Asymmetric uniform designs based on mixture discrepancy

ABSTRACT

Efficient experimental design is crucial in the study of scientific problems. The uniform design is one of the most widely used approaches. The discrepancies have played an important role in quasi-Monte Carlo methods and uniform design. Zhou et al. [17] proposed a new type of discrepancy, mixture discrepancy ($\mathcal{M}\mathcal{D}$), and showed that $\mathcal{M}\mathcal{D}$ may be a better uniformity measure than other discrepancies. In this paper, we discuss in depth the $\mathcal{M}\mathcal{D}$ as the uniformity measure for asymmetric mixed two and three levels U-type designs. New analytical expression based on row distance and new lower bound of the $\mathcal{M}\mathcal{D}$ are given for asymmetric levels designs. Using the new formulation and the new lower bound as the benchmark, we can implement a new version of the fast local search heuristic threshold accepting. By this search heuristic, we can obtain mixed two and three levels U-type designs with low discrepancy.

KEYWORDS:

• Mixture discrepancy
• uniform design
• U-type design
• asymmetric levels design
• lower bound
• TA algorithm

MSC:

• 62K05
• 62K15

Acknowledgments

The authors would like to thank the referees for constructive suggestions leading to improvements in the presentation of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.