# demo/HenonEx.r In CollocInfer: Collocation Inference for Dynamic Systems

```#### 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)

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,