RandomFields (version 3.1.12)

RMchoquet: Schoenberg's representation for the classes psi_d and $psi_{\infty}$ in d=2

Description

RMchoquet is a isotropic covariance model. The corresponding covariance function only depends on the angle $0 \le \theta \le \pi$ between two points on the sphere and is given for d=2 by $$\psi(\theta) = \sum_{n=0}^{\infty} b_{n,2}/(n+1)*P_n(cos(\theta)),$$ where $$\sum_{n=0}^{\infty} b_{n,d}=1$$ 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.

Value

• RMchoquet returns an object of class RMmodel

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 $[0,\pi]$, with $\psi(0)=1$ and such that the associated isotropic function $$h(x,y)=\psi(theta) with cos(\theta)=$$ $$for x,y in {x in R^d: ||x|| = 1}$$ 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).

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