Visualization of Gausian and Laplace noise

Gaussian noise: \(\mathcal{N}(\mu, \sigma^2)\)

Gaussian_noise_generator <- function(mu, sd, n.noise){
  # generate n noise points
  noise.points <- rnorm(n = n.noise, mean = mu, sd = sd)
  noise.density <- dnorm(noise.points, mean = mu, sd = sd)
  
  # plot Gaussian distribution
  Gaussian.x <- seq(-5, 5, length=100)
  Gaussian.y <- dnorm(x = Gaussian.x, mean = mu, sd = sd)
  
  Gaussian.plot <- ggplot(data = data.frame('x' = Gaussian.x, 'y' = Gaussian.y), aes(x, y)) + geom_line() + scale_x_continuous(limits = c(-5, 5), breaks = seq(-5, 5)) + theme_bw()
  noise.plot <- Gaussian.plot + geom_point(data=data.frame('x' = noise.points, 'y' = noise.density), aes(x, y), size = 3, alpha = 0.3)
  
  return(list(noise = noise.points,
              plot = noise.plot))
}

Generate Gaussian distributed noise data points Specifically, noise ~ \(\mathcal{N}(0, 1)\)

set.seed(1)
n = 300
Gaussian.noise <- Gaussian_noise_generator(mu = 0, sd = 1, n.noise = n)
#Gaussian.noise.points$noise

Plot noise

Gaussian.noise$plot + labs(x = 'x', y = 'p(x)')

Gaussian.noise$plot + scale_y_log10() + labs(x = 'x', y = 'log( p(x) )')

Laplace noise: \(Laplace(\mu, b)\)

rlaplace <- function (n = 1, m = 0, s = 1){
    if (any(s <= 0)) 
        stop("s must be positive")
    q <- runif(n)
    ifelse(q < 0.5, s * log(2 * q) + m, -s * log(2 * (1 - q)) + m)
}
dlaplace <- function (y, m = 0, s = 1, log = FALSE) {
    if (any(s <= 0)) 
        stop("s must be positive")
    density <- -abs(y - m)/s - log(2 * s)
    if (!log) 
        density <- exp(density)
    density
}
Laplace_noise_generator <- function(mu, b, n.noise){
  # generate n noise points
  noise.points <- rlaplace(n = n.noise, m = mu, s = b)
  noise.density <- dlaplace(noise.points, m = mu, s = b)
  
  # plot distribution
  background.distr.x <- seq(-5, 5, length=100)
  background.distr.y <- rmutil::dlaplace(background.distr.x, m = mu, s = b)
  
  background.plot <- ggplot(data = data.frame('x' = background.distr.x, 'y' = background.distr.y), aes(x, y)) + geom_line() + scale_x_continuous(limits = c(-5, 5), breaks = seq(-5, 5)) + theme_bw()
  noise.plot <- background.plot + geom_point(data=data.frame('x' = noise.points, 'y' = noise.density), aes(x, y), size = 3, alpha = 0.3)
  
  return(list(noise = noise.points,
              plot = noise.plot))
}

Generate Laplace distributed noise data points Specifically, noise ~ \(Laplace(0, 1)\)

set.seed(1)
n <- 300
Laplace.noise <- Laplace_noise_generator(mu = 0, b = 1, n.noise = n)
#Laplace.noise$noise

Plot noise

Laplace.noise$plot + labs(x = 'x', y = 'p(x)')

Laplace.noise$plot + scale_y_log10() + labs(x = 'x', y = 'log( p(x) )')