ifs (version 0.1.10)

IFSM: IFSM operator

Description

IFSM operator

Usage

IFSM(x, cf, a, s, k = 2)

Value

The value of the approximate target function.

Arguments

x

where to approximate the function

cf

the vector of coefficients \(phi_i\)

s

the vector of coefficients \(s_i\) in: \(w_i = s_i *x + a_i\)

a

the vector of coefficients \(a_i\) in: \(w_i = s_i *x + a_i\)

k

number of iterations, default = 2

Author

S. M. Iacus

Details

This operator is intended to approximate a function on L2[0,1]. If `u' is simulated, then the IFSM can be used to simulate a IFSM version of `u'.

References

Iacus, S.M, La Torre, D. (2005) IFSM representation of Brownian motion with applications to simulation, forthcoming.

Examples

require(ifs)

set.seed(123)
n <- 50
dt <- 1/n
t <- (1:n)*dt
Z <- rnorm(n)
B <- sqrt(dt)*cumsum(Z)

ifsm.w.maps() -> maps
a <- maps$a
s <- maps$s

ifsm.setQF(B, s, a) -> QF
ifsm.cf(QF$Q,QF$b,QF$L1,QF$L2,s)-> SOL
psi <- SOL$psi

t1 <- seq(0,1,length=250)
as.numeric(sapply(t1, function(x) IFSM(x,psi,a,s,k=5))) -> B.ifsm
old.mar <- par()$mar
old.mfrow <- par()$mfrow
par(mfrow=c(2,1))
par(mar=c(4,4,1,1))
plot(t1,B.ifsm,type="l",xlab="time",ylab="IFSM")
plot(t,B,col="red",type="l",xlab="time",ylab="Euler scheme")
par(mar=old.mar)
par(mfrow=old.mfrow)