locquadvar {FieldSim}R Documentation

Estimation of the multifractional function of the multifractalional Brownian field by the localized quadratic variations method

Description

The function locquadvar yields the estimation of the multifractional function at point t of the multifractional Brownian field by the localized quadratic variations method.

Usage

locquadvar(Z,t,h)

Arguments

Z a matrix of size (n x n). Z is the matrix associated with the sample path of one fractal Brownian field. n must be of the form 2^{J}+1 where J is a positive integer. So Z[i,j] is the value of the process at the point ((i-1)2^{-J},(j-1)2^{-J}).
t a vector of length 2. t is the point of the square [0,1] x [0,1] where multifractional function must be estimated.
h a positive real. h defines the neighborhood used to estimate multifractional function via the following form ((i-1)2^{-J},(j-1)2^{-J}), i,j=1,...,n such that abs(2^{-iJ}-t[1])<=h and abs(2^{-jJ}-t[2])<=h.

Details

The multifractional function at point t of the multi-fractal Brownian field is estimated by the procedure described in Lacaux (2004).

Value

A list with the following components:

H a real in ]0,1[ that represents the estimate of the multifractional function at point t.

Author(s)

Alexandre Brouste (http://ljk.imag.fr/membres/Alexandre.Brouste) and Sophie Lambert-Lacroix (http://ljk.imag.fr/membres/Sophie.Lambert).

References

C. Lacaux (2004). Real Harmonizable Multifractal Levy Motions. Annales Institut Henri Poincaré, Prob. Stat., *40* 259-277.

See Also

fieldsim.

Examples

# load FieldSim library
library(FieldSim)

#Field simulation
F<-function(x){
H1<-(0.4*x[1]+0.5)
H2<-(0.4*x[3]+0.5)
out<-list(H1=H1,H2=H2,H=(H1+H2)/2)
return(out)
}

R<-function(x){
a<-F(x)
alpha<-a$H
alpha1<-a$H1
alpha2<-a$H2

C2D(alpha)^2/(2*C2D(alpha1)*C2D(alpha2))*((x[1]^2+x[2]^2)^(alpha)+(x[3]^2+x[4]^2)^(alpha)-((x[1]-x[3])^2+(x[2]-x[4])^2)^(alpha))
}

res <- fieldsim(R,Elevel=1,Rlevel=5,nbNeighbor=4)

# Estimation
locquadvar(Z=res$Z,t=c(0.5,0.5),h=0.125)

[Package FieldSim version 1.1 Index]