Last updated: 2018-05-30

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File	Version	Author	Date	Message
Rmd	3ecb2bb	Dongyue	2018-05-30	covariate iterative

An iterative version of the algorithm described here. The iterative version is 1. estimate \(\mu\) and \(\sigma\) from residual using smash.gaus; 2. estimate \(\beta\) and set residual=\(Y-X\hat\beta\).

Experiments

We show the performance of smashgen(iterative version) when covariates exist. The signal-to-noise ratio(SNR) is fixed at 2. The length of sequence is \(n=256\).

simu_study_x=function(mu,beta,snr=2,nsimu=100,filter.number=1,family='DaubExPhase',seed=1234){
  set.seed(1234)
  n=length(mu)
  p=length(beta)
  X=matrix(rnorm(n*p,0,1),nrow=n,byrow = T)
  cte=X%*%beta
  sd.noise=sd(mu)/snr
  sd.noise=ifelse(sd.noise==0,mean(mu),sd.noise)
  mse.mu=c()
  mse.beta=c()
  mse.mu.iter=c()
  mse.beta.iter=c()
  for(i in 1:nsimu){
    y=cte+mu+rnorm(n,0,sd.noise)
    #########
    s.out=smash.gaus.x(X,y,filter.number=filter.number,family=family)
    mu.hat=s.out$mu.hat
    beta.hat=s.out$beta.hat
    mse.mu[i]=mse(mu,mu.hat)
    mse.beta[i]=mse(beta,beta.hat)
    #########
    s.out.iter=smash.gaus.x(X,y,filter.number=filter.number,family=family,niter=30)
    mu.hat.iter=s.out.iter$mu.hat
    beta.hat.iter=s.out.iter$beta.hat
    mse.mu.iter[i]=mse(mu,mu.hat.iter)
    mse.beta.iter[i]=mse(beta,beta.hat.iter)
  }
  return(list(est=list(mu.hat=mu.hat,beta.hat=beta.hat,y=y,mu.hat.iter=mu.hat.iter,beta.hat.iter=beta.hat.iter),err=data.frame(mse.mu=mse.mu,mse.mu.iter=mse.mu.iter,mse.beta=mse.beta,mse.beta.iter=mse.beta.iter)))
}

Step trend

library(smashrgen)
library(ggplot2)

n=256

mu=c(rep(1,64),rep(2,64),rep(5,64),rep(1,64))
beta=c(1,-2,-3,4,5)
beta=beta/norm(beta,'2')
result=simu_study_x(mu,beta)
par(mfrow=c(1,2))
boxplot(result$err[,1:2],main='Estimation of mu',ylab='MSE')
#boxplot(result$err$mse.mu.iter,main='Estimate of mu, iterative',ylab='MSE')

boxplot(result$err[,3:4],main='Estimation of beta',ylab='MSE')

#boxplot(result$err$mse.beta.iter,main='Estimate of beta, iterative',ylab='MSE')

plot(result$est$y,col='gray80',ylab = '',main='Estimated mu')
lines(mu)
lines(result$est$mu.hat,col=4)

plot(result$est$y,col='gray80',ylab = '',main='Estimated mu, iterative')
lines(mu)
lines(result$est$mu.hat.iter,col=4)

plot(beta,result$est$beta.hat,xlab = 'True beta', ylab = 'Beta hat',main='Beta')
abline(0,1)

plot(beta,result$est$beta.hat.iter,xlab = 'True beta', ylab = 'Beta hat',main='Beta, iterative')
abline(0,1)

Wave

f=function(x){return(0.5 + 0.2*cos(4*pi*x) + 0.1*cos(24*pi*x))}
mu=f((1:n)/n)

result=simu_study_x(mu,beta,filter.number = 8,family='DaubLeAsymm')
par(mfrow=c(1,2))
boxplot(result$err[,1:2],main='Estimation of mu',ylab='MSE')
#boxplot(result$err$mse.mu.iter,main='Estimate of mu, iterative',ylab='MSE')

boxplot(result$err[,3:4],main='Estimation of beta',ylab='MSE')

#boxplot(result$err$mse.beta.iter,main='Estimate of beta, iterative',ylab='MSE')

plot(result$est$y,col='gray80',ylab = '',main='Estimated mu')
lines(mu)
lines(result$est$mu.hat,col=4)

plot(result$est$y,col='gray80',ylab = '',main='Estimated mu, iterative')
lines(mu)
lines(result$est$mu.hat.iter,col=4)

plot(beta,result$est$beta.hat,xlab = 'True beta', ylab = 'Beta hat',main='Beta')
abline(0,1)

plot(beta,result$est$beta.hat.iter,xlab = 'True beta', ylab = 'Beta hat',main='Beta, iterative')
abline(0,1)

Parabola

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:n/n)

result=simu_study_x(mu,beta,filter.number = 8,family='DaubLeAsymm')
par(mfrow=c(1,2))
boxplot(result$err[,1:2],main='Estimation of mu',ylab='MSE')
#boxplot(result$err$mse.mu.iter,main='Estimate of mu, iterative',ylab='MSE')

boxplot(result$err[,3:4],main='Estimation of beta',ylab='MSE')

#boxplot(result$err$mse.beta.iter,main='Estimate of beta, iterative',ylab='MSE')

plot(result$est$y,col='gray80',ylab = '',main='Estimated mu')
lines(mu)
lines(result$est$mu.hat,col=4)

plot(result$est$y,col='gray80',ylab = '',main='Estimated mu, iterative')
lines(mu)
lines(result$est$mu.hat.iter,col=4)

plot(beta,result$est$beta.hat,xlab = 'True beta', ylab = 'Beta hat',main='Beta')
abline(0,1)

plot(beta,result$est$beta.hat.iter,xlab = 'True beta', ylab = 'Beta hat',main='Beta, iterative')
abline(0,1)

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    stringi_1.1.6      
[37] Rmosek_8.0.69       lazyeval_0.2.1      munsell_0.4.3      
[40] doParallel_1.0.11   pscl_1.4.9          truncnorm_1.0-7    
[43] SQUAREM_2017.10-1   R.oo_1.21.0

This reproducible R Markdown analysis was created with workflowr 1.0.1

Smoothing with covariates, iterative algorithm

Dongyue Xie

May 30, 2018

Experiments

Step trend

Wave

Parabola

Session information