card: Cardioid distribution

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

Cardioid distribution

Description

Density, distribution function, and random generation for the cardioid distribution.

Usage

dcard(x, mu = 0, rho, log = FALSE)

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

rcard(n, mu = 0, rho)

Arguments

x, q	vector of angles in radians.
mu	mean direction in radians.
rho	concentration parameter, must be in the interval from 0 to 1/2.
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 cardioid distribution has density

f(x;\,\mu,\rho) = \frac{1 + 2\rho\cos(x - \mu)}{2\pi}, \qquad 0 \le \rho \le \frac{1}{2},

and is the circular uniform for \rho = 0. With the circle cut open at \mu - \pi, its distribution function is

F(q) = \frac{q - \mu + \pi}{2\pi} + \frac{\rho}{\pi}\sin(q - \mu), \quad \mu - \pi \le q < \mu + \pi.

The origin can be changed with from, exactly as for pwrpcauchy, 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

dcard gives the density, pcard gives the distribution function, and rcard generates random deviates in [-\pi, \pi).

References

Jammalamadaka, S. R. and SenGupta, A. (2001) Topics in Circular Statistics. World Scientific.

See Also

wrpcauchy, wrpnorm, vm

Examples

set.seed(1)
x <- rcard(10, 0, 0.3)
d <- dcard(x, 0, 0.3)
p <- pcard(x, 0, 0.3)

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