非高斯随机过程的短期极值估计:复合Hermite模型
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摘要
Hermite模型自20世纪80年代后期开始被广泛应用于非高斯随机过程的短期极值估计。当随机过程的非高斯性很强时,尤其是偏度很大时,常用的3阶Hermite模型不足以表征出极值分布的尾端特征。工程中,样本统计矩的不确定性使得更高阶的Hermite模型不宜使用。基于此,该文提出了同时基于中心矩与线性矩的复合Hermite模型,有效地将Hermite模型由3阶拓展到4阶。该文以对数正态模型作为非线性系统的研究对象,对比分析了在解析条件下和在使用蒙特卡洛模拟获得样本数据条件下,各类Hermite模型与传统的Gumbel法以及平均条件穿越率(ACER)法用于极值分析的表现。结果表明,对于大偏度强非高斯随机过程的极值预测,复合Hermite模型具有更好的精确度和鲁棒性。
The Hermite model has been widely used in estimating the short term extrema of non-Gaussian processes since late 1980s.When the non-Gaussianity of a process is very strong,especially with a large skewness,the commonly used cubic Hermite model has its limited capacity to capture the characteristics of the tail distribution of the extreme value.However,higher-order models are not recommended for engineering use due to the uncertainty in moments.In this paper,a hybrid use of ordinary central moments(C-moments) and linear moments(L-moments) is proposed to construct Hermite models up to quartic order.A lognormal function is chosen as the original nonlinear system for validating the performance of hybrid Hermite models.Both analytical solutions and numerical solutions using Monte-Carlo simulations are investigated.The comparative study involves the conventional Gumbel method and the averaged conditional exceedance rate(ACER) method.The results show that the proposed hybrid Hermite models render better accuracy and higher robustness in estimating the extreme value.
The Hermite model has been widely used in estimating the short term extrema of non-Gaussian processes since late 1980s.When the non-Gaussianity of a process is very strong,especially with a large skewness,the commonly used cubic Hermite model has its limited capacity to capture the characteristics of the tail distribution of the extreme value.However,higher-order models are not recommended for engineering use due to the uncertainty in moments.In this paper,a hybrid use of ordinary central moments(C-moments) and linear moments(L-moments) is proposed to construct Hermite models up to quartic order.A lognormal function is chosen as the original nonlinear system for validating the performance of hybrid Hermite models.Both analytical solutions and numerical solutions using Monte-Carlo simulations are investigated.The comparative study involves the conventional Gumbel method and the averaged conditional exceedance rate(ACER) method.The results show that the proposed hybrid Hermite models render better accuracy and higher robustness in estimating the extreme value.
引文
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[2]Winterstein S R,Ude T C,Marthinsen T.Volterra models of ocean structures:extreme and fatigue reliability[J].Journal of Engineering Mechanics,1994,120(6):1369-1385.
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[14]Winterstein S R.Non-normal responses and fatigue damage[J].Journal of Engineering Mechanics,1985,111(10):1291-1295.
[15]Winterstein S R,Kashef T.Moment-based load and response models with wind engineering applications[J].Journal of Solar Energy Engineering,2000,122(3):122-128.
[16]Ding J,Chen X.Moment-Based Translation model for hardening non-gaussian response processes[J].Journal of Engineering Mechanics,2016,142(2):06015006-1.
[17]庄翔,董欣,郑毅敏,等.矩形高层建筑非高斯风压时程峰值因子计算方法[J].工程力学,2017,34(7):177-185,223.Zhuang Xiang,Dong Xin,Zheng Yimin,et al.Peak factor estimation methods of non-Gaussian wind pressures on a rectangular high-rise building[J].Engineering Mechanics,2017,34(7):177-185,223.(in Chinese)
[18]Cianetti F,Palmieri M,Morettini G,et al.Correction formula approach to evaluate fatigue damage induced by non-Gaussian stress state[J].Procedia Structural Integrity,2018,8:390-398.
[19]Hosking J R M.L-moments:analysis and estimation of distributions using linear combinations of order statistics[J].Journal of the Royal Statistical Society,1990,52(1):105-124.
[20]Gini C.Variabilitàe mutabilità:contributo allo studio delle distribuzioni e delle relazioni statistiche[J].Studi Economico-Giuridici dell’Universita di Cagliari,1912,3:1-158.
[21]Kaigh W D,Driscoll M F.Numerical and graphical data summary using o-statistics[J].American Statistician,1987,41(1):25-32.
[22]Royston P.Which measures of skewness and kurtosis are best?[J].Statistics in Medicine,1992,11(3):333-343.
[23]Najafian G.Comparison of three different methods of moments for derivation of probability distribution parameters[J].Applied Ocean Research,2010,32(3):298-307.
[24]Xiao L,Lu H,Xin L,et al.Probability analysis of wave run-ups and air gap response of a deepwater semisubmersible platform using lh-moments estimation method[J].Journal of Waterway Port Coastal&Ocean Engineering,2015,142(2):4015019-1.
[25]田玉基,杨庆山.非高斯风压时程峰值因子的简化计算式[J].建筑结构学报,2015,36(3):20-28.Tian Yuji,Yang Qingshan.Reduced formula of peak factor for non-Gaussian wind pressure history[J].Journal of Building Structures,2015,36(3):20-28.(in Chinese)
[26]DNV-RP-C205,Environmental conditions and environmental loads[S].Norway:Det Norske Veritas,2010.
[27]Newland D E.An introduction to random vibrations and spectral analysis[M].Longman,1984.
[28]Zheng X Y,Moan T,Quek S T.Non-gaussian random wave simulation by two-dimensional fourier transform and linear oscillator response to morison force[J].Journal of Offshore Mechanics&Arctic Engineering,2007,129(4):327-334.
[29]Masters F,Gurley K R.Non-Gaussian simulation:Cumulative distribution function map-based spectral correction[J].Journal of Engineering Mechanics,2003,129(12):1418-1428.
[30]Grigoriu M.Simulation of stationary non-gaussian translation processes[J].Journal of Engineering Mechanics,1998,124(2):121-126.
[31]Technical and Research Bulletin 5-5A,Guidelines for site specific assessment of mobile jack-up units[S].Jersey City,New York:SNAME(The Society of Naval Architects and Marine Engineers),2008.
[32]Davenport A G.Note on the Distribution of the Largest Value of a Random Function with Application to Gust Loading[J].Proceedings of the Institute of Civil Engineers,1964,28(2):187-196.
[33]Zheng X Y,Liaw C Y.Response cumulant analysis of a linear oscillator driven by Morison force[J].Applied Ocean Research,2004,26(3/4):154-161.
[2]Winterstein S R,Ude T C,Marthinsen T.Volterra models of ocean structures:extreme and fatigue reliability[J].Journal of Engineering Mechanics,1994,120(6):1369-1385.
[3]Winterstein S R,Torhaug R.Extreme jack-up response:simulation and nonlinear analysis methods[J].Journal of Offshore Mechanics&Arctic Engineering,1996,118(2):103-108.
[4]Gumbel E J.Statistics of extremes[M].NewYork:Columbia University Press,1958.
[5]王飞,全涌,顾明.基于广义极值理论的非高斯风压极值计算方法[J].工程力学,2013,30(2):44-49.Wang Fei,Quan Yong,Gu Ming.An extreme-value estimation method of non-Gaussian wind pressure based on generalized extreme value theory[J].Engineering Mechanics,2013,30(2):44-49.(in Chinese)
[6]Coles S.An Introduction to Statistical modeling of extreme values[J].Technometrics,2001,44(4):107-111.
[7]Simiu E,Heckert N A.Extreme wind distribution tails:a“peaks over threshold”approach[J].Journal of Structural Engineering,1996,122(5):539-547.
[8]Naess A,Clausen P H.Combination of the peaks-over-threshold and bootstrapping methods for extreme value prediction[J].Structural Safety,2001,23(4):315-330.
[9]Naess A,Gaidai O,Karpa O.Estimation of extreme values by the average conditional exceedance rate method[J].Journal of Probability&Statistics,2013,2013(3):1-15.
[10]N?ss A,Gaidai O.Estimation of extreme values from sampled time series[J].Structural Safety,2009,31(4):325-334.
[11]Naess A,Gaidai O.Monte carlo methods for estimating the extreme response of dynamical systems[J].Journal of Engineering Mechanics,2008,134(8):628-636.
[12]Ding J,Chen X.Assessment of methods for extreme value analysis of non-Gaussian wind effects with short-term time history samples[J].Engineering Structures,2014,80:75-88.
[13]Winterstein S R,Mackenzie C A.Extremes of nonlinear vibration:comparing models based on moments,l-moments,and maximum entropy[J].Journal of Offshore Mechanics&Arctic Engineering,2013,135(2):21602-1.
[14]Winterstein S R.Non-normal responses and fatigue damage[J].Journal of Engineering Mechanics,1985,111(10):1291-1295.
[15]Winterstein S R,Kashef T.Moment-based load and response models with wind engineering applications[J].Journal of Solar Energy Engineering,2000,122(3):122-128.
[16]Ding J,Chen X.Moment-Based Translation model for hardening non-gaussian response processes[J].Journal of Engineering Mechanics,2016,142(2):06015006-1.
[17]庄翔,董欣,郑毅敏,等.矩形高层建筑非高斯风压时程峰值因子计算方法[J].工程力学,2017,34(7):177-185,223.Zhuang Xiang,Dong Xin,Zheng Yimin,et al.Peak factor estimation methods of non-Gaussian wind pressures on a rectangular high-rise building[J].Engineering Mechanics,2017,34(7):177-185,223.(in Chinese)
[18]Cianetti F,Palmieri M,Morettini G,et al.Correction formula approach to evaluate fatigue damage induced by non-Gaussian stress state[J].Procedia Structural Integrity,2018,8:390-398.
[19]Hosking J R M.L-moments:analysis and estimation of distributions using linear combinations of order statistics[J].Journal of the Royal Statistical Society,1990,52(1):105-124.
[20]Gini C.Variabilitàe mutabilità:contributo allo studio delle distribuzioni e delle relazioni statistiche[J].Studi Economico-Giuridici dell’Universita di Cagliari,1912,3:1-158.
[21]Kaigh W D,Driscoll M F.Numerical and graphical data summary using o-statistics[J].American Statistician,1987,41(1):25-32.
[22]Royston P.Which measures of skewness and kurtosis are best?[J].Statistics in Medicine,1992,11(3):333-343.
[23]Najafian G.Comparison of three different methods of moments for derivation of probability distribution parameters[J].Applied Ocean Research,2010,32(3):298-307.
[24]Xiao L,Lu H,Xin L,et al.Probability analysis of wave run-ups and air gap response of a deepwater semisubmersible platform using lh-moments estimation method[J].Journal of Waterway Port Coastal&Ocean Engineering,2015,142(2):4015019-1.
[25]田玉基,杨庆山.非高斯风压时程峰值因子的简化计算式[J].建筑结构学报,2015,36(3):20-28.Tian Yuji,Yang Qingshan.Reduced formula of peak factor for non-Gaussian wind pressure history[J].Journal of Building Structures,2015,36(3):20-28.(in Chinese)
[26]DNV-RP-C205,Environmental conditions and environmental loads[S].Norway:Det Norske Veritas,2010.
[27]Newland D E.An introduction to random vibrations and spectral analysis[M].Longman,1984.
[28]Zheng X Y,Moan T,Quek S T.Non-gaussian random wave simulation by two-dimensional fourier transform and linear oscillator response to morison force[J].Journal of Offshore Mechanics&Arctic Engineering,2007,129(4):327-334.
[29]Masters F,Gurley K R.Non-Gaussian simulation:Cumulative distribution function map-based spectral correction[J].Journal of Engineering Mechanics,2003,129(12):1418-1428.
[30]Grigoriu M.Simulation of stationary non-gaussian translation processes[J].Journal of Engineering Mechanics,1998,124(2):121-126.
[31]Technical and Research Bulletin 5-5A,Guidelines for site specific assessment of mobile jack-up units[S].Jersey City,New York:SNAME(The Society of Naval Architects and Marine Engineers),2008.
[32]Davenport A G.Note on the Distribution of the Largest Value of a Random Function with Application to Gust Loading[J].Proceedings of the Institute of Civil Engineers,1964,28(2):187-196.
[33]Zheng X Y,Liaw C Y.Response cumulant analysis of a linear oscillator driven by Morison force[J].Applied Ocean Research,2004,26(3/4):154-161.