FhNEx4.R
In CollocInfer: Collocation Inference for Dynamic Systems

# An example file fitting the FitzHugh-Nagumo equations to data in the
# new R Profiling Code. This will eventually be interfaced with the new R 
# "Partially Observed Markov Process (pomp)" class of objects. 

#################################################
####  Define list fhn containing rhs functions ##
#################################################

fhn = make.fhn()

#################################################
####   Data Generation (1) 41 time values #######
#################################################

times = seq(0,20,0.5)
pars  = c(0.2,0.2,3)
names(pars) = c('a','b','c')
x0 = c(-1,1)
names(x0) = c('V','R')
y = lsoda(x0,times=times,func=make.fhn()$fn.ode,pars)
y = y[,2:3]
data = y + 0.2*array(rnorm(82),dim(y))
                    
################################################
####   Data Generation (2) 401 time values  ####
################################################

times = seq(0,20,0.05)
pars  = c(0.2,0.2,3)
names(pars) = c('a','b','c')
x0 = c(-1,1)
names(x0) = c('V','R')
z=matrix(rep(0,80002),,2)
z[1,1]=-1
z[1,2]=1
w=matrix(rnorm(80000),,2)
for (i in 2:40001)
{ 
  z[i,1] = z[i-1,1]+.0005*3*(z[i-1,1]- z[i-1,1]^3/3+z[i-1,2])+ 
           .011*w[i-1,1]
  z[i,2] = z[i-1,2]-.0005*  (z[i-1,1]-.2+.2* z[i-1,2])/3 + 
           .011*w[i-1,2]
}
y= matrix(rep(0,802),,2)
for (i in 0:400)
y[i+1,]=z[i*100+1,]
data = y + 0.05*array(rnorm(802),dim(y))

###############################
####  Basis Object      #######
###############################

knots  = seq(-0.1,20.1,0.2)
norder = 3
nbasis = length(knots) + norder - 2
range  = c(-0.1,20.1)

bbasis = create.bspline.basis(range=range,nbasis=nbasis,
	norder=norder,breaks=knots)

# Initial values for coefficients will be obtained by smoothing

fd.data = array(data,c(dim(data)[1],1,dim(data)[2]))

DEfd = data2fd(fd.data,times,bbasis,fdnames=list(NULL,NULL,c('V','R')) )

coefs = DEfd$coefs[,1,]

####################################
####  Optimization Control Lists ###
####################################

control=list()                 
control$trace  = 0
control$maxit  = 1000
control$maxtry = 10
control$reltol = 1e-6
control$meth   = "BFGS"

control.in = control
control.in$reltol      = 1e-12
control.in$print.level = 0
control.in$iterlim     = 1000

control.out = control
control.out$trace = 2

#################################
### Initial Parameter Guesses ###
#################################

profile.obj = LS.setup(pars=pars,fn=make.fhn(),lambda=10000,
                       times=times,fd.obj=DEfd)
lik = profile.obj$lik
proc= profile.obj$proc

pres = ParsMatchOpt(pars,coefs,proc)

npars = pres$pars

#############################################################
### If We Only Observe One State, We Can Re-Smooth Others ### 
#############################################################

tcoefs = coefs
tcoefs[,2] = 0

fres = FitMatchOpt(coefs=tcoefs,which=2,pars=pars,proc)

ncoefs = fres$coefs

###############################
#### Parameter Optimization ###
###############################

spars = c(0.2,0.2,2)          # Perturbed parameters
names(spars)=names(pars)
lambda = 10000

### SSE Shortcuts ####

Ires1	= Smooth.LS(make.fhn(),data,t,pars=spars,coefs,bbasis,lambda=lambda,
  in.meth='nlminb')
  
Ores1 = Profile.LS(make.fhn(),data=data,times=t,pars=spars,coefs=coefs,
  basisvals=bbasis,lambda=lambda,in.meth='nlminb',out.meth='ProfileGN')
#	control.in=control.in,control.out=control.out)
	

### SSE with ProfileErr ###


Ires2 = inneropt(data,times=times,pars,coefs,lik,proc,in.meth='nlminb') #,control.in)

Ores2 = outeropt(data=data,times=times,pars=pars,coefs=coefs,lik=lik,proc=proc,
  in.meth="nlminb",out.meth="optim")#,control.in=control.in,control.out=control.out)


### Multinorm ####

var = c(1,0.01)

Ires3 = Smooth.multinorm(make.fhn(),data,t,pars=spars,coefs,bbasis,var=var,in.meth='nlminb')#,control.in=control.in)

Ores3 = Profile.multinorm(make.fhn(),data,t,pars=spars,coefs,bbasis,var=var, 
  out.meth='nlminb',in.meth='nlminb')#,control.in=control.in,control.out=control.out)



# Lets look at the result

DEfd = fd(Ores1$coefs,bbasis)   # Data and reconstructed trajectory

par(mfrow=c(2,1))
 plotfit.fd(data,t,DEfd)
 

traj = as.matrix(Ores1$proc$bvals$bvals%*%Ores1$coefs)    # Look at how well the
colnames(traj) = Ores1$proc$more$names                    # derivative of the
dtraj = as.matrix(Ores1$proc$bvals$dbvals%*%Ores1$coefs)  # trajectory fits the 
ftraj = Ores1$proc$more$fn(t,traj,Ores1$pars)             # right hand side. 

matplot(dtraj,type='l',col=2)
matplot(ftraj,type='l',col=4,add=TRUE) 
 
 
Profile.covariance(pars=Ores1$pars,times=t,data=data,coefs=Ores1$coefs,
  lik=Ores1$lik,proc=Ores1$proc)
  
  
#################################################################
#### We can also estimate derivatives by finite differencing ####
#### if we don't feel like calculating them analytically.    ####
#################################################################

Ires1	= Smooth.LS(make.fhn()$fn,data,t,pars=spars,fd.obj=DEfd,lambda=lambda,
  in.meth='nlminb',control.in=control.in)
  

Ores1 = Profile.LS(make.fhn()$fn,data=data,times=t,pars=spars,fd.obj=DEfd,
  lambda=lambda,in.meth='nlminb',out.meth='nls',control.in=control.in,control.out=control.out)