# Prediction of MAG Welding Formation Based on Fuzzy Interpolation Reasoning
## 3 The Fuzzy Interpolation Methods
### Single Anteced with Triangular Fuzzy Sets  
Suppose that \(A_{1}\Rightarrow B_{1}\) and \(A_{2}\Rightarrow B_{2}\) are two adjacent fuzzy rules on universe \(X\) and \(Y\). Assume that the observation value is \(A^{*}=(a_{1}^{*},a_{2}^{*},a_{3}^{*})\), and \(A_{1}<A^{*}<A_{2}\). Rule 1: If \(x_{1}\) is \(A_{1}\), then \(y\) is \(B_{1}\)  
Rule 2: If \(x_{1}\) is \(A_{2}\), then \(y\) is \(B_{2}\)Observation: If X is \(A^{*}\)  
where \(A_{i}=(a_{i1},a_{i2},a_{i3})\), \(B_{i}=(b_{i1},b_{i2},b_{i3})\), \(i=1,2\). The proposed fuzzy interpolation reasoning method is presented as follows. (Step 1) Construct a new intermediate fuzzy set \(A^{\prime}\) by the distance factor \(\lambda_{D}\) and \(0\leq\lambda_{D}\leq 1\). The distance \(D(A_{1},A^{*})\) and \(D(A_{1},A_{2})\) between \(A_{1}\) and \(A^{*}\), \(A_{1}\) and \(A_{2}\) can be calculated as follows:  
\[D(A_{1},A^{*})=\left[\frac{1}{3}{\sum_{i=1}^{3}{(a_{i}^{*}-a_{1i})^{2}}}\right] ^{1/2} \tag{23}\]  
\[D(A_{1},A_{2})=\left[\frac{1}{3}{\sum_{i=1}^{3}{(a_{2i}-a_{1i})^{2}}}\right] ^{1/2} \tag{24}\]  
then, the distance factor \(\lambda_{D}\) can be calculated as follows:  
\[\lambda_{D}=\frac{D(A_{1},A^{*})}{D(A_{1},A_{2})} \tag{25}\]  
Based on the value of \(\lambda_{D}\), each characteristic value of \(A^{\prime}\) can be calculated as follows:  
\[a_{1}^{\prime}=(1-\lambda_{D})a_{11}+\lambda_{D}a_{21} \tag{26}\]  
\[a_{2}^{\prime}=(1-\lambda_{D})a_{12}+\lambda_{D}a_{22} \tag{27}\]  
\[a_{3}^{\prime}=(1-\lambda_{D})a_{13}+\lambda_{D}a_{23} \tag{28}\]  
The intermediate fuzzy set \(A^{\prime}\) and observation \(A^{*}\) have the same distance from \(A_{1}\). \[D(A_{1},A^{*})=D(A_{1},A^{\prime}) \tag{29}\]  
The consequent part \(B^{\prime}\) corresponding to the antecedent of the intermediate rule \(A^{\prime}\) can be constructed in the same way:  
\[B^{\prime}=(1-\lambda_{D})B_{1}+\lambda_{D}B_{2} \tag{30}\]  
(Step 2) Find the width similarity of observed value \(A^{*}\) and the intermediate \(A^{\prime}\). Suppose there is a certain degree of similarity between the antecedent of observed value \(A^{*}\) and the intermediate \(A^{\prime}\), in which case it can be intuitively believed that the same degree of similarity exists between the consequent part \(B^{*}\) and \(B^{\prime}\). Therefore, the membership function can be divided into left and right parts. According to the observation value \(A^{*}\) and intermediate fuzzy set \(A^{\prime}\) shown in Figure 2, the left and right similarity ratio \(S^{L}(A^{*},A^{\prime})\) and \(S^{R}(A^{*},A^{\prime})\) of antecedent \(A^{*}\) and \(A^{\prime}\) can be calculated by the following formula:  
\[S^{L}(A^{*},A^{\prime})=\frac{a_{2}^{*}-a_{1}^{*}}{a_{2}^{\prime}-a_{1}^{ \prime}} \tag{31}\]  
\[S^{R}(A^{*},A^{\prime})=\frac{a_{3}^{*}-a_{2}^{*}}{a_{3}^{*}-a_{2}^{\prime}} \tag{32}\](Step 3) Find the parameter values of \(B^{*}\) by the distance and width similarity. According to step 1, it can be seen that the intermediate fuzzy set \(A^{*}\) and observation \(A^{\prime}\) have the same distance from \(A_{1}\). Then, in the same way, it can be judged that \(B^{*}\) and \(B^{\prime}\) have the same distance from \(B_{1}\), and according to the width similarity, the following equation can be obtained:  
\[\left\{ \begin{array}{c}D(B^{*},B_{1})=D(B^{\prime},B_{1})\\ S^{L}(A^{*},A^{\prime})=S^{L}(B^{*},B^{\prime})\\ S^{R}(A^{*},A^{\prime})=S^{R}(B^{*},B^{\prime})\end{array} \right. \tag{33}\]  
By substituting (30) and (31) into Equation (32), the following formula can be obtained:  
\[\left\{ \begin{array}{c}(b_{1}^{*}-b_{11})^{2}+(b_{2}^{*}-b_{12})^{2}+(b_{3}^ {*}-b_{13})^{2}=\\ (b_{1}^{\prime}-b_{11})^{2}+(b_{2}^{\prime}-b_{12})^{2}+(b_{3}^{\prime}-b_{13 })^{2}\\ b_{2}^{*}=b_{1}^{*}+S^{L}(A^{*},A^{\prime})\cdot(b_{2}^{\prime}-b_{1}^{\prime })\\ b_{3}^{*}=b_{2}^{*}+S^{R}(A^{*},A^{\prime})\cdot(b_{3}^{\prime}-b_{2}^{\prime })\end{array} \right. \tag{34}\]  
According to Equation (33), the reasoning consequence \(B_{1}^{*}\) and \(B_{2}^{*}\) can be obtained.