Estimates the two parameters of the cardioid distribution by maximum likelihood estimation.
cardioid(lmu = extlogitlink(min = 0, max = 2*pi), lrho = extlogitlink(min = -0.5, max = 0.5), imu = NULL, irho = 0.3, nsimEIM = 100, zero = NULL)
Parameter link functions applied to the mu
and rho parameters, respectively.
The two-parameter cardioid distribution has a density that can be written as
f(y;mu,rho) = (1 + 2*rho*cos(y-mu)) / (2*pi)
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 ρ 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)).
An object of class
The object is used by modelling functions such as
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
Fisher scoring using simulation is used.
T. W. Yee
Jammalamadaka, S. R. and SenGupta, A. (2001). Topics in Circular Statistics, Singapore: World Scientific.
CircStats and circular currently have a lot more R functions for circular data than the VGAM package.
## 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)
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