## 2 Materials and Methods
### Fuzzy Sets and Similarity Measures
#### 2.2.1 New Similarity Measure for Fuzzy Sets  
Let \(X=\{x_{1},x_{2},\ldots,x_{n}\}\) be a universal set. In this section, for all \(A,B\in F(X)\), the similarity measure in Equation (2) can be considered. **Definition 3**. : _For all \(A\in F(X)\), the real function \(S_{H}:F(X)\times F(X)\rightarrow[0,1]\) is defined as follows:_  
\[S_{H}(A,B)=\frac{1}{n}\sum_{i=1}^{n}\frac{2-2|\mu  
[PERSON] and [PERSON] proposed the following in 1993 [14]:  
\[S_{PK1}(A,B)=1-\underset{x_{i}\in X}{\max}\{|\mu_{A}(x_{i})-\mu_{B}(x_{i})|\}, \tag{4}\]  
\[S_{PK2}(A,B)=1-\frac{\sum\limits_{i=1}^{n}\lvert\mu_{A}(x_{i})-\mu_{B}(x_{i}) \rvert}{\sum\limits_{i=1}^{n}\lvert\mu_{A}(x_{i})+\mu_{B}(x_{i})\rvert}, \tag{5}\]  
\[S_{PK3}(A,B)=1-\sum\limits_{i=1}^{n}\frac{\lvert\mu_{A}(x_{i})-\mu_{B}(x_{i}) \rvert}{n} \tag{6}\]  
\[S_{PK4}(A,B)=\frac{\underset{x_{i}\in X}{\max}(\mu_{A}(x_{i})\times\mu_{B}(x_{ i}))}{\max\left(\sum\limits_{i=1}^{n}(\mu_{A}(x_{i}))^{2},\sum\limits_{i=1}^{n}( \mu_{B}(x_{i}))^{2}\right)}, \tag{7}\]  
\[S_{PK5}(A,B)=\frac{\sum\limits_{i=1}^{n}\min(\mu_{A}(x_{i}),\mu_{B}(x_{i}))}{ \sum\limits_{i=1}^{n}\max(\mu_{A}(x_{i}),\mu_{B}(x_{i}))}. \tag{8}\]  
[PERSON] et al. proposed the following in 2002 [15]:  
\[S_{RP1}(A,B)=1-\frac{\underset{x_{i}\in X}{\max}\min(\mu_{A}(x_{i}),\mu_{B}(x_{ i}))}{\underset{x_{i}\in X}{\max}\max(\mu_{A}(x_{i}),\mu_{B}(x_{i}))}, \tag{9}\]  
\[S_{RP2}(A,B)=\min\{\sqrt{\frac{\sum\limits_{i=1}^{n}\mu_{A}(x_{i})\mu_{B}(x_{ i})}{\max}(\mu_{A}(x_{i}),\mu_{B}(x_{i}))^{2}},\sqrt{\frac{\sum\limits_{i=1}^{n}(1- \mu_{A}(x_{i}))(1-\mu_{B}(x_{i}))}{\max}}\}. \tag{10}\]  
[PERSON] proposed the following in 1997 [16]:  
\[S_{\text{W1}}(A,B)=\sum\limits_{i=1}^{n}\frac{\min(\mu_{A}(x_{i}),\mu_{B}(x_{ i}))}{\max(\mu_{A}(x_{i}),\mu_{B}(x_{i}))}, \tag{11}\]  
\[S_{\text{W2}}(A,B)=\frac{\sum\limits_{i=1}^{n}\left[\mu_{A}(x_{i})\mu_{B}(x_{ i})+(1-\mu_{A}(x_{i}))(1-\mu_{B}(x_{i}))\right]}{\sqrt{\sum\limits_{i=1}^{n} \left[(\mu_{A}(x_{i}))^{2}+(1-\mu_{A}(x_{i}))^{2}\right]}\sqrt{\sum\limits_{i=1 }^{n}\left[(\mu_{B}(x_{i}))^{2}+(1-\mu_{B}(x_{i}))^{2}\right]}} \tag{12}\]  
[PERSON] et al. proposed the following in 1994 [17]:  
\[S_{LS}(A,B)=\underset{x_{i}\in X}{\max}\min(\mu_{A}(x_{i}),\mu_{B}(x_{i})). \tag{13}\]  
[PERSON] et al. proposed the following in 2014 [18]:  
\[S_{PK2}(A,B)=1-\frac{\sum\limits_{i=1}^{n}\lvert\mu_{A}(x_{i})-\mu_{B}(x_{i}) \rvert}{\sum\limits_{i=1}^{n}\lvert\mu_{A}(x_{i})+\mu_{B}(x_{i})\rvert}, \tag{5}\]0\[S_{PK2}(A,B)=1-\frac{\sum\limits_{i=1}^{n}\lvert\mu_{A}(x_{i})-\mu_{B}(x_{i}) \rvert}{\sum\limits_{i=1}^{n}\lvert\mu_{A}(x_{i})+\mu_{B}(x_{i})\rvert}, \tag{5}\]1  
\[S_{PK2}(A,B)=1-\frac{\sum\limits_{i=1}^{n}\lvert\mu_{A}(x_{i})-\mu_{B}(x_{i}) \rvert}{\sum\limits_{i=1}^{n}\lvert\mu_{A}(x_{i})+\mu_{B}(x_{i})\rvert}, \tag{5}\]2  
\[S_{PK2}(A,B)=1-\frac{\sum\limits_{i=1}^{n}\lvert\mu_{A}(x_{i})-\mu_{B}(x_{i}) \rvert}{\sum\limits_{i=1}^{n}\lvert\mu_{A}(x_{i})+\mu_{B}(x_{i})\rvert}, \tag{5}\]3  
In pattern recognition problems, the similarity measures are used to classify the sample \(B\) into a class of given patterns \(A=\{A_{1},A_{2},\ldots,A_{m}\}\). Given a set of m patterns \(A=\{A_{1},A_{2},\ldots,A_{m}\}\), where \(A_{i}\) is a fuzzy set on the universal set \(X\), a new sample \(B\) also belongs to a fuzzy set on the universal set \(X\). **Question:** What class does \(B\) belong to? To solve this problem, it is possible to use the similarity measures according to the two steps in Algorithm 1. ```
Step 1. Calculate all similarity measures \(s(B,A_{i})\) from sample \(B\) to patterns \(A_{i}\) for all \(i=1,2,\ldots,m\). Step 2. Classify sample \(B\) into pattern \(A_{i}\) with \(s(A_{i},B)=\max\limits_{i=1,2,\ldots,m}\{s(A_{i},B)\}\). ```  
**Algorithm 1** Solving pattern recognition problems using similarity measures. Below, we consider two examples to compare the similarity measures. **Example 1**.