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In this vignette, we assume a simple parametric model of the form

$$ y_i = \mu_i + \epsilon_i, $$

where $\epsilon_i \sim N(0,\sigma_i^2)$, and both $\mu$ and $\sigma$ are unknowns. We use the "smash" procedure (SMoothing via Adaptive SHrinkage) to estimate both the mean and the variances. Here we present a brief demonstration of the method.

We first look at mean estimation, which is our primary focus. A sample mean function is presented, as well as a couple of different variance functions. Our method is compared against a few other simple methods.





In [2]:



    


library(MASS)
library(smashr)
library(EbayesThresh)
library(wavethresh)



    


















    






WaveThresh: R wavelet software, release 4.6.8, installed

Copyright Guy Nason and others 1993-2016

Note: nlevels has been renamed to nlevelsWT


















In [3]:



    


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    <- 1024
t    <- 1:n/n
mu.s <- spike.f(t)



    











In [4]:



    


mse    <- function(x, y) mean((x - y)^2)
l2norm <- function(x) sum(x^2)
mise   <- function(x, y, r)
  10000 * mean(apply(x - rep(1, r) %o% y, 1, l2norm)/l2norm(y))
sig.est.func <- function(x, n)
  sqrt(2/(3 * (n - 2)) * sum((1/2 * x[1:(n - 2)] - x[2:(n - 1)] +
  1/2 * x[3:n])^2))



    











In [5]:



    


waveti.u <- function(x, filter.number = 10, family = "DaubLeAsymm",
                     min.level = 3, noise.level) {
    TT = length(x)
    thresh = noise.level * sqrt(2 * log(TT))
    x.w = wavethresh::wd(x, filter.number, family, type = "station")
    x.w.t = threshold(x.w, levels = (min.level):(x.w$nlevels - 1),
                      policy = "manual", value = thresh, type = "hard")
    x.w.t.r = AvBasis(convert(x.w.t))
    return(x.w.t.r)
}



    











In [6]:



    


waveti.ebayes <- function(x, filter.number = 10, family = "DaubLeAsymm",
                          min.level = 3, noise.level) {
    n = length(x)
    J = log2(n)
    x.w = wd(x, filter.number, family, type = "station")
    for (j in min.level:(J - 1)) {
        x.pm = ebayesthresh(accessD(x.w, j), sdev = noise.level)
        x.w = putD(x.w, j, x.pm)
    }
    mu.est = AvBasis(convert(x.w))
    return(mu.est)
}



    











In [7]:



    


mu.t <- (1 + mu.s)/5
rsnr <- sqrt(1)
var1 <- rep(1, n)
var2 <- (1e-04 + 4 * (exp(-550 * (t - 0.2)^2) + exp(-200 * (t - 0.5)^2) +
          exp(-950 * (t - 0.8)^2)))/1.35



    











In [ ]:



    


set.seed(327)
sigma.ini <- sqrt(var1)
sigma.t   <- sigma.ini/mean(sigma.ini) * sd(mu.t)/rsnr^2
X.s       <- matrix(rnorm(10 * n, mu.t, sigma.t), nrow = 10, byrow = TRUE)
mu.est    <- apply(X.s, 1, smash.gaus)
mu.est.tivar.ash <- apply(X.s, 1, ti.thresh, method = "smash")
mu.est.tivar.mad <- apply(X.s, 1, ti.thresh, method = "rmad")
mu.est.ti        <- matrix(0, 10, n)
mu.est.ti.ebayes <- matrix(0, 10, n)
for (i in 1:10) {
  sig.est = sig.est.func(X.s[i, ], n)
  mu.est.ti[i, ] = waveti.u(X.s[i, ], noise.level = sig.est)
  mu.est.ti.ebayes[i, ] = waveti.ebayes(X.s[i, ], noise.level = sig.est)
}



    


















    






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Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm
Due to absence of package REBayes, switching to EM algorithm

















In [ ]:



    


cat("SMASH:",mise(t(mu.est), mu.t, 10),"\n")
cat("TI thresholding with variance estimated from smash:",
    mise(t(mu.est.tivar.ash), mu.t, 10),"\n")
cat("TI thresholding with variance estimated from running MAD:",
    mise(t(mu.est.tivar.mad), mu.t, 10),"\n")
cat("TI thresholding with constant variance (estimated):",
	mise(mu.est.ti, mu.t, 10),"\n")
cat("EBayes with constant variance (estimated):",
	mise(mu.est.ti.ebayes, mu.t, 10),"\n")



    











In [ ]:

Content source: saketkc/notebooks

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