BetaDist: The beta distribution

In IPMbook: Functions and Data for the Book 'Integrated Population Models'

The beta distribution

Description

Density, distribution function, quantile function and random number generation for the beta distribution with parameters mean and sd OR mode and concentration. These are wrappers for stats::dbeta, etc. getBeta*Par returns the shape parameters.

Usage

dbeta2(x, mean, sd)
pbeta2(q, mean, sd, lower.tail=TRUE, log.p=FALSE)
qbeta2(p, mean, sd, lower.tail=TRUE, log.p=FALSE)
rbeta2(n, mean, sd)
getBeta2Par(mean, sd)

dbeta3(x, mode, concentration)
pbeta3(q, mode, concentration, lower.tail=TRUE, log.p=FALSE)
qbeta3(p, mode, concentration, lower.tail=TRUE, log.p=FALSE)
rbeta3(n, mode, concentration)
getBeta3Par(mode, concentration)

Arguments

x	vector of parameter values.
q	vector of quantiles.
p	vector of probabilities.
n	number of random draws required.
mean	mean of the beta distribution; cannot be 0 or 1.
sd	standard deviation of the beta distribution; this must be less than sqrt(mean * (1-mean)), larger values will return NA, with a warning.
mode	mode of the beta distribution; may be 0 or 1.
concentration	concentration of the beta distribution; concentration = 2 is uniform, and the distribution becomes narrower as concentration increases. It is sometimes referred to as 'sample size', but best thought of as sample size + 2.
lower.tail	logical; if TRUE (default), cumulative probabilities up to x, otherwise, above x.
log.p	logical; if TRUE, probabilities p are given as log(p).

Value

dbeta* gives the density, pbeta* gives the distribution function, qbeta* gives the quantile function, and rbeta* generates random deviates.

getBeta*Par returns a 2-column matrix with the shape parameters corresponding to mean and sd OR mode and concentration.

Author(s)

Mike Meredith

See Also

See the stats functions dbeta, pbeta, qbeta, rbeta.

Examples

# Plot some curves with dbeta2
xx <- seq(0, 1, length.out=101)
plot(xx, dbeta2(xx, 0.4, 0.12), xlab="x", ylab="Probability density",
  main="Beta curves with mean = 0.4", type='l', lwd=2)
lines(xx, dbeta2(xx, 0.4, 0.24), col='darkgreen', lwd=2)
lines(xx, dbeta2(xx, 0.4, 0.28), col='red', lwd=2)
lines(xx, dbeta2(xx, 0.4, 0.36), col='blue', lwd=2)
abline(v=0.4, lty=3, lwd=2)
legend('topright', paste("sd =", c(0.12,0.24, 0.28, 0.36)), lwd=2,
  col=c('black', 'darkgreen', 'red', 'blue'), bty='n')

# Get shape and rate parameters for mean = 0.4 and sd = c(0.12,0.24, 0.28, 0.36, 0.49)
# The last value for sd is too big and will produce NAs and a warning
getBeta2Par(mean = 0.4, sd = c(0.12,0.24, 0.28, 0.36, 0.49))

# The parameterisation with mean and sd doesn't seem intuitive,
#   let's try mode and concentration.
# This does not allow 'bathtub' curves, which are bimodal.
plot(xx, dbeta3(xx, 0.4, 16), xlab="x", ylab="Probability density",
  main="Beta curves with mode = 0.4", type='l', lwd=2)
lines(xx, dbeta3(xx, 0.4, 8), col='darkgreen', lwd=2)
lines(xx, dbeta3(xx, 0.4, 4), col='red', lwd=2)
lines(xx, dbeta3(xx, 0.4, 2), col='blue', lwd=2)
abline(v=0.4, lty=3, lwd=2)
legend('topright', , lwd=2, paste("concentration =", c(16, 8, 4, 2)),
  col=c('black', 'darkgreen', 'red', 'blue'), bty='n')

# The mode can be at 0 or 1:
plot(xx, dbeta3(xx, 1, 16), xlab="x", ylab="Probability density",
  main="Beta curves with mode = 1", type='l', lwd=2)
lines(xx, dbeta3(xx, 1, 8), col='darkgreen', lwd=2)
lines(xx, dbeta3(xx, 1, 4), col='red', lwd=2)
lines(xx, dbeta3(xx, 1, 2), col='blue', lwd=2)
legend('topleft', paste("concentration =", c(16, 8, 4, 2)), lwd=2,
  col=c('black', 'darkgreen', 'red', 'blue'), bty='n')

# Cumulative plots with pbeta3
plot(xx, pbeta3(xx, 0.4, 16), xlab="x", ylab="Cumulative probability",
  main="Beta curves with mode = 0.4", type='l', lwd=2)
lines(xx, pbeta3(xx, 0.4, 8), col='darkgreen', lwd=2)
lines(xx, pbeta3(xx, 0.4, 4), col='red', lwd=2)
lines(xx, pbeta3(xx, 0.4, 2), col='blue', lwd=2)
abline(v=0.4, lty=3, lwd=2)
legend('topleft', paste("concentration =", c(16, 8, 4, 2)), lwd=2,
  col=c('black', 'darkgreen', 'red', 'blue'), bty='n')

# Generate random draws and plot a histogram
rnd <- rbeta3(1e5, 0.4, 8)
hist(rnd, freq=FALSE)
# Add the curve:
lines(xx, dbeta3(xx, 0.4, 8), col='darkgreen', lwd=2)

# Get shape and rate parameters for mode = 0.4 and concentration = c(2, 4, 8, 16)
getBeta3Par(mode = 0.4, concentration = c(2, 4, 8, 16))

IPMbook documentation built on Aug. 22, 2023, 1:07 a.m.