## 3 Methodology
### Data Imputation
#### 3.2.2 An application of Pearson's Correlation Coefficient (PCC) Analysis  
The PCC plays a key role in statistical research. This method assesses a relationship between two variables or data sets [95; 96]. An analysis shows a value in a range \(-1\) to \(1\). Various aspects of PCC data analysis relate to this area of research. As shown in Table 2, the interpretation of the PCC differs; for example, the range for weak relationships is less than \(0.3\) in Psychology, \(0.2\) in Politics and \(0.5\) in Medicine. Regarding this research, values in the range \(\pm 0.1\)-\(\pm 0.3\) refer to a weak relation, \(\pm 0.3\)-\(\pm 0.5\) refer to a moderate relation and \(\pm 0.5\)-\(\pm 1.0\) refer to a strong relation [96]. A number over zero implies both variables are positively correlated, while a value below zero indicates that both factors are negatively correlated. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Interpretation**} & \multicolumn{4}{c}{**Areas of Study**} \\ \cline{2-5}  & **Engineering [97]** & **Medicine [97]** & **Politics [97]** & **Psychology [97]** \\ \hline Perfect & & \(-1\) (negative) or \(1\) (positive) & \\ Strong & \(\pm 0.5\)–\(\pm 1.0\) & \(\pm 0.5\)–\(\pm 1.0\) & \(\pm 0.5\)–\(\pm 1.0\) & \(\pm 0.5\)–\(\pm 1.0\) \\ Moderate & \(\pm 0.3\)–\(\pm 0.5\) & \(\pm 0.3\)–\(\pm 0.5\) & \(\pm 0.3\)–\(\pm 0.5\) & \(\pm 0.3\)–\(\pm 0.5\) \\ Weak & \(\pm 0.1\)–\(\pm 0.3\) & \(\pm 0.1\)–\(\pm 0.3\) & \(\pm 0.1\)–\(\pm 0.3\) & \(\pm 0.1\)–\(\pm 0.3\) \\ None & & & 0 & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The interpretation of PCC’s value in engineering, medicine, politics and psychology [97]. Figure 2: The overall flowchart of the data imputation stage allows four different options (KNN, NNM, Mean, Median) to calculate the most-fittest model. This study investigates how societal conditions are connected to HSR expansion by utilising the PCC analysis, as illustrated in Equation (2). \[\mathrm{P}=\frac{\sum(X_{i}-x)(Y_{i}-y)}{\sqrt{\sum(X_{i}-x)^{2}\sum(Y_{i}-y)^{2}}} \tag{2}\]  
where: \(\mathrm{P}=\) correlation coefficient, \(X_{i}\) = social component values \(i\), \(x\) = the average of the social variables' values \(i\); \(Y_{i}\) = the overall distance of the HSR network in year \(i\); \(y\) = the average value of the HSR network's length. With respect to the broad benefits of PCC, many studies in railway research have adopted this method as the main methodology. Some authors use PCC to improve the efficiency of railway operation. It has been involved in improvements in safety performance in the long-term. Some scholar applied PCC to verify safety conditions at railway crossings in the event of extreme weather. The research revealed that some indicators should be adjusted to assess the conditions of railway crossings such as fatigue spots, an irregular contact ratio [98]. The PCC has also been broadly applied for the enhancement of technical services of railway. [PERSON] and [PERSON] focused on traffic loads and track parameters with the aim of evaluating vertical and lateral rail wear [99]. Traffic loads are unpredictable information based on uncertain passenger numbers and volumes of materials. Another study focuses on correlations to track geometry, especially the correlation between track stiffness and variable settlement. By using PCC, precise measurement of vertical track stiffness was obtained, providing better upkeep and environmental sustainability [100]. Additionally, PCC has been used in urban planning for categorisation. Areas surrounding train stations have seen better management and long-term improvements due to these findings. Some scholar used PCC to analyse 1700 Swiss railway stations using contextual factors such as density and purpose for use of rail services [101]. PCC clearly offers classification in the context of the system structure, with the outcome providing strategic planning for railway stations to support a fluctuating level of passenger volume.