## 2 Data and Methods
### Methodology
#### 2.3.3 MLR Analysis  
The MLR analysis is a statistical approach with only one dependent and many independent variables. MLR provides insights into the strength and direction of the relationships between the independent variables and the dependent variable [56; 57; 58]. In this approach, a relationship between the main function (\(\gamma\)) and the independent variable of \(x_{i}\) is defined as  
\[Y=f(x_{i}) \tag{7}\]  
When \(Y\) is defined as a linear function, the relationship is called the linear regression. Similarly, if \(Y\) is defined as a nonlinear function of \(x_{i}\), it is called the nonlinear regression [58].  
The MLR approach delivers suitable predictive models for various surface and subsurface geoscience applications [58]. Consequently, in this research, the MLR approach was utilized to estimate the \(V_{s}\). The general form of the approach is  
\[Y=a_{0}+a_{1}x_{1}+\cdots+a_{n}x_{n}+C, \tag{8}\]  
where \(x_{1},x_{2},x_{3},\ldots,x_{n}\) are the independent variables, \(Y\) represents the dependent variable, and \(a_{0},a_{1},a_{2},a_{3},\ldots,a_{n}\) are regression coefficients. The coefficients can be interpreted to understand the effect of each independent variable while holding others constant. Such coefficients are calculated by the least square method. Furthermore, the parameter of \(C\) is a real number.  
In the MLR analysis, the correlation coefficient, \(R^{2}\), serves as a fitness indicator of the extracted relationship between the \(Y\) and independent variables. The corresponding mathematical formula is  
\[R^{2}=\frac{\sum_{i=1}^{n}\left(\hat{Y}_{i}-\overline{Y}\right)^{2}}{\sum_{i=1 }^{n}\left(Y_{i}-\overline{Y}\right)^{2}}=1-\frac{\sum_{i=1}^{n}\left(Y_{i}- \widehat{Y}_{i}\right)^{2}}{\sum_{i=1}^{n}\left(Y_{i}-\overline{Y}\right)^{2 }}, \tag{9}\]  
where \(\hat{Y}_{i}\) and \(Y_{i}\) represent the calculated value and real value of the \(i\)th sample of the dependent parameter, respectively. In addition, \(\overline{Y}\) indicates the mean of the dependent parameter. When \(R^{2}\) is close to 1, it means that there is a good correlation between the independent and dependent variables. Nevertheless, when \(R^{2}\) approaches 0, it means that the fitness of the function is low. More technical details are available in the research published by [PERSON] et al. [56].  
An advantage of MLR is its simplicity, as it allows for the incorporation of multiple independent variables to capture complex relationships. Nevertheless, this flexibility can become a disadvantage when dealing with a large number of predictors, as MLR may be prone to overfitting. Overfitting occurs when the model fits the training data too closely, capturing noise and idiosyncrasies rather than the underlying patterns. Including too many predictors relative to the sample size can lead to a highly flexible model that performs well  
\begin{table}
\begin{tabular}{c c c c} \hline
**Correlation** & **Formula** & **Units** \\ \hline [PERSON] et al., 1998 [18] & \(V_{s}=-0.05509\)\(V_{p}{}^{2}+1.0168\)\(V_{p}-1.0305\) & (2) & \(V_{p}\) (km/s) and \(V_{s}\) (km/s) \\ [PERSON], 1969 [15] & \(V_{s}=0.937562\)\(V_{p}{}^{0.82}\) & (3) & \(V_{p}\) (kt/s) and \(V_{s}\) (kt/s) \\ [PERSON] et al., 2010 [27] & \(V_{s}=1.09913326\)\(V_{p}{}^{0.92}\) & (4) & \(V_{p}\) (m/s) and \(V_{s}\) (m/s) \\ [PERSON], 1963 [14] & \(V_{s}=V_{p}/1.9\) & (5) & \(V_{p}\) (ft/ us) and \(V_{s}\) (ft/ us) \\ [PERSON] and [PERSON], 1993 [23] & \(V_{s}=199\)\((\gamma)^{2.84}\) & (6) & \(V_{s}\) (m/s); \(\gamma\) is density (g/cm\({}^{3}\)) \\ \hline \end{tabular}
\end{table}
Table 2: Empirical correlations related to estimation of \(V_{s}\) in limestone reservoirs.  
on the training data but fails to generalize effectively to new, unseen data. Regularization techniques, such as ridge regression or lasso regression, can be employed in MLR to address overfitting by imposing constraints on the coefficients, preventing them from reaching extreme values and promoting a more parsimonious model that generalizes better to new observations.