Statistical Analysis Of Spearman’s Footrule And Gini’s Gamma

Correlation analysis started in the pioneering period of statistics and it is still an important research topic in the fields of statistics and signal processing,It has also been widely used in various fields.The correlation coefficient is an indicator of the strength of relationship between two random variables,Among the many correlation coefficients,Pearson product moment correlation coefficient,Spearman’s rho,and Kendall’s tau are perhaps the most widely used.The theoretical research shows that the Pearson product moment correlation coefficient is the optimality algorithm when dealing with linear correlations,it is simple and efficient,but when a monotonic nonlinear transformation occurs in the sample,the Pearson product moment correlation coefficient may give the wrong conclusions,and scholars have proved that it is very sensitive to impulsive noise.In the real world,even if modern electronic equipment is very precise,there will still be some noise interference.Under these conditions,Spearman’s rho and Kendall’s tau could be employed.Previous studies have shown that although Spearman’s rho and Kendall’s tau do not perform as well as Pearson product moment correlation coefficients when dealing with linear correlations,but they are more robust against impulse noise in the data.Due to the excellent work of researchers,the relevant statistical characteristics of these three classical correlation coefficients are relatively complete.Besides the above three classic correlation coefficients,Spearman’s footrule and Gini ’s gamma are two relatively unknown correlation coefficients that remained largely neglected from researchers,but in recent years,due to their natural characteristics,it is quickly gaining popularity in applications,but their theoretical research is still insufficient.In order to fill the gaps in related research and analyze the performance of these two correlation coefficients,this paper have analyzed and investigated systematica lly the properties of these two correlation coefficients.In order to investigate the properties of Spearman’s footrule and Gini’s gamma,this paper first theoretically derives the expressions of the expectations and variances of these two correlation coefficients under the Binary Gaussian Model and the Monte Carlo simulation experiments were used to verify the accuracy of the theoretical derivation results,based on these theoretical results,we give their relative efficiency to the optimality correlation coefficient under the Binary Gaussian Model which is Pearson product moment correlation coefficient,and employ that as a performance index compare with Spearman’s rho and Kendall’s tau.Because the Pearson product moment correlation coefficient is very sensitive to impulsive noise,this paper employ the Contaminated Gaussian Model to simulate the scene of impulsive noise in the sample,and theoretically derived the expectation expressions of Spearman’s rho and Gini’s gamma under this model,simulation experiments results shows the correctness of the results,and theoretical results also shows that they are robust against impulse noise.In order to further analyze the properties of each correlation coefficient and explore their applicable scenarios,this paper employ the Sensitivity Ratio as an indicator to compare the performance of several correlation coefficients.Simulation experiments results shows that the Spearman’s footrule and the Gini’s gamma correlation coefficient are performs better than the Spearman’s rho and Kendall’s tau in several occasions.Therefore,the research conclusions of this paper shows that when the sample is contaminated by outliers,that is,the sample are affected by impulse noise,besides the Spearman’s rho and Kendall’s tau,the Spearman’s footrule and Gini’s gamma are strong competitors and could also be an option.