Abstract
We present a model for representing stationary multivariate time-series input processes with marginal distributions from the Johnson translation system and an autocorrelation structure specified through some finite lag. We then describe how to generate data accurately to drive computer simulations. The central idea is to transform a Gaussian vector autoregressive process into the desired multivariate time-series input process that we presume as having a VARTA (Vector-Autoregressive-To-Anything) distribution. We manipulate the autocorrelation structure of the Gaussian vector autoregressive process so that we achieve the desired autocorrelation structure for the simulation input process. We call this the correlation-matching problem and solve it by an algorithm that incorporates a numerical-search procedure and a numerical-integration technique. An illustrative example is included.
- Berntsen, J. and Espelid, T. O. 1991. Error estimation in automatic quadrature routines. ACM Trans. Math. Soft. 17, 233--252.]] Google ScholarDigital Library
- Biller, B. and Nelson, B. L. 2002. Parameter estimation for ARTA processes. In Proceedings of the 2002 Winter Simulation Conference, E. Yucesan, C. H. Chen, J. L. Snowdon, and J. M. Charnes, Eds. Institute of Electrical and Electronics Engineers, Piscataway, N.J., 255--262.]] Google ScholarDigital Library
- Biller, B. and Nelson, B. L. 2003a. Fitting time-series input processes for simulation. Tech. rep., Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh, PA.]]Google Scholar
- Biller, B. and Nelson, B. L. 2003b. On the performance of the ARTA fitting algorithm for stochastic simulation. Tech. Rep., Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh, Pa.]]Google Scholar
- Biller, B. and Nelson, B. L. 2003c. Online companion to "modeling and generating multivariate time-series input processes using a vector autoregressive technique. Tech. Rep., Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Ill.]]Google Scholar
- Blum, E. K. 1972. Numerical Analysis and Computation Theory and Practice. Addison-Wesley, Reading, Mass.]]Google Scholar
- Cambanis, S. and Masry, E. 1978. On the reconstruction of the covariance of stationary Gaussian processes observed through zero-memory nonlinearities. IEEE Trans. Inf. Theory 24, 485--494.]]Google ScholarDigital Library
- Cario, M. C. and Nelson, B. L. 1996. Autoregressive to anything: Time-series input processes for simulation. Oper. Res. Lett. 19, 51--58.]]Google ScholarDigital Library
- Cario, M. C. and Nelson, B. L. 1998. Numerical methods for fitting and simulating autoregressive-to-anything processes. INFORMS J. Comput. 10, 72--81.]]Google ScholarDigital Library
- Cario, M. C., Nelson, B. L., Roberts, S. D., and Wilson, J. R. 2001. 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
- Chen, H. 2001. Initialization for NORTA: Generation of random vectors with specified marginals and correlations. INFORMS J. Comput. 13, 312--331.]]Google ScholarCross Ref
- Clemen, R. T. and Reilly, T. 1999. Correlations and copulas for decision and risk analysis. Manage. Sci. 45, 208--224.]]Google Scholar
- Cook, R. D. and Johnson, M. E. 1981. A family of distributions for modeling non-elliptically symmetric multivariate data. J. Roy. Stat. Soc. B 43, 210--218.]]Google Scholar
- Cools, R. 1994. The subdivision strategy and reliability in adaptive integration revisited. Tech. Rep., Department of Computer Science, Katholieke University Leuven, Leuven, Belgium.]]Google Scholar
- Cools, R., Laurie, D., and Pluym, L. 1997. Algorithm 764: Cubpack++: A C++ package for automatic two-dimensional cubature. ACM Trans. Math. Softw. 23, 1--15.]] Google ScholarDigital Library
- Devroye, L. 1986. Non-Uniform Random Variate Generation. Springer-Verlag, New York.]]Google Scholar
- Genz, A. 1992. Statistics applications of subregion adaptive multiple numerical integration. In Numerical Integration---Recent Developments, Software, and Applications. 267--280.]]Google Scholar
- Ghosh, S. and Henderson, S. G. 2002. Chessboard distributions and random vectors with specified marginals and covariance matrix. Oper. Res. 50, 820--834.]]Google ScholarDigital Library
- Gleser, L. J. and Moore, D. S. 1983. The effect of dependence on chi-squared test and empiric distribution tests of fit. Ann. Stat. 11, 1100--1108.]]Google ScholarCross Ref
- Gross, D. and Juttijudata, M. 1997. Sensitivity of output performance measures to input distributions in queueing simulation modeling. In Proceedings of the 1997 Winter Simulation Conference, D. H. Withers, B. L. Nelson, S. Andradóttir, and K. J. Healy, Eds. Institute of Electrical and Electronics Engineers, Piscataway, N.J., 296--302.]] Google ScholarDigital Library
- Hill, I. D., Hill, R., and Holder, R. L. 1976. Fitting Johnson curves by moments. Appl. Stat. 25, 180--189.]]Google ScholarCross Ref
- Hill, R. R. and Reilly, C. H. 1994. Composition for multivariate random vectors. In Proceedings of the 1994 Winter Simulation Conference, D. A. Sadowski, A. F. Seila, J. D. Tew, and S. Manivannan, Eds. Institute of Electrical and Electronics Engineers, Piscataway, N.J., 332--339.]] Google ScholarDigital Library
- Johnson, M. E. 1987. Multivariate Statistical Simulation. Wiley, New York.]] Google ScholarDigital Library
- Johnson, N. L. 1949a. Systems of frequency curves generated by methods of translation. Biometrika 36, 149--176.]]Google ScholarCross Ref
- Johnson, N. L. 1949b. Bivariate distributions based on simple translation systems. Biometrika 36, 297--304.]]Google ScholarCross Ref
- Krommer, A. R. and Ueberhuber, C. W. 1994. Numerical Integration on Advanced Computer Systems. Springer-Verlag, New York.]]Google Scholar
- Kruskal, W. 1958. Ordinal measures of association. J. Ameri. Stat, Assoc. 53, 814--861.]]Google ScholarCross Ref
- Laurie, D. P. 1994. Null rules and orthogonal expansions. In Proceedings of the International Conference on Special Functions, Approximation, Numerical Quadrature and Orthogonal Polynomials, R. V. Zahar, Ed., Birkhävser, 359--370.]] Google ScholarDigital Library
- Law, A. M. and Kelton, W. D. 2000. Simulation Modeling and Analysis. McGraw-Hill, New York.]] Google ScholarDigital Library
- Lewis, P. A. W., McKenzie, E., and Hugus, D. K. 1989. Gamma processes. Commun. Stat. Stoch. Models 5, 1--30.]]Google ScholarCross Ref
- 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
- Livny, M., Melamed, B., and Tsiolis, A. K. 1993. The impact of autocorrelation on queueing systems. Manage. Sci. 39, 322--339.]]Google ScholarDigital Library
- 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
- Lutkepohl, H. 1993. Introduction to Multiple Time Series Analysis. Springer-Verlag, New York.]]Google Scholar
- Mardia, K. V. 1970. A translation family of bivariate distributions and Frèchet's bounds. Sankhya A 32, 119--122.]]Google Scholar
- Melamed, B. 1991. TES: A class of methods for generating autocorrelated uniform variates. ORSA J. Comput. 3, 317--329.]]Google ScholarCross Ref
- Melamed, B., Hill, J. R., and Goldsman, D. 1992. The TES methodology: Modeling empirical stationary time series. In Proceedings of the 1992 Winter Simulation Conference, R. C. Crain, J. R. Wilson, J. J. Swain, and D. Goldsman, Eds. Institute of Electrical and Electronics Engineers, Piscataway, N.J., 135--144.]] Google ScholarDigital Library
- Moore, D. S. 1982. The effect of dependence on chi-squared tests of fit. Ann. Stat. 4, 357--369.]]Google Scholar
- Nelsen, R. B. 1998. An Introduction to Copulas. Springer-Verlag, New York.]] Google ScholarDigital Library
- Nelson, B. L. and Yamnitsky, M. 1998. Input modeling tools for complex problems. In Proceedings of the 1998 Winter Simulation Conference, J. S. Carson, M. S. Manivannan, D. J. Medeiros, and E. F. Watson, Eds. Institute of Electrical and Electronics Engineers, Piscataway, N.J., 105--112.]] Google ScholarDigital Library
- Pritsker, A. A. B., Martin, D. L., Reust, J. S., Wagner, M. A. F., Wilson, J. R., Kuhl, M. E., Roberts, J. P., Daily, O. P., Harper, A. M., Edwards, E. B., Bennett, L., Burdick, J. F., and Allen, M. D. 1995. Organ transplantation policy evaluation. In Proceedings of the 1995 Winter Simulation Conference, W. R. Lilegdon, D. Goldsman, C. Alexopoulos, and K. Kang, Eds. Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1314--1323.]] Google ScholarDigital Library
- Rabinowitz, P. and Richter, N. 1969. Perfectly symmetric two-dimensional integration formulas with minimal number of points. Math. Comput. 23, 765--799.]]Google ScholarCross Ref
- Robinson, I. and Doncker, E. D. 1981. Algorithm 45: Automatic computation of improper integrals over a bounded or unbounded planar region. Computing 27, 253--284.]]Google ScholarCross Ref
- Song, W. T., Hsiao, L., and Chen, Y. 1996. Generating pseudorandom time series with specified marginal distributions. European Journal of Operational Research 93, 1--12.]]Google Scholar
- Stephens, M. A. 1974. EDF statistics for goodness of fit and some comparisons. J. Amer. Stat. Assoc. 69, 730--737.]]Google ScholarCross Ref
- Stroud, A. H. 1971. Approximate Calculation of Multiple Integrals. Prentice-Hall, Englewood Cliffs. N.J.]]Google Scholar
- Swain, J. J., Venkatraman, S., and Wilson, J. R. 1988. Least-squares estimation of distribution functions in Johnson's translation system. J. Stat. Comput. Simul. 29, 271--297.]]Google ScholarCross Ref
- Tchen, A. H. 1980. Inequalities for distributions with given marginals. Ann. Probab. 8, 814--827.]]Google ScholarCross Ref
- Tong, Y. L. 1990. The Multivariate Normal Distribution. Springer-Verlag, New York.]]Google Scholar
- Ware, P. P., Page, T. W., and Nelson, B. L. 1998. Automatic modeling of file system workloads using two-level arrival processes. ACM Trans. Model. Comput. Simul. 8, 305--330.]] Google ScholarDigital Library
- Whitt, W. 1976. Bivariate distributions with given marginals. Ann. Stat. 4, 1280--1289.]]Google ScholarCross Ref
- Willemain, T. R. and Desautels, P. A. 1993. A method to generate autocorrelated uniform random numbers. J. Stat. Comput. Simul. 45, 23--31.]]Google ScholarCross Ref
Index Terms
- Modeling and generating multivariate time-series input processes using a vector autoregressive technique
Recommendations
Spatio-temporal heterogeneous graph using multivariate earth observation time series: Application for drought forecasting
AbstractAccurate forecasting is required for the effective risk management of drought disasters. Many machine learning- and deep learning-based models have been proposed for drought forecasting, however, they cannot handle the temporal and/or ...
Highlights- A novel HetSPGraph approach with three layers is proposed for drought forecasting.
A comparison of multivariate and univariate time series approaches to modelling and forecasting emergency department demand in Western Australia
The model identification process for VARMA, ARMA and Winters method.Display Omitted VARMA, ARMA and Winters methods are used extensively for planning and management.Multivariate VARMA model is a reliable tool for predicting ED demand by category.It ...
Time series forecasting by a seasonal support vector regression model
The support vector regression (SVR) model is a novel forecasting approach and has been successfully used to solve time series problems. However, the applications of SVR models in a seasonal time series forecasting has not been widely investigated. This ...
Comments