Alternatives: Basis functions for the emulator mean

Overview

The process of building an emulator of a simulator involves first specifying prior beliefs about the simulator and then updating this using a training sample of simulator runs. Prior specification may be either using the fully Bayesian approach in the form of a Gaussian process or using the Bayes linear approach in the form of first and second order moments. The basics of building an emulator using these two approaches are set out in the two core threads (ThreadCoreGP, ThreadCoreBL).

In either approach it is necessary to specify a mean function and covariance function. Choice of the form of the emulator prior mean function is addressed in AltMeanFunction. We consider here the additional problem of specifying the forms of the basis functions \(h(x)\) when the form of the mean function is linear.

The Nature of the Alternatives

In order to specify an appropriate form of the global trend component, we typically first identify a set of potential basis functions appropriate to the simulator in question, and then select which elements of that set would best describe the simulator’s mean behaviour via the methods described above. There are a huge number of possible choices for the specific forms of \(h(x)\). Common choices include:

  • Monomials - to capture simple large-scale effects
  • Orthogonal polynomials - to exploit computational simplifications in subsequent calculations
  • Fourier/trigonometric functions - to adequately represent periodic output
  • A very fast approximate version of the simulator - to capture the gross simulator behaviour using an existing model

Choosing the Alternatives

When the mean function takes a linear form, an appropriate choice of the basis functions \(h(x)\) is required. This is particularly true for a Bayes linear emulator as the emphasis is placed on a detailed structural representation of the simulator’s mean behaviour.

Typically, there are two primary methods for determining an appropriate collection of trend basis functions:

  1. prior information about the model can be used directly to specify an appropriate form for the mean function;
  2. if the number of available model evaluations is very large, then we can empirically determine the form of the mean function from this data alone.

Expert information about \(h(x)\) can be derived from a variety of sources including, but not limited to, the following:

  • knowledge and experience with the computer simulator and its outputs;
  • beliefs about the behaviour of the actual physical system that the computer model simulates;
  • experience with similar computer models such as previous versions of the same simulator or alternative models for the same system;
  • series expansions of the generating equations underlying the computer model (or an appropriately simplified model form);
  • fast approximate versions of the computer model derived from simplifications to the current simulator.

If the prior information is weak relative to the available number of model evaluations and the computer model is inexpensive to evaluate, then we may choose instead to determine the form of the trend directly from the model evaluations. This empirical approach is reasonable since any Bayesian posterior would be dominated by the large volume of data. Thus in such situations it is reasonable to apply standard statistical modelling techniques. Empirical construction of the emulator mean is a similar problem to traditional regression model selection. In this case, methods such as stepwise model selection could be applied given a set of potential trend basis functions. However, using empirical methods to identify the form of the emulator mean function requires many more model evaluations than are required to fit an emulator with known form and hence is only applicable if a substantial number of simulator runs is available. Empirical construction of emulators using cheap simulators is a key component of multilevel emulation which will be described in a subsequent version of the Toolkit.

Often, the majority of the global variation of the output from a computer simulator \(f(x)\) can be attributed to a relatively small subset, \(x_A\), of the input quantities called the active inputs. In such cases, the emulator mean is considered to be a function of only the active inputs combined with a modified form of the covariance. Using this active input approach can make substantial computational savings; see the discussion page on active and inactive inputs (DiscActiveInputs) for further details. If the simulator has a high-dimensional input space then the elicitation of information about potential active inputs possibly after a suitable transformation of the input space, then and the form of at least some of the model effects can be very helpful in emulator construction (see Craig et al. 1998).

References

Craig, P. S., Goldstein, M., Seheult, A. H., and Smith, J. A. (1998) “Constructing partial prior specifications for models of complex physical systems,” Applied Statistics, 47:1, 37–53