R: Statistical inference of ordinary differential equations...

simode {simode}

R Documentation

Statistical inference of ordinary differential equations using separable integral-matching

Description

Estimating the parameters of an ODE system in two stages: 1) Estimate the parameters using separable integral-matching, 2) Estimate the parameters using nonlinear least squares starting from the values obtained in stage 1.

Usage

simode(
  equations,
  pars,
  time,
  obs,
  obs_sets = 1,
  nlin_pars = NULL,
  likelihood_pars = NULL,
  fixed = NULL,
  start = NULL,
  lower = NULL,
  upper = NULL,
  im_method = c("separable", "non-separable"),
  decouple_equations = F,
  gen_obs = NULL,
  calc_nll = NULL,
  simode_ctrl = simode.control(),
  ...
)

Arguments

`equations`	Named vector. The equations describing the ODE system. Each element of the vector should contain a character representation of the right-hand side of an equation, and should be named according to the left-hand side of the equation (i.e., the variable name). An equation can contain parameters appearing in `pars`, variables appearing in the equations names, observed non-modeled variables appearing in `obs`, and/or any function of 't', which is a reserved symbol for the time domain.
`pars`	The names of the parameters and initial conditions to be estimated. An initial condition name for a certain variable is the name given to the relevant equation in `equations` (e.g., if an equation is named 'x' than its initial condition should be named 'x' as well). Note: The symbol 't' is reserved for the time domain and cannot be used as a parameter name.
`time`	Time points of the observations. Either a vector, if the same time points were used for observing all variables, or a list of vectors the length of `obs`, of which each element is the length of the relevant element in `obs`.
`obs`	Named list. The observations. When `obs_sets=1`, `obs` should contain a list of vectors with observations of either a variable described by one of the equations (named according to the relevant equation name) or a non-modeled variable appearing in one of the equations. Each observations vector should be the length of the relevant time vector. When `obs_sets>1`, `obs` should contain a list, where each list member is a list that fits the description in the case of `obs_sets=1`.
`obs_sets`	Number of observations sets. When `obs_sets>1`, the function will fit the set of observations according to the value of `obs_sets_fit` in `simode_ctrl`.
`nlin_pars`	Names of parameters or initial conditions that will be estimated in stage 1 using nonlinear least squares optimization. The parameter names in `nlin_pars` must appear in `pars`.
`likelihood_pars`	Names of likelihood parameters not appearing in the ODE system, which are needed for the the user-defined function `calc_nll`. The parameter names in `likelihood_pars` must appear in `pars`.
`fixed`	Named vector. Fixed values for one or more of the ODE system parameters or initial conditions. Parameters in this list will not be estimated.
`start`	Named vector. Starting values for optimization of parameters/initial conditions. Must contain starting values for all the parameters in `nlin_pars` and `likelihood_pars`. If im_method="non-seperable", can optionally contain starting values for any other parameter/initial condition.
`lower`	Named vector. Lower bounds for any parameter/initial condition.
`upper`	Named vector. Upper bounds for any parameter/initial condition.
`im_method`	The method to use for integral-matching. Default "separable" means that linear parameters are estimated directly while "non-separable" means that linear parameters are estimated using nonlinear least squares optimization. If none of the parameters are linear then the default can be used.
`decouple_equations`	Whether to fit each equation separately in the integral-matching stage.
`gen_obs`	A user-defined function for completing missing observations (see Details).
`calc_nll`	A user-defined function for calculating negative log-likelihood for the model (see Details).
`simode_ctrl`	Various control parameters. See `simode.control`.
`...`	Additional arguments passed to `optim`, `gen_obs` and `calc_nll`

Details

gen_obs can be used in cases of a partially observed system, for which observations of the missing variables can be generated given values for the system parameters. The function will be called during the optimization using integral-matching.
It must be defined as gen_obs <- function(equations, pars, x0, time, obs, ...), where:

equations the ODE equations
pars the parameter values
x0 the initial conditions
time the timing of the observations (vector or list)
obs the observations
... additional parameters passed from the call to simode

The function should return a list with two items:

time the vector or list of time points of the observations
obs the list of observations with the newly generated observations

calc_nll allows the user to pass his own likelihood function to be used in the optimization in the second stage (if not defined, the default nonlinear least squares optimization will be used). The likelihood function will also be used in a following call to profile, for the calculation of likelihood profiles. It must be defined as calc_nll <- function(pars, time, obs, model_out, ...), where:

pars the parameter values
time the timing of the observations (vector or list)
obs the observations
model_out the model output returned from a call to solve_ode. If time is a list with possibly different times for each variable then model_out will contain a union of all these times.
... additional parameters passed from the call to simode

The function should return the negative log-likelihood.

Value

If obs_sets=1, the function returns a simode object containing the parameter estimates after integral-matching (stage 1) and after nonlinear least squares optimization (stage 2). If obs_sets>1 and obs_sets_fit!="together" in simode_ctrl, the function returns a list.simode object which is a list of simode objects the length of obs_sets.

References

Dattner & Klaassen (2015). Optimal Rate of Direct Estimators in Systems of Ordinary Differential Equations Linear in Functions of the Parameters, Electronic Journal of Statistics, Vol. 9, No. 2, 1939-1973.

Dattner, Miller, Petrenko, Kadouriz, Jurkevitch & Huppert (2017). Modelling and Parameter Inference of Predator-prey Dynamics in Heterogeneous Environments Using The Direct Integral Approach, Journal of The Royal Society Interface 14.126: 20160525.

Examples


## =================================================
## Predator-Prey Lotka-Volterra model
## =================================================

## generate model equations and parameters (X=Prey,Y=Predator)
pars <- c('alpha','beta','gamma','delta')
vars <- c('X','Y')
eq_X <- 'alpha*X-beta*X*Y'
eq_Y <- 'delta*X*Y-gamma*Y'
equations <- c(eq_X,eq_Y)
names(equations) <- vars
x0 <- c(0.9,0.9)
names(x0) <- vars
theta <- c(2/3,4/3,1,1)
names(theta) <- pars

## generate observations
n <- 50
time <- seq(0,25,length.out=n)
model_out <- solve_ode(equations,theta,x0,time)
x_det <- model_out[,vars]
set.seed(1000)
sigma <- 0.05
obs <- list()
for(i in 1:length(vars)) {
  obs[[i]] <- pmax(0, rnorm(n,x_det[,i],sigma))
}
names(obs) <- vars

## estimate model parameters with known initial conditions
simode_fit1 <- simode(equations=equations, pars=pars, fixed=x0, time=time, obs=obs)
plot(simode_fit1, type='fit', time=seq(0,25,length.out=100), pars_true=theta, mfrow=c(2,1))
plot(simode_fit1, type='est', pars_true=theta)


## estimate model parameters and initial conditions
simode_fit2 <- simode(equations=equations, pars=c(pars,vars), time=time, obs=obs)
plot(simode_fit2, type='fit', time=seq(0,25,length.out=100), pars_true=c(theta,x0), mfrow=c(2,1))
plot(simode_fit2, type='est', pars_true=c(theta,x0))

profiles_fit2 <- profile(simode_fit2,step_size=0.01,max_steps=50)
plot(profiles_fit2,mfrow=c(2,3))
ci_fit2 <- confint(profiles_fit2)
ci_fit2
plot(ci_fit2,pars_true=c(theta,x0),legend=T)

[Package simode version 1.2.2 Index]