Modeling and generating multivariate time-series input processes using a vector autoregressive technique

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.

[1]  Averill M. Law,et al.  Simulation Modeling and Analysis , 1982 .

[2]  A. Genz.  Statistics Applications of Subregion Adaptive Multiple Numerical Integration , 1992 .

[3]  Ronald Cools,et al.  Algorithm 764: Cubpack++: a C++ package for automatic two-dimensional cubature , 1994, TOMS.

[4]  Robert J. Beaver,et al.  An Introduction to Probability Theory and Mathematical Statistics , 1977 .

[5]  Terje O. Espelid,et al.  Error estimation in automatic quadrature routines , 1991, TOMS.

[6]  Charles H. Reilly,et al.  Composition for multivariate random vectors , 1994 .

[7]  M. E. Johnson,et al.  A Family of Distributions for Modelling Non‐Elliptically Symmetric Multivariate Data , 1981 .

[8]  Benjamin Melamed,et al.  TES: A Class of Methods for Generating Autocorrelated Uniform Variates , 1991, INFORMS J. Comput..

[9]  Barry L. Nelson,et al.  Advanced input modeling: parameter estimation for ARTA processes , 2002, WSC '02.

[10]  I. D. Hill,et al.  Fitting Johnson Curves by Moments , 1976 .

[11]  N L JOHNSON,et al.  Bivariate distributions based on simple translation systems. , 1949, Biometrika.

[12]  C. R. Rao,et al.  Linear Statistical Inference and its Applications , 1968 .

[13]  Calyampudi R. Rao,et al.  Linear Statistical Inference and Its Applications. , 1975 .

[14]  P. A. P. Moran,et al.  An introduction to probability theory , 1968 .

[15]  David E. Booth,et al.  Multivariate statistical inference and applications , 1997 .

[16]  Barry L. Nelson,et al.  Automatic modeling of file system workloads using two-level arrival processes , 1998, TOMC.

[17]  Donald Gross,et al.  Sensitivity of output performance measures to input distributions in queueing simulation modeling , 1997, WSC '97.

[18]  S. Geer,et al.  Least Squares Estimation , 2005 .

[19]  Huifen Chen,et al.  Initialization for NORTA: Generation of Random Vectors with Specified Marginals and Correlations , 2001, INFORMS J. Comput..

[20]  N. L. Johnson,et al.  Systems of frequency curves generated by methods of translation. , 1949, Biometrika.

[21]  Barry L. Nelson,et al.  Numerical Methods for Fitting and Simulating Autoregressive-to-Anything Processes , 1998, INFORMS J. Comput..

[22]  Dirk P. Laurie,et al.  Null rules and orthogonal expansions , 1994 .

[23]  Barry L. Nelson,et al.  Autoregressive to anything: Time-series input processes for simulation , 1996, Oper. Res. Lett..

[24]  David Goldsman,et al.  The TES methodology: modeling empirical stationary time series , 1992, WSC '92.

[25]  Charles H. Reilly,et al.  Composition for multivariate random variables , 1994, Proceedings of Winter Simulation Conference.

[26]  P. Billingsley,et al.  Probability and Measure , 1980 .

[27]  L. Devroye.  Non-Uniform Random Variate Generation , 1986 .

[28]  E. Masry,et al.  On the reconstruction of the covariance of stationary Gaussian processes observed through zero-memory nonlinearities , 1978, IEEE Trans. Inf. Theory.

[29]  Barry L. Nelson,et al.  Fitting Time-Series Input Processes for Simulation , 2005, Oper. Res..

[30]  David S. Moore,et al.  The Effect of Dependence on Chi Squared Tests of Fit , 1982 .

[31]  G. C. Tiao,et al.  An introduction to multiple time series analysis. , 1993, Medical care.

[32]  Song Wheyming Tina,et al.  Generating pseudo-random time series with specified marginal distributions , 1996 .

[33]  W. Kruskal.  Ordinal Measures of Association , 1958 .

[34]  Barry L. Nelson,et al.  Parameter estimation for ARTA processes , 2002, Proceedings of the Winter Simulation Conference.

[35]  R. Nelsen.  An Introduction to Copulas , 1998 .

[36]  Bill Ravens,et al.  An Introduction to Copulas , 2000, Technometrics.

[37]  Patrick Billingsley,et al.  Probability and Measure. , 1986 .

[38]  A. Tchen.  Inequalities for distributions with given marginals , 1976 .

[39]  W. Whitt.  Bivariate Distributions with Given Marginals , 1976 .

[40]  Janet S. Reust,et al.  Organ transplantation policy evaluation , 1995, Winter Simulation Conference Proceedings, 1995..

[41]  Miron Livny,et al.  The Impact of Autocorrelation on Queuing Systems , 1993 .

[42]  David S. Moore,et al.  The Effect of Dependence on Chi-Squared and Empiric Distribution Tests of Fit , 1983 .

[43]  Shane G. Henderson,et al.  Chessboard Distributions and Random Vectors with Specified Marginals and Covariance Matrix , 2002, Oper. Res..

[44]  A. Stroud.  Approximate calculation of multiple integrals , 1973 .

[45]  E. Lehmann.  Elements of large-sample theory , 1998 .

[46]  J. J. Swain,et al.  Least-squares estimation of distribution functions in johnson's translation system , 1988 .

[47]  E. Blum.  Numerical analysis and computation theory and practice , 1972 .

[48]  Philip M. Lurie,et al.  An Approximate Method for Sampling Correlated Random Variables From Partially-Specified Distributions , 1998 .

[49]  Helmut Lütkepohl,et al.  Introduction to multiple time series analysis , 1991 .

[50]  BillerBahar,et al.  Modeling and generating multivariate time-series input processes using a vector autoregressive technique , 2003 .

[51]  R. Clemen,et al.  Correlations and Copulas for Decision and Risk Analysis , 1999 .

[52]  M. Stephens.  EDF Statistics for Goodness of Fit and Some Comparisons , 1974 .

[53]  R.Thomas Willemain,et al.  A method to generate autocorrelated uniform random numbers , 1993 .

[54]  Elise de Doncker,et al.  Algorithm 45. Automatic computation of improper integrals over a bounded or unbounded planar region , 2005, Computing.

[55]  Barry L. Nelson,et al.  Input modeling tools for complex problems , 1998, 1998 Winter Simulation Conference. Proceedings (Cat. No.98CH36274).

[56]  Joseph L. Hammond,et al.  Generation of Pseudorandom Numbers with Specified Univariate Distributions and Correlation Coefficients , 1975, IEEE Transactions on Systems, Man, and Cybernetics.

[57]  Christoph W. Ueberhuber,et al.  Numerical Integration on Advanced Computer Systems , 1994, Lecture Notes in Computer Science.

[58]  Philip Rabinowitz,et al.  Perfectly symmetric two-dimensional integration formulas with minimal numbers of points , 1969 .