# simul.far.wiener: FAR(1) process simulation with Wiener noise In far: Modelization for Functional AutoRegressive Processes

## Description

Simulation of a FAR(1) process using a Wiener noise.

## Usage

 ```1 2``` ```simul.far.wiener(m=64, n=128, d.rho=diag(c(0.45, 0.9, 0.34, 0.45)), cst1=0.05, m2=NULL) ```

## Arguments

 `m` Integer. Number of discretization points. `n` Integer. Number of observations. `d.rho` Numerical matrix. Expression of the first bloc of the linear operator in the Karhunen-Loève basis. `cst1` Numeric. Perturbation coefficient on the linear operator. `m2` Integer. Length of the Karhunen-Loève expansion (2`m` by default).

## Details

This function simulate a FAR(1) process with a Wiener noise. As for the `simul.wiener`, the function use the Karhunen-Loève expansion of the noise. The FAR(1) process, defined by its linear operator (see `far` for more details), is computed in the Karhunen-Loève basis then projected in the natural basis. The parameters given in input (`d.rho` and `cst1`) are expressed in the Karhunen-Loève basis.

The linear operator, expressed in the Karhunen-Loève basis, is of the form:

\code{d.rho} 0 \cr 0 eps.rho

Where `d.rho` is the matrix provided in ths 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`.

## Value

A `fdata` object containing one variable ("var") which is a FAR(1) process of length `n` with `m` discretization points.

J. Damon

## References

Pumo, B. (1992). Estimation et Prévision de Processus Autoregressifs Fonctionnels. Applications aux Processus à Temps Continu. PhD Thesis, University Paris 6, Pierre et Marie Curie.

`fdata`, `far` , `simul.far.wiener`.

## Examples

 ```1 2 3 4 5 6``` ``` far1 <- simul.far.wiener(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) ```

far documentation built on May 2, 2019, 9:28 a.m.