simul.far | R Documentation |
Simulation of a FAR process using a Gram-Schmidt basis.
simul.far(m=12, n=100, base=base.simul.far(24, 5), d.rho=diag(c(0.45, 0.9, 0.34, 0.45)), alpha=diag(c(0.5, 0.23, 0.018)), cst1=0.05)
m |
Integer. Number of discretization points. |
n |
Integer. Number of observations. |
base |
A functional basis expressed as a matrix, as the matrix
created by |
d.rho |
Numerical matrix. Expression of the first bloc of the linear operator in the Gram-Schmidt basis. |
alpha |
Numerical matrix. Expression of the first bloc of the covariance operator in the Gram-Schmidt basis. |
cst1 |
Numeric. Perturbation coefficient on the linear operator. |
This function simulate a FAR(1) process with a strong white noise.
The simulation is realized in two steps.
First step, the function compute a FAR(1) process Tn in a
functional space (that we call in the sequel H) using a simple
equation and the d.rho
, alpha
and cst
parameters.
Second step, the process Tn is projected in the canonical
basis using the base
linear projector.
The base
basis need to be a orthonormal basis wide enought. In the
contrary, the function use the orthonormalization
function
to make it so. Notice that the size of this matrix corresponds to the
dimension of the "modelization space" H (let's call it
m2). Of course, the larger m2
the better the
functionnal approximation is. Whatever, keep in mind that
m2
=2m
is a good compromise, in order to avoid the memory
limits.
In H, the linear operator rho is expressed as:
\code{d.rho} 0 \cr 0 eps.rho
Where d.rho
is the matrix provided in the call, the two 0 are
in fact two blocks of 0, and eps.rho is a diagonal matrix having on
his diagonal the terms:
(eps(k+1), eps(k+2), …, eps(\code{m2}))
where
eps(i)= \code{cst1}/(i^2)+(1-\code{cst1})/exp(i)
and k is the length of the d.rho
diagonal.
The d.rho
matrix can be viewed as the information and the
eps.rho matrix as a perturbation. In this logic, the norm of eps.rho
need to be smaller than the one of d.rho
.
In H, CT, the covariance operator of Tn, is defined by:
m2 * \code{alpha} 0 \cr 0 eps.alpha
Where alpha
is the matrix provided in the call, the two 0 are
in fact two blocks of 0, and eps.alpha is a diagonal matrix having on
his diagonal the terms:
(epsilon(k+1), epsilon(k+2), …, epsilon(\code{m2}))
where
epsilon(i)= \code{cst1}/i
A fdata
object containing one variable ("var") which is a
FAR(1) process of length n
with p
discretization
points.
To simulate Tn, the function creates a white noise En having the following covariance operator:
CT - rho * CT t(CT)
where t(.) is the transposition operator. Tn is the computed using the equation:
Tn+1 = rho * Tn + En
J. Damon, S. Guillas
simul.far.sde
, simul.far.wiener
,
simul.farx
, simul.wiener
,
base.simul.far
.
far1 <- simul.far(m=64,n=100) summary(far1) print(far(far1,kn=4)) par(mfrow=c(2,1)) plot(far1,date=1) plot(select.fdata(far1,date=1:5),whole=TRUE,separator=TRUE)
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