## 2 Materials and Methods
### Support Vector Machine  
[PERSON], [PERSON], and [PERSON] were the first to formulate and introduce the support vector machine (SVM) [40]. In the case of non-separable data, to accommodate errors for certain objects \(i\), the \"ideal boundary\" must be introduced:  
\[\left\{\begin{array}{l}\text{minimize}\Big{(}\frac{1}{2}|\delta|^{2}+C{\sum_{ i=1}^{n}\xi_{i}}\Big{)}\\ \text{under the constraints}\ y_{i}(b+\delta\cdot x_{i})+\xi_{i}\geq 1\text{ and }\xi_{i}\geq 0\end{array}\right. \tag{1}\]  
where \(C\) is the penetrating parameter; \(\delta\) and \(b\) are, respectively, the normal vector and the bias of the hyperplane; and each \(\xi_{i}\) refers to the distance within object \(i\) and the respective margin hyperplane [41; 42].  
Data are implicitly mapped to a higher-dimensional space through mercer kernels, which can be broken down into a dot product to learn nonlinearly separable functions  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Data Set** & **Min Value** & **Max Value** & **Mean** & **Standard** \\ \hline \multirow{2}{*}{\(D_{10}\) (mm)} & Training & 0.010 & 33.900 & 4.857 & 9.179 \\ \cline{2-6}  & Testing & 0.010 & 33.900 & 2.887 & 7.453 \\ \hline \multirow{2}{*}{\(D_{30}\) (mm)} & Training & 0.560 & 42.400 & 8.465 & 10.577 \\ \cline{2-6}  & Testing & 0.560 & 42.400 & 5.442 & 9.050 \\ \hline \multirow{2}{*}{\(D_{60}\) (mm)} & Training & 1.200 & 80.100 & 19.287 & 15.135 \\ \cline{2-6}  & Testing & 1.200 & 50.000 & 14.252 & 10.349 \\ \hline \multirow{2}{*}{\(D_{90}\) (mm)} & Training & 2.600 & 100.000 & 40.386 & 22.018 \\ \cline{2-6}  & Testing & 2.600 & 99.000 & 38.091 & 24.289 \\ \hline \multirow{2}{*}{\(C_{C}\)} & Training & 0.100 & 22.270 & 2.199 & 3.075 \\ \cline{2-6}  & Testing & 0.100 & 22.270 & 3.226 & 4.492 \\ \hline \multirow{2}{*}{\(C_{U}\)} & Training & 1.360 & 1040.000 & 53.324 & 156.064 \\ \cline{2-6}  & Testing & 1.470 & 1040.000 & 134.510 & 294.958 \\ \hline \multirow{2}{*}{GM} & Training & 0.200 & 6.000 & 2.788 & 1.243 \\ \cline{2-6}  & Testing & 0.200 & 6.000 & 3.365 & 1.331 \\ \hline \multirow{2}{*}{FM} & Training & 3.000 & 8.800 & 6.250 & 1.261 \\ \cline{2-6}  & Testing & 3.000 & 8.800 & 5.709 & 1.374 \\ \hline \multirow{2}{*}{\(R\)} & Training & 1.000 & 6.000 & 4.364 & 0.910 \\ \cline{2-6}  & Testing & 1.000 & 5.000 & 4.182 & 1.131 \\ \hline \multirow{2}{*}{\(UCS_{min}\) (MPa)} & Training & 1.000 & 250.000 & 75.045 & 39.230 \\ \cline{2-6}  & Testing & 1.000 & 100.000 & 68.273 & 32.444 \\ \hline \multirow{2}{*}{\(UCS_{max}\) (MPa)} & Training & 5.000 & 400.000 & 170.682 & 88.010 \\ \cline{2-6}  & Testing & 5.000 & 250.000 & 159.545 & 87.957 \\ \hline \multirow{2}{*}{\(\gamma\) (KN/m\({}^{3}\))} & Training & 9.320 & 38.900 & 20.766 & 4.605 \\ \cline{2-6}  & Testing & 9.320 & 38.900 & 20.932 & 5.854 \\ \hline \multirow{2}{*}{\(\sigma_{n}\) (MPa)} & Training & 0.002 & 4.205 & 0.729 & 0.780 \\ \cline{2-6}  & Testing & 0.021 & 3.223 & 0.756 & 0.816 \\ \hline \multirow{2}{*}{\(\tau\) (MPa)} & Training & 0.005 & 3.921 & 0.660 & 0.662 \\ \cline{2-6}  & Testing & 0.024 & 2.492 & 0.668 & 0.619 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Statistical parameters of the training and testing data sets.  
\(K\big{(}x_{i},x_{j}\big{)}=\varphi(x_{i})\cdot\varphi(x_{j})\)[42]. The kernel of the radial basis function (RBF) that is used widely is listed below:  
\[K\big{(}x_{i},x_{j}\big{)}=\exp\Bigl{(}-\sigma\|x_{i}-x_{j}\|^{2}\Bigr{)} \tag{2}\]  
where \(\sigma\) is the kernel parameter.