Kato and Jones Density Function
Density and random generation for the Kato and Jones distribution.
rkatojones(n, mu, nu, r, kappa, control.circular=list()) dkatojones(x, mu, nu, r, kappa)
x |
the angular value the density must be computed in. |
n |
number of observations. |
mu |
the Mobius 'mu' parameter. The object is coerced to class |
nu |
the Mobius 'nu' parameter. The object is coerced to class |
r |
the Mobius 'r' parameter. It must be in [0,1). |
kappa |
the positive vonMises parameter. |
control.circular |
the attribute of the resulting object. |
The Kato and Jones distribution has density
f(x)= [1-r^2]/[{2π\mathcal I_0(κ)}{1+r^2-2r\cos(x -γ)}] exp[{κ{ ξ\cos(x-η)-2r\cosν }} / {1+r^2-2r\cos(x -γ)}],
for 0 <= x < 2 π, where γ=μ+ν, ξ={r^4+2r^2 cos(2ν)+1}^{1/2} and η=μ+arg[ r^2{cos(2ν)+i sin(2ν)}+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.
The density.
dkatojones gives the density and rkatojones generates random deviates.
Federico Rotolo
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.
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")) )
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