Wind and Structures

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Translation method: a historical review and its application to simulation of non-Gaussian stationary processes

  • Choi, Hang (Institute of Environmental Studies, Graduate School of Frontier Sciences,The University of Tokyo) ;
  • Kanda, Jun (Institute of Environmental Studies, Graduate School of Frontier Sciences,The University of Tokyo)
  • Received : 2003.04.30
  • Accepted : 2003.09.03
  • Published : 2003.10.25
⟨ Previous Next ⟩

Abstract

A number of methods based on various ideas have been proposed for simulating the non-Gaussian stationary process. However, these methods have some limitations. This paper reviewed several simulation methods based on the translation method using logarithmic and polynomial functions, which have emerged in the history of statistics and in the field of civil engineering. The applicability of each method is discussed from the viewpoint of the reproducibility of higher order statistics of the object function in the simulated sample functions, and examined using pressure signals measured from wind tunnel experiments for various shapes of buildings. The parameter estimation methods, i.e. the method of moments and quantile plot, are also reviewed, and the useful aspects of each method are discussed. Additionally, a simple worksheet for parameter estimation is derived based on the method of moment for practical application, and the accuracy is discussed comparing with a set of previously proposed formulae.

Keywords

References

  1. Aitchison, A. and Brown, J.A.C. (1957), The Lognormal Distribution with Special Reference to Its Uses in Economics, Cambridge Univ. press.
  2. Ammon, D. (1990), "Approximation and generation of Gaussian and non-Gaussian stationary processes", Struct.Safety, 8, 153-160. https://doi.org/10.1016/0167-4730(90)90037-P
  3. Bowley, A.L. (1928), F. Y. Edgeworth's Contributions to Mathematical Statistics, Roy. Stat. Soc. London.
  4. Calderone, I. and Melbourne, W.H. (1993), "The behavior of glass under wind loading", J. Wind Eng. Ind.Aerodyn., 48, 81-94. https://doi.org/10.1016/0167-6105(93)90282-S
  5. Calderone, I., Cheung, J.C.K. and Melbourne, W.H. (1994), "The full-scale significance, on glass claddingpanels, of data obtained from wind tunnel measurements of pressure fluctuations on building cladding", J.Wind Eng. Ind. Aerodyn., 53, 247-259. https://doi.org/10.1016/0167-6105(94)90029-9
  6. Capinski, M. and Kopp, E. (1999), Measure, Integral and Probability, Springer-Verlag.
  7. Choi, H. and Kanda, J. (2001), "Simulations for non-Gaussian processes based on polynomial transformation",Proc. Wind Hazard Mitigation in Urban Areas, Atsugi, Japan, 118-137.
  8. Choi, H. and Kanda, J. (2002), "Simulation method of non-Gaussian stationary processes based on thetranslation method", Proc. 17th Natl. Symp. Wind Eng. JAWE, 125-130 (in Japanese).
  9. Choi, H. and Kanda, J. (2003), "Normality and convergency of simulated stationary random processes byspectral representation", J. Struct. Const. Eng., AIJ, 568, 51-58.
  10. Cornish, E.A. and Fisher, R.A. (1937), "Moments and cumulants in the specification of distributions", Rev. Intl.Stat. Inst., 5, 307-322.
  11. Crandall, S.H. (1980), "Non-Gaussian closure for random vibration of non-linear oscillators", Int. J. Non-LinearMech., 15, 303-313. https://doi.org/10.1016/0020-7462(80)90015-3
  12. Deodatis, G. and Micaletti, R.C. (2001), "Simulation of highly skewed non-Gaussian stochastic processes", J.Eng. Mech., ASCE, 127(12), 1284-1295. https://doi.org/10.1061/(ASCE)0733-9399(2001)127:12(1284)
  13. Doob, J.L. (1994), Measure Theory, Springer-Verlag.
  14. Draper, N.R. and Tierney, D.E. (1972), "Regions of positive and unimodal series expansion of the Edgeworthand Gram-Charlier approximations", Biometrika, 59(2), 463-465. https://doi.org/10.1093/biomet/59.2.463
  15. Edgeworth, F.Y. (1892), "The law of error and correlated averages", Philosophical Magazine, 5th Ser., 34, 429-438, 518-526. https://doi.org/10.1080/14786449208620355
  16. Edgeworth, F.Y. (1898), "On the representation of statistics by mathematical formulae (Part I)", J. Roy. Stat. Soc.,61(4), 670-700.
  17. Edgeworth, F.Y. (1914), "On the use of analytical geometry to represent certain kinds of statistics", J. Roy. Stat.Soc., 77(3), 300-312, 415-432, 653-671, 724-749, 838-852. https://doi.org/10.2307/2339725
  18. Edgeworth, F.Y. (1916), "On the mathematical representation of statistical data", J. Roy. Stat. Soc., 79(4), 455-500. https://doi.org/10.2307/2341002
  19. Edgeworth, F.Y. (1917), "On the mathematical representation of statistical data", J. Roy. Stat. Soc., 80(1), 65-83,266-288, 411-437. https://doi.org/10.2307/2340696
  20. Edgeworth, F.Y. (1924), "Untried methods of presenting frequency", J. Roy. Stat. Soc., 87(4), 571-594. https://doi.org/10.2307/2341462
  21. Edgeworth, F.Y. (1926), "Mr. Rhodes' curve and the method of adjustment", J. Roy. Stat. Soc., 89(1), 129-143. https://doi.org/10.2307/2341487
  22. Fisher, A. (1936), Mathematical Theory of Probabilities and Its Application to Frequency Curves and StatisticalMethods, 2nd Ed., Macmillan.
  23. Fisher, R.A. and Cornish, E.A. (1960), "The percentile points of distributions having known cumulants",Technometrics, 2, 209-225. https://doi.org/10.1080/00401706.1960.10489895
  24. Fleishman, A.I. (1978), "A method for simulating non-normal distributions", Psychometrika, 43(4), 521-532. https://doi.org/10.1007/BF02293811
  25. Gaddum, J.H. (1945), "The lognormal distribution", Nature, London, 156, 463-466. https://doi.org/10.1038/156463a0
  26. Galton, F. (1879), "The geometric mean, in vital and social statistics", Proc. Roy Soc.London, 29, 365-367. https://doi.org/10.1098/rspl.1879.0060
  27. Gaver, D.P. and Lewis, P.A.W. (1980), "First-order autoregressive Gamma sequences and point processes", Adv.Appl. Prob., 12, 727-745. https://doi.org/10.2307/1426429
  28. Gioffre, M., Gusella, V. and Grigoriu, M. (1999), "Analysis of non-Gaussian stochastic field of wind pressure",Stochastic Structural Dynamics, Balkema, 567-574.
  29. Gioffre, M., Gusella, V. and Grigoriu, M. (2000), "Simulation of non-Gaussian field applied to wind pressurefluctuations", Prob. Eng. Mech., 15, 339-345. https://doi.org/10.1016/S0266-8920(99)00035-1
  30. Gioffre, M., Guesella, V. and Grigoriu, M. (2001a), "Non-Gaussian wind pressure on prismatic buildings. I:Stochastic field", J. Struct. Eng., ASCE, 127(9), 981-989. https://doi.org/10.1061/(ASCE)0733-9445(2001)127:9(981)
  31. Gioffre, M., Guesella, V. and Grigoriu, M. (2001b), "Non-Gaussian wind pressure on prismatic buildings. II:Numerical simulation", J. Struct. Eng., 127(9), 990-995. https://doi.org/10.1061/(ASCE)0733-9445(2001)127:9(990)
  32. Grigoriu, M. (1984a), "Crossing of non-Gaussian translation processes", J. Eng. Mech., ASCE, 110(4), 610-620. https://doi.org/10.1061/(ASCE)0733-9399(1984)110:4(610)
  33. Grigoriu, M. (1984b), "Extremes of wave forces", J. Eng. Mech., ASCE, 110(12), 1731-1742. https://doi.org/10.1061/(ASCE)0733-9399(1984)110:12(1731)
  34. Grigoriu, M. (1993), "On the spectral representation method in simulation", Prob. Eng. Mech., 8, 75-90. https://doi.org/10.1016/0266-8920(93)90002-D
  35. Grigoriu, M. (1995), "Applied non-Gaussian processes; examples, theory, simulation, linear random vibration,and MATLAB solutions", PTR Prentice Hall.
  36. Grigoriu, M. (2002), "Stochastic calculus: Applications in science and engineering", Birkhauser.
  37. Gurley, K.R, Kareem, A. and Tognarelli, M.A. (1996), "Simulation of a class of non-normal random processes",Int. J. Non-Linear Mech., 31(5), 601-617. https://doi.org/10.1016/0020-7462(96)00025-X
  38. Gurley, K.R., Tognarelli, M.A. and Kareem, A. (1997a), "Analysis and simulation tools for wind engineering",Prob. Eng. Mech., 12(1), 9-31. https://doi.org/10.1016/S0266-8920(96)00010-0
  39. Gurley, K.R. and Kareem, A. (1997b), "Analysis interpretation modeling and simulation of unsteady wind andpressure data", J. Wind Eng. Ind. Aerodyn., 69-71, 657-669. https://doi.org/10.1016/S0167-6105(97)00195-5
  40. Gurley, K.R. and Kareem, A. (1998), "A conditional simulation of non-normal velocity/pressure field", J. Wind Eng. Ind. Aerodyn., 77&78, 39-51.
  41. Hald, A. (1952), Statistical Theory with Engineering Applications, John Wiley & Sons.
  42. Hald, A. (1998), A History of Mathematical Statistics from 1750 to 1930, Wiley-Interscience.
  43. Halmos, P.R. and von Neumann, J. (1942), "Operation methods in classical mechanics", Annal. Math. (2nd Ser.),43(2), 332-350.
  44. Hill, I.D., Hill, R. and Holder, R.L. (1976), "Algorithm AS99: Fitting Johnson curves by moments", Appl. Stat.,25(2), 180-189. https://doi.org/10.2307/2346692
  45. Johnson, N.L. (1949), "System of frequency curves generated by methods of translation", Biometrika, 36(1/2),149-176. https://doi.org/10.1093/biomet/36.1-2.149
  46. Johnson, N.L., Nixon, E., Amos, D.E. and Pearson, E.S. (1963), "Table of percentage points of Pearson curves,for given $\sqrt{\beta_{1}}$ and $\sqrt{\beta_{2}}$, expressed in standard measure", Biometrika, 50(3/4), 459-498.
  47. Johnson, N.L. (1965), "Tables to facilitate fitting $S_{U}$ frequency curves", Biometrika, 52, 547-558.
  48. Johnson, N.L., Kotz, S. and Balakrishnan, N. (1994), Continuous Univariate Distributions, 1, 2nd Ed., Wiley-Interscience.
  49. Kaneko, S. and Matsumoto M. (1984), Special functions, Baihuukan Pub. (in Japanese).
  50. Kapteyn, J.C. (1903), "Skew frequency curves in biology and statistics", Groningen.
  51. Kendall, M. and Stuart, A. (1977), The Advanced Theory of Statistics (Vol. 1 Distribution theory), 4th Ed.,Charles Griffin & Co.Ltd..
  52. Kendall, M.G. (1968), "Studies in the history of probability and statistics -XIX Francis Ysidro Edgeworth (1845-1926)", Biometrika, 55(2), 269-275.
  53. Kim, H.Y. and Park, J.-K. (2001), personal communication.
  54. Koning, A.G. and Does, R.J.M.M. (1988), "Algorithm AS234: Approximating the percentage points of simplelinear rank statistics with Cornish-Fisher expansions", Appl. Statist., 37, 278-284. https://doi.org/10.2307/2347354
  55. Kotz, S., Balakrishnan, N. and Johnson, N.L. (2000), Continuous Multivariate Distributions: Vol. 1: Models andapplications, Wiley-Interscience.
  56. Kumar, K.S. and Stathopoulos, T. (1997), "Computational simulation of fluctuating wind pressures on lowbuilding roofs", J. Wind Eng. Ind. Aerodyn., 69-71, 485-495. https://doi.org/10.1016/S0167-6105(97)00179-7
  57. Kumar, K.S. and Stathopoulos, T. (1998), "Fatigue analysis of roof cladding under simulated wind loading", J.Wind Eng. Ind. Aerodyn., 77&78, 171-183.
  58. Kumar, K.S. and Stathopoulos, T. (1999), "Synthesis of non-Gaussian wind pressure time series on low buildingroofs", Eng. Struct., 21, 1086-1100. https://doi.org/10.1016/S0141-0296(98)00069-8
  59. Kumar, K.S. and Stathopoulos, T. (2000), "Discussion of Digital generation of surface fluctuations with spikyfeatures", J. Wind Eng. Ind. Aerodyn., 84, 257-260. https://doi.org/10.1016/S0167-6105(99)00120-8
  60. Kuznetsov, P.I., Stratonovich, R.L. and Tikhonov, V.I. (1965), "The transmission of random functions throughnon-linear systems", Proc. Symposium on Nonlinear transformations of stochastic processes (Ed.Kuznetsov et al., Translated/Edited by J.Wise and D.C. Cooper), Pergamon press, 29-58.
  61. LeCam, L. (1986), "The central limit theorem around 1935", Stat. Sci., 1(1), 78-91. https://doi.org/10.1214/ss/1177013818
  62. Lee, Y-S. and Lin, T-K. (1992), "Algorithm AS269: High order Cornish-Fisher expansion", Appl. Statist., 41(1),233-240. https://doi.org/10.2307/2347649
  63. Leslie, D.C.M. (1959), "Determination of parameters in the Johnson system of probability distributions",Biometrika, 46(1/2), 229-231. https://doi.org/10.1093/biomet/46.1-2.229
  64. Li, Q.S., Calderone, I. and Melbourne, W.H. (1999), "Probabilistic characteristics of pressure fluctuations inseparated and reattaching flows over various free-stream turbulence", J. Wind Eng. Ind. Aerodyn., 82, 125-145. https://doi.org/10.1016/S0167-6105(98)00214-1
  65. Maharam, D. (1966), "On the Radon-Nikodym derivatives of measurable transformations", Trans. Am. Math.Soc., 122(1), 229-248.
  66. Maharam, D. (1969), "Invariant measures and Radon-Nikodym derivatives", Trans. Am. Math. Soc., 135, 223-248. https://doi.org/10.1090/S0002-9947-1969-0232914-7
  67. Matui, G., Suda, K. and Higuchi, K. (1982), "Full-scale measurement of wind pressures acting on high risebuilding of rectangular plan", J. Wind Eng. Ind. Aerodyn., 10, 267-286. https://doi.org/10.1016/0167-6105(82)90002-2
  68. McAlister, D. (1879), "The law of the geometric mean", Proc. Roy. Soc. London, 29, 367-376. https://doi.org/10.1098/rspl.1879.0061
  69. Okada, H. and Ha, Y.C. (1992), "Comparison of wind tunnel and full scale pressure measurement tests on theTexas Tech Building", J. Wind Eng. Ind. Aerodyn., 41-44, 1601-1612.
  70. Pearson, K. (1894), "Contributions to the mathematical theory of evolution", Phil. Trans. Roy. Soc. London, Ser.A, 185(1), 71-110. https://doi.org/10.1098/rsta.1894.0003
  71. Pearson, K. (1895), "Contributions to the mathematical theory of evolution -II. Skew variation in homogeneous material", Phil. Trans. Roy. Soc. London, Ser. A, 186(1), 343-414. https://doi.org/10.1098/rsta.1895.0010
  72. Pearson, K. (1901), "Mathematical contributions to the theory of evolution -X. Supplement to a Memoir on skewvariation", Phil. Trans. Roy. Soc. London, Ser. A, 192, 443-459.
  73. Pearson, K. (1904), "Mathematical contributions to the theory of evolution -XIV. On the general theory of skewcorrelation and non-linear regression", Drapers's company Research Memoirs, Biometric series, Dulau and Co.
  74. Pearson, K. (1905), "Das Fehlergesetz und Seine Verallgemeinerungen Durch Fechner und Pearson. ARejoinder", Biometrika, 4(1/2), 169-212
  75. Pearson, K. (1906), "A rejoinder to Professor Kapteyn", Biometrika, 5(1/2), 168-171.
  76. Pearson, E.S. (1967), "Studies in the history of probability and statistics, XVII: Some reflexions on continuity inthe development of mathematical statistics, 1885-1920", Biometrika, 54(3/4), 341-355.
  77. Peterka, J.A. (1983), "Selection of local peak pressure coefficients for wind tunnel studies of buildings", J. WindEng. Ind. Aerodyn., 13, 477-488. https://doi.org/10.1016/0167-6105(83)90166-6
  78. Popescu, R., Deodatis, G. and Prevost, J.H. (1998), "Simulation of homogeneous nonGaussian stochastic vectorfields", Prob. Eng. Mech., 13(1), 1-13. https://doi.org/10.1016/S0266-8920(97)00001-5
  79. Puig, H., Poirion, F. and Soize, C. (2002), "Non-Gaussian simulation using Hermite polynomial expansion:covergences and algorithms", Prob. Eng. Mech., 17, 253-264. https://doi.org/10.1016/S0266-8920(02)00010-3
  80. Rubinstein, R.Y. (1981), Simulation and the Monte Carlo Method, Wiley series in Probability and MathematicalStatistics, John Wiley & Sons.
  81. Sakamoto, S. and Ghanem, R. (2002), "Polynomial chaos decomposition for the simulation of non-Gaussiannonstationary stochastic processes", J. Eng. Mech., ASCE, 128(2), 190-201. https://doi.org/10.1061/(ASCE)0733-9399(2002)128:2(190)
  82. Seong, S.H. and Peterka, J.A. (1993), "Computer simulation of non-Gaussian wind pressure fluctuation", Proc.7th U.S. Natl. Conf. Wind Eng., Los Angeles, 623-632.
  83. Seong, S.H. and Peterka, J.A. (1997), "Computer simulation of non-Gaussian multiple wind pressure timeseries", J. Wind Eng. Ind. Aerodyn., 72, 95-105. https://doi.org/10.1016/S0167-6105(97)00243-2
  84. Seong, S.H. and Peterka, J.A. (1998) "Digital generation of surface pressure fluctuations with spiky features", J.Wind Eng. Ind. Aerodyn., 73, 181-192. https://doi.org/10.1016/S0167-6105(97)00283-3
  85. Seong, S.H. and Peterka, J.A. (2001), "Experiments of Fourier phases for synthesis of non-Gaussian spikes inturbulence time series", J. Wind Eng. Ind. Aerodyn., 89, 421-443. https://doi.org/10.1016/S0167-6105(00)00073-8
  86. Shinozuka, M. and Deodatis, G. (1991), "Simulation of stochastic processes by spectral representation", Appl.Mech. Rev., 44(4), 191-204. https://doi.org/10.1115/1.3119501
  87. Stigler, S.M. (1978), "Francis Ysidro Edgeworth, Statistician", J. Roy. Stat. Soc., Ser. A, 141(3), 287-322. https://doi.org/10.2307/2344804
  88. Stigler, S.M. (1986), The history of statistics: The measurement of Uncertainty before 1900, The Belknap pressof Harvard Univ. press.
  89. Stigler, S.M. (1999), Statistics on the table: The history of statistical concepts and methods, Harvard Univ. press.
  90. Tamura, Y. (2001), personal communication.
  91. Yajima, K. (2002), Lebesgue Integral and Function Analysis, Asakura-shoten Pub. (in Japanese).
  92. Tognarelli, M.A., Zhao, J. and Kareem, A. (1997a), "Equivalent statistical cubicization for systems and forcingnonlinearities", J. Eng. Mech., ASCE, 123(8), 890-893. https://doi.org/10.1061/(ASCE)0733-9399(1997)123:8(890)
  93. Tognarelli, M.A. Zhao, J., Rao, K.B. and Kareem, A. (1997b), "Equivalent statistical quadratization andcubicization for nonlinear systems", J. Eng. Mech., ASCE, 123(5), 512-532. https://doi.org/10.1061/(ASCE)0733-9399(1997)123:5(512)
  94. Wilk, M.B. and Gnanadesikan, R. (1968), "Probability plotting methods for analysis of data", Biometrika, 55(1), 1-17.
  95. Winterstein, S.R. (1988), "Nonlinear vibration models for extremes and fatigue", J. Eng. Mech., ASCE, 114(10),1772-1790.
  96. Wheeler, R.E. (1980), "Quantile estimators of Johnson curve parameters", Biometrika, 67(3), 725-728. https://doi.org/10.1093/biomet/67.3.725
  97. Xu, Y.L. (1995), "Model- and full-scale comparison of fatigue-related characteristics of wind pressures on theTexas Tech Building", J. Wind Eng. Ind. Aerodyn., 58, 147-173. https://doi.org/10.1016/0167-6105(95)00012-7
  98. Yamazaki, F. and Shinozuka, M. (1988), "Digital generation of non-Gaussian stochastic fields", J. Eng. Mech.,ASCE, 114(7), 1183-1197. https://doi.org/10.1061/(ASCE)0733-9399(1988)114:7(1183)
  99. Yang, J.N. (1973), "On the normality and accuracy of simulated random processes", J. Sound Vib., 26(3), 417-428. https://doi.org/10.1016/S0022-460X(73)80196-8
  100. Yim, J.Z., Lin, J-G. and Huang, W-P. (1999), "The statistical properties of wind field in Keelung, Taiwan", Proc.Wind Engineering into 21st Century, Larsen, Belkema, 385-390.
  101. Zar, J.H. (1978), "Approximations for the percentage points of the chi-squared distribution", Appl. Stat., 27(3),280-290. https://doi.org/10.2307/2347163

Cited by

  1. Simulation of non-Gaussian stochastic processes by amplitude modulation and phase reconstruction vol.18, pp.6, 2014, https://doi.org/10.12989/was.2014.18.6.693
  2. SIMULATION OF EXTERNAL PRESSURE FIELDS AND INTERNAL PRESSURES IN A LOW-RISE BUILDING MODEL IN TURBULENCE BOUNDARY LAYER vol.73, pp.633, 2003, https://doi.org/10.3130/aijs.73.1927