## 2 Materials and Methods
### Fuzzy Sets and Similarity Measures
#### 2.2.1 New Similarity Measure for Fuzzy Sets
: _Given a set of three patterns \(A=\{A_{1},A_{2},A_{3}\}\), \(A_{i}\) (\(i=1,2,3\)) is a fuzzy set on the universal set \(X=\{x_{1},x_{2},x_{3},x_{4}\}\): \(A_{1}=\{(x_{1},0.2),(x_{2},0.3),(x_{3},0.1),(x_{4},0.5)\}\), \(A_{2}=\{(x_{1},0.25),(x_{2},0.4),(x_{3},0.1),(x_{4},0.6)\}\), and \(A_{3}=\{(x_{1},0.5),(x_{2},0.3),(x_{3},0.5),(x_{4},0.4)\}\)._  
_New sample \(B=\{(x_{1},0.2),(x_{2},0.3),(x_{3},0.5),(x_{4},0.6)\}\) also belongs to a fuzzy set on the universal set \(X\)._  
**Question:** What class does \(B\) belong to? Result: Using Algorithm 1, the results shown in Table 2 can be obtained. In this case, it can be seen that our proposed similarity measure (Equation (2)), as well as 10 other similarity measures, assigned sample \(B\) to pattern \(A_{3}\). On the other hand, three measures (Equations (7), (9) and (13)) assigned sample \(B\) to pattern \(A_{2}\), whereas Equation (15) assigned sample \(B\) to pattern \(A_{1}\). Lastly, measure \(S_{PK3}\) (Equation (6)) failed to classify sample \(B\). In Example 1, we can see that several measures were feasible, with most assigning sample \(B\) to pattern \(A_{3}\). In Example 2, several measures failed to classify sample B, whereas the majority (including the proposed measure) assigned sample \(B\) to pattern \(A_{2}\). **Example 2**. : _Given a set of three patterns \(A=\{A_{1},A_{2},A_{3}\}\), \(A_{i}\) (\(i=1,2,3\)) is a fuzzy set on the universal set \(X=\{x_{1},x_{2},x_{3},x_{4}\}\): \(A_{1}=\{(x_{1},0.2),(x_{2},0.3),(x_{3},0.1),(x_{4},0.5)\}\), \(A_{2}=\{(x_{1},0.5),(x_{2},0.4),(x_{3},0.4),(x_{4},0.5)\}\), and \(A_{3}=\{(x_{1},0.5),(x_{2},0.3),(x_{3},0.5),(x_{4},0.4)\}\)._  
_New sample \(B=\{(x_{1},0.5),(x_{2},0.5),(x_{3},0.8),(x_{4},0.9)\}\) also belongs to a fuzzy set on the universal set \(X\)._  
**Question:** What class does \(B\) belong to? Result: Using Algorithm 1, the results shown in Table 3 can be obtained. In this case, it can be seen that the measures \(S_{PK4}\) (Equation (7)), \(S_{RP1}\) (Equation (9)), \(S_{LS}\) (Equation (13)), and \(S_{PM2}\) (Equation (15)) failed to classify sample \(B\). However, the proposed similarity measure (Equation (2)), as well as eleven other similarity measures, assigned sample \(B\) to pattern \(A_{2}\). Examples 1 and 2 demonstrate the feasibility of our proposed similarity measure through application to classification problems. It was able to match or surpass the performance of the other measures identified in the literature when applied to recognition problems (Examples 1 and 2). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Similarity Measures**} & \multicolumn{2}{c}{**The Similarity Measures for Classifying Sample \(B\) in Patterns \(A_{i}\)**} & \multirow{2}{*}{**Classification Results**} \\ \cline{2-2} \cline{4-4}  & \(s(A_{1},B)\) & \(s(A_{1},B)\) & \(s(A_{3},B)\) & \\ \hline Equation (3) & 0.9150 & 0.9138 & **0.9350** & \(A_{3}\) \\ Equation (4) & 0.6000 & 0.6000 & **0.7000** & \(A_{3}\) \\ Equation (5) & 0.8148 & 0.8136 & **0.8485** & \(A_{3}\) \\ Equation (6) & **0.8750** & 0.8625 & **0.8750** & Null \\ Equation (7) & 0.4054 & **0.4865** & 0.3333 & \(A_{2}\) \\ Equation (8) & 0.6875 & 0.6857 & **0.7368** & \(A_{3}\) \\ Equation (9) & 0.8333 & 1 & 0.8333 & \(A_{2}\) \\ Equation (10) & 0.6486 & 0.6967 & **0.7158** & \(A_{3}\) \\ Equation (11) & 0.7583 & 0.6875 & **0.7667** & \(A_{3}\) \\ Equation (12) & 0.9318 & 0.9285 & **0.9414** & \(A_{3}\) \\ Equation (13) & 0.5000 & **0.6000** & 0.5000 & \(A_{2}\) \\ Equation (14) & 0.6875 & 0.7500 & **0.8235** & \(A_{3}\) \\ Equation (15) & **1** & 0.8889 & 0.8235 & \(A_{1}\) \\ Equation (16) & 0.6875 & 0.7500 & **0.8750** & \(A_{3}\) \\ Equation (17) & 0.6875 & 0.7500 & **0.8235** & \(A_{3}\) \\ Equation (2) & 0.9243 & 0.9179 & **0.9281** & \(A_{3}\) \\ (proposed measure) & & & & \\ \hline \hline \end{tabular} The maximum similarity measures are indicated in bold.  
\end{table} 
Table 2: Classification results for Example 1.