Last updated: 2018-10-02

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We see that smash.gaus still gives similar mean estimations given inaccurate variance(to ones given accurate variance) in previous section. We now try some more examples on this.

spike mean

library(smashr)
n = 2^9
t = 1:n/n
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))
mu.s = spike.f(t)

# Scale the signal to be between 0.2 and 0.8.
mu.t = (1 + mu.s)/5
plot(mu.t, type = "l",main='main function')

Expand here to see past versions of unnamed-chunk-1-1.png:
Version Author Date
e7afb4d Dongyue Xie 2018-10-02 

# Create the baseline variance function. (The function V2 from Cai &
# Wang 2008 is used here.)
var.fn = (1e-04 + 4 * (exp(-550 * (t - 0.2)^2) +
  exp(-200 * (t - 0.5)^2) + exp(-950 * (t - 0.8)^2)))/1.35+1
#plot(var.fn, type = "l")

# Set the signal-to-noise ratio.
rsnr = sqrt(3)
sigma.t = sqrt(var.fn)/mean(sqrt(var.fn)) * sd(mu.t)/rsnr^2
set.seed(12345)
y=mu.t+rnorm(n,0,sigma.t)
plot(y,col='grey80',main='n=512,SNR=3')
lines(smash.gaus(y))
lines(smash.gaus(y,mean(sigma.t)),col=2)
lines(smash.gaus(y,sigma.t),col=3)
lines(mu.t,col='grey80')
legend('topleft',c('smashu','smash.mean.sigma','smash.true.sigma','true mean'),col=c(1,2,3,'grey80'),lty=c(1,1,1,1))

Expand here to see past versions of unnamed-chunk-2-1.png:
Version Author Date
aacfc46 Dongyue Xie 2018-10-02 

plot(sqrt(smash.gaus(y,joint=T)$var.res),type='l',ylim=c(0.02,0.05),main='sigma')
lines(rep(mean(sigma.t),n),col=2)
lines(sigma.t,col=3)

Expand here to see past versions of unnamed-chunk-2-2.png:
Version Author Date
aacfc46 Dongyue Xie 2018-10-02 

mean(sigma.t)
[1] 0.03185823
mean(sqrt(smash.gaus(y,joint=T)$var.res))
[1] 0.02924347

Decrease n to 256:

n = 64
t = 1:n/n
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))
mu.s = spike.f(t)

# Scale the signal to be between 0.2 and 0.8.
mu.t = (1 + mu.s)/5
#plot(mu.t, type = "l",main='main function')

# Create the baseline variance function. (The function V2 from Cai &
# Wang 2008 is used here.)
var.fn = (1e-04 + 4 * (exp(-550 * (t - 0.2)^2) +
  exp(-200 * (t - 0.5)^2) + exp(-950 * (t - 0.8)^2)))/1.35+1
#plot(var.fn, type = "l")

# Set the signal-to-noise ratio.
rsnr = sqrt(3)
sigma.t = sqrt(var.fn)/mean(sqrt(var.fn)) * sd(mu.t)/rsnr^2

set.seed(12345)
y=mu.t+rnorm(n,0,sigma.t)
plot(y,col='grey80',main='n=256,SNR=3')
lines(smash.gaus(y))
lines(smash.gaus(y,mean(sigma.t)),col=2)
lines(smash.gaus(y,sigma.t),col=3)
lines(mu.t,col='grey80')
legend('topleft',c('smashu','smash.mean.sigma','smash.true.sigma','true mean'),col=c(1,2,3,'grey80'),lty=c(1,1,1,1))

Expand here to see past versions of unnamed-chunk-3-1.png:
Version Author Date
e7afb4d Dongyue Xie 2018-10-02 

plot(sqrt(smash.gaus(y,joint=T)$var.res),type='l',ylim=c(0.01,0.08),main='sigma')
lines(rep(mean(sigma.t),n),col=2)
lines(sigma.t,col=3)

Expand here to see past versions of unnamed-chunk-3-2.png:
Version Author Date
e7afb4d Dongyue Xie 2018-10-02 

mean(sigma.t)
[1] 0.03423118
mean(sqrt(smash.gaus(y,joint=T)$var.res))
[1] 0.07396342

Decrease n to 64:

n = 64
t = 1:n/n
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))
mu.s = spike.f(t)

# Scale the signal to be between 0.2 and 0.8.
mu.t = (1 + mu.s)/5
#plot(mu.t, type = "l",main='main function')

# Create the baseline variance function. (The function V2 from Cai &
# Wang 2008 is used here.)
var.fn = (1e-04 + 4 * (exp(-550 * (t - 0.2)^2) +
  exp(-200 * (t - 0.5)^2) + exp(-950 * (t - 0.8)^2)))/1.35+1
#plot(var.fn, type = "l")

# Set the signal-to-noise ratio.
rsnr = sqrt(3)
sigma.t = sqrt(var.fn)/mean(sqrt(var.fn)) * sd(mu.t)/rsnr^2

set.seed(12345)
y=mu.t+rnorm(n,0,sigma.t)
plot(y,col='grey80',main='n=64,SNR=3')
lines(smash.gaus(y))
lines(smash.gaus(y,mean(sigma.t)),col=2)
lines(smash.gaus(y,sigma.t),col=3)
lines(mu.t,col='grey80')
legend('topleft',c('smashu','smash.mean.sigma','smash.true.sigma','true mean'),col=c(1,2,3,'grey80'),lty=c(1,1,1,1))

Expand here to see past versions of unnamed-chunk-4-1.png:
Version Author Date
e7afb4d Dongyue Xie 2018-10-02 

plot(sqrt(smash.gaus(y,joint=T)$var.res),type='l',ylim=c(0.01,0.08),main='sigma')
lines(rep(mean(sigma.t),n),col=2)
lines(sigma.t,col=3)

Expand here to see past versions of unnamed-chunk-4-2.png:
Version Author Date
e7afb4d Dongyue Xie 2018-10-02 

mean(sigma.t)
[1] 0.03423118
mean(sqrt(smash.gaus(y,joint=T)$var.res))
[1] 0.07396342

Decrease SNR:

n = 512
t = 1:n/n
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))
mu.s = spike.f(t)

# Scale the signal to be between 0.2 and 0.8.
mu.t = (1 + mu.s)/5
#plot(mu.t, type = "l",main='main function')

# Create the baseline variance function. (The function V2 from Cai &
# Wang 2008 is used here.)
var.fn = (1e-04 + 4 * (exp(-550 * (t - 0.2)^2) +
  exp(-200 * (t - 0.5)^2) + exp(-950 * (t - 0.8)^2)))/1.35+1
#plot(var.fn, type = "l")

# Set the signal-to-noise ratio.
rsnr = sqrt(1)
sigma.t = sqrt(var.fn)/mean(sqrt(var.fn)) * sd(mu.t)/rsnr^2

set.seed(12345)
y=mu.t+rnorm(n,0,sigma.t)
plot(y,col='grey80',main='n=512,SNR=1')
lines(smash.gaus(y))
lines(smash.gaus(y,mean(sigma.t)),col=2)
lines(smash.gaus(y,sigma.t),col=3)
lines(mu.t,col='grey80')
legend('topleft',c('smashu','smash.mean.sigma','smash.true.sigma','true mean'),col=c(1,2,3,'grey80'),lty=c(1,1,1,1))

plot(sqrt(smash.gaus(y,joint=T)$var.res),type='l',ylim=c(0.07,0.15),main='sigma')
lines(rep(mean(sigma.t),n),col=2)
lines(sigma.t,col=3)

mean(sigma.t)
[1] 0.0955747
mean(sqrt(smash.gaus(y,joint=T)$var.res))
[1] 0.09278895

Summary

  1. smash.gaus is more sensitive to the scale of variance instead of the shape of variance.
  2. As long as n is large enough so that smash can give good estimation of scale of variance and roughly satifactory shape, then smash.gaus can still give similar mean estimation.

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] smashr_1.2-0

loaded via a namespace (and not attached):
 [1] Rcpp_0.12.18      knitr_1.20        whisker_0.3-2    
 [4] magrittr_1.5      workflowr_1.1.1   REBayes_1.3      
 [7] MASS_7.3-50       pscl_1.5.2        doParallel_1.0.14
[10] SQUAREM_2017.10-1 lattice_0.20-35   foreach_1.4.4    
[13] ashr_2.2-7        stringr_1.3.1     caTools_1.17.1.1 
[16] tools_3.5.1       parallel_3.5.1    grid_3.5.1       
[19] data.table_1.11.6 R.oo_1.22.0       git2r_0.23.0     
[22] iterators_1.0.10  htmltools_0.3.6   assertthat_0.2.0 
[25] yaml_2.2.0        rprojroot_1.3-2   digest_0.6.17    
[28] Matrix_1.2-14     bitops_1.0-6      codetools_0.2-15 
[31] R.utils_2.7.0     evaluate_0.11     rmarkdown_1.10   
[34] wavethresh_4.6.8  stringi_1.2.4     compiler_3.5.1   
[37] Rmosek_8.0.69     backports_1.1.2   R.methodsS3_1.7.1
[40] truncnorm_1.0-8

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