Soumyadip Ghosh
Shane G. Henderson
Abstract
The NORTA method is a fast general-purpose method for generating samples of a random vector with given marginal distributions and given correlation matrix. It is known that there exist marginal distributions and correlation matrices that the NORTA method cannot match, even though a random vector with the prescribed qualities exists. We investigate this problem as the dimension of the random vector increases. Simulation results show that the problem rapidly becomes acute, in the sense that NORTA fails to work with an increasingly large proportion of correlation matrices. Simulation results also show that if one is willing to settle for a correlation matrix that is "close" to the desired one, then NORTA performs well with increasing dimension. As part of our analysis, we develop a method for sampling correlation matrices uniformly (in a certain precise sense) from the set of all such matrices. This procedure can be used more generally for sampling uniformly from the space of all symmetric positive definite matrices with diagonal elements fixed at positive values.
- Alfakih, A. and Wolkowicz, H. 2000. Matrix completion problems. In Handbook of Semidefinite Programming: Theory, Algorithms and Applications, H. Wolkowicz, R. Saigal, and L. Vandenberghe, Eds. Kluwer, Boston, Mass. 533--545.Google Scholar
- Cario, M. C. and Nelson, B. L. 1997. Modeling and generating random vectors with arbitrary marginal distributions and correlation matrix. Tech. Rep., Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Ill.Google Scholar
- Clemen, R. T. and Reilly, T. 1999. Correlations and copulas for decision and risk analysis. Manage. Sci. 45, 208--224. Google ScholarDigital Library
- Das, S. R., Fong, G., and Geng, G. 2001. Impact of correlated default risk on credit portfolios. J. Fixed Inc. 11, 3, 9--19.Google ScholarCross Ref
- Devroye, L. 1986. Non-uniform Random Variate Generation. Springer-Verlag, New York.Google Scholar
- Edelman, A. 1989. Eigenvalues and condition numbers of random matrices. Ph.D. dissertation, Dep. Mathematics, MIT, Cambridge, Mass.Google Scholar
- Frobenius, F. G. 1968. Gessammelte Abhandlungen. Vol. 3. Springer, Berlin, Germany.Google Scholar
- Ghosh, S. and Henderson, S. G. 2001. Chessboard distributions. In Proceedings of the 2001 Winter Simulation Conference, B. A. Peters, J. S. Smith, D. J. Medeiros, and M. W. Rohrer, Eds. IEEE, Piscataway, N.J., 385--393. Google ScholarDigital Library
- Ghosh, S. and Henderson, S. G. 2002a. Chessboard distributions and random vectors with specified marginals and covariance matrix. Oper. Res. 50, 5, 820--834. Google ScholarDigital Library
- Ghosh, S. and Henderson, S. G. 2002b. Properties of the NORTA method in higher dimensions. In Proceedings of the 2002 Winter Simulation Conference, E. Yücesan, C. H. Chen, J. L. Snowdon, and J. M. Charnes, Eds. IEEE, Piscataway, N.J., 263--269. Google ScholarDigital Library
- Guttman, L. 1946. Enlargement methods for computing the inverse matrix. Ann. Math. Stat. 17, 336--343.Google ScholarCross Ref
- Henderson, S. G. 2001. Mathematics for simulation. In Proceedings of the 2001 Winter Simulation Conference, B. A. Peters, J. S. Smith, D. J. Medeiros, and M. W. Rohrer, Eds. IEEE, Piscataway, N.J., 83--94. Google ScholarDigital Library
- Henderson, S. G., Chiera, B. A., and Cooke, R. M. 2000. Generating "dependent" quasi-random numbers. In Proceedings of the 2000 Winter Simulation Conference, J. A. Joines, R. R. Barton, K. Kang, and P. A. Fishwick, Eds. IEEE, Piscataway, N.J., 527--536. Google ScholarDigital Library
- Hill, R. R. and Reilly, C. H. 2000. The effects of coefficient correlation structure in two-dimensional knapsack problems on solution procedure performance. Manage. Sci. 46, 302--317. Google ScholarDigital Library
- Iman, R. and Conover, W. 1982. A distribution-free approach to inducing rank correlation among input variables. Commun. Stat. Simulat. Comput. 11, 311--334.Google ScholarCross Ref
- Joe, H. 1997. Multivariate Models and Dependence Concepts. Chapman & Hall, London, England.Google Scholar
- Johnson, C. R. 1990. Matrix completion problems: A survey. Proc. Symp. Appl. Math. 40, 171--198.Google ScholarCross Ref
- Johnson, M. E. 1987. Multivariate Statistical Simulation. Wiley, New York. Google ScholarDigital Library
- Kendall, M. G. 1961. A Course in the Geometry of n Dimensions. Charles Griffin and Co., London, England.Google Scholar
- Kruskal, W. 1958. Ordinal measures of associaton. J. Amer. Stat. Assoc. 53, 814--861.Google ScholarCross Ref
- Kurowicka, D. and Cooke, R. M. 2001. Conditional, partial and rank correlation for the elliptical copula; Dependence modelling in uncertainty analysis. Tech. Rep., Delft University of Technology, Mekelweg 4, 2628CD Delft, Netherlands.Google Scholar
- Law, A. M. and Kelton, W. D. 2000. Simulation Modeling and Analysis, 3rd ed. McGraw-Hill, New York. Google ScholarDigital Library
- Li, S. T. and Hammond, J. L. 1975. Generation of pseudorandom numbers with specified univariate distributions and correlation coefficients. IEEE Trans. Syst. Man, and Cybernet. 5, 557--561.Google ScholarCross Ref
- Lurie, P. M. and Goldberg, M. S. 1998. An approximate method for sampling correlated random variables from partially-specified distributions. Manage. Sci. 44, 203--218. Google ScholarDigital Library
- Mardia, K. V. 1970. A translation family of bivariate distributions and Fréchet's bounds. Sankhya A32, 119--122.Google Scholar
- Marsaglia, G. and Olkin, I. 1984. Generating correlation matrices. SIAM J. Sci. Stat. Comput. 5, 470--475.Google ScholarCross Ref
- Ouellette, D. V. 1981. Schur complements and statistics. Lin. Alg. Appl. 36, 187--295.Google ScholarCross Ref
- Schmeiser, B. and Lal, R. 1982. Bivariate gamma random vectors. Oper. Res. 30, 355--374.Google ScholarDigital Library
- Whitt, W. 1976. Bivariate distributions with given marginals. Ann. Stat. 4, 1280--1289.Google ScholarCross Ref
- Wolkowicz, H., Saigal, R., and Vandenberghe, L., Eds. 2000. Handbook of Semidefinite Programming: Theory, Algorithms and Applications. Kluwer, Boston, Mass.Google Scholar
Index Terms
- Behavior of the NORTA method for correlated random vector generation as the dimension increases
Recommendations
Corrigendum: Behavior of the NORTA method for correlated random vector generation as the dimension increases
This note corrects an error in Ghosh and Henderson [2003].
Corrigendum: Behavior of the NORTA method for correlated random vector generation as the dimension increases
This note corrects an error in Ghosh and Henderson [2003].
Generating random correlation matrices based on vines and extended onion method
We extend and improve two existing methods of generating random correlation matrices, the onion method of Ghosh and Henderson [S. Ghosh, S.G. Henderson, Behavior of the norta method for correlated random vector generation as the dimension increases, ACM ...
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