Random and Deterministic Digit Permutations of the Halton Sequence

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 23)


The Halton sequence is one of the classical low-discrepancy sequences. It is effectively used in numerical integration when the dimension is small, however, for larger dimensions, the uniformity of the sequence quickly degrades. As a remedy, generalized (scrambled) Halton sequences have been introduced by several researchers since the 1970s. In a generalized Halton sequence, the digits of the original Halton sequence are permuted using a carefully selected permutation. Some of the permutations in the literature are designed to minimize some measure of discrepancy, and some are obtained heuristically.In this paper, we investigate how these carefully selected permutations differ from a permutation simply generated at random. We use a recent genetic algorithm, test problems from numerical integration, and a recent randomized quasi-Monte Carlo method, to compare generalized Halton sequences with randomly chosen permutations, with the traditional generalized Halton sequences. Numerical results suggest that the random permutation approach is as good as, or better than, the “best” deterministic permutations.


  1. 1.
    E. I. Atanassov, On the discrepancy of the Halton sequences, Math. Balkanica, New Series 18 (2004) 15–32.Google Scholar
  2. 2.
    E. Braaten, G. Weller, An improved low-discrepancy sequence for multidimensional quasi-monte Carlo integration, Journal of Computational Physics 33 (1979) 249–258.Google Scholar
  3. 3.
    H. Chaix, H. Faure, Discrepance et diaphonie en dimension un, Acta Arithmetica LXIII (1993) 103–141.Google Scholar
  4. 4.
    H. Chi, M. Mascagni, T. Warnock, On the optimal { H}alton sequence, Mathematics and Computers in Simulation 70 (2005) 9–21.Google Scholar
  5. 5.
    H. Faure, Good permutations for extreme discrepancy, Journal of Number Theory 42 (1992) 47–56.Google Scholar
  6. 6.
    H. Faure, C. Lemieux, Generalized Halton sequences in 2008: A comparative study, ACM Transactions on Modeling and Computer Simulation 19 (2009) 15:1–31.Google Scholar
  7. 7.
    Y. Goncharov, G. Ökten, M. Shah, Computation of the endogenous mortgage rates with randomized quasi-Monte Carlo simulations, Mathematical and Computer Modelling 46 (2007) 459–481.Google Scholar
  8. 8.
    L. Kocis, W. J. Whiten, Computational investigations of low-discrepancy sequences, ACM Transactions on Mathematical Software 23 (1997) 266–294.Google Scholar
  9. 9.
    J. Matoušek, On the L2-discrepancy for anchored boxes, Journal of Complexity 14 (1998) 527–556.Google Scholar
  10. 10.
    H. Niederreiter. Random Number Generation and Quasi-Monte Carlo Methods, SIAM, Philadelphia, 1992.Google Scholar
  11. 11.
    L. Kuipers and H. Niederreiter Uniform Distribution of Sequences, Dover Publications, Mineola, NY, 2006.Google Scholar
  12. 12.
    G. Ökten, Generalized von Neumann-Kakutani transformation and random-start scrambled Halton sequences, Journal of Complexity 25 (2009) 318–331.Google Scholar
  13. 13.
    M. Shah, A genetic algorithm approach to estimate lower bounds of the star discrepancy, Monte Carlo Methods and Applications, Monte Carlo Methods Appl. 16 (2010) 379–398.Google Scholar
  14. 14.
    M. Shah, Quasi-Monte Carlo and Genetic Algorithms with Applications to Endogenous Mortgage Rate Computation, Ph.D. Dissertation, Department of Mathematics, Florida State University, 2008.Google Scholar
  15. 15.
    E. Thiémard, An algorithm to compute bounds for the star discrepancy, Journal of Complexity 17 (2001) 850–880.Google Scholar
  16. 16.
    M. Gnewuch, Bracketing numbers for axis-parallel boxes and applications to geometric discrepancy, Journal of Complexity 24 (2008), 154–172.Google Scholar
  17. 17.
    B. Tuffin, A New Permutation Choice in Halton Sequences, in: H. Niederreiter, P. Hellekalek, G. Larcher, and P. Zinterhof, editors, Monte Carlo and Quasi-Monte Carlo Methods 1996, Vol 127, Springer Verlag, New York, 1997, pp 427–435.Google Scholar
  18. 18.
    B. Vandewoestyne, R. Cools, Good permutations for deterministic scrambled Halton sequences in terms of L 2-discrepancy, Journal of Computational and Applied Mathematics 189 (2006) 341–361.Google Scholar
  19. 19.
    T. T. Warnock, Computational Investigations of Low-discrepancy Point Sets II, in: Harald Niederreiter and Peter J.-S. Shiue, editors, Monte Carlo and quasi-Monte Carlo methods in scientific computing, Springer, New York, 1995, pp. 354–361.Google Scholar
  20. 20.
    X. Wang, F. J. Hickernell, Randomized { H}alton sequences, Mathematical and Computer Modelling 32 (2000) 887–899.Google Scholar
  21. 21.
    I. Radovic, I. M. Sobol’, R. F. Tichy, Quasi-Monte Carlo methods for numerical integration: Comparison of different low-discrepancy sequences, Monte Carlo Methods Appl. 2 (1996) 1–14.Google Scholar
  22. 22.
    R. E. Caflisch, W. Morokoff, A. B. Owen, Valuation of mortgage backed securities using { B}rownian bridges to reduce effective dimension, Journal of Computational Finance 1 (1997) 27–46.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsFlorida State UniversityTallahasseeUSA

Personalised recommendations