locquadvar {FieldSim} | R Documentation |
The function locquadvar
yields the estimation of the multifractional function at point t
of the multifractional Brownian field by the localized quadratic variations method.
locquadvar(Z,t,h)
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. |
The multifractional function at point t
of the multi-fractal Brownian field
is estimated by the procedure described in Lacaux (2004).
A list with the following components:
H |
a real in ]0,1[ that represents the estimate of
the multifractional function at point t . |
Alexandre Brouste (http://ljk.imag.fr/membres/Alexandre.Brouste) and Sophie Lambert-Lacroix (http://ljk.imag.fr/membres/Sophie.Lambert).
C. Lacaux (2004). Real Harmonizable Multifractal Levy Motions. Annales Institut Henri Poincaré, Prob. Stat., *40* 259-277.
# 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)