VGAM (version 1.0-2)

cardioid: Cardioid Distribution Family Function

Description

Estimates the two parameters of the cardioid distribution by maximum likelihood estimation.

Usage

cardioid(lmu = extlogit(min = 0, max = 2*pi), lrho = extlogit(min = -0.5, max = 0.5), imu = NULL, irho = 0.3, nsimEIM = 100, zero = NULL)

Arguments

lmu, lrho

Parameter link functions applied to the $mu$ and $rho$ parameters, respectively. See Links for more choices.

imu, irho

Initial values. A NULL means an initial value is chosen internally. See CommonVGAMffArguments for more information.

nsimEIM, zero

See CommonVGAMffArguments for more information.

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, rrvglm and vgam.

Warning

Numerically, this distribution can be difficult to fit because of a log-likelihood having multiple maximums. The user is therefore encouraged to try different starting values, i.e., make use of imu and irho.

Details

The two-parameter cardioid distribution has a density that can be written as $$f(y;\mu,\rho) = \frac{1}{2\pi} \left(1 + 2\, \rho \cos(y - \mu) \right) $$ where $0 < y < 2*pi$, $0 < mu < 2*pi$, and $-0.5 < rho < 0.5$ is the concentration parameter. The default link functions enforce the range constraints of the parameters.

For positive $\rho$ the distribution is unimodal and symmetric about $mu$. The mean of $Y$ (which make up the fitted values) is $ pi + (rho/pi) ((2*pi-mu)*sin(2*pi-mu) + cos(2*pi-mu) - mu*sin(mu) - cos(mu))$.

References

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

See Also

rcard, extlogit, vonmises.

CircStats and circular currently have a lot more R functions for circular data than the VGAM package.

Examples

## Not run: 
# cdata <- data.frame(y = rcard(n = 1000, mu = 4, rho = 0.45))
# fit <- vglm(y ~ 1, cardioid, data = cdata, trace = TRUE)
# coef(fit, matrix=TRUE)
# Coef(fit)
# c(with(cdata, mean(y)), head(fitted(fit), 1))
# summary(fit)
# ## End(Not run)