Smoothing Methods

N. Frerebeau

2024-07-26

library(alkahest)

## Simulate data
set.seed(12345)
x <- seq(-4, 4, length = 100)
y <- dnorm(x)
z <- y + rnorm(100, mean = 0, sd = 0.01) # Add some noise

## Plot raw data
plot(x, z, type = "l", xlab = "", ylab = "", las = 1)
lines(x, y, type = "l", lty = 2, col = "red")

Simulated data.

1. Rectangular smoothing

unweighted <- smooth_rectangular(x, z, m = 3)

par(mar = c(3, 3, 1, 1) + 0.1, las = 1)
layout(matrix(c(1, 2), nrow = 2, ncol = 1), heights = c(2, 1))

plot(unweighted, type = "l", xlab = "", ylab = "") 
lines(x, y, type = "l", lty = 2, col = "red")

plot(x, y - unweighted$y, ylim = c(-0.03, 0.03), type = "l", xlab = "", ylab = "")
abline(h = 0, lty = 2)

Rectangular smoothing.

2. Triangular smoothing

weighted <- smooth_triangular(x, z, m = 5)

par(mar = c(3, 3, 1, 1) + 0.1, las = 1)
layout(matrix(c(1, 2), nrow = 2, ncol = 1), heights = c(2, 1))

plot(weighted, type = "l", xlab = "", ylab = "")
lines(x, y, type = "l", lty = 2, col = "red")

plot(x, y - weighted$y, ylim = c(-0.03, 0.03), type = "l", xlab = "", ylab = "")
abline(h = 0, lty = 2)

Triangular smoothing.

3. Loess smoothing

loess <- smooth_loess(x, z, span = 0.2)

par(mar = c(3, 3, 1, 1) + 0.1, las = 1)
layout(matrix(c(1, 2), nrow = 2, ncol = 1), heights = c(2, 1))

plot(loess, type = "l", xlab = "", ylab = "")
lines(x, y, type = "l", lty = 2, col = "red")

plot(x, y - loess$y, ylim = c(-0.03, 0.03), type = "l", xlab = "", ylab = "")
abline(h = 0, lty = 2)

Loess smoothing.

4. Savitzky-Golay filter

savitzky <- smooth_savitzky(x, z, m = 21, p = 2)

par(mar = c(3, 3, 1, 1) + 0.1, las = 1)
layout(matrix(c(1, 2), nrow = 2, ncol = 1), heights = c(2, 1))

plot(savitzky, type = "l", xlab = "", ylab = "")
lines(x, y, type = "l", lty = 2, col = "red")

plot(x, y - savitzky$y, ylim = c(-0.03, 0.03), type = "l", xlab = "", ylab = "")
abline(h = 0, lty = 2)

Savitzky–Golay filter.

5. Whittaker smoothing

whittaker <- smooth_whittaker(x, z, lambda = 1000, d = 3, sparse = TRUE)

par(mar = c(3, 3, 1, 1) + 0.1, las = 1)
layout(matrix(c(1, 2), nrow = 2, ncol = 1), heights = c(2, 1))

plot(whittaker, type = "l", xlab = "", ylab = "")
lines(x, y, type = "l", lty = 2, col = "red")

plot(x, y - whittaker$y, ylim = c(-0.03, 0.03), type = "l", xlab = "", ylab = "")
abline(h = 0, lty = 2)

Whittaker smoothing.