## 3 Materials and Methods
### Methodology
#### 3.2.4 LSM Based on the Graph Theory and Multi-Modal Classification  
This study presents a new landslide susceptibility mapping method based on the graph theory and multi-modal classification. The presented method contains four main steps: (a) feature extraction from each data modality; (b) graph construction based on features of each data modality; (c) nonlinear graph fusion; and (d) landslide susceptibility mapping. Figure 9 illustrates the flowchart of the proposed method. This study groups 15 causative factors into 4 data modalities: (1) seismic modal, including peak ground acceleration, distance to fault, seismic intensity, and lithology; (2) terrain modal, including elevation, slope, aspect, profile curvature, and landform; (3) land cover modal, including land use, soil, and distance to river; and (4) hydrological modal, including distance to river, topographic wetness index, and annual average rainfall. Figure 8: Importance of 15 causative factors. After the feature extraction, this study applies the graph theory to represent attribute values of slope units (SUs). For each modal data, we build a graph according to its features. Suppose the study area contains n SUs and each SU has I features derived from m modalities. Using features from \(i\)-th modality, we construct a graph \(G^{i}=(V^{i},E^{i}),i=1,\ldots,m\) according to model relations between n SUs, where \(V^{i}\) represents n SUs and \(E^{i}\) are weighted by how similar SUs are. \(W^{i}\) is a \(n\times n\) similarity matrix that denotes edge weights, \(W^{i}(a,b)\) represents the similarity between SU \(a\) and SU \(b\) in \(i\)-th modality. This study calculates the pairwise Euclidean distance and then converts it to similarity matrix by the exponential kernel. This study applies random forest to calculate the similarity between pairs of subjects. The similarity matrix provides a consistent measure for pairwise similarity of SUs, and thus finds a way to incorporate information from multiple modalities with a unified measure. The graph construction generates a similarity matrix \(W^{i}\) corresponding to \(i\)-th modality. These similarity matrices are then combined to make full use of the complementary information between different modalities. Since different modalities are not necessarily linearly related, a nonlinear graph fusion is utilized for fusion. The similarity matrix \(W^{i}\) is normalized as follows:  
\[\widetilde{W}^{i}=\frac{W^{i}}{D^{i}}, \tag{1}\]  
Figure 9: Workflow of LSM based on the graph theory and multi-modal classification. where \(D^{i}(a,a)=\sum_{b=1}^{n}W^{i}(a,b)\) is the diagonal matrix and \(\sum_{b=1}^{n}\widetilde{W}^{i}(a,b)=1\). The occurrence of self-similarity on the diagonal of \(W^{i}\) renders the normalization algorithm in Equation (1) unstable. To tackle this issue, the study normalizes \(W^{i}\) as:  
\[\widetilde{w}^{i}(a,b)=\left\{ \begin{array}{l}\frac{w^{i}(a,b)}{2\sum_{b=1}^ {n}w^{i}(a,b)},\;\text{b}\
eq\text{a}\\ 1/2\,,\text{b}=\text{a}\end{array} \right., \tag{2}\]  
The normalization in Equation (2) not only speeds up the convergence rate, but also guarantees the full rank of the final graph. The sparse matrix \(\widehat{S}^{i}\) of \(W^{i}\) is calculated by k-nearest neighbors (KNN) to measure local affinity as:  
\[\widehat{s}^{i}(a,b)=\left\{ \begin{array}{l}\frac{\widetilde{w}^{i}(a,b)}{ \sum_{b\in k-NN(a)}\widetilde{w}^{i}}\;\text{if}\;\text{b}\in\text{k}-\text{NN }(\text{a})\\ 0\;\text{otherwise}\end{array} \right., \tag{3}\]  
To keep strong connections and discard weak connections between SUs, the weighted edges between non-neighborhood SUs are set to 0. This is beneficial to improve the robustness to the noise of similarity measures. Given features from m modalities, based on Equations (2) and (3), we obtain m normalized matrices \(\widetilde{W}^{i}\) and m sparse matrices \(\widehat{S}^{i}\).