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## Example Diagnostics -- Learning the FitzHugh-Nagumo Equations
#
# Demonstration estimation of function functions.
library('CollocInfer')
# First Create Some data from the FitzHugh-Nagumo equations. We will first estimate
# a linear differential equation to mimic this, and then we will try to work
# out how to fix it up
t = 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')
y = lsoda(x0,t,make.fhn()$fn.ode,pars)
y = y[,2:3]
data = y + 0.2*matrix(rnorm(802),401,2)
# Now define a basis object
knots = seq(0,20,0.2)
norder = 3
nbasis = length(knots) + norder - 2
range = c(0,20)
bbasis = create.bspline.basis(range=range,nbasis=nbasis,
norder=norder,breaks=knots)
# Initial values for coefficients will be obtained by smoothing
DEfd = smooth.basis(t,data,fdPar(bbasis,1,0.5))
coefs = DEfd$fd$coefs
names(coefs) = c('V','R')
# Starting parameter estimates correspond to circular motion
spars = c(0,1,-1,0)
# Now set up some profiling; make.genlin() produces a linear ODE using parameters
# in row-wise order in the equaiton Dx = Ax
out = LS.setup(coefs=coefs,pars=spars,times=t,fn=make.genlin(),basisvals=bbasis,
lambda=c(100,100),names=c('V','R'))
lik = out$lik
proc = out$proc
# With this we can run model-based smoothing
Ires = inneropt(data,times=t,spars,coefs,lik,proc)
# and profiling
Ores = outeropt(data=data,times=t,pars=spars,coefs=Ires$coefs,lik=lik,proc=proc,
in.meth="nlminb",out.meth="nlminb")
# There is some clear lack of fit here
out2 = CollocInferPlots(Ores$coefs,Ores$pars,lik,proc,times=t,data=data)
# And we will attempt to relate the mis-match in Dx and f(x) to the value of x
# to see how we might be able to fix them up.
par(mfrow=c(2,2))
plot( out2$traj[,1], out2$dtraj[,1]-out2$ftraj[,1],type='l')
plot( out2$traj[,2], out2$dtraj[,1]-out2$ftraj[,1],type='l')
plot( out2$traj[,1], out2$dtraj[,2]-out2$ftraj[,2],type='l')
plot( out2$traj[,2], out2$dtraj[,2]-out2$ftraj[,2],type='l')
## Now set up empirical forcing functions to get a more precise handle on this
# by examining the equations Dx = f(x) + Psi(t) P where f(x) is the equation we
# estimated above, Psi is a new basis system and we will estimate the coefficients
# P as parameters via the same method.
# The additional functions g(t) = Psi(t) P are known as "empirical forcing
# functions", hence the "f" in front of various objects employed below.
# First we need a set of basis functions at which to evaluate the forcing
# functions and their evaluation at the quadrature points
fbasis = create.bspline.basis(range=range,nbasis=23,norder=4)
fbvals = eval.basis(proc$more$qpts,fbasis)
# Now we can call the steup functions, in this case we need to give a lot
# to the more object. In particular, we start off with the first more
# element of the proc object we just used
procmore = proc$more
# and we have to add to this the estimated parameters
procmore$p = Ores$pars
# and which elements to add the forcing functions to
procmore$which = 1:2
# and the evalution of fbasis at the quadrature points
procmore$psi = fbvals
# Now we call the setup function
out3 = LS.setup(coefs=Ores$coefs,pars=rep(0,fbasis$nbasis),times=t,
fn=make.diagnostics(),basisvals=bbasis,
lambda=c(100,100),names=c('V','R'),more=procmore)
lik2 = out3$lik
proc2 = out3$proc
# With this we can run model-based smoothing
Ires2 = inneropt(data,times=t,rep(0,2*fbasis$nbasis),Ores$coefs,lik2,proc2)
# and profiling
Ores2 = outeropt(data=data,times=t,pars=rep(0,2*fbasis$nbasis),coefs=Ires2$coefs,
lik=lik2,proc=proc2,in.meth="nlminb",out.meth="nlminb")
# Usual CollocInferPlots
out4 = CollocInferPlots(Ores2$coefs,Ores2$pars,lik2,proc2,times=t,data=data)
# Now we can look at the values of the time-varying forcing functions. These
# values are given by combining the forcing function basis and the parameters
fvals = eval.basis(out4$timevec,fbasis)%*%matrix(Ores2$pars,fbasis$nbasis,2)
# which we can plot
matplot(out4$timevec,fvals,type='l')
# And relate to the estimated trajectories where the mising cubic term can
# be discerned.
par(mfrow=c(2,1))
matplot(out4$traj[,1],fvals,type='l')
matplot(out4$traj[,2],fvals,type='l')
## Manual set-up.
# This is the equivalent analysis to what we have done above, but
# we create the lik and proc objects manually -- we believe that the structures
# we work with should not be hidden from the user.
# Usual meta-parameters; quadrature points, weights and knots
lambda = c(100,100)
qpts = knots
qwts = rep(1/length(knots),length(knots))
qwts = qwts%*%t(lambda)
weights = array(1,dim(data))
# Now I define a measurement process log likelihood along with some
# additional features: in this case it's squared error.
varnames = c('V','R')
parnames = c('a','b','c')
likmore = make.id()
likmore$weights = weights
lik = make.SSElik()
lik$more = likmore
lik$bvals = eval.basis(t,bbasis)
# Proc is a process log likelihood -- in this case treated as squared
# discrepancy from the ODE definition.
procmore = make.genlin()
procmore$names = varnames
procmore$parnames = parnames
procmore$more = list(mat=matrix(0,2,2),sub= matrix(c(1,1,1,1,2,2,2,1,3,2,2,4),4,3,byrow=TRUE))
procmore$weights = qwts
procmore$qpts = qpts
proc = make.SSEproc()
proc$more = procmore
proc$bvals = list(bvals=eval.basis(procmore$qpts,bbasis,0),
dbvals = eval.basis(procmore$qpts,bbasis,1))
spars = c(0,1,-1,0)
Ires = inneropt(data,times=t,spars,coefs,lik,proc,in.meth='nlminb')
Ores = outeropt(data=data,times=t,pars=spars,coefs=Ires$coefs,lik=lik,proc=proc,
in.meth="nlminb",out.meth="nlminb")
traj = as.matrix(proc$bvals$bvals %*% Ores$coefs)
dtraj = as.matrix(proc$bvals$dbvals %*% Ores$coefs)
ftraj = dtraj - proc$more$fn(proc$more$qpts,dtraj,Ores$pars,proc$more$more)
par(mfrow=c(2,2))
for(i in 1:2){
for(j in 1:2){
plot(traj[,i],ftraj[,j],type='l')
}
}
## Now we estimate some forcing functions
fbasis = create.bspline.basis(range=range,nbasis=23,norder=4)
dproc = make.SSEproc()
dproc$more = make.diagnostics()
dproc$more$qpts = procmore$qpts
dproc$more$weights = procmore$weights
dproc$more$more = procmore
dproc$more$more$p = Ores$pars
dproc$more$more$which = 1:2
dproc$more$more$psi = eval.basis(procmore$qpts,fbasis)
dproc$bvals = list(bvals=eval.basis(procmore$qpts,bbasis,0),
dbvals = eval.basis(procmore$qpts,bbasis,1))
dpars = rep(0,2*fbasis$nbasis)
dOres = outeropt(data=data,times=t,pars=dpars,coefs=Ires$coefs,lik=lik,proc=dproc,
in.meth="nlminb",out.meth="nlminb")
# Trajectories
force = dproc$more$more$psi %*% matrix(dOres$par,fbasis$nbasis,2)
traj = dproc$bvals$bvals %*% dOres$coefs
plot(traj[,1],force[,1],type='l')
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