FieldSim (version 2.1)

locquadvar: 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

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])

Value

• A list with the following components:
• H
  a real in $]0,1[$ that represents the estimate of the multifractional function at point t.

encoding

latin1

Details

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

References

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

See Also

fieldsim.

Examples

# load FieldSim library
library(FieldSim)

#Field simulation
F<-function(y){0.4*y+0.5}

R<-function(x,Fun=F){
H1<-Fun(x[1])
H2<-Fun(x[3])
alpha<-1/2*(H1+H2)
C2D(alpha)^2/(2*C2D(H1)*C2D(H2))*((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)