## 3 Results and Analyses
### Results of K-Means Cluster Analysis  
Acoustic emission signals constitute high-dimensional datasets where strong parameter correlations may systematically bias analytical outcomes. Specifically, such correlations can distort both the eigenvalue distribution of principal component analysis covariance matrices and the directional orientation of K-means clustering results toward dominant variables. Consequently, the direct application of unprocessed AE data for clustering risks introducing bias into the analysis. This limitation necessitates a preliminary correlation analysis of AE parameters prior to dimensionality reduction or clustering operations.  
The Pearson correlation coefficient is commonly used to assess the linear correlation degree in multidimensional data, calculated using the following formula:  
\[r=\frac{Cov(X,Y)}{\sigma_{X}\sigma_{Y}} \tag{3}\]  
where \(r\) is the Pearson correlation coefficient, which is bounded between \(-1\) and \(1\), with values approaching \(0\) indicating progressively weaker linear associations between the paired datasets. \(X\) and \(Y\) are two datasets for which the correlation coefficient needs to be calculated. \(Cov(X,Y)\) denotes the covariance of \(X\) and \(Y\), and \(\sigma_{X},\sigma_{Y}\) represent the standard deviations of \(X\) and \(Y\), respectively.  
Figure 10 presents the Pearson correlation matrix; there is a significant correlation between the duration and count of the captured acoustic emission signals for both C-1 and C-2 specimens. This is determined by the calculation method of counting. The difference is that the counting and amplitude of the C-1 sample also exhibit strong correlations, which may be due to the presence of a large number of long duration acoustic emission signals. Given the comprehensive analysis of counts in preceding sections, this section retains duration as a key predictive feature while excluding its collinear counterpart to mitigate multicollinearity risks. Therefore, amplitude, duration, energy, rise count, and rise time are used as feature parameters for PCA for facultative analysis.  
Prior to conducting K-means clustering, PCA was employed to mitigate dimensionality while preserving critical feature relationships. As detailed in Table 1, the first four principal components (PC1-PC4) collectively capture 95.5%-96.76% of the total explained variance, with PC1 and PC2 accounting for 69.5%-76.36% of the explained variance. To balance computational efficiency with information retention, we retained these four components that jointly exceed the 90% cumulative variance threshold. Subsequent K-means clustering  
Figure 10: Pearson correlation matrix.  
utilized the component loading matrix (Table 2) derived from PCA, ensuring optimal cluster configuration through dimension-reduced feature space optimization.  
To visually demonstrate the inter-cluster distribution characteristics, Figure 11 projects the clustering results into the two-dimensional feature space spanned by principal components PC1 and PC2. The energy and amplitude ranges, along with the mean values for each cluster, were calculated as shown in Table 3. Cluster 1 displays the lowest energy and amplitude levels, which may be related to the formation of microcracks in the cement matrix. Cluster 2 shows lower energy and amplitude values, likely associated with slippage between coarse/fine aggregates and cement mortar. Cluster 3 demonstrates relatively high energy and amplitude, attributed to acoustic emission signals generated by the failure of the interfacial transition zone in the concrete. Cluster 4 exhibits high energy and amplitude characteristics, which correspond to the fracture of coarse aggregates in the reinforced concrete.