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Uniform Design in Computer and Physical Experiments

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The Grammar of Technology Development

Abstract

Computer experiments have been widely used in various fields of industry, system engineering, and others because many physical phenomena are difficult or even impossible to study by conventional experimental methods. Design and modeling of computer experiments have become a hot topic since late Seventies of the Twentieth Century. Almost in the same time two different approaches are proposed for design of computer experiments: Latin hypercube sampling (LHS) and uniform design (UD). The former is a stochastic approach and the latter is a deterministic one. A uniform design is a low-discrepancy set in the sense of the discrepancy, the latter is a measure of uniformity. The uniform design can be used for computer experiments and also for physical experiments when the underlying model is unknown. In this paper we review some developments of the uniform design in the past years. More precisely, review and discuss relationships of fractional factorial designs including orthogonal arrays, supersaturated designs and uniform designs. Some basic knowledge of the uniform design with a demonstration example will be given.

A keynote speech in International Workshop on The Grammar of Technology Development, January 17–18, 2005, Tokyo, Japan. The authors would thank the invitation.

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References

  1. Booth, K.H.V. and Cox, D.R. (1962) Some systematic supersaturated designs, Technometric, 4, 489–495.

    Article  MATH  MathSciNet  Google Scholar 

  2. Box, G.E.P., Hunter, E.P. and Hunter, J.S. (1978) Statistics for Experimenters, Wiley, New York.

    MATH  Google Scholar 

  3. Dey, A. and Mukerjee, R. (1999) Fractional Factorial Plans, John Wiley, New York.

    MATH  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  5. Fang, K.T. and Ge, G.N. (2004) A sensitive algorithm for detecting the inequivalence of Hadamard matrices, Math. Computation, 73, 843–851.

    Article  MATH  MathSciNet  Google Scholar 

  6. Fang, K.T., Ge, G.N. and Liu, M.Q. (2002) Uniform supersaturated design and its construction, Science in China, Ser. A, 45, 1080–1088.

    Article  MATH  MathSciNet  Google Scholar 

  7. Fang, K.T., Ge, G.N, and Liu, M.Q. (2003) Construction of optimal supersaturated designs by the packing method, Science in China (Series A), 47, 128–143.

    MathSciNet  Google Scholar 

  8. Fang, K.T., Ge, G.N., Liu, M. and Qin, H. (2004a) Construction of uniform designs via super-simple resolvable t-designs, Utilitas Math., 66 15–32.

    MATH  MathSciNet  Google Scholar 

  9. Fang, K.T., Ge, G.N, Liu, M.Q. and Qin, H. (2004b) Combinatorial construction for optimal supersaturated designs, Discrete Math., 279, 191–202.

    Article  MATH  MathSciNet  Google Scholar 

  10. Fang, K.T., Li, R. and Sudjianto, A. (2005) Design and Modeling for Computer Experiments, Chapman & Hall/CRC Press, London.

    Google Scholar 

  11. Fang, K.T. and Lin, D.K.J. (2003) Uniform experimental design and its application in industry. Handbook on Statistics 22: Statistics in Industry, Eds. by R. Khattree and C.R. Rao, Elsevier, North-Holland, 131–170.

    Google Scholar 

  12. Fang, K.T., Lin, D.K.J. and Liu, M.Q. (2003) Optimal mixed-level supersaturated design, Metrika, 58, 279–291.

    Article  MATH  MathSciNet  Google Scholar 

  13. Fang, K.T., Lin, D.K.J., Winker, P. and Zhang, Y. (2000) Uniform design: Theory and Applications, Technometrics, 42, 237–248.

    Article  MATH  MathSciNet  Google Scholar 

  14. Fang, K.T. and Ma, C.X. (2000) The usefulness of uniformity in experimental design, in New Trends in Probability and Statistics, Vol. 5, T. Kollo, E.-M. Tiit and M. Srivastava, Eds., TEV and VSP, The Netherlands, 51–59.

    Google Scholar 

  15. Fang, K.T. and Mukerjee, R. (2000) A connection between uniformity and aberration in regular fractions of two-level factorials, Biometrika, 87, 193–198.

    Article  MATH  MathSciNet  Google Scholar 

  16. Fang, K.T. and Qin, H. (2004) Uniformity pattern and related criteria for two-level factorials, Science in China Ser. A. Mathematics, 47, 1–12.

    Article  MathSciNet  Google Scholar 

  17. Fang, K.T. and Winker, P. (1998) Uniformity and orthogonality, Technical Report MATH-175, Hong Kong Baptist University.

    Google Scholar 

  18. Fries, A. and Hunter, W.G. (1980) Minimum aberration 2k-p designs, Technometrics, 22, 601–608.

    Article  MATH  MathSciNet  Google Scholar 

  19. Hedayat, A.S., Sloane, N.J.A. and Stufken, J. (1999) Orthogonal Arrays: Theory and Applications, Springer, New York.

    MATH  Google Scholar 

  20. Hickernell, F.J. (1998) Lattice rules: how well do they measure up? in Random and Quasi-Random Point Sets, Eds. P. Hellekalek and G. Larcher, Springer-Verlag, 106–166.

    Google Scholar 

  21. Hickernell, F.J. (1999) Goodness-of-Fit Statistics, Discrepancies and Robust Designs, Statist. Probab. Lett., 44, 73–78.

    Article  MATH  MathSciNet  Google Scholar 

  22. Hickernell, F.J. and Liu, M.Q. (2002) Uniform designs limit aliasing, Biometrika, 389, 893–904.

    Article  MathSciNet  Google Scholar 

  23. Liang, Y.Z., Fang, K.T. and Xu, Q.S. (2001) Uniform design and its applications in chemistry and chemical engineering, Chemometrics and Intelligent Laboratory Systems, 58, 43–57.

    Article  Google Scholar 

  24. Lin, D.K.L. (1993) A new class of supersaturated designs, Technometrics, 35, 28–31.

    Article  Google Scholar 

  25. Lin, D.K.J. (2000) Recent developments in supersaturated designs, in Statistical Process Monitoring and Optimization, Eds. by S.H. Park and G.G. Vining, Marcel Dekker, Chapter 18.

    Google Scholar 

  26. Ma, C.X. and Fang, K.T. (2001) A note on generalized aberration in factorial designs, Metrika, 53, 85–93.

    Article  MATH  MathSciNet  Google Scholar 

  27. Ma, C.X., Fang, K.T. and Lin, D.K.J. (2001) On isomorphism of fractional factorial designs, J. Complexity, 17, 86–97.

    Article  MATH  MathSciNet  Google Scholar 

  28. Ma, C.X., Fang, K.T. and Lin, D.K.J. (2003) A note on uniformity and orthogonality, J. Statist. Plan. Inf., 113, 323–334.

    Article  MATH  MathSciNet  Google Scholar 

  29. Marshall, A.W. and Olkin, I. (1979) Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.

    MATH  Google Scholar 

  30. Niederreiter, H. (1992) Random Number Generation and Quasi-Monte Carlo Methods, SIAM CBMS-NSF Regional Conference Series in Applied Mathematics, Philadelphia.

    Google Scholar 

  31. Nguyen, N.-K. (1996) An algorithmic approach to constructing supersaturated designs, Technometrics, 38, 69–73.

    Article  MATH  Google Scholar 

  32. Sacks, J., Welch, W.J., Mitchell, T.J. and Wynn, H.P. (1989) Design and analysis of computer experiments. Statistical Science. 4, 409–435.

    Article  MATH  MathSciNet  Google Scholar 

  33. Santner, T.J., Williams, B. and Notz, W.I. (2001) The Design and Analysis of Computer Experiments, Springer, New York.

    Google Scholar 

  34. Saltelli, A. and Chan, K. and Scott, E.M. (2000) Sensitivity Analysis, Wiley, New York.

    MATH  Google Scholar 

  35. Stein, M.L. (1999) Interpolation of Spatial Data, Some Theory for Kriging, Springer, New York.

    MATH  Google Scholar 

  36. Wang, Y. and Fang, K.T. (1981) A note on uniform distribution and experimental design, KeXue TongBao, 26, 485–489.

    MathSciNet  MATH  Google Scholar 

  37. Wiens, D.P. (1991) Designs for approximately linear regression: two optimality properties of uniform designs, Statist. & Prob. Letters., 12, 217–221.

    Article  MATH  MathSciNet  Google Scholar 

  38. Winker, P. and Fang, K.T. (1997) Application of Threshold accepting to the evaluation of the discrepancy of a set of points, SIAM Numer Analysis, 34, 2038–2042.

    MathSciNet  Google Scholar 

  39. Winker, P. and Fang, K.T. (1998) Optimal U-type design, in Monte Carlo and Quasi-Monte Carlo Methods 1996, eds. by H. Niederreiter, P. Zinterhof and P. Hellekalek, Springer, 436–448.

    Google Scholar 

  40. Wu, C.F.J. and Hamada, M. (2000) Experiments: Planning, Analysis, and Parameter Design Optimization, Wiley, New York.

    MATH  Google Scholar 

  41. Xie, M.Y. and Fang, K.T. (2000) Admissibility and minimaxity of the uniform design in nonparametric regression model, J. Statist. Plan. Inference, 83 101–111.

    Article  MATH  MathSciNet  Google Scholar 

  42. Xu, H. and Wu, C.F.J. (2001) Generalized minimum aberration for asymmetrical fractional factorial designs. Ann. Statist., 29: 1066–77.

    Article  MATH  MathSciNet  Google Scholar 

  43. Yamada, S. and Lin, D.K.J. (1997) Supersaturated design including an orthogonal base, Canad. J. Statist., 25, 203–213.

    Article  MATH  MathSciNet  Google Scholar 

  44. Yamada, S. and Lin, D.K.J. (1999) Three-level supersaturated designs. Statist. Probab. Lett. 45, 31–39.

    Article  MATH  MathSciNet  Google Scholar 

  45. Yamada, S., Ikebe, Y.T., Hashiguchi, H. and Niki, N. (1999) Construction of three-level supersaturated design, J. Statist. Plann. Inference, 81, 183–193.

    Article  MATH  MathSciNet  Google Scholar 

  46. Yamada, S. and Matsui, T. (2002) Optimality of mixed-level supersaturated designs, J. Statist. Plann. Infer. 104, 459–468.

    Article  MATH  MathSciNet  Google Scholar 

  47. Zhang, A.J., Fang, K.T., Li, R. and Sudjianto, A. (2005) Majorization framework balanced lattice designs, The Annals of Statistics, 33, 2837–2853.

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China

    Kai-Tai Fang

  2. Department of Supply Chain and Information Systems, The Pennsylvania State University, University Park, PA, USA

    Dennis K. J. Lin

Authors
  1. Kai-Tai Fang
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  2. Dennis K. J. Lin
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Editors and Affiliations

  1. Graduate School of Business Sciences, University of Tsukuba, 3-29-1 Otsuka, Bunkyo-ku, Tokyo, 112-0012, Japan

    Hiroe Tsubaki Ph.D. (Professor) & Shu Yamada Ph.D. (Professor) (Professor) &  (Professor)

  2. Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya, Aichi, 466-8555, Japan

    Ken Nishina Ph.D. (Professor) (Professor)

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Fang, KT., Lin, D.K.J. (2008). Uniform Design in Computer and Physical Experiments. In: Tsubaki, H., Yamada, S., Nishina, K. (eds) The Grammar of Technology Development. Springer, Tokyo. https://doi.org/10.1007/978-4-431-75232-5_8

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  • DOI: https://doi.org/10.1007/978-4-431-75232-5_8

  • Publisher Name: Springer, Tokyo

  • Print ISBN: 978-4-431-75231-8

  • Online ISBN: 978-4-431-75232-5

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