Highest Density Region (HDR) Levels
Compute the levels of Highest Density Regions (HDRs) for any density and probability levels.
hdrlevels(density, prob)
density |
A vector of density values computed on a set of (observed) evaluation points. |
prob |
A vector of probability levels in the range [0,1]. |
From Hyndman (1996), let f(x) be the density function of a random variable X. Then the 100(1-α)\% HDR is the subset R(f_α) of the sample space of X such that
R(f_α) = {x : f(x) ≥ f_α }
where f_α is the largest constant such that Pr( X \in R(f_α)) ≥ 1-α
The function returns a vector of density values corresponding to HDRs at given probability levels.
L. Scrucca
Rob J. Hyndman (1996) Computing and Graphing Highest Density Regions. The American Statistician, 50(2):120-126.
# Example: univariate Gaussian
x <- rnorm(1000)
f <- dnorm(x)
a <- c(0.5, 0.25, 0.1)
(f_a <- hdrlevels(f, prob = 1-a))
plot(x, f)
abline(h = f_a, lty = 2)
text(max(x), f_a, labels = paste0("f_", a), pos = 3)
mean(f > f_a[1])
range(x[which(f > f_a[1])])
qnorm(1-a[1]/2)
mean(f > f_a[2])
range(x[which(f > f_a[2])])
qnorm(1-a[2]/2)
mean(f > f_a[3])
range(x[which(f > f_a[3])])
qnorm(1-a[3]/2)
# Example 2: univariate Gaussian mixture
set.seed(1)
cl <- sample(1:2, size = 1000, prob = c(0.7, 0.3), replace = TRUE)
x <- ifelse(cl == 1,
rnorm(1000, mean = 0, sd = 1),
rnorm(1000, mean = 4, sd = 1))
f <- 0.7*dnorm(x, mean = 0, sd = 1) + 0.3*dnorm(x, mean = 4, sd = 1)
a <- 0.25
(f_a <- hdrlevels(f, prob = 1-a))
plot(x, f)
abline(h = f_a, lty = 2)
text(max(x), f_a, labels = paste0("f_", a), pos = 3)
mean(f > f_a)
# find the regions of HDR
ord <- order(x)
f <- f[ord]
x <- x[ord]
x_a <- x[f > f_a]
j <- which.max(diff(x_a))
region1 <- x_a[c(1,j)]
region2 <- x_a[c(j+1,length(x_a))]
plot(x, f, type = "l")
abline(h = f_a, lty = 2)
abline(v = region1, lty = 3, col = 2)
abline(v = region2, lty = 3, col = 3)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.