Sharp lower bounds of various uniformity criteria for constructing uniform designs
- 87 Downloads
Several techniques are proposed for designing experiments in scientific and industrial areas in order to gain much effective information using a relatively small number of trials. Uniform design (UD) plays a significant role due to its flexibility, cost-efficiency and robustness when the underlying models are unknown. UD seeks its design points to be uniformly scattered on the experimental domain by minimizing the deviation between the empirical and theoretical uniform distribution, which is an NP hard problem. Several approaches are adopted to reduce the computational complexity of searching for UDs. Finding sharp lower bounds of this deviation (discrepancy) is one of the most powerful and significant approaches. UDs that involve factors with two levels, three levels, four levels or a mixture of these levels are widely used in practice. This paper gives new sharp lower bounds of the most widely used discrepancies, Lee, wrap-around, centered and mixture discrepancies, for these types of designs. Necessary conditions for the existence of the new lower bounds are presented. Many results in recent literature are given as special cases of this study. A critical comparison study between our results and the existing literature is provided. A new effective version of the fast local search heuristic threshold accepting can be implemented using these new lower bounds. Supplementary material for this article is available online.
KeywordsBalanced design Uniform design Discrepancy Lower bound
Mathematics Subject Classification62K05 62K15
The authors greatly appreciate helpful suggestions of the three reviewers, the associate editor and the Editor-in-Chief Professor Werner G. Müller that significantly improved the paper. This work was partially supported by the UIC Grants (Nos. R201810 and R201912), the Zhuhai Premier Discipline Grant and the National Natural Science Foundation of China No. 11871237.
- Fang KT (1980) The uniform designs: application of number-theoretic methods in experimental design. Acta Math Appl Sin 3:363–372Google Scholar
- Fang KT, Hickernell FJ (1995) The uniform design and its applications, Bulletin of the International Statistical Institute, 50th Session, Book 1, pp 333–349. International Statistical Institute, BeijingGoogle Scholar
- Fang KT, Ma CX, Mukerjee R (2002) Uniformity in fractional factorials. In: Fang KT, Hickernell FJ, Niederreiter H (eds) Monte Carlo and quasi-Monte Carlo methods in scientific computing. Springer, BerlinGoogle Scholar
- Winker P, Fang KT (1997) Optimal U-type designs. In: Niederreiter H, Hellekalek P, Larcher G, Zinterhof P (eds) Monte Carlo and quasi-Monte Carlo methods. Springer, New York, pp 436–488Google Scholar