Simulating observations in the Lighthouse problem in R

cumdensity.svg

density.svg

lighthouse.R

library(ggplot2)

# Metaparameters.
d  <- 5   # Distance of the lighthouse from the coast.
x0 <- 4   # Perpendicular position of the lighthouse along the coast.
set.seed(1987)

# Simulate the data.
N <- 1e4
phi <- runif(N, -pi / 2, pi / 2)
x <- d * tan(phi) + x0
df_sims <- tibble(x = x)

# Plot the distribution.
ggplot(df_sims, aes(x, after_stat(density))) +
  geom_density(colour = "sienna3", linewidth = 1) +
  geom_histogram(binwidth = 1, center = .5, fill = "slategrey", colour = "white") +
  geom_vline(xintercept = x0, colour = "goldenrod1", linetype = 2, linewidth = .8) +
  xlim(-40, 50) +
  theme_minimal()

# Distribution of absolute deviations from the median.
absolute_deviation <- abs(x - median(x))
mad_obs <- median(absolute_deviation)
df_absdev <- tibble(absolute_deviation = absolute_deviation)

ggplot(df_absdev, aes(absolute_deviation, after_stat(density))) +
  geom_histogram(binwidth = 1, center = .5, fill = "sienna3", colour = "white") +
  geom_vline(xintercept = mad_obs, colour = "slategrey", linetype = 2, linewidth = 1) +
  annotate("text", x = mad_obs + 1.3, y = 0.114,
           label = "Median absolute deviation",
           colour = "slategrey", size = 3.5, hjust = 0) +
  xlim(0, 60) +
  theme_minimal()

ggplot(df_absdev, aes(absolute_deviation, after_stat(y))) +
  stat_ecdf(colour = 'sienna3') +
  geom_vline(xintercept = mad_obs, colour = "slategrey", linetype = 2, linewidth = 1) +
  annotate("text", x = mad_obs + 1.3, y = 0.114,
           label = "Median absolute deviation",
           colour = "slategrey", size = 3.5, hjust = 0) +
  scale_x_log10() +
  ylab("Cumulative density") +
  theme_minimal()

mad.svg