hypersim {FieldSim}R Documentation

Random hyperboloïd indexed field simulation by the mifieldsim method on visual grid

Description

The function hypersim yields discretization of sample path of a Gaussian hyperboloïd indexed field following the procedure described in Brouste et al. (2009).

Usage

hypersim(R,Ne=100,Nr=100,Ng=100,nbNeighbor=4)

Arguments

R a covariance function (defined on the hyperboloïd) of a Random hyperboloïd indexed field to simulate.
Ne a positive integer. Ne is the number of simulation points associated with the uniform distributed discretization of the hyperboloïd for the first step of the algorithm (Accurate simulation step)
Nr a positive integer. Nr is the number of simulation points associated with the uniform distributed discretization of the sphere for the second step of the algorithm (Refined simulation step).
Ng a positive integer. Nr is the number of simulation points associated with the visual grid discretization of the hyperboloïd for the third step of the algorithm (Visual refined simulation step).
nbNeighbor a positive integer. nbNeighbor must be between 1 and 32. nbNeighbor is the number of neighbors to use in the second step of the algorithm.

Details

The function hypersim yields discretization of sample path of a Gaussian hyperboloïd indexed field associated with the covariance function given by R.

Value

A list with the following components:

X the vector of length at more Ne+Nr+Nr^2 containing the discretization of the x axis.
Y the vector of length at more Ne+Nr+Nr^2 containing the discretization of the y axis.
Z the vector of length at more Ne+Nr+Nr^2 containing the discretization of the z axis.
W the vector of length at more Ne+Nr+Nr^2 containing the value of the simulated field at point (X[n],Y[n],Z[n])
W1 the matrice of size Ng^2 that give values of the simulated hyperboloïd indexed field at the points of the visual grid
time the CPU time

Author(s)

Alexandre Brouste (http://subaru.univ-lemans.fr/sciences/statist/pages_persos/Brouste/) and Sophie Lambert-Lacroix (http://ljk.imag.fr/membres/Sophie.Lambert).

References

A. Brouste, J. Istas and S. Lambert-Lacroix (2009). On simulation of manifold indexed fractional Gaussian fields.

See Also

hypersimgrid,fieldsim,spheresim.

Examples

# Load FieldSim library
library(FieldSim)

d<-function(x){    #Distance on the hyperboloïd
u <- -x[1]*x[4]-x[2]*x[5]+x[3]*x[6]
if (u<1){u<-1}
acosh(u)
}

#Example 1: Hyperboloïd indexed Brownian fractional Field with RH1 covariance function

RH1<-function(x){
H<-0.45          # H can vary from 0 to 0.5
1/2*(d(c(0,0,1,x[1:3]))^{2*H}+d(c(0,0,1,x[4:6]))^{2*H}-d(x)^{2*H})
}

resh1<- hypersim(RH1,Ne=100,Nr=1000,Ng=50,nbNeighbor=4)

library(rgl)
library(RColorBrewer)
printhyper(resh1)

#Example 2: Hyperboloïd indexed Brownian Field with RH4 covariance function

RH4<-function(x){
H<-0.45
1/(1+d(x)^{2*H})
}

resh4<- hypersim(RH4,Ne=100,Nr=1000,Ng=50,nbNeighbor=4)
printhyper(resh4)

[Package FieldSim version 2.1 Index]