INTRODUCTION

Since 1990 Hong Kong Baptist University (HKBU) has been steadily building a research program in quasi-Monte Carlo methods. This booklet serves to describe our activities and invite collaboration with like-minded scholars.

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HISTORY

The roots of our research efforts can be traced back to the eminent Chinese mathematicians Loo-Keng Hua and Yuan Wang who made significant contributions to the field of quasi-Monte Carlo or number-theoretic methods in the 1960s and 1970s. They summarized their results in the monograph, Applications of Number Theory to Numerical Analysis, published in 1981. Around that time Yuan Wang and Kai-Tai Fang, both of the Chinese Academy of Sciences, pioneered the use of quasi-Monte Carlo methods for design of experiments. They found that experimental designs based on low discrepancy sets could often lead to an understanding of the underlying physical process using far fewer experiments than traditional orthogonal designs. After achieving success in the field of experimental design Prof. Fang also explored the use of quasi-Monte Carlo methods to solve other problems in statistics.

In 1990 Prof. Fang joined the Mathematics Department of Hong Kong Baptist University and introduced quasi-Monte Carlo methods to the local community. Prof. Wang also visited HKBU and together with Prof. Fang authored the monograph Number-Theoretic Methods in Statistics, published in 1994. Fred Hickernell began collaborating with Prof. Fang in the early 1990s. In 1992 the University established the Statistics Research and Consultancy Centre (SRCC), with Prof. Fang as its director. Much of the research and scholarly activities of the Centre are related to quasi-Monte Carlo methods.

Our present quasi-Monte Carlo methods team also includes Profs. Fang and Hickernell, Drs. Yiu-Wing Leung and Gang Wei, visitors, research assistants, and postgraduate students. Activities are funded by research grants from the Hong Kong Research Grants Council and Hong Kong Baptist University.

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RESEARCH AREAS

At HKBU our research covers several aspects of the theory and applications of quasi-Monte Carlo methods and related disciplines including:

Publications arising out of this research appear below.

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COLLABORATIONS

Our research efforts have benefited substantially from scholars who have visited us and collaborated with us. They have brought new ideas and approaches to our research problems, helped us to organize international conferences, and welcomed our postgraduate students to visit and work with them. Our collaborators come from all over the world and include

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CONFERENCES ORGANIZED

The Mathematics Department and Statistics Research and Consultancy Centre have organized and hosted a number of international conferences related to quasi-Monte Carlo methods. These include

In addition, Profs. Fang and Hickernell have been involved in organizing other conferences with invited talks or special sessions devoted to quasi-Monte Carlo methods.

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POSTGRADUATE STUDENTS

A number of MPhil and PhD students have at HKBU and the Chinese Academy of Sciences (CAS) have engaged in quasi-Monte Carlo research. These include:

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PUBLICATIONS

The list below includes books, book chapters, articles, and conference papers related to quasi-Monte Carlo methods. A number of recent articles have appeared or been accepted to appear in Biometrika, various IEEE Transactions, the Journal of Complexity, Mathematics of Computation, the SIAM Journal on Numerical Analysis, the SIAM Journal on Scientific Computing, Statistica Sinica, Statistical Sciences, and Technometrics. Some publications in the list below have been very influential. For example, according to the Science Citation Index the monograph Number Theoretic Methods in Statistics has been cited 55 times.

  1. K. T. Fang, Experimental design by uniform distribution, Acta Math. Appl. Sinica 3 (1980), 363-372.
  2. Y. Wang and K. T. Fang, A note on uniform distribution and experimental design, Chinese Sci. Bull. 26 (1981), 485-489.
  3. K. T. Fang and Yuan K. H., A unified approach to maximum likelihood estimation, Chinese J. Appl. Probab. Statist. 7 (1990), 412-41.
  4. Y. Wang and K. T. Fang, Number theoretic methods in applied statistics, Chinese Ann. Math. Ser. B 11 (1990), 41-55.
  5. ____________, Number theoretic methods in applied statistics (II), Chinese Ann. Math. Ser. B 11 (1990), 384-394.
  6. ____________, A sequential number-theoretic methods for optimization and its applications to regression analysis, Lecture Notes in Contemporary Math., Longman, London, (1990), 139-156.
  7. K. T. Fang and Y. Wang, A sequential algorithm for solving a system of nonlinear equations, J. Comput. Math. 9 (1991), 9-16.
  8. ____________, Applications of quasi-random sequence in statistics, Proceedings of Asian Mathematical Conference 1990 (Z. Li, et al, ed.), World Scientific, Singapore, (1992), 135-139.
  9. K. T. Fang, K. H. Yuan, and P. M. Bentler, Applications of sets of points uniformly distributed on sphere to test multinormality and robust estimation, Probability and Statistics (Z. P. Jiang, et al, ed.), World Scientific, Singapore, (1992), 56-73.
  10. K. T. Fang and H. N. Zhen, A maximum symmetric differences principle and its applications in uniform design, Chinese J. Appl. Probab. Statist. 8 (1992), 10-18.
  11. Y. Wang and K. T. Fang, A sequential number-theoretic methods for optimization and its applications in statistics, The Development of Statistics: Recent Contributions from China, Longman, London, (1992), 139-156.
  12. K. T. Fang and G. Wei, The distribution of a class the first hitting time, Acta Math. Appl. Sinica 15 (1993), 460-467.
  13. K. T. Fang and J. T. Zhang, A new algorithm for calculation of estimates of parameters of nonlinear regression modeling, Acta Math. Appl. Sinica 16 (1993), 366-37.
  14. K. T. Fang, Uniform design and design tables, Science Press, Beijing, 1994, (in Chinese).
  15. K. T. Fang, P. M. Bentler, and K. H. Yuan, Applications of number-theoretic methods to quantizers of elliptically contoured distributions, Multivariate Analysis and Its Applications, IMS Lecture Notes Monograph Ser., (1994), 211-225.
  16. K. T. Fang and J. K. Li, Some new results on uniform design, Chinese Sci. Bull. 9 (1994), 416-428, English version in 40 (1995), 268-272.
  17. K. T. Fang and Y. Wang, Number-theoretic methods in statistics, Chapman and Hall, London, (1994).
  18. K. T. Fang, Y. Wang, and P. M. Bentler, Some applications of number-theoretic methods in statistics, Statist. Sci. 9 (1994), 416-428.
  19. K. T. Fang and F. J. Hickernell, The uniform design and its applications, Bull. Inst. Internat. Statist., 50thSession, Book 1 (1995), 333-349.
  20. F. J. Hickernell, A comparison of random and quasirandom points for multidimensional quadrature, Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (H. Niederreiter and P. J. S. Shiue, eds.), vol. 106, Springer-Verlag, New York, 1995, 213-227.
  21. W. C. Shiu, S. L. Ma, and K. T. Fang, On the rank of cyclic Latin squares, Linear and Multilinear Algebra 40 (1995), 183-188.
  22. Y. Wang, D. K. Lin, and K. T. Fang, Designing outer array points, J. Quality Technology 27 (1995), 226-241.
  23. L. X. Zhu, K. T. Fang, and J. T. Zhang, A projection NT-type test for spherical symmetry of a multivariate distribution, New Trends in Probability and Statistics, vol. 3 - Multivariate Statistics and Matrices in Statistics (M. Tiit, T. Kollo, and H. Niemi, eds.), VSP - TEV, Utrecht, The Netherlands, (1995), 109 - 122.
  24. K. T. Fang and F. J. Hickernell, Discussion of the papers by Atkinson, and Bates et al, J. Roy. Statist. Soc. B 58 (1996), 103.
  25. F. J. Hickernell, The mean square discrepancy of randomized nets, ACM Trans. Model. Comput. Simul. 6 (1996), 274-296.
  26. ____________, Quadrature error bounds with applications to lattice rules, SIAM J. Numer. Anal. 33 (1996), 1995-2016.
  27. Y. Z. Liang and K. T. Fang, Robust multivariate calibration algorithm based on least median squares and sequential number theoretic optimization method, Analyst Chemistry 121 (1996), 1025-1029.
  28. Y. Wang and K. T. Fang, Uniform design of experiments with mixtures, Sci. China Ser. A 39 (1996), 264-275.
  29. K. T. Fang and R. Z. Li, Some methods for generating both an NT-net and the uniform distribution on a stiefel manifold and their applications, Comput. Statist. Data Anal. 24 (1997), 29-46.
  30. K. T. Fang and Y. Wang, Number-theoretic methods, Encyclopedia of Statistics, Update vol. 2, Wiley, New York, (1997), 993-998.
  31. F. J. Hickernell and H. S. Hong, Computing multivariate normal probabilities using rank-1 lattice sequences, Proceedings of the Workshop on Scientific Computing, (Hong Kong) (G. H. Golub, S. H. Lui, F. T. Luk, and R. J. Plemmons, eds.), Springer-Verlag, Singapore, (1997), 209-215.
  32. F. J. Hickernell and Y. X. Yuan, A simple multi-start algorithm for global optimization, Oper. Res. Trans. 1 (1997), no. 2, 1-11.
  33. F. J. Hickernell, R. X. Yue, and F. S. Hickernell, Statistical modeling for the optimal deposition of sputtered piezoelectric films, IEEE Trans. Ultrason., Ferroelectrics & Frequency Control 44 (1997), 615-623.
  34. S. Kotz, K. T. Fang, and J. J. Liang, On multivariate vertical density representation and its application to random number generation, Statistics 30 (1997), 163-180.
  35. A. W. M. Lee, W. F. Chan, F. S. Y. Yuen, P. K. Tse, Y. Z. Liang, and K. T. Fang, An example of a sequential uniform design: application in capillary electrophoresis, Chemometrics and Intelligent Laboratory Systems 39 (1997), 11-18.
  36. P. Winker and K. T. Fang, Application of threshold accepting to the evaluation of the discrepancy of a set of points, SIAM J. Numer. Anal. 34 (1997), 2038-2042.
  37. L. Zhang, Y. Z. Liang, R. Q. Yu, and K. T. Fang, Sequential number-theoretic optimization (SNTO) method applied to chemical quantitative analysis, J. Chemometrics 11 (1997), 267-281.
  38. L. X. Zhu, K. T. Fang, and R. Z. Li, A new approach for testing symmetry of a high-dimensional distribution, Bull. Hong Kong Math. Soc. 1 (1997), 35-46.
  39. K. T. Fang, Z. Zheng, and W. Lu, Discrepancy with respect to Kaplan-Meier estimator, Commun. Statist.-Simula 27 (1998), 329-344.
  40. F. J. Hickernell, A generalized discrepancy and quadrature error bound, Math. Comp. 67 (1998), 299-322.
  41. ____________, Lattice rules: How well do they measure up?, Random and Quasi-Random Point Sets, Springer Lecture Notes in Statistics, vol. 138, Springer-Verlag, New York, (1998), 109-166.
  42. G. L. Tian and K. T. Fang, Stochastic representation and uniform designs for mixture-amount experiments and for mixture experiments under order restrictions, Sci. China 28 (1998), no. 12, 1087-1101.
  43. P. Winker and K. T. Fang, Optimal U-type design, Monte Carlo and Quasi-Monte Carlo Methods 1996 (H. Niederreiter, P. Zinterhof, and P. Hellekalek, eds.), Springer, (1998), 436-448.
  44. L. Zhang, Y. Z. Liang, J. H. Jiang, R. Q. Yu, and K. T. Fang, Uniform design applied to nonlinear multivariate calibration by ANN, Analytic Chimica Acta 370 (1998), 65-77.
  45. K. T. Fang and C. X. Ma, Applications of uniformity to factorial designs, J. Chinese Statist. Assoc. (1999), to appear, (in Chinese).
  46. ____________, Connections and comparisons between uniform design and orthogonal design, Symposium on the Theory of Uniform Design and Its Applications, Hong Kong Baptist University, 1999, 8-40, (in Chinese).
  47. K. T. Fang, C. X. Ma, and J. K. Li, Recent development of orthogonal factorial designs and their application -- applications of regression analysis to orthogonal designs, Appl. Statist. & Management 18 (1999), no. 2, 244-49, (in Chinese).
  48. ____________, Recent development of orthogonal factorial designs and their applications (II) - uniformly orthogonal designs, Appl. Statist. & Management 18 (1999), no. 3, 43-51, (in Chinese).
  49. ____________, Recent development of orthogonal factorial designs and their applications (III) - D-optimality of orthogonal designs, Appl. Statist. & Management 18 (1999), no. 4, 43-52, (in Chinese).
  50. ____________, Recent development of orthogonal factorial designs and their applications (IV) - projection properties of orthogonal designs, Appl. Statist. & Management 18 (1999), no. 4, 35-43, 5, (in Chinese).
  51. 51 K. T. Fang, W. C. Shiu, and J. X. Pan, Uniform designs based on Latin squares, Statist. Sinica 9 (1999), 905-912.
  52. 52 K. T. Fang and Z. H. Yang, On uniform design of experiments with restricted mixtures and generation of uniform distribution on some domains, Statist. Probab. Lett. 46 (1999), 113-120.
  53. 53 K. T. Fang and Z. K. Zheng, A two-stage algorithm of numerical evaluation of integrals in number-theoretic methods, J. Comput. Math. 17 (1999), 285-292.
  54. F. J. Hickernell, What affects the accuracy of quasi-Monte Carlo quadrature?, Monte Carlo and Quasi-Monte Carlo Methods 1998 (H. Niederreiter and J. Spanier, eds.), Springer-Verlag, Berlin, (1999), 16-55.
  55. F. J. Hickernell and H. S. Hong, The asymptotic efficiency of randomized nets for quadrature, Math. Comp. 68 (1999), 767-791.
  56. C. X. Ma and K. T. Fang, The usefulness of repeated experiments in experimental designs, Symposium on the Theory of Uniform Design and Its Applications, Hong Kong Baptist University, 1999, 104-114, (in Chinese).
  57. G. L. Tian and K. T. Fang, Uniform design for mixture-amount experiments and for mixture experiments under order restrictions, Sci. China Ser. A 42 (1999), no. 5, 456-470.
  58. P. Winker and K. T. Fang, Randomness and quasi-Monte Carlo approaches, some remarks on fundamentals and applications in statistics and econometrics, Jahrbücher für Nationalökonomie and Statistics 218 (1999), 215-228.
  59. R. X. Yue, Variance of quadrature over scrambled unions of nets, Statist. Sinica 9 (1999), 451-473.
  60. R. X. Yue and F. J. Hickernell, Robust optimal designs for fitting linear models with misspecification, Statist. Sinica 9 (1999), 1053-1069.
  61. Q. Zhang and Y. W. Leung, An orthogonal genetic algorithm for multimedia multicast routing, IEEE Trans. Evolutionary Comput. 3 (1999), no. 1, 53-62.
  62. K. T. Fang, S. Deng, and C. X. Ma, A sequential optimization algorithm for evaluation of the form - position errors, (2000), submitted for publication, (in Chinese).
  63. K. T. Fang, Z. Geng, and G. L. Tian, Statistical inference for truncated dirichlet distribution and its application in misclassification, Biometrical J. 8 (2000), 137-152.
  64. K. T. Fang, D. J. Lin, and C. X. Ma, On construction of meulti-level supersaturated designs, J. Statist. Plann. Inference 6 (2000), 129-141.
  65. K. T. Fang, D. K. J. Lin, P. Winker, and Y. Zhang, Uniform design: Theory and applications, Technometrics 42 (2000), 237-248.
  66. K. T. Fang and C. X. Ma, Wrap-around L2-discrepancy of radome sampling, Latin hypercube and uniform designs, J. Complexity (2000), to appear.
  67. K. T. Fang and R. Mukerjee, A connection between uniformity and aberration in regular fractions of two-level factorials, Biometrika 87 (2000), 193-198.
  68. F. J. Hickernell and H. S. Hong, Quasi-Monte Carlo methods and their randomizations, Proceedings of the IMS Workshop on Applied Probability, 2000, to appear.
  69. F. J. Hickernell and H. Wozniakowski, Integration and approximation in arbitrary dimensions, Adv. Comput. Math. 12 (2000), 25-58.
  70. Y. W. Leung and Y. P. Wang, Multiobjective programming using uniform design and genetic algorithm, IEEE Trans. System Man Cybernet 30 (2000), no. 3, 293-304.
  71. ____________, An orthogonal genetic algorithm with quantization for global numerical optimization, IEEE Trans. Evolutionary Comput. 4 (2000), no. 4.
  72. C. X. Ma, K. T. Fang, and E. Liski, A new approach in constructing orthogonal and nearly orthogonal arrays, Metrika 50 (2000), 255-268.
  73. M. Y. Xie and K. T. Fang, Admissibility and minimaxity of the uniform design in nonparametric regression model, J. Statist. Plann. Inference 83 (2000), 101-111.
  74. Q. S. Xu, Y. Z. Liang, and K. T. Fang, The effects of different experimental designs on parameter estimation in the kinetics of a reversible chemical reaction, Chemometrics and Intelligent laboratory Systems 52 (2000), 155-166.
  75. K. T. Fang, C. X. Ma, and P. Winker, Centered L2-discrepancy of random sampling and Latin hypercube design, and construction of uniform designs, Math. Comp. (2001), to appear.
  76. K. T. Fang, Z. H. Yang, and S. Kotz, Generation of multivariate distributions by vertical density representation, Statistics (2001), to appear.
  77. F. J. Hickernell, H. S. Hong, P. L'écuyer, and C. Lemieux, Extensible lattice sequences for quasi-Monte Carlo quadrature, SIAM J. Sci. Comput. 22 (2001), 1117-1138, to appear.
  78. F. J. Hickernell and H. Wozniakowski, Tractability of multivariate integration for periodic functions, J. Complexity (2001), to appear.
  79. F. J. Hickernell and R. X. Yue, The mean square discrepancy of scrambled (t,s)-sequences, SIAM J. Sci. Comput. 38 (2001), 1089-1112, to appear.
  80. J. J. Liang, K. T. Fang, F. J. Hickernell, and R. Z. Li, Testing multivariate uniformity and its applications, Math. Comp. (2001), to appear.
  81. C. X. Ma, K. T. Fang, and D. K. J. Lin, On isomorphism of fractional factorial designs, J. Complexity (2001), to appear.
  82. X. Wang and F. J. Hickernell, Randomized Halton sequences, Math. Comput. Modelling (2001), to appear.
  83. R. X. Yue and F. J. Hickernell, Integration and approximation based on scramble sampling in arbitrary dimensions, J. Complexity (2001), to appear.

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PREPRINTS AND TECHNICAL REPORTS

The list below contains manuscripts that have not been published, including a number are under review for publication.
  1. K. T. Fang and J. T. Zhang, Criteria for the uniform design, Tech. Report 002, Department of Mathematics, Hong Kong Baptist University, 1991.
  2. K. T. Fang, Y. Wang, and H. L. Wong, A new method for generating the uniform distribution on the unit sphere, Tech. Report 015, Department of Mathematics, Hong Kong Baptist University, 1992.
  3. F. J. Hickernell and K. T. Fang, Combining quasirandom search and Newton-like methods for nonlinear equations, Tech. Report 037, Department of Mathematics, Hong Kong Baptist University, 1993.
  4. L. X. Zhu, K. T. Fang, and R. Z. Li, A projection NT-Type test of multinormality based on the skewness and kurtosis indices, Tech. Report 052, Department of Mathematics, Hong Kong Baptist University, 1994.
  5. K. T. Fang and W. Li, A global optimum algorithm on two factor uniform design, Tech. Report 095, Department of Mathematics, Hong Kong Baptist University, 1995.
  6. K. T. Fang and M. Y. Xie, Orthogonality of the uniform type design under general Fourier regression models, Tech. Report 102, Department of Mathematics, Hong Kong Baptist University, 1995.
  7. M. Y. Xie and K. T. Fang, Orthogonal array: Uniformity and D-optimality of wavelet regression models, Tech. Report 131, Department of Mathematics, Hong Kong Baptist University, 1996.
  8. K. T. Fang, M. Y. Xie, and Y. Wang, The usefulness of uniformity of experimental points over the domain, Tech. Report 145, Department of Mathematics, Hong Kong Baptist University, 1997.
  9. C. Ma and K. T. Fang, Applications of uniformity to orthogonal fractional factorial designs, Tech. Report 193, Department of Mathematics, Hong Kong Baptist University, 1998.
  10. K. T. Fang and C. X. Ma, Relationship between uniformity and aberration, constructions of uniform designs in regular fractions 3s-1, Tech. Report 243, Department of Mathematics, Hong Kong Baptist University, 1999.
  11. ____________, Some connections between uniformity orthogonality and aberration in regular fractional factorial designs, Tech. Report 248, Department of Mathematics, Hong Kong Baptist University, 1999.
  12. K. T. Fang, C. X. Ma, and R. Mukerjee, Uniformity in fractional factorials, Tech. Report 274, Department of Mathematics, Hong Kong Baptist University, 2000.
  13. F. J. Hickernell and M. Q. Liu, Uniform designs limit aliasing, Tech. Report 275, Department of Mathematics, Hong Kong Baptist University, 2000.
  14. F. J. Hickernell, M. Q. Liu, and C. Y. Yam, Discrepancy measures of uniformity, Tech. Report 269, Department of Mathematics, Hong Kong Baptist University, 2000, (in Chinese).
  15. F. J. Hickernell and X. Wang, The error bounds and tractability of quasi-Monte Carlo algorithms in infinite dimension, Tech. Report 273, Department of Mathematics, Hong Kong Baptist University, 2000.
  16. F. J. Hickernell and H. Wozniakowski, The price of pessimism for multidimensional quadrature, Tech. Report 268, Department of Mathematics, Hong Kong Baptist University, 2000.
  17. M. Q. Liu and K. T. Fang, Some results on resolvable incomplete block designs, Tech. Report 280, Department of Mathematics, Hong Kong Baptist University, 2000.
  18. M. Q. Liu and F. J. Hickernell, Connections between uniformity and E(s2)-optimality in 2-level supersaturated designs, Tech. Report 289, Department of Mathematics, Hong Kong Baptist University, 2000.
  19. 19 C. X. Ma and K. T. Fang, A note on generalized aberration in factorial designs, Tech. Report 281, Department of Mathematics, Hong Kong Baptist University, 2000.

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OPPORTUNITIES FOR COOPERATION

As mentioned above we value collaborations with other scholars and research groups. We want to learn from the expertise of others and involve others in research problems that interest us. Cooperation may take place in several ways, including the following.

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CONTACT INFORMATION

To find out more about our work on quasi-Monte Carlo methods or opportunities for cooperation please contact either Prof. Kai-Tai Fang or Prof. Fred J. Hickernell. We look forward to hearing from you.

Prof. Kai-Tai Fang, Director
Statistics Research and Consultancy Centre
Hong Kong Baptist University
Kowloon Tong, Hong Kong SAR, China
Phone: +852 2339 7025, Fax: +852 2339 5811
Email: ktfang@hkbu.edu.hk
Web: http://www.math.hkbu.edu.hk/~ktfang.html

Prof. Fred J. Hickernell, Head
Department of Mathematics
Hong Kong Baptist University
Kowloon Tong, Hong Kong SAR, China
Phone: +852 2339 7015, Fax: +852 2339 5811
Email: fred@hkbu.edu.hk
Web: http://www.math.hkbu.edu.hk/~fred

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