R: The Multiquadric Family Covariance Model on th Sphere

RMmultiquad {RandomFields} | R Documentation

## 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 ≤ 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

    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, cberreth@mail.uni-mannheim.de, Martin Schlather

### 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")

[Package RandomFields version 3.1.50 Index]