## 3 Results and Discussion
### Multiple Linear Regression  
Multiple linear regression examines the simultaneous effect of several independent variables on a dependent variable. Various combinations of rock index properties were used to predict the KIC of the studied rocks. In addition, for verifying the accuracy of the results, statistical accuracy controllers, including root mean squared error (RMSE), VIF, R\({}^{2}\), R\({}^{2}\) Adjust, and \(p\)-Value, were used. Initially, the \(p\)-value index was used to control the overall validity of the proposed models. A proposed model was considered to be statistically correct if its \(p\)-value was less than 0.05. The closer to zero, the more efficient the model. Then, the validity of each of the dependent variables was determined using the VIF criterion and the \(p\)-value. The value of the VIF criterion for each variable used in the model should be less than 10. The closer the RMSE indicator is to zero, the more functional the model. The most reliable models according to the statistical controllers are presented in Table 9. From this table, it can be clearly seen that by taking into account the petrographic characteristics of the studied limestones, their fracture toughness can be successfully estimated by means of their index properties. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Class** & **Dependent** & **Independent** & **Vialable** & **VIF** & \(p\)**-Value** & **Predictive Model** & **RMSE** & **R2 (\%)** & **Adj. R2** & \(p\)**-Value** \\ \hline \multirow{9}{*}{KIC-Set} & Constant & & 0.000 & \multirow{2}{*}{KIC-Set} & 0.1557 & 85.22 & 84.08 & 0.000 \\  & & V\({}_{\rm F,Set}\) (Km/s) & 1.43 & 0.000 & \multirow{2}{*}{3.123 \(\rm k_{50\,Sat}-0.924\)} & \multirow{2}{*}{0.1934} & \multirow{2}{*}{78.07} & \multirow{2}{*}{75.43} & \multirow{2}{*}{0.000} \\  & & I\({}_{\rm S00\,Sat}\) (MPa) & 1.43 & 0.000 & & & & \\ \cline{2-10}  & Constant & & 0.000 & \multirow{2}{*}{KIC-Set} & 0.1934 & 78.07 & 75.43 & 0.000 \\  & & I\({}_{\rm S00\,Sat}\) (MPa) & 1.43 & 0.000 & & & & \\ \cline{2-10}  & Constant & & 0.000 & \multirow{2}{*}{KIC-Set} & 0.1934 & 78.07 & 75.43 & 0.000 \\  & & I\({}_{\rm S00\,Sat}\) (MPa) & 1.54 & 0.001 & & & & \\ \cline{2-10}  & Constant & & 0.0118 & \multirow{2}{*}{KIC-Set} & 0.1934 & 78.07 & 75.43 & 0.000 \\ All Data & & V\({}_{\rm S}\) (Km/s) & 1.43 & 0.011 & \multirow{2}{*}{KIC-Set} & 0.1540 & & & \\  & p\({}_{\rm S00\,Sat}\) (Dm) & 1.32 & 0.030 & \multirow{2}{*}{p\({}_{\rm S00\,Sat}-0.337\)} & \multirow{2}{*}{0.1786} & \multirow{2}{*}{82.03} & \multirow{2}{*}{79.04} & \multirow{2}{*}{0.000} \\  & p\({}_{\rm S00\,Sat}\) (MPa) & 1.48 & 0.030 & & & & & \\ \cline{2-10}  & I\({}_{\rm S00\,Sat}\) (MPa) & 1.54 & 0.000 & & & & & \\ \cline{2-10}  & & V\({}_{\rm S}\) (Km/s) & 9.06 & 0.008 & \multirow{2}{*}{KIC-Set} & 0.1936 & 96.30 & 95.87 & 0.000 \\  & & I\({}_{\rm S00\,Sat}\) (MPa) & 8.69 & 0.000 & & & & & \\ \cline{2-10}  & & \({}_{\rm S00\,Sat}\) (\(\Omega\).m) & 1.61 & 0.001 & \multirow{2}{*}{KIC-Set} & 0.00024 & \multirow{2}{*}{0.2182} & \multirow{2}{*}{95.11} & \multirow{2}{*}{94.75} & \multirow{2}{*}{0.000} \\  & I\({}_{\rm S00\,Sat}\) (MPa) & 1.61 & 0.000 & & & & & \\ \hline \multirow{9}{*}{KIC-Dy} & V\({}_{\rm VIPO}\) (Km/s) & 1.18 & 0.001 & \multirow{2}{*}{KIC-Dy} & \multirow{2}{*}{0.116301 V\({}_{\rm F,Dy}+\)} & \multirow{2}{*}{0.1778} & \multirow{2}{*}{70.00} & \multirow{2}{*}{69.72} & \multirow{2}{*}{0.000} \\  & I\({}_{\rm S00\,Dy}\) (MPa) & 1.18 & 0.000 & & & & & \\ \cline{2-10}  & Constant & & 0.000 & \multirow{2}{*}{KIC-Dy} & \multirow{2}{*}{-0.03318 n + 0.2757} & \multirow{2}{*}{0.1677} & \multirow{2}{*}{73.56} & \multirow{2}{*}{71.60} & \multirow{2}{*}{0.000} \\  & I\({}_{\rm S00\,Dy}\) (MPa) & 1.14 & 0.000 & & & & & \\ \cline{2-10}  & Constant & & 0.000 & \multirow{2}{*}{KIC-Dy} & \multirow{2}{*}{-0.0701 W.A. +} & \multirow{2}{*}{0.1711} & \multirow{2}{*}{72.48} & \multirow{2}{*}{70.45} & \multirow{2}{*}{0.000} \\  & W.A.