Copula理论与极值统计的应用
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摘要
Copula用来描述多元随机变量的一维边缘分布与其联合分布之间的函数关系.Copula的精妙结构和潜在价值使其理论得到较快发展,成为近年来一个热门关键词.通过Copula函数,可以捕捉到变量间非线性、非对称以及分布尾部的相关关系,这让它在极值分析和对极端现象的预测中大有作为.本文主要研究Copula的理论与极值统计的应用.
CopulaC(u, v)的上下界问题是Copula理论的基本问题之一,如果已知C(u,v)在某个点处的函数值,它的界要变窄.本文证明了由Copula及其生存、对偶和伴随Copula组成的集合在函数的复合运算下构成了Klein四元群,求解了群中元素的界以及它们在某个点处的函数值给定时的界.计算了某个点的值给定时Copula的界CU和CL的不可交换性度量,构造了四个具有最大不可交换性的Copula,揭示了它们的结构特点和性质.给出了在一般的LP距离下不可交换性度量的计算公式.
FGM Copula由于其简单的结构和优良的分析性质在建模时被广泛应用.本文从Copula的构造方法出发,研究了FGM Copula的生成与拓展.通过证明形如:C_θ(u, v) = uv +θf(u)g(v)的二元函数是一个Copula的充要条件,构造了一种新型广义FGM Copula: C(u,v) = uv +θu~av~b(1 - u~m)~c(1 - v~n)~d,并研究了它的关联性度量和相关性质.所提出的Copula涵盖了文献中许多特殊类型的广义FGM Copula,是它们的推广和综合,取得了广义FGM Copula理论研究的新进展.
在深入研究一元极值和Copula理论的基础上,本文建立了一元和二元阈值模型,分别探讨了它们在公交车调度和泥石流沟地貌要素的相关性分析中的应用.先根据一元阈值模型建立了公交车客容量的分布函数,由此可以优化公交车的发车方案.然后将一元超阈值分布与Logistic Copula相结合,构造二元阈值模型,分析了流域面积和流域高差两个地貌要素的极值相关性.另外,利用多元极值Copula研究了多元极值分布函数的相关序理论.
Copulas are functions that joint multivariate distribution functions to their one–dimensional marginal distribution functions. The wonderful structure and latent valueof Copulas make their theory develop quickly, and become a popular keyword.By wayof Copula, we can catch the nonlinear and asymmetric dependence between randomvariables and the tail dependence of the distributions, which make Copula give greatplay to analysis of extremes and forecast of extremely phenomenon. This paper mainlytalks about the theory of Copula and the applications of Statistics of Extremes.
The upper and lower bounds are the fundamental theory of Copula. When wepossess information about the values of a Copula at points in its domain of definition,the bounds can often be narrowed. We prove the set of Copula, survival Copula, dualof a Copula and the co–Copula together with the composition operation being a Kleinquaternion group, then we give the Fr′echet–Hoe ding bounds and the bounds when wepossess the values of a Copula at some points. We calculate the nonexchangeabilitymeasures of Copula CU and CL, the bounds of Copulas when we possess the valuesof some points, construct four Copulas with maximum nonexchangeability, reveal thestructure feature and property of the four Copulas, and give the calculating formula ofnonexchangeability for the LP distance.
FGM Copulas have been widely used in modeling primarily because of their simpleanalytical form. This paper studies the generation and continuation of FGM Copulasbeginning with the methods of constructing Copulas. We construct a new class of generalizedFGM Copulas:C(u; v) = uv+θu~av~b(1-u~m)~c(1-v~n)~d, by proving the necessary andsucient condition when C_θ(u; v) = uv +θf (u)g(v) are Copulas, and study the associationmeasures and dependence properties of these new Copulas,which including manyother types of generalized FGM Copulas appeared in literatures. We make progress ofthe theory of generalized FGM Copulas.
Bases on the further study of univariate Extreme and Copula theory, this paperfounds the univariate and bivariate threshold model, studies their applications in busdispatch and the dependence relationships between relief factors of debris flow. Firstly,we found the distribution of bus capacity according to threshold model, and fromthis the optimizing bus scheduling can be given. Secondly, combining the univariatethreshold distribution with Logistic Copula, we construct bivariate threshold model to analyze the extreme dependence between drainage area and drainage height differenceof relief factors of debris flow.Moreover, we study the dependence order of multivariateextreme value distributions using extreme value Copula.
CopulaC(u, v)的上下界问题是Copula理论的基本问题之一,如果已知C(u,v)在某个点处的函数值,它的界要变窄.本文证明了由Copula及其生存、对偶和伴随Copula组成的集合在函数的复合运算下构成了Klein四元群,求解了群中元素的界以及它们在某个点处的函数值给定时的界.计算了某个点的值给定时Copula的界CU和CL的不可交换性度量,构造了四个具有最大不可交换性的Copula,揭示了它们的结构特点和性质.给出了在一般的LP距离下不可交换性度量的计算公式.
FGM Copula由于其简单的结构和优良的分析性质在建模时被广泛应用.本文从Copula的构造方法出发,研究了FGM Copula的生成与拓展.通过证明形如:C_θ(u, v) = uv +θf(u)g(v)的二元函数是一个Copula的充要条件,构造了一种新型广义FGM Copula: C(u,v) = uv +θu~av~b(1 - u~m)~c(1 - v~n)~d,并研究了它的关联性度量和相关性质.所提出的Copula涵盖了文献中许多特殊类型的广义FGM Copula,是它们的推广和综合,取得了广义FGM Copula理论研究的新进展.
在深入研究一元极值和Copula理论的基础上,本文建立了一元和二元阈值模型,分别探讨了它们在公交车调度和泥石流沟地貌要素的相关性分析中的应用.先根据一元阈值模型建立了公交车客容量的分布函数,由此可以优化公交车的发车方案.然后将一元超阈值分布与Logistic Copula相结合,构造二元阈值模型,分析了流域面积和流域高差两个地貌要素的极值相关性.另外,利用多元极值Copula研究了多元极值分布函数的相关序理论.
Copulas are functions that joint multivariate distribution functions to their one–dimensional marginal distribution functions. The wonderful structure and latent valueof Copulas make their theory develop quickly, and become a popular keyword.By wayof Copula, we can catch the nonlinear and asymmetric dependence between randomvariables and the tail dependence of the distributions, which make Copula give greatplay to analysis of extremes and forecast of extremely phenomenon. This paper mainlytalks about the theory of Copula and the applications of Statistics of Extremes.
The upper and lower bounds are the fundamental theory of Copula. When wepossess information about the values of a Copula at points in its domain of definition,the bounds can often be narrowed. We prove the set of Copula, survival Copula, dualof a Copula and the co–Copula together with the composition operation being a Kleinquaternion group, then we give the Fr′echet–Hoe ding bounds and the bounds when wepossess the values of a Copula at some points. We calculate the nonexchangeabilitymeasures of Copula CU and CL, the bounds of Copulas when we possess the valuesof some points, construct four Copulas with maximum nonexchangeability, reveal thestructure feature and property of the four Copulas, and give the calculating formula ofnonexchangeability for the LP distance.
FGM Copulas have been widely used in modeling primarily because of their simpleanalytical form. This paper studies the generation and continuation of FGM Copulasbeginning with the methods of constructing Copulas. We construct a new class of generalizedFGM Copulas:C(u; v) = uv+θu~av~b(1-u~m)~c(1-v~n)~d, by proving the necessary andsucient condition when C_θ(u; v) = uv +θf (u)g(v) are Copulas, and study the associationmeasures and dependence properties of these new Copulas,which including manyother types of generalized FGM Copulas appeared in literatures. We make progress ofthe theory of generalized FGM Copulas.
Bases on the further study of univariate Extreme and Copula theory, this paperfounds the univariate and bivariate threshold model, studies their applications in busdispatch and the dependence relationships between relief factors of debris flow. Firstly,we found the distribution of bus capacity according to threshold model, and fromthis the optimizing bus scheduling can be given. Secondly, combining the univariatethreshold distribution with Logistic Copula, we construct bivariate threshold model to analyze the extreme dependence between drainage area and drainage height differenceof relief factors of debris flow.Moreover, we study the dependence order of multivariateextreme value distributions using extreme value Copula.
引文
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