Skip to main content
Log in

An Adjusted Gray Map Technique for Constructing Large Four-Level Uniform Designs

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

A uniform experimental design (UED) is an extremely used powerful and efficient methodology for designing experiments with high-dimensional inputs, limited resources and unknown underlying models. A UED enjoys the following two significant advantages: (i) It is a robust design, since it does not require to specify a model before experimenters conduct their experiments; and (ii) it provides uniformly scatter design points in the experimental domain, thus it gives a good representation of this domain with fewer experimental trials (runs). Many real-life experiments involve hundreds or thousands of active factors and thus large UEDs are needed. Constructing large UEDs using the existing techniques is an NP-hard problem, an extremely time-consuming heuristic search process and a satisfactory result is not guaranteed. This paper presents a new effective and easy technique, adjusted Gray map technique (AGMT), for constructing (nearly) UEDs with large numbers of four-level factors and runs by converting designs with s two-level factors and n runs to (nearly) UEDs with 2t−1s four-level factors and 2tn runs for any t ≥ 0 using two simple transformation functions. Theoretical justifications for the uniformity of the resulting four-level designs are given, which provide some necessary and/or sufficient conditions for obtaining (nearly) uniform four-level designs. The results show that the AGMT is much easier and better than the existing widely used techniques and it can be effectively used to simply generate new recommended large (nearly) UEDs with four-level factors.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Fang K T, The uniform design: Application of number-theoretic methods in experimental design, Acta Math. Appl. Sinica, 1980, 3: 363–372.

    MathSciNet  Google Scholar 

  2. Wang Y and Fang K T, A note on uniform distribution and experimental design, Chin. Sci. Bull., 1981, 26: 485–489.

    MathSciNet  MATH  Google Scholar 

  3. Hickernell F J, A generalized discrepancy and quadrature error bound, Math. Comp., 1998, 67: 299–322.

    Article  MathSciNet  MATH  Google Scholar 

  4. Hickernell F J, Lattice Rules: How Well Do They Measure Up? Random and Quasi-Random Point Sets, Eds. by Hellekalek P and Larcher G, Springer, New York, 1998.

  5. Zhou Y D, Ning J H, and Song X B, Lee discrepancy and its applications in experimental designs, Statist. Probab. Lett., 2008, 78: 1933–1942.

    Article  MathSciNet  MATH  Google Scholar 

  6. Elsawah A M, Designing uniform computer sequential experiments with mixture levels using Lee discrepancy, Journal of Systems Science and Complexity, 2019, 32(2): 681–708.

    Article  MathSciNet  MATH  Google Scholar 

  7. Elsawah A M, An appealing technique for designing optimal large experiments with three-level factors, J. Computational and Applied Mathematics, 2021, 384: 113164.

    Article  MathSciNet  MATH  Google Scholar 

  8. Lan W G, Wong M K, Chen N, et al., Four-level orthogonal array design as a chemometric approach to the optimization of polarographic reaction system for phosphorus determination, Talanta, 1994, 41(11): 1917–1927.

    Article  Google Scholar 

  9. Edmondson R N, Agricultural response surface experiments in view of four- level factorial designs, Biometrics, 1991, 47(4): 1435–1448.

    Article  Google Scholar 

  10. Ankenman B E, Design of experiments with two- and four-level factors, J. Quality Technol., 1999, 31(4): 363–375.

    Article  Google Scholar 

  11. Phadke M S, Design optimization case studies, AT & T Techn. J., 1986, 65: 51–68.

    Article  Google Scholar 

  12. Elsawah A M, Constructing optimal router bit life sequential experimental designs: New results with a case study, Commun. Statist. Simul. Comput., 2019, 48(3): 723–752.

    Article  MathSciNet  MATH  Google Scholar 

  13. Elsawah A M, Designing optimal large four-level experiments: A new technique without recourse to optimization softwares, Communications in Mathematics and Statistics, 2022, 10: 623–652.

    Article  MathSciNet  MATH  Google Scholar 

  14. Bettonvil B and Kleijnen J P C, Searching for important factors in simulation models with many factors: Sequential bifurcation, European J. Oper. Res., 1996, 96: 180–194.

    Article  MATH  Google Scholar 

  15. Kleijnen J P C, Ham G V, and Rotmans J, Techniques for sensitivity analysis of simulation models: A case study of the CO2 greenhouse effect, Simulation, 1992, 58(6): 410–417.

    Article  Google Scholar 

  16. Kleijnen J P C, Bettonvil B, and Persson F, Screening for the important factors in large discrete-event simulation: Sequential bifurcation and its applications, Screening Methods for Experimentation in Industry, Drug Discovery, and Genetics, Eds. by Dean A and Lewis S, Springer, New York, 2006.

    Google Scholar 

  17. Morris M D, Factorial sampling plans for preliminary computational experiments, Technometrics, 1991, 33: 161–174.

    Article  Google Scholar 

  18. Phoa F K H and Xu H, Quarter-fraction factorial designs constructed via quaternary codes, Ann. Statist., 2009, 37: 2561–2581.

    Article  MathSciNet  MATH  Google Scholar 

  19. Phoa F K H, A code arithmetic approach for quaternary code designs and its application to (1/64)th fraction, Ann. Statist., 2012, 40: 3161–3175.

    Article  MathSciNet  MATH  Google Scholar 

  20. Chatterjee K, Ou Z, Phoa F K H, et al., Uniform four-level designs from two-level designs: A new look, Statist. Sinica, 2017, 27: 171–186.

    MathSciNet  MATH  Google Scholar 

  21. Elsawah A M and Fang K T, New results on quaternary codes and their Gray map images for constructing uniform designs, Metrika, 2018, 81(3): 307–336.

    Article  MathSciNet  MATH  Google Scholar 

  22. Hu L, Ou Z, and Li H, Construction of four-level and mixed-level designs with zero Lee discrepancy, Metrika, 2020, 83: 129–139

    Article  MathSciNet  MATH  Google Scholar 

  23. Winke P and Fang K T, Optimal U-Type Designs. Monte Carlo and Quasi-Monte Carlo Methods, Eds. by Niederreiter H, Hellekalek P, Larcher G, and Zinterhof P, Springer, New York, 1997.

  24. Fang K T, Ke X, and Elsawah A M, Construction of uniform designs via an adjusted threshold accepting algorithm, J. Complexity, 2017, 43: 28–37.

    Article  MathSciNet  MATH  Google Scholar 

  25. Elsawah A M and Qin H, Optimum mechanism for breaking the confounding effects of mixed-level designs, Computational Statistics, 2017, 32(2): 781–802.

    Article  MathSciNet  MATH  Google Scholar 

  26. Yang F, Zhou Y D, and Zhang X R, Augmented uniform designs, J. Statist. Plan. Infer., 2017, 182: 61–73.

    Article  MathSciNet  MATH  Google Scholar 

  27. Elsawah A M, Constructing optimal asymmetric combined designs via Lee discrepancy, Statist. Probab. Lett. 2016, 118: 24–31.

    Article  MathSciNet  MATH  Google Scholar 

  28. Tang Y and Xu H, An effective construction method for multi-level uniform designs, J. Statist. Plan. Infer., 2013, 143: 1583–1589.

    Article  MathSciNet  MATH  Google Scholar 

  29. Elsawah A M, Fang K T, and Ke X, New recommended designs for screening either qualitative or quantitative factors, Statistical Papers, 2021, 62: 267–307.

    Article  MathSciNet  MATH  Google Scholar 

  30. Yang F, Zhou Y D, and Zhang A J, Mixed-level column augmented uniform designs, J. Complexity, 2019, 53: 23–39.

    Article  MathSciNet  MATH  Google Scholar 

  31. Elsawah A M, Fang K T, He P, et al., Optimum addition of information to computer experiments in view of uniformity and orthogonality, Bulletin of the Malaysian Math. Sci. Soc., 2019, 42(2): 803–826.

    Article  MathSciNet  MATH  Google Scholar 

  32. Fang K T and Hickernell F J, The uniform design and its applications, Bull. Inst. Int. Stat., 1995, 1: 333–349.

    Google Scholar 

  33. Elsawah A M and Qin H, A new strategy for optimal foldover two-level designs, Statist. Probab. Lett., 2015, 103: 116–126.

    Article  MathSciNet  MATH  Google Scholar 

  34. Elsawah A M, Multiple doubling: A simple effective construction technique for optimal two-level experimental designs, Statistal Papers, 2021, 62(6): 2923–2967.

    Article  MathSciNet  MATH  Google Scholar 

  35. Mukerjee R and Wu C F J, On the existence of saturated and nearly saturated asymmetrical orthogonal arrays, Ann. Statist., 1995, 23(6): 2102–2115.

    Article  MathSciNet  MATH  Google Scholar 

  36. Cheng C S, Deng L Y, and Tang B, Generalized minimum aberration and design efficiency for nonregular fractional factorial designs, Statist. Sinica, 2002, 12: 991–1000.

    MathSciNet  MATH  Google Scholar 

  37. Elsawah A M, Building some bridges among various experimental designs, J. Korean Statist. Soc., 2020, 49: 55–81.

    Article  MathSciNet  MATH  Google Scholar 

  38. Xu H, Minimum moment aberration for nonregular designs and supersaturated designs, Statist Sinica, 2003, 13: 691–708.

    MathSciNet  MATH  Google Scholar 

  39. Elsawah A M, Fang K T, He P, et al., Sharp lower bounds of various uniformity criteria for constructing uniform designs, Statistal Papers, 2021, 62: 1461–1482.

    Article  MathSciNet  MATH  Google Scholar 

  40. Cheng C S, Projection Properties of Factorial Designs for Factor Screening Screening, Eds. by Dean A and Lewis S, Springer, New York, 2006.

  41. Sun F, Wang Y, and Xu H, Uniform projection designs, Ann. Statist., 2019, 47(1): 641–661.

    Article  MathSciNet  MATH  Google Scholar 

  42. Elsawah A M, Tang Y, and Fang K T, Constructing optimal projection designs, Statistics, 2019, 53(6): 1357–1385.

    Article  MathSciNet  MATH  Google Scholar 

  43. Elsawah A M and Qin H, An effective approach for the optimum addition of runs to three-level uniform designs, J. Korean Statist. Soc., 2016, 45(4): 610–622.

    Article  MathSciNet  MATH  Google Scholar 

  44. Weng L C, Elsawah A M, and Fang K T, Cross-entropy loss for recommending efficient fold-over technique, Journal of Systems Science and Complexity, 2021, 34(1): 402–439.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. M. Elsawah.

Additional information

Elsawah’s work was supported by the UIC Research Grants with No. of (R201912 and R202010); the Curriculum Development and Teaching Enhancement with No. of (UICR0400046-21CTL); the Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science, BNU-HKBU United International College with No. of (2022B1212010006); and Guangdong Higher Education Upgrading Plan (2021–2025) with No. of (UICR0400001-22).

This paper was recommended for publication by HE Xu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Elsawah, A.M., Vishwakarma, G.K., Mohamed, H.S. et al. An Adjusted Gray Map Technique for Constructing Large Four-Level Uniform Designs. J Syst Sci Complex 36, 433–456 (2023). https://doi.org/10.1007/s11424-023-1144-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-023-1144-x

Keywords

Navigation