# Application of Soft Computing Techniques for Predicting Thermal Conductivity of Rocks
## 4 Results
### GEP Model  
The GEP model was developed through the GenXPro Tools software. This software extracts the significant features of datasets, including a high number of variables, and finds relationships among them with high accuracy. Similar to the process used in Section 4.1, the same training and testing subsets were used for the GEP model's development. To develop a model for TC prediction by GEP, a simple equation in the form of \(TC=f(P\text{-}wave,\,\,\text{}proosity,\,\,\text{}UCS,\,\text{}density)\) was first proposed. To acquire the best setting for the GEP model generation in the initial step, function sets and fitness functions were then chosen from the study of [PERSON] et al. [69]. Afterward, several trial-and-error procedures were carried out to obtain the best settings. Having utilized these settings, the GEP-developed models obtained minimal error percentages. Table 5 presents the GEP software settings for the model generation during this study. The procedure of GEP modeling was illustrated in Section 3. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{**Data Status**} & \multicolumn{2}{c}{**R2**} & \multicolumn{2}{c}{**RMSE**} \\ \cline{3-6}  & & **Training** & **Testing** & **Training** & **Testing** \\ \hline NLMR & 65\% of all for training & & & & \\  & 35\% of all for testing & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Resulting performance indices’ values of the proposed NLMR model. \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{1}{c}{**Parameter**} & \multicolumn{1}{c}{**Value**} \\ \hline Number of chromosomes & 30 \\ Head size & 8 \\ Number of genes & 4 \\ Linking function & Addition \\ Fitness function & RMSE \\ Mutation rate & 0.044 \\ Inversion rate & 0.1 \\ One-point recombination rate & 0.3 \\ Two-point recombination rate & 0.3 \\ Gene recombination rate & 0.1 \\ Insertion sequences transposition rate & 0.1 \\ Root insertion sequence transposition rate & 0.1 \\ Gene transposition rate & 0.1 \\ Function set & \(+,-,\times,\,\times,\,\div,\,\sqrt{x},\,e^{x},\,\sin,\cos\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameters of the GEP model. The prediction of the GEP performance models was evaluated by both R-square and RMSE. Several models were developed to find a better model with the lowest RMSE and highest R-square. Table 6 gives the prediction performance of the selected GEP model. The developed models are presented in terms of expression trees (ETs) or as computer codes. However, these presentations should be interpreted as a form of a mathematical equation. The extraction of the mathematical equation from ETs is an easy task. The ETs are read from left to right and bottom to the top. The ET of each gene of the GEP model is shown in Figure (a)a-d, and the mathematical equation of each gene is presented as Equations (5)-(8). The genes' equations were linked, and the GEP model was generated using Equation (9). \[\text{SubET1}=\cos\sqrt[3]{\phi\times\cos(\frac{\rho+\text{UCS}-\text{V}_{ \text{p}}}{10.61-\rho})^{2}} \tag{5}\]  
\[\text{SubET2}=(\cos(\cos(0.973-\frac{\text{V}_{\text{p}}+\text{UCS}}{\rho-\phi} )))^{2} \tag{6}\]  
\[\text{SubET3}=\frac{1}{\sqrt[3]{\phi^{2}}}+\frac{1}{\phi^{3}} \tag{7}\]  
\[\text{SubET4}=\cos(\cos(-1-\sqrt[3]{\tan(2\rho)-\text{UCS}+1.71})) \tag{8}\]  
\[\lambda=\text{SubET1}+\text{SubET2}+\text{SubET3}+\text{SubET4} \tag{9}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{**Data Status**} & \multicolumn{2}{c}{**R2**} & \multicolumn{2}{c}{**RMSE**} \\  & & **Training** & **Testing** & **Training** & **Testing** \\ \hline \multirow{2}{*}{**GEP**} & 65\% of all for training & \multirow{2}{*}{0.95} & \multirow{2}{*}{0.90} & \multirow{2}{*}{0.17} & \multirow{2}{*}{0.22} \\  & 35\% of all for testing & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Resulting performance indices’ values of the proposed GEP model.