## 2 Materials and Methods
### The Neuro-Fuzzy Transformation with Minimum Entropy Principle Model
#### 2.2.2 Fuzzy Membership Function with Minimum Entropy Principle  
Fuzzy logic is based on uncertainty and an unsharp boundary that can be applied in some real-world applications. A difference between Boolean logic and fuzzy logic is that Boolean logic is a set of two values, completely true or 1 and completely false or 0. On the other hand, fuzzy logic is a fuzzy set including an infinite value between partial false or 0 and partial true or 1. Values in a fuzzy set called membership values are calculated by an FMF. This research selected trapezoidal functions as the FMF. Each input feature can include one or more FMFs and the number of FMFs of each input feature is two to five functions. To find the optimal positions of the changing slope on the FMF, the minimum entropy principle (MEP) was used. This method finds the minimum value of entropy, which is an uncertainty of the data. The high entropy value means that there is a high probability that cannot divide data between classes. To find MEP, the threshold (\(x\)) in a range between \(X_{1}\) and \(X_{2}\) was calculated by Equations (1)-(3). This threshold divided data into two sides: the left side in \([X_{1},\,x]\) as side \(p\) is calculated by Equation (1) and the right side in \([x,\,X_{2}]\) as side \(q\) is calculated by Equation (2). Then, \(x\) was gradually adjusted to the values between  
Figure 2: Scatter plot with three classes of PM\({}_{10}\) in the LPA-Met station: (**a**) RH and (**b**) CO.  
and \(X_{2}\) to find the minimum entropy from Equation (3) and its value is the lowest entropy of data divided into two sides in ranges \([X_{1},x]\) and \([x,X_{2}]\)[34]. \[S_{p}(x)=-\sum_{i=1}^{2}p_{i}(x)\ln p_{i}(x) \tag{1}\]  
\[S_{q}(x)=-\sum_{i=1}^{2}q_{i}(x)\ln q_{i}(x) \tag{2}\]  
\[S(x)=p(x)S_{p}(x)+q(x)S_{q}(x) \tag{3}\]  
where \(S(x)\) denotes the entropy value of \(x\) in range \(X_{1}\) and \(X_{2}\); \(p(x)\) and \(q(x)\) denote probabilities that all samples are in range \([X_{1},x]\) and \([x,X_{2}]\), respectively; \(p(x)+q(x)=1\), \(p_{i}(x)\) and \(q_{i}(x)\) denote conditional probabilities that class \(i\) sample is in range \([X_{1},x]\) and \([x,X_{2}]\), respectively. After finding the minimum entropy as \(x_{min}\), this value was used to determine the positions of the changing slope on the trapezoidal function by applying MEP again to find \(x_{L}\) and \(x_{H}\). The \(x_{L}\) is a threshold with the minimum entropy in the range \([X_{1},x_{L}]\) and \([x_{L},x_{min}]\), while \(x_{H}\) is a threshold with the minimum entropy in range \([x_{min},x_{H}]\) and \([x_{H},X_{2}]\). Finally, \(x_{L}\) and \(x_{H}\) are separate points of the trapezoidal function. Next, the FMF was applied and each parameter has three to five new input features from the membership values. Dataset-I from the ETL process applied FMF with MEP to create Dataset-III. For example, the CO in Figure 2b was applied to the MEP twice. The first MEP was used to divide between Class 1 and Class 2, while the second MEP was used to divide between Class 2 and Class 3. The first MEP results showed that \(x_{L}\) and \(x_{H}\) were 0.75 and 1.05 with the minimum entropy values 0.5165 and 0.6584, respectively. In addition, the second MEP results showed that \(x_{L}\) and \(x_{H}\) were 1.15 and 1.45 with the minimum entropy values 0.5595 and 0.4230, respectively. This feature was divided into three FMFs. The membership values of each membership function were calculated from Equations (4)-(6) for low, medium, and high, respectively, where \(\mu\) denotes the membership value and \(x\) denotes an input feature. In addition, Figure 3 shows a graph of three trapezoidal membership functions of the CO.  
\[\mu_{Low}(x)=max\bigg{(}min\bigg{(}1,\frac{1.05-x}{1.05-0.75}\bigg{)},0\bigg{)} \tag{4}\]  
\[\mu_{Medium}(x)=max\bigg{(}min\bigg{(}\frac{1.05-x}{1.05-0.75},1,\frac{x-1.15 }{1.45-1.15}\bigg{)},0\bigg{)} \tag{5}\]  
\[\mu_{High}(x)=max\bigg{(}min\bigg{(}1,\frac{x-1.15}{1.45-1.15}\bigg{)},0\bigg{)} \tag{6}\]  
Figure 3: Fuzzy membership function for CO in the LPA-Met station.