Procedure: Iterate the single step emulator using an approximation approach

Description and Background

This page is concerned with task of emulating a dynamic simulator, as set out in the variant thread for dynamic emulation (ThreadVariantDynamic).

We have an emulator for the single step function \(w_t=f(w_{t-1},a_t,\phi)\), and wish to predict the full time series \(w_1,\ldots,w_T\) for a specified initial state variable \(w_0\), time series of forcing variables \(a_1,\ldots,a_T\) and simulator parameters \(\phi\). It is not possible to analytically derive a distribution for \(w_1,\ldots,w_T\) if \(f(\cdot)\) is modelled as a Gaussian Process, so here we use an approximation based on the normal distribution to estimate the marginal distribution of each of \(w_1,\ldots,w_T\).

Inputs

  • An emulator for the single step function \(w_t=f(w_{t-1},a_t,\phi)\), formulated as a GP or t-process conditional on hyperparameters, plus a set of hyperparameter values \(\theta^{(1)},\ldots,\theta^{(s)}\).
  • An initial value for the state variable \(w_0\).
  • The values of the forcing variables \(a_1,\ldots,a_T\).
  • The values of the simulator parameters \(\phi\).

Outputs

  • Approximate marginal distributions for each of \(w_1,\ldots,w_T\). The distribution of each \(w_t\) is approximated by a normal distribution with a specified mean and variance.

Procedure

For a single choice of emulator hyperparameters \(\theta\), we approximate the marginal distribution of \(w_t\) by the normal distribution \(N_r(\mu_t,V_t)\)

We have

\[\begin{split}\mu_1 &= m^*(w_0,a_1,\phi), \\ V_1 &= v^*\{(w_0,a_1,\phi),(w_0,a_1,\phi)\}.\end{split}\]

The mean and variance are defined recursively:

\[\begin{split}\mu_{t+1} &= \textrm{E}[m^*(w_t,a_{t+1},\phi)|f(D),\theta], \\ V_{t+1} &= \textrm{E}[v^*\{(w_t,a_{t+1},\phi),(w_t,a_{t+1},\phi)\}|f(D),\theta] + \textrm{Var}[m^*(w_t,a_{t+1},\phi)|f(D),\theta],\end{split}\]

where the expectations and variances are taken with respect to \(w_{t}\), where \(w_{t} \sim N_r(\mu_{t},V_{t})\)

Explicit formulae for \(\mu_{t+1}\) and \(V_{t+1}\) can be derived in the case of a linear mean and a separable Gaussian covariance function. The procedure for calculating \(\mu_{t+1}\) and \(V_{t+1}\) is described in ProcUpdateDynamicMeanAndVariance. Otherwise, we can use simulation to estimate \(\mu_{t+1}\) and \(V_{t+1}\). A simulation procedure is given in ProcApproximateUpdateDynamicMeanandVariance.

Integrating out the emulator hyperparameters

Assuming we have \(s>1\), we can integrate out the emulator hyperparameters to obtain the unconditional mean and variance of \(w_{t}\) using Monte Carlo estimation. In the following procedure, we define \(N\) to be the number of Monte Carlo iterations, and for notational convenience, we suppose that \(N\le s\). For discussion of the choice of \(N\), including the case \(N>s\), see the discussion page on Monte Carlo estimation, sample sizes and emulator hyperparameter sets (DiscMonteCarlo).

  1. For \(i=1,2,\ldots,N\) fix the hyperparameters at the value \(\theta^{(i)}\), and calculate the corresponding mean and variance of \(w_t\), which we denote by \(\mu_t^{(i)}\) and \(V_t^{(i)}\).

  2. Estimate \(\textrm{E}[w_t|f(D)]\) by

    \[\hat{E}_t=\frac{1}{N}\sum_{i=1}^N \mu_t^{(i)}.\]

  3. Estimate \(\textrm{Var}[w_t|f(D)]\) by

    \[\frac{1}{N}\sum_{i=1}^N V_t^{(i)}+ \frac{1}{N-1}\sum_{i=1}^N \left(\mu_t^{(i)}-\hat{E}_t\right)^2.\]