The Multiquadric Family Covariance Model on the Sphere
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.
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
|
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 Schlather, schlather@math.uni-mannheim.de, https://www.wim.uni-mannheim.de/schlather/
Gneiting, T. (2013) Strictly and non-strictly positive definite functions on spheres Bernoulli, 19(4), 1327-1349.
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")
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