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\name{KatoJones}
\alias{rkatojones}
\alias{dkatojones}
\title{Kato and Jones Density Function}
\description{Density and random generation for the Kato and Jones distribution.}
\usage{
rkatojones(n, mu, nu, r, kappa, control.circular=list())
dkatojones(x, mu, nu, r, kappa)
}
\arguments{
\item{x}{the angular value the density must be computed in.}
\item{n}{number of observations.}
\item{mu}{the Mobius 'mu' parameter. The object is coerced to class \code{\link{circular}}.}
\item{nu}{the Mobius 'nu' parameter. The object is coerced to class \code{\link{circular}}.}
\item{r}{the Mobius 'r' parameter. It must be in [0,1).}
\item{kappa}{the positive vonMises parameter.}
\item{control.circular}{the attribute of the resulting object.}
}
\details{The Kato and Jones distribution has density
\deqn{
f(x)= \frac{1-r^2}{2\pi\mathcal I_0(\kappa)}
\exp\left[
\frac{\kappa\{ \xi\cos(x-\eta)-2r\cos\nu \}}
{1+r^2-2r\cos(x -\gamma)}
\right]\\
\phantom{\exp[]} \times \frac1{1+r^2-2r\cos(x -\gamma)},
}{%
f(x)= [1-r^2]/[{2\pi\mathcal I_0(\kappa)}{1+r^2-2r\cos(x -\gamma)}]
exp[{\kappa\{ \xi\cos(x-\eta)-2r\cos\nu \}} / {1+r^2-2r\cos(x -\gamma)}],
}
for \eqn{0 \le x < 2\pi}{0 <= x < 2 \pi},
where \eqn{\gamma=\mu+\nu}, \eqn{\xi=\{r^4+2r^2\cos(2\nu)+1\}^{1/2}}{\xi={r^4+2r^2 cos(2\nu)+1}^{1/2}}
and \eqn{\eta=\mu+\arg[ r^2\{\cos(2\nu)+i\sin(2\nu)\}+1 ]}{\eta=\mu+arg[ r^2{cos(2\nu)+i sin(2\nu)}+1 ]}.
Original code for random generation is by Kato, S. and Jones, M.C. and can be found at the address
http://pubs.amstat.org/doi/suppl/10.1198/jasa.2009.tm08313/suppl_file/t08-313code.txt.
}
\value{The density.
\code{dkatojones} gives the density and \code{rkatojones} generates random deviates.
}
\references{Kato , S. and Jones, M.C. (2010). A family of distributions on the circle with links to, and applications arising from,
Mobius transformation. J. Am. Statist. Assoc. 105, 249-262.}
\author{Federico Rotolo}
\examples{
data1 <- rkatojones(n=100, mu=circular(0), nu=circular(pi/4), r=.2, kappa=1)
plot(data1)
data1 <- rkatojones(n=100, mu=circular(pi/3), nu=circular(pi), r=.7, kappa=2.3)
plot(data1)
ff <- function(x) dkatojones(x, mu=circular(pi/3), nu=circular(pi), r=.7, kappa=2.3)
curve.circular(ff, join=TRUE, xlim=c(-1, 1), ylim=c(-1.2, 1.2),
main="Density of a KatoJones Distribution",
xlab=expression(paste(mu,"=",pi,"/3, ",nu,"=",pi,", r=0.7, ",kappa,"=2.3"))
)
}
\keyword{distribution}
\keyword{circle}
\keyword{circular}
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