Last updated: 2025-12-19

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Knit directory: fsusie-experiments/analysis/

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Load the wavethresh package as well as a few other packages used this demo:

library(wavethresh)
# Loading required package: MASS
# WaveThresh: R wavelet software, release 4.7.2, installed
# Copyright Guy Nason and others 1993-2022
# Note: nlevels has been renamed to nlevelsWT
library(ggplot2)
library(cowplot)

Set the seed for reproducibility:

set.seed(1)

Simulate a set of signals with ARMA noise:

n <- 1024
m <- 100
SNR <- 2
v <- DJ.EX()
x <- seq(1,n)/n
ssig <- sd(v$bumps)
sigma <- ssig/SNR
Y <- matrix(0,m,n)
for (i in 1:m) {
  e <- arima.sim(n = n,model = list(ar = 0.99,ma = 1))
  e <- sigma*e/sqrt(var(e))
  y <- v$bumps + e
  Y[i,] <- y
}

This is the first noisy signal:

y <- Y[1,]
pdat <- data.frame(x = x,y = y)
ggplot(pdat,aes(x = x,y = y)) +
  geom_line() +
  theme_cowplot(font_size = 12)

Version Author Date
0338444 Peter Carbonetto 2025-12-19

Now let’s compute the discrete wavelet transform (DWT) for each signal:

D <- matrix(0,m,n-1)
for (i in 1:m) {
  y <- Y[i,]
  D[i,] <- wd(y)$D
}

One way to illustrate the decorrelating effect of the wavelet transform is to compare the autocorrelation for the original signal and for the wavelet coefficients (WCs). Let’s try this on the first noisy signal:

par(mfrow = c(1,2))
y <- Y[1,]
d <- D[1,]
acf(y)
acf(d)

Version Author Date
0338444 Peter Carbonetto 2025-12-19

Observe that the autocorrelation is mostly near zero for the WCs (right).

Another way to illustrate the decorrelating effect of the DWT is to compute the sample correlations:

cY <- cor(Y)
cD <- cor(D)
p1 <- ggplot(data.frame(x = cY[upper.tri(cY)]),aes(x = x)) +
  geom_histogram(fill = "black",color = "white",bins = 64) +
  xlim(-1,1) +
  labs(x = "correlation",title = "data space") +
  theme_cowplot(font_size = 12)
p2 <- ggplot(data.frame(x = cD[upper.tri(cD)]),aes(x = x)) +
  geom_histogram(fill = "black",color = "white",bins = 64) +
  xlim(-1,1) +
  labs(x = "correlation",title = "wavelet space") +
  theme_cowplot(font_size = 12)
plot_grid(p1,p2,nrow = 2,ncol = 1)

Version Author Date
0338444 Peter Carbonetto 2025-12-19

Indeed, the strong correlations in the original data space are removed in the wavelet space.


sessionInfo()
# R version 4.3.3 (2024-02-29)
# Platform: aarch64-apple-darwin20 (64-bit)
# Running under: macOS 15.7.1
# 
# Matrix products: default
# BLAS:   /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRblas.0.dylib 
# LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.11.0
# 
# locale:
# [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
# 
# time zone: America/Chicago
# tzcode source: internal
# 
# attached base packages:
# [1] stats     graphics  grDevices utils     datasets  methods   base     
# 
# other attached packages:
# [1] cowplot_1.1.3    ggplot2_3.5.2    wavethresh_4.7.2 MASS_7.3-60.0.1 
# 
# loaded via a namespace (and not attached):
#  [1] gtable_0.3.6       jsonlite_2.0.0     dplyr_1.1.4        compiler_4.3.3    
#  [5] promises_1.3.3     tidyselect_1.2.1   Rcpp_1.1.0         stringr_1.5.1     
#  [9] git2r_0.33.0       dichromat_2.0-0.1  later_1.4.2        jquerylib_0.1.4   
# [13] scales_1.4.0       yaml_2.3.10        fastmap_1.2.0      R6_2.6.1          
# [17] labeling_0.4.3     generics_0.1.4     workflowr_1.7.1    knitr_1.50        
# [21] tibble_3.3.0       rprojroot_2.0.4    RColorBrewer_1.1-3 bslib_0.9.0       
# [25] pillar_1.11.0      rlang_1.1.6        cachem_1.1.0       stringi_1.8.7     
# [29] httpuv_1.6.14      xfun_0.52          fs_1.6.6           sass_0.4.10       
# [33] cli_3.6.5          withr_3.0.2        magrittr_2.0.3     digest_0.6.37     
# [37] grid_4.3.3         lifecycle_1.0.4    vctrs_0.6.5        evaluate_1.0.4    
# [41] glue_1.8.0         farver_2.1.2       whisker_0.4.1      rmarkdown_2.29    
# [45] tools_4.3.3        pkgconfig_2.0.3    htmltools_0.5.8.1