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#### Henon map examples
#
# This example demonstrates the use of CollocInfer on a discrete-time system.
# The Henon Map is a classical dynamical system that exhibits chaos.
# It's equations are given as
#
# x[t+1] = 1 - a*x[t]^2 + y[t]
# y[t+1] = b*x[t]
# To ensure reproducibility
set.seed((2004*2007)/2014)
###############################
#### Data Generation #######
###############################
hpars = c(1.4,0.3)
ntimes = 200
x = c(-1,1)
X = matrix(0,ntimes+20,2)
X[1,] = x
for(i in 2:(ntimes+20)){ X[i,] = make.Henon()$ode(i,X[i-1,],hpars,NULL) }
X = X[20+1:ntimes,]
par(mar=c(5,5,1,1))
plot(X,xlab='x',ylab='y',cex.lab=2.5,cex.axis=2.5,cex=1.5)
Y = X + 0.05*matrix(rnorm(ntimes*2),ntimes,2)
t = 1:ntimes
coefs = as.matrix(Y)
###############################
#### Optimization Control ###
###############################
control=list(trace = 0,maxit = 1000,maxtry = 10,reltol = 1e-6,meth = "BFGS")
control.in = control
control.in$reltol = 1e-12
control.out = control
control.out$trace = 2
###############################
#### Optimization ###
###############################
hpars2 = c(1.3,0.4) # Perturbed parameters
names(hpars2)=names(hpars)
lambda = 10000
### SSE for discrete process####
Ires1 = Smooth.LS(make.Henon(),data=Y,times=t,pars=hpars2,coefs=coefs,basisvals=NULL,
lambda=lambda,in.meth='nlminb',control.in=control.in,discrete=TRUE)
Ores1 = Profile.LS(make.Henon(),data=Y,t,pars=hpars2,coefs,basisvals=NULL,
lambda=lambda,in.meth='nlminb',out.meth='nls',control.in=control.in,
control.out=control.out,discrete=TRUE)
# Add some plots
parest = Ores1$pars
Fit = Ores1$coef
X11()
par(mfrow=c(2,1),mar=c(5,5,1,1))
plot(t,Fit[,1],type='l',ylab='x',xlab='time',lty=2,cex.lab=2.5,cex.axis=2.5,lwd=3)
points(t,Y[,1],col=2,pch=8,cex=1.5)
plot(t,Fit[,2],type='l',lty=2,ylab='y',xlab='time',cex.lab=2.5,cex.axis=2.5,lwd=3)
points(t,Y[,2],col=2,cex=1.5,pch=8)
# Also compare prediction and new estimate -- equivalent of derivative plots
pred = Ores1$proc$more$fn(t,Fit,Ores1$pars,Ores1$proc$more$more)
X11()
par(mfrow=c(2,1),mar=c(5,5,1,1))
plot(t,pred[,1],type='l',ylab='x',xlab='time',cex.lab=2.5,cex.axis=2.5,lwd=3)
points(t[-ntimes],Fit[-1,1],col=2,type='l',cex=1.5,pch=8)
plot(t,pred[,2],type='l',ylab='y',xlab='time',cex.lab=2.5,cex.axis=2.5,lwd=3)
points(t[-ntimes],Fit[-1,2],col=2,type='l',cex=1.5,pch=8)
# and look at the deviation
X11()
par(mar=c(5,5,1,1))
matplot(t[-ntimes],Fit[-1,]-pred[-ntimes,],xlab='time',ylab='Model Residuals',cex.lab=2.5,cex.axis=2.5,pch=c('x','y'),cex=1.5)
### ProfileErr with LSproc ####
profile.obj = LS.setup(pars=hpars2,coefs=coefs,fn=make.Henon(),basisvals=NULL,
lambda=lambda,times=t,discrete=TRUE)
lik = profile.obj$lik
proc= profile.obj$proc
Ires2 = inneropt(data=Y,times=t,pars=hpars2,coefs,lik,proc,in.meth='nlminb',control.in)
Ores2 = outeropt(data=Y,times=t,pars=hpars2,coefs=coefs,lik=lik,proc=proc,
in.meth="nlminb",out.meth="nlminb",control.in=control.in,control.out=control.out)
### Dproc with the multinorm functions
var = c(1,0.01)
Ires3 = Smooth.multinorm(make.Henon(),data=Y,t,pars=hpars2,coefs,basisvals=NULL,
var=var,in.meth='nlminb',control.in=control.in,discrete=TRUE)
Ores3 = Profile.multinorm(fn=make.Henon(),data=Y,times=t,pars=hpars2,coefs=coefs,
basisvals=NULL,var=var,fd.obj=NULL,more=NULL,quadrature=NULL,in.meth='nlminb',
out.meth='optim',control.in=control.in,control.out=control.out,eps=1e-6,
active=NULL,,discrete=TRUE)
Ores3$pars
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