Wrapped normal distribution

Density, distribution function, and random generation for the wrapped normal distribution.

Usage

dwrpnorm(x, mu = 0, sigma, log = FALSE)

pwrpnorm(q, mu = 0, sigma, from = NULL, lower.tail = TRUE, log.p = FALSE)

rwrpnorm(n, mu = 0, sigma)

Arguments

x, q

vector of angles in radians.

mu

mean direction in radians.

sigma

standard deviation of the normal distribution before wrapping, must be positive.

log, log.p

logical; if TRUE, probabilities/ densities \(p\) are returned as \(\log(p)\).

from

origin, in radians, at which the circle is cut open for the distribution function. If NULL (default), it is set to mu - pi.

lower.tail

logical; if TRUE (default), probabilities are \(P[X \le x]\), otherwise \(P[X > x]\).

n

number of random values to return.

Value

dwrpnorm gives the density, pwrpnorm gives the distribution function, and rwrpnorm generates random deviates in \([-\pi, \pi)\).

Details

This implementation allows for automatic differentiation with RTMB.

The wrapped normal is the distribution of \(Y \bmod 2\pi\) for a normal \(Y\) with mean \(\mu\) and standard deviation \(\sigma\), with density $$f(x;\,\mu,\sigma) = \sum_{k=-\infty}^{\infty} \frac{1}{\sigma}\,\phi\!\left(\frac{x - \mu + 2\pi k}{\sigma}\right) = \frac{1}{2\pi}\left(1 + 2\sum_{k=1}^{\infty} e^{-k^2\sigma^2/2} \cos(k(x - \mu))\right).$$ For \(\sigma < 2\) the first sum is used with \(|k| \le 3\), otherwise the second with \(k \le 4\); the terms left out are below \(10^{-20}\).

The distribution function cuts the circle open at \(\mu - \pi\) by default, or at from, exactly as pwrpcauchy does, where the choice of origin and the one-step-ahead (OSA) residuals are explained. As there, OSA residuals via RTMB::oneStepPredict with method = "cdf" cut the circle at the fixed origin \(-\pi\).

See also

wrpcauchy, vm, card; cjw() for circular-linear copulas.

Examples

set.seed(1)
x <- rwrpnorm(10, 0, 0.8)
d <- dwrpnorm(x, 0, 0.8)
p <- pwrpnorm(x, 0, 0.8)