Last updated: 2018-05-06

workflowr checks: (Click a bullet for more information)

  • ✖ R Markdown file: uncommitted changes The R Markdown is untracked by Git. To know which version of the R Markdown file created these results, you’ll want to first commit it to the Git repo. If you’re still working on the analysis, you can ignore this warning. When you’re finished, you can run wflow_publish to commit the R Markdown file and build the HTML.

  • ✔ 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: a214777

    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:    analysis/figure/
    Ignored:    log/

Untracked files:
    Untracked:  analysis/nuggetunkown.Rmd

Unstaged changes:
    Modified:   analysis/index.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.

Examine the performance of smash-gen(unknown \(\sigma\)) under different simulation settings. When \(\sigma\) is unkown, the algorithm interatively estiamte \(\mu_t\) and \(s_t^2+\sigma^2\) in each interation.

#' smash generaliation function, unknown \sigma.

smash.gen.us=function(x,family='DaubExPhase',niter=100,tol=1e-2){
  mu=c()
  s=c()
  mu=rbind(mu,rep(mean(x),length(x)))
  s=rbind(s,rep(1/mu[1],length(x)))
  y=log(mean(x))+(x-mean(x))/mean(x)
  for(i in 1:niter){
    mu.hat=try(smash.gaus(y))
    if(class(mu.hat)=='try-error'){
      message('Algorithm diverges')
      break
    }
    mu=rbind(mu,mu.hat)
    #update m and s_t
    s=rbind(s,1/mu.hat)
    #update y
    mt=exp(mu.hat)
    y=log(mt)+(x-mt)/mt
    #y=log(mu.hat)+(x-mu.hat)/mu.hat
    if(norm(mu.hat-mu[i,],'2')<tol){
      break
    }
  }
  if(class(mu.hat)=='try-error'){
    return(list(mu.hat=mu[i,],mu=mu,s=s))
  }else{
    return(list(mu.hat=mu.hat,mu=mu,s=s))
  }
}

simu_study=function(m,sigma,seed=1234,
                    niter=100,family='DaubExPhase',tol=1e-2,
                    reflect=FALSE){
  set.seed(seed)
  lamda=exp(m+rnorm(length(m),0,sigma))
  x=rpois(length(m),lamda)
  #fit data
  smash.out=smash.poiss(x,reflect=FALSE)
  smash.gen.out=smash.gen.us(x,niter=niter,family = family,tol=tol)
  return(list(smash.out=smash.out,smash.gen.out=exp(smash.gen.out$mu.hat),smash.gen.est=smash.gen.out,x=x))
}

Simulation 1: Constant trend Poisson nugget

\(\sigma=0.01\)

library(smashr)
m=rep(3,256)
simu.out=simu_study(m,0.01)

#par(mfrow = c(1,2))

plot(simu.out$x,col = "gray80" ,ylab = '')
lines(simu.out$smash.gen.out, col = "red", lwd = 2)
lines(exp(m))
legend("topright", # places a legend at the appropriate place
       c("truth","smash-gen"), # puts text in the legend
       lty=c(1,1), # gives the legend appropriate symbols (lines)
       lwd=c(1,1),
       cex = 1,
       col=c("black","red", "blue"))

plot(simu.out$x,col = "gray80" ,ylab = '')
lines(simu.out$smash.out, col = "blue", lwd = 2)
lines(exp(m))
legend("topright", 
       c("truth", "smash"), 
       lty=c(1,1), 
       lwd=c(1,1),
       cex = 1,
       col=c("black", "blue"))

\(\sigma=0.1\)

simu.out=simu_study(m,0.1)

#par(mfrow = c(1,2))

plot(simu.out$x,col = "gray80" ,ylab = '')
lines(simu.out$smash.gen.out, col = "red", lwd = 2)
lines(exp(m))
legend("topright", 
       c("truth","smash-gen"), 
       lty=c(1,1), 
       lwd=c(1,1),
       cex = 1,
       col=c("black","red", "blue"))

plot(simu.out$x,col = "gray80" ,ylab = '')
lines(simu.out$smash.out, col = "blue", lwd = 2)
lines(exp(m))
legend("topright", 
       c("truth", "smash"), 
       lty=c(1,1), 
       lwd=c(1,1),
       cex = 1,
       col=c("black", "blue"))

\(\sigma=0.5\)

simu.out=simu_study(m,0.5)

#par(mfrow = c(1,2))

plot(simu.out$x,col = "gray80" ,ylab = '')
lines(simu.out$smash.gen.out, col = "red", lwd = 2)
lines(exp(m),col='gray80')
legend("topright", 
       c("truth","smash-gen"), 
       lty=c(1,1), 
       lwd=c(1,1),
       cex = 1,
       col=c("black","red", "blue"))

plot(simu.out$x,col = "gray80" ,ylab = '')
lines(simu.out$smash.out, col = "blue", lwd = 2)
lines(exp(m),col='gray80')
legend("topright", 
       c("truth", "smash"), 
       lty=c(1,1), 
       lwd=c(1,1),
       cex = 1,
       col=c("black", "blue"))

\(\sigma=1\)

simu.out=simu_study(m,1)

#par(mfrow = c(1,2))

plot(simu.out$x,col = "gray80" ,ylab = '')
lines(simu.out$smash.gen.out, col = "red", lwd = 2)
lines(exp(m),col='black')
legend("topleft", 
       c("truth","smash-gen"), 
       lty=c(1,1), 
       lwd=c(1,1),
       cex = 1,
       col=c("black","red", "blue"))

plot(simu.out$x,col = "gray80" ,ylab = '')
lines(simu.out$smash.out, col = "blue", lwd = 2)
lines(exp(m),col='black')
legend("topleft", 
       c("truth", "smash"), 
       lty=c(1,1), 
       lwd=c(1,1),
       cex = 1,
       col=c("black", "blue"))

Simulation 2: Step trend

\(\sigma=0.01\)

m=c(rep(3,128), rep(5, 128), rep(6, 128), rep(4, 128))
simu.out=simu_study(m,0.01)

#par(mfrow = c(1,2))

plot(simu.out$x,col = "gray80" ,ylab = '')
lines(simu.out$smash.gen.out, col = "red", lwd = 2)
lines(exp(m))
legend("topright", 
       c("truth","smash-gen"), 
       lty=c(1,1), 
       lwd=c(1,1),
       cex = 1,
       col=c("black","red", "blue"))

plot(simu.out$x,col = "gray80" ,ylab = '')
lines(simu.out$smash.out, col = "blue", lwd = 2)
lines(exp(m))
legend("topright", 
       c("truth", "smash"), 
       lty=c(1,1), 
       lwd=c(1,1),
       cex = 1,
       col=c("black", "blue"))

\(\sigma=0.1\)

simu.out=simu_study(m,0.1)

#par(mfrow = c(1,2))

plot(simu.out$x,col = "gray80" ,ylab = '')
lines(simu.out$smash.gen.out, col = "red", lwd = 2)
lines(exp(m))
legend("topright", 
       c("truth","smash-gen"), 
       lty=c(1,1), 
       lwd=c(1,1),
       cex = 1,
       col=c("black","red", "blue"))

plot(simu.out$x,col = "gray80" ,ylab = '')
lines(simu.out$smash.out, col = "blue", lwd = 2)
lines(exp(m))
legend("topright", 
       c("truth", "smash"), 
       lty=c(1,1), 
       lwd=c(1,1),
       cex = 1,
       col=c("black", "blue"))

\(\sigma=0.5\)

simu.out=simu_study(m,0.5)

#par(mfrow = c(1,2))

plot(simu.out$x,col = "gray80" ,ylab = '')
lines(simu.out$smash.gen.out, col = "red", lwd = 2)
lines(exp(m))
legend("topright", 
       c("truth","smash-gen"), 
       lty=c(1,1), 
       lwd=c(1,1),
       cex = 1,
       col=c("black","red", "blue"))

plot(simu.out$x,col = "gray80" ,ylab = '')
lines(simu.out$smash.out, col = "blue", lwd = 2)
lines(exp(m))
legend("topright", 
       c("truth", "smash"), 
       lty=c(1,1), 
       lwd=c(1,1),
       cex = 1,
       col=c("black", "blue"))

\(\sigma=1\)

simu.out=simu_study(m,1)

#par(mfrow = c(1,2))

plot(simu.out$x,col = "gray80" ,ylab = '')
lines(simu.out$smash.gen.out, col = "red", lwd = 2)
lines(exp(m))
legend("topleft", 
       c("truth","smash-gen"), 
       lty=c(1,1), 
       lwd=c(1,1),
       cex = 1,
       col=c("black","red", "blue"))

plot(simu.out$x,col = "gray80" ,ylab = '')
lines(simu.out$smash.out, col = "blue", lwd = 2)
lines(exp(m))
legend("topleft", 
       c("truth", "smash"), 
       lty=c(1,1), 
       lwd=c(1,1),
       cex = 1,
       col=c("black", "blue"))

Simulation 3: Oscillating Poisson nugget

Low Oscillating Poisson nugget

\(\sigma=0.01\)

m=c()
for(k in 1:8){
    m=c(m, rep(1,16), rep(5, 16))
}
simu.out=simu_study(m,0.01)

#par(mfrow = c(1,2))

plot(simu.out$x,col = "gray80" ,ylab = '',main='smash-gen')
lines(simu.out$smash.gen.out, col = "red", lwd = 2)
lines(exp(m))

plot(simu.out$x,col = "gray80" ,ylab = '',main='smash')
lines(simu.out$smash.out, col = "blue", lwd = 2)
lines(exp(m))

\(\sigma=0.1\)

simu.out=simu_study(m,0.1)

#par(mfrow = c(1,2))

plot(simu.out$x,col = "gray80" ,ylab = '',main='smash-gen')
lines(simu.out$smash.gen.out, col = "red", lwd = 2)
lines(exp(m))

plot(simu.out$x,col = "gray80" ,ylab = '',main='smash')
lines(simu.out$smash.out, col = "blue", lwd = 2)
lines(exp(m))

\(\sigma=0.5\)

simu.out=simu_study(m,0.5)

#par(mfrow = c(1,2))

plot(simu.out$x,col = "gray80" ,ylab = '',main='smash-gen')
lines(simu.out$smash.gen.out, col = "red", lwd = 2)
lines(exp(m))

plot(simu.out$x,col = "gray80" ,ylab = '',main='smash')
lines(simu.out$smash.out, col = "blue", lwd = 2)
lines(exp(m))

\(\sigma=1\)

simu.out=simu_study(m,1)

#par(mfrow = c(1,2))

plot(simu.out$x,col = "gray80" ,ylab = '',main='smash-gen')
lines(simu.out$smash.gen.out, col = "red", lwd = 2)
lines(exp(m))

plot(simu.out$x,col = "gray80" ,ylab = '',main='smash')
lines(simu.out$smash.out, col = "blue", lwd = 2)
lines(exp(m))

Fast Oscillating Poisson nugget

\(\sigma=0.01\)

m=c()
for(k in 1:32){
    m=c(m, c(1,5))
}
simu.out=simu_study(m,0.01)

#par(mfrow = c(1,2))

plot(simu.out$x,col = "gray80" ,ylab = '',main='smash-gen')
lines(simu.out$smash.gen.out, col = "red", lwd = 2)
lines(exp(m))

plot(simu.out$x,col = "gray80" ,ylab = '',main='smash')
lines(simu.out$smash.out, col = "blue", lwd = 2)
lines(exp(m))

\(\sigma=0.1\)

simu.out=simu_study(m,0.1)

#par(mfrow = c(1,2))

plot(simu.out$x,col = "gray80" ,ylab = '',main='smash-gen')
lines(simu.out$smash.gen.out, col = "red", lwd = 2)
lines(exp(m))

plot(simu.out$x,col = "gray80" ,ylab = '',main='smash')
lines(simu.out$smash.out, col = "blue", lwd = 2)
lines(exp(m))

\(\sigma=0.5\)

simu.out=simu_study(m,0.5)

#par(mfrow = c(1,2))

plot(simu.out$x,col = "gray80" ,ylab = '',main='smash-gen')
lines(simu.out$smash.gen.out, col = "red", lwd = 2)
lines(exp(m))

plot(simu.out$x,col = "gray80" ,ylab = '',main='smash')
lines(simu.out$smash.out, col = "blue", lwd = 2)
lines(exp(m))

\(\sigma=1\)

simu.out=simu_study(m,1)
Algorithm diverges
#par(mfrow = c(1,2))

plot(simu.out$x,col = "gray80" ,ylab = '',main='smash-gen')
lines(simu.out$smash.gen.out, col = "red", lwd = 2)
lines(exp(m))

plot(simu.out$x,col = "gray80" ,ylab = '',main='smash')
lines(simu.out$smash.out, col = "blue", lwd = 2)
lines(exp(m))

Simulation 4: Polynomial curve Poisson nugget

\(\sigma=0.01\)

m = seq(-1,1,length.out = 256)
m = m^3-2*m+1
simu.out=simu_study(m,0.01)

#par(mfrow = c(1,2))

plot(simu.out$x,col = "gray80" ,ylab = '')
lines(simu.out$smash.gen.out, col = "red", lwd = 2)
lines(exp(m))
legend("topright", 
       c("truth","smash-gen"), 
       lty=c(1,1), 
       lwd=c(1,1),
       cex = 1,
       col=c("black","red", "blue"))

plot(simu.out$x,col = "gray80" ,ylab = '')
lines(simu.out$smash.out, col = "blue", lwd = 2)
lines(exp(m))
legend("topright", 
       c("truth", "smash"), 
       lty=c(1,1), 
       lwd=c(1,1),
       cex = 1,
       col=c("black", "blue"))

\(\sigma=0.1\)

m = seq(-1,1,length.out = 256)
m = m^3-2*m+1
simu.out=simu_study(m,0.1)

#par(mfrow = c(1,2))

plot(simu.out$x,col = "gray80" ,ylab = '')
lines(simu.out$smash.gen.out, col = "red", lwd = 2)
lines(exp(m))
legend("topright", 
       c("truth","smash-gen"), 
       lty=c(1,1), 
       lwd=c(1,1),
       cex = 1,
       col=c("black","red", "blue"))

plot(simu.out$x,col = "gray80" ,ylab = '')
lines(simu.out$smash.out, col = "blue", lwd = 2)
lines(exp(m))
legend("topright", 
       c("truth", "smash"), 
       lty=c(1,1), 
       lwd=c(1,1),
       cex = 1,
       col=c("black", "blue"))

\(\sigma=0.5\)

m = seq(-1,1,length.out = 256)
m = m^3-2*m+1
simu.out=simu_study(m,0.5)

#par(mfrow = c(1,2))

plot(simu.out$x,col = "gray80" ,ylab = '')
lines(simu.out$smash.gen.out, col = "red", lwd = 2)
lines(exp(m))
legend("topright", 
       c("truth","smash-gen"), 
       lty=c(1,1), 
       lwd=c(1,1),
       cex = 1,
       col=c("black","red", "blue"))

plot(simu.out$x,col = "gray80" ,ylab = '')
lines(simu.out$smash.out, col = "blue", lwd = 2)
lines(exp(m))
legend("topright", 
       c("truth", "smash"), 
       lty=c(1,1), 
       lwd=c(1,1),
       cex = 1,
       col=c("black", "blue"))

\(\sigma=1\)

m = seq(-1,1,length.out = 256)
m = m^3-2*m+1
simu.out=simu_study(m,1)

#par(mfrow = c(1,2))

plot(simu.out$x,col = "gray80" ,ylab = '')
lines(simu.out$smash.gen.out, col = "red", lwd = 2)
lines(exp(m))
legend("topright", 
       c("truth","smash-gen"), 
       lty=c(1,1), 
       lwd=c(1,1),
       cex = 1,
       col=c("black","red", "blue"))

plot(simu.out$x,col = "gray80" ,ylab = '')
lines(simu.out$smash.out, col = "blue", lwd = 2)
lines(exp(m))
legend("topright", 
       c("truth", "smash"), 
       lty=c(1,1), 
       lwd=c(1,1),
       cex = 1,
       col=c("black", "blue"))

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] smashr_1.1-1

loaded via a namespace (and not attached):
 [1] Rcpp_0.12.16        knitr_1.20          whisker_0.3-2      
 [4] magrittr_1.5        workflowr_1.0.1     REBayes_1.3        
 [7] MASS_7.3-47         pscl_1.4.9          doParallel_1.0.11  
[10] SQUAREM_2017.10-1   lattice_0.20-35     foreach_1.4.3      
[13] ashr_2.2-7          stringr_1.3.0       caTools_1.17.1     
[16] tools_3.4.0         parallel_3.4.0      grid_3.4.0         
[19] data.table_1.10.4-3 R.oo_1.21.0         git2r_0.21.0       
[22] iterators_1.0.8     htmltools_0.3.5     assertthat_0.2.0   
[25] yaml_2.1.19         rprojroot_1.3-2     digest_0.6.13      
[28] Matrix_1.2-9        bitops_1.0-6        codetools_0.2-15   
[31] R.utils_2.6.0       evaluate_0.10       rmarkdown_1.8      
[34] wavethresh_4.6.8    stringi_1.1.6       compiler_3.4.0     
[37] Rmosek_8.0.69       backports_1.0.5     R.methodsS3_1.7.1  
[40] truncnorm_1.0-7

This reproducible R Markdown analysis was created with workflowr 1.0.1