## 3 Algorithms Used in Correlation Mining Analyses
### Machine Learning-Based Algorithms
#### 3.3.1 Linear Regression Method  
In statistics, linear regression is a method used to model the relationship between a response (target) variable and one or more descriptive variables (features). This technique is frequently employed to uncover linear relationships between variables. It operates under the assumption that descriptive variables, which have values independent of each other, can effectively predict the response variable, whose value is neither dependent on these descriptive variables nor under the researcher's control. The primary objective of regression analysis is to identify and quantify the linear relationship between the target and features, facilitating predictions and insights based on the data.  
The correlation coefficient is a statistical measure used to evaluate the relationship between variables, ranging in value from \(-1\) to \(+1\). A correlation coefficient close to either \(-1\) or \(+1\) indicates a strong relationship between variables. A coefficient near \(+1\) signifies a direct relationship, meaning that as one variable increases, the other also increases. Conversely, a coefficient near \(-1\) signifies an inverse relationship, where an increase in one variable corresponds to a decrease in the other. The form of the linear regression model is represented by the below equation:  
\[Y=\beta_{1}\cdot X+\beta_{0}+\epsilon \tag{7}\]  
where \(X\) and \(Y\) represent the independent and dependent variables, respectively. Also, the parameters of the linear model are \(\epsilon\): error, \(\beta_{0}\): distance from the origin or intercept, and \(\beta_{1}\): coefficient or line slope.  
The line slope and distance from the origin in a linear regression model respectively indicate the sensitivity of the dependent variable to the independent variables and the value of the dependent variable when the independent variable equals zero [58; 59; 60]. These parameters play crucial roles in understanding the relationship between variables and predicting outcomes based on the model. The slope indicates the rate of change in the dependent variable for a unit change in the independent variable, providing insights intothe strength and direction of the relationship. Meanwhile, the distance from the origin represents the intercept, which determines the value of the dependent variable when the independent variable is absent or equals zero, providing a reference point for the regression line.  
As shown in Figure 6a, linear regression models the relationship between the outcome variable and explanatory variables using a linear function. In the current case, the features include the longitudinal levels of railway tracks, which came from the track geometry recording car. The vertical rail deflection as well as ballast stiffness are targeted as dependent variables. Linear regression estimates the coefficients (slope and intercept) of a linear equation that best fits the data. The model aims to minimize the difference between predicted and actual vertical ballast stiffness. The input data should be organized as a matrix, where each row corresponds to a measurement and columns represent the features. The algorithm learns the optimal coefficients by minimizing the sum of squared errors between predicted and actual deflections.  
Figure 6: Machine learning-based algorithms used in current study: (**a**) linear regression algorithm, (**b**) decision tree algorithm, and (**c**) random forest algorithm.