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

## Description

`RMmultiquad` is a 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.

## Usage

 `1` ```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.

## Details

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`.

## Value

`RMmultiquad` returns an object of class `RMmodel`

## Author(s)

Christoph Berreth, [email protected], Martin Schlather

## References

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

`RMmodel`, `RFsimulate`, `RFfit`, `RMchoquet`, `spherical models`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20``` ```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") ```