Last updated: 2026-02-19

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Here we give two simple examples to illustrate the decorrelating (“biwhitening”) effect of the wavelet transform. These examples use synthetic signals not generated using wavelets.

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

library(wavethresh)
library(reshape)
library(ggplot2)
library(cowplot)

Set the seed for reproducibility:

set.seed(1)

1. Bumps example

Simulate a set of signals using the “bumps” test function from Donoho & Johnson, 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 was the first noisy signal looks like:

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
78ad4fd Peter Carbonetto 2026-02-19
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-hand plot).

Another way to illustrate the decorrelating effect of the DWT is to compute the sample correlations. Compare the sample correlations in the original data space to the sample correlations in the wavelet space:

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
78ad4fd Peter Carbonetto 2026-02-19
6bc1fb1 Peter Carbonetto 2025-12-19
0338444 Peter Carbonetto 2025-12-19

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

2. Gaussian process example

In the second example, we simulate a small data set from a Gaussian process (GP) with a Ornstein-Uhlenbeck (O-H) kernel:

n <- 2000
x <- seq(1,32)
d <- as.matrix(dist(x))
Sigma <- exp(-d/30)
Y <- mvrnorm(n,rep(0,32),Sigma)

In a GP model, the correlations between the points “decay” with distance:

pdat <- melt(cor(Y))
colnames(pdat) <- c("x","y","cor")
ggplot(pdat,aes(x = x,y = y,fill = cor)) +
  geom_tile(color = "white") +
  scale_fill_gradient(low = "white", high = "red") +
  theme_cowplot(font_size = 12)

Version Author Date
78ad4fd Peter Carbonetto 2026-02-19

Now perform the discrete wavelet transform on these data:

D <- t(apply(Y,1,function (x) wd(x)$D))

Examining the first signal, the autocorrelation is very strong in the original data space, and almost completely vanishes after applying the DWT:

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

Version Author Date
78ad4fd Peter Carbonetto 2026-02-19

Likewise, the original correlations are very strong, whereas the correlations among the WCs are mostly zero or close to zero:

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
78ad4fd Peter Carbonetto 2026-02-19 

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_4.0.1    reshape_0.8.9    wavethresh_4.7.2
# [5] 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     
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# [49] htmltools_0.5.8.1