Advertisement

Designing Uniform Computer Sequential Experiments with Mixture Levels Using Lee Discrepancy

  • A M Elsawah
Article
  • 8 Downloads

Abstract

Computer experiments are constructed to simulate the behavior of complex physical systems. Uniform designs have good performance in computer experiments from several aspects. In practical use, the experimenter needs to choose a small size uniform design at the beginning of an experiment due to a limit of time, budget, resources, and so on, and later conduct a follow up experiment to obtain precious information about the system, that is, a sequential experiment. The Lee distance has been widely used in coding theory and its corresponding discrepancy is an important measure for constructing uniform designs. This paper proves that all the follow up designs of a uniform design are uniform and at least two of them can be used as optimal follow up experimental designs. Thus, it is not necessary that the union of any two uniform designs yields a uniform sequential design. Therefore, this article presents a theoretical justification for choosing the best follow up design of a uniform design to construct a uniform sequential design that involves a mixture of ω ≥ 1 factors with βk ≥ 2, 1 ≤ kω levels. For illustration of the usage of the proposed results, a closer look is given at using these results for the most extensively used six particular cases, three symmetric and three asymmetric designs, which are often met in practice.

Keywords

Computer experiment Lee discrepancy Lee distance lower bound sequential design uniform design 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The author greatly appreciates helpful suggestions of the Editor and the referees that significantly improved the paper. The author also thanks Prof. Kai-Tai Fang for his guidance and support.

References

  1. [1]
    Fang K T and Li R, Uniform design for computer experiments and its optimal properties, Int. J. Materials and Product Technology, 2006, 25(1/2/3): 198–210.CrossRefGoogle Scholar
  2. [2]
    Simpson T W, Lin D K J, and Chen W, Sampling strategies for computer experiments: Design and analysis, Int. J. Relability and Applications, 2001, 2(3): 209–240.Google Scholar
  3. [3]
    Fang K T, The uniform design: Application of number-theoretic methods in experimental design, Acta Math. Appl. Sinica, 1980, 3: 363–372.MathSciNetGoogle Scholar
  4. [4]
    Wang Y and Fang K T, A note on uniform distribution and experimental design. Chin. Sci. Bull., 1981, 26: 485–489.MathSciNetzbMATHGoogle Scholar
  5. [5]
    Hickernell F J, A generalized discrepancy and quadrature error bound, Math. Comp., 1998, 67: 299–322.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Hickernell F J, Lattice Rules: How Well Do They Measure Up? Random and Quasi-Random Point Setsq, Eds. by Hellekalek P and Larcher G, Springer, New York, 1998.Google Scholar
  7. [7]
    Roth R M, Introduction to Coding Theory, Cambridge University Press, Cambridge, UK, 2006.CrossRefzbMATHGoogle Scholar
  8. [8]
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Ji Y B, Alaerts G, Xu C J, et al., Sequential uniform designs for fingerprints development of Ginkgo biloba extracts by capillary electrophoresis, J. Chromatography, 2006, A1128: 273–281.CrossRefGoogle Scholar
  10. [10]
    Tong C, Refinement strategies for stratified sampling methods, Reliability Engineering and System Safety, 2006, 91: 1257–1265.CrossRefGoogle Scholar
  11. [11]
    Durrieu G and Briollais L, Sequential design for microarray experiments, J. Amer. Statist. Association, 2009, 104: 650–660.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Loeppky J L, Moore L M, and Williams B J, Batch sequential designs for computer experiments, J. Statist. Plann. Inference, 2010, 140: 1452–1464.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Cheong K T W, Htay K, Tan R H C, et al., Identifying combinatorial growth inhibitory effects of various plant extracts on leukemia cells through systematic experimental design, Amer. J. Plant Sci., 2012, 3: 1390–1398.CrossRefGoogle Scholar
  14. [14]
    Elsawah A M, Constructing optimal router bit life sequential experimental designs: New results with a case study, Commun. Stat. Simulat. Comput., 2017, http://dx.doi.org/10.1080/03610918.2017.1397164.Google Scholar
  15. [15]
    Bullington K E, Hool J N, and Maghsoodloo S, A simple method for obtaining resolution IV designs for use with Taguchi orthogonal arrays, J. Qual. Technol., 1990, 22(4): 260–264.CrossRefGoogle Scholar
  16. [16]
    Li W and Lin D K J, Optimal foldover plans for two-level fractional factorial designs, Technometrics, 2003, 45: 142–149.MathSciNetCrossRefGoogle Scholar
  17. [17]
    Box, G E P, Hunter W G, and Hunter J S, Statistics for Experiments, John Wiley and Sons, New York, 1978.zbMATHGoogle Scholar
  18. [18]
    Montgomery D C and Runger G C, Foldover of 2k-p resolution IV experimental designs, J. Qual. Technol., 1996, 28: 446–450.CrossRefGoogle Scholar
  19. [19]
    Li W, Lin D K J, and Ye K Q, Optimal foldover plans for non-regular orthogonal designs, Technometrics, 2003, 45: 347–351.MathSciNetCrossRefGoogle Scholar
  20. [20]
    Li P F, Liu M Q, and Zhang R C, Choice of optimal initial designs in sequential experiments, Metrika, 2005, 61(2): 127–135.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Wang B, Robert G M, and John F B, A note on the selection of optimal foldover plans for 16- and 32-run fractional factorial designs, J. Stat. Plan. Inference, 2010, 140: 1497–1500.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Fang K T, Lin D K J, and Qin H, A note on optimal foldover design, Statist. Probab. Lett., 2003, 62: 245–250.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    Elsawah A M and Qin H, An efficient methodology for constructing optimal foldover designs in terms of mixture discrepancy, J. Korean Statist. Soc., 2016, 45: 77–88.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Elsawah A M and Qin H, Optimum mechanism for breaking the confounding effects of mixed-level designs, Comput. Stat., 2017, 32(2): 781–802.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    Ou Z and Qin H, Optimal foldover plans of asymmetric factorials with minimum wrap-around L 2-discrepancy, Stat. Papers, 2017, DOI: 10.1007/s00362-017-0892-x.Google Scholar
  26. [26]
    Elsawah A M, A closer look at de-aliasing effects using an efficient foldover technique, Statistics, 2017, 51(3): 532–557.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    Elsawah A M, A powerful and efficient algorithm for breaking the links between aliased effects in asymmetric designs, Aust. N. Z. J. Stat., 2017, 59(1): 17–41.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    Elsawah A M, Choice of optimal second stage designs in two-stage experiments, Comput. Stat., 2018, 33(2): 933–965MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    Elsawah A M and Qin H, A new strategy for optimal foldover two-level designs, Statist. Probab. Lett., 2015, 103: 116–126.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    Elsawah A M and Qin H, New lower bound for centered L 2-discrepancy of four-level U-type designs, Statist. Probab. Lett., 2014, 93: 65–71.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    Elsawah A M and Qin H, A new look on optimal foldover plans in terms of uniformity criteria, Commun. Stat. Theory Methods, 2017, 46(4): 1621–1635.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    Elsawah A M, Constructing optimal asymmetric combined designs via Lee discrepancy, Statist. Probab. Lett., 2016, 118: 24–31.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    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.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceZagazig UniversityZagazigEgypt
  2. 2.Division of Science and TechnologyBNU-HKBU United International CollegeZhuhaiChina

Personalised recommendations