# Application of Artificial Neural Networks in Assessing Mining Subsidence Risk
## 4 Analysis and Discussion
### Analysis B
#### 4.2.2 Indirect Influence Factor Group  
An interaction matrix is a tool used to represent correlations among multiple components simultaneously. The given components are aligned horizontally in the matrix so that their correlations are expressed in parallel links rather than in one-to-one relations. The approach is based on the concept that every factor of an actual location is related to every other factor, albeit to varying degrees [48]. [PERSON] [49] positioned subject components as diagonal entries and represented their interactions in the clockwise direction. As shown in Figure 11, the major components A and B are placed as diagonal entries, and the components are linked in the clockwise direction. Accordingly, the components to the right of A and above B represent the sensitivity of A toward B. Conversely, the components to the left of B and below A represent the sensitivity of B toward A. The significance of each component of the matrix can be expressed with partial derivatives. Although interactions can be represented with the matrix in a simple manner, the critical question is how the sensitivity of each component should be determined for best use of the concept for analysis and application of field data [49; 50]. Although only two factors were used to explain the concept in Figure 11, the interaction matrix can be constructed for multiple factors based on the identical concept. Although the matrix is mathematically capable of expressing the correlations among all the factors in diagonal entries, during the initial stage, [PERSON] [49] constructed each component of the matrix, indicating sensitivity by independently considering the linear relationship between linked components (A and B in Figure 11) and intuitively assigning values to the matrix. Based on the concept that two or more components on the matrix diagonal are related to each other with varying degrees of influence, [PERSON] and [PERSON] [51] proposed an algorithm for constructing an interaction matrix for which all of the component values of the matrix are fully coupled. However, each component of the interaction matrix is still determined based on the intuition of experts, making its objective assessment difficult. [PERSON] and [PERSON] [50] proposed constructing an interaction matrix by  
Figure 11: Principle of interaction matrix (modified after [49]). \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multicolumn{3}{c}{**Depth (m, F\({}_{\text{a}}\))**} & \multicolumn{3}{c}{**Height (m, F\({}_{\text{b}}\))**} & \multicolumn{3}{c}{**Correction**} & \multicolumn{2}{c}{**Distance (m, F\({}_{\text{c}}\))**} & \multicolumn{2}{c}{**Correction**} \\ \hline
**Coal** & **Other** & \multicolumn{3}{c}{**Coal**} & \multicolumn{3}{c}{**Other**} & \multicolumn{3}{c}{**Factor**} & \multicolumn{2}{c}{**Coal and Other**} & \multicolumn{2}{c}{**Factor**} \\ \hline \(<\) & 1000 & \(<\) & 1000 & 10 & \(<\) & 1000 & \(<\) & 1000 & 1.0 & \(<\) & 1000 & 0.1 \\ \(<=\) & 60 & \(<=\) & 30 & 10 & \(<\) & 27 & \(<=\) & 4  
using the ANN. To do so, they introduced the concept of the relative strength effect (RES) and an ANN-based calculation algorithm to assess the linkage strength among the components. For this study, a RES value between 0 and 1 was used to represent the mutual relationship between components. Figure 12 shows the architecture of the ANN specially organized for sensitivity analysis, which consists of the same number of nodes for one input layer, one output layer, and two hidden layers. Once training is complete, a partial derivative of an input node can be calculated for each output node using the ANN sensitivity analytical system [39]. All five components of the interaction matrix can then be assigned with the calculated partial derivatives. Figure 13 shows an interaction matrix constructed using this process. Every value on the diagonal calculated by the trained ANN is 1.0, which is natural because, in theory, a partial derivative of an identical component should always be 1.0.