A Statistical Method for Emulation of Computer Models with Invariance-Preserving Properties, with Application to Structural Energy Prediction

Statistical design and analysis of computer experiments is a growing area in statistics. Computer models with structural invariance properties now appear frequently in materials science, physics, biology and other fields. These properties are consequences of dependency on structural geometry, and cannot be accommodated by standard statistical emulation methods. In this paper, we propose a statistical framework for building emulators to preserve invariance. The framework uses a weighted complete graph to represent the geometry and introduces a new class of function, called the relabeling symmetric functions, associated with the graph. We establish a characterization theorem of the relabeling symmetric functions, and propose a nonparametric kernel method for estimating such functions. The effectiveness of the proposed method is illustrated by examples from materials science.