.. _ProcExploreFullSimulatorDesignRegion:

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 :ref:`emulating<DefEmulator>` a
:ref:`dynamic simulator<DefDynamic>`, as set out in the variant
thread on dynamic emulation
(:ref:`ThreadVariantDynamic<ThreadVariantDynamic>`).

We have an emulator for the :ref:`single step
function<DefSingleStepFunction>` :math:`w_t=f(w_{t-1},a_t,\phi)`
given an initial assessment of the input region of interest
:math:`\mathcal{X}_{single}`, and some training data. However, the
'correct' region :math:`\mathcal{X}_{single}` depends on the input
region of interest :math:`\mathcal{X}_{full}` for the full simulator,
and the single step function :math:`f(\cdot)`. Here, we use simulation to make
an improved assessment of :math:`\mathcal{X}_{single}` given
:math:`\mathcal{X}_{full}` and an emulator for :math:`f(.)`.

Inputs
------

-  An emulator for the single step function :math:`w_t=f(w_{t-1},a_t,\phi)`,
   formulated as a :ref:`GP<DefGP>` or
   :ref:`t-process<DefTProcess>` conditional on hyperparameters,
   :ref:`training inputs<DefTrainingSample>` :math:`D` and
   training outputs :math:`f(D)`.
-  A set :math:`\{\theta^{(1)},\ldots,\theta^{(s)}\}` of emulator
   hyperparameter values.
-  The input region of interest :math:`\mathcal{X}_{full}` for the full
   simulator.

Outputs
-------

-  A set of :math:`N` simulated joint time series
   :math:`\{(w_{0}^{(i)},a_1^{(i)}),\ldots, (w_{T-1}^{(i)},a_T^{(i)})\}` for
   :math:`i=1,\ldots,N`.

Procedure
---------

#. Choose a set of design points :math:`\{x_1^{full},\ldots,x_N^{full}\}`
   from :math:`\mathcal{X}_{full}`, with :math:`x_i^{full}=(w_0^{(i)},a_1^{(i)},
   \ldots,a_T^{(i)},\phi^{(i)})`. Both
   :math:`N` and the design points can be chosen following the
   principles in the alternatives page on training sample design
   (:ref:`AltCoreDesign<AltCoreDesign>`), but see additional comments
   at the end. Then for :math:`i=1,\ldots,N`:
#. For :math:`x_i^{full}` and :math:`\theta^{(i)}` , generate one random
   time series :math:`w_1^{(i)},\ldots,w_T^{(i)}` using the simulation
   method given in the procedure page
   :ref:`ProcExactIterateSingleStepEmulator<ProcExactIterateSingleStepEmulator>`
   (use :math:`R=1` within this procedure). Note that we have
   assumed :math:`N\le s` here. If it is not possible to increase
   :math:`s` and we need :math:`N>s`, then we suggest cycling round
   the set :math:`\{\theta^{(1)},\ldots,\theta^{(s)}\}` for each
   iteration :math:`i`.
#. Organise the forcing inputs from :math:`x_i^{full}` and simulated state
   variables into a joint time series :math:`(w_{t-1}^{(i)},a_t^{(i)})`
   for :math:`t=1,\ldots,T`.

Additional Comments
-------------------

Since generating a single time series :math:`w_1^{(i)},\ldots,w_T^{(i)}`
should be a relatively quick procedure, we can use a larger value of
:math:`N` than might normally be considered in
:ref:`AltCoreDesign<AltCoreDesign>`. The main aim here is to
establish the boundaries of :math:`\mathcal{X}_{single}`, and so we
should check that any such assessment is stable for increasing values of
:math:`N`.