| Title: | Iterated Function Systems |
|---|---|
| Description: | Iterated Function Systems Estimator as in Iacus and La Torre (2005) <doi:10.1155/JAMDS.2005.33>. |
| Authors: | S. M. Iacus |
| Maintainer: | Stefano M. Iacus <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 0.1.10 |
| Built: | 2025-02-13 05:27:46 UTC |
| Source: | https://github.com/siacus/ifs |
Distribution function estimator based on sample quantiles.
ifs(x, p, s, a, k = 5) ifs.flex(x, p, s, a, k = 5, f = NULL) IFS(y, k = 5, q = 0.5, f = NULL, n = 512, maps = c("quantile", "wl1", "wl2"))ifs(x, p, s, a, k = 5) ifs.flex(x, p, s, a, k = 5, f = NULL) IFS(y, k = 5, q = 0.5, f = NULL, n = 512, maps = c("quantile", "wl1", "wl2"))
x |
where to estimate the distribution function |
p |
the vector of coefficients |
s |
the vector of coefficients |
a |
the vector of coefficients |
k |
number of iterations, default = 5 |
y |
a vector of sample observations |
q |
the proportion of quantiles to use in the construction of the
estimator, default = 0.5. The number of quantiles is the
|
f |
the starting point in the space of distribution functions |
n |
the number of points in which to calculate the IFS |
maps |
type of affine maps |
This estimator is intended to estimate the continuous distribution function of a random variable on [0,1]. The estimator is a continuous function not everywhere differentiable.
The estimated value of the distribution function for ifs and ifs.flex or
a list of ‘x’ and ‘y’ coordinates of the IFS(x) graph for IFS.
It is asymptotically as good as the empirical distribution function
(see Iacus and La Torre, 2001).
This function is called by IFS. If you need to call the function
several times, you should better use ifs providing the
points and coefficients once instead of IFS.
Empirical evidence shows that the IFS-estimator is better than the edf (even
for very small samples) in the sup-norm metric. It is also better in the MSE
sense outside of the distribution's tails if the sample quantiles are used
as points.
S. M. Iacus
Iacus, S.M, La Torre, D. (2005) Approximating distribution functions by iterated function systems, Journal of Applied Mathematics and Decision Sciences, 1, 33-46.
require(ifs) y<-rbeta(50,.5,.1) # uncomment if you want to test the normal distribution # y<-sort(rnorm(50,3,1))/6 IFS.est <- IFS(y) xx <- IFS.est$x tt <- IFS.est$y ss <- pbeta(xx,.5,.1) # uncomment if you want to test the normal distribution # ss <- pnorm(6*xx-3) par(mfrow=c(2,1)) plot(ecdf(y),xlim=c(0,1),main="IFS estimator versus EDF") lines(xx,ss,col="blue") lines(xx,tt,col="red") # calculates MSE ww <- ecdf(y)(xx) mean((ww-ss)^2) mean((tt-ss)^2) plot(xx,(ww-ss)^2,main="MSE",type="l",xlab="x",ylab="MSE(x)") lines(xx,(tt-ss)^2,col="red")require(ifs) y<-rbeta(50,.5,.1) # uncomment if you want to test the normal distribution # y<-sort(rnorm(50,3,1))/6 IFS.est <- IFS(y) xx <- IFS.est$x tt <- IFS.est$y ss <- pbeta(xx,.5,.1) # uncomment if you want to test the normal distribution # ss <- pnorm(6*xx-3) par(mfrow=c(2,1)) plot(ecdf(y),xlim=c(0,1),main="IFS estimator versus EDF") lines(xx,ss,col="blue") lines(xx,tt,col="red") # calculates MSE ww <- ecdf(y)(xx) mean((ww-ss)^2) mean((tt-ss)^2) plot(xx,(ww-ss)^2,main="MSE",type="l",xlab="x",ylab="MSE(x)") lines(xx,(tt-ss)^2,col="red")
Distribution function estimator based on inverse Fourier transform of ans IFSs.
ifs.FT(x, p, s, a, k = 2) ifs.setup.FT(m, p, s, a, k = 2, cutoff) ifs.pf.FT(x,b,nterms) ifs.df.FT(x,b,nterms) IFS.pf.FT(y, k = 2, n = 512, maps=c("quantile","wl1","wl2")) IFS.df.FT(y, k = 2, n = 512, maps=c("quantile","wl1","wl2"))ifs.FT(x, p, s, a, k = 2) ifs.setup.FT(m, p, s, a, k = 2, cutoff) ifs.pf.FT(x,b,nterms) ifs.df.FT(x,b,nterms) IFS.pf.FT(y, k = 2, n = 512, maps=c("quantile","wl1","wl2")) IFS.df.FT(y, k = 2, n = 512, maps=c("quantile","wl1","wl2"))
x |
where to estimate the function |
p |
the vector of coefficients |
s |
the vector of coefficients |
a |
the vector of coefficients |
m |
the vector of sample moments |
k |
number of iterations, default = 2 |
y |
a vector of sample observations |
n |
the number of points in which to calculate the estimator |
maps |
type of affine maps |
b |
the Fourier coefficients |
nterms |
the number of significant Fourier coefficients after the cutoff |
cutoff |
cutoff used to determine how many Fourier coefficients are needed |
This estimator is intended to estimate the continuous distribution function, the charateristic function (Fourier transform) and the density function of a random variable on [0,1].
The estimated value of the Fourier transform for ifs.FT, the estimated value
of the distribution function for ifs.pf.FT and the estimated value
of the density function for ifs.df.FT.
A list of ‘x’ and ‘y’ coordinates plus the Fourier coefficients and the number of
significant coefficients of the distribution function estimator for IFS.pf.FT
and the density function for IFS.df.FT.
The function ifs.setup.FT return a list of Fourier coefficients and the number
of significant coefficients.
Details of this tecnique can be found in Iacus and La Torre, 2002.
S. M. Iacus
Iacus, S.M, La Torre, D. (2005) Approximating distribution functions by iterated function systems, Journal of Applied Mathematics and Decision Sciences, 1, 33-46.
require(ifs) nobs <- 100 y<-rbeta(nobs,2,4) # uncomment if you want to test the normal distribution # y<-sort(rnorm(nobs,3,1))/6 IFS.est <- IFS(y) xx <- IFS.est$x tt <- IFS.est$y ss <- pbeta(xx,2,4) # uncomment if you want to test the normal distribution # ss <- pnorm(6*xx-3) par(mfrow=c(3,1)) plot(ecdf(y),xlim=c(0,1),main="IFS estimator versus EDF") lines(xx,ss,col="blue") lines(IFS.est,col="red") IFS.FT.est <- IFS.pf.FT(y) xxx <- IFS.FT.est$x uuu <- IFS.FT.est$y sss <- pbeta(xxx,2,4) # uncomment if you want to test the normal distribution # sss <- pnorm(6*xxx-3) lines(IFS.FT.est,col="green") # calculates MSE ww <- ecdf(y)(xx) mean((ww-ss)^2) mean((tt-ss)^2) mean((uuu-sss)^2) plot(xx,(ww-ss)^2,main="MSE",type="l",xlab="x",ylab="MSE(x)") lines(xx,(tt-ss)^2,col="red") lines(xxx,(uuu-sss)^2,col="green") plot(IFS.df.FT(y),type="l",col="green",ylim=c(0,3),main="IFS vs Kernel") lines(density(y),col="blue") curve(dbeta(x,2,4),0,1,add=TRUE) # uncomment if you want to test the normal distribution # curve(6*dnorm(x*6-3,0,1),0,1,add=TRUE)require(ifs) nobs <- 100 y<-rbeta(nobs,2,4) # uncomment if you want to test the normal distribution # y<-sort(rnorm(nobs,3,1))/6 IFS.est <- IFS(y) xx <- IFS.est$x tt <- IFS.est$y ss <- pbeta(xx,2,4) # uncomment if you want to test the normal distribution # ss <- pnorm(6*xx-3) par(mfrow=c(3,1)) plot(ecdf(y),xlim=c(0,1),main="IFS estimator versus EDF") lines(xx,ss,col="blue") lines(IFS.est,col="red") IFS.FT.est <- IFS.pf.FT(y) xxx <- IFS.FT.est$x uuu <- IFS.FT.est$y sss <- pbeta(xxx,2,4) # uncomment if you want to test the normal distribution # sss <- pnorm(6*xxx-3) lines(IFS.FT.est,col="green") # calculates MSE ww <- ecdf(y)(xx) mean((ww-ss)^2) mean((tt-ss)^2) mean((uuu-sss)^2) plot(xx,(ww-ss)^2,main="MSE",type="l",xlab="x",ylab="MSE(x)") lines(xx,(tt-ss)^2,col="red") lines(xxx,(uuu-sss)^2,col="green") plot(IFS.df.FT(y),type="l",col="green",ylim=c(0,3),main="IFS vs Kernel") lines(density(y),col="blue") curve(dbeta(x,2,4),0,1,add=TRUE) # uncomment if you want to test the normal distribution # curve(6*dnorm(x*6-3,0,1),0,1,add=TRUE)
IFSM operator
IFSM(x, cf, a, s, k = 2)IFSM(x, cf, a, s, k = 2)
x |
where to approximate the function |
cf |
the vector of coefficients |
s |
the vector of coefficients |
a |
the vector of coefficients |
k |
number of iterations, default = 2 |
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’.
The value of the approximate target function.
S. M. Iacus
Iacus, S.M, La Torre, D. (2005) IFSM representation of Brownian motion with applications to simulation, forthcoming.
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()$mfrow par(mfrow=c(2,1)) par(mar=c(4,4,1,1)) plot(t1,B.ifsm,type="l",xlab="time",ylab="IFSM") plot(t,B,col="red",type="l",xlab="time",ylab="Euler scheme") par(mar=old.mar) par(mfrow=old.mfrow)require(ifs) set.seed(123) n <- 50 dt <- 1/n t <- (1:n)*dt Z <- rnorm(n) B <- sqrt(dt)*cumsum(Z) ifsm.w.maps() -> maps a <- maps$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)
Tool function to construct and find the solution of the minimization
problem involving the quadratic form . Not an optimal one.
You can provide one better then this.
ifsm.cf(Q, b, d, l2, s, mu=1e-4)ifsm.cf(Q, b, d, l2, s, mu=1e-4)
Q |
the matrix |
b |
the vector |
d |
the L1 norm of the target function |
l2 |
the L2 norm of the target function |
s |
the vector |
mu |
tolerance |
A list
cf |
the vector of the coefficients to be plugged into the IFSM |
delta |
the collage distance at the solution |
Iacus, S.M, La Torre, D. (2005) IFSM representation of Brownian motion with applications to simulation, forthcoming.
Tool function to construct the quadratic form to be
minimized under some constraint depending on l1. This is used to construct
the IFSM operator.
ifsm.setQF(u, s, a)ifsm.setQF(u, s, a)
u |
the vector of values of the target function u |
s |
the vector of coefficients |
a |
the vector of coefficients |
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’.
List of elements
Q |
the matrix of the quadratic form |
b |
the matrix of the quadratic form |
L1 |
the L1 norm of the target function |
L2 |
the L2 norm of the target function |
M1 |
the integral of the target function |
S. M. Iacus
Iacus, S.M, La Torre, D. (2005) IFSM representation of Brownian motion with applications to simulation, forthcoming.
This is called before calling ifsm.setQF to prepare the
parameters to be passed in ifsm.setQF.
ifsm.w.maps(M=8)ifsm.w.maps(M=8)
M |
M is such that |
A list of
a |
the vector of the coefficents ‘a’ in the maps |
s |
the vector of the coefficents ‘s’ in the maps |
S. M. Iacus
Tool function to construct and find the solution of the minimization
problem involving the quadratic form . Not an optimal one.
You can provide one better then this.
ifsp.cf(Q,b)ifsp.cf(Q,b)
Q |
the matrix |
b |
the vector |
p |
the vector of the coefficients to be plugged into the IFS |
Iacus, S.M, La Torre, D. (2005) Approximating distribution functions by iterated function systems, Journal of Applied Mathematics and Decision Sciences, 1, 33-46.
Tool function to construct the quadratic form to be minimized
to construct the IFSP operator.
ifsp.setQF(m, s, a, n = 10)ifsp.setQF(m, s, a, n = 10)
m |
the vector of the sample or true moments of the target function |
s |
the vector of coefficients |
a |
the vector of coefficients |
n |
number of parameter to use in the IFSP operator, default = 10 |
This operator is intended to approximate a continuous distribution function of a random variable on [0,1]. If moments are estimated on a random sample, then the IFSP operator is an estimator of the distribution function of the data.
Q |
the matrix of the quadratic form |
b |
the matrix of the quadratic form |
S. M. Iacus
Iacus, S.M, La Torre, D. (2005) Approximating distribution functions by iterated function systems, Journal of Applied Mathematics and Decision Sciences, 1, 33-46.
This is called before calling ifsp.setQF to prepare the
parameters to be passed in ifsp.setQF.
ifsp.w.maps(y, maps = c("quantile","wl1","wl2"), qtl)ifsp.w.maps(y, maps = c("quantile","wl1","wl2"), qtl)
y |
the vector of the sample observations |
maps |
type of maps: quantile, wl1 or wl2 |
qtl |
instead of passing the data y you can pass a vector of quantiles |
m |
the vector of the empirical moments |
a |
the vector of the coefficents ‘a’ in the maps |
s |
the vector of the coefficents ‘s’ in the maps |
n |
the number of maps |
S. M. Iacus