Last updated: 2018-10-02

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    Rmd ba36f40 Dongyue Xie 2018-10-02 smash robust

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')

# 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))

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)

mean(sigma.t)
mean(sqrt(smash.gaus(y,joint=T)$var.res))

Decrease n:

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))

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)

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 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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