## 2 Materials and Methods
### Fuzzifier Value  
IT2 PFCM is expressed as the sum of the weights of FCM and PCM method. Therefore, it is clustered in the direction of minimizing PFCM objective function, as follows in Equation (2).  
In Equation (3), \(u_{ik}\) represents a membership value where the input pattern k belongs to cluster  
\[J_{m,\eta}(U,T,V;X)\ =\ \sum_{k=1}^{n}\sum_{i=1}^{c}(\underset{ik}{\text{ }}{u}\quad^{m}+\underset{ik}{\text{ }}{b}t_{ik}^{\eta})\times\|x_{k}-v_{i}\|^{2}+\sum_{i=1}^{c}\gamma_{i}\sum_{ k=1}^{n}((1-t_{ik})^{\eta}), \tag{2}\]  
\[\sum_{i=1}^{c}u_{ik}=1,0\leq u_{ik},t_{ik}\leq 1,m=>1,\eta>1,\gamma_{i}>0. \tag{3}\]  
\(x_{i}\) is the \(k\)-th input pattern, and \(v_{i}\) is the center value of the \(i\)-th cluster. m is a constant representing the degree of fuzziness and satisfies the condition of \(\text{m}\ \in(1,\infty)\).tik represents typicality that the input pattern \(k\) belongs to cluster \(i\), which is a feature of PFCM method using absolute distance. \(\gamma_{i}\) is a scale defining point, where typicality of the \(i\)-th cluster is 0.5, and the value is an arbitrary number.  
For PFCM cluster method, in above Equation (5) the objective function should be minimized with respect to the membership function \(u_{ik}\). Membership that is obtained by Equation (2). To draw m,you must create the lowest and highest membership functions using the primary membership function. The highest and lowest membership functions of PFCM according to m are as follows.  
\[\overline{u}_{ik}=(\frac{1}{\sum_{j=1}^{1}\left(\frac{d_{ik}}{d_{ij}}\right)^{ \frac{2}{m_{1}-1}}}), \tag{4}\]  
\[\overline{u}_{ik}=(\frac{1}{\sum_{j-1}^{1}\left(\frac{d_{ik}}{d_{ij}}\right)^{ \frac{2}{m_{2}-1}}}), \tag{5}\]  
As shown in Equation (6), the lowest and highest membership values where \(m_{1}\) and \(m_{2}\) are representing objective function, the value \(\gamma_{i}\) also changes according to the lowest and highest membership functions. Using \(\gamma_{i}\), the lowest and highest typicality is,  
\[\gamma=\frac{\log\left(\frac{1}{u_{j}}\right)-\log\left(1+\sum_{k=2}^{C} \left(\frac{d_{ij}}{d_{ik}}\right)^{\frac{2}{m_{old}}-1}\right)}{\log\left( \frac{d_{ij}}{d_{ik}}\right)} \tag{6}\]  
\[m_{jnew}=1+\frac{2}{\gamma} \tag{7}\]  
For updating the center value, as shown in Equation (7), the type reduction process of changing type-2 fuzzy takes up type-1 using the K-means algorithm that it is performed, and the center value of each cluster is updated.  
\[m_{1}=\left(\sum_{i=1}^{N}m_{1i}\right)/N,\;\;m_{2}=\left(\sum_{i=1}^{N}m_{2i} \right)/N \tag{8}\]  
Table 1 shows the symbol and original values of IT2-PFCM.  
Figure 4 shows that histograms and Footprint of Uncertainty (FOU) examples are determined by class and dimension. Upper membership function (UMF) histogram and lower membership function (LMF) histogram are obtained according to class and dimension. A new membership function from the Gaussian Curve Fitting (GF-F) method can be applied to calculate the adaptive fuzzifier value.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Explanation** \\ \hline \(C\) & Bulky cluster \\ \(v\) & Cluster center \\ \(v_{i}\),\(V\) & Cluster prototype \\ \(m\) & Fuzzifier value \\ \(u\) & Membership function \\ \(U\) & Partition matric \\ \(d_{ik}\)/\(d_{ij}\) & Euclidean distance value \\ \(\frac{8}{A}\) & Threshold of fuzzifier constant \\ \(\widetilde{A}\) & Secondary membership degree \\ \(J\) & PFCM objective function \\ \(x\) & Input pattern \\ \(t_{R}\) & Represents typicality, the input pattern k belongs to cluster i \\ \(\gamma i\) & Scale defining point where typicality of the i-th cluster is 0.5 \\ \(X_{i}\) & Input space \\ \(\Phi\left(X_{j}\right)\) & Kernel property space \\ \(K\) & Input space for kernel \\ \(S\) & Number of kernels \\ \hline \hline \end{tabular}
\end{table}
Table 1: Explanation of symbols.