## 2 Materials and Methods
### Introduction to Common Clustering Methods  
This section introduces five common clustering methods including k-means, fuzzy c-means (FCM), Gaussian mixture model (GMM), self-organizing mapping (SOM), and hierarchical model clustering to analyze acoustic emission signals. The following five clustering methods are introduced separately:  
(1) K-means clustering:  
The main steps of the k-means clustering are: first, the number of clustering categories k (k cluster centers) is artificially determined, and then the similarity between each object and the cluster center are analyzed according to the distance function, and the sample points are classified into the most similar clusters. Next, the centroid of each cluster (the cluster center) is recalculated and this process is repeated until the centroid no longer changes. There are two phases of the k-means clustering algorithm [40]:  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Name** & **Number** & **Function** & **Specification** \\ \hline Micro-II Digital AE & 1 & Processing AE signals & \begin{tabular}{c} Data acquisition and transfer efficiency: 40 MB/s; \\ Filter bandwidth: 3 kHz to \\ 3 MHz. \\ Peak sensitivity: 75 dB with \\ reference 1 V/(m/s); \\ Working temperature range: \\ \(-65\) to \(177\)\({}^{\circ}\)C; \\ Operating frequency: \\ 35–100 kHz. \\ \end{tabular}  \\ AE preamplifier & 8 & Amplify AE signals & Model: 2/4/6 preamplifier \\ Electronic universal testing machine & 1 & \begin{tabular}{c} Complete four-point \\ bending load \\ Record AE data and \\ loading data \\ Transmitting \\ AE signals \\ \end{tabular}  \\ Computer & 2 & \ \begin{tabular}{c} Complete four-point \\ bending load \\ Record AE data and \\ loading data \\ Transmitting \\ AE signals \\ \end{tabular}  \\ Data cable & several &
\begin{tabular}{c} **(b)** \\ **(c)** \\ **(b)** \\ **(b)** \\ **(d)** \\ **(e)** \\ **(f)** \\ **(g)** \\ **(h  
Distribution phase: Use the nearest cluster center to assign each datum to the same cluster. \[C_{i}^{(t)}=\left\{A_{n}:\left\|A_{n}-m_{i}^{(t)}\right\|^{2}\leq\left\|A_{n}-m_{ j}^{(t)}\right\|^{2}\forall j,1\leq j\leq k\right\}\] (1) among them \(C_{i}^{(t)}\) is the cluster \(i\), \(m_{i}^{(t)}\) is the center of cluster \(i\), and \(A_{n}\) represents the data points. (b) Update phase: Recalculate the cluster center and treat it as the average of the whole cluster. \[m_{i}^{(t+1)}=\frac{1}{\left|C_{i}^{(t)}\right|}\sum_{A_{j}\in C_{i}^{(t)}}A_{j}\] (2)  
Next, iterate through the above phases until the cluster no longer changes. Due to the initial randomly selected clustering center playing a vital role on the results of the entire calculation, k-means clustering cannot converge to the optimal solution to a large extent [43]. (2) FCM clustering  
The main idea of the fuzzy c-means (FCM) clustering follows. Assume a sample set and divide it into c fuzzy groups, and then find the cluster center of each group. Comparing with k-means clustering, this clustering method integrates the essence of fuzzy theory, thereby providing more flexible clustering results. Each object is not directly assigned to a specific cluster, but a weight is assigned to each object and each cluster. If the weight is larger, the object is closer to this cluster. This method attempts to find the minimum of the following objective functions [44]:  
\[J(A;C)=\sum\limits_{i=1}^{k}\sum\limits_{j=1}^{n}{(\gamma_{ij})}^{\partial}{ \left\|A_{j}-C_{i}\right\|^{2}} \tag{3}\]  
\[D_{\bar{\eta}}^{2}=\left\|A_{j}-C_{i}\right\|^{2}=(A_{j}-C_{i})^{T}(A_{j}-C_{i}) \tag{4}\]  
Through comparison, the difference can be found between these functions and the function of k-means clustering: the fuzzy degree of clustering is determined by the fuzzy coefficient (\(\partial\)) and the member value (\(\gamma_{ij}\)) together.