## 4 Methodology  
A regression analysis is a statistical technique employed for analyzing empirical relationships between a dependent variable and one or more independent variables. This technique can be primarily applied to infer a relationship between the dependent and independent variable in a fixed dataset. It can also help in the visualization of variation of the values of the dependent variable, when one of the independent variables is adjusted while keeping the other independent variables constant. A particular type of regression analysis, the non-linear regression technique involves the modelling of the observed dependent variables using a mathematical function that combines one or more independent variables in a non-linear function. Power functions, exponential functions, logarithmic functions, and trigonometric functions are several typical examples of non-linear regression functions. In this research, a power function was employed to correlate the estimated shear wave velocities, to measure the SPT-N values. This non-linear function is widely utilized in rapid evaluation of V\({}_{\text{s}}\) and associated geotechnical parameters. The power function has a basic form of  
\[E(x,y)=ax^{b} \tag{1}\]  
where \(x\) and \(y\) are two variables, with \(x\) as the independent variable and where \(y\) is the response variable, while \(a\) and \(b\) are parameters of the power function [49]. \(a\) is a constant controlling the amplitude of the function, and \(b\) controls the relationship curve. This research examined the correlation based upon corrected and uncorrected SPT-N values and the V\({}_{\text{s}}\), irrespective of the depth, overburden pressure, geologic age of the deposit, and fines content. Applying the natural logarithm function to Equation (1) transforms the non-linear function into a linear regression equation with the form  
\[Y^{\prime}=b_{0}+b_{1}X^{\prime} \tag{2}\]  
where \(Y^{\prime}=\ln y\), \(b_{0}=\ln a\), \(b_{1}=b\), and \(X^{\prime}=\ln x\). The parameters of regression \(b_{0}\) and \(b_{1}\) are determined using the least squares method  
\[b_{1}=\frac{\sum_{i=1}^{n}x_{i}y_{i}-\frac{1}{n}\ \sum_{i=1}^{n}x_{i}\ \sum_{i=1}^{n}y_{i}}{ \sum_{i=1}^{n}x_{i}^{2}-\frac{1}{n}\ {\left(\sum_{i=1}^{n}x_{i}\right)}^{2}} \tag{3}\]  
\[b_{0}=\frac{1}{n}\ \sum_{i=1}^{n}y_{i}-\frac{b_{1}}{n}\sum_{i=1}^{n}x_{i} \tag{4}\]  
with \(x_{i}\) substituted by \(\ln x\), \(y_{i}\) by \(\ln y\), and \(n\) as the number of data points. When the values of \(b_{0}\) and \(b_{1}\) are determined, then the original parameters of the power function at Equation (1), \(a\) and \(b\), can be also be calculated. To measure the quality of the non-linear regression analysis, various statistical parameters need to be determined as well. The residual (\(\varepsilon_{i}\)) of the regression analysis is the difference from the observed value (\(y_{i}\)) of the dependent variable in comparison to its predicted value (\(y_{i}\)'), using the regression function (Equation (5)). \[\varepsilon_{i}=y_{i}-y_{i}^{*} \tag{5}\]  
The total variability (\(S_{YY}\)) of the dependent variable (\(y_{i}\)) is determined from two components: the (1) variability in the observations of the \(y_{i}\), which is accounted for by the regression function (\(SS_{R}\)); and the (2) residual variation, which is left unexplained by the regression function (\(SS_{E}\)) [49]. The method of determining \(S_{YY}\) is shown in Equations (6) and (7). In these equations, \(\overline{y}\) is the average of the observed values of the dependent variable. \[S_{YY}=SS_{R}+SS_{E} \tag{6}\] \[\sum_{i=1}^{n}(y_{i}-\overline{y})^{2}=\sum_{i=1}^{n}(y_{i}^{*}- \overline{y})^{2}+\sum_{i=1}^{n}(y_{i}-y_{i}^{*})^{2} \tag{7}\]  
The coefficient of regression (\(R^{2}\)) is the fraction of the total variability in the dependent variable that is accounted for by the regression function.