RMmultiquad: The Multiquadric Family Covariance Model on the Sphere
In RandomFields: Simulation and Analysis of Random Fields

Description Usage Arguments Details Value Author(s) References See Also Examples

RMmultiquad is an isotropic covariance model. The corresponding covariance function, the multiquadric family, only depends on the angle 0 ≤ θ ≤ π between two points on the sphere and is given by

ψ(θ) = (1 - δ)^{2*τ} / (1 + delta^2 - 2*δ*cos(θ))^{τ},

where 0 < δ < 1 and τ > 0.

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

`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.

Special cases (cf. Gneiting, T. (2013), p.1333) are known for fixed parameter τ=0.5 which leads to the covariance function called 'inverse multiquadric'

ψ(θ) = (1 - δ) / √( 1 + delta^2 - 2*δ*cos(θ) )

and for fixed parameter τ=1.5 which gives the covariance function called 'Poisson spline'

ψ(θ) = (1 - δ)^{3} / (1 + delta^2 - 2*δ*cos(θ))^{1.5}.

For a more general form, see RMchoquet.

RMmultiquad returns an object of class RMmodel.

Christoph Berreth, \martin

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

RMmodel, RFsimulate, RFfit, RMchoquet, spherical models

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")