Sharp lower bounds of various uniformity criteria for constructing uniform designs

  • A. M. ElsawahEmail author
  • Kai-Tai Fang
  • Ping He
  • Hong Qin
Regular Article


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.


Balanced design Uniform design Discrepancy Lower bound 

Mathematics Subject Classification

62K05 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.

Supplementary material

362_2019_1143_MOESM1_ESM.pdf (278 kb)
Supplementary material 1 (pdf 277 KB)


  1. Androulakis E, Drosou K, Koukouvinos C, Zhou YD (2016) Measures of uniformity in experimental designs: a selective overview. Commun Stat Theory Method 45(13):3782–3806MathSciNetCrossRefGoogle Scholar
  2. Bates RA, Buck RJ, Riccomagno E, Wynn HP (1996) Experimental design and observation for large systems. J R Stat Soc Ser B 58:77–94MathSciNetzbMATHGoogle Scholar
  3. Chatterjee K, Li Z, Qin H (2012a) Some new lower bounds to centered and wrap-round \(L_2\)-discrepancies. Stat Probab Lett 82(7):1367–73CrossRefGoogle Scholar
  4. Chatterjee K, Qin H, Na Zou (2012b) Lee discrepancy on two and three mixed level factorials. Sci China 55(3):663–670MathSciNetCrossRefGoogle Scholar
  5. Elsawah AM (2016) Constructing optimal asymmetric combined designs via Lee discrepancy. Stat Probab Lett 118:24–31MathSciNetCrossRefGoogle Scholar
  6. Elsawah AM (2017a) A closer look at de-aliasing effects using an efficient foldover technique. Statistics 51(3):532–557MathSciNetCrossRefGoogle Scholar
  7. Elsawah AM (2017b) A powerful and efficient algorithm for breaking the links between aliased effects in asymmetric designs. Aust NZ J Stat 59(1):17–41MathSciNetCrossRefGoogle Scholar
  8. Elsawah AM (2019a) Building some bridges among various experimental designs. J Korean Stat Soc. CrossRefGoogle Scholar
  9. Elsawah AM (2019b) Constructing optimal router bit life sequential experimental designs: new results with a case study. Commun Stat Simul Comput 48(3):723–752MathSciNetCrossRefGoogle Scholar
  10. Elsawah AM, Qin H (2014) New lower bound for centered \(L_2\)-discrepancy of four-level \(U\)-type designs. Stat Probab Lett 93:65–71CrossRefGoogle Scholar
  11. Elsawah AM, Qin H (2015a) A new strategy for optimal foldover two-level designs. Stat Probab Lett 103:116–126MathSciNetCrossRefGoogle Scholar
  12. Elsawah AM, Qin H (2015b) Mixture discrepancy on symmetric balanced designs. Stat Probab Lett 104:123–132MathSciNetCrossRefGoogle Scholar
  13. Elsawah AM, Fang KT (2017) New foundations for designing U-optimal follow-up experiments with flexible levels. Stat Pap. CrossRefGoogle Scholar
  14. Elsawah AM, Qin H (2017) Optimum mechanism for breaking the confounding effects of mixed-level designs. Comput Stat 32(2):781–802MathSciNetCrossRefGoogle Scholar
  15. Elsawah AM, Fang KT (2018) New results on quaternary codes and their Gray map images for constructing uniform designs. Metrika 81(3):307–336MathSciNetCrossRefGoogle Scholar
  16. Elsawah AM, Fang KT (2019) A catalog of optimal foldover plans for constructing U-uniform minimum aberration four-level combined designs. J Appl Stat 46(7):1288–1322MathSciNetCrossRefGoogle Scholar
  17. Elsawah AM, Fang KT, Ke X (2019) New recommended designs for screening either qualitative or quantitative factors. Stat Pap. CrossRefGoogle Scholar
  18. Fang KT (1980) The uniform designs: application of number-theoretic methods in experimental design. Acta Math Appl Sin 3:363–372Google Scholar
  19. 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
  20. Fang KT, Mukerjee R (2000) A connection between uniformity and aberration in regular fractions of two-level factorials. Biometrika 87:93–198MathSciNetCrossRefGoogle Scholar
  21. Fang KT, Lin DKJ, Winker P, Zhang Y (2000) Uniform design: theory and application. Technometrics 42:237–248MathSciNetCrossRefGoogle Scholar
  22. 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
  23. Fang KT, Li R, Sudjianto A (2006a) Design and modeling for computer experiments. CRC Press, New YorkzbMATHGoogle Scholar
  24. Fang KT, Maringer D, Tang Y, Winker P (2006b) Lower bounds and stochastic optimization algorithms for uniform designs with three or four levels. Math Comput 75:859–878MathSciNetCrossRefGoogle Scholar
  25. Fang KT, Ke X, Elsawah AM (2017) Construction of uniform designs via an adjusted threshold accepting algorithm. J Complex 43:28–37MathSciNetCrossRefGoogle Scholar
  26. Fang KT, Tang Y, Yin J (2005) Lower bounds for the wrap-around \(L_2\)-discrepancy of symmetrical uniform designs. J Complex 21:757–771CrossRefGoogle Scholar
  27. Fang KT, Wang Y (1994) Number-theoretic methods in statistics. Chapman and Hall, LondonCrossRefGoogle Scholar
  28. Hu L, Chatterjee K, Liu J, Ou Z (2018) New lower bound for Lee discrepancy of asymmetrical factorials. Stat Pap. CrossRefGoogle Scholar
  29. Hickernell FJ (1998a) A generalized discrepancy and quadrature error bound. Math Comput 67:299–322MathSciNetCrossRefGoogle Scholar
  30. Hickernell FJ (1998b) Lattice rules: how well do they measure up? In: Hellekalek P, Larcher G (eds) Random and quasi-random point sets. Lecture notes in statistics, vol 138. Springer, New York, pp 109–166CrossRefGoogle Scholar
  31. Hickernell FJ, Liu M (2002) Uniform designs limit aliasing. Biometrika 89:893–904MathSciNetCrossRefGoogle Scholar
  32. Ke X, Zhang R, Ye HJ (2015) Two- and three-level lower bounds for mixture \(L_2\)-discrepancy and construction of uniform designs by threshold accepting. J Complex 31:741–753CrossRefGoogle Scholar
  33. Liang YZ, Fang KT, Xu QS (2001) Uniform design and its applications in chemistry and chemical engineering. Chemom Intell Lab Syst 58:43–57CrossRefGoogle Scholar
  34. Phadke MS (1986) Design optimization case studies. AT T Tech J 65:51–68CrossRefGoogle Scholar
  35. Wang Y, Fang KT (1981) A not on uniform distribution and experimental design. Chin Sci Bull 26:485–489MathSciNetzbMATHGoogle Scholar
  36. 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
  37. Yang F, Zhou Y-D, Zhang X-R (2017) Augmented uniform designs. J Stat Plan Inference 182:64–737MathSciNetCrossRefGoogle Scholar
  38. Zhang Q, Wang Z, Hu J, Qin H (2015) A new lower bound for wrap-around \(L_2\)-discrepancy on two and three mixed level factorials. Stat Probab Lett 96:133–140CrossRefGoogle Scholar
  39. Zhou YD, Ning JH (2008) Lower bounds of the wrap-around \(L_2\)-discrepancy and relationships between MLHD and uniform design with a large size. J Stat Plan Inference 138:2330–2339CrossRefGoogle Scholar
  40. Zhou YD, Ning JH, Song XB (2008) Lee discrepancy and its applications in experimental designs. Stat Probab Lett 78:1933–1942MathSciNetCrossRefGoogle Scholar
  41. Zhou YD, Fang KF, Ning JH (2013) Mixture discrepancy for quasi-random point sets. J Complex 29:283–301MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Division of Science and TechnologyBNU-HKBU United International CollegeZhuhaiChina
  2. 2.Department of Mathematics, Faculty of ScienceZagazig UniversityZagazigEgypt
  3. 3.The Key Lab of Random Complex Structures and Data AnalysisThe Chinese Academy of SciencesBeijingChina
  4. 4.Department of Statistics, Faculty of Mathematics and StatisticsCentral China Normal UniversityWuhanChina
  5. 5.Department of Statistics, School of Statistics and MathematicsZhongnan University of Economics and LawWuhanChina

Personalised recommendations