## 2 Data Processing and Analysis Procedure
### Data Processing Methods
#### 2.2.2 Correlation Analysis Between Long-Wave Signals and XRA Data  
When investigating the relationship between low-frequency signal strength and X-ray flux (XRA) during solar storms, correlation analysis is a crucial method for uncovering the inherent connection between the two variables. Given that solar storms can cause diverse ionospheric disturbances, which may manifest as signal patterns ranging from linear to nonlinear relationships, the application of multiple correlation metrics is essential for obtaining a comprehensive understanding of the dynamic interactions between long-wave signal strength and XRA. In this study, three correlation indicators were used for a thorough analysis:  
1. Pearson Correlation Coefficient  
The Pearson correlation coefficient is the most widely used metric for quantifying the linear relationship between two variables and is defined as follows:  
\[r=\frac{\sum\limits_{i=1}^{N}{(x_{i}-\overline{x})(y_{i}-\overline{y})}}{\sqrt {\sum\limits_{i=1}^{N}{(x_{i}-\overline{x})^{2}}}\sqrt{\sum\limits_{i=1}^{N}{ (y_{i}-\overline{y})^{2}}}} \tag{2}\]  
where \(x_{i}\) and \(y_{i}\) represent the values of long-wave signal strength and X-ray flux, respectively, \(\overline{x}\) and \(\overline{y}\) are their mean values. The Pearson correlation coefficient r ranges from \([-1,1]\), where a value of 1 indicates a perfect positive correlation, \(-1\) indicates a perfect negative correlation, and 0 indicates no linear relationship. In the context of this study, the Pearson coefficient quantifies the degree of linear dependence between X-ray flux (XRA) and low-frequency signal strength. A strong positive or negative r suggests that changes in XRA have a direct and proportional impact on ionospheric electron density, which subsequently alters signal propagation characteristics. This makes Pearson correlation particularly valuable for identifying clear, linear relationships during solar flare events. 2. Spearman Rank Correlation CoefficientThe Spearman rank correlation is used to assess the monotonic relationship between two variables and is defined as follows:  
\[r_{s}=1-\frac{6\Sigma d_{i}^{2}}{n(n^{2}-1)} \tag{3}\]  
where \(d_{i}\) represents the difference in rankings between the two variables, and \(n\) is the number of data points. The Spearman correlation is particularly suitable for analyzing non-linear but monotonic relationships. During solar storms, the relationship between XRA and long-wave signal strength often deviates from linearity, particularly during low-intensity events. As a rank-based metric, the Spearman correlation is well-suited for capturing these intricate nonlinear associations, offering robust insights into ionospheric response patterns under conditions dominated by weak disturbances or elevated noise levels. Its ability to detect subtle yet consistent trends make it particularly advantageous for analyzing complex datasets where linear correlation methods may fail to reveal underlying relationships. (3) Kendall Tau Correlation Coefficient  
The Kendall Tau coefficient is used to assess the degree of concordance between two variables, and it is defined as follows:  
\[\tau=\frac{(N_{c}-N_{d})}{\frac{1}{2}n(n-1)} \tag{4}\]  
where \(N_{c}\) represents the number of concordant pairs and \(N_{d}\) denotes the number of discordant pairs. The Kendall Tau correlation coefficient is a robust and stable measure of association, particularly suited for scenarios involving small sample sizes or datasets with significant variability. Unlike the Pearson correlation coefficient, which primarily assesses linear relationships, Kendall Tau focuses on the consistency of trend changes between variables rather than their absolute values. Additionally, compared to the Spearman correlation coefficient, Kendall Tau provides a more accurate reflection of trend alignment in smaller datasets and is especially effective for analyzing high-noise or dynamically fluctuating data.