R: Schoenberg's representation for the classes psi_d and...

RMchoquet {RandomFields} | R Documentation

## Schoenberg's representation for the classes psi_d and psi_{∞} in d=2

### Description

`RMchoquet` is a isotropic covariance model. The corresponding covariance function only depends on the angle 0 ≤ θ ≤ π between two points on the sphere and is given for d=2 by

ψ(θ) = ∑_{n=0}^{∞} b_{n,2}/(n+1)*P_n(cos(θ)),

where

∑_{n=0}^{∞} b_{n,d}=1

and P_n is the Legendre Polynomial of integer order n >= 0.

### Usage

    RMchoquet(b)

### Arguments

`b` | a numerical vector of weights in (0,1), such that sum(b)=1.

### Details

By the results (cf. Gneiting, T. (2013), p.1333) of Schoenberg and others like Menegatto, Chen, Sun, Oliveira and Peron, the class psi_d of all realvalued funcions on [0,π], with ψ(0)=1 and such that the associated isotropic function

h(x,y)=ψ(theta) with cos(θ)=<x,y>

for x,y in {x in R^d: ||x|| = 1}

is (strict) positive definit is represented by this covariance model. The model can be interpreted as Choquet representation in terms of extremal members, which are non-strictly positive definite.

Special cases are the multiquadric famiy (see `RMmultiquad`) and the model of the sine power function (see `RMsinepower`).

### Value

`RMchoquet` returns an object of class `RMmodel`

### Author(s)

Christoph Berreth, cberreth@mail.uni-mannheim.de

### References

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

- Schoenberg, I.J. (1942) Positive definite functions on spheres. Duke Math.J.,9, 96-108.

- Menegatto, V.A. (1994) Strictly positive definite kernels on the Hilbert sphere. Appl. Anal., 55, 91-101.

- Chen, D., Menegatto, V.A., and Sun, X. (2003) A necessary and sufficient condition for strictly positive definite functions on spheres. Proc. Amer. Math. Soc.,131, 2733-2740.

- Menegatto, V.A., Oliveira, C.P. and Peron, A.P. (2006) Strictly positive definite kernels on subsets of the complex plane. Comput. Math. Appl., 51, 1233-1250.

### See Also

`RMmodel`, `RFsimulate`, `RFfit`, `spherical models`, `RMmultiquad`, `RMsinepower`

### Examples

    ## todo
    RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set
    ##                   RFoptions(seed=NA) to make them all random again

    #b =
    #model <- RMchoquet(b=b)
    #x <- seq(0, 10, 0.02)
    #plot(model)
    #plot(RFsimulate(model, x=x))

[Package RandomFields version 3.1.50 Index]