Study on Linear Correlation Coefficient and Nonlinear Correlation Coefficient in Mathematical Statistics

Abstract

In the two-dimensional or multidimensional experimental data in the traditional statistics, there is usually a linear relationship, or a similar linear relationship between independent variables and the dependent variable. Commonly the linear correlation coefficient is used to measure the degree of linear between the independent variables and the dependent variable. However, for the two-dimensional or multidimensional experimental data, there may be a simple linear relationship between independent variables and the dependent variable, or a simple non-linear relationship, or both linear and non-linear relationship. Article ^[1] found that the traditional correlation coefficient (linear correlation coefficient) r is only suitable for simple linear relationship. On the basis of article ^[1], this article discusses the linear correlation coefficient r, analyzes nonlinear correlation coefficient r_nl, and gives a new definition of the correlation coefficient R. The new correlation coefficient R can not only describe the case of a simple linear relationship, but also describe the case of a simple nonlinear relationship and the case of both simple linear relationship and nonlinear relationship. That is to say, the new correlation coefficient R can describe the internal law of any experimental data.