Mean and variance functions used to simulate Gaussian data

Set up environment

Load the ggplot2 and cowplot packages, and the functions definining the mean and variances used to simulate the data.

library(ggplot2)
library(cowplot)
source("../code/signals.R")

Generate the ground-truth signals

Here, n specifies the length of the signals.

n = 1024
t = 1:n/n

Some code.

mu.s = spike.f(t)

More code.

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 = 2.97/5 * c(4, 5, 3, 4, 5, 4.2, 2.1, 4.3, 3.1, 5.1, 4.2)
wth = c(0.005, 0.005, 0.006, 0.01, 0.01, 0.03, 0.01, 0.01, 0.005, 0.008, 0.005)
mu.b = rep(0, n)
for (j in 1:length(pos))
  mu.b = mu.b + hgt[j]/((1 + (abs(t - pos[j])/wth[j]))^4)

mu.dop     = dop.f(t)
mu.dop     = 3/(max(mu.dop) - min(mu.dop)) * (mu.dop - min(mu.dop))
mu.dop.var = 10 * dop.f(t)
mu.dop.var = mu.dop.var - min(mu.dop.var)

sig = ((2 * t + 0.5) * (t <= 0.15)) +
      ((-12 * (t - 0.15) + 0.8) * (t > 0.15 & t <= 0.2)) +
      0.2 * (t > 0.2 & t <= 0.5) + 
      ((6 * (t - 0.5) + 0.2) * (t > 0.5 & t <= 0.6)) +
      ((-10 * (t - 0.6) + 0.8) * (t > 0.6 & t <= 0.65)) +
      ((-0.5 * (t - 0.65) + 0.3) * (t > 0.65 & t <= 0.85)) +
      ((2 * (t - 0.85) + 0.2) * (t > 0.85))
mu.ang = 3/5 * ((5/(max(sig) - min(sig))) * sig - 1.6) - 0.0419569

heavi = 4 * sin(4 * pi * t) - sign(t - 0.3) - sign(0.72 - t)
mu.hs = heavi/sqrt(var(heavi)) * 1 * 2.99/3.366185
mu.hs = mu.hs - min(mu.hs)

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))
mu.blk = rep(0, n)
for (j in 1:length(pos))
  mu.blk = mu.blk + (1 + sign(t - pos[j])) * (hgt[j]/2)

mu.cblk = mu.blk
mu.cblk[mu.cblk < 0] = 0

Define the Blip mean function.

mu.blip = (0.32 + 0.6 * t +
           0.3 * exp(-100 * (t - 0.3)^2)) * (t >= 0 & t <= 0.8) +
  (-0.28 + 0.6 * t + 0.3 * exp(-100 * (t - 1.3)^2)) * (t > 0.8 & t <= 1)

Define the Corner mean function.

mu.cor = 623.87 * t^3 * (1 - 2 * t) * (t >= 0 & t <= 0.5) +
         187.161 * (0.125 - t^3) * t^4 * (t > 0.5 & t <= 0.8) +
         3708.470441 * (t - 1)^3 * (t > 0.8 & t <= 1)
mu.cor = (0.6/(max(mu.cor) - min(mu.cor))) * mu.cor
mu.cor = mu.cor - min(mu.cor) + 0.2

Define the rest of the mean functions.

mu.sp   = (1 + mu.s)/5
mu.bump = (1 + mu.b)/5
mu.blk  = 0.2 + 0.6 * (mu.blk - min(mu.blk))/max(mu.blk - min(mu.blk))
mu.ang  = (1 + mu.ang)/5
mu.dop  = (1 + mu.dop)/5

Define the variance functions.

var1 = rep(1, n)
var2 = (1e-02 + 4 * (exp(-550 * (t - 0.2)^2) +
                     exp(-200 * (t - 0.5)^2) +
                     exp(-950 * (t - 0.8)^2)))
var3 = (1e-02 + 2 * mu.dop.var)
var4 = 1e-02 + mu.b
var5 = 1e-02 + 1 * (mu.cblk - min(mu.cblk))/max(mu.cblk)
var1 = var1/2
var2 = var2/max(var2)
var3 = var3/max(var3)
var4 = var4/max(var4)
var5 = var5/max(var5)

Plot the signal means

These plots show each of the mean functions used to generate the simulated data sets.

plot_grid(qplot(t,mu.sp,  geom="line",xlab="",ylab="",main="Spikes (sp)"),
          qplot(t,mu.bump,geom="line",xlab="",ylab="",main="Bumps (bump)"),
          qplot(t,mu.blk, geom="line",xlab="",ylab="",main="Blocks (blk)"),
          qplot(t,mu.ang, geom="line",xlab="",ylab="",main="Angles (ang)"),
          qplot(t,mu.dop, geom="line",xlab="",ylab="",main="Doppler (dop)"),
          qplot(t,mu.blip,geom="line",xlab="",ylab="",main="Blip (blip)"),
          qplot(t,mu.cor, geom="line",xlab="",ylab="",main="Corner (cor)"),
          nrow = 4,ncol = 2)

Plot the signal variances

These are rescaled in the simulations to achieve the desired signal-to-noise ratios (see the paper for a more detailed explanation).

plot(t,var1,xlab = "",ylab = "",ylim = c(0,1),type = "l",
     main = "Constant variance function")

plot(t,var2,xlab = "",ylab = "",type = "l",
     main = "Triple exponential variance function")

plot(t,var3,xlab = "",ylab = "",type = "l",main = "Doppler variance function")

plot(t,var4,xlab = "",ylab = "",type = "l",main = "Bumps variance function")

plot(t,var5,xlab = "",ylab = "",type = "l",main = "Clipped variance function")

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_3.1.0

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.0        compiler_3.4.3    pillar_1.2.1     
 [4] git2r_0.23.0      plyr_1.8.4        workflowr_1.1.1  
 [7] bindr_0.1.1       R.methodsS3_1.7.1 R.utils_2.6.0    
[10] tools_3.4.3       digest_0.6.17     evaluate_0.11    
[13] tibble_1.4.2      gtable_0.2.0      pkgconfig_2.0.2  
[16] rlang_0.2.2       yaml_2.2.0        bindrcpp_0.2.2   
[19] withr_2.1.2       stringr_1.3.1     dplyr_0.7.6      
[22] knitr_1.20        rprojroot_1.3-2   grid_3.4.3       
[25] tidyselect_0.2.4  glue_1.3.0        R6_2.2.2         
[28] rmarkdown_1.10    purrr_0.2.5       magrittr_1.5     
[31] whisker_0.3-2     backports_1.1.2   scales_0.5.0     
[34] htmltools_0.3.6   assertthat_0.2.0  colorspace_1.4-0 
[37] labeling_0.3      stringi_1.2.4     lazyeval_0.2.1   
[40] munsell_0.4.3     R.oo_1.21.0