warpfun(kkkrkg, bbb, peod, eps, fixlens, y_no, testData, witer)

## Not run: 
require(mvtnorm)
noise = 0.1  
SEED = 19537
set.seed(SEED)
## Define ode function, we use lotka-volterra model in this example. 
## we have two ode states x[1], x[2] and four ode parameters alpha, beta, gamma and delta.
LV_fun = function(t,x,par_ode){
  alpha=par_ode[1]
  beta=par_ode[2]
  gamma=par_ode[3]
  delta=par_ode[4]
  as.matrix( c( alpha*x[1]-beta*x[2]*x[1] , -gamma*x[2]+delta*x[1]*x[2] ) )
}
## Define the gradient of ode function against ode parameters 
## df/dalpha,  df/dbeta, df/dgamma, df/ddelta where f is the differential equation.
LV_grlNODE= function(par,grad_ode,y_p,z_p) { 
alpha = par[1]; beta= par[2]; gamma = par[3]; delta = par[4]
dres= c(0)
dres[1] = sum( -2*( z_p[1,]-grad_ode[1,])*y_p[1,]*alpha ) 
dres[2] = sum( 2*( z_p[1,]-grad_ode[1,])*y_p[2,]*y_p[1,]*beta)
dres[3] = sum( 2*( z_p[2,]-grad_ode[2,])*gamma*y_p[2,] )
dres[4] = sum( -2*( z_p[2,]-grad_ode[2,])*y_p[2,]*y_p[1,]*delta)
dres
}

## create a ode class object
kkk0 = ode$new(2,fun=LV_fun,grfun=LV_grlNODE)
## set the initial values for each state at time zero.
xinit = as.matrix(c(0.5,1))
## set the time interval for the ode numerical solver.
tinterv = c(0,6)
## solve the ode numerically using predefined ode parameters. alpha=1, beta=1, gamma=4, delta=1.
kkk0$solve_ode(c(1,1,4,1),xinit,tinterv) 

## Add noise to the numerical solution of the ode model and use it as the noisy observation.
n_o = max( dim( kkk0$y_ode) )
t_no = kkk0$t
y_no =  t(kkk0$y_ode) + rmvnorm(n_o,c(0,0),noise*diag(2))

## create a ode class object by using the simulation data we created from the Ode numerical solver.
## If users have experiment data, they can replace the simulation data with the experiment data.
## set initial value of Ode parameters.
init_par = rep(c(0.1),4)
init_yode = t(y_no)
init_t = t_no
kkk = ode$new(1,fun=LV_fun,grfun=LV_grlNODE,t=init_t,ode_par= init_par, y_ode=init_yode )

## The following examples with CPU or elapsed time > 10s

## Use function 'rkg' to estimate the Ode parameters.
ktype ='rbf'
rkgres = rkg(kkk,y_no,ktype)
bbb = rkgres$bbb

###### warp all ode states
peod = c(6,5.3) ## the guessing period
eps= 1          ## the uncertainty level of period

###### learn the initial value of the hyper parameters of the warping basis function
fixlens=warpInitLen(peod,eps,rkgres)

kkkrkg = kkk$clone() ## make a copy of ode class objects
##learn the warping function, warp data points and do gradient matching in the warped time domain.
www = warpfun(kkkrkg,bbb,peod,eps,fixlens,y_no,kkkrkg$t)

dtilda= www$dtilda  ## gradient of warping function
bbbw = www$bbbw      ## interpolation in warped time domain
resmtest = www$wtime  ## warped time points
##display the results of parameter estimation using gradient matching in the warped time domain.
www$wkkk$ode_par       

## End(Not run)