# Prediction of MAG Welding Formation Based on Fuzzy Interpolation Reasoning
## 2 Overview of Fuzzy Interpolation
### HS Method [20]  
The HS method is a representative method based on intermediate rule-interpolation inference. The main process of the HS method is as follows: let \(A\) be a triangular fuzzy set, and \(A=(a_{1},a_{2},a_{3})\). Then the representative value \(Rep(A)\) of \(A\) can be defined by  
\[Rep(A)=\frac{a_{1}+a_{2}+a_{3}}{3} \tag{11}\]  
The first step of the HS method is to construct a new fuzzy set \(A^{\prime}\) by the distance ratio of representative value \(\lambda_{Rep}\), which has the same representative value of \(A^{*}\). Let  
\[\begin{array}{ll}\lambda_{Rep}&=\frac{d(A_{1}A^{*})}{d(A_{1},A_{2})}\\ &=\frac{d(Rep(A_{1}),Rep(A^{*}))}{d(Rep(A_{1}),Rep(A_{2}))}\end{array} \tag{12}\]  
Then, the fuzzy set \(A^{\prime}(a_{1}^{\prime},a_{2}^{\prime},a_{3}^{\prime})\) can be obtained, and the parameters of \(A^{\prime}\) are determined as follows:  
\[a_{1}^{\prime}=(1-\lambda_{Rep})a_{11}+\lambda_{Rep}a_{21} \tag{13}\]\[a_{2}^{\prime}=(1-\lambda_{Rep})a_{12}+\lambda_{Rep}a_{22} \tag{14}\]  
\[a_{3}^{\prime}=(1-\lambda_{Rep})a_{13}+\lambda_{Rep}a_{23} \tag{15}\]  
The fuzzy set \(B^{\prime}(b_{1}^{\prime},b_{2}^{\prime},b_{3}^{\prime})\) can be obtained the same way, and the parameters of \(B^{\prime}\) are determined as follows:  
\[B^{\prime}=(1-\lambda_{Rep})B_{1}+\lambda_{Rep}B_{2} \tag{16}\]  
The HS method uses the following two transformations:  
(1) Scale transformation: Scale operation from \(A^{\prime}\) to \(\hat{A}^{\prime}\) by the scale rate \(s\), where the scale rate \(s\) is calculated as follows:  
\[s=\frac{a_{3}-a_{1}}{a_{3}^{\prime}-a_{1}^{\prime}} \tag{17}\]  
Each characteristic value of \(\hat{A}^{\prime}\) can be calculated as follows:  
\[\hat{a}_{1}^{\prime}=\frac{(1+2s)a_{1}^{\prime}+(1-s)a_{2}^{\prime}+(1-s)a_{3 }^{\prime}}{3} \tag{18}\]  
\[\hat{a}_{2}^{\prime}=\frac{(1-s)a_{1}^{\prime}+(1+2s)a_{2}^{\prime}+(1-s)a_{3 }^{\prime}}{3} \tag{19}\]  
\[\hat{a}_{3}^{\prime}=\frac{(1-s)a_{1}^{\prime}+(1-s)a_{2}^{\prime}+(1+2s)a_{3 }^{\prime}}{3} \tag{20}\]  
\(B^{\prime}\) can be converted to \(\hat{B}^{\prime}\) the same way.  
(2) Move transformation: The move operation shifts the position of \(\hat{A}^{\prime}\) to become the same as that of \(A^{*}\), and the move ratio \(m\) can be calculated as follows:  
\[m=\begin{cases}\frac{3\left(a_{1}-\hat{a}_{1}^{\prime}\right)}{\hat{a}_{2}^{ \prime}-\hat{a}_{1}^{\prime}},&if\ a_{1}\geq\hat{a}_{1}^{\prime}\\ \frac{3\left(a_{1}-\hat{a}_{1}^{\prime}\right)}{\hat{a}_{3}^{\prime}-\hat{a}_{ 2}^{\prime}},&if\ a_{1}\leq\hat{a}_{1}^{\prime}\end{cases} \tag{21}\]  
Then, move transformation is applied to construct consequence \(B^{*}=(b_{1}^{*},b_{2}^{*},b_{3}^{*})\):  
\[\left\{\begin{array}{ll}b_{1}^{*}=\hat{b}_{1}^{\prime}+m\cdot\gamma\\ b_{2}^{*}=\hat{b}_{2}^{\prime}-2m\cdot\gamma&\gamma=\left\{\begin{array}{ll} \frac{\hat{b}_{2}-\hat{b}_{1}^{\prime}}{3}&if\ m\geq 0\\ \frac{b_{3}^{*}-\hat{b}_{2}^{\prime}}{3}&otherwise\end{array}\right.\end{array}\right. \tag{22}\]  
Finally, the HS method obtains the fuzzy interpolative reasoning consequence \(B^{*}\) by transforming \(B^{\prime}\) with the same scale rate \(s\) and the same moving ratio \(m\) between the fuzzy sets \(A^{*}\) and \(A^{\prime}\). For more details, refer to [20].