论文标题
调整Pearson的$ R $以测量任意单调依赖
Adjust Pearson's $r$ to Measure Arbitrary Monotone Dependence
论文作者
论文摘要
皮尔逊（Pearson）的R是最广泛使用的相关系数，传统上被认为是仅捕获线性依赖性的，导致其在涉及非线性关系的上下文中灰心。但是，最近的研究提出了这一观念的挑战，表明不应将皮尔逊的R排除在测量非线性单调关系的先验中。皮尔逊（Pearson）的R本质上是一种缩放的协方差，植根于著名的Cauchy-Schwarz不平等。我们的发现表明，不同的缩放界限产生不同捕获范围的系数，有趣的是，更紧密的边界实际上扩大了这些范围。我们得出比Cauchy-Schwarz的不等式更严格的不等式，利用它来完善Pearson的R，并提出了一个新的相关系数，即重新排列的相关性。该系数能够捕获线性和非线性的任意单调关系。它在线性场景中恢复为皮尔逊的R。模拟实验和现实生活调查表明，重排相关性在测量非线性单调依赖性方面比三个经典相关系数更为准确，并且其他最近提出的依赖性措施。
Pearson's r, the most widely-used correlation coefficient, is traditionally regarded as exclusively capturing linear dependence, leading to its discouragement in contexts involving nonlinear relationships. However, recent research challenges this notion, suggesting that Pearson's r should not be ruled out a priori for measuring nonlinear monotone relationships. Pearson's r is essentially a scaled covariance, rooted in the renowned Cauchy-Schwarz Inequality. Our findings reveal that different scaling bounds yield coefficients with different capture ranges, and interestingly, tighter bounds actually expand these ranges. We derive a tighter inequality than Cauchy-Schwarz Inequality, leverage it to refine Pearson's r, and propose a new correlation coefficient, i.e., rearrangement correlation. This coefficient is able to capture arbitrary monotone relationships, both linear and nonlinear ones. It reverts to Pearson's r in linear scenarios. Simulation experiments and real-life investigations show that the rearrangement correlation is more accurate in measuring nonlinear monotone dependence than the three classical correlation coefficients, and other recently proposed dependence measures.
