KGode (version 1.0.4)

crossv: The 'crossv' function

Description

This function is used to estimate the weighting parameter for ode regularisation using cross validation.

Usage

crossv(lam, kkk, bbb, crtype, y_no, woption, resmtest, dtilda, fold)

Value

return list containing :

• lam - scalar containing the optimised weighting parameter.

• ress -vector containing the cross validation error for all choices of weighting parameter.

Arguments

lam

vector containing different choices of the weighting parameter of ode regularisation.

kkk

'ode' class object containing all information about the odes.

bbb

list of 'rkhs' class object containing the interpolation for all ode states.

crtype

character containing the optimisation scheme type. User can choose 'i' or '3'. 'i' is for fast iterative scheme and '3' for optimising the ode parameters and interpolation coefficients simultaneously.

y_no

matrix(of size n_s*n_o) containing noisy observations. The row(of length n_s) represent the ode states and the column(of length n_o) represents the time points.

woption

character containing the indication of using warping. If the warping scheme is done before using the ode regularisation, user can choose 'w' otherwise just leave this option empty.

resmtest

vector(of length n_o) containing the warped time points. This variable is only used if user want to combine warping and the ode regularisation.

dtilda

vector(of length n_o) containing the gradient of warping function. This variable is only used if user want to combine warping and the ode regularisation.

fold

scalar indicating the folds of cross validation.

Author

Mu Niu

Details

Arguments of the 'crossv' function are list of weighting parameter for ode regularisation, 'ode' class objects, 'rkhs' class objects, noisy observation, type of regularisation scheme, option of warping and the gradient of warping function. It return the interpolation for each of the ode states. The ode parameters are estimated using gradient matching, and the results are stored in the 'ode' class as the ode_par attribute.

Examples

# \dontshow{
  ##examples for checks: executable in < 5 sec together with the examples above not shown to users
  ### define ode 
  toy_fun = function(t,x,par_ode){
       alpha=par_ode[1]
      as.matrix( c( -alpha*x[1]) )
   }

   toy_grlNODE= function(par,grad_ode,y_p,z_p) { 
       alpha = par[1]
       dres= c(0)
       dres[1] = sum( 2*( z_p-grad_ode)*y_p*alpha ) #sum( -2*( z_p[1,2:lm]-dz1)*z1*alpha ) 
       dres
   }

  t_no = c(0.1,1,2,3,4,8)
  n_o = length(t_no)   
  y_no =  matrix( c(exp(-t_no)),ncol=1  )
  ######################## create and initialise ode object #########################################
 init_par = rep(c(0.1))
 init_yode = t(y_no)
 init_t = t_no

 kkk = ode$new(1,fun=toy_fun,grfun=toy_grlNODE,t=init_t,ode_par= init_par, y_ode=init_yode )

 ##### standard gradient matching
 ktype='rbf'
 rkgres = rkg(kkk,(y_no),ktype)
 bbb = rkgres$bbb

############ gradient matching + ode regularisation
crtype='i'
lam=c(1e-4,1e-5)
lamil1 = crossv(lam,kkk,bbb,crtype,y_no)
lambdai1=lamil1[[1]]
res = third(lambdai1[1],kkk,bbb,crtype)
## display ode parameters
res$oppar
# }
if (FALSE) {
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

############# gradient matching + third step
crtype='i'
## using cross validation to estimate the weighting parameters of the ode regularisation 
lam=c(1e-4,1e-5)
lamil1 = crossv(lam,kkk,bbb,crtype,y_no)
lambdai1=lamil1[[1]]
}