Spikes demo

Set up environment

We begin by loading the ashr, smashr, scales, ggplot2 and cowplot packages, as well as some additional functions used in the code below.

library(ashr)
library(smashr)
library(scales)
library(ggplot2)
suppressMessages(library(cowplot))
source("../code/misc.R")

Define the “spikes” mean function

Variable mu.sp defines the mean signal.

n <- 1024
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.sp <- spike.f(t)
mu.sp <- (1 + mu.sp)/5

Simulate data

Simulate the noisy observations from the normal distribution with means that are given by the “spikes” function, and with spatially-structured variances.

pos <- c(0.1,0.13,0.15,0.23,0.25,0.4,0.44,0.65,0.76,0.78,0.81)
hgt <- 2.88/5 * c(4,-5,3,-4,5,-4.2,2.1,4.3,-3.1,2.1,-4.2)
sig.cb <- rep(0,length(t))
for (j in 1:length(pos)) 
  sig.cb <- sig.cb + (1 + sign(t - pos[j])) * (hgt[j]/2)
sig.cb[sig.cb < 0] <- 0
sig.cb <- 0.1 + (sig.cb - min(sig.cb))/max(sig.cb)
rsnr   <- sqrt(3)
sig.cb <- sig.cb/mean(sig.cb) * sd(mu.sp)/rsnr^2
x.sim  <- rnorm(n,mu.sp,sig.cb)

Plot the simulated data set.

par(cex.axis = 1,cex.lab = 1.25)
plot(mu.sp,type = 'l',ylim = c(-0.05,1),xlab = "position",
     ylab = "",lwd = 1.7,xaxp = c(0,1024,4),yaxp = c(0,1,4))
lines(mu.sp + 2*sig.cb,col = "darkorange",lty = 5,lwd = 1.8)
lines(mu.sp - 2*sig.cb,col = "darkorange",lty = 5,lwd = 1.8)
points(x.sim,cex = 0.7,pch = 16,col = "darkblue")

This plot shows the “spikes” mean function (the black line), +/- 2 standard deviations (the yellow lines), and the simulated data (the blue points).

Run SMASH and TI thresholding

Now we apply SMASH, as well as the translation invariant (TI) thresholding method, to the “spikes” data. Here we run the TI thresholding method twice—once when the variance is assumes to be constant (“homoskedastic”), and once when it is estimated using the RMAD algorithm. The first method yields very poor estimates, so they are not shown in the plots below.

The SMASH method is able to estimate the variance.

sig.est  <- sqrt(2/(3 * (n - 2)) *
              sum((1/2 * x.sim[1:(n-2)] - x.sim[2:(n-1)] + x.sim[3:n])^2/2))
mu.smash <- smash(x.sim,family = "DaubLeAsymm",filter.number = 8)
mu.ti.homo <- ti.thresh(x.sim,sigma = sig.est,family = "DaubLeAsymm",
                        filter.number = 8)
mu.ti    <- ti.thresh(x.sim,method = "rmad",family = "DaubLeAsymm",
                      filter.number = 8)

It should take at the very most a few minutes to run SMASH and TI thresholding.

Get the (true) wavelet coefficients and their variances.

wc.sim     <- titable(x.sim)$difftable
wc.var.sim <- titable(sig.cb^2)$sumtable
wc.true    <- titable(mu.sp)$difftable

Get the shrunken estimates of the wavelet coefficients.

wc.sim.shrunk <- vector("list",10)
wc.pres       <- vector("list",10)
for(j in 0:(log2(n) - 1)){
  wc.sim.shrunk[[j+1]] <-
    ash(wc.sim[j+2,],sqrt(wc.var.sim[j+2,]),prior = "nullbiased",
        pointmass = TRUE,mixsd = NULL,mixcompdist = "normal",
        gridmult = 2,df = NULL)$result
  wc.pres[[j+1]] <- 1/sqrt(wc.var.sim[j+2,])
}

Summarize results

Plot the distribution of the observed wavelet coefficients at a coarser scale (scale = 1, orange) and at a finer scale (scale = 7, dark blue).

par(cex.axis = 1,cex.lab = 1.25)
hist(wc.sim[4,],breaks = 2,xlab = "observed wavelet coefficients",
     xlim = c(-25,25),ylim = c(0,600),col = "darkblue",xaxp = c(-25,25,10),
     yaxp = c(0,600,6),main = "")
hist(wc.sim[10,],breaks = 40,add = TRUE,col = "darkorange")

Observe that the scale = 7 WCs (dark blue) are much more concentrated near zero because the signal is smoother at this finer scale.

Plot the observed wavelet coefficients (at scales 1 and 7 only) vs. the “shrunken” wavelet coefficients estimated by adaptive shrinkage.

par(cex.axis = 1,cex.lab = 1.25)
plot(c(),c(),xlab = "observed wavelet coefficients",
     ylab = "shrunken wavelet coefficients",
     xlim = c(-2.5,2.5),ylim = c(-2.5,2.5))
abline(0,1,lty = 1,col = "gray",lwd = 1)
points(wc.sim[10,],wc.sim.shrunk[[9]]$PosteriorMean,pch = 20,cex = 1,
       col = "darkorange")
points(wc.sim[4,],wc.sim.shrunk[[3]]$PosteriorMean,pch = 20,cex = 1,
       col = "darkblue")

Observe that the scale = 7 WCs (dark blue) are strongly shrunk toward zero, whereas the scale = 1 WCs (orange) are not shrunk nearly as much; SMASH infers from the data that the scale = 7 WCs are heavily concentrated around zero, and consequently SMASH shrinks them more strongly.

Plot the observed wavelet coefficients (at scale 7 only) vs. the “shrunken” wavelet coefficients estimated by adaptive shrinkage in order to see how the amount of shrinkage depends on the standard error (s.e.) in the observations.

wc.sig.3 <- 1/wc.pres[[3]]
p <- ggplot(data.frame(observed = wc.sim[4,],
                       shrunken = wc.sim.shrunk[[3]]$PosteriorMean,
                       se       = wc.sig.3),
            aes(x = observed,y = shrunken,col = se)) +
  geom_point(na.rm = TRUE) +
  xlim(c(-1,1)) +
  ylim(c(-1,1)) +
  scale_color_gradientn(colors = c("deepskyblue","darkblue")) +
  theme_cowplot()
print(p)

From this plot, it is evident that SMASH shrinks WCs differently depending on their precision; specifically, WCs that are less precise—i.e., higher standard error (s.e.)—are shrunk more strongly toward zero.

Finally, we plot the ground-truth signal (the mean function, drawn as a black line) and the signals recovered by TI thresholding (light blue line) and SMASH (the red line).

par(cex.axis = 1)
plot(mu.sp,type = "l",col = "black",lwd = 3,xlab = "position",ylab = "",
     ylim = c(-0.05,1),xaxp = c(0,1024,4),yaxp = c(0,1,4))
lines(mu.ti,col = "dodgerblue",lwd = 3)
lines(mu.smash,col = "orangered",lwd = 3)

In this example, the smoothed signal from SMASH is noticeably more accurate than using TI thresholding (in which the variance is estimated by running the median absolute deviation, or “RMAD”, method). The TI thresholding estimate shows notable artifacts.

Session information

sessionInfo()
# R version 3.4.3 (2017-11-30)
# 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/3.4/Resources/lib/libRblas.0.dylib
# LAPACK: /Library/Frameworks/R.framework/Versions/3.4/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] cowplot_0.9.3 ggplot2_3.0.0 scales_0.5.0  smashr_1.2-0  ashr_2.2-9   
# 
# loaded via a namespace (and not attached):
#  [1] wavethresh_4.6.8  tidyselect_0.2.4  purrr_0.2.5      
#  [4] lattice_0.20-35   Rmosek_8.0.69     colorspace_1.4-0 
#  [7] htmltools_0.3.6   yaml_2.2.0        rlang_0.2.1      
# [10] R.oo_1.21.0       pillar_1.2.1      glue_1.2.0       
# [13] withr_2.1.2       R.utils_2.6.0     REBayes_1.3      
# [16] bindrcpp_0.2.2    foreach_1.4.4     plyr_1.8.4       
# [19] bindr_0.1.1       stringr_1.3.0     munsell_0.4.3    
# [22] gtable_0.2.0      workflowr_1.1.1   R.methodsS3_1.7.1
# [25] caTools_1.17.1    codetools_0.2-15  evaluate_0.10.1  
# [28] labeling_0.3      knitr_1.20        pscl_1.5.2       
# [31] doParallel_1.0.11 parallel_3.4.3    Rcpp_0.12.18     
# [34] backports_1.1.2   truncnorm_1.0-8   digest_0.6.16    
# [37] stringi_1.1.7     dplyr_0.7.5       grid_3.4.3       
# [40] rprojroot_1.3-2   tools_3.4.3       bitops_1.0-6     
# [43] magrittr_1.5      lazyeval_0.2.1    tibble_1.4.2     
# [46] whisker_0.3-2     pkgconfig_2.0.1   MASS_7.3-48      
# [49] Matrix_1.2-12     SQUAREM_2017.10-1 data.table_1.11.4
# [52] mixSQP_0.1-17     assertthat_0.2.0  rmarkdown_1.9    
# [55] iterators_1.0.9   R6_2.2.2          git2r_0.21.0     
# [58] compiler_3.4.3