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| Format: | Preprint |
| Veröffentlicht: |
2022
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| Online-Zugang: | https://arxiv.org/abs/2205.04571 |
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| _version_ | 1866914970724204544 |
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| author | Ai, Xinbo |
| author_facet | Ai, Xinbo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_04571 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Adjust Pearson's $r$ to Measure Arbitrary Monotone Dependence Ai, Xinbo Other Statistics 62H20 G.3 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. |
| title | Adjust Pearson's $r$ to Measure Arbitrary Monotone Dependence |
| topic | Other Statistics 62H20 G.3 |
| url | https://arxiv.org/abs/2205.04571 |