Theory and applications of stratification criteria based on space-filling pattern and projection pattern

Space-filling designs are crucial for computer experiments. The quality of a space-filling design can be appropriately reflected by its stratification properties. In a recent paper, Tian and Xu (Biometrika 109(2):489–501, 2022) introduced the concept of a space-filling pattern to properly characterize a design’s stratification properties on various grids. In this study, we generalize the space-filling pattern using arbitrary orthonormal contrasts. We also propose a new pattern called the two-dimensional projection pattern to capture the stratification properties of balanced designs in two dimensions more comprehensively. We derive some theoretical results for both patterns and show that they are easier to compute and apply to a wider range of designs. We further show the use of the two patterns in constructing space-filling designs based on existing strong orthogonal arrays.