VGAM (version 0.7-5)

cardioid: Cardioid Distribution Family Function

Description

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

Usage

cardioid(lmu="elogit", lrho="elogit",
         emu=if(lmu=="elogit") list(min=0, max=2*pi) else list(),
         erho=if(lmu=="elogit") list(min=-0.5, max=0.5) else list(),
         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.

emu, erho

List. Extra argument for each of the link functions. See earg in Links for general information.

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 maxima. 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, elogit, vonmises.

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

Examples

y = rcard(n=1000, mu=4, rho=0.45)
fit = vglm(y ~ 1, cardioid, trace=TRUE) 
coef(fit, matrix=TRUE)
Coef(fit)
c(mean(y), fitted(fit)[1,])
summary(fit)