1), # gives the legend appropriate symbols (lines)
       lwd=c(1,1),
       cex = 1,
       col=c("black","blue"))
plot(result$est$x/ntri,col='gray80',ylab='',main='smashgen: unknown variance')
lines(exp(p)/(1+exp(p)),col=1)
lines(result$est$smashu.out,col=4)
legend("topright", # places a legend at the appropriate place
       c("truth","smashgen"), # puts text in the legend
       lty=c(1,1), # gives the legend appropriate symbols (lines)
       lwd=c(1,1),
       cex = 1,
       col=c("black","blue"))

mean(result$err$smashgen)
[1] 8.957221e-05
mean(result$err$smashgen.smashu)
[1] 9.913624e-05
ggplot(df2gg(result$err),aes(x=method,y=MSE))+geom_boxplot(aes(fill=method))+labs(x='')

\(\sigma=1\)

result=simu_study(p,1,ntri)
par(mfrow=c(1,2))
plot(result$est$x/ntri,col='gray80',ylab='',main='smashgen: known variance')
lines(exp(p)/(1+exp(p)),col=1)
lines(result$est$smashgen.out,col=4)
plot(result$est$x/ntri,col='gray80',ylab='',main='smashgen: unknown variance')
lines(exp(p)/(1+exp(p)),col=1)
lines(result$est$smashu.out,col=4)

mean(result$err$smashgen)
[1] 0.0007841879
mean(result$err$smashgen.smashu)
[1] 0.0008206814
ggplot(df2gg(result$err),aes(x=method,y=MSE))+geom_boxplot(aes(fill=method))+labs(x='')

\(\sigma=1\), small prob

p=rep(-3,128)
result=simu_study(p,1,ntri)
par(mfrow=c(1,2))
plot(result$est$x/ntri,col='gray80',ylab='',main='smashgen: known variance')
lines(exp(p)/(1+exp(p)),col=1)
lines(result$est$smashgen.out,col=4)
plot(result$est$x/ntri,col='gray80',ylab='',main='smashgen: unknown variance')
lines(exp(p)/(1+exp(p)),col=1)
lines(result$est$smashu.out,col=4)

mean(result$err$smashgen)
[1] 2.175465e-05
mean(result$err$smashgen.smashu)
[1] 2.726665e-05
ggplot(df2gg(result$err),aes(x=method,y=MSE))+geom_boxplot(aes(fill=method))+labs(x='')

Step trend

\(\sigma=0.1\)

p=c(rep(-2,128), rep(0, 128), rep(2, 128), rep(-2, 128))
set.seed(111)
ntri=rpois(512,30)
result=simu_study(p,0.1,ntri)

par(mfrow=c(1,2))
plot(result$est$x/ntri,col='gray80',ylab='',main='smashgen: known variance')
lines(exp(p)/(1+exp(p)),col=1)
lines(result$est$smashgen.out,col=4)
plot(result$est$x/ntri,col='gray80',ylab='',main='smashgen: unknown variance')
lines(exp(p)/(1+exp(p)),col=1)
lines(result$est$smashu.out,col=4)

mean(result$err$smashgen)
[1] 0.0003595882
mean(result$err$smashgen.smashu)
[1] 0.0004350061
ggplot(df2gg(result$err),aes(x=method,y=MSE))+geom_boxplot(aes(fill=method))+labs(x='')

\(\sigma=1\)

p=c(rep(-2,128), rep(0, 128), rep(2, 128), rep(-2, 128))
set.seed(111)
ntri=rpois(512,30)
result=simu_study(p,1,ntri)

par(mfrow=c(1,2))
plot(result$est$x/ntri,col='gray80',ylab='',main='smashgen: known variance')
lines(exp(p)/(1+exp(p)),col=1)
lines(result$est$smashgen.out,col=4)
plot(result$est$x/ntri,col='gray80',ylab='',main='smashgen: unknown variance')
lines(exp(p)/(1+exp(p)),col=1)
lines(result$est$smashu.out,col=4)

mean(result$err$smashgen)
[1] 0.001850991
mean(result$err$smashgen.smashu)
[1] 0.001842285
ggplot(df2gg(result$err),aes(x=method,y=MSE))+geom_boxplot(aes(fill=method))+labs(x='')

\(\sigma=1\), more variant ntri

p=c(rep(-2,128), rep(0, 128), rep(2, 128), rep(-2, 128))
set.seed(111)
ntri=rpois(512,50)-20
result=simu_study(p,1,ntri)

par(mfrow=c(1,2))
plot(result$est$x/ntri,col='gray80',ylab='',main='smashgen: known variance')
lines(exp(p)/(1+exp(p)),col=1)
lines(result$est$smashgen.out,col=4)
plot(result$est$x/ntri,col='gray80',ylab='',main='smashgen: unknown variance')
lines(exp(p)/(1+exp(p)),col=1)
lines(result$est$smashu.out,col=4)

mean(result$err$smashgen)
[1] 0.001875976
mean(result$err$smashgen.smashu)
[1] 0.001847042
ggplot(df2gg(result$err),aes(x=method,y=MSE))+geom_boxplot(aes(fill=method))+labs(x='')

Bumps

\(\sigma=0.1\)

m=seq(0,1,length.out = 256)
h = c(4, 5, 3, 4, 5, 4.2, 2.1, 4.3, 3.1, 5.1, 4.2)
w = c(0.005, 0.005, 0.006, 0.01, 0.01, 0.03, 0.01, 0.01, 0.005,0.008,0.005)
t=c(.1,.13,.15,.23,.25,.4,.44,.65,.76,.78,.81)
f = c()
for(i in 1:length(m)){
  f[i]=sum(h*(1+((m[i]-t)/w)^4)^(-1))
}
p=f-3

set.seed(111)
ntri=rpois(256,30)
result=simu_study(p,0.1,ntri)

par(mfrow=c(1,2))
plot(result$est$x/ntri,col='gray80',ylab='',main='smashgen: known variance')
lines(exp(p)/(1+exp(p)),col=1)
lines(result$est$smashgen.out,col=4)
plot(result$est$x/ntri,col='gray80',ylab='',main='smashgen: unknown variance')
lines(exp(p)/(1+exp(p)),col=1)
lines(result$est$smashu.out,col=4)

mean(result$err$smashgen)
[1] 0.001821751
mean(result$err$smashgen.smashu)
[1] 0.005024706
ggplot(df2gg(result$err),aes(x=method,y=MSE))+geom_boxplot(aes(fill=method))+labs(x='')

\(\sigma=1\)

set.seed(111)
ntri=rpois(256,30)
result=simu_study(p,1,ntri)

par(mfrow=c(1,2))
plot(result$est$x/ntri,col='gray80',ylab='',main='smashgen: known variance')
lines(exp(p)/(1+exp(p)),col=1)
lines(result$est$smashgen.out,col=4)
plot(result$est$x/ntri,col='gray80',ylab='',main='smashgen: unknown variance')
lines(exp(p)/(1+exp(p)),col=1)
lines(result$est$smashu.out,col=4)

mean(result$err$smashgen)
[1] 0.0124867
mean(result$err$smashgen.smashu)
[1] 0.01233137
ggplot(df2gg(result$err),aes(x=method,y=MSE))+geom_boxplot(aes(fill=method))+labs(x='')

\(\sigma=1\), more variant ntri

set.seed(111)
ntri=rpois(256,50)-20
result=simu_study(p,1,ntri)

par(mfrow=c(1,2))
plot(result$est$x/ntri,col='gray80',ylab='',main='smashgen: known variance')
lines(exp(p)/(1+exp(p)),col=1)
lines(result$est$smashgen.out,col=4)
plot(result$est$x/ntri,col='gray80',ylab='',main='smashgen: unknown variance')
lines(exp(p)/(1+exp(p)),col=1)
lines(result$est$smashu.out,col=4)

mean(result$err$smashgen)
[1] 0.01249222
mean(result$err$smashgen.smashu)
[1] 0.01226534
ggplot(df2gg(result$err),aes(x=method,y=MSE))+geom_boxplot(aes(fill=method))+labs(x='')

\(\sigma=1\), smaller ntri

set.seed(111)
ntri=rpois(256,20)-9
result=simu_study(p,1,ntri)

par(mfrow=c(1,2))
plot(result$est$x/ntri,col='gray80',ylab='',main='smashgen: known variance')
lines(exp(p)/(1+exp(p)),col=1)
lines(result$est$smashgen.out,col=4)
plot(result$est$x/ntri,col='gray80',ylab='',main='smashgen: unknown variance')
lines(exp(p)/(1+exp(p)),col=1)
lines(result$est$smashu.out,col=4)

mean(result$err$smashgen)
[1] 0.02209255
mean(result$err$smashgen.smashu)
[1] 0.01565017
ggplot(df2gg(result$err),aes(x=method,y=MSE))+geom_boxplot(aes(fill=method))+labs(x='')

Spike mean

\(\sigma=0.1\)

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
p = spike.f(t)*2-2

set.seed(111)
ntri=rpois(256,20)-9
result=simu_study(p,0.1,ntri)

par(mfrow=c(1,2))
plot(result$est$x/ntri,col='gray80',ylab='',main='smashgen: known variance')
lines(exp(p)/(1+exp(p)),col=1)
lines(result$est$smashgen.out,col=4)
plot(result$est$x/ntri,col='gray80',ylab='',main='smashgen: unknown variance')
lines(exp(p)/(1+exp(p)),col=1)
lines(result$est$smashu.out,col=4)

mean(result$err$smashgen)
[1] 0.003198372
mean(result$err$smashgen.smashu)
[1] 0.003931562
ggplot(df2gg(result$err),aes(x=method,y=MSE))+geom_boxplot(aes(fill=method))+labs(x='')

\(\sigma=1\)

set.seed(111)
ntri=rpois(256,20)-9
result=simu_study(p,1,ntri)

par(mfrow=c(1,2))
plot(result$est$x/ntri,col='gray80',ylab='',main='smashgen: known variance')
lines(exp(p)/(1+exp(p)),col=1)
lines(result$est$smashgen.out,col=4)
plot(result$est$x/ntri,col='gray80',ylab='',main='smashgen: unknown variance')
lines(exp(p)/(1+exp(p)),col=1)
lines(result$est$smashu.out,col=4)

mean(result$err$smashgen)
[1] 0.01327303
mean(result$err$smashgen.smashu)
[1] 0.009787627
ggplot(df2gg(result$err),aes(x=method,y=MSE))+geom_boxplot(aes(fill=method))+labs(x='')

Session information

sessionInfo()
R version 3.4.0 (2017-04-21)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 10 x64 (build 16299)

Matrix products: default

locale:
[1] LC_COLLATE=English_United States.1252 
[2] LC_CTYPE=English_United States.1252   
[3] LC_MONETARY=English_United States.1252
[4] LC_NUMERIC=C                          
[5] LC_TIME=English_United States.1252    

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] ggplot2_2.2.1    smashrgen_0.1.0  wavethresh_4.6.8 MASS_7.3-47     
[5] caTools_1.17.1   ashr_2.2-7       smashr_1.1-5    

loaded via a namespace (and not attached):
 [1] Rcpp_0.12.16        plyr_1.8.4          compiler_3.4.0     
 [4] git2r_0.21.0        workflowr_1.0.1     R.methodsS3_1.7.1  
 [7] R.utils_2.6.0       bitops_1.0-6        iterators_1.0.8    
[10] tools_3.4.0         digest_0.6.13       tibble_1.3.3       
[13] evaluate_0.10       gtable_0.2.0        lattice_0.20-35    
[16] rlang_0.1.2         Matrix_1.2-9        foreach_1.4.3      
[19] yaml_2.1.19         parallel_3.4.0      stringr_1.3.0      
[22] knitr_1.20          REBayes_1.3         rprojroot_1.3-2    
[25] grid_3.4.0          data.table_1.10.4-3 rmarkdown_1.8      
[28] magrittr_1.5        whisker_0.3-2       backports_1.0.5    
[31] scales_0.4.1        codetools_0.2-15    htmltools_0.3.5    
[34] assertthat_0.2.0    colorspace_1.3-2    labeling_0.3       
[37] stringi_1.1.6       Rmosek_8.0.69       lazyeval_0.2.1     
[40] munsell_0.4.3       doParallel_1.0.11   pscl_1.4.9         
[43] truncnorm_1.0-7     SQUAREM_2017.10-1   R.oo_1.21.0

This reproducible R Markdown analysis was created with workflowr 1.0.1