# Hybrid Fuzzy Regression Analysis Using the \(F\)-Transform
## 1 Introduction  
Regression analysis is one of the most useful and popular sub-parts of supervised learning, and success stories have been reported in many areas. Regression analysis assumes that the relevant experimental data are accurate, and numerous statistical treatments have been proposed to model accurately defined data [1]. However, there are cases in which this assumption is unrealistic, because data used in regression analysis consist of observations that might be imprecise such as \"about 50 years old\" or \"almost fully cured\". In regression analysis, we sometimes encounter imprecise data with non-sharp boundary, linguistic data, and incomplete information. The fuzzy set theory introduced by [PERSON] [2] has been used as a means to solve this problem. The fuzzy regression model is needed to estimate the statistical relationship among variables when independent variables, dependent variables or regression coefficients are expressed as a fuzzy set. Since the fuzzy regression model introduced by [PERSON] [3], numerous modified or extended fuzzy regression models have been proposed, and many successful applications have been presented [2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15]. One goal of the fuzzy regression analysis is to discover statistical relationships between given fuzzy values. The general approach, as discussed by many authors, is to attempt to induce their relationship from center and spread of fuzzy values using the same function. In the fuzzy regression model, the response functions for center and spread may not be the same. [PERSON] and [PERSON] [6] described a fuzzy regression model whose response function for center and spread is an exponential function and a linear function, respectively. In the fuzzy regression model, the response functions for center and spread may not be the same. [PERSON] and [PERSON] [6] described a fuzzy regression model whose response function for center and spread is an exponential function and a linear function, respectively. If the response function for center and spread does not match, it may be more effective to assume the response function for center and spread in an independent form, and to estimate the assumed response function. In addition, in fuzzy regression models, the center or spread of observation of the response function is often obtained in repeated numbers. If the results of the survey or the quality of the product is expressed as fuzzy numbers, the spread of the response function may repeat a finite value as examples presented in [PERSON] [8] and [PERSON] et al. [14; 15]. In this case, it may be more efficient to use regression models for categorical data instead of continuous data. In the ordinary regression analysis, the categorical regression analysis, logistic regression analysis or discriminant analysis is used to analyze the categorical response variable. If the spreads of the fuzzy data are repeated by the same number and the center of the fuzzy data varies, it may be more efficient to estimate the fuzzy regression model by independently estimating the spread and center instead of applying the equivalent function. This is our motivation. The proposed hybrid estimation algorithm is a modified and extended new version of the algorithm introduced in [PERSON] et al. [16]. We propose an algorithm that constructs a fuzzy regression model when the response function for the spread and the center of the dependent variable do not match. To estimate a regression model for the center of the dependent variable with continuous data, the least absolute deviations (LAD) estimation method is used that is not sensitive to an outlier. In addition, a discriminant analysis is used to estimate the response function for the spread of the dependent variable expressed as repeated numbers by finite number. The \(F\)-transform introduced by [PERSON] [17] has been studied and found useful in many applications [18; 19; 20; 21; 22].