# Hybrid Fuzzy Regression Analysis Using the \(F\)-Transform
## 4 Numerical Examples and Comparison Studies  
In this section, we illustrate two examples to compare the performance of the proposed method and the existing methods which assume the same model for center and spread. [PERSON] and [PERSON] estimated the fuzzy regression model by minimizing the sum of the square of the left and right endpoints of the \(\alpha\)-level of residuals [4]. [PERSON] constructed the fuzzy regression model by minimizing the sum of the square of residuals of the end points of support and the center [7]. Two measures are used to compare the performance of the estimated fuzzy regression model. One is the \(D(Y,\hat{Y})\) based on the difference between the estimated value and observed value, and the other is \(I(Y,\hat{Y})\) comparing the overlapped area. The performance measure \(D(Y,\hat{Y})\) comparing the difference between the predicted value and observed value and is given as follows [5]. \[D(Y,\hat{Y})=\sum_{i=1}^{n}m_{d}(Y_{i},\hat{Y}_{i}),\]  
where  
\[m_{d}(Y_{i},\hat{Y}_{i})=\frac{\int_{-\infty}^{\infty}\Big{|}\mu_{Y_{i}}(x)- \mu_{\hat{Y}_{i}}(x)\Big{|}dx}{\int_{-\infty}^{\infty}\big{|}\mu_{Y_{i}}(x) \big{|}dx}+h_{d}(Y_{i}(0),\hat{Y}_{i}(0))\]  
and \(h_{d}(Y_{i}(0),\hat{Y}_{i}(0))=\inf\{\inf\{|a-b|:a\in Y_{i}(0)\}:b\in\hat{Y}_{i }(0)\}\). The more efficient model has \(m_{d}\) value closer to zero. The performance measure \(I(Y,\hat{Y})\) comparing the overlapped area between the predicted value and observed value is given as follows [4]. \[I(Y,\hat{Y})=\sum_{i=1}^{n}c(Y_{i},\hat{Y}_{i}),\]  
where  
\[c(Y_{i},\hat{Y}_{i})=\frac{\int_{-\infty}^{\infty}Min\{\mu_{Y_{i}}(x),\mu_{ \hat{Y}_{i}}(x)\}dx}{\int_{-\infty}^{\infty}\mu_{Y_{i}}(x)dx+\int_{-\infty}^{ \infty}\mu_{\hat{Y}_{i}}(x)dx-\int_{-\infty}^{\infty}Min\{\mu_{Y_{i}}(x),\mu_{ \hat{Y}_{i}}(x)\}dx}\]  
and \(Min\) means the minimum value. The more efficient method has the smaller value of \(D(Y,\hat{Y})\) and the larger value of \(I(Y,\hat{Y})\). To show the efficiency of the proposed estimation method, we use examples used by [PERSON] [8] and [PERSON] et al. [14]. To compare the performance of the proposed algorithm with the existing methods, the [PERSON] and Taheri method [4] and the Diamond method [7], which are based on least squares estimation, are used. **Example 1**. : _[PERSON] [8] and [PERSON] [4] consider a multiple fuzzy regression model. Table 1 shows the performance data for 30 quality Roman restaurants._  
1. Estimate the center of the dependent variable using the least absolute deviation method. The estimated center is given by the following equation: \[\hat{m}_{Y_{i}}=0.070486+0.418084x_{1}+0.500537x_{2}\]
2. From Table 1, we obtain \(S_{I_{Y}}=\{0,0.25,0.5,0.75\}\) which is the set of left spreads of dependent variable. Define the universe discourse \(S^{l}\) as \([max\{0,S_{I_{Y_{min}}}-c_{1}\},S_{I_{Y_{max}}}+c_{2}]=[0,0.8]\), where \(S_{I_{Y_{min}}}=0\) and \(S_{I_{Y_{max}}}=0.75\) with proper constant \(c_{1}\) and \(c_{2}\). 3. Estimate the left spread of dependent variable using the F-transform and [PERSON]'s linear discriminant analysis. The given data shows that the spreads of 30 samples are expressed in only four numbers. The basic functions of \(S^{l}\) can be given as follows: \[L_{k}=\left\{ \begin{array}{ll}(0,0,0.2)_{T}&k=1,\\ (0.2(k-1),0.2,0.2)_{T}&2\leq k\leq 4,\\ (0.8,0.2,0)_{T}&k=5.\end{array} \right.\] Using the defined fuzzy partition and Equation (2), the F-transform is obtained as follows: \[F_{k}[S_{I_{Y}}]=[F^{l}_{1},F^{l}_{2},F^{l}_{3},F^{l}_{4},F^{l}_{5}]=[0,0.25,0. 42742,0.55357,0.75].\]  
Since the number of the overlapped basic function is two, each of the data is grouped two times. The first one is represented by \(g_{1}\) and the Fisher linear discriminant functions which classify the given data using independent variables \(l_{X_{1}}\) and \(l_{X_{2}}\) are as follows:  
\[\left\{