Procedure: Explore the full simulator design region to identify a suitable single step function design region¶
Description and Background¶
This page is concerned with task of emulating a dynamic simulator, as set out in the variant thread on dynamic emulation (ThreadVariantDynamic).
We have an emulator for the single step function \(w_t=f(w_{t-1},a_t,\phi)\) given an initial assessment of the input region of interest \(\mathcal{X}_{single}\), and some training data. However, the ‘correct’ region \(\mathcal{X}_{single}\) depends on the input region of interest \(\mathcal{X}_{full}\) for the full simulator, and the single step function \(f(\cdot)\). Here, we use simulation to make an improved assessment of \(\mathcal{X}_{single}\) given \(\mathcal{X}_{full}\) and an emulator for \(f(.)\).
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, training inputs \(D\) and training outputs \(f(D)\).
- A set \(\{\theta^{(1)},\ldots,\theta^{(s)}\}\) of emulator hyperparameter values.
- The input region of interest \(\mathcal{X}_{full}\) for the full simulator.
Outputs¶
- A set of \(N\) simulated joint time series \(\{(w_{0}^{(i)},a_1^{(i)}),\ldots, (w_{T-1}^{(i)},a_T^{(i)})\}\) for \(i=1,\ldots,N\).
Procedure¶
- Choose a set of design points \(\{x_1^{full},\ldots,x_N^{full}\}\) from \(\mathcal{X}_{full}\), with \(x_i^{full}=(w_0^{(i)},a_1^{(i)}, \ldots,a_T^{(i)},\phi^{(i)})\). Both \(N\) and the design points can be chosen following the principles in the alternatives page on training sample design (AltCoreDesign), but see additional comments at the end. Then for \(i=1,\ldots,N\):
- For \(x_i^{full}\) and \(\theta^{(i)}\) , generate one random time series \(w_1^{(i)},\ldots,w_T^{(i)}\) using the simulation method given in the procedure page ProcExactIterateSingleStepEmulator (use \(R=1\) within this procedure). Note that we have assumed \(N\le s\) here. If it is not possible to increase \(s\) and we need \(N>s\), then we suggest cycling round the set \(\{\theta^{(1)},\ldots,\theta^{(s)}\}\) for each iteration \(i\).
- Organise the forcing inputs from \(x_i^{full}\) and simulated state variables into a joint time series \((w_{t-1}^{(i)},a_t^{(i)})\) for \(t=1,\ldots,T\).
Additional Comments¶
Since generating a single time series \(w_1^{(i)},\ldots,w_T^{(i)}\) should be a relatively quick procedure, we can use a larger value of \(N\) than might normally be considered in AltCoreDesign. The main aim here is to establish the boundaries of \(\mathcal{X}_{single}\), and so we should check that any such assessment is stable for increasing values of \(N\).