Abstract
It is often desirable to build a statistical emulator of a complex computer simulator in order to perform analysis which would otherwise be computationally infeasible. We propose methodology to model multivariate output from a computer simulator taking into account output structure in the responses. The utility of this approach is demonstrated by applying it to a chemical and biological hazard prediction model. Predicting the hazard area that results from an accidental or deliberate chemical or biological release is imperative in civil and military planning and also in emergency response. The hazard area resulting from such a release is highly structured in space and we therefore propose the use of a thin-plate spline to capture the spatial structure and fit a Gaussian process emulator to the coefficients of the resultant basis functions. We compare and contrast four different techniques for emulating multivariate output: dimension-reduction using (i) a fully Bayesian approach with a principal component basis, (ii) a fully Bayesian approach with a thin-plate spline basis, assuming that the basis coefficients are independent, and (iii) a "plug-in" Bayesian approach with a thin-plate spline basis and a separable covariance structure; and (iv) a functional data modeling approach using a tensor-product (separable) Gaussian process. We develop methodology for the two thin-plate spline emulators and demonstrate that these emulators significantly outperform the principal component emulator. Further, the separable thin-plate spline emulator, which accounts for the dependence between basis coefficients, provides substantially more realistic quantification of uncertainty, and is also computationally more tractable, allowing fast emulation. For high resolution output data, it also offers substantial predictive and computational advantages over the tensor-product Gaussian process emulator.
Highlights
The simulation of scientific and engineering systems via complex mathematical models has become a common method of gaining knowledge about processes where physical experimentation is infeasible or unaffordable
Building an emulator or surrogate for the computer model trained on a—usually small—set of simulator evaluations has become standard practice; see, for example, the seminal paper of Sacks et al [29], Kennedy, Anderson, and O’Hagan [19], who presented a number of case studies of such computer experiments, and the book-length treatments of Santner, Williams, and Notz [30], Fang, Li, and Sudjianto [9], and Forrester, Sobester, and Keane [10]
Motivated by a simulator of chemical and biological dispersion, the contribution of this paper is to propose the use of thin-plate splines as basis functions for multivariate data with spatial structure, and to develop the necessary methodology for their application
Summary
The simulation of scientific and engineering systems via complex mathematical models has become a common method of gaining knowledge about processes where physical experimentation is infeasible or unaffordable. Encapsulated in computer codes or simulators, many of these models require substantial computing time to evaluate the response for a given set of inputs. For even moderately expensive simulators, the computational resources required to perform, for example, Monte Carlo inference may be prohibitive in practice. WOODS building an emulator or surrogate for the computer model trained on a—usually small—set of simulator evaluations has become standard practice; see, for example, the seminal paper of Sacks et al [29], Kennedy, Anderson, and O’Hagan [19], who presented a number of case studies of such computer experiments, and the book-length treatments of Santner, Williams, and Notz [30], Fang, Li, and Sudjianto [9], and Forrester, Sobester, and Keane [10]. Cheap emulators allow for real-time decision making and greater scientific understanding through, for example, sensitivity and uncertainty analyses
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