Last updated: 2018-05-17

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    Rmd 996d337 Dongyue 2018-05-14 poisson data unkown variance

Simulations of Poisson nugget effect(unkown).

Previously, we have studies the methods to estimate unknown \(\sigma\) in the model \(Y_t=\mu_t+N(0,\sigma^2)+N(0,s_t^2)\). Here, we apply mle and smashu methods and compare them with smash as well as smashgen with known \(sigma\). The measure of accuracy is mean square error. Plots are also given as visual aid.

library(smashrgen)
library(ggplot2)
#' Simulation study comparing smash and smashgen

simu_study=function(m,sigma,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()
  for(k in 1:nsimu){
    lamda=exp(m+rnorm(length(m),0,sigma))
    x=rpois(length(m),lamda)
    #fit data
    smash.out=smash.poiss(x)
    smashgen.out=smash_gen(x,dist_family = 'poisson',sigma = sigma)
    smashu.out=smash_gen(x,dist_family = 'poisson',y_var_est = 'smashu')
    mle.out=smash_gen(x,dist_family = 'poisson',y_var_est = 'mle')
    smash.err[k]=mse(exp(m),smash.out)
    smashgen.err[k]=mse(exp(m),smashgen.out)
    smashgen.smashu.err[k]=mse(exp(m),smashu.out)
    smashgen.mle.err[k]=mse(exp(m),mle.out)
  }
  return(list(est=list(smash.out=smash.out,smashgen.out=smashgen.out,smashu.out=smashu.out,mle.out=mle.out,x=x),err=data.frame(smash=smash.err,smashgen=smashgen.err,
              smashgen.smashu=smashgen.smashu.err,smashgen.mle=smashgen.mle.err)))
}

Simulation 1: Constant trend Poisson nugget

\(\sigma=0.1\)

m=rep(1,128)
result=simu_study(m,0.1)
par(mfrow=c(2,2))
plot(result$est$x,col='gray80',ylab='',main='smash')
lines(exp(m),col=1)
lines(result$est$smash.out,col=4)
plot(result$est$x,col='gray80',ylab='',main='smashgen: known sigma')
lines(exp(m),col=1)
lines(result$est$smashgen.out,col=4)
plot(result$est$x,col='gray80',ylab='',main='smashgen: (sigma^2+s_t^2) from smash')
lines(exp(m),col=1)
lines(result$est$smashu.out,col=4)
plot(result$est$x,col='gray80',ylab='',main='smashgen: sigma from mle')
lines(exp(m),col=1)
lines(result$est$mle.out,col=4)

mean(result$err$smash)
[1] 0.0318433
mean(result$err$smashgen)
[1] 0.02987159
mean(result$err$smashgen.smashu)
[1] 0.03133639
mean(result$err$smashgen.mle)
[1] 0.02987841
ggplot(df2gg(result$err),aes(x=method,y=MSE))+geom_boxplot(aes(fill=method))+labs(x='')

\(\sigma=1\)

m=rep(1,128)
result=simu_study(m,1)
par(mfrow=c(2,2))
plot(result$est$x,col='gray80',ylab='',main='smash')
lines(exp(m),col=1)
lines(result$est$smash.out,col=4)
plot(result$est$x,col='gray80',ylab='',main='smashgen: known sigma')
lines(exp(m),col=1)
lines(result$est$smashgen.out,col=4)
plot(result$est$x,col='gray80',ylab='',main='smashgen: (sigma^2+s_t^2) from smash')
lines(exp(m),col=1)
lines(result$est$smashu.out,col=4)
plot(result$est$x,col='gray80',ylab='',main='smashgen: sigma from mle')
lines(exp(m),col=1)
lines(result$est$mle.out,col=4)

mean(result$err$smash)
[1] 29.73513
mean(result$err$smashgen)
[1] 0.233901
mean(result$err$smashgen.smashu)
[1] 0.2236465
mean(result$err$smashgen.mle)
[1] 0.288378
#ggplot(df2gg(result$err),aes(x=method,y=MSE))+geom_boxplot(aes(fill=method))+labs(x='')

\(\sigma=2\)

m=rep(1,128)
result=simu_study(m,2)
par(mfrow=c(2,2))
plot(result$est$x,col='gray80',ylab='',main='smash')
lines(exp(m),col=1)
lines(result$est$smash.out,col=4)
plot(result$est$x,col='gray80',ylab='',main='smashgen: known sigma')
lines(exp(m),col=1)
lines(result$est$smashgen.out,col=4)
plot(result$est$x,col='gray80',ylab='',main='smashgen: (sigma^2+s_t^2) from smash')
lines(exp(m),col=1)
lines(result$est$smashu.out,col=4)
plot(result$est$x,col='gray80',ylab='',main='smashgen: sigma from mle')
lines(exp(m),col=1)
lines(result$est$mle.out,col=4)

mean(result$err$smash)
[1] 8372.27
mean(result$err$smashgen)
[1] 0.4991542
mean(result$err$smashgen.smashu)
[1] 0.5850626
mean(result$err$smashgen.mle)
[1] 0.5626063
#ggplot(df2gg(result$err),aes(x=method,y=MSE))+geom_boxplot(aes(fill=method))+labs(x='')

Simulation 2: Step trend

\(\sigma=0.1\)

m=c(rep(3,128), rep(5, 128), rep(6, 128), rep(3, 128))
result=simu_study(m,0.1)
par(mfrow=c(2,2))
plot(result$est$x,col='gray80',ylab='',main='smash')
lines(exp(m),col=1)
lines(result$est$smash.out,col=4)
plot(result$est$x,col='gray80',ylab='',main='smashgen: known sigma')
lines(exp(m),col=1)
lines(result$est$smashgen.out,col=4)
plot(result$est$x,col='gray80',ylab='',main='smashgen: (sigma^2+s_t^2) from smash')
lines(exp(m),col=1)
lines(result$est$smashu.out,col=4)
plot(result$est$x,col='gray80',ylab='',main='smashgen: sigma from mle')
lines(exp(m),col=1)
lines(result$est$mle.out,col=4)

mean(result$err$smash)
[1] 368.4722
mean(result$err$smashgen)
[1] 29.20088
mean(result$err$smashgen.smashu)
[1] 31.90063
mean(result$err$smashgen.mle)
[1] 36.42519
#ggplot(df2gg(result$err),aes(x=method,y=MSE))+geom_boxplot(aes(fill=method))+labs(x='')

\(\sigma=1\)

m=c(rep(3,128), rep(5, 128), rep(6, 128), rep(3, 128))
result=simu_study(m,1)
par(mfrow=c(2,2))
plot(result$est$x,col='gray80',ylab='',main='smash')
lines(exp(m),col=1)
lines(result$est$smash.out,col=4)
plot(result$est$x,col='gray80',ylab='',main='smashgen: known sigma')
lines(exp(m),col=1)
lines(result$est$smashgen.out,col=4)
plot(result$est$x,col='gray80',ylab='',main='smashgen: (sigma^2+s_t^2) from smash')
lines(exp(m),col=1)
lines(result$est$smashu.out,col=4)
plot(result$est$x,col='gray80',ylab='',main='smashgen: sigma from mle')
lines(exp(m),col=1)
lines(result$est$mle.out,col=4)

mean(result$err$smash)
[1] 223491.6
mean(result$err$smashgen)
[1] 1658.668
mean(result$err$smashgen.smashu)
[1] 1650.308
mean(result$err$smashgen.mle)
[1] 1660.029
#ggplot(df2gg(result$err),aes(x=method,y=MSE))+geom_boxplot(aes(fill=method))+labs(x='')

Simulation 3: 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))
}
m=f
result=simu_study(m,0.1)
par(mfrow=c(2,2))
plot(result$est$x,col='gray80',ylab='',main='smash')
lines(exp(m),col=1)
lines(result$est$smash.out,col=4)
plot(result$est$x,col='gray80',ylab='',main='smashgen: known sigma')
lines(exp(m),col=1)
lines(result$est$smashgen.out,col=4)
plot(result$est$x,col='gray80',ylab='',main='smashgen: (sigma^2+s_t^2) from smash')
lines(exp(m),col=1)
lines(result$est$smashu.out,col=4)
plot(result$est$x,col='gray80',ylab='',main='smashgen: sigma from mle')
lines(exp(m),col=1)
lines(result$est$mle.out,col=4)

mean(result$err$smash)
[1] 23.25972
mean(result$err$smashgen)
[1] 35.17066
mean(result$err$smashgen.smashu)
[1] 276.9639
mean(result$err$smashgen.mle)
[1] 36.6836
#ggplot(df2gg(result$err),aes(x=method,y=MSE))+geom_boxplot(aes(fill=method))+labs(x='')

\(\sigma=1\)

result=simu_study(m,1)
par(mfrow=c(2,2))
plot(result$est$x,col='gray80',ylab='',main='smash')
lines(exp(m),col=1)
lines(result$est$smash.out,col=4)
plot(result$est$x,col='gray80',ylab='',main='smashgen: known sigma')
lines(exp(m),col=1)
lines(result$est$smashgen.out,col=4)
plot(result$est$x,col='gray80',ylab='',main='smashgen: (sigma^2+s_t^2) from smash')
lines(exp(m),col=1)
lines(result$est$smashu.out,col=4)
plot(result$est$x,col='gray80',ylab='',main='smashgen: sigma from mle')
lines(exp(m),col=1)
lines(result$est$mle.out,col=4)

mean(result$err$smash)
[1] 6462.494
mean(result$err$smashgen)
[1] 480.1025
mean(result$err$smashgen.smashu)
[1] 471.428
mean(result$err$smashgen.mle)
[1] 856.6609
#ggplot(df2gg(result$err),aes(x=method,y=MSE))+geom_boxplot(aes(fill=method))+labs(x='')

Simulation 4: 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
m = spike.f(t)

result=simu_study(m,0.1)
par(mfrow=c(2,2))
plot(result$est$x,col='gray80',ylab='',main='smash')
lines(exp(m),col=1)
lines(result$est$smash.out,col=4)
plot(result$est$x,col='gray80',ylab='',main='smashgen: known sigma')
lines(exp(m),col=1)
lines(result$est$smashgen.out,col=4)
plot(result$est$x,col='gray80',ylab='',main='smashgen: (sigma^2+s_t^2) from smash')
lines(exp(m),col=1)
lines(result$est$smashu.out,col=4)
plot(result$est$x,col='gray80',ylab='',main='smashgen: sigma from mle')
lines(exp(m),col=1)
lines(result$est$mle.out,col=4)

mean(result$err$smash)
[1] 0.6011156
mean(result$err$smashgen)
[1] 12306.79
mean(result$err$smashgen.smashu)
[1] 2931.977
mean(result$err$smashgen.mle)
[1] 12630.5
#ggplot(df2gg(result$err),aes(x=method,y=MSE))+geom_boxplot(aes(fill=method))+labs(x='')

\(\sigma=1\)

result=simu_study(m,1)
par(mfrow=c(2,2))
plot(result$est$x,col='gray80',ylab='',main='smash')
lines(exp(m),col=1)
lines(result$est$smash.out,col=4)
plot(result$est$x,col='gray80',ylab='',main='smashgen: known sigma')
lines(exp(m),col=1)
lines(result$est$smashgen.out,col=4)
plot(result$est$x,col='gray80',ylab='',main='smashgen: (sigma^2+s_t^2) from smash')
lines(exp(m),col=1)
lines(result$est$smashu.out,col=4)
plot(result$est$x,col='gray80',ylab='',main='smashgen: sigma from mle')
lines(exp(m),col=1)
lines(result$est$mle.out,col=4)

mean(result$err$smash)
[1] 34.61266
mean(result$err$smashgen)
[1] 7991947410
mean(result$err$smashgen.smashu)
[1] 4418267884
mean(result$err$smashgen.mle)
[1] 13168337929
#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

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