RandomFields (version 3.1.12)

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 $\theta \in [0,\pi]$ 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 $\delta \in (0,1)$ 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
StartExample()
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")
FinalizeExample()

FinalizeExample()