4.1 APPROXIMATING THE COMPUTATIONAL MODEL

Representing the input/output relationships in a model with a statistical surrogate (or emulator) and using a reduced-order model are two broad methods effectively used to reduce the computational cost of model exploration. For instance, a reduced-order model (Section 4.1.2) or an emulator (Section 4.1.1) can be used to stand in place of the computer model when a sensitivity analysis is being conducted or uncertainty is propagating across the computer model (see Section 4.2 and the example on electromagnetic interference phenomena in Section 4.5). Of course, as with any approximation, there is a reduction in the accuracy of the estimates obtained, and the trade-off between accuracy and cost needs to be considered by the analyst.

4.1.1 Computer Model Emulation

In settings in which the simulation model is computationally expensive, an emulator can be used in its place. The computer model is generally viewed as a black box, and constructing the emulator can be thought of as a type of response-surface modeling exercise (e.g., Box and Draper, 2007). That is, the aim is to establish an approximation to the input-output map of the model using a limited number of calls of the simulator.

Many possible parametric and nonparametric regression techniques can provide good approximations to the computer-model response surface. For example, there are those that interpolate between model runs such as GP models (Sacks et al., 1989; Gramacy and Lee, 2008) or Lagrange interpolants (e.g., see Lin et al., 2010). Approaches that do not interpolate the simulations, but which have been used to stand in place of the computer models, include polynomial regression (Box and Draper, 2007), multivariate adaptive regression splines (Jin et al., 2000), projection pursuit (see Ben-Ari and Steinberg, 2007, for a comparison with several methods), radial basis functions (Floater and Iske, 1996), support vector machines (Clarke et al., 2003), and neural networks (Hayken, 1998), to name only a few. When the simulator has a stochastic or noisy response (Iooss and Ribatet, 2007), the situation is similar to the sampling of noisy physical systems in which random error is included in the statistical model, though the variability is likely to also depend on the inputs. In this case, any of the above models can be specified so that the randomness in the simulator response is accounted for in the emulation of the simulator.

Some care must be taken when emulating deterministic computer models if one is interested in representing the uncertainty (e.g., a standard deviation or a prediction interval) in predictions at unsampled inputs. To deal with the difference from the usual noisy settings, Sacks et al. (1989) proposed modeling the response from a computer code as a realization of a GP, thereby providing a basis for UQ (e.g., prediction interval estimation) that most other methods (e.g., polynomial regression) fail to do. A correlated stochastic process model with probability distribution more general than that of the GP could also be used for this interpolation task. A significant benefit of the Gaussian model is the persistence of the tractable Gaussian form following conditioning of the process at the sampled points and the representation of uncertainty at unsampled inputs.

Consider, for example, the behavior of the prediction intervals in Figure 4.1. Figure 4.1(a) shows a GP fit to deterministic computer-model output, and Figure 4.1(b) shows the same data fit using ordinary least squares regression with the set of Legendre polynomials. Both representations emulate the computer model output fairly well, but the GP has some obvious advantages. Notice that the fitted GP model passes through the observed points, thereby perfectly representing the deterministic computational model at the sampled inputs. In addition, the prediction uncertainty disappears entirely at sites for which simulation runs have been conducted (the prediction is the simulated response). Furthermore, the resulting prediction intervals reflect the uncertainty one would expect from a deterministic computer model—zero predictive uncertainty at the observed input points, small predictive uncertainty close to these points, and larger uncertainty farther away from the observed input points.

In spite of the aforementioned advantages, GP and related models do have shortcomings. For example, they are challenging to implement for large ensemble sizes. Many response-surface methods (e.g., polynomial regression or multivariate adaptive regression splines) can handle much larger sample sizes than the GP can and are computationally faster. Accordingly, adapting these approaches so that they can have the same sort of inferential advantages, as shown in Figure 4.1, as those of the GP in the deterministic setting is a topic of ongoing and future research.



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