On Figures of Merit for Randomly-Shifted Lattice Rules

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

Abstract

Randomized quasi-Monte Carlo (RQMC) can be seen as a variance reduction method that provides an unbiased estimator of the integral of a function f over the s-dimensional unit hypercube, with smaller variance than standard Monte Carlo (MC) under certain conditions on f and on the RQMC point set. When f is smooth enough, the variance converges faster, asymptotically, as a function of the number of points n, than the MC rate of \(\mathcal{O}(1/n)\). The RQMC point sets are typically constructed to minimize a given parameterized measure of discrepancy between their empirical distribution and the uniform distribution. These parameters can give different weights to the different subsets of coordinates (or lower-dimensional projections) of the points, for example. The ideal parameter values (to minimize the variance) depend very much on the integrand f and their choice (or estimation) is far from obvious in practice. In this paper, we survey this question for randomly-shifted lattice rules, an important class of RQMC point sets, and we explore the practical issues that arise when we want to use the theory to construct lattices for applications. We discuss various ways of selecting figures of merit and for estimating their ideal parameters (including the weights), we examine how they can be implemented in practice, and we compare their performance on examples inspired from real-life problems. In particular, we look at how much improvement (variance reduction) can be obtained, on some examples, by constructing the points based on function-specific figures of merit compared with more traditional general-purpose lattice-rule constructions.

Keywords

Monte Carlo Fourier Coefficient Product Weight Dual Lattice Asian Option 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

This research has been supported by NSERC-Canada grant No. ODGP0110050 and a Canada Research Chair to the first author. Computations were performed using the facilities of the Réseau québécois de calcul haute performance (RQCHP).

References

  1. 1.
    Avramidis, A.N., Wilson, J.R.: Integrated variance reduction strategies for simulation. Operations Research 44, 327–346 (1996)Google Scholar
  2. 2.
    Avramidis, A.N., Wilson, J.R.: Correlation-induction techniques for estimating quantiles in simulation experiments. Operations Research 46(4), 574–591 (1998)Google Scholar
  3. 3.
    Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups, 3rd edn. Grundlehren der Mathematischen Wissenschaften 290. Springer-Verlag, New York (1999)Google Scholar
  4. 4.
    Cools, R., Nuyens, D.: A Belgian view on lattice rules. In: A. Keller, S. Heinrich, H. Niederreiter (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2006, pp. 3–21. Springer-Verlag, Berlin (2008)Google Scholar
  5. 5.
    Cranley, R., Patterson, T.N.L.: Randomization of number theoretic methods for multiple integration. SIAM Journal on Numerical Analysis 13(6), 904–914 (1976)Google Scholar
  6. 6.
    Dick, J., Sloan, I.H., Wang, X., Wozniakowski, H.: Liberating the weights. Journal of Complexity 20(5), 593–623 (2004)Google Scholar
  7. 7.
    Dick, J., Sloan, I.H., Wang, X., Wozniakowski, H.: Good lattice rules in weighted Korobov spaces with general weights. Numerische Mathematik 103, 63–97 (2006)Google Scholar
  8. 8.
    Efron, B., Stein, C.: The jackknife estimator of variance. Annals of Statistics 9, 586–596 (1981)Google Scholar
  9. 9.
    Elmaghraby, S.: Activity Networks. Wiley, New York (1977)Google Scholar
  10. 10.
    Hickernell, F.J.: A generalized discrepancy and quadrature error bound. Mathematics of Computation 67(221), 299–322 (1998)Google Scholar
  11. 11.
    Hickernell, F.J.: Lattice rules: How well do they measure up? In: P. Hellekalek, G. Larcher (eds.) Random and Quasi-Random Point Sets, Lecture Notes in Statistics, vol. 138, pp. 109–166. Springer-Verlag, New York (1998)Google Scholar
  12. 12.
    Hickernell, F.J.: Obtaining O(N  − 2 + ε) convergence for lattice quadrature rules. In: K.T. Fang, F.J. Hickernell, H. Niederreiter (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2000, pp. 274–289. Springer-Verlag, Berlin (2002)Google Scholar
  13. 13.
    L’Ecuyer, P.: Good parameters and implementations for combined multiple recursive random number generators. Operations Research 47(1), 159–164 (1999)Google Scholar
  14. 14.
    L’Ecuyer, P.: Quasi-Monte Carlo methods with applications in finance. Finance and Stochastics 13(3), 307–349 (2009)Google Scholar
  15. 15.
    L’Ecuyer, P., Lemieux, C.: Variance reduction via lattice rules. Management Science 46(9), 1214–1235 (2000)Google Scholar
  16. 16.
    L’Ecuyer, P., Lemieux, C.: Recent advances in randomized quasi-Monte Carlo methods. In: M. Dror, P. L’Ecuyer, F. Szidarovszky (eds.) Modeling Uncertainty: An Examination of Stochastic Theory, Methods, and Applications, pp. 419–474. Kluwer Academic, Boston (2002)Google Scholar
  17. 17.
    L’Ecuyer, P., Munger, D., Tuffin, B.: On the distribution of integration error by randomly-shifted lattice rules. Electronic Journal of Statistics 4, 950–993 (2010)Google Scholar
  18. 18.
    Lemieux, C.: Monte Carlo and Quasi-Monte Carlo Sampling. Springer-Verlag, New York, NY (2009)Google Scholar
  19. 19.
    Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods, SIAM CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 63. SIAM, Philadelphia, PA (1992)Google Scholar
  20. 20.
    Owen, A.B.: Latin supercube sampling for very high-dimensional simulations. ACM Transactions on Modeling and Computer Simulation 8(1), 71–102 (1998)Google Scholar
  21. 21.
    Sloan, I.H., Joe, S.: Lattice Methods for Multiple Integration. Clarendon Press, Oxford (1994)Google Scholar
  22. 22.
    Sloan, I.H., Kuo, F.Y., Joe, S.: On the step-by-step construction of quasi-Monte Carlo rules that achieve strong tractability error bounds in weighted Sobolev spaces. Mathematics of Computation 71, 1609–1640 (2002)Google Scholar
  23. 23.
    Sloan, I.H., Rezstov, A.: Component-by-component construction of good lattice rules. Mathematics of Computation 71, 262–273 (2002)Google Scholar
  24. 24.
    Sloan, I.H., Woźniakowski, H.: When are quasi-Monte Carlo algorithms efficient for high-dimensional integrals. Journal of Complexity 14, 1–33 (1998)Google Scholar
  25. 25.
    Sobol’, I.M., Myshetskaya, E.E.: Monte Carlo estimators for small sensitivity indices. Monte Carlo Methods and Applications 13(5–6), 455–465 (2007)Google Scholar
  26. 26.
    Wang, X.: Constructing robust good lattice rules for computational finance. SIAM Journal on Scientific Computing 29(2), 598–621 (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Département d’Informatique et de Recherche OpérationnelleUniversité de MontréalMontréalCanada

Personalised recommendations