## 5 Modelling Based on EPR Method  
### Developing Prediction Models for \(\Delta H\) and CSR  
The dataset described in Section 4 was employed to develop EPR models. The development of the EPR models was undertaken by assuming the influence of the four variables sand content (SC), normal stress (\(\sigma_{n}\)), shear stress level (SL), and relative density (Dr) on the collapse settlement (\(\Delta\)H) and coefficient of stress release (CSR). Due to the small changes in the percentage of clay content, this variable was not considered in the modelling, and the initial moisture was also assumed to be constant. The overall procedure for developing the predictive models for \(\Delta\)H and CSR is depicted in Figure 3.  
After normalization and partitioning of data, the modelling process was completed using the EPR method. This process is based on inner functions, such as the exponential, natural logarithm, sinusoidal, tangent hyperbolic, and secant hyperbolic. The EPR-MOGA program, a MATLAB tool, was employed to develop the EPR models. The best model for each of the functions and structures was selected based on the smallest error in the modelling. The structure of EPR model for the prediction of the collapse settlement was developed to be similar to Equation (17) in the secant hyperbolic and natural and exponential logarithms, Equation (18) in the tangent hyperbolic, and Equation (19) in the sinusoidal model. Furthermore, to predict the coefficient of stress release, the structure of EPR model was assumed to be similar to Equation (17) in the secant hyperbolic and exponential models, Equation (18) in tangent hyperbolic and natural logarithm models, and Equation (19) in the sinusoidal model. \[Y=sum(a_{i}\times X_{1}\times X_{2}\times f(X_{1})\times f(X_{2}))+a_{0} \tag{17}\]  
\[Y=sum(a_{i}\times X_{1}\times X_{2}\times f(X_{1}\times X_{2}))+a_{0} \tag{18}\]  
\[Y=\sin(sum(a_{i}\times X_{1}\times X_{2})+a_{0}) \tag{19}\]  
Figure 3: Overall procedure for developing EPR models. The other details related to the development of the EPR predictive models are shown in Tables 3 and 4. The representative range shown in the tables determined the linearity or nonlinearity of the model, as well as the direct or reverse dependence between the input variables and output variable. The optimal models for the different functions for predicting collapse settlement are presented in Equations (20)-(24), and those for predicting the coefficient of stress release are shown in Equations (25)-(29). **Secant hyperbolic**  
\[ \begin{array}{c}\Delta H=4.3437\cdot\frac{1}{sech(SC)^{0.5}}+0.36122\cdot \frac{1}{sech(\sigma_{n}).sech(SL)}-0.064019\cdot Dr^{0.5}.\frac{sech(SC)^{0.5} sech(Dr)}{sech(SL)}\\ \qquad\qquad-0.46804\cdot SC.\frac{1}{sech(SC).sech(\sigma_{n})}+0.68941\cdot SC.\sigma_{n}.sech(\sigma_{n})^{0.5}-4.6782\end{array}  \tag{20}\]  
**Tangent hyperbolic**  
\[ \begin{array}{c}\Delta H=-1.8072\cdot\tanh\Bigl{(}\frac{1}{5\lambda}\Bigr{)} -0.8547\cdot\tanh\Bigl{(}\frac{1}{5\lambda}\Bigr{)}+0.34027\cdot SL^{0.5}. \tanh\Bigl{(}\frac{SC.\sigma_{n}}{SL^{0.5}}\Bigr{)}\\ \qquad\qquad-0.1282\cdot SL.\tanh\Bigl{(}\frac{SC^{0.5}Dr^{0.5}}{SL}\Bigr{)}+ 0.24921\cdot\sigma_{n}.\tanh\Bigl{(}\frac{1}{SC.\sigma_{n}.SL^{0.5}}\Bigr{)}+ 2.6271\end{array}  \tag{21}\]  
**Natural logarithm**  
\[ \begin{array}{c}\Delta H=-2.9734\cdot Ln(SC+1)+0.33875\cdot SL.Ln(SL+1)+0.24 728\cdot\sigma_{n}.Ln(\sigma_{n}+1)^{0.5}\\ \qquad\qquad+0.26894\cdot SC^{0.5}.Ln(\sigma_{n}+1)^{0.5}+2.4068\cdot SC-0.025 247\end{array}  \tag{22}\]  
**Exponential**  
\[ \begin{array}{c}\Delta H=+0.20777\cdot\exp\left(0.5\cdot\sigma_{n}-SL\right) +0.43164\cdot\exp\left(SC\right)-0.028606\cdot Dr^{0.5}.\exp\left(SL\right)+ \\ \qquad\qquad\qquad 0.12536\cdot SL^{0.5}.\exp\left(0.5\cdot\sigma_{n}+SL\right)-0. 37924\cdot SC^{0.5}.\exp\left(-\sigma_{n}\right)-0.63544\end{array}  \tag{23}\]  
**Sinusoidal**  
\[\Delta H=\sin\Bigl{(}+0.15549\cdot SL+0.2798\cdot\sigma_{n}-0.88635\cdot SC^{ 0.5}+1.3442\cdot SC+0.52595\cdot SC.\sigma_{n}.SL-0.056354\Bigr{)} \tag{24}\]