Generalized Integrated Brownian Fields for Simulation Metamodeling

In operations research, stochastic simulations are often used to model complex systems. Simulation runs can be time-consuming to execute, especially when there are many scenarios that need to be evaluated, or the scenarios to be evaluated cannot be anticipated in advance of when the results are needed. Simulation metamodels are statistical models built using the simulation output at a small set of scenarios and can be used to predict the value of the response surface for any scenario, simulated or not. Thus, simulation metamodels can provide support for real-time decision making and sensitivity analysis. Gaussian process (GP) regression is a popular technique for metamodeling; GP regression represents the unknown response surface as the realization of a Gaussian random field (GRF). Specifying the proper GRF is crucial for effective metamodeling. In “Generalized Integrated Brownian Fields for Simulation Metamodeling”, Salemi, Staum, and Nelson propose a novel class of GRFs called generalized integrated Brownian fields. These GRFs have several desirable properties, including differentiability that can be customized in each coordinate direction, no mean reversion, and the Markov property. These properties are shown to lead to better metamodels than those obtained from standard choices for the GRF.

We introduce a novel class of Gaussian random fields (GRFs), called generalized integrated Brownian fields (GIBFs), focusing on the use of GIBFs for Gaussian process regression in deterministic and stochastic simulation metamodeling. We build GIBFs from the well-known Brownian motion and discuss several of their properties, including differentiability that can differ in each coordinate, no mean reversion, and the Markov property. We explain why we desire to use GRFs with these properties and provide formal definitions of mean reversion and the Markov property for real-valued, differentiable random fields. We show how to use GIBFs with stochastic kriging, covering trend modeling and parameter fitting, discuss their approximation capability, and show that the resulting metamodel also has differentiability that can differ in each coordinate. Last, we use several examples to demonstrate superior prediction capability as compared with the GRFs corresponding to the Gaussian and Matérn covariance functions.

The e-companion is available at https://doi.org/10.1287/opre.2018.1804.