基于正则Vine copula的相依建模及软件开发

There are different or even complex dependence structures in multivariate data, among which some data variables possess features such as upper and/or lower tail dependence. In statistical modeling, we try to look for such a model which is simple and robust, and can be as much as possible to reveal the characteristics of data. The ordinary copula method has some disadvantages in modeling high-dimensional data, which can not provide wide range of dependence structures of variables. The newly developed vine copulas with graphic structures can overcome the above disadvantages of ordinary copulas in modeling high-dimensional data, and have receive more attention of the people. In this project, we will investigate properties of regular vine (R-Vine) copulas from the view of statistical dependence, exploit graphical structures of R-vines to study model selection, parameters estimations, goodness-of-fit testing algorithms and sampling techniques, develop a statistical Python package pyvine for R-Vine modeling, apply the modeling methods to several real data sets and compare fits with other known methods, and try to model longitudinal data using a R-Vine decomposition of serial dependence. This project will enable one to understand the model, grasp and apply the model.

多维数据中变量之间会呈现出不同的甚至复杂的相依结构，一些变量之间有时也会呈现出上或下尾相依性。统计建模就是要寻找这样的一个简单且稳健的模型，该模型能够尽可能多地揭示数据的特质。通常Copula 方法在对高维数据相依建模中存在一些不足，不能很好地把握模型变量之间的相依结构。新近发展起来的带有图结构的Vine Copula 可以克服通常 Copula 在多维建模方面的不足，因此备受人们的关注。本项目将从统计相依的角度研究 R-Vine 模型性质；充分利用R-Vine图的结构研究其模型选择、参数估计、拟合优度检验以及抽样技术；开发适应于R-Vine 建模的统计软件包pyvine；将R-Vine 的建模方法应用于几个实际数据，并与已有的建模方法作对比，尝试用R-Vine 的思想来对纵向数据内部相依结构建模。本项目的研究将有助于人们更好地认识模型、把握模型和应用模型。

多维数据中变量之间会呈现出不同的甚至复杂的相依结构，一些变量之间有时也会呈现出上或下尾相依性。统计建模就是要寻找这样的一个简单且稳健的模型，该模型能够尽可能多地揭示数据的特质。通常Copula 方法在对高维数据相依建模中存在一些不足，不能很好地把握模型变量之间的相依结构。新近发展起来的带有图结构的Vine Copula 可以克服通常 Copula 在多维建模方面的不足，因此备受人们的关注。本项目从统计相依的角度研究 R-Vine 模型性质；充分利用了R-Vine图的结构来开发适应于R-Vine 建模的统计软件包 pyvine；已将R-Vine 的建模方法应用于几个实际数据。另外，我们在多维极值分布的谱测度与相依性之间的关系、相依不确定性、基于距离的波动性度量，以及椭球分布随机向量的随机比较方面开展了研究。本项目的研究将有助于人们更好地认识模型、把握模型和应用模型。