# Soil Depth Prediction Model Using Terrain Attributes in Gangwon-do, South Korea
## 2 Methodology
### Statistics Analysis
#### 2.2.3 Multi-Linear Regression Analysis  
Multiple regression analysis is one of the statistical methods used to establish and model the relationship between two or more independent variables for one dependent variable. If the data are linearly related, the multi-linear regression (MLR) of the observation '\(y\)' for '\(n\)' independent variables '\(x\)' is as follows:  
\[y=\underline{x}\cdot\beta+\varepsilon=\beta_{0}+\beta_{1}x_{1}\cdots\beta_{n} x_{n}+\varepsilon \tag{2}\]  
where \(\beta_{0}\) represents the intercept, and \(\beta_{1}\cdots\beta_{n}\) represent the regression coefficients. \(\varepsilon\) is an error that is not explained by the model and is assumed to follow a normal distribution with \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Case No.**} & \multirow{2}{*}{**Parameter**} & \multicolumn{2}{c}{**Igneous Rock**} & \multicolumn{2}{c}{**Metamorphic Rock**} & \multicolumn{2}{c}{**Sedimentary Rock**} \\ \cline{3-8}  & & **Tolerance** & **VIF** & **Tolerance** & **VIF** & **Tolerance** & **VIF** \\ \hline \multirow{8}{*}{1} & SLOPE & 0.012 & 82.070 & 0.010 & 96.059 & 0.270 & 3.706 \\  & TWI & 0.002 & 591.425 & 0.002 & 523.360 & 0.021 & 46.671 \\  & STI * & 0.001 & 1166.374 & 0.001 & 1089.255 & 0.027 & 37.516 \\  & SPI & 0.001 & 1836.403 & 0.000 & 2307.441 & 0.061 & 16.270 \\  & CURV & 0.731 & 1.368 & 0.700 & 1.429 & 0.937 & 1.068 \\  & SCA ** & 0.015 & 66.595 & 0.013 & 76.471 & 0.032 & 30.870 \\ \hline \multirow{8}{*}{2} & SLOPE & 0.121 & 8.287 & 0.142 & 7.048 & 0.270 & 3.699 \\  & TWI & 0.016 & 63.426 & 0.018 & 56.247 & 0.027 & 37.110 \\  & STI * & 0.062 & 16.075 & 0.068 & 14.708 & 0.055 & 18.134 \\  & CURV & 0.738 & 1.355 & 0.740 & 1.352 & 0.941 & 1.063 \\  & SCA ** & 0.015 & 66.008 & 0.014 & 72.581 & 0.032 & 30.814 \\ \hline \multirow{8}{*}{3} & SLOPE & 0.141 & 7.075 & 0.167 & 5.975 & 0.271 & 3.695 \\  & TWI & 0.266 & 3.763 & 0.230 & 4.351 & 0.538 & 1.859 \\  & STI * & 0.239 & 4.181 & 0.221 & 4.516 & 0.384 & 2.604 \\  & CURV & 0.779 & 1.284 & 0.749 & 1.335 & 0.961 & 1.040 \\ \hline \multirow{8}{*}{4} & SLOPE & 0.141 & 7.074 & 0.172 & 5.809 & 0.272 & 3.678 \\  & TWI & 0.282 & 3.545 & 0.254 & 3.939 & 0.545 & 1.833 \\ \cline{1-1}  & STI * & 0.244 & 4.107 & 0.221 & 4.515 & 0.389 & 2.573 \\ \hline \multirow{2}{*}{5} & SLOPE & 0.577 & 1.733 & 0.639 & 1.566 & 0.556 & 1.798 \\  & TWI & 0.577 & 1.733 & 0.639 & 1.566 & 0.556 & 1.798 \\ \hline \multirow{2}{*}{6} & SLOPE & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\ \hline \hline \end{tabular}  
* * In(ST), ** In(SCA).  
\end{table} 
Table 2: The result of multi-collinearity analysis. a mean of zero and a variance of \(\sigma^{2}\). Estimates of regression coefficients can be obtained by \(\hat{\beta}=\big{(}\underline{x}^{T}\underline{x}\big{)}^{-1}\underline{x}^{T}y\). The selection of appropriate independent variables for multi-regression analysis is one of the critical issues. Generally, the methods of selecting independent variables include forward selection, backward elimination, stepwise method, and so on. However, the regression equation does not necessarily include the selected variables. If multi-collinearity exists between the independent variables, another model can be selected without a separate analysis. [PERSON] and [PERSON] [53] described the existing selection method and its shortcomings. They stated that there are many advantages of choosing a suitable model by evaluating the combination of statistics used in model selection and independent variables used in the model. In this study, the adjusted coefficient of determination (\(R_{adj}^{2}\)) and RMSE are used to determine the number of variables to be used in the soil depth prediction model. The coefficient of determination is a statistic that indicates the model's suitability for a given dataset. Therefore, the large value of the coefficient of determination means that the proportion of changes that the independent variable can explain is large.