Statistical learning with vine copulas

针对多个变量数据的统计学习方法不仅要对每个变量的行为单独进行充分建模，还要考虑到它们之间的依赖性。Copula方法非常适合，因为它通过将单独的边际函数与描述依赖关系的Copula连接起来来构建模型。这种基于Copula的模型在统计学习中的应用的一个障碍是它们在高维中缺乏灵活性。然而，最近的葡萄藤Copula类被证明是适合于在高维的依赖建模，因为它们是在独立的二元Copula块的帮助下构建的。进一步的基于vine copula的模型可以捕获不对称的尾部依赖。在金融、保险和工程领域的风险管理中，都可以观察到这种情况。标准的相关模型，如多变量高斯分布或学生t分布，不能适应不对称的尾部。 本项目希望利用这些优势来构建和实现一个基于vine copula的统计学习工具箱，以应对具有挑战性的高维应用。特别是，我们将研究基于藤的分位数回归方法的估计和选择。此外，我们将通过设计具有藤蔓成分的新型混合模型来实现聚类和分类任务。我们将发展统计理论，以便对条件分位数的预测进行不确定性评估，以及对新数据进行聚类和分类分配。比较研究将证明更现实和可解释的建模的优势。

Statistical learning methods for data on many variables have to not only adequately model the behavior of each variable separately but also to allow for dependence between them. The copula approach is highly suitable since it builds models by joining separate marginal functions with a copula describing the dependence. An obstacle for the application of such copula based models in statistical learning was their lack of flexibility in high dimensions. The class of vine copulas however has recently shown to be suitable for dependence modeling in high dimensions, since they are constructed with the help of independent bivariate copula blocks. Further vine copula based models can capture asymmetric tail dependence. These are observed in risk management in finance, insurance and engineering. Standard dependence models such as the multivariate Gaussian or Student t distribution cannot accommodate asymmetric tails. This project wants to harvest these advantages to build and implement a vine copula based statistical learning toolbox for challenging high dimensional applications. In particular, we will investigate the estimation and selection of vine-based quantile regression methods. Further, we will approach clustering and classification tasks by designing novel mixture models with vine components. We will develop statistical theory to allow for uncertainty assessment of prediction of conditional quantiles as well as for the cluster and classification assignment of new data. Comparison studies will demonstrate the advantages of more realistic and interpretable modeling.