Alternatives: Dynamic Emulation Approaches

Overview

This page discusses two approaches for building an emulator of a dynamic simulator. Dynamic simulators produce multivariate outputs: a time series of state variables \(w_1,\ldots,w_T\). The strategy for emulating such a simulator presented in the variant thread for dynamic emulation (ThreadVariantDynamic) is to construct an emulator of the single step function \(w_t=f(w_{t-1},a_t,\phi)\). An alternative is to treat the simulator like any other multivariate output simulator, and construct an emulator directly as in the variant thread for the analysis of a simulator with multiple outputs (ThreadVariantMultipleOutputs). Here, we consider the relative merits of the two approaches.

The Nature of the Alternatives

  1. Build an emulator of the single step function \(w_t=f(w_{t-1},a_t,\phi)\). If we wish to obtain a joint distribution of \(w_1,\ldots,w_T\) given \(a_1,\ldots,a_T\) and \(\phi\), we cannot do so directly. We must either use simulation methods (ProcExactIterateSingleStepEmulator) or an approximation approach (ProcApproximateIterateSingleStepEmulator).
  2. Build an emulator of the full times series simulator \((w_1,\ldots,w_T)=f_{full}(w_0,a_1,\ldots,a_T,\phi)\). This simulator and corresponding emulator have a much larger input and output space, but the emulator will directly give us the joint distribution of \(w_1,\ldots,w_T\) given \(w_0,a_1,\ldots,a_T\) and \(\phi\). We may only be interested in some subvector or function of \((w_1,\ldots,w_T)\), which could be emulated directly, reducing the dimension of the output space.

Choosing the Alternatives

We argue that emulating the single step function is preferable under any of the following circumstances:

  1. We wish to emulate the outputs over a range of different series of forcing inputs \(a_1,\ldots,a_T\). This may be because we are uncertain about the value of a ‘true’ forcing input series, or we may just wish to investigate how the outputs vary as \(a_1,\ldots,a_T\) varies. In a multivariate output emulator, the inputs are \(w_0, \phi, a_1,\ldots,a_T\). Hence the input dimension is large if \(T\) is large, and building any emulator becomes increasingly difficult as the number of inputs increases. In the single step emulator, the inputs are \(w_{t-1}, \phi, a_t\), and so the number of emulator inputs is fixed regardless of the value of \(T\).
  2. We are uncertain about the maximum \(T\) of interest. A ‘black box’ emulator of the function \((w_{1}\ldots,w_{T})=f_{full}(w_0, \phi, a_1,\ldots,a_T)\) cannot predict \(w_{T+t}\) for any \(t>0\). It is, however, possible to extrapolate if
    • the forcing variables are fixed and are not treated as emulator inputs
    • \(t\) is treated as an additional simulator/emulator input
    • The prior mean function is a good approximation of the relationship between \(w_{t}\) and \(t\) in the simulator. See the alternatives page on emulator prior mean function (AltMeanFunction)
  3. We wish to increase the complexity of the simulator single step function. As the computational expense of the single step function increases, it may become increasingly impractical to obtain sufficient training runs of the full times series simulator, depending on the value of \(T\).