Uniform Projection Designs and Strong Orthogonal Arrays

We explore the connections between uniform projection designs and strong orthogonal arrays of strength 2+ in this article. Both of these classes of designs are suitable designs for computer experiments and space-filling in two-dimensional margins, but they are motivated by different considerations. Uniform projection designs are introduced by Sun, Wang, and Xu to capture two-dimensional uniformity using the centered L2-discrepancy whereas strong orthogonal arrays of strength 2+ are brought forth by He, Cheng, and Tang as they achieve stratifications in two-dimensions on finer grids than ordinary orthogonal arrays. We first derive a new expression for the centered L2-discrepancy, which gives a decomposition of the criterion into a sum of squares where each square measures one aspect of design uniformity. This result is not only insightful in itself but also allows us to study strong orthogonal arrays in terms of the discrepancy criterion. More specifically, we show that strong orthogonal arrays of strength 2+ are optimal or nearly optimal under the uniform projection criterion.