Description Usage Arguments Details Value Note Author(s) See Also Examples
Simulation of functional data with exogenous variables using a GramSchmidt basis.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16  simul.farx(m=12,n=100,base=base.simul.far(24,5),
base.exo=base.simul.far(24,5),
d.a=matrix(c(0.5,0),nrow=1,ncol=2),
alpha.conj=matrix(c(0.2,0),nrow=1,ncol=2),
d.rho=diag(c(0.45,0.90,0.34,0.45)),
alpha=diag(c(0.5,0.23,0.018)),
d.rho.exo=diag(c(0.45,0.90,0.34,0.45)),
cst1=0.05)
theoretical.coef(m=12,base=base.simul.far(24,5),
base.exo=NULL,
d.rho=diag(c(0.45,0.90,0.34,0.45)),
d.a=NULL,
d.rho.exo=NULL,
alpha=diag(c(0.5,0.23,0.018)),
alpha.conj=NULL,
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 
base.exo 
A functional basis expressed as a matrix, as the matrix
created by 
d.rho 
Numerical matrix. Part of the linear operator in the GramSchmidt basis (see details for more informations). 
d.a 
Numerical matrix. Part of the linear operator in the GramSchmidt basis (see details for more informations). 
d.rho.exo 
Numerical matrix. Part of the linear operator in the GramSchmidt basis (see details for more informations). 
alpha 
Numerical matrix. Part of the linear operator in the GramSchmidt basis (see details for more informations). 
alpha.conj 
Numerical matrix. Part of the linear operator in the GramSchmidt basis (see details for more informations). 
cst1 
Numeric. Perturbation coefficient on the linear operator. 
The simul.farx
function simulates a FARX(1) process with one
endogeneous variable, one exogeneous variable and a strong white
noise. To do so, the function uses the fact that a FARX(1) model can
be seen as a FAR(1) model in a wider space. Therefore, the method is
very similar to the one used by the function simul.far
.
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 given parameters. Tn is of the form (T1n,T2n) where T1n and T2n are respectively the endogeneous and the exogeneous parts of the process.
Second step, the process Tn is projected in the canonical
basis using the base
and base.exo
linear projectors to
give the endogeneous (Xn) and the exogeneous
(Zn) variables respectively.
Those two basis need to be orthonormal and 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=m12+m22). 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:
d.rho.mod \code{d.a} \cr 0 d.rho.exo.mod
Where d.rho.mod and d.rho.exo.mod are modified version of the provided
d.rho
and d.rho.exo
respectively to avoid 0 on their
diagonal. More precisely, the 0 on their diaginals are replaced by:
(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 position in the d.rho
or d.r.ho.exo
diagonal.
In H, CT, the covariance operator of Tn, is defined by:
alpha.mod alpha.conj.mod \cr t(alpha.conj.mod) alpha.exo
Where alpha.mod and alpha.exo.mod are modified versions of
m12 * alpha
and m22 *
alpha.conj
respectively to avoid 0 on their diagonal. More
precisely, the 0 on their diaginals are replaced by:
(epsilon(k+1), epsilon(k+2), …, epsilon(\code{m2b}))
where
epsilon(i)= \code{cst1}/i
alpha.exo is a matrix representation of the covariance operator of T2n and is obtained by inverting the following relation:
alpha.conj.mod = d.rho.exo.mod * alpha.conj.mod * t(d.rho.mod) +% d.rho.exo.mod * mod.alpha * t(\code{d.a})
The theoretical.coef
function is provided to help the user
making comparison. Calling this function with the same parameters that
where used in a simulation (realized with simul.farx
or
simul.far
), we obtain the parameters used internaly by the
function to make the simulation. Those values can therefore be
compared to those obtained with the estimation function far
(examples are provided below).
A fdata
object containing two variables ("X" the endogeous
variable, and "Z" the exogeneous variable) which is a FARX(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.far
, simul.wiener
.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31  # Simulation of a FARX process
data1 < simul.farx(m=10,n=400,base=base.simul.far(20,5),
base.exo=base.simul.far(20,5),
d.a=matrix(c(0.5,0),nrow=1,ncol=2),
alpha.conj=matrix(c(0.2,0),nrow=1,ncol=2),
d.rho=diag(c(0.45,0.90,0.34,0.45)),
alpha=diag(c(0.5,0.23,0.018)),
d.rho.exo=diag(c(0.45,0.90,0.34,0.45)),
cst1=0.0)
# Modelisation of the FARX process (joined and separate)
model1 < far(data1,k=4,joined=TRUE)
model2 < far(data1,k=c(3,1),joined=FALSE)
# Calculation of the theoretical coefficients
coef.theo < theoretical.coef(m=10,base=base.simul.far(20,5),
base.exo=base.simul.far(20,5),
d.a=matrix(c(0.5,0),nrow=1,ncol=2),
alpha.conj=matrix(c(0.2,0),nrow=1,ncol=2),
d.rho=diag(c(0.45,0.90,0.34,0.45)),
alpha=diag(c(0.5,0.23,0.018)),
d.rho.exo=diag(c(0.45,0.90,0.34,0.45)),
cst1=0.0)
# Joined coefficient
round(coef(model1),2)
coef.theo$rho.T
# Separate coefficient
round(coef(model2),2)
coef.theo$rho.X.Z

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