The ifs Package. December 28, 2005

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1 The if Pckge December 28, 2005 Verion Title Iterted Function Sytem Author S. M. Icu Mintiner S. M. Icu Iterted Function Sytem Licene GPL Verion 2 or lter. R topic documented: IFSM if.ft if ifm.cf ifm.etqf ifm.w.mp ifp.cf ifp.etqf ifp.w.mp Index 12 IFSM IFSM opertor IFSM opertor IFSM(x, cf,,, k = 2) 1

2 2 IFSM x where to pproximte the function cf the vector of coefficient phi i the vector of coefficient i in: w i = i x + i the vector of coefficient i in: w i = i x + i k number of itertion, defult = 2 Detil Thi opertor i intended to pproximte function on L2[0,1]. If u i imulted, then the IFSM cn be ued to imulte IFSM verion of u. The vlue of the pproximte trget function. Author() S. M. Icu Reference Icu, S.M, L Torre, D. (2005) IFSM repreenttion of Brownin motion with ppliction to imultion, forthcoming. Exmple require(if) et.eed(123) n <- 50 dt <- 1/n t <- (1:n)*dt Z <- rnorm(n) B <- qrt(dt)*cumum(z) ifm.w.mp() -> mp <- mp$ <- mp$ ifm.etqf(b,, ) -> QF ifm.cf(qf$q,qf$b,qf$l1,qf$l2,)-> SOL pi <- SOL$pi t1 <- eq(0,1,length=250).numeric(pply(t1, function(x) IFSM(x,pi,,,k=5))) -> B.ifm old.mr <- pr()$mr old.mfrow <- pr()$mfrow pr(mfrow=c(2,1)) pr(mr=c(4,4,1,1)) plot(t1,b.ifm,type="l",xlb="time",ylb="ifsm") plot(t,b,col="red",type="l",xlb="time",ylb="euler cheme") pr(mr=old.mr) pr(mfrow=old.mfrow)

3 if.ft 3 if.ft IFS etimtor Ditribution function etimtor bed on invere Fourier trnform of n IFS. if.ft(x, p,,, k = 2) if.etup.ft(m, p,,, k = 2, cutoff) if.pf.ft(x,b,nterm) if.df.ft(x,b,nterm) IFS.pf.FT(y, k = 2, n = 512, mp=c("quntile","wl1","wl2")) IFS.df.FT(y, k = 2, n = 512, mp=c("quntile","wl1","wl2")) x p m where to etimte the function the vector of coefficient p i the vector of coefficient i in: w i = i x + i the vector of coefficient i in: w i = i x + i the vector of mple moment k number of itertion, defult = 2 y n mp b Detil nterm cutoff vector of mple obervtion the number of point in which to clculte the etimtor type of ffine mp the Fourier coefficient the number of ignificnt Fourier coefficient fter the cutoff cutoff ued to determine how mny Fourier coefficient re needed Thi etimtor i intended to etimte the continuou ditribution function, the chrteritic function (Fourier trnform) nd the denity function of rndom vrible on [0,1]. The etimted vlue of the Fourier trnform for if.ft, the etimted vlue of the ditribution function for if.pf.ft nd the etimted vlue of the denity function for if.df.ft. A lit of x nd y coordinte plu the Fourier coefficient nd the number of ignificnt coefficient of the ditribution function etimtor for IFS.pf.FT nd the denity function for IFS.df.FT. The function if.etup.ft return lit of Fourier coefficient nd the number of ignificnt coefficient. Note Detil of thi tecnique cn be found in Icu nd L Torre, 2002.

4 4 if.ft Author() S. M. Icu Reference Icu, S.M, L Torre, D. (2005) Approximting ditribution function by iterted function ytem, Journl of Applied Mthemtic nd Deciion Science, 1, ecdf Exmple require(if) nob <- 100 y<-rbet(nob,2,4) # uncomment if you wnt to tet the norml ditribution # y<-ort(rnorm(nob,3,1))/6 IFS.et <- IFS(y) xx <- IFS.et$x tt <- IFS.et$y <- pbet(xx,2,4) # uncomment if you wnt to tet the norml ditribution # <- pnorm(6*xx-3) pr(mfrow=c(3,1)) plot(ecdf(y),xlim=c(0,1),min="ifs etimtor veru EDF") line(xx,,col="blue") line(ifs.et,col="red") IFS.FT.et <- IFS.pf.FT(y) xxx <- IFS.FT.et$x uuu <- IFS.FT.et$y <- pbet(xxx,2,4) # uncomment if you wnt to tet the norml ditribution # <- pnorm(6*xxx-3) line(ifs.ft.et,col="green") # clculte MSE ww <- ecdf(y)(xx) men((ww-)^2) men((tt-)^2) men((uuu-)^2) plot(xx,(ww-)^2,min="mse",type="l",xlb="x",ylb="mse(x)") line(xx,(tt-)^2,col="red") line(xxx,(uuu-)^2,col="green")

5 if 5 plot(ifs.df.ft(y),type="l",col="green",ylim=c(0,3),min="ifs v Kernel") line(denity(y),col="blue") curve(dbet(x,2,4),0,1,dd=true) # uncomment if you wnt to tet the norml ditribution # curve(6*dnorm(x*6-3,0,1),0,1,dd=true) if IFS etimtor Ditribution function etimtor bed on mple quntile. if(x, p,,, k = 5) if.flex(x, p,,, k = 5, f = NULL) IFS(y, k = 5, q = 0.5, f = NULL, n = 512, mp = c("quntile", "wl1", "wl2")) x where to etimte the ditribution function p the vector of coefficient p i the vector of coefficient i in: w i = i x + i the vector of coefficient i in: w i = i x + i k number of itertion, defult = 5 y vector of mple obervtion q the proportion of quntile to ue in the contruction of the etimtor, defult = 0.5. The number of quntile i the q * length(y). f n mp the trting point in the pce of ditribution function the number of point in which to clculte the IFS type of ffine mp Detil Thi etimtor i intended to etimte the continuou ditribution function of rndom vrible on [0,1]. The etimtor i continuou function not everywhere differentible. The etimted vlue of the ditribution function for if nd if.flex or lit of x nd y coordinte of the IFS(x) grph for IFS. Note It i ymptoticlly good the empiricl ditribution function (ee Icu nd L Torre, 2001). Thi function i clled by IFS. If you need to cll the function everl time, you hould better ue if providing the point nd coefficient once inted of IFS. Empiricl evidence how tht the IFS-etimtor i better thn the edf (even for very mll mple) in the up-norm metric. It i lo better in the MSE ene outide of the ditribution til if the mple quntile re ued point.

6 6 ifm.cf Author() S. M. Icu Reference Icu, S.M, L Torre, D. (2005) Approximting ditribution function by iterted function ytem, Journl of Applied Mthemtic nd Deciion Science, 1, ecdf Exmple require(if) y<-rbet(50,.5,.1) # uncomment if you wnt to tet the norml ditribution # y<-ort(rnorm(50,3,1))/6 IFS.et <- IFS(y) xx <- IFS.et$x tt <- IFS.et$y <- pbet(xx,.5,.1) # uncomment if you wnt to tet the norml ditribution # <- pnorm(6*xx-3) pr(mfrow=c(2,1)) plot(ecdf(y),xlim=c(0,1),min="ifs etimtor veru EDF") line(xx,,col="blue") line(xx,tt,col="red") # clculte MSE ww <- ecdf(y)(xx) men((ww-)^2) men((tt-)^2) plot(xx,(ww-)^2,min="mse",type="l",xlb="x",ylb="mse(x)") line(xx,(tt-)^2,col="red") ifm.cf Clculte the min prmeter of the IFSM opertor Tool function to contruct nd find the olution of the minimiztion problem involving the qudrtic form x Qx + b x. Not n optiml one. You cn provide one better then thi.

7 ifm.etqf 7 ifm.cf(q, b, d, l2,, mu=1e-4) Q b d l2 mu the mtrix Q of x Qx + b x the vector b of x Qx + b x the L1 norm of the trget function the L2 norm of the trget function the vector in: w i = i x + i tolernce A lit cf delt the vector of the coefficient to be plugged into the IFSM the collge ditnce t the olution Reference Icu, S.M, L Torre, D. (2005) IFSM repreenttion of Brownin motion with ppliction to imultion, forthcoming. IFSM ifm.etqf Set up the qudrtic form for the IFSM Tool function to contruct the qudrtic form x Qx+b x+l2 to be minimized under ome contrint depending on l1. Thi i ued to contruct the IFSM opertor. ifm.etqf(u,, ) u the vector of vlue of the trget function u the vector of coefficient i in: w i = i x + i the vector of coefficient i in: w i = i x + i Detil Thi opertor i intended to pproximte function on L2[0,1]. If u i imulted, then the IFSM cn be ued to imulte IFSM verion of u.

8 8 ifm.w.mp Lit of element Q b L1 L2 M1 the mtrix of the qudrtic form the mtrix of the qudrtic form the L1 norm of the trget function the L2 norm of the trget function the integrl of the trget function Author() S. M. Icu Reference Icu, S.M, L Torre, D. (2005) IFSM repreenttion of Brownin motion with ppliction to imultion, forthcoming. IFSM ifm.w.mp Set up the prmeter for the mp of the IFSM opertor Thi i clled before clling ifm.etqf to prepre the prmeter to be ped in ifm.etqf. ifm.w.mp(m=8) M M i uch tht um(2ˆ(1:m)) mp re creted A lit of the vector of the coefficent in the mp the vector of the coefficent in the mp Author() S. M. Icu IFSM

9 ifp.cf 9 ifp.cf Clculte the min prmeter of the IFS etimtor Tool function to contruct nd find the olution of the minimiztion problem involving the qudrtic form x Qx + b x. Not n optiml one. You cn provide one better then thi. ifp.cf(q,b) Q b the mtrix Q of x Qx + b x the vector b of x Qx + b x p the vector of the coefficient to be plugged into the IFS Reference Icu, S.M, L Torre, D. (2005) Approximting ditribution function by iterted function ytem, Journl of Applied Mthemtic nd Deciion Science, 1, if ifp.etqf Set up the qudrtic form for the IFSP Tool function to contruct the qudrtic form x Qx + b x to be minimized to contruct the IFSP opertor. ifp.etqf(m,,, n = 10) m the vector of the mple or true moment of the trget function the vector of coefficient i in: w i = i x + i the vector of coefficient i in: w i = i x + i n number of prmeter to ue in the IFSP opertor, defult = 10

10 10 ifp.w.mp Detil Thi opertor i intended to pproximte continuou ditribution function of rndom vrible on [0,1]. If moment re etimted on rndom mple, then the IFSP opertor i n etimtor of the ditribution function of the dt. Q b the mtrix of the qudrtic form the mtrix of the qudrtic form Author() S. M. Icu Reference Icu, S.M, L Torre, D. (2005) Approximting ditribution function by iterted function ytem, Journl of Applied Mthemtic nd Deciion Science, 1, if ifp.w.mp Set up the prmeter for the mp of the IFSP opertor Thi i clled before clling ifp.etqf to prepre the prmeter to be ped in ifp.etqf. ifp.w.mp(y, mp = c("quntile","wl1","wl2"), qtl) y mp qtl the vector of the mple obervtion type of mp: quntile, wl1 or wl2 inted of ping the dt y you cn p vector of quntile m n the vector of the empiricl moment the vector of the coefficent in the mp the vector of the coefficent in the mp the number of mp Author() S. M. Icu

11 ifp.w.mp 11 if

12 Index Topic ditribution if.ft, 2 Topic mic IFSM, 1 ifm.cf, 6 ifm.etqf, 7 ifm.w.mp, 8 ifp.cf, 9 ifp.etqf, 9 ifp.w.mp, 10 Topic nonprmetric if, 5 ecdf, 4, 6 IFS, 5 IFS (if), 5 if, 5, 9 11 IFS.df.FT (if.ft), 2 if.df.ft (if.ft), 2 if.ft, 2 IFS.pf.FT (if.ft), 2 if.pf.ft (if.ft), 2 if.etup.ft (if.ft), 2 IFSM, 1, 7, 8 ifm.cf, 6 ifm.etqf, 7 ifm.w.mp, 8 ifp.cf, 9 ifp.etqf, 9 ifp.w.mp, 10 12