Package 'pCODE' reference manual

Title:	Estimation of an Ordinary Differential Equation Model by Parameter Cascade Method
Description:	An implementation of the parameter cascade method Ramsay, J. O., Hooker,G., Campbell, D., and Cao, J. (2007) <doi:10.1111/j.1467-9868.2007.00610.x> for estimating ordinary differential equation models with missing or complete observations. It combines smoothing method and profile estimation to estimate any non-linear dynamic system. The package also offers variance estimates for parameters of interest based on either bootstrap or Delta method.
Authors:	Haixu Wang [aut, cre], Jiguo Cao [aut]
Maintainer:	Haixu Wang <[email protected]>
License:	GPL
Version:	0.9.4
Built:	2024-11-14 06:01:13 UTC
Source:	https://github.com/alex-haixuw/pcode

Bootstrap variance estimator of structural parameters.

Description

Obtaining an estimate of variance for structural parameters by bootstrap method.

Usage

bootsvar(data, time, ode.model,par.names,state.names, likelihood.fun = NULL,
       par.initial, basis.list, lambda = NULL,bootsrep,controls = NULL)
bootsvar(data, time, ode.model,par.names,state.names, likelihood.fun = NULL,
       par.initial, basis.list, lambda = NULL,bootsrep,controls = NULL)

Arguments

`data`	A data frame or a matrix contain observations from each dimension of the ODE model.
`time`	A vector contain observation times or a matrix if time points are different between dimensions.
`ode.model`	An R function that computes the time derivative of the ODE model given observations of states variable and structural parameters.
`par.names`	The names of structural parameters defined in the 'ode.model'.
`state.names`	The names of state variables defined in the 'ode.model'.
`likelihood.fun`	A likelihood function passed to PCODE in case of that the error termsdevtools::document()do not have a Normal distribution.
`par.initial`	Initial value of structural parameters to be optimized.
`basis.list`	A list of basis objects for smoothing each dimension's observations. Can be the same or different across dimensions.
`lambda`	Penalty parameter.
`bootsrep`	Bootstrap sample to be used for estimating variance.
`controls`	A list of control parameters. Same as the controls in `pcode`.

Value

boots.var The bootstrap variance of each structural parameters.

Numeric estimation of variance of structural parameters by Delta method.

Description

Obtaining variance of structural parameters by Delta method.

Usage

deltavar(data, time, ode.model,par.names,state.names,
        likelihood.fun, par.initial, basis.list, lambda,stepsize,y_stepsize,controls)
deltavar(data, time, ode.model,par.names,state.names,
        likelihood.fun, par.initial, basis.list, lambda,stepsize,y_stepsize,controls)

Arguments

`data`	A data frame or a matrix contain observations from each dimension of the ODE model.
`time`	A vector contain observation times or a matrix if time points are different between dimensions.
`ode.model`	An R function that computes the time derivative of the ODE model given observations of states variable and structural parameters.
`par.names`	The names of structural parameters defined in the 'ode.model'.
`state.names`	The names of state variables defined in the 'ode.model'.
`likelihood.fun`	A likelihood function passed to PCODE in case of that the error termsdevtools::document()do not have a Normal distribution.
`par.initial`	Initial value of structural parameters to be optimized.
`basis.list`	A list of basis objects for smoothing each dimension's observations. Can be the same or different across dimensions.
`lambda`	Penalty parameter.
`stepsize`	Stepsize used in estimating partial derivatives with respect to structural parameters for the Delta method.
`y_stepsize`	Stepsize used in estimating partial derivatives with respect to observations for the Delta method.
`controls`	A list of control parameters. Same as the controls in `pcode`.

Value

par.var The variance of structural parameters obtained by Delta method.

Inner objective function (Single dimension version)

Description

An objective function combines the sum of squared error of basis expansion estimates and the penalty controls how those estimates fail to satisfies the ODE model

Usage

innerobj(basis_coef, ode.par, input, derive.model,NLS)
innerobj(basis_coef, ode.par, input, derive.model,NLS)

Arguments

`basis_coef`	Basis coefficients for interpolating observations given a basis object.
`ode.par`	Structural parameters of the ODE model.
`input`	Contains dependencies for the optimization, including observations, penalty parameter lambda, and etc..
`derive.model`	The function defines the ODE model and is the same as the ode.model in 'pcode'
`NLS`	Default is `TRUE` so the function returns vector of residuals, and otherwise returns sum of squared errors.

Value

residual.vec

Vector of residuals and evaluation of penalty function on quadrature points for approximating the integral.

Inner objective function (likelihood and multiple dimension version)

Description

An objective function combines the likelihood or loglikelihood of errors from each dimension of state variables and the penalty controls how the state estimates fail to satisfy the ODE model.

Usage

innerobj_lkh(basis_coef, ode.par, input, derive.model, likelihood.fun)
innerobj_lkh(basis_coef, ode.par, input, derive.model, likelihood.fun)

Arguments

`basis_coef`	Basis coefficients for interpolating observations given a basis boject.
`ode.par`	Structural parameters of the ODD model.
`input`	Contains dependencies for the optimization, including observations, ode penalty, and etc..
`derive.model`	The function defines the ODE model and is the same as the ode.model in 'pcode'.
`likelihood.fun`	The likelihood or loglikelihood function of the errors.

Value

obj.eval The evaluation of the inner objective function.

Inner objective function (Likelihood and Single dimension version)

Description

An objective function combines the likelihood or loglikelihood of errors from each dimension of state variables and the penalty controls how the state estimates fail to satisfy the ODE model.

Usage

innerobj_lkh_1d(basis_coef, ode.par, input, derive.model, likelihood.fun)
innerobj_lkh_1d(basis_coef, ode.par, input, derive.model, likelihood.fun)

Arguments

`basis_coef`	Basis coefficients for interpolating observations given a basis boject.
`ode.par`	Structural parameters of the ODD model.
`input`	Contains dependencies for the optimization, including observations, ode penalty, and etc..
`derive.model`	The function defines the ODE model and is the same as the ode.model in 'pcode'.
`likelihood.fun`	The likelihood or loglikelihood function of the errors.

Value

obj.eval The evaluation of the inner objective function.

Inner objective function (multiple dimension version)

Description

An objective function combines the sum of squared error of basis expansion estimates and the penalty controls how those estimates fail to satisfies the ODE model

Usage

innerobj_multi(basis_coef, ode.par, input, derive.model,NLS)
innerobj_multi(basis_coef, ode.par, input, derive.model,NLS)

Arguments

`basis_coef`	Basis coefficients for interpolating observations given a basis object.
`ode.par`	Structural parameters of the ODE model.
`input`	Contains dependencies for the optimization, including observations, penalty parameter lambda, and etc..
`derive.model`	The function defines the ODE model and is the same as the `ode.model` in `pcode`.
`NLS`	Default is `TRUE` so the function returns vector of residuals, and otherwise returns sum of squared errors.

Value

residual.vec

Vector of residuals and evaluation of penalty function on quadrature points for approximating the integral.

Inner objective function (multiple dimension version with unobserved state variables)

Description

An objective function combines the sum of squared error of basis expansion estimates and the penalty controls how those estimates fail to satisfies the ODE model

Usage

innerobj_multi_missing(basis_coef, ode.par, input, derive.model,NLS)
innerobj_multi_missing(basis_coef, ode.par, input, derive.model,NLS)

Arguments

`basis_coef`	Basis coefficients for interpolating observations given a basis object.
`ode.par`	Structural parameters of the ODE model.
`input`	Contains dependencies for the optimization, including observations, penalty parameter lambda, and etc..
`derive.model`	The function defines the ODE model and is the same as the ode.model in 'pcode'
`NLS`	Default is `TRUE` so the function returns vector of residuals, and otherwise returns sum of squared errors.

Value

residual.vec

Vector of residuals and evaluation of penalty function on quadrature points for approximating the integral.

Optimizer for non-linear least square problems

Description

Obtain the solution to minimize the sum of squared errors of the defined function fun by levenberg-marquardt method. Adapted from PRACMA package.

Usage

nls_optimize(fun, x0, ..., options,verbal)
nls_optimize(fun, x0, ..., options,verbal)

Arguments

`fun`	The function returns the vector of weighted residuals.
`x0`	The initial value for optimization.
`...`	Parameters to be passed for `fun`
`options`	Additional optimization controls.
`verbal`	Default = 1 for printing iteration and other for suppressing

Value

par

The solution to the non-linear least square problem, the same size as x0

Optimizer for non-linear least square problems (for inner objective functions)

Description

Obtain the solution to minimize the sum of squared errors of the defined function fun by levenberg-marquardt method. Adapted from PRACMA package.

Usage

nls_optimize.inner(fun, x0, ..., options)
nls_optimize.inner(fun, x0, ..., options)

Arguments

`fun`	The function returns the vector of weighted residuals.
`x0`	The initial value for optimization.
`...`	Parameters to be passed for `fun`
`options`	Additional optimization controls.

Value

par

The solution to the non-linear least square problem, the same size as x0

Outter objective function (Single dimension version)

Description

An objective function of the structural parameter computes the measure of fit.

Usage

outterobj(ode.parameter, basis.initial, derivative.model, inner.input, NLS)
outterobj(ode.parameter, basis.initial, derivative.model, inner.input, NLS)

Arguments

`ode.parameter`	Structural parameters of the ODE model.
`basis.initial`	Initial values of the basis coefficients for nonlinear least square optimization.
`derivative.model`	The function defines the ODE model and is the same as the ode.model in 'pcode'
`inner.input`	Input that will be passed to the inner objective function. Contains dependencies for the optimization, including observations, penalty parameter lambda, and etc..
`NLS`	Default is `TRUE` so the function returns vector of residuals, and otherwise returns sum of squared errors.

Value

residual

Vector of residuals and evaluation of penalty function on quadrature points for approximating the integral.

Outter objective function (likelihood and multiple dimension version)

Description

An objective function of the structural parameter computes the measure of fit.

Usage

outterobj_lkh(ode.parameter, basis.initial, derivative.model, likelihood.fun, inner.input)
outterobj_lkh(ode.parameter, basis.initial, derivative.model, likelihood.fun, inner.input)

Arguments

`ode.parameter`	Structural parameters of the ODE model.
`basis.initial`	Initial values of the basis coefficients for nonlinear least square optimization.
`derivative.model`	The function defines the ODE model and is the same as the ode.model in 'pcode'
`likelihood.fun`	The likelihood or loglikelihood function of the errors.
`inner.input`	Input that will be passed to the inner objective function. Contains dependencies for the optimization, including observations, penalty parameter lambda, and etc..

Value

neglik The negative of the likelihood or the loglikelihood function that will be passed further to the 'optim' function.

Outter objective function (likelihood and single dimension version)

Description

An objective function of the structural parameter computes the measure of fit.

Usage

outterobj_lkh_1d(ode.parameter, basis.initial,
                        derivative.model, likelihood.fun, inner.input)
outterobj_lkh_1d(ode.parameter, basis.initial,
                        derivative.model, likelihood.fun, inner.input)

Arguments

`ode.parameter`	Structural parameters of the ODE model.
`basis.initial`	Initial values of the basis coefficients for nonlinear least square optimization.
`derivative.model`	The function defines the ODE model and is the same as the ode.model in 'pcode'
`likelihood.fun`	The likelihood or loglikelihood function of the errors.
`inner.input`	Input that will be passed to the inner objective function. Contains dependencies for the optimization, including observations, penalty parameter lambda, and etc..

Value

neglik The negative of the likelihood or the loglikelihood function that will be passed further to the 'optim' function.

Outter objective function (multiple dimension version with unobserved state variables)

Description

An objective function of the structural parameter computes the measure of fit for the basis expansion.

Usage

outterobj_multi_missing(ode.parameter, basis.initial, derivative.model, inner.input, NLS)
outterobj_multi_missing(ode.parameter, basis.initial, derivative.model, inner.input, NLS)

Arguments

`ode.parameter`	Structural parameters of the ODE model.
`basis.initial`	Initial values of the basis coefficients for nonlinear least square optimization.
`derivative.model`	The function defines the ODE model and is the same as the ode.model in 'pcode'
`inner.input`	Input that will be passed to the inner objective function. Contains dependencies for the optimization, including observations, penalty parameter lambda, and etc..
`NLS`	Default is `TRUE` so the function returns vector of residuals, and otherwise returns sum of squared errors.

Value

residual

Vector of residuals and evaluation of penalty function on quadrature points for approximating the integral.

Outter objective function (multiple dimension version)

Description

An objective function of the structural parameter computes the measure of fit for the basis expansion.

Usage

outterobj_multi_nls(ode.parameter, basis.initial, derivative.model, inner.input, NLS)
outterobj_multi_nls(ode.parameter, basis.initial, derivative.model, inner.input, NLS)

Arguments

`ode.parameter`	Structural parameters of the ODE model.
`basis.initial`	Initial values of the basis coefficients for nonlinear least square optimization.
`derivative.model`	The function defines the ODE model and is the same as the `ode.model` in `pcode`.
`inner.input`	Input that will be passed to the inner objective function. Contains dependencies for the optimization, including observations, penalty parameter lambda, and etc..
`NLS`	Default is `TRUE` so the function returns vector of residuals, and otherwise returns sum of squared errors.

Value

residual

Vector of residuals and evaluation of penalty function on quadrature points for approximating the integral.

Parameter Cascade Method for Ordinary Differential Equation Models

Description

Obtain estimates of both structural and nuisance parameters of an ODE model by parameter cascade method.

Usage

pcode(data, time, ode.model, par.names, state.names,
             likelihood.fun, par.initial, basis.list,lambda,controls)
pcode(data, time, ode.model, par.names, state.names,
             likelihood.fun, par.initial, basis.list,lambda,controls)

Arguments

`data`	A data frame or a matrix contain observations from each dimension of the ODE model.
`time`	A vector contain observation times or a matrix if time points are different between dimensions.
`ode.model`	An R function that computes the time derivative of the ODE model given observations of states variable and structural parameters.
`par.names`	The names of structural parameters defined in the 'ode.model'.
`state.names`	The names of state variables defined in the 'ode.model'.
`likelihood.fun`	A likelihood function passed to PCODE in case of that the error terms do not have a Normal distribution.
`par.initial`	Initial value of structural parameters to be optimized.
`basis.list`	A list of basis objects for smoothing each dimension's observations. Can be the same or different across dimensions.
`lambda`	Penalty parameter for controling the fidelity of interpolation.
`controls`	A list of control parameters. See Details.

Details

The controls argument is a list providing addition inputs for the nonlinear least square optimizer or general optimizer optim:

nquadpts: Determine the number of quadrature points for approximating an integral. Default is 101.
smooth.lambda: Determine the smoothness penalty for obtaining initial value of nuisance parameters.
tau: Initial value of Marquardt parameter. Small values indicate good initial values for structural parameters.
tolx: Tolerance for parameters of objective functions. Default is set at 1e-6.
tolg: Tolerance for the gradient of parameters of objective functions. Default is set at 1e-6.
maxeval: The maximum number of evaluation of the outter optimizer. Default is set at 20.

Value

`structural.par`	The structural parameters of the ODE model.
`nuisance.par`	The nuisance parameters or the basis coefficients for interpolating observations.

Examples

library(fda)
library(deSolve)
library(MASS)
library(pracma)
#Simple ode model example
#define model parameters
model.par   <- c(theta = c(0.1))
#define state initial value
state       <- c(X     = 0.1)
#Define model for function 'ode' to numerically solve the system
ode.model <- function(t, state,parameters){
 with(as.list(c(state,parameters)),
      {
        dX <- theta*X*(1-X/10)
        return(list(dX))
      })
}
#Observation time points
times <- seq(0,100,length.out=101)
#Solve the ode model
desolve.mod <- ode(y=state,times=times,func=ode.model,parms = model.par)
#Prepare for doing parameter cascading method
#Generate basis object for interpolation and as argument of pcode
#21 konts equally spaced within [0,100]
knots <- seq(0,100,length.out=21)
#order of basis functions
norder <- 4
#number of basis funtions
nbasis <- length(knots) + norder - 2
#creating Bspline basis
basis  <- create.bspline.basis(c(0,100),nbasis,norder,breaks = knots)
#Add random noise to ode solution for simulating data
nobs  <- length(times)
scale <- 0.1
noise <- scale*rnorm(n = nobs, mean = 0 , sd = 1)
observation <- desolve.mod[,2] + noise
#parameter estimation
pcode(data = observation, time = times, ode.model = ode.model,
                     par.initial = 0.1, par.names = 'theta',state.names = 'X',
                     basis.list = basis, lambda = 1e2)

library(fda)
library(deSolve)
library(MASS)
library(pracma)
#Simple ode model example
#define model parameters
model.par   <- c(theta = c(0.1))
#define state initial value
state       <- c(X     = 0.1)
#Define model for function 'ode' to numerically solve the system
ode.model <- function(t, state,parameters){
 with(as.list(c(state,parameters)),
      {
        dX <- theta*X*(1-X/10)
        return(list(dX))
      })
}
#Observation time points
times <- seq(0,100,length.out=101)
#Solve the ode model
desolve.mod <- ode(y=state,times=times,func=ode.model,parms = model.par)
#Prepare for doing parameter cascading method
#Generate basis object for interpolation and as argument of pcode
#21 konts equally spaced within [0,100]
knots <- seq(0,100,length.out=21)
#order of basis functions
norder <- 4
#number of basis funtions
nbasis <- length(knots) + norder - 2
#creating Bspline basis
basis  <- create.bspline.basis(c(0,100),nbasis,norder,breaks = knots)
#Add random noise to ode solution for simulating data
nobs  <- length(times)
scale <- 0.1
noise <- scale*rnorm(n = nobs, mean = 0 , sd = 1)
observation <- desolve.mod[,2] + noise
#parameter estimation
pcode(data = observation, time = times, ode.model = ode.model,
                     par.initial = 0.1, par.names = 'theta',state.names = 'X',
                     basis.list = basis, lambda = 1e2)

Parameter Cascade Method for Ordinary Differential Equation Models (Single dimension version)

Description

Obtain estiamtes of structural parameters of an ODE model by parameter cascade method.

Usage

pcode_1d(data, time, ode.model, par.initial,par.names, basis,lambda,controls = list())
pcode_1d(data, time, ode.model, par.initial,par.names, basis,lambda,controls = list())

Arguments

`data`	A data frame or a vector contains observations from the ODE model.
`time`	The vector contain observation times.
`ode.model`	Defined R function that computes the time derivative of the ODE model given observations of states variable.
`par.initial`	Initial value of structural parameters to be optimized.
`par.names`	The names of structural parameters defined in the 'ode.model'.
`basis`	A basis objects for smoothing observations.
`lambda`	Penalty parameter.
`controls`	A list of control parameters. See ‘Details’.

Value

`structural.par`	The structural parameters of the ODE model.
`nuisance.par`	The nuisance parameters or the basis coefficients for interpolating observations.

pcode_lkh (likelihood and multiple dimension version)

Description

Obtain estimates of both structural and nuisance parameters of an ODE model by parameter cascade method.

Usage

pcode_lkh(data, likelihood.fun, time, ode.model, par.names,
                 state.names, par.initial, basis.list, lambda, controls)
pcode_lkh(data, likelihood.fun, time, ode.model, par.names,
                 state.names, par.initial, basis.list, lambda, controls)

Arguments

`data`	A data frame or a matrix contain observations from each dimension of the ODE model.
`likelihood.fun`	A function computes the likelihood or the loglikelihood of the errors.
`time`	A vector contains observation ties or a matrix if time points are different between dimesion.
`ode.model`	An R function that computes the time derivative of the ODE model given observations of states variable and structural parameters.
`par.names`	The names of structural parameters defined in the 'ode.model'.
`state.names`	The names of state variables defined in the 'ode.model'.
`par.initial`	Initial value of structural parameters to be optimized.
`basis.list`	A list of basis objects for smoothing each dimension's observations. Can be the same or different across dimensions.
`lambda`	Penalty parameter.
`controls`	A list of control parameters. See ‘Details’.

Details

The controls argument is a list providing addition inputs for the nonlinear least square optimizer:

nquadpts Determine the number of quadrature points for approximating an integral. Default is 101.
smooth.lambda Determine the smoothness penalty for obtaining initial value of nuisance parameters.
tau Initial value of Marquardt parameter. Small values indicate good initial values for structural parameters.
tolx Tolerance for parameters of objective functions. Default is set at 1e-6.
tolg Tolerance for the gradient of parameters of objective functions. Default is set at 1e-6.
maxeval The maximum number of evaluation of the optimizer. Default is set at 20.

Value

`structural.par`	The structural parameters of the ODE model.
`nuisance.par`	The nuisance parameters or the basis coefficients for interpolating observations.

Parameter Cascade Method for Ordinary Differential Equation Models (likelihood and Single dimension version)

Description

Obtain estimates of both structural and nuisance parameters of an ODE model by parameter cascade method.

Usage

pcode_lkh_1d(data, likelihood.fun, time, ode.model, par.names,
                    state.names, par.initial, basis.list, lambda, controls)
pcode_lkh_1d(data, likelihood.fun, time, ode.model, par.names,
                    state.names, par.initial, basis.list, lambda, controls)

Arguments

`data`	A data frame or a matrix contain observations from each dimension of the ODE model.
`likelihood.fun`	A function computes the likelihood or the loglikelihood of the errors.
`time`	A vector contains observation ties or a matrix if time points are different between dimesion.
`ode.model`	An R function that computes the time derivative of the ODE model given observations of states variable and structural parameters.
`par.names`	The names of structural parameters defined in the 'ode.model'.
`state.names`	The names of state variables defined in the 'ode.model'.
`par.initial`	Initial value of structural parameters to be optimized.
`basis.list`	A list of basis objects for smoothing each dimension's observations. Can be the same or different across dimensions.
`lambda`	Penalty parameter.
`controls`	A list of control parameters. See ‘Details’.

Details

The controls argument is a list providing addition inputs for the nonlinear least square optimizer:

nquadpts Determine the number of quadrature points for approximating an integral. Default is 101.
smooth.lambda Determine the smoothness penalty for obtaining initial value of nuisance parameters.
tau Initial value of Marquardt parameter. Small values indicate good initial values for structural parameters.
tolx Tolerance for parameters of objective functions. Default is set at 1e-6.
tolg Tolerance for the gradient of parameters of objective functions. Default is set at 1e-6.
maxeval The maximum number of evaluation of the optimizer. Default is set at 20.

Value

`structural.par`	The structural parameters of the ODE model.
`nuisance.par`	The nuisance parameters or the basis coefficients for interpolating observations.

Parameter Cascade Method for Ordinary Differential Equation Models with missing state variable

Description

Obtain estiamtes of both structural and nuisance parameters of an ODE model by parameter cascade method when the dynamics are partially observed.

Usage

pcode_missing(data, time, ode.model, par.names, state.names,
                     likelihood.fun,par.initial, basis.list,lambda,controls)
pcode_missing(data, time, ode.model, par.names, state.names,
                     likelihood.fun,par.initial, basis.list,lambda,controls)

Arguments

`data`	A data frame or a matrix contain observations from each dimension of the ODE model.
`time`	A vector contain observation times or a matrix if time points are different between dimensions.
`ode.model`	An R function that computes the time derivative of the ODE model given observations of states variable and structural parameters.
`par.names`	The names of structural parameters defined in the 'ode.model'.
`state.names`	The names of state variables defined in the 'ode.model'.
`likelihood.fun`	A likelihood function passed to PCODE in case of that the error termsdevtools::document()do not have a Normal distribution.
`par.initial`	Initial value of structural parameters to be optimized.
`basis.list`	A list of basis objects for smoothing each dimension's observations. Can be the same or different across dimensions.
`lambda`	Penalty parameter.
`controls`	A list of control parameters. See Details.

Details

The controls argument is a list providing addition inputs for the nonlinear least square optimizer or general optimizer optim:

nquadpts Determine the number of quadrature points for approximating an integral. Default is 101.
smooth.lambda Determine the smoothness penalty for obtaining initial value of nuisance parameters.
tau Initial value of Marquardt parameter. Small values indicate good initial values for structural parameters.
tolx Tolerance for parameters of objective functions. Default is set at 1e-6.
tolg Tolerance for the gradient of parameters of objective functions. Default is set at 1e-6.
maxeval The maximum number of evaluation of the optimizer. Default is set at 20.

Value

`structural.par`	The structural parameters of the ODE model.
`nuisance.par`	The nuisance parameters or the basis coefficients for interpolating observations.

Evaluate basis objects over observation times and quadrature points

Description

Calculate all basis functions over observation time points and store them as columns in a single matrix for each dimension. Also include first and second order derivative. Repeat over quadrature points.

Usage

prepare_basis(basis, times, nquadpts)
prepare_basis(basis, times, nquadpts)

Arguments

`basis`	A basis object.
`times`	The vector contain observation times for corresponding dimension.
`nquadpts`	Number of quadrature points will be used later for approximating integrals.

Value

`Phi.mat`	Evaluations of all basis functions stored as columns in the matrix.
`Qmat`	Evaluations of all basis functions over quadrature points stored as columns in the matrix.
`Q.D1mat`	Evaluations of first order derivative all basis functions over quadrature points stored as columns in the matrix.
`Q.D2mat`	Evaluations of second order derivative all basis functions over quadrature points stored as columns in the matrix.
`quadts`	Quadrature points.
`quadwts`	Quadrature weights.

Find optimial penalty parameter lambda by cross-validation.

Description

Obtain the optimal sparsity parameter given a search grid based on cross validation score with replications.

Usage

tunelambda(data, time, ode.model, par.names, state.names,
                  par.initial, basis.list,lambda_grid,cv_portion,kfolds, rep,controls)
tunelambda(data, time, ode.model, par.names, state.names,
                  par.initial, basis.list,lambda_grid,cv_portion,kfolds, rep,controls)

Arguments

`data`	A data frame or matrix contrain observations from each dimension of the ODE model.
`time`	The vector contain observation times or a matrix if time points are different between dimensions.
`ode.model`	Defined R function that computes the time derivative of the ODE model given observations of states variable.
`par.names`	The names of structural parameters defined in the 'ode.model'.
`state.names`	The names of state variables defined in the 'ode.model'.
`par.initial`	Initial value of structural parameters to be optimized.
`basis.list`	A list of basis objects for smoothing each dimension's observations. Can be the same or different across dimensions.
`lambda_grid`	A search grid for finding the optimial sparsity parameter lambda.
`cv_portion`	A number indicating the proportion of data will be saved for doing cross validation. Default is set at 5 as minimum.
`kfolds`	A number indicating the number of folds the data should be seprated into.
`rep`	A integer controls the number of replication of doing cross-validation for each penalty parameter.
`controls`	A list of control parameters. See ‘Details’.

Value

`lambda_grid`	The original input vector of a search grid for the optimal lambda.
`cv.score`	The matrix contains the cross validation score for each lambda of each replication

Package 'pCODE'

Help Index

Bootstrap variance estimator of structural parameters.

Description

Usage

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Value

Numeric estimation of variance of structural parameters by Delta method.

Description

Usage

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Value

Inner objective function (Single dimension version)

Description

Usage

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Value

Inner objective function (likelihood and multiple dimension version)

Description

Usage

Arguments

Value

Inner objective function (Likelihood and Single dimension version)

Description

Usage

Arguments

Value

Inner objective function (multiple dimension version)

Description

Usage

Arguments

Value

Inner objective function (multiple dimension version with unobserved state variables)

Description

Usage

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Value

Optimizer for non-linear least square problems

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Value

Optimizer for non-linear least square problems (for inner objective functions)

Description

Usage

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Value

Outter objective function (Single dimension version)

Description

Usage

Arguments

Value

Outter objective function (likelihood and multiple dimension version)

Description

Usage

Arguments

Value

Outter objective function (likelihood and single dimension version)

Description

Usage

Arguments

Value

Outter objective function (multiple dimension version with unobserved state variables)

Description

Usage

Arguments

Value

Outter objective function (multiple dimension version)

Description

Usage

Arguments

Value

Parameter Cascade Method for Ordinary Differential Equation Models

Description

Usage

Arguments

Details

Value

Examples

Parameter Cascade Method for Ordinary Differential Equation Models (Single dimension version)