相关系数研究综述

doi:10.3969/j.issn.1007-7162.2012.03.002

广东工业大学学报 ›› 2012, Vol. 29 ›› Issue (3): 12-17.doi: 10.3969/j.issn.1007-7162.2012.03.002

A Review on Correlation Coefficients

Xu Wei-chao

School of Automation, Guangdong University of Technology, Guangzhou 510006, China

Received:2012-06-03 Online:2012-09-20 Published:2012-09-20

摘要/Abstract

摘要： 相关系数是表征两个随机变量之间统计关系强弱的统计量,在几乎所有科学与技术领域都获得了广泛应用.本文以二元高斯分布为基本模型, 对文献中常见的5种相关系数的统计特性作相对全面的回顾与总结,并探讨了其适用的场合.具体结论如下:（1）当样本满足二元高斯分布时, 皮尔逊积距相关系数是最佳选择;（2）当样本中存在轻微的单调非线性畸变时, 序统计量相关系数比较适用;（3）当样本中存在严重的单调非线性畸变时, 斯皮尔曼秩次相关系数或肯德尔秩次相关系数是合适的选择;（4）当只有一路信号中存在单调非线性畸变时,基尼相关是最佳选择;（5）当样本中存在脉冲干扰时, 斯皮尔曼秩次相关系数或肯德尔秩次相关系数是合适的选择.

Abstract: As statistics characterizing the strength of statistical relationship between two random variables,correlation coefficients have found wide applications in nearly all science and technology fields.This paper attempts to provide a detailed review on 5 commonly used correlation coeffcients in the literature, including a relatively detailed discussion on their statistical properties under the fundamental bivariate normal model and their suitable application scenarios. Specifically,(1) if the sample follows normal distribution, then the Pearsons Product Moment Correlation Coefficient is optimal;(2) if the monotone nonlinear distortion in the sample is small, then the Order Statistics Correlation Coefficient should be used;(3) if the monotone nonlinear distortion in the sample is severe, then the Spearmans rho or Kendalls tau are feasible;(4) if only one channel is distorted by monotone nonlinearity, then the Gini Correlation is best choice; and (5) if there exsits impulsive noise in the sample, then the Spearmans rho or Kendalls tau should be employed.

Key words: Pearson‘s Product Moment Correlation Coefficient (PPMCC); Spearman’s Rho (SR); Kendall‘s Tau (KT);Order Statistics Correlation Coefficient (OSCC); Gini Correlation (GC); Bivariate No

徐维超. 相关系数研究综述[J]. 广东工业大学学报, 2012, 29(3): 12-17.

Xu Wei-chao. A Review on Correlation Coefficients[J]. Journal of Guangdong University of Technology, 2012, 29(3): 12-17.

参考文献

［1］ Gibbons J D, Chakraborti S. Nonparametric Statistical Inference［M］. 3rd. New York: M Dekker, 1992.

［2］ Fisher R A. Statistical Methods, Experimental Design,and Scientific Inference［M］. New York: Oxford UniversityPress, 1990.

［3］ Fisher R A. On the ‘probable error’ of a coefficient of correlation deduced from a small sample［J］. Metron, 1921,1:332.

［4］ Fieller E C, Hartley H O, Pearson E S. Tests for rank correlation coefficients［J］. Biometrika, 1957,44(3/4):470481.

［5］ Kendall M, Gibbons J D. Rank Correlation Methods［M］.5th. New York: Oxford University Press, 1990.

［6］ Fisher R. Frequency distribution of the values of the correlation coefficient in samples from an indefinitely large population［J］. Biometrika, 1915, 10(4):507521.

［7］ Tumanski S. Principles of Electrical Measurement［M］.New York: Taylor & Francis, 2006.

［8］ Stein D. Detection of random signals in Gaussian mixture noise［J］. IEEE Trans Inf Theory, 1995,41(6):17881801.

［9］ Chen R, Wang X, Liu J. Adaptive joint detection and decoding in flatfading channels via mixture Kalman filtering［J］. IEEE Trans Inf Theory, 2000, 46(6):20792094.

［10］〖ZK（#〗Reznic Z, Zamir R, Feder M. Joint sourcechannel coding of a Gaussian mixture source over the Gaussian broadcast channel［J］. IEEE Trans Inf Theory, 2002,48(3):776781.

［11］ Shevlyakov G L, Vilchevski N O. Robustness in Data Analysis: criteria and methods［M］. Modern probability and statistics. Utrecht: VSP, 2002.

［12］ Schechtman E, Yitzhaki S. A measure of association base on Ginis mean difference［J］. Commun StatistTheor Meth, 1987,16(1):207231.

［13］ Xu W, Chang C, Hung Y S, et al. Order Statistics Correlation Coefficient as a Novel Association Measurement With Applications to Biosignal Analysis［J］. IEEE Trans Signal Process, 2007, 55(12):55525563.

［14］ Xu W, Chang C, Hung Y S, et al. Asymptotic Properties of Order Statistics Correlation Coefficient in the Normal Cases［J］. IEEE Trans Signal Process, 2008,56(6):22392248.

［15］ Xu W, Hung Y S, Niranjan M, et al. Asymptotic Mean and Variance of Gini Correlation for Bivariate Normal Samples［J］. IEEE Trans Signal Process, 2010,58(2):522534.

［16］ Mari D D, Kotz S. Correlation and Dependence［M］. London:Imperial College Press, 2001.

［17］ Hoeffding W. A Class of Statistics with Asymptotically Normal Distribution［J］. Ann Math Stat, 1948,19(3):293325.

［18］ Daniels H E. The Relation Between Measures of Correlation in the Universe of Sample Permutations［J］.Biometrika,1944, 33(2):129135.

［19］ Hotelling H. New light on the correlation coefficient and its transforms［J］. Journal of the Royal Statistical Society, Series B(Methodological), 1953, 15(2):193232.

［20］ Moran P A P. Rank Correlation and ProductMoment Correlation［J］. Biometrika, 1948, 35(1/2):203206.

［21］ Esscher F. On a method of determining correlation from the ranks of the variates［J］. Skand Aktuar, 1924,7:201219.

［22］ David F N, Mallows C L. The variance of Spearmans rho in normal samples［J］. Biometrika, 1961, 48(1/2):1928.

［23］ Stuart A, Ord J K. Kendalls Advanced Theory of Statistics: Volume 2 Classical Inference and Relationship［M］.5th. London: Edward Arnold, 1991.

［24］ Xu W, Hou Y, Hung Y, et al. Comparison of Spearmans rho and Kendalls tau in Normal and Contaminated Normal Models［DB/OL］. (2010-12-31). http://arxiv.org/abs/1011.2009/.

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