In this article, we show that the recently introduced ordinal pattern dependence fits into the axiomatic framework of general multivariate dependence measures, i.e., measures of dependence between two multivariate random objects. Furthermore, we consider multivariate generalizations of established univariate dependence measures like Kendall's $\tau$, Spearman's $\rho$ and Pearson's correlation coefficient. Among these, only multivariate Kendall's $\tau$ proves to take the dynamical dependence of random vectors stemming from multidimensional time series into account. Consequently, the article focuses on a comparison of ordinal pattern dependence and multivariate Kendall's $\tau$ in this context. To this end, limit theorems for multivariate Kendall's $\tau$ are established under the assumption of near-epoch dependent data-generating time series. We analyze how ordinal pattern dependence compares to multivariate Kendall's $\tau$ and Pearson's correlation coefficient on theoretical grounds. Additionally, a simulation study illustrates differences in the kind of dependencies that are revealed by multivariate Kendall's $\tau$ and ordinal pattern dependence.
翻译:在此篇文章中, 我们显示, 最近引入的 odin 模式依赖性符合一般多变量依赖性措施, 即两个多变量随机对象之间依赖性的度量。 此外, 我们考虑对已有的单变量依赖性措施, 如Kendall $tau$、 Spearman $\rho$和 Pearson 相关系数等, 进行多种变量的概括化概括化。 其中, 只有多变量 Kendall 的 $\tau$ 和 Pearson 相关系数, 才能将来自多层面时间序列的随机矢量的动态依赖性( 即两个多变量随机对象之间的依赖性, 即两个多变量随机对象之间的依赖性度。 因此, 文章侧重于在此背景下比较 。 此外, 一项模拟研究表明, 多变量的肯达尔 $ 和 美元 模式的模型显示差异, 以理论角度显示 。