driftJp {sdetorus}

Drift for the JP diffusion

Description

Drift for the Langevin diffusion associated to the Jones and Pewsey (JP) family of circular distributions.

Usage

driftJp(x, alpha, mu, psi)

Arguments

x

vector with the evaluation points for the drift.

alpha

strength of the drift.

mu

unconditional mean of the diffusion.

psi

shape parameter, see details.

Details

Particular interesting choices for the shape parameter are:

• psi = -1: gives the Wrapped Cauchy as stationary density.

• psi = 0: is the sinusoidal drift of the vM diffusion.

• psi = 1: gives the Cardioid as stationary density.

See Section 2.2.3 in García-Portugués et al. (2019) for details.

Value

A vector of the same length as x containing the drift.

References

García-Portugués, E., Sørensen, M., Mardia, K. V. and Hamelryck, T. (2019) Langevin diffusions on the torus: estimation and applications. Statistics and Computing, 29(2):1–22. doi:10.1007/s11222-017-9790-2

Jones, M. C. and Pewsey, A. (2005). A family of symmetric distributions on the circle. Journal of the American Statistical Association, 100(472):1422–1428. doi:10.1198/016214505000000286

Examples

x <- seq(-pi, pi, l = 200)
plot(x, x, type = "n", ylab = "drift")
for (i in 0:20) {
  lines(x, driftJp(x = x, alpha = 1, mu = 0, psi = -1 + 2 * i / 20),
        col = rainbow(21)[i + 1])
}