## 3 Modeling Approaches
### Broad Learning System (BLS)  
The BLS is a model based on the Random Vector Functional-Link Neural Network (RVFLNN), which does not rely on deep structures. The RVFLNN was first proposed by [PERSON] and [PERSON] in 1994 [44]. The basic structure of the RVFLNN is illustrated in Figure 4, where \(XeR^{m\times n}\) represents the input data, \(h\) represents the hidden nodes and \(\omega_{h}\) represents the weight between the enhancement layer and the input layer. The sum of all these weights can be denoted as \(W_{h}=\left[\omega_{h_{1}},\omega_{h_{2}},\cdots,\omega_{h_{j}}\right]\), and the sum of all these enhancement nodes can be denoted as \(H=\left[h_{1},h_{3},\cdots,h_{j}\right]\). The combination of \(H\) with \(X\), as the input to the prediction process, is denoted as \(A\). \(Y\epsilon R^{m\times c}\) represents the output, and \(W_{o}\) represents the weight between \(A\) and the output layer. Therefore, the RVFLNN can be formulated as  
\[Y=\left[X,h_{1},h_{2},\cdots,h_{j}\right]W_{o}=\sum_{i=1}^{j}\left[X\big{|} \xi\big{(}X\omega_{h_{i}}+\beta_{i}\big{)}\right]W_{o}=AW_{2} \tag{1}\]  
where \(A=\left[X|\xi(XW_{h}+\beta)\right]\), \(h_{j}=\xi\big{(}X\omega_{h_{i}}+\beta_{i}\big{)}\). The weight \(W_{h}\) and the bias \(\beta\) of the enhancement nodes are initialized randomly. Only the weight \(W_{o}\) in the whole model needs to be obtained by training. The network typically selects the non-linear activation function \(\xi\) as either \(sig(x)\) or \(tanh(x)\).  
The structure of the BLS is based on the RVFLNN, with some improvements to the input. The BLS no longer uses the input data \(X\) to obtain the enhancement layer and the output directly, Instead, the input \(X\in R^{m\times n}\) is first processed to obtain the feature nodes, and then the enhancement nodes are obtained from the feature nodes. The feature nodes and enhancement nodes collectively form the input \(A\) to the prediction process. The process from \(X\) to the map's features and the process from the map's features to the enhancement nodes can be expressed by following equations, respectively  
\[Z_{i}=\phi(XW_{ei}+\beta_{ei}),i=1,2,\cdots,n \tag{2}\]  
\[H_{j}=\xi\big{(}Z^{n}W_{h_{j}}+\beta_{h_{j}}\big{)},j=1,2\cdots,m \tag{3}\]  
In these formulae, \(n\) groups of map features make up the feature node layer, denoted as \(Z^{n}=\left[Z_{1},Z_{2},\cdots,Z_{n}\right]\). The enhancement layer, denoted as \(H^{m}=\left[H_{1},H_{2},\cdots,H_{m}\right]\), consists of a set of \(m\) enhancement nodes. The weights and biases corresponding to the map's features and the enhancement nodes are randomly generated.  
Figure 4: Structural diagram of the RVFLNN.  
The weight between the output and A is defined as \(W^{n}\), and the output results can be formulated as  
\[Y=[Z_{1},\cdots,Z_{n}|H_{1},\cdots,H_{m}]W^{n}=[Z_{n}|H_{m}]W^{n}=AW^{n} \tag{4}\]  
where \(W^{n}=[Z^{n}|H^{m}]^{+}Y=A^{+}Y\) is the result that needs to be obtained by training and is the key to making predictions. \(W^{n}\) can be quickly calculated by the following equation.  
\[A^{+}=\underset{\lambda\to 0}{\text{lim}}\Big{(}\lambda I+AA^{T}\Big{)}^{ -1}A^{T} \tag{5}\]  
In this study, 160 data samples, each containing 12 impact factor values, were used as input to the model, denoted by \(X\). The feature nodes generated from the input were denoted by \(Z\), and the enhanced nodes generated from the feature nodes were denoted by \(H\). Together, they constituted the prediction input \(A\). The landslide conditions of the 160 pieces of data were replaced by 0 and 1 as the target output \(Y\). The \(A^{+}\) required for the prediction was eventually obtained by calculating the pseudo-inverse. Figure 5 illustrates the architecture of the BLS model.