联结(copula)函数在概率地震危险性分析中的应用
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摘要
介绍了一种成熟的已广泛应用于金融领域的估计不确定性的方法——copula函数方法,并推广了n重的Frank’s copulas;在实际应用中,本文采用了俞言祥和美国西部由Boore等人给出的两个衰减模型,针对其中存在的模型不确定性以及它们之间的相互依赖性,构造出概率意义上联合的copula分布函数,并将其应用于实例分析.结果表明,对比于传统逻辑树中所用的线性结合方法,copula将两者带来的概率分布写成一个联合概率分布,能够很好地考虑双方不尽相同的意见.另外,由于copula函数可采用各种各样的边际分布函数来获得联合概率分布,且在金融风险投资评估已有大量的应用,因此在现代地震危险性评估中将有着广泛的前景.
This paper introduces a new mature mathematical technique used to deal with models’ uncertainty in financial risk analysis-copula joint function, and generalize it to the n-dimensional Frank’s copula. In addition, we used a two-dimensional copula joint probabilistic function as an example to illustrate the uncertainty treatment at low probability level between two different attenuation models derived from Yu and Boore et al, and the result shows a better estimation than that given by linear weight technique usually used in the traditional logic-tree method. In light of widespread application in the risk analysis in the financial investment and insurance assessment, we believe that the copula-based method will have a potential application in the seismic hazard analysis.
This paper introduces a new mature mathematical technique used to deal with models’ uncertainty in financial risk analysis-copula joint function, and generalize it to the n-dimensional Frank’s copula. In addition, we used a two-dimensional copula joint probabilistic function as an example to illustrate the uncertainty treatment at low probability level between two different attenuation models derived from Yu and Boore et al, and the result shows a better estimation than that given by linear weight technique usually used in the traditional logic-tree method. In light of widespread application in the risk analysis in the financial investment and insurance assessment, we believe that the copula-based method will have a potential application in the seismic hazard analysis.
引文
胡聿贤.2006.地震工程学[M].北京:地震出版社:354--355.
鄢家全.2005.地震危险性分析的困惑与希望[J].国际地震动态,(6):14--24.
俞言祥.2002.长周期地震动衰减关系研究[D].北京:中国地震局地球物理研究所:110--119.
张尧庭.2002.连接函数(copula)技术与金融风险分析[J].统计研究,(4):48--51.
Bernreuter D L,Savy J B,Mensing R W,Chen J C.1989.Seismic Hazard Characterization of69Nuclear Sites Eastofthe Rocky Mountains[R].Washington D C:U.S.Nuclear Regulatory Commission.Vol.1--8,NUREG/CR5250,UCID-21517.
Boore D M,Joyner WB,Fumal T E.1997.Equations for esti mating horizontal response spectra and peak accelerationfrom Western North American earthquakes:Asummary of recent work[J].SeismRes Lett,68(1):128--153.
Bouye E,Durrleman V,Nikeghbali A.2000.Copulas for Finance-a Reading Guide and Some Application[R].Paris:Groupe de Recherche Operationnelle,Credit Lyonnais,working paper.
Budnitz RJ,Apostolakis G,Boore D M,Cluff KJ.1997.Recommendations for Probabilistic Seismic Hazard Analysis:Guidance on Uncertainty and Use of Experts[R].Washington D C:Report Prepared for the U.S.Nuclear Regula-tory Commission,the U.S.Depart ment of Energy,and the Electric Power Research Institute by the Senior SeismicHazard Analysis Committee(SSHAC),Report No.NUREG/CR-6372.
Clayton D G.1978.A model for associationin bivariate life tables andits applicationin epidemiological studies of familialtendency in chronic disease incidence[J].Biometrika,65:141--151.
Clemen R T,Winkler R L.1999.Combining probability distributions fromexperts in risk analysis[J].Risk Analysis,19:187--203.
Embrechts P,McNeal AJ,Straumann D.2002.Correlation and dependence in risk management:Properties and pitfalls[C]∥M A H Demster ed.Risk Management:Value at Risk and Beyond.Cambridge:Cambridge University Press:176--223.
Frank MJ.1979.On the si multaneous associativity ofF(x,y)andx+y-F(x,y)[J].Aeq Math,19:194--226.
Gumbel E J.1960.Ditributions des valeurs extr毢mes en plusiers di mensions[J].Pub Inst Stat Univ Paris,9:171--173.
Kendall M G.1938.Anew measure of rank correlation[J].Biometrika,30:81--93.
Kruskal W H.1958.Ordinal measures of association[J].J Amer Statist Assoc,53:814--861.
Kramer S L.2003.Geotechnical Earthquake Engineering[M].Upper Saddle River,NJ:Prentice Hall:137.
Morgan T Page,Carlson J M.2006.Methodologies for earthquake hazard assessment:Model Uncertainty and the WG-CEP-2002forecast[J].Bull Seism Soc Amer,96:1624--1633.
Nelsen R B.1998.An Introduction to Copulas[M].New York:Springer-Verlag:37--43.
R啨nyi A.1959.On measures of dependence[J].Acta Math Acad Sci Hungar,10:441--451.
Rockinger M,Jondeau E.2002.Conditional Dependency of Financial Series:An Application of Copulas[R].Paris:HEC School of Management.
Shafer G.1976.A Mathematical Theory of Evidence[M].Princeton,NewJersey:Princeton University Press:10--40.
Sklar A.1959.Fonctions de r啨partition劋ndi mentions et leurs marges[J].Pub Inst Stat Univ Paris,8:229--231.
WGCEP(Working Group on California Earthquake Probabilities).2003.Earthquake probabilities in the San FranciscoBay Region:2002—2031[R]∥Working Group on California Earthquake Probabilities.Tech.Rep.Open-file Re-port.[s.l.]:USGS:03-214.
鄢家全.2005.地震危险性分析的困惑与希望[J].国际地震动态,(6):14--24.
俞言祥.2002.长周期地震动衰减关系研究[D].北京:中国地震局地球物理研究所:110--119.
张尧庭.2002.连接函数(copula)技术与金融风险分析[J].统计研究,(4):48--51.
Bernreuter D L,Savy J B,Mensing R W,Chen J C.1989.Seismic Hazard Characterization of69Nuclear Sites Eastofthe Rocky Mountains[R].Washington D C:U.S.Nuclear Regulatory Commission.Vol.1--8,NUREG/CR5250,UCID-21517.
Boore D M,Joyner WB,Fumal T E.1997.Equations for esti mating horizontal response spectra and peak accelerationfrom Western North American earthquakes:Asummary of recent work[J].SeismRes Lett,68(1):128--153.
Bouye E,Durrleman V,Nikeghbali A.2000.Copulas for Finance-a Reading Guide and Some Application[R].Paris:Groupe de Recherche Operationnelle,Credit Lyonnais,working paper.
Budnitz RJ,Apostolakis G,Boore D M,Cluff KJ.1997.Recommendations for Probabilistic Seismic Hazard Analysis:Guidance on Uncertainty and Use of Experts[R].Washington D C:Report Prepared for the U.S.Nuclear Regula-tory Commission,the U.S.Depart ment of Energy,and the Electric Power Research Institute by the Senior SeismicHazard Analysis Committee(SSHAC),Report No.NUREG/CR-6372.
Clayton D G.1978.A model for associationin bivariate life tables andits applicationin epidemiological studies of familialtendency in chronic disease incidence[J].Biometrika,65:141--151.
Clemen R T,Winkler R L.1999.Combining probability distributions fromexperts in risk analysis[J].Risk Analysis,19:187--203.
Embrechts P,McNeal AJ,Straumann D.2002.Correlation and dependence in risk management:Properties and pitfalls[C]∥M A H Demster ed.Risk Management:Value at Risk and Beyond.Cambridge:Cambridge University Press:176--223.
Frank MJ.1979.On the si multaneous associativity ofF(x,y)andx+y-F(x,y)[J].Aeq Math,19:194--226.
Gumbel E J.1960.Ditributions des valeurs extr毢mes en plusiers di mensions[J].Pub Inst Stat Univ Paris,9:171--173.
Kendall M G.1938.Anew measure of rank correlation[J].Biometrika,30:81--93.
Kruskal W H.1958.Ordinal measures of association[J].J Amer Statist Assoc,53:814--861.
Kramer S L.2003.Geotechnical Earthquake Engineering[M].Upper Saddle River,NJ:Prentice Hall:137.
Morgan T Page,Carlson J M.2006.Methodologies for earthquake hazard assessment:Model Uncertainty and the WG-CEP-2002forecast[J].Bull Seism Soc Amer,96:1624--1633.
Nelsen R B.1998.An Introduction to Copulas[M].New York:Springer-Verlag:37--43.
R啨nyi A.1959.On measures of dependence[J].Acta Math Acad Sci Hungar,10:441--451.
Rockinger M,Jondeau E.2002.Conditional Dependency of Financial Series:An Application of Copulas[R].Paris:HEC School of Management.
Shafer G.1976.A Mathematical Theory of Evidence[M].Princeton,NewJersey:Princeton University Press:10--40.
Sklar A.1959.Fonctions de r啨partition劋ndi mentions et leurs marges[J].Pub Inst Stat Univ Paris,8:229--231.
WGCEP(Working Group on California Earthquake Probabilities).2003.Earthquake probabilities in the San FranciscoBay Region:2002—2031[R]∥Working Group on California Earthquake Probabilities.Tech.Rep.Open-file Re-port.[s.l.]:USGS:03-214.
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