Measuring Linear Correlation Between Random Vectors

Giovanni Puccetti
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引用次数: 8

Abstract

We introduce a new scalar coefficient to measure linear correlation between random vectors which preserves all the relevant properties of Pearson’s correlation in arbitrary dimensions. The new measure and its bounds are derived from a mass transportation approach in which the expected inner product of two random vectors is taken as a measure of their covariance and then standardized by the maximal attainable value given their marginal covariance matrices. In several simulative studies we show the limiting distribution of the empirical estimator of the newly defined index and of the corresponding rank correlation. A comparative study shows that our proposed correlation, though derived from a novel approach, behaves similarly to some of the multivariate dependence notions recently introduced in the literature. Throughout the paper, we also give some auxiliary results of independent interest in matrix analysis and mass transportation theory, including an improvement to the Cauchy-Schwarz inequality for positive definite covariance matrices.
测量随机向量之间的线性相关性
我们引入了一个新的标量系数来度量随机向量之间的线性相关性,它保留了任意维的Pearson相关的所有相关性质。新测度及其界是由一种质量传递方法导出的,该方法以两个随机向量的期望内积作为其协方差的测度,然后用给定其边际协方差矩阵的最大可得值进行标准化。在一些模拟研究中,我们证明了新定义的指数的经验估计量和相应的秩相关的极限分布。一项比较研究表明,我们提出的相关性虽然来自一种新的方法,但其行为与最近在文献中引入的一些多变量依赖概念相似。在整个论文中,我们还给出了一些在矩阵分析和质量传递理论中独立感兴趣的辅助结果,包括对正定协方差矩阵的Cauchy-Schwarz不等式的改进。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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