Design and Analysis of Computer Experiments with both Numeral and Distributional Inputs

Abstract

Nowadays stochastic computer simulations with both numeral and distributional inputs are widely used to mimic complex systems which contain a great deal of uncertainty. This article studies the design and analysis issues of such computer experiments. First, we provide preliminary results concerning the Wasserstein distance in probability measure spaces. To handle the product space of the Euclidean space and the probability measure space, we prove that, through the mapping from a point in the Euclidean space to the mass probability measure at this point, the Euclidean space can be isomorphic to the subset of the probability measure space, which consists of all the mass measures, with respect to the Wasserstein distance. Therefore, the product space can be viewed as a product probability measure space. We derive formulas of the Wasserstein distance between two components of this product probability measure space. Second, we use the above results to construct Wasserstein distance-based space-filling criteria in the product space of the Euclidean space and the probability measure space. A class of optimal Latin hypercube-type designs in this product space are proposed. Third, we present a Wasserstein distance-based Gaussian process model to analyze data from computer experiments with both numeral and distributional inputs. Numerical examples and real applications to a metro simulation are presented to show the effectiveness of our methods.

KEYWORDS:

• Gaussian process model
• Metro simulation
• Mixed inputs
• Space-filling design
• Wasserstein distance

Acknowledgments

We thank the editors and referees for constructive comments which lead to a significant improvement of this article.

Additional information

Funding

Xiong’s work is partially supported by National Key R&D Program of China (grant nos. 2021YFA1000300, 2021YFA1000301, and 2021YFA1000303) and the National Natural Science Foundation of China (grant no. 12171462). Cui’s work is partially supported by the National Natural Science Foundation of China (grant nos. 12171234, 11790272), the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), and the Fundamental Research Funds for the Central Universities.