wrpnorm: Wrapped normal distribution

In RTMBdist: Distributions Compatible with Automatic Differentiation by 'RTMB'

Wrapped normal distribution

Description

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.

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.

Value

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

RTMBdist documentation built on Oct. 31, 2026, 5:07 p.m.