# Highest Density Region (HDR) Levels Compute the levels of Highest Density Regions (HDRs) for any density and probability levels. ```r hdrlevels(density, prob) ``` ## Arguments - `density`: A vector of density values computed on a set of (observed) evaluation points. - `prob`: A vector of probability levels in the range $[0,1]$. ## Returns The function returns a vector of density values corresponding to HDRs at given probability levels. ## Details From Hyndman (1996), let $f(x)$ be the density function of a random variable $X$. Then the $100(1-\alpha)\%$ HDR is the subset $R(f_\alpha)$ of the sample space of $X$ such that $$ R(f_\alpha) = {x : f(x) \ge f_\alpha } $$ where $f_\alpha$ is the largest constant such that c("$\n$", "$Pr( X \\in R(f_\\alpha)) \\ge 1-\\alpha\n$") ## See Also `plot.densityMclust` ## References Rob J. Hyndman (1996) Computing and Graphing Highest Density Regions. **The American Statistician**, 50(2):120-126. ## Author(s) L. Scrucca ## Examples ```r # 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) ```

hdrlevels function