Last updated: 2019-02-27

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Introduction

Consider estimating spatially-structured \(\mu_t\) from Poisson sequence: \[Y_t\sim Poisson(\lambda_t).\] We assume that \(\lambda_t\) satisfies \[\log(\lambda_t)=X_t'\beta+\mu_t\], where \(X_t\) are \(p-\)dimensional covaraites and \(\beta\) is unknown coefficients.

For Bionomial sequence: \[Y_t\sim Binomial(n_t,p_t).\] Assume that \[logit(p_t)=X_t'\beta+\mu_t\], where \(\mu_t\) has smooth structure.

Method

Let \(\tilde\lambda_t=Y_t\). Define \(\tilde Y_t=\log(\tilde\lambda_t)+\frac{Y_t-\tilde\lambda_t}{\tilde\lambda_t}\) and apply smash.gaus allowing covariates method to \(\tilde Y_t\), which gives \(\hat\mu_t\) and \(\hat\beta\). The recovered smooth mean structure is given by \(\exp(\hat\mu_t)\).

Similarly to Binomial data, \(\tilde Y_t=logit(\tilde p_t)+\frac{Y_t/n_t-\tilde p_t}{\tilde p_t(1-\tilde p_t)}\).

Simulations

Data generation

Poisson sequence: Given \(\mu_t, t=1,2,\dots,T\), \(\lambda_t=\exp(\mu_t+X_t'\beta+N(0,\sigma^2))\), generate \(Y_t\sim Poisson(\lambda_t)\).

Bionimial sequence(\(p\)): Given \(\mu_t, t=1,2,\dots,T\),\(p_t=logit(\mu_t+X_t'\beta+N(0,\sigma^2))\), generate \(Y_t\sim Binomial(n_t,p_t)\), where \(n_t\) is given.

For each case, we run 3 times of simulation and plot the fitted curve.

Poisson Seq

The length of sequence \(T\) is set to be 256, covariates \(X_t\) are generate from \(N(0,I_{p\times p})\), and \(\beta\) is chosen to be \((1,2,-3,-4,5)\) then normalized to have unit norm.

simu_study_poix=function(mu,beta,snr=3,nsimu=3,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)
  Xbeta=X%*%beta
  mu.est=c()
  beta.est=c()
  y.data=c()
  sigma=sqrt(var(Xbeta)/snr)
  for(s in 1:nsimu){
    lambda=exp(mu+Xbeta+rnorm(n,0,sigma))
    yt=rpois(n,lambda)
    fit=smash_gen_x_lite(yt,X,wave_family = family,filter.number = filter.number,dist_family = 'poisson')
    mu.est=rbind(mu.est,fit$mu.est)
    beta.est=rbind(beta.est,fit$beta.est)
    y.data=rbind(y.data,yt)
  }
  return(list(mu.est=mu.est,beta.est=beta.est,y=y.data,X=X,sigma=sigma))
}
beta=c(1,2,-3,-4,5)
beta=beta/norm(beta,'2')

Step

mu=c(rep(2,64), rep(5, 64), rep(6, 64), rep(2, 64))
result=simu_study_poix(mu,beta)
plot(result$mu.est[1,],type='l',col=2,ylab = '',main='Estimated mean function(mu), 3 runs')
lines(result$mu.est[2,],col=3)
lines(result$mu.est[3,],col=4)
lines(exp(mu),lty=2)

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Version Author Date
c039b80 Dongyue 2018-05-28 

plot(log(result$mu.est[1,]),type='l',col=2,ylab = '',main='Estimated mean function(log mu), 3 runs')
lines(log(result$mu.est[2,]),col=3)
lines(log(result$mu.est[3,]),col=4)
lines(mu,lty=2)

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Version Author Date
c039b80 Dongyue 2018-05-28 

plot(beta,result$beta.est[1,],col=2,pch=1,xlab = 'True beta',ylab = 'Estimated beta')
lines(beta,result$beta.est[2,],col=3,pch=2,type='p')
lines(beta,result$beta.est[3,],col=4,pch=3,type='p')
abline(0,1,lty=2)

Expand here to see past versions of unnamed-chunk-2-3.png:
Version Author Date
c039b80 Dongyue 2018-05-28

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))
}
mu=f*1.2

result=simu_study_poix(mu,beta)
plot(exp(mu),lty=2,ylab = '',main='Estimated mean function(mu), 3 runs',type = 'l')
lines(result$mu.est[1,],type='l',col=2)
lines(result$mu.est[2,],col=3)
lines(result$mu.est[3,],col=4)

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Version Author Date
c039b80 Dongyue 2018-05-28 

plot(mu,lty=2,ylab = '',main='Estimated mean function(log mu), 3 runs',type = 'l')
lines(log(result$mu.est[1,]),type='l',col=2)
lines(log(result$mu.est[2,]),col=3)
lines(log(result$mu.est[3,]),col=4)

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Version Author Date
c039b80 Dongyue 2018-05-28 

plot(beta,result$beta.est[3,],col=4,pch=3,xlab = 'True beta',ylab = 'Estimated beta')
lines(beta,result$beta.est[2,],col=3,pch=2,type='p')
lines(beta,result$beta.est[1,],col=2,pch=1,type='p')
abline(0,1,lty=2)

Expand here to see past versions of unnamed-chunk-3-3.png:
Version Author Date
c039b80 Dongyue 2018-05-28

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(1:256/256)
mu=mu*7

result=simu_study_poix(mu,beta,filter.number = 8,family = 'DaubLeAsymm')
plot(result$mu.est[1,],type='l',col=2,ylab = '',main='Estimated mean function(mu), 3 runs')
lines(result$mu.est[2,],col=3)
lines(result$mu.est[3,],col=4)
lines(exp(mu),lty=2)

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Version Author Date
c039b80 Dongyue 2018-05-28 

plot(log(result$mu.est[1,]),type='l',col=2,ylab = '',main='Estimated mean function(log mu), 3 runs')
lines(log(result$mu.est[2,]),col=3)
lines(log(result$mu.est[3,]),col=4)
lines(mu,lty=2)

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Version Author Date
c039b80 Dongyue 2018-05-28 

plot(beta,result$beta.est[1,],col=2,pch=1,xlab = 'True beta',ylab = 'Estimated beta')
lines(beta,result$beta.est[2,],col=3,pch=2,type='p')
lines(beta,result$beta.est[3,],col=4,pch=3,type='p')
abline(0,1,lty=2)

Expand here to see past versions of unnamed-chunk-4-3.png:
Version Author Date
c039b80 Dongyue 2018-05-28

wave

f=function(x){return(0.5 + 2*cos(4*pi*x) + 2*cos(24*pi*x))}
mu=f(1:256/256)
mu=mu-min(mu)

result=simu_study_poix(mu,beta,filter.number = 8,family = 'DaubLeAsymm')
plot(exp(mu),lty=2,ylab = '',main='Estimated mean function(mu), 3 runs',type = 'l')
lines(result$mu.est[1,],type='l',col=2)
lines(result$mu.est[2,],col=3)
lines(result$mu.est[3,],col=4)

Expand here to see past versions of unnamed-chunk-5-1.png:
Version Author Date
c039b80 Dongyue 2018-05-28 

plot(mu,lty=2,ylab = '',main='Estimated mean function(log mu), 3 runs',type = 'l')
lines(log(result$mu.est[1,]),type='l',col=2)
lines(log(result$mu.est[2,]),col=3)
lines(log(result$mu.est[3,]),col=4)

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Version Author Date
c039b80 Dongyue 2018-05-28 

plot(beta,result$beta.est[1,],col=2,pch=1,xlab = 'True beta',ylab = 'Estimated beta')
lines(beta,result$beta.est[2,],col=3,pch=2,type='p')
lines(beta,result$beta.est[3,],col=4,pch=3,type='p')
abline(0,1,lty=2)

Expand here to see past versions of unnamed-chunk-5-3.png:
Version Author Date
c039b80 Dongyue 2018-05-28

Binomial Seq

The length of sequence \(T\) is set to be 256, covariates \(X_t\) are generate from \(N(0,I_{p\times p})\), and \(\beta\) is chosen to be \((1,2,-3,-4,5)\) then normalized to have unit norm. \(n_t\) is from Poisson(50).

Step

mu=c(rep(-3,64), rep(0, 64), rep(3, 64), rep(-3, 64))
result=simu_study_binomx(mu,beta,ntri)
plot(result$mu.est[1,],type='l',col=2,ylab = '',main='Estimated mean function(p), 3 runs')
lines(result$mu.est[2,],col=3)
lines(result$mu.est[3,],col=4)
lines(logistic(mu),lty=2)

Expand here to see past versions of unnamed-chunk-7-1.png:
Version Author Date
c039b80 Dongyue 2018-05-28 

plot(logit(result$mu.est[1,]),type='l',col=2,ylab = '',main='Estimated mean function(logit p), 3 runs')
lines(logit(result$mu.est[2,]),col=3)
lines(logit(result$mu.est[3,]),col=4)
lines(mu,lty=2)

Expand here to see past versions of unnamed-chunk-7-2.png:
Version Author Date
c039b80 Dongyue 2018-05-28 

plot(beta,result$beta.est[3,],col=4,pch=3,xlab = 'True beta',ylab = 'Estimated beta')
lines(beta,result$beta.est[2,],col=3,pch=2,type='p')
lines(beta,result$beta.est[1,],col=2,pch=1,type='p')
abline(0,1,lty=2)

Expand here to see past versions of unnamed-chunk-7-3.png:
Version Author Date
c039b80 Dongyue 2018-05-28

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))
}
mu=f-3

result=simu_study_binomx(mu,beta,ntri)
plot(logistic(mu),lty=2,type='l',ylab = '',main='Estimated mean function(p), 3 runs')
lines(result$mu.est[2,],col=3)
lines(result$mu.est[3,],col=4)
lines(result$mu.est[1,],col=2)

Expand here to see past versions of unnamed-chunk-8-1.png:
Version Author Date
c039b80 Dongyue 2018-05-28 

plot(mu,lty=2,type='l',ylab = '',main='Estimated mean function(logit p), 3 runs',ylim=c(-4,3))
lines(logit(result$mu.est[2,]),col=3)
lines(logit(result$mu.est[3,]),col=4)
lines(logit(result$mu.est[1,]),col=2)

Expand here to see past versions of unnamed-chunk-8-2.png:
Version Author Date
c039b80 Dongyue 2018-05-28 

plot(beta,result$beta.est[2,],col=3,pch=2,xlab = 'True beta',ylab = 'Estimated beta')
lines(beta,result$beta.est[3,],col=4,pch=3,type='p')
lines(beta,result$beta.est[1,],col=2,pch=1,type='p')
abline(0,1,lty=2)

Expand here to see past versions of unnamed-chunk-8-3.png:
Version Author Date
c039b80 Dongyue 2018-05-28

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(1:256/256)
mu=(mu-min(mu))*10-3

result=simu_study_binomx(mu,beta,ntri,filter.number = 8,family = 'DaubLeAsymm')
plot(result$mu.est[1,],type='l',col=2,ylab = '',main='Estimated mean function(p), 3 runs')
lines(result$mu.est[2,],col=3)
lines(result$mu.est[3,],col=4)
lines(logistic(mu),lty=2)

Expand here to see past versions of unnamed-chunk-9-1.png:
Version Author Date
c039b80 Dongyue 2018-05-28 

plot(logit(result$mu.est[1,]),type='l',col=2,ylab = '',main='Estimated mean function(logit p), 3 runs')
lines(logit(result$mu.est[2,]),col=3)
lines(logit(result$mu.est[3,]),col=4)
lines(mu,lty=2)

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Version Author Date
c039b80 Dongyue 2018-05-28 

plot(beta,result$beta.est[3,],col=4,pch=3,xlab = 'True beta',ylab = 'Estimated beta',ylim=c(-0.5,0.85))
lines(beta,result$beta.est[2,],col=3,pch=2,type='p')
lines(beta,result$beta.est[1,],col=2,pch=1,type='p')
abline(0,1,lty=2)

Expand here to see past versions of unnamed-chunk-9-3.png:
Version Author Date
c039b80 Dongyue 2018-05-28

wave

f=function(x){return(0.5 + 2*cos(4*pi*x) + 2*cos(24*pi*x))}
mu=f(1:256/256)

result=simu_study_binomx(mu,beta,ntri,filter.number = 8,family = 'DaubLeAsymm')
plot(logistic(mu),lty=2,type='l',ylab = '',main='Estimated mean function(p), 3 runs')
lines(result$mu.est[2,],col=3)
lines(result$mu.est[3,],col=4)
lines(result$mu.est[1,],col=2)

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Version Author Date
c039b80 Dongyue 2018-05-28 

plot(mu,lty=2,type='l',ylab = '',main='Estimated mean function(logit p), 3 runs')
lines(logit(result$mu.est[2,]),col=3)
lines(logit(result$mu.est[3,]),col=4)
lines(logit(result$mu.est[1,]),col=2)

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Version Author Date
c039b80 Dongyue 2018-05-28 

plot(beta,result$beta.est[2,],col=3,pch=2,xlab = 'True beta',ylab = 'Estimated beta',ylim=c(-0.7,0.6))
lines(beta,result$beta.est[3,],col=4,pch=3,type='p')
lines(beta,result$beta.est[1,],col=2,pch=1,type='p')
abline(0,1,lty=2)

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Version Author Date
c039b80 Dongyue 2018-05-28

Session information

sessionInfo()
R version 3.5.1 (2018-07-02)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.6

Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] smashrgen_0.1.0  wavethresh_4.6.8 MASS_7.3-51.1    caTools_1.17.1.1
[5] ashr_2.2-7       smashr_1.2-0    

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.0        compiler_3.5.1    git2r_0.23.0     
 [4] workflowr_1.1.1   R.methodsS3_1.7.1 R.utils_2.7.0    
 [7] bitops_1.0-6      iterators_1.0.10  tools_3.5.1      
[10] digest_0.6.18     evaluate_0.11     lattice_0.20-35  
[13] Matrix_1.2-14     foreach_1.4.4     yaml_2.2.0       
[16] parallel_3.5.1    stringr_1.3.1     knitr_1.20       
[19] REBayes_1.3       rprojroot_1.3-2   grid_3.5.1       
[22] data.table_1.11.6 rmarkdown_1.10    magrittr_1.5     
[25] whisker_0.3-2     backports_1.1.2   codetools_0.2-15 
[28] htmltools_0.3.6   assertthat_0.2.0  stringi_1.2.4    
[31] Rmosek_8.0.69     doParallel_1.0.14 pscl_1.5.2       
[34] truncnorm_1.0-8   SQUAREM_2017.10-1 R.oo_1.22.0

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