## 4 Results and Discussion
### Localization Results Using Linear Regression  
Before inputting all 15 AE features into the linear regression model, feature analysis was performed by computing the correlation coefficient. The correlation coefficient is a numerical value that measures the strength and direction of a linear relationship between two variables. It is a descriptive value that is used heavily in statistics to make conclusions about data. The coefficient can range from \(-1\) to \(1\), and the closer the coefficient is to either \(-1\) or \(1\), the stronger the relationship between variables. We utilized this technique to identify potentially relevant features and subsequently plot a Pearson correlation heatmap (Figure 6) for visualization purposes. Upon examining the correlations between the AE feature and the impact location (rib number), several significant relationships are observed. The strongest negative correlation is with amplitude (\(-0.9302\)), suggesting an inverse relationship between it and the rib number. As amplitude increases, the rib number tends to decrease, indicating that amplitude is a critical factor in determining the impact location. Other features, such as duration (\(-0.7022\)), count (\(-0.6845\)), and average frequency (\(-0.7199\)), also exhibit strong negative correlations with the rib number, highlighting their potential correlation with the impact locations. Conversely, rise time (\(0.3463\)) demonstrates a moderate positive correlation with rib number, implying that higher rise time values correspond to higher rib number. When looking at the five frequency-related parameters used (reverberation frequency, average frequency, peak frequency, initial frequency, and frequency centroid), the reverberation frequency and the average frequency are the parameters of more prominent relevance. The reverberation frequency is defined as the frequency of a signal after reflection from various structural boundaries, and it conveys a wealth of information about the geometry of the structure and the AE wave propagation path. This information becomes indispensable when the task is to locate the AE source with a single sensor [45; 46]. The average frequency, another frequency parameter with high correlation, contains the reverberation frequency. Thus, both the reverberation frequency and the average frequency have a high correlation with the impact location. In contrast, the peak and initial frequencies and frequency centroid contain no information about the reverberation. These parameters encapsulate different sides of the AE signal and, unlike reverberant frequencies, have no propagation path or structural boundary effects. Therefore, they have a much lower correlation with the impact location. Figure 6: Heat map of correction coefficients. The correlation coefficients presented between the features and the label offer critical insights into the relationships that may be leveraged when selecting features for a linear regression model. In developing an accurate and reliable model, it is essential to select features that exhibit a strong correlation with the label, while minimizing multicollinearity among the selected features. The absolute values of the correlation coefficients between the impact location and the 15 AE features are shown in Figure 7. All correlation coefficients are arranged in ascending order. For the linear regression model in this paper, the learning rate is set as 0.1, and the max iteration is 1000. Following the ranking of features based on their correlations, a backward elimination process was employed, iteratively removing the feature with the lowest ranking, and subsequently utilizing the remaining features in the linear regression model. The RMSE and R\({}^{2}\) values were computed at each step to evaluate the performance of the model. Tables 2 and 3 display the coefficients of all features used in constructing the linear regression equation, as well as the corresponding R\({}^{2}\) and RMSE values obtained when each feature is sequentially deleted.