Optimal U—Type Designs

  • Peter Winker
  • Kai-Tai Fang
Part of the Lecture Notes in Statistics book series (LNS, volume 127)


Designs with low discrepancy are of interest in many areas of statistical work. U—type designs are among the most widely studied design classes. In this paper a heuristic global optimization algorithm, Threshold Accepting, is used to find optimal U—type designs (uniform designs) or at least good approximations to uniform designs. As the evaluation of the discrepancy of a given point set is performed by an exact algorithm, the application presented here is restricted to small numbers of experiments in low dimensional spaces. The comparison with known optimal results for the two—factor uniform design and good designs for three to five factors shows a good performance of the algorithm.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    I. Althöfer, and K.-U. Koschnick (1991). On the Convergence of ‘Threshold Accepting’. Applied Mathematics and Optimization, 24, 183–195.CrossRefzbMATHMathSciNetGoogle Scholar
  2. [2]
    R. Bates, R. Buck, E. Riccomagno, and H. P. Wynn (1996). Experimental Design and Observation for Large Systems. Journal of the Royal Statistical Society B 57, 77–94.MathSciNetGoogle Scholar
  3. [3]
    P. Bundschuh, and Y. C. Zhu (1993). A method for exact calculation of the discrepancy of low-dimensional finite point set (I). Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 63, 115–133.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [4]
    L. De Clerk (1986). A method for exact calculation of the stardiscrepancy of plane sets applied to the sequences of Hammersley. Monatshefte für Mathematik, 101, 261–278.CrossRefGoogle Scholar
  5. [5]
    G. Dueck, and T. Scheuer (1990). Threshold Accepting: A General Purpose Algorithm Appearing Superior to Simulated Annealing. Journal of Computational Physics, 90, 161–175.CrossRefzbMATHMathSciNetGoogle Scholar
  6. [6]
    K.-T. Fang, and F.J. Hickernell (1995). The uniform design and its applications, Bulletin of the International Statistical Institute, 50th session, book 1, 11, pp. 49–65.zbMATHGoogle Scholar
  7. [[7]]
    K.-T. Fang, and Y. Wang (1994). Applications of Number Theoretic Methods in Statistics. Chapman and Hall, London.CrossRefGoogle Scholar
  8. [8]
    Y. Fang (1995). Relationships between Uniform and Orthogonal Designs. The 3rd ICSA Statistical Conference, Beijing.Google Scholar
  9. [9]
    B. L. Fox. Simulated Annealing: Folklore,Facts, and Directions. In: H. Niederreiter and P. J.-S. Shiue (eds.). Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing. Lecture Notes in Statistics 106, Springer, New York 1995.Google Scholar
  10. [10]
    L. K. Hua, and Y. Wang (1981). Applications of Number Theory to Numerical Analysis. Springer, Berlin.zbMATHGoogle Scholar
  11. [11]
    S. Kirkpatrick, C. Gelatt, and M. Vecchi (1983). Optimization by Simulated Annealing. Science, 220, 671–680.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [12]
    W. Li, and K.-T. Fang (1995). A Global Optimum Algorithm on Two Factor Uniform Design. In: K.-T. Fang and F. J. Hickernell (eds.). Proceedings Workshop on Quasi-Monte Carlo Methods and Their Applications. Hong Kong Baptist University, 189–201.Google Scholar
  13. [13]
    H. Niederreiter, (1973). Application of diophantine approximations to numerical integration. In: C. F. Osgood (ed.). Diophantine Approximations and Its Applications. Academic Press, New York, 129–199.Google Scholar
  14. [14]
    H. Niederreiter, (1992a). Random Number Generation and Quasi-Monte Carlo Methods. SIAM CBMS-NSF Regional Conference Series in Applied Mathematics, Philadelphia.CrossRefzbMATHGoogle Scholar
  15. [15]
    H. Niederreiter, (1992b). Lattice Rules for Multiple Integration. In: Marti, K. (ed.). Stochastic Optimization. Lecture Notes in Economics and Mathematical Systems 379, Springer-Verlag, Berlin.Google Scholar
  16. [16]
    P. Winker, and K.-T. Fang (1997). Application of Threshold Accepting to the Evaluation of the Discrepancy of a Set of Points. SIAM Journal on Numerical Analysis, forthcoming.Google Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Peter Winker
    • 2
  • Kai-Tai Fang
    • 1
  1. 1.Dept. of Economics and StatisticsUniversity of KonstanzKonstanzGermany
  2. 2.Dept. of MathematicsHong Kong Baptist UniversityKowloon Tong Hong KongChina

Personalised recommendations