Alternatives: Iterating single step emulators¶
Overview¶
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 \(w_t=f(w_{t-1},a_t,\phi)\) in a dynamic simulator, and wish to iterate the emulator to obtain the distribution of \(w_1,\ldots,w_T\) given \(w_0,a_1,\ldots,a_T\) and \(\phi\). For a Gaussian Process emulator, it is not possible to do so analytically, and so we consider two alternatives: a simulation based approach and an approximation based on the normal distribution.
The Nature of the Alternatives¶
- Recursively simulate random outputs from the single step emulator, adding the simulated outputs to the training data at each iteration to obtain an exact draw from the distribution of \(w_1,\ldots,w_T\) (ProcExactIterateSingleStepEmulator).
- Approximate the distribution of \(w_t=f(w_{t-1},a_t,\phi)\) at each time step with a normal distribution. The mean and variance of \(w_t\) for each \(t\) are computed recursively. (ProcApproximateIterateSingleStepEmulator).
Choosing the Alternatives¶
The simulation method can be computationally intensive, but has the advantage of producing exact draws from the distribution of \(w_1,\ldots,w_T\). The approximation method is computationally faster to implement, and so is to be preferred if the normal approximation is sufficiently accurate. The approximation is likely to be accurate when
- uncertainty about \(f(\cdot)\) is small, and
- the simulator is approximately linear over any small part of the input space.
As always, it is important to validate the emulator (see the procedure page on validating a GP emulator (ProcValidateCoreGP)), but particular attention should be paid to emulator uncertainty in addition to emulator predictions.
Additional comments¶
If the approximation method is to be used for many different choices \(w_0,a_1,\ldots,a_T\) and \(\phi\), it is recommended that both methods are tried for a subset of these choices, to test the accuracy of the approximation.