Last updated: 2018-05-08

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Gaussian models are not robust to outliers so the smash-gen algorithm does not converge when the nugget effect is large. One solution might be setting the very highest resolution wavelet coefficients to 0.

library(smashr)
library(wavethresh)

Warning: package 'wavethresh' was built under R version 3.4.3

Loading required package: MASS

WaveThresh: R wavelet software, release 4.6.8, installed

Copyright Guy Nason and others 1993-2016

Note: nlevels has been renamed to nlevelsWT

#' smash generaliation function(set the highest resolution wavelet coeffs to 0)


#' @param x: a vector of observations
#' @param sigma: standard deviations, scalar.
#' @param family: choice of wavelet basis to be used, as in wavethresh.
#' @param niter: number of iterations for IRLS
#' @param tol: tolerance of the criterion to stop the iterations
#' @param robust: whether set the highest resolution wavelet coeffs to 0

smash.gen=function(x,sigma,family='DaubExPhase',filter.number = 1, niter=30,tol=1e-2,robust=FALSE){
  mu=c()
  s=c()
  y=c()
  munorm=c()
  mu=rbind(mu,rep(mean(x),length(x)))
  s=rbind(s,rep(1/mu[1],length(x)))
  y0=log(mean(x))+(x-mean(x))/mean(x)
  #######set the highest resolution wavelet coeffs to 0
  if(robust){
    wds=wd(y0,family = family,filter.number = filter.number)
    wtd=threshold(wds, levels = wds$nlevels-1,  policy="manual",value = Inf) 
    y=rbind(y,wr(wtd))
  }else{
    y=rbind(y,y0)
  }
  for(i in 1:niter){
    vars=ifelse(s[i,]<0,1e-8,s[i,])
    mu.hat=smash.gaus(y[i,],sigma=sqrt(vars))#mu.hat is \mu_t+E(u_t|y)
    
    mu=rbind(mu,mu.hat)
    munorm[i]=norm(mu.hat-mu[i,],'2')
    if(munorm[i]<tol){
      break
    }
    #update m and s_t
    mt=exp(mu.hat)
    s=rbind(s,1/mt)
    y=rbind(y,log(mt)+(x-mt)/mt)
    
    
  }
  mu.hat=smash.gaus(y[i,],sigma = sqrt(sigma^2+ifelse(s[i,]<0,1e-8,s[i,])))
  return(list(mu.hat=mu.hat,mu=mu,s=s,y=y,munorm=munorm))
}

#' Simulation study comparing smash and smashgen

simu_study=function(m,sigma,seed=1234,
                    niter=30,family='DaubExPhase',tol=1e-2,
                    reflect=FALSE,robust=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=reflect)
  smash.gen.out=smash.gen(x,sigma=sigma,niter=niter,family = family,tol=tol,robust=robust)
  return(list(smash.out=smash.out,smash.gen.out=exp(smash.gen.out$mu.hat),smash.gen.est=smash.gen.out,x=x,loglik=smash.gen.out$loglik))
}

m=c(rep(3,128), rep(5, 128), rep(6, 128), rep(3, 128))


par(mfrow = c(1,2))
simu.out=simu_study(m,1,seed=2132)
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"))

simu.out=simu_study(m,1,seed=2132,robust = T)
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"))

#bumps
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))
}

par(mfrow = c(1,2))
simu.out=simu_study(f,1)
plot(simu.out$x,col = "gray80" ,ylab = '')
lines(simu.out$smash.gen.out, col = "red", lwd = 2)
lines(exp(f))
legend("topright", 
       c("truth","smash-gen"), 
       lty=c(1,1), 
       lwd=c(1,1),
       cex = 1,
       col=c("black","red", "blue"))
simu.out=simu_study(f,1,robust = T)
plot(simu.out$x,col = "gray80" ,ylab = '')
lines(simu.out$smash.gen.out, col = "red", lwd = 2)
lines(exp(f))
legend("topright", 
       c("truth","smash-gen"), 
       lty=c(1,1), 
       lwd=c(1,1),
       cex = 1,
       col=c("black","red", "blue"))

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] wavethresh_4.6.8 MASS_7.3-47      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] pscl_1.4.9          doParallel_1.0.11   SQUAREM_2017.10-1  
[10] lattice_0.20-35     foreach_1.4.3       ashr_2.2-7         
[13] stringr_1.3.0       caTools_1.17.1      tools_3.4.0        
[16] parallel_3.4.0      grid_3.4.0          data.table_1.10.4-3
[19] R.oo_1.21.0         git2r_0.21.0        iterators_1.0.8    
[22] htmltools_0.3.5     assertthat_0.2.0    yaml_2.1.19        
[25] rprojroot_1.3-2     digest_0.6.13       Matrix_1.2-9       
[28] bitops_1.0-6        codetools_0.2-15    R.utils_2.6.0      
[31] evaluate_0.10       rmarkdown_1.8       stringi_1.1.6      
[34] compiler_3.4.0      Rmosek_8.0.69       backports_1.0.5    
[37] R.methodsS3_1.7.1   truncnorm_1.0-7    

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