## 3 Results and Discussion of Data Analysis
### Correlation Coefficient Analysis and Its Relationship with Low-Frequency Signal Response  
In calculating the correlation coefficients, this study first aligns the low-frequency signal strength with the XRA data based on the lag time obtained from the analysis in Section 2.2, ensuring optimal correlation alignment. The Pearson, Spearman, and Kendall Tau correlation coefficients are then computed. This approach ensures that the correlation coefficients accurately reflect the relationship between the XRA and signal strength variations after correcting for lag time. Lag time alignment not only removes the interference caused by temporal misalignment between the signals but also enhances the accuracy of the correlation analysis. This method is widely used in the literature to address the issue of underestimating correlation in dynamic systems due to signal delays. For instance, [PERSON] et al. (2001) demonstrated that lag alignment improves the capture of the relationship between VLF signals in polar regions and solar activity ([PERSON] et al. (2017), while [PERSON] et al. (2017) highlighted that lag correction aids in more accurately quantifying the impact of solar flares on ionospheric disturbances ([PERSON] et al., 2017). Figure 5 shows the distribution of lag times and the three correlation coefficients for each sample, while Table 1 provides the statistical results of the three correlation coefficients across different time intervals. The selection of a 5-min interval for lag segmentation is based primarily on the time-scale characteristics of ionospheric disturbances caused by solar flares and the anomaly detection features of signal receiving terminals. [PERSON] et al. (2005) pointed out that C-class solar flares typically induce significant changes in ionospheric electron density within a few minutes ([PERSON] et al., 2011). [PERSON] et al. (2011) further emphasized that a 5-min interval effectively captures dynamic changes within this time scale while maintaining a balance between analytical precision and interpretability, avoiding the amplification of random noise from overly small segments or the loss of subtle variations with excessively large segments ([PERSON] et al., 2011). Additionally, [PERSON] (1974) noted that significant fluctuations within a 5-min period are often regarded as anomalous signals from the perspective of signal receiving terminals, potentially reflecting rapid disturbances caused by solar activity ([PERSON] et al., 2011). From the figure, it is evident that in the lag intervals (\(-5,0\)) and (\(0,5\)), the average Pearson correlation coefficient is significantly positive, indicating a strong linear relationship between XRA and signal strength during these periods. In contrast, negative correlations are observed in the earlier negative lag intervals. For example, for a sample with a lag of \(-144\) s (approximately \(-2.4\) min), the Pearson correlation coefficient is \(0.75\), which lies within the high positive correlation range (\(-5,0\)), demonstrating that near the peak of the flare, XRA is strongly linearly associated with changes in signal strength. However, it is important to note that this is just one sample case, and such a high correlation may not be representative of the overall trend. These results suggest that the most significant positive correlation between XRA and signal strength occurs within the lag intervals (\(-5,0\)) and (\(0,5\)). This indicates that during C-class flare events, an increase in XRA directly causes changes in the ionospheric electron density, which in turn alters the propagation characteristics of low-frequency skywave signals. The high linear correlation observed during this period underscores the significant direct impact of C-class flares on low-frequency signals. The Spearman correlation coefficient reaches its peak value in the lag interval (\(-5\), \(0\)), highlighting a highly significant monotonic relationship between signal strength and XRA.