Simulation of Non-Gaussian Stochastic Process with Target Power Spectral Density and Lower-Order Moments
Publication: Journal of Engineering Mechanics
Volume 138, Issue 5
Abstract
In this paper, a direct simulation algorithm is presented for the generation of a class of non-Gaussian stochastic processes according to target lower-order moments and prescribed power spectral density (PSD) function. The proposed algorithm is to expand the autoregressive (AR) model and the autoregressive moving average (ARMA) model, which are available to generate Gaussian random process, to simulate directly non-Gaussian stochastic process. The coefficients of the AR or ARMA model are determined based on the prescribed PSD function. It is well known that outputting stochastic process is also non-Gaussian if inputting white noise is non-Gaussian. But the skewness and kurtosis of the outputting non-Gaussian random process are not identical to these of inputting non-Gaussian white noise. In this paper, the relationships of lower-order moments such as skewness and kurtosis between output and input are analyzed and close to linear transformations. To corroborate the feasibility and correctness of the present methodology, numerical examples involving simulation of fluctuating wind pressures are taken into consideration. Numerical results indicate that the skewness and kurtosis of generated wind pressures based on the AR or ARMA model closely match their targets. In addition, the PSD and correlation functions of simulated samples also show considerably good agreement with prescribed functions. Therefore, the proposed algorithm is effective to simulate directly the class of non-Gaussian stochastic process.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The authors acknowledge the financial contributions received from the National Science Foundation of China with Grant Nos. 50578092 and 11162005. The authors thank the reviewers for their careful, unbiased, and constructive suggestions that significantly improved the quality of this paper.
References
Bocchini, P., and Deodatis, G. (2008). “Critical review and latest developments of a class of simulation algorithms for strongly non-Gaussian random fields.” Prob. Eng. Mech.PEMEEX, 23(4), 393–407.
Brockwell, P. J., and Davis, R. A. (1987). Time series: Theory and methods, Springer-Verlag, New York.
Cai, G. Q., and Lin, Y. K. (1996). “Generation of non-Gaussian stationary stochastic processes.” Phys. Rev. E.PHRVAO, 54(1), 299–303.
Chen, L., and Letchford, C. W. (2005). “Simulation of multivariate stationary Gaussian stochastic processes: Hybrid spectral representation and proper orthogonal decomposition approach.” J. Eng. Mech.JENMDT, 131(8), 801–808.
Chen, X. Z., and Huang, G. Q. (2009). “Evaluation of peak resultant response for wind-excited tall buildings.” Eng. Struct.ENSTDF, 31(4), 858–868.
Deodatis, G. (1996). “Simulation of ergodic multivariate stochastic processes.” J. Eng. Mech.JENMDT, 122(8), 778–787.
Deodatis, G., and Micaletti, R. C. (2001). “Simulation of highly skewed non-Gaussian stochastic processes.” J. Eng. Mech.JENMDT, 127(12), 1284–1295.
Deodatis, G., Popescu, R., and Prevost, J. H. (1995). “Simulation of stochastic processes and fields for Monte Carlo simulation applications: Some recent developments.” Proc., 1995 ASME Design Engineering Technical Conf., ASME, New York, 955–965.
Gersch, W., and Yonemoto, J. (1977). “Synthesis of multivariate random vibration systems: A two-stage least squares ARMA model approach.” J. Sound Vib.JSVIAG, 52(4), 553–565.
Grigoriu, M. (1993). “On the spectral representation method in simulation.” Prob. Eng. Mech.PEMEEX, 8(2), 75–90.
Grigoriu, M. (1995). Applied non-Gaussian processes: Examples, theory, simulation, linear random vibration, and MATLAB solutions, Prentice Hall, Inc., Englewood Cliffs, NJ.
Grigoriu, M. (1998). “Simulation of stationary non-Gaussian translation processes.” J. Eng. Mech.JENMDT, 124(2), 121–126.
Grigoriu, M. (2000). “Non-Gaussian models for stochastic mechanics.” Prob. Eng. Mech.PEMEEX, 15(1), 15–23.
Grigoriu, M. (2004). “Spectral representation for a class of non-Gaussian processes.” J. Eng. Mech.JENMDT, 130(5), 541–546.
Grigoriu, M., Ruiz, S. E., and Rosenblueth, E. (1988). “The Mexico earthquake of September 19, 1985-Nonstationary models of seismic ground acceleration.” Earthquake SpectraEASPEF, 4(3), 551–568.
Gurley, K. R. (1997). “Modeling and simulation of non-Gaussian processes.” Ph.D. thesis, Dept. of Civil Engineering and Geological Sciences, Univ. of Notre Dame, Notre Dame, IN.
Gurley, K. R., and Kareem, A. (1997). “Modeling of PDFs of non-Gaussian system response.” Proc., 7th Int. Conf. on Structure Safety and Reliability, Balkema, Leiden, Netherlands, 811–818.
Gurley, K. R., and Kareem, A. (1998a). “A conditional simulation of non-normal velocity/pressure fields.” J. Wind Eng. Ind. Aerodyn.JWEAD6, 77–78, 39–51.
Gurley, K. R., and Kareem, A. (1998b). “Simulation of non-Gaussian processes.” Proc., 3rd Int. Conf. on Computational Stochastic Mechanics, Balkema, Leiden, Netherlands, 11–20.
Gurley, K. R., and Kareem, A. (1998c). “Simulation of correlated non-Gaussian pressure fields.” MeccanicaMECCB9, 33, 309–317.
Gurley, K. R., Kareem, A., and Tognarelli, M. A. (1996). “Simulation of a class of non-normal random processes.” Int. J. Non-linear Mech.IJNMAG, 31(5), 601–617.
Gurley, K. R., Tognarelli, M. A., and Kareem, A. (1997). “Analysis and simulation tools for wind engineering.” Prob. Eng. Mech.PEMEEX, 12(1), 9–31.
Hill, I. D., Hill, R., and Holder, R. L. (1976). “Fitting Johnson curves by moments.” Appl. Statist.APSTAG, 25(2), 180–189.
Kareem, A., and Li, Y. (1992). “Digital simulation of wind load effects.” Proc., 6th ASCE Specialty Conf. on Probabilistic Mechanics, and Structural and Geotechnical Reliability, ASCE, Reston, VA, 284–287.
Kozin, F. (1988). “Autoregressive moving-average models of earthquake records.” Prob. Eng. Mech.PEMEEX, 3(2), 58–63.
Li, L. B., Phoon, K. K., and Quek, S. T. (2007). “Comparison between Karhunen-Loève expansion and translation-based simulation of non-Gaussian process.” Comput. Struct.CMSTCJ, 85(5-6), 264–276.
Li, Y., and Kareem, A. (1991). “Simulation of multivariate nonstationary random processes by FFT.” J. Eng. Mech.JENMDT, 117(5), 1037–1058.
Masters, F., and Gurley, K. R. (2003). “Non-Gaussian simulation: Cumulative distribution function map-based spectral correction.” J. Eng. Mech.JENMDT, 129(12), 1418–1428.
Mignolet, M. P., and Spanos, P. D. (1987). “Recursive simulation of stationary multivariate random processes-Part I.” J. Appl. Mech.JAMCAV, 54(3), 674–680.
Mignolet, M. P., and Spanos, P. D. (1990). “MA to ARMA modeling of wind.” J. Wind Eng. Ind. Aerodyn.JWEAD6, 36(1), 429–438.
Novak, D., Stoyanoff, S., and Herda, H. (1995). “Error assessment for wind histories generated by autoregressive method.” Struct. Saf., 17(2), 79–90.
Phoon, K. K., Huang, S. P., and Quek, S. T. (2002). “Simulation of second-order processes using Karhunen-Loève expansion.” Comput. Struct.CMSTCJ, 80(12), 1049–1060.
Piccolo, D. (1990). Introduzione all analisi delle serie storiche, La Nuova Italia Scientifica, Roma.
Popescu, R., Deodatis, G., and Nobahar, A. (2005). “Effects of random heterogeneity of soil properties on bearing capacity.” Prob. Eng. Mech.PEMEEX, 20(4), 324–341.
Popescu, R., Deodatis, G., and Prevost, J. H. (1998). “Simulation of homogenous non-Gaussian stochastic vector fields.” Prob. Eng. Mech.PEMEEX, 13(1), 1–13.
Puig, B., and Akian, J. L. (2004). “Non-Gaussian simulation using Hermite polynomials expansion and maximum entropy principle.” Prob. Eng. Mech.PEMEEX, 19(4), 293–305.
Rossi, R., Lazzari, M., and Vitaliani, R. (2004). “Wind field simulation for structural engineering purposes.” Int. J. Numer. Methods Eng.IJNMBH, 61(5), 738–763.
Sakamoto, S., and Ghanem, R. (2002). “Simulation of multidimensional non-Gaussian non-stationary random fields.” Prob. Eng. Mech.PEMEEX, 17(2), 167–176.
Shinozuka, M., and Jan, C. M. (1972). “Digital simulation of random processes and its application.” J. Sound Vib.JSVIAG, 25(1), 111–128.
Spanos, P. D., and Mignolet, M. P. (1987). “Recursive simulation of stationary multivariate random processes—Part II.” J. Appl. Mech.JAMCAV, 54(3), 681–687.
Stefanou, G. (2009). “The stochastic finite element method: Past, present, and future.” Comput. Methods Appl. Mech. Eng.CMMECC, 198, 1031–1051.
Yamazaki, F. (1988). “Digital generation of non-Gaussian stochastic fields.” J. Eng. Mech.JENMDT, 114(7), 1183–1197.
Yeh, C. H., and Wen, Y. K. (1990). “Modeling of nonstationary ground motion and analysis of inelastic structural response.” Struct. Saf., 8(1–4), 281–298.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 138 • Issue 5 • May 2012
Pages: 391 - 404
Copyright
© 2012. American Society of Civil Engineers.
History
Received: Jun 24, 2010
Accepted: Nov 2, 2011
Published online: Nov 4, 2011
Published in print: May 1, 2012
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.