## 3 Study Methods and Evaluation Process
### Determination of Index Weight
#### 3.2.3 Examples of Index Weight Calculation  
Fourteen partitions are selected as a sample set from the evaluation results of five regions. In terms of the criteria layer factors in this paper, the decision table is constructed where the geotechnical characteristics (\(B_{1}\)), geomorphology (\(B_{2}\)), geological structure (\(B_{3}\)) and hydrogeology (\(B_{4}\)) are considered as the conditional attributes, and the actual classification of underground space suitability is considered as the decision attribute (\(D\)). Table 3 shows the discretization results of the sample data.  
The calculation process of index weight is as follows:  
1. According to the process shown in Figure 3a, the sample data are processed from three aspects. (1) Removing duplicate samples: the fifth group and the tenth group are duplicate samples, thus the tenth group is removed; (2) Removing contradictory samples: the level of each conditional attribute of the third group is two, one, two and two, respectively, which are all higher than the level three of decision attribute. It is not consistent with the reality, so let us dispose of the third group. Similarly, the seventh group is removed; (3) Cleaning conflict samples: the sixth group and the first group have the same level of conditional attributes, but different levels of decision attributes are contradictory, so the sixth group is removed.  
Through data pre-processing, sample data \(x_{1}\), \(x_{2}\), \(x_{4}\), \(x_{5}\), \(x_{8}\), \(x_{9}\), \(x_{11}\), \(x_{12}\), \(x_{13}\) and \(x_{14}\) are retained to form the decision table. (2) Calculation according to rough set theory and conditional entropy.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**No. of Sample (X)** & \(B_{1}\) & \(B_{2}\) & \(B_{3}\) & \(B_{4}\) & \(D\) \\ \hline \(x_{1}\) & 1 & 2 & 1 & 2 & 2 \\ \(x_{2}\) & 2 & 3 & 2 & 4 & 3 \\ \(x_{3}\) & 2 & 1 & 2 & 2 & 3 \\ \(x_{4}\) & 2 & 3 & 3 & 3 & 3 \\ \(x_{5}\) & 3 & 3 & 2 & 3 & 3 \\ \(x_{6}\) & 1 & 2 & 1 & 2 & 1 \\ \(x_{7}\) & 3 & 2 & 2 & 3 & 4 \\ \(x_{8}\) & 2 & 5 & 3 & 5 & 3 \\ \(x_{9}\) & 3 & 3 & 2 & 4 & 3 \\ \(x_{10}\) & 3 & 3 & 2 & 3 & 3 \\ \(x_{11}\) & 3 & 4 & 3 & 5 & 4 \\ \(x_{12}\) & 4 & 4 & 4 & 3 & 4 \\ \(x_{13}\) & 3 & 5 & 3 & 5 & 4 \\ \(x_{14}\) & 4 & 5 & 4 & 4 & 4 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Sample data discretization of criterion layer.  
\[U/(B) =\{\{x_{1}\},\{x_{2}\},\{x_{4}\},\{x_{5}\},\{x_{8}\},\{x_{9}\},\{x_{ 11}\},\{x_{12}\},\{x_{13}\},\{x_{14}\}\}\] \[U/(D)=\{\{x_{1}\},\{x_{2},x_{4},x_{5},x_{8},x_{9}\},\{x_{11},x_{12},x_{13},x_{14}\}\}=\{D_{1},D_{2},D_{3}\}\] \[U/(B_{1})=\{\{x_{1}\},\{x_{2},x_{4},x_{8}\},\{x_{5},x_{9},x_{11},x _{13}\},\{x_{12},x_{14}\}\}\] \[U/(B-B_{1})=\{\{x_{1}\},\{x_{2},x_{9}\},\{x_{4}\},\{x_{5}\},\{x_{8 },x_{13}\},\{x_{11}\},\{x_{12},\{x_{14}\}\}\] \[X_{1}\cap D_{1}=\{x_{1}\},X_{1}\cap D_{2}=\Phi,X_{1}\cap D_{3}=\Phi\] \[\ldots\] \[X_{10}\cap D_{1}=\Phi,X_{10}\cap D_{2}=\Phi,X_{10}\cap D_{3}=\{x_ {14}\}\] \[\therefore I(D|B)=\left(\frac{1}{10}\right)^{2}\left[\frac{1}{1} \times\left(1-\frac{1}{1}\right)+\frac{1}{7}\times\left(1-\frac{1}{7}\right)+ \frac{1}{7}\times\left(1-\frac{1}{7}\right)\right]\times 10=0\]  
Similarly, \(I(D|B_{1})=0.08\) and \(I(D|B-B_{1})=0.02\) can be calculated.  
According to Equation (7), the importance of condition attribute \(B_{1}\) is calculated as follows:  
\[\begin{array}{c}New\;\;\mathit{Sig}(B_{1})=I(D|B-B_{1})-I(D|B)=0.02\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Based on the above decision table construction, sample data processing and the calculation process of attribute weight, the conditional entropy, importance degree and attribute weight of other factors in the index layer are calculated, respectively. The calculation results are shown in Table 6.