基于Copula函数的土体抗剪强度参数二维分布模型
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摘要
传统的土体抗剪强度参数二维正态分布模型不足以表征抗剪强度参数的非正态分布特性。为此,提出了基于Copula函数的土体抗剪强度参数二维非正态分布模型建模方法。收集整理了小浪底水利枢纽工程中土体抗剪强度参数的三组试验数据,采用A-D检验法识别出抗剪强度参数的最优边缘分布模型。基于Copula函数建立了土体抗剪强度参数的二维分布模型,并采用最小平方欧氏距离法识别出表征抗剪强度参数间相关性的最优Copula函数。在此基础上,系统地比较了基于Copula函数的二维非正态分布模型与二维正态分布模型的优缺点。结果表明:提出的基于Copula函数的土体抗剪强度参数二维分布模型能够有效地表征抗剪强度参数的统计特性;与抗剪强度参数二维正态分布模型相比,基于最优Copula函数和最优边缘分布函数建立的抗剪强度参数二维非正态分布模型在适用性和拟合能力方面均具有一定的优势;此外,常用的Gaussian copula并不一定是表征抗剪强度参数间相关性的最优Copula函数。
This paper aims to propose a new method for modeling the bivariate non-normal distribution of soil shear strength parameter using Copula,due to the limitation of bivariate normal distribution to show the non-normal statistical characteristic.First,three datasets of soil triaxial shear strength from Xiaolangdi Hydraulic Project are collected,and the A-D method is used to identify the best-fit marginal distributions model.Then,the method for dependence modeling of soil shear strength parameter using Copula is briefly introduced,with the Least Square Euclidean Distances method used to identify the best-fit Copula underlying the collected shear strength parameter.Finally,the advantage and disadvantage to reproduce the bivariate model of soil shear strength parameter using the proposed method and the traditional bivariate normal distribution is compared systematically.The results indicate that it is an effective way to show the non-normal statistical characteristic of soil shear strength parameter using Copula.Compared with the traditional bivariate normal distribution,the proposed Copula-based method can provide a more general way of modeling the bivariate non-normal distribution of shear strength parameter in isolation from their marginal distributions,which results in more accurate results.Furthermore,the Gaussian copula,often adopted out of experience,is not always the best-fit Copula for modeling the dependence structure underlying the shear strength parameter.
This paper aims to propose a new method for modeling the bivariate non-normal distribution of soil shear strength parameter using Copula,due to the limitation of bivariate normal distribution to show the non-normal statistical characteristic.First,three datasets of soil triaxial shear strength from Xiaolangdi Hydraulic Project are collected,and the A-D method is used to identify the best-fit marginal distributions model.Then,the method for dependence modeling of soil shear strength parameter using Copula is briefly introduced,with the Least Square Euclidean Distances method used to identify the best-fit Copula underlying the collected shear strength parameter.Finally,the advantage and disadvantage to reproduce the bivariate model of soil shear strength parameter using the proposed method and the traditional bivariate normal distribution is compared systematically.The results indicate that it is an effective way to show the non-normal statistical characteristic of soil shear strength parameter using Copula.Compared with the traditional bivariate normal distribution,the proposed Copula-based method can provide a more general way of modeling the bivariate non-normal distribution of shear strength parameter in isolation from their marginal distributions,which results in more accurate results.Furthermore,the Gaussian copula,often adopted out of experience,is not always the best-fit Copula for modeling the dependence structure underlying the shear strength parameter.
引文
[1]陈炜韬,王玉锁,王明年,等.黏土质隧道围岩抗剪强度参数的概率分布及优化实例[J].岩石力学与工程学报,2006,25(S2):3782-3787.
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[10]Grimaldi S,Serinaldi F.Asymmetric copula in multi-variate flood frequency analysis[J].Advances in Wa-ter Resources,2006,29(8):1155-1167.
[11]Zhang L,Singh V P.Bivariate rainfall frequency dis-tributions using archimedean copulas[J].Journal ofHydrology,2007,332(1-2):93-109.
[12]Zhang Q,Chen Y D,Chen X,et al.Copula-based a-nalysis of hydrological extremes and implications ofhydrological behaviors in the Pearl River basin,China[J].Journal of Hydrologic Engineering,2011,16(7):598-607.
[13]唐小松,李典庆,周创兵,等.基于Copula函数的基桩荷载-位移双曲线概率分析[J].岩土力学,2012,33(1):171-178.
[14]Uzielli M,Mayne P W.Load-displacement uncertaintyof vertically loaded shallow footings on sands andeffects on probabilistic settlement[J].Georisk,2012,6(1):50-69.
[15]于清波.土石坝坝坡稳定可靠度研究及其工程应用[D].南京:河海大学,2006.
[16]王奇峰,王碧.小浪底高土石坝心墙土料应用探讨[J].西北水电,2001,(1):65-69.
[2]陈立宏,陈祖煜,刘金梅.土体抗剪强度指标的概率分布类型研究[J].岩土力学,2005,26(1):37-40.
[3]范明桥.粘性填筑土强度指标φ,c的概率特性[J].水利水运科学研究,2000,(1):49-53.
[4]赵宇飞,杨钧.岩土强度参数二维正态分布拟合研究[J].华北水利水电学院学报,2006,27(2):79-81.
[5]Nataf A.Détermination des distributions de probabilitédont les marges sont données[J].Comptes Rendus del’Académie des Sciences,1962,225:42-43.
[6]Nelsen R B.An Introduction to Copulas[M].NewYork:Springer Verlag,2006.
[7]Sklar A.Fonctions de répartitionàn dimensions etleurs marges[J].Publications de l’Institut de Statis-tique de l’Universitéde Paris,1959,8:229-231.
[8]Kole E,Koedijk K,Verbeek M.Selecting copulas forrisk management[J].Journal of Banking and Fi-nance,2007,31(8):2405-2423.
[9]张尧庭.连接函数(Copula)技术与金融风险分析[J].统计研究,2002,(4):48-51.
[10]Grimaldi S,Serinaldi F.Asymmetric copula in multi-variate flood frequency analysis[J].Advances in Wa-ter Resources,2006,29(8):1155-1167.
[11]Zhang L,Singh V P.Bivariate rainfall frequency dis-tributions using archimedean copulas[J].Journal ofHydrology,2007,332(1-2):93-109.
[12]Zhang Q,Chen Y D,Chen X,et al.Copula-based a-nalysis of hydrological extremes and implications ofhydrological behaviors in the Pearl River basin,China[J].Journal of Hydrologic Engineering,2011,16(7):598-607.
[13]唐小松,李典庆,周创兵,等.基于Copula函数的基桩荷载-位移双曲线概率分析[J].岩土力学,2012,33(1):171-178.
[14]Uzielli M,Mayne P W.Load-displacement uncertaintyof vertically loaded shallow footings on sands andeffects on probabilistic settlement[J].Georisk,2012,6(1):50-69.
[15]于清波.土石坝坝坡稳定可靠度研究及其工程应用[D].南京:河海大学,2006.
[16]王奇峰,王碧.小浪底高土石坝心墙土料应用探讨[J].西北水电,2001,(1):65-69.