椭圆偏相关图中的依赖性
Dependence in elliptical partial correlation graphs
摘要：
高斯模型具有强大的属性，便于研究和解释图形模型。具体地说，它减少了条件独立性和确定偏相关及其符号的正关联研究。当高斯性不支持偏相关时，图是图形模型的一种有用的放松，但不清楚它们包含什么信息（除了明显缺乏线性关联）。我们研究椭圆和transelliptical分布之间的高斯和其他家庭，更灵活，但要么不嵌入强大的属性或不导致简单的解释中间地带。我们刻画了零偏相关在椭圆族和transelliptic copula模型中的意义，并表明它保留了高斯情形下的大部分依赖结构。关于正相关性，我们证明了学习（反）椭圆图形模型的不可能结果，包括对于所有维度都是二阶多变量全正的椭圆分布必须是高斯分布。然后，我们将展示如何解释正偏相关作为一个放松，并获得重要的属性相关的忠实性和辛普森悖论。我们说明了transelliptical模型的潜力，研究尾部依赖的标准普尔500指数的数据，并积极改善正则化推理。
英文摘要：
The Gaussian model equips strong properties that facilitate studying and interpreting graphical models. Specifically it reduces conditional independence and the study of positive association to determining partial correlations and their signs. When Gaussianity does not hold partial correlation graphs are a useful relaxation of graphical models, but it is not clear what information they contain (besides the obvious lack of linear association). We study elliptical and transelliptical distributions as middle-ground between the Gaussian and other families that are more flexible but either do not embed strong properties or do not lead to simple interpretation. We characterize the meaning of zero partial correlations in the elliptical family and transelliptical copula models and show that it retains much of the dependence structure from the Gaussian case. Regarding positive dependence, we prove impossibility results to learn (trans)elliptical graphical models, including that an elliptical distribution that is multivariate totally positive of order two for all dimensions must be essentially Gaussian. We then show how to interpret positive partial correlations as a relaxation, and obtain important properties related to faithfulness and Simpson's paradox. We illustrate the transelliptical model potential to study tail dependence in S&P500 data, and of positivity to improve regularized inference.
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