Deterministic construction methods for asymmetrical uniform designs

Abstract

Asymmetrical (mixed-level) uniform designs are useful for both computer and physical experiments. However, constructing these designs is often challenging due to their complex asymmetrical structure. In this paper, we propose novel methods for constructing uniform designs with mixed two-, three-, and four/nine-levels. Our construction methods are deterministic, allowing us to circumvent the complexity associated with stochastic algorithms. We evaluate uniformity using the wrap-around $L_2$- and Lee discrepancies. We establish useful analytic relationships between uniformity and aberration, and derive new general lower bounds for discrepancies that are tighter than those currently available in the literature. These new benchmarks can effectively measure the uniformity of asymmetrical designs. Additionally, we provide examples demonstrating the efficacy of our construction methods and the relevance of the newly obtained lower bounds. Finally, through simulations, we show that the designs produced using our methods perform well in constructing statistical surrogate models.