# Hybrid Fuzzy Regression Analysis Using the \(F\)-Transform
## 4 Numerical Examples and Comparison Studies
\begin{array}{ll}g_{12}=68.95679+8.57909l_{X_{1}}+11.23412l_{X_{2}} \\ g_{13}=-54.63401+7.71719l_{X_{1}}+9.92721l_{X_{2}}\\ g_{14}=-68.25801+8.59155l_{X_{1}}+11.12702l_{X_{2}}\\ g_{15}=-64.67950+8.022077l_{X_{1}}+11.13042l_{X_{2}}\end{array} \right.\]  
where \(l_{X_{1}}=S_{I_{X_{1}}}-m_{X_{1}}\) and \(l_{X_{2}}=S_{I_{X_{2}}}-m_{X_{2}}\) are not spreads but endpoints of the triangular fuzzy set. Using the Fisher's discriminant scores obtained by the above the Fisher linear discriminant functions, \(g_{1}\) is predicted as \(\hat{g_{1}}\). By similar method, the second one is represented by \(g_{2}\) and \(g_{2}\) is predicted as \(\hat{g_{2}}\). The results are presented in Table 2. In Table 2, the symbol \(m.g_{1}\) (or \(m.g_{2}\)) represents the membership degree to the group \(g_{1}\) (or \(g_{2}\)) of \(s_{ly}\). From Table 2 and Equation (3), the left spreads are predicted. For instance, \(\hat{s}_{I_{Y_{1}}}=\frac{F^{l}_{1}.m.g_{1}+F^{l}_{2}.m.g_{2}}{m.g_{1}+m.g_{2}} =\frac{0.42742.075+0.25.025}{0.75+0.25}=0.38306\). By same method, the right spreads are also predicted and the predicted spreads are presented in Table 3. Table 4 shows the final values estimated by the proposed hybrid algorithm. The symbols \(l_{y}\) and \(r_{y}\) denote the left endpoint and the right endpoint of the dependent variable, respectively. Table 6 shows data on the satisfaction study, and the data \((m,s)_{T}\) in Table 6 is represented by a symmetric triangular fuzzy number with central \(m\) with width \(s\). The number of data on the life satisfaction in Table 6 is 106 but the size of the set of the spread \(S_{s}=\{0,0.5,1,1.5,2.5,3.5,4,4.5,5\}\) is only nine. This means that the same value is expressed repeatedly. In this case, it may be more efficient to classify the spread of a dependent variable using discriminant analysis. The fuzzy partition of \(S_{s}\) for discriminant analysis can be defined as follows:  
\[S_{k}=\left\{ \begin{array}{ll}(0,0,1.25)_{T}&k=1,\\ ((k-1)1.25,1.25,1.25)_{T}&2\leq k\leq 5,\\ (6.25,1.25,0)_{T}&k=6.\end{array} \right.\]  
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:  
\[\left\{ \begin{array}{ll}g_{12}=-166.828+1.12827l_{X_{1}}+1.69757l_{X_{1}}+0. 63796l_{X_{1}}+0.40676l_{X_{1}}\\ g_{13}=-160.58558+1.08448l_{X_{1}}+1.69049l_{X_{1}}+0.662188l_{X_{1}}+0.40055l_{ X_{1}}\\ g_{14}=-163.4525+1.10958l_{X_{1}}+1.67704l_{X_{1}}+0.59332l_{X_{1}}+0.44547l_{X_{1}} \\ g_{15}=-165.96681+1.09384l_{X_{1}}+1.69994l_{X_{1}}+0.62236l_{X_{1}}+0.45089l_{ X_{1}}\end{array} \right.\] \begin{table}
\begin{tabular}{c c c c c} \hline
**obs** & \(s_{ly}\) & \(\hat{s}_{ly}\) & \(s_{ry}\) & \(\hat{s}_{ry}\) \\ \hline  
\hline \end{tabular}
\end{table} 
Table 3: The predicted spreads based on the inverse \(F\)-transform. where \(l_{X_{i}}=5_{I_{X_{i}}}-m_{X_{i}}(i=1,2,3,4)\) are not spreads but endpoints of the triangular fuzzy set. And the result of LAD estimation for the centers \(\{(m_{X_{i1}},\cdots,m_{X_{i4}},m_{Y_{i}}):i=1,\cdots,106\}\) of dependent and independent variables is as follows:  
\[m_{\hat{Y}_{i}}=0.0039+0.2005m_{X_{i1}}+0.2501m_{X_{i2}}+0.3500m_{X_{i3}}+0.1992m _{X_{i4}}\]  
The life satisfaction estimated using fuzzy partition, discriminant analysis, and LAD estimation is given in Table 6. The results of the performance measure for life satisfaction \(Y_{i}\) is presented in Table 7.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{5}{c}{**Level of Satisfaction**} \\ \cline{2-7}  & \(\mathbf{X_{i1}}\) & \(\mathbf{X_  
Examples 1 and 2 show that the proposed hybrid estimation algorithm may be more efficient if the spread of a given dependent variable is expressed as a repeated number. That is, when the number of spreads of the dependent variable is smaller than the sample size, the proposed hybrid estimation algorithm is effective.