RandomFields (version 3.1.16)

RMmultiquad: The Multiquadric Family Covariance Model on th Sphere

Description

RMmultiquad is a isotropic covariance model. The corresponding covariance function, the multiquadric family, only depends on the angle $0 \le \theta \le \pi$ between two points on the sphere and is given by $$\psi(\theta) = (1 - \delta)^{2*\tau} / (1 + delta^2 - 2*\delta*cos(\theta))^{\tau}$$ where $0 < \delta < 1$ and $\tau > 0$.

Usage

RMmultiquad(delta, tau, var, scale, Aniso, proj)

Arguments

delta

a numerical value in $(0,1)$

tau

a numerical value greater than $0$

var,scale,Aniso,proj

optional arguments; same meaning for any RMmodel. If not passed, the above covariance function remains unmodified.

Value

RMmultiquad returns an object of class RMmodel

Details

Special cases (cf. Gneiting, T. (2013), p.1333) are known for fixed parameter $\tau=0.5$ which leads to the covariance function called 'inverse multiquadric'$$\psi(\theta) = (1 - \delta) / \sqrt( 1 + delta^2 - 2*\delta*cos(\theta) )$$ and for fixed parameter $\tau=1.5$ which gives the covariance function called 'Poisson spline' $$\psi(\theta) = (1 - \delta)^{3} / (1 + delta^2 - 2*\delta*cos(\theta))^{1.5}$$ For a more general form, see RMchoquet.

References

Gneiting, T. (2013) Strictly and non-strictly positive definite functions on spheres Bernoulli, 19(4), 1327-1349.

See Also

RMmodel, RFsimulate, RFfit, RMchoquet, spherical models

Examples

RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set
##                   RFoptions(seed=NA) to make them all random again

RFoptions(coord_system="sphere")
model <- RMmultiquad(delta=0.5, tau=1)
plot(model, dim=2)

## the following two pictures are the same

x <- seq(0, 0.12, 0.01)
z1 <- RFsimulate(model, x=x, y=x)
plot(z1)

x2 <- x * 180 / pi
z2 <- RFsimulate(model, x=x2, y=x2, coord_system="earth")
plot(z2)

stopifnot(all.equal(as.array(z1), as.array(z2)))

RFoptions(coord_system="auto")