## 2 Materials and Methods
### Introduction to Common Clustering Methods
The minimum value of the function can be obtained if [44]:  
\[\gamma_{\bar{\eta}}=\frac{1}{\sum\limits_{m=1}^{k}{\left(\frac{D_{\bar{\eta}} }{D_{mj}}\right)^{\frac{2}{\alpha-1}}}};1\leq i\leq k,1\leq j\leq n \tag{5}\]  
\[C_{i}=\frac{\sum\limits_{j=1}^{n}{(\gamma_{ij})}^{\partial}A_{j}}{\sum\limits _{j=1}^{n}{(\gamma_{ij})}^{\partial}};1\leq i\leq k \tag{6}\]  
There are three stages in the FCM clustering algorithm:  
1. Determine cluster centers according to Equations (5) and (6). (b) Calculate the distance by Equation (4). (c) Update the objective function according to Equation (3). The FCM clustering has better performance than the k-means clustering, but some limitations remain. For example, the determination of the initial cluster center and member values have a great impact on the clustering results [45], which may lead to the limitation of FCM clustering at local minimum values. (3) GMM clustering GMM is an abbreviation for Gaussian mixture model (or mixed Gaussian model). According to many phenomena in nature, Gaussian (normal) distribution is followed; nevertheless, the reasons for affecting a distribution are multiple, and each influence factor may determine one. A single effect is a Gaussian distribution; a combination of multiple effects is a multiple Gaussian distribution. Therefore, the Gaussian mixture model can also be understood as a large model with K single models. The Gaussian distribution is generally combined with the EM algorithm as its likelihood estimation algorithm. The probability model GMM can be expressed as [46]:  
\[\mathrm{p}(x)=\sum_{k=1}^{K}w_{k}g(A|\mu_{k},\sum_{k}) \tag{7}\]  
\[\mathrm{g}(A\bigg{|}\mu_{k},\sum_{k})=\frac{1}{\sqrt{(2\pi)^{k}|\sum_{k}|}} \exp(\frac{-\frac{1}{2}(A-\mu_{k})^{T}}{\sum_{k}(A-\mu_{k})}) \tag{8}\]  
\[\sum_{k=1}^{K}w_{k}=1 \tag{9}\]  
where \(w_{k}\) is the mixed weight function, \(A\) are the data, and the mean vector \(\mu_{k}\) and the covariance matrix \(\sum_{k}\) are represented by a Gaussian density function. There are three steps in the GMM clustering algorithm [46]:  
1. First, the algorithm estimates two quantities: the mean vector and the covariance matrix. It then calculates all the data and all the blending weight functions. 2. New parameters are calculated using membership and data. If \(N_{k}=\sum_{wik}\), the new blending weight can be expressed as: \[\mathrm{w}_{k}^{new}=\frac{N_{k}}{N};1\leq k\leq K\] (10)  
The new mean vector and covariance matrix can be expressed as:  
\[m_{i}^{(t+1)}=\frac{1}{\left|C_{i}^{(t)}\right|}\sum_{A_{j}\in C_{i}^{(t)}}A_{j}\]0  
\[m_{i}^{(t+1)}=\frac{1}{\left|C_{i}^{(t)}\right|}\sum_{A_{j}\in C_{i}^{(t)}}A_{j}\]1
3. Repeat the above two steps to meet the stopping criteria. 4. SOM clustering  
The self-organizing mapping (SOM) neural network is a self-organizing and competitive network, and it is also a type of unsupervised learning model. SOM is unique in that it adopts a competitive learning approach to optimize the network. The weight vector of the \"winning neuron\" of the output layer and its neighboring neurons are calculated according to the following Equation [47]:  
\[m_{i}^{(t+1)}=\frac{1}{\left|C_{i}^{(t)}\right|}\sum_{A_{j}\in C_{i}^{(t)}}A_{j}\]2  
\[m_{i}^{(t+1)}=\frac{1}{\left|C_{i}^{(t)}\right|}\sum_{A_{j}\in C_{i}^{(t)}}A_{j}\]3  
(5) Hierarchical model clustering  
The hierarchical clustering method refers to the hierarchical decomposition of the data set until a certain condition is met. The traditional hierarchical clustering algorithm mainly has two parts: agglomerative and divisive hierarchical clustering [48]. (a) Agglomerative hierarchical clustering  
A bottom-up method which starts by determining each object as a single cluster, and these clusters are gradually merged according to certain criteria (the similarity measurement between the two clusters), and the clustering process repeats until all objects meet the number of clusters.