Modeling and Generating Multivariate Time Series with Arbitrary Marginals Using a Vector Autoregress

We present a model for representing stationary multivariate time series with arbitrary marginal distributions and autocorrelation structures and describe how to generate data quickly and 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 correlation structure of the Gaussian vector autoregressive process so that we achieve the desired correlation structure for the simulation input process. For the purpose of computational efficiency, we provide a numerical method, which incorporates a numerical-search procedure and a numerical-integration technique, for solving this correlation-matching problem.

[1]  A. Cohen.  An Introduction to Probability Theory and Mathematical Statistics , 1979 .

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

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

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

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

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

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

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

[9]  R. V. M. Zahar.  Approximation and Computation: A Festschrift in Honor of Walter Gautschi , 1994 .

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

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

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

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

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

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

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

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

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

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

[20]  Ian T. Jolliffe,et al.  Introduction to Multiple Time Series Analysis , 1993 .

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

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

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

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

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

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

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

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

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

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

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

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

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

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