Solving ordinary (ODEs) differntial equations with a specified time squence.

Description

The function computes numerical solutions for a system of ordinary differential equations given an initial state, parameters, and a sequence of time points t_1, ..., t_T.

Usage

dyn_solve(odefun, times, param, init, method)

Arguments

Details

This function numerically integrates a system of ODEs using solvers available in the deSolve package. The ODE system is defined by the user through odefun, which specifies the rate of change for each state variable.

The solver computes the values of the state variables at each time step, producing a trajectory of the system's evolution over time.

The ODE system must be written in first-order form:

\frac{dy}{dt} = f(t, y, p)

where: - y is the vector of state variables, - t is time, - p is a set of model parameters, - f is a function returning the derivatives.

This function simplifies the process of solving ODEs by managing input formatting and output structure, making it easier to extract and analyze results.

For details, see "ode".

Value

A list containing:

• times: copy of times

• <state variable>: Numeric vectors of computed values for each state variable.

Examples

### Lotka-Volterra equations ###
LVmod0D <- function(Time, State, Pars) {
  with(as.list(c(State, Pars)),{
    IngestC <- rI * P * C
    GrowthP <- rG * P * (1 - P/K)
    MortC <- rM * C

    dP <- GrowthP - IngestC
    dC <- IngestC * AE - MortC
    return(list(c(dP, dC)))
  })
}

### Define parameters ###
pars <- c(rI = 1.5, rG = 1.5, rM = 2, AE = 1, K = 10)

### Define time sequence ###
times <- seq(0, 30, by = 0.01)

### Initial conditions ###
state <- c(P = 1, C = 2)

### Solve ODE ###
dyn_solve(LVmod0D, times, pars, state)

Fit Gaussian process (GP) emulators for One-step-ahead approach

Description

The function fits GP emulators for each state variable by solving an ODE model one step ahead and using the generated outputs as training data.

Usage

dynemu_GP(odefun, times, param, X)

Arguments

Details

Given an initial set of input conditions, this function: 1. Solves the ODE system for a short time interval using dyn_solve. 2. Fits GP emulators to model the next-step evolution of each state variable.

The ODE system is defined by the user through odefun, and the GP emulators are trained using the generated one-step-ahead outputs.

Value

A list of GP emulators, each corresponding to a state variable.

Examples

library(lhs)
### Lotka-Volterra equations ###
LVmod0D <- function(Time, State, Pars) {
  with(as.list(c(State, Pars)), {
    IngestC <- rI * P * C
    GrowthP <- rG * P * (1 - P/K)
    MortC <- rM * C

    dP <- GrowthP - IngestC
    dC <- IngestC * AE - MortC
    return(list(c(dP, dC)))
  })
}

### Define parameters ###
pars <- c(rI = 1.5, rG = 1.5, rM = 2, AE = 1, K = 10)

### Define time sequence ###
times <- seq(0, 30, by = 0.01)

### Initial conditions ###
set.seed(1)
d <- 2
n <- 12*d
Design <- maximinLHS(n, d) * 5 # Generate LHS samples in range [0,5]
colnames(Design) <- c("P", "C")

### Fit GP emulators ###
fit.dyn <- dynemu_GP(LVmod0D, times, pars, Design)
print(fit.dyn)

Predictive posterior computation via exact closed-form expression for one-steap-ahead Gaussian process (GP) emulators

Description

The function computes the predictive posterior distribution (mean and variance) for GP emulators using closed-form expression, given uncertain inputs.

Usage

dynemu_exact(mean, var, dynGP)

Arguments

Details

Given a trained set of GP emulators 'dynGP' fitted using dynemu_GP, this function: 1. Computes the closed-form predictive posterior mean and variance for each state variable. 2. Incorporates input uncertainty by integrating over the input distribution via exact computation.

The computation follows a closed-form approach, leveraging the posterior distributions of Linked GP.

Value

A list containing:

• predy: A single-row matrix of predictive mean values, with each column for corresponding state variable.

• predsig2: A single-row matrix of predictive variance values, with each column for corresponding state variable.

Examples

library(lhs)
### Lotka-Volterra equations ###
LVmod0D <- function(Time, State, Pars) {
  with(as.list(c(State, Pars)), {
    IngestC <- rI * P * C
    GrowthP <- rG * P * (1 - P/K)
    MortC <- rM * C
    
    dP <- GrowthP - IngestC
    dC <- IngestC * AE - MortC
    return(list(c(dP, dC)))
  })
}

### Define parameters ###
pars <- c(rI = 1.5, rG = 1.5, rM = 2, AE = 1, K = 10)

### Define time sequence ###
times <- seq(0, 30, by = 0.01)

### Initial conditions ###
set.seed(1)
d <- 2
n <- 12*d
Design <- maximinLHS(n, d) * 5 # Generate LHS samples in range [0,5]
colnames(Design) <- c("P", "C")

### Fit GP emulators ###
fit.dyn <- dynemu_GP(LVmod0D, times, pars, Design)

### Define uncertain inputs ###
xmean <- c(P = 1, C = 2)
xvar <- c(P = 1e-5, C = 1e-5)

### Compute the next point ###
dynemu_exact(xmean, xvar, fit.dyn)

Predictive posterior computation via Monte Carlo (MC) approximation for one-steap-ahead Gaussian process (GP) emulators

Description

The function computes the predictive posterior distribution (mean and variance) for GP emulators using MC method, given uncertain inputs.

Usage

dynemu_mc(mean, var, dynGP, mc.sample)

Arguments

Details

Given a trained set of GP emulators 'dynGP' fitted using dynemu_GP, this function: 1. Computes the MC-approximated predictive posterior mean and variance for each state variable. 2. Incorporates input uncertainty by integrating over the input distribution via MC sampling.

Unlike dynemu_exact, which provides a closed-form solution, this function uses MC sampling to approximate the posterior.

Value

A list containing:

• predy: A single-row matrix of predictive mean values, with each column for corresponding state variable.

• predsig2: A single-row matrix of predictive variance values, with each column for corresponding state variable.

References

H. Mohammadi, P. Challenor, and M. Goodfellow (2019). Emulating dynamic non-linear simulators using Gaussian processes. Computational Statistics & Data Analysis, 139, 178-196; doi:10.1016/j.csda.2019.05.006

Examples

library(lhs)
### Lotka-Volterra equations ###
LVmod0D <- function(Time, State, Pars) {
  with(as.list(c(State, Pars)), {
    IngestC <- rI * P * C
    GrowthP <- rG * P * (1 - P/K)
    MortC <- rM * C

    dP <- GrowthP - IngestC
    dC <- IngestC * AE - MortC
    return(list(c(dP, dC)))
  })
}

### Define parameters ###
pars <- c(rI = 1.5, rG = 1.5, rM = 2, AE = 1, K = 10)

### Define time sequence ###
times <- seq(0, 30, by = 0.01)

### Initial conditions ###
set.seed(1)
d <- 2
n <- 12*d
Design <- maximinLHS(n, d) * 5 # Generate LHS samples in range [0,5]
colnames(Design) <- c("P", "C")

### Fit GP emulators ###
fit.dyn <- dynemu_GP(LVmod0D, times, pars, Design)

### Define uncertain inputs ###
xmean <- c(P = 1, C = 2)
xvar <- c(P = 1e-5, C = 1e-5)

### Define number of MC samples ###
mc.sample <- 1000

### Compute the next point ###
dynemu_mc(xmean, xvar, fit.dyn, mc.sample)

Predictive posterior computation for dynamic systems using one-steap-ahead approach by Gaussian process (GP) emulators

Description

The function computes the predictive posterior distribution (mean and variance) at all time steps for dynamic systems modeled by GP emulators. It can use either the exact computation or Monte Carlo (MC) approximation to compute the predictive posterior.

Usage

dynemu_pred(dynGP, times, xnew, method="Exact", mc.sample=NULL, trace=TRUE)

Arguments

Details

Given a trained set of GP emulators 'dynGP' fitted using dynemu_GP, this function: 1. Computes the predictive posterior mean and variance for each state variable at all time steps. 2. The function can either: - Use the exact computation for a closed-form solution by method="Exact". - Use the MC approximation method to generate samples and estimate the posterior by method="MC".

Value

A list containing:

• times: The time sequence vector.

• mu: A matrix of predictive mean values for each state variable. Each row corresponds to a time step, and each column corresponds to a state variable.

• sig2: A matrix of predictive variance values for each state variable. Each row corresponds to a time step, and each column corresponds to a state variable.

Examples

library(lhs)
### Lotka-Volterra equations ###
LVmod0D <- function(Time, State, Pars) {
  with(as.list(c(State, Pars)), {
    IngestC <- rI * P * C
    GrowthP <- rG * P * (1 - P/K)
    MortC <- rM * C

    dP <- GrowthP - IngestC
    dC <- IngestC * AE - MortC
    return(list(c(dP, dC)))
  })
}

### Define parameters ###
pars <- c(rI = 1.5, rG = 1.5, rM = 2, AE = 1, K = 10)

### Define time sequence ###
times <- seq(0, 30, by = 0.01)

### Initial conditions ###
set.seed(1)
d <- 2
n <- 12*d
Design <- maximinLHS(n, d) * 5 # Generate LHS samples in range [0,5]
colnames(Design) <- c("P", "C")

### Fit GP emulators ###
fit.dyn <- dynemu_GP(LVmod0D, times, pars, Design)

### Define initial test values ###
state <- c(P = 1, C = 2)

### Generate test set ###
test <- dyn_solve(LVmod0D, times, pars, state)

### Define number of MC samples ###
mc.sample <- 1000

### Prediction ###
pred.exact.dyn <- dynemu_pred(fit.dyn, times, state, method="Exact")
pred.exact.mu <- pred.exact.dyn$mu
pred.exact.sig2 <- pred.exact.dyn$sig2

pred.mc.dyn <- dynemu_pred(fit.dyn, times, state, method="MC", trace=TRUE)
pred.mc.mu <- pred.mc.dyn$mu
pred.mc.sig2 <- pred.mc.dyn$sig2

### Compute RMSE ###
sqrt(mean((pred.exact.mu[,1]-test$P)^2)) # 0.03002421
sqrt(mean((pred.exact.mu[,2]-test$C)^2)) # 0.02291742

sqrt(mean((pred.mc.mu[,1]-test$P)^2)) # 0.03050849
sqrt(mean((pred.mc.mu[,2]-test$C)^2)) # 0.02330542