## 4 Results and Discussion
### Importance of Each Variable Based on Random Forest  
RF algorithm was used to examine the effects of input variables on aspect ratio and crack density. The 130 data set including compressional wave velocity, shear wave velocity, total porosity, crack porosity, equant porosity, derived during 13 weathering processes in 10 rocks, was used. The selected input variables are normalized through the technique of min-max scaling, as shown in Equation (13),  
Figure 10: Change ratios for (**a**) aspect ratio and (**b**) crack density. The 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 denote samples albite 1, albite 2, albite 3, albite 4, quartz 1, quartz 2, quartz 3, biotite, calcite, and chlorite, respectively.  
to remove the bias that occurred by magnitude and unit of variables, and thus the data set was normalized in the range of \(-1\) to \(1\).  
\[\text{RF}_{\text{min-max scaling}}=\left(\frac{\text{IV}-\text{IV}_{\text{min}}}{ \text{IV}_{\text{max}}-\text{IV}_{\text{min}}}\right)\left(\overline{\text{IV} _{\text{max}}}-\overline{\text{IV}_{\text{min}}}\right)+\overline{\text{IV}_{ \text{min}}} \tag{13}\]  
where IV denotes the input variable. The \(\text{IV}_{\text{max}}\) and \(\text{IV}_{\text{min}}\) mean the maximum and minimum values of input variable. The \(\overline{\text{IV}_{\text{max}}}\) and \(\overline{\text{IV}_{\text{min}}}\) are \(-1\) and \(1\), respectively.  
The number of random data set and tree were determined to \(20\) and \(10\) under the condition that the accuracy is \(0.876\). The accuracy was calculated with true positive (TP), false positive (FP), false negative (FN), and true negative (TN) through Equation (14).  
\[\text{Accuracy}=\frac{\left(\text{TP}+\text{TN}\right)}{\left(\text{TP}+ \text{TN}+\text{FP}+\text{FN}\right)} \tag{14}\]  
The important score of each variable for estimating aspect ratio is calculated by using Equation (8) and it was addressed in Figure 11. The highest affecting parameters to determine aspect ratio are crack density (\(0.32\) point), crack density (\(0.35\) point), total porosity (\(0.37\) point), crack density (\(0.27\) point), shear wave velocity (\(0.32\) point), equant porosity (\(0.37\) point), crack porosity (\(0.18\) point), crack density (\(0.16\) point), crack porosity (\(0.45\) point) and crack porosity (\(0.45\) point) for albite \(1\), albite \(2\), albite \(3\), albite \(4\), quartz \(1\), quartz \(2\), quartz \(3\), biotite, calcite, and chlorite, respectively. The important scores were calculated considering overall characteristics of the variables since there are various variables to derive the aspect ratio. Although the factors with high scores were slightly different for characterization of each rock, the factors that had the highest influence on the aspect ratio through averaged points were in the order of crack porosity (\(0.20\) point) \(>\) total porosity (\(0.19\) point) \(>\) shear wave (\(0.16\) point) \(>\) crack density (\(0.15\) point) \(>\) equant porosity (\(0.12\) point) \(>\) compressional wave (\(0.10\) point).  
The GRG was selected to find the correlation between pairwise variables of aspect ratio, crack density, compressional wave velocity, shear wave velocity, total porosity, crack porosity and equant porosity. The Figure 12 shows the GRG value in each panel and, the mean, minimum and maximum values of GRG are \(0.52\), \(0.99\), and \(0.1\), respectively. It indicates the crack porosity is the most significant effect on both crack density and aspect ratio. It shows that the accuracy of crack porosity is important to obtain a reliable aspect ratio and crack density.  
Figure 11: Important scores of each variable.