Connecting U-type Designs Before and After Level Permutations and Expansions

In both physical and computer experiments, U-type designs, including Latin hypercube designs, are commonly used. Two major approaches for evaluating U-type designs are orthogonality and space-filling criteria. Level permutations and level expansions are powerful tools for generating good U-type designs under the above criteria in the literature. In this paper, we systematically study the theoretical properties of U-type designs before and after level permutations and expansions. We establish the relationships between the initial designs’ generalized word-length patterns (GWLP) and the generated designs’ orthogonal and space-filling properties. Our findings generalize the existing results and provide theoretical justifications for the current level permutation and expansion algorithms.