Last updated: 2018-05-21

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We have shown that smashgen-Poisson outperforms smash when smoothing Poisson data with nugget effect. One natural question is: is this true for Poisson data(no nugget effect)?

One limitation of smash.pois is that it can only use an analogue of the Haar wavelet transform to the Poisson data. So it may lose power when dealing with wavelet whose signal is better captured by more complex basis functions.

In this analysis, we try to address the above two questions.

We have seen in the previous analysis that Symmlet 8 gives better decomposition for smoother functions, while for functions with spike or sharp changes, the two basis have similar results.

simu_study_basis=function(mu,nsimu=100,seed=1234,niter=1,robust=F){
  n=length(mu)
  set.seed(seed)
  smash.err=c()
  gen.haar.err=c()
  gen.sym.err=c()
  for(iter in 1:nsimu){
    y=rpois(n,mu)
    smash.out=smash.poiss(y)
    gen.haar.out=smash_gen(y,niter = niter,robust=robust)
    gen.sym.out=smash_gen(y,wave_family = 'DaubLeAsymm',filter.number = 8,niter=niter,robust=robust)
    smash.err[iter]=mse(smash.out,mu)
    gen.haar.err[iter]=mse(gen.haar.out,mu)
    gen.sym.err[iter]=mse(gen.sym.out,mu)
  }
  return(list(est=data.frame(smash=smash.out,smashgen.haar=gen.haar.out,smashgen.sym=gen.sym.out),err=data.frame(smash=smash.err,smashgen.haar=gen.haar.err,smashgen.sym=gen.sym.err)))
}

library(smashrgen)

mu=DJ.EX(256,signal = 2)$heavi
mu=mu-min(mu)+0.3
plot(mu,type='l')

result=simu_study_basis(mu)
boxplot(result$err)

mu=DJ.EX(256,signal = 20)$heavi
mu=mu-min(mu)+0.1
#plot(mu,type='l')
result=simu_study_basis(mu)
boxplot(result$err)

mu=DJ.EX(256,signal = 2)$doppler
mu=mu-min(mu)+0.1
plot(mu,type='l')

result=simu_study_basis(mu)
boxplot(result$err)

mu=DJ.EX(256,signal = 20)$doppler
mu=mu-min(mu)+0.1
#plot(mu,type='l')
result=simu_study_basis(mu)
boxplot(result$err)

r=function(x,c){return((x-c)^2*(x>c)*(x<=1))}
f=function(x){return(0.8 − 30*r(x,0.1) + 60*r(x, 0.2) − 30*r(x, 0.3) +
500*r(x, 0.35) − 1000*r(x, 0.37) + 1000*r(x, 0.41) − 500*r(x, 0.43) +
7.5*r(x, 0.5) − 15*r(x, 0.7) + 7.5*r(x, 0.9))}
mu=f(1:256/256)
mu=mu*10-1.9
plot(mu,type = 'l')

result=simu_study_basis(mu)
boxplot(result$err)

r=function(x,c){return((x-c)^2*(x>c)*(x<=1))}
f=function(x){return(0.8 − 30*r(x,0.1) + 60*r(x, 0.2) − 30*r(x, 0.3) +
500*r(x, 0.35) − 1000*r(x, 0.37) + 1000*r(x, 0.41) − 500*r(x, 0.43) +
7.5*r(x, 0.5) − 15*r(x, 0.7) + 7.5*r(x, 0.9))}
mu=f(1:256/256)
mu=mu*120-23
#plot(mu,type = 'l')
result=simu_study_basis(mu)
boxplot(result$err)

mu=c(rep(1,64), rep(3, 64), rep(6, 64), rep(1, 64))
result=simu_study_basis(mu)
boxplot(result$err)

mu=c(rep(1,64), rep(30, 64), rep(80, 64), rep(1, 64))
result=simu_study_basis(mu)
boxplot(result$err)

f=function(x){return(0.5 + 0.2*cos(4*pi*x) + 0.1*cos(24*pi*x))}
mu=f(1:256/256)
mu=mu*10-2
plot(mu,type='l')

result=simu_study_basis(mu)
boxplot(result$err)

mu=f(1:256/256)
mu=mu*130-29
#plot(mu,type='l')
result=simu_study_basis(mu)
boxplot(result$err)

g=function(x){return((1 − cos(pi*x))/2)}
f=function(x){return(0.3*sin(3*pi*(g(g(g(g(x)))) + x) + 0.5))}
mu=f(1:256/256)
mu=mu*10+3.1
plot(mu,type='l')

result=simu_study_basis(mu)
boxplot(result$err)

mu=f(1:256/256)
mu=mu*200+60.1
#plot(mu)
result=simu_study_basis(mu)
boxplot(result$err)

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] smashrgen_0.1.0  wavethresh_4.6.8 MASS_7.3-47      caTools_1.17.1  
[5] ashr_2.2-7       smashr_1.1-5    

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

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