# Hybrid Fuzzy Regression Analysis Using the \(F\)-Transform
## 3 Fuzzy Regression Based on the \(F\)-Transform and Discriminant Analysis  
General fuzzy linear regression model proposed by [PERSON] [3] and applied in many fields is formulated as follows:  
\[Y(\mathbf{X}_{i})=F(\mathbf{A},\mathbf{X_{i}})+E_{i},\quad i=1\cdots,n\]  
where \(\mathbf{X}_{i}=(X_{ij})_{(1\times(p+1))}\) is the fuzzy input, \(\mathbf{A}=(A_{j})_{(1\times(p+1))}\) is the fuzzy coefficients, \(F(\mathbf{A},\mathbf{X_{i}})\) is the known response function, \(Y(\mathbf{X}_{i})\) is the fuzzy output and \(E_{i}\) is the fuzzy error. The \(\alpha\)-level set of the general fuzzy regression model is given as follows:  
\[Y(\mathbf{X_{i}})_{\alpha}=F(\mathbf{A_{\alpha}},\mathbf{X_{i}}_{\alpha})+E_{ i_{a}}\ \ (*).\]  
The center of the above model(*) is represented as follows:  
\[m_{y_{i}}=f(a_{0},a_{1},\cdots,a_{p},x_{i0},x_{i1}\cdots x_{ip})+e_{i}.\]  
And the models for the left and right spread of the model (*) are as follows:  
\[(s_{I_{Y_{i}}})_{\alpha}=f_{l}((s_{I_{A}})_{\alpha},(s_{I_{X_{i}}})_{\alpha}) +(s_{I_{E_{i}}})_{\alpha}\ \text{and}\ (s_{r_{Y_{i}}})_{\alpha}=f_{r}((s_{r_{A}})_{\alpha},(s_{r_{X_{i}}})_{\alpha}) +(s_{r_{E_{i}}})_{\alpha}.\]  
In particular, if the response function is linear, the fuzzy regression model is formulated as follows:  
\[Y_{i}=A_{0}+A_{1}X_{i1}+\cdots+A_{p}X_{ip}+E_{i},\quad i=1\cdots,n\]  
where \(Y_{i}=(m_{Y_{i}},s_{I_{Y_{i}}},s_{r_{Y_{i}}})_{T}\), \(A_{i}=(m_{A_{i}},s_{I_{A_{i}}},s_{r_{A_{i}}})_{T}\) and \(X_{ik}=(m_{X_{ik}},s_{I_{X_{ik}}},s_{r_{X_{ik}}})_{T}\) are fuzzy numbers. We estimate the \(\alpha\)-level set of the proposed fuzzy regression model using by the least squares method which minimizes the sum of squared residuals  
\[\sum_{i=1}^{n}[(s_{I_{Y_{i}}})_{\alpha^{*}}-f_{l}((s_{I_{A}})_{\alpha^{*}},(s _{I_{X_{i}}})_{\alpha^{*}})]^{2}.\]  
In fuzzy regression analysis, many researchers have used the above least squares method. However, there are two parts to consider in a general fuzzy regression model. The first is that the response function for the left spread \(f_{l}(\cdot)\) and the response function for the right spread \(f_{r}(\cdot)\) are not the same. [PERSON] and [PERSON] [6] proposed a general fuzzy regression model whose response function to the center is exponential and the response function to the spread is linear. The second is that the size of spread or center set of the dependent variable may be very small than the number of samples. In fuzzy analysis, if the response functions of the spread and center are not match, it may be more efficient to construct the fuzzy regression model by independently estimating the spread and center instead of applying the equivalent function. In addition, if the spreads of dependent and independent variables are expressed in several repetitive numbers as shown in Table 1 presented by [PERSON] [8], it may be more efficient to use statistical methods for categorical data than statistical methods for continuous variables. In Table 1 presented by [PERSON] [8], the number of samples is 30 but the spread of the response variable is expressed in only 4 values. In general, the spreads of the fuzzy data in a fuzzy sample vary greatly. However, the spreads of fuzzy data obtained from quality of products, or the preference or surveys, consists of repeating numbers. If the spread is repeated by the same number and the center changes, it may be more efficient to estimate the fuzzy regression model by estimating the spread and center separately. Especially for fuzzy regression models, it may be more efficient to estimate the spread using categorical data analysis methods when the number of spreads of the dependent variable is small due to iterations. Discriminant analysis [23,24] is one of the techniques that are used to predict the probability of belonging to a given category based on one or multiple independent variables when the dependent variable is categorical and the independent variable is interval in nature. The categorical variable means that the dependent variable is divided into several categories.