Novel Fuzzy Correlation Coefficient and Variable Selection Method for Fuzzy Regression Analysis Based on Distance Approach

Abstract

In data analysis, analyzing the relationships between the variables such as correlation analysis and regression analysis are very important. Correlation analysis and regression analysis are not only very important in analyzing the influence relationship and causal relationship of variables but also serve as the basis for statistical analysis. Furthermore, they are essential and important as basic analysis for machine learning analysis such as deep learning. This is because in analyzing the input and output in deep learning, variables with high correlation are selected first, and in analyzing the causal relationship, it is basic to first conduct basic analysis such as regression analysis. Especially, when data are observed as fuzzy data with ambiguous information, it is difficult to propose unique methods for those analyses due to its complexity. However, the application of fuzzy theory to correlation analysis for data with such ambiguous information has not been an effective study, and several studies have been conducted in cases where the data is not general fuzzy data or interval estimation. As a result, the effectiveness of the fuzzy theory was not highlighted. In particular, the variable selection method for selecting important variables in multiple regression analysis is a very important and essential process in regression analysis. A variable that is significant in simple regression analysis may not be significant in multiple regression analysis due to its relationship with other variables. Therefore, not all variables that affect the dependent variable can be used as independent variables in multiple regression analysis. Therefore, multiple regression analysis goes through the process of excluding some variables. But until now, the process of fuzzy multiple regression analysis has not been applied without a variable selection method and the significance of important variables has not been emphasized that much. In this paper, a fuzzy correlation coefficient and multiple fuzzy regression analysis using variable section method are proposed. For this, first defuzzification and fuzzy ordering are defined. And then fuzzy correlation coefficient is proposed using ${\varvec{L}}_{2}$ distance. Next, fuzzy sum of squares are defined for F-statistics to test the significance of the regression model. Using this F-statistics, fuzzy R², and fuzzy RMSE, several variable selection methods are proposed based on distance approach. For the data analysis, foreign exchange reserve data and house price of South Korea have been applied which are important indicators for economic crisis. The financial data is mostly recorded as closing values, but the closing values cannot be the representative of the given period of time. Therefore, we can deal with the financial data as fuzzy data which have some fluctuation that can be considered as vagueness that the data originally include. We have used foreign exchange reserve data and house price data with several financial variables. And the proposed fuzzy correlation coefficient and variable selection for fuzzy regression analysis are applied to these financial data.