本文研究了干旱发生的联合概率、条件概率和重现期等干旱特征。以陕西省西安站月降水为例,应用Meta-Gaussian Copula和Student t Copula构造了干旱历时、干旱烈度和烈度峰值的联合概率分布,并进行了多变量分布拟合优质评价及拟合检验,...本文研究了干旱发生的联合概率、条件概率和重现期等干旱特征。以陕西省西安站月降水为例,应用Meta-Gaussian Copula和Student t Copula构造了干旱历时、干旱烈度和烈度峰值的联合概率分布,并进行了多变量分布拟合优质评价及拟合检验,在此基础上计算了联合分布的重现期以及2变量和3变量情形下的条件概率与条件重现期。研究表明,Meta-Gaussian Copula可以描述干旱历时、干旱烈度和烈度峰值三者的联合分布。由于多元联合分布可以考虑到多个变量之间的不同组合,能够求得不同干旱历时、干旱烈度或烈度峰值下的条件概率和条件重现期,因而能够更加全面客观地反映干旱的特征。展开更多
The question of how to choose a copula model that best fits a given dataset is a predominant limitation of the copula approach, and the present study aims to investigate the techniques of goodness-of-fit tests for mul...The question of how to choose a copula model that best fits a given dataset is a predominant limitation of the copula approach, and the present study aims to investigate the techniques of goodness-of-fit tests for multi-dimensional copulas. A goodness-of-fit test based on Rosenblatt＇s transformation was mathematically expanded from two dimensions to three dimensions and procedures of a bootstrap version of the test were provided. Through stochastic copula simulation, an empirical application of historical drought data at the Lintong Gauge Station shows that the goodness-of-fit tests perform well, revealing that both trivariate Gaussian and Student t copulas are acceptable for modeling the dependence structures of the observed drought duration, severity, and peak. The goodness-of-fit tests for multi-dimensional copulas can provide further support and help a lot in the potential applications of a wider range of copulas to describe the associations of correlated hydrological variables. However, for the application of copulas with the number of dimensions larger than three, more complicated computational efforts as well as exploration and parameterization of corresponding copulas are required.展开更多
A general method was proposed to evaluate the distribution function of （C1｜C2 ） . Some examples were presented to validate the application of the method. Then the sufficient and necessary condition for that the dis...A general method was proposed to evaluate the distribution function of （C1｜C2 ） . Some examples were presented to validate the application of the method. Then the sufficient and necessary condition for that the distribution function of （ C1 ｜ C2 ） is uniform was proved.展开更多
Based on the trivariate reduction technique two different trivariate Bernoulli mixtures of univariate uniform distributions and their associated trivariate copulas with bivariate linear Spearman marginal copulas are c...Based on the trivariate reduction technique two different trivariate Bernoulli mixtures of univariate uniform distributions and their associated trivariate copulas with bivariate linear Spearman marginal copulas are considered. Mathematical characterizations of these Bernoulli mixture models are obtained. Since Bernoulli mixture trivariate reduction copulas are not compatible with all valid grade correlation coefficients, and there exist linear Spearman compatible non-Bernoulli mixture trivariate copulas, one can ask when there exists at all a trivariate copula with given linear Spearman marginal copulas. Based on a known concordance ordering compatibility criterion, a set of grade correlation inequalities, which must necessarily be satisfied for compatibility, is derived. The existence question for trivariate copulas with compatible linear Spearman marginal copulas is settled in the main result, which states that this set of inequalities is also sufficient for compatibility. The constructive proof makes use of two new classes of trivariate copulas that are obtained from the Bernoulli mixture trivariate copulas through a natural parametric extension. Finally, the obtained classes of trivariate copulas are compared with another class that contains as special case some trivariate copulas with linear Spearman marginal copulas. Since the latter class is incompatible with some type of linear Spearman copulas, the new classes of trivariate copulas build a richer class in this respect. Moreover, in contrast to the mentioned class, which requires in general 11 different elementary copulas in the defining convex linear combination, the new classes require at most five of them, which results in a more parsimonious parametric modelling.展开更多
文摘本文研究了干旱发生的联合概率、条件概率和重现期等干旱特征。以陕西省西安站月降水为例,应用Meta-Gaussian Copula和Student t Copula构造了干旱历时、干旱烈度和烈度峰值的联合概率分布,并进行了多变量分布拟合优质评价及拟合检验,在此基础上计算了联合分布的重现期以及2变量和3变量情形下的条件概率与条件重现期。研究表明,Meta-Gaussian Copula可以描述干旱历时、干旱烈度和烈度峰值三者的联合分布。由于多元联合分布可以考虑到多个变量之间的不同组合,能够求得不同干旱历时、干旱烈度或烈度峰值下的条件概率和条件重现期,因而能够更加全面客观地反映干旱的特征。
基金supported by the Program of Introducing Talents of Disciplines to Universities of the Ministry of Education and State Administration of the Foreign Experts Affairs of China (the 111 Project, Grant No.B08048)the Special Basic Research Fund for Methodology in Hydrology of the Ministry of Sciences and Technology of China (Grant No. 2011IM011000)
文摘The question of how to choose a copula model that best fits a given dataset is a predominant limitation of the copula approach, and the present study aims to investigate the techniques of goodness-of-fit tests for multi-dimensional copulas. A goodness-of-fit test based on Rosenblatt＇s transformation was mathematically expanded from two dimensions to three dimensions and procedures of a bootstrap version of the test were provided. Through stochastic copula simulation, an empirical application of historical drought data at the Lintong Gauge Station shows that the goodness-of-fit tests perform well, revealing that both trivariate Gaussian and Student t copulas are acceptable for modeling the dependence structures of the observed drought duration, severity, and peak. The goodness-of-fit tests for multi-dimensional copulas can provide further support and help a lot in the potential applications of a wider range of copulas to describe the associations of correlated hydrological variables. However, for the application of copulas with the number of dimensions larger than three, more complicated computational efforts as well as exploration and parameterization of corresponding copulas are required.
文摘A general method was proposed to evaluate the distribution function of （C1｜C2 ） . Some examples were presented to validate the application of the method. Then the sufficient and necessary condition for that the distribution function of （ C1 ｜ C2 ） is uniform was proved.
文摘Based on the trivariate reduction technique two different trivariate Bernoulli mixtures of univariate uniform distributions and their associated trivariate copulas with bivariate linear Spearman marginal copulas are considered. Mathematical characterizations of these Bernoulli mixture models are obtained. Since Bernoulli mixture trivariate reduction copulas are not compatible with all valid grade correlation coefficients, and there exist linear Spearman compatible non-Bernoulli mixture trivariate copulas, one can ask when there exists at all a trivariate copula with given linear Spearman marginal copulas. Based on a known concordance ordering compatibility criterion, a set of grade correlation inequalities, which must necessarily be satisfied for compatibility, is derived. The existence question for trivariate copulas with compatible linear Spearman marginal copulas is settled in the main result, which states that this set of inequalities is also sufficient for compatibility. The constructive proof makes use of two new classes of trivariate copulas that are obtained from the Bernoulli mixture trivariate copulas through a natural parametric extension. Finally, the obtained classes of trivariate copulas are compared with another class that contains as special case some trivariate copulas with linear Spearman marginal copulas. Since the latter class is incompatible with some type of linear Spearman copulas, the new classes of trivariate copulas build a richer class in this respect. Moreover, in contrast to the mentioned class, which requires in general 11 different elementary copulas in the defining convex linear combination, the new classes require at most five of them, which results in a more parsimonious parametric modelling.
