## 3 Fuzzy-BLS Model for Slope Risk Assessment  
### Fuzzy-BLS Risk Assessment Model Structure  
A fuzzy broad learning system (Fuzzy-BLS) is a fuzzy neural network model proposed by [PERSON] and [PERSON] [21] in 2018, combining the advantages of fuzzy logic and a generalized learning system to deal with uncertainty and ambiguity. As landslides are characterized by multifactorial effects, nonlinearities, variable features, and large time lags, slope risk assessment faces strong uncertainties. Our Fuzzy-BLS was obtained by replacing the feature layer of a broad learning system with a Takagi-Sugeno-Kang fuzzy subsystem. Unlike traditional neural networks with a multilayer structure, a Fuzzy-BLS has a flat structure consisting only of a fuzzy subsystem layer and an augmentation layer. It employs the k-means algorithm to cluster the input data to reduce computational complexity. This allows the network to be computed quickly, and it can be arranged in parallel in the network without imposing a large computational burden. The structure of our Fuzzy-BLS is shown in Figure 4 and consists of four main parts: inputs, fuzzy subsystems, enhanced nodes, and outputs. Figure 3: Ranking of indicator weights. Slopes of different risk classes differ in key assessment metrics, and a Fuzzy-BLS aims to extract these differences and translate them into differences in the output of a neural network to assess slope risk classes. In the slope landslide risk evaluation dataset, \(\{F,\Theta\}\), \(F\) represents the input F\({}_{1}\)-F\({}_{10}\) key evaluation indicators of the Fuzzy-BLS model, and \(\Theta\) is the output of the Fuzzy-BLS model. \(\Theta\) is an M \(\times\) 4 matrix, which can be expressed in Equation (9) as follows:  
\[\Theta= \begin{bmatrix}\Theta_{1}\\ \Theta_{2}\\ \vdots\\ \Theta_{f}\\ \Theta_{g}\end{bmatrix} = \begin{bmatrix}\theta_{1-A}&\theta_{1-B}&\theta_{1-C }&\theta_{1-D}\\ \theta_{2-A}&\theta_{2-B}&\theta_{2-C}&\theta_{2-D}\\ \vdots&\vdots&\vdots&\vdots\\ \theta_{f-A}&\theta_{f-B}&\theta_{f-C}&\theta_{f-D}\\ \theta_{g-A}&\theta_{g-B}&\theta_{g-C}&\theta_{g-D}\end{bmatrix}  \tag{9}\]  
where \(\theta_{i-A}\), \(\theta_{i-B}\), \(\theta_{i-C}\), and \(\theta_{i-D}\) are values between 0 and 1, representing the probability that the slope is at one of the four risk levels, attention, warning, alert, and alarm, and satisfies \(\theta_{i-A}+\theta_{i-B}+\theta_{i-C}+\theta_{i-D}=1\). If there are \(n\) fuzzy subsystems in the network, the intermediate output, \(z^{i}_{pk}\), of the \(k\)-th subsystem of the \(x_{pt}\) training data sample can be expressed in Equation (10) as follows:  
\[z^{i}_{pk}=\sum_{i=1}^{M}\alpha^{i}_{kt}x_{pt} \tag{10}\]  
where \(\alpha^{i}_{kt}\) is a parameter whose value is initialized by a uniform distribution between [0, 1]. The activation strength of the \(k\)-th fuzzy rule in the \(i\)-th fuzzy subsystem can be expressed as follows:  
\[\tau^{i}_{pk}=\prod_{t=1}^{M}\mu^{i}_{kt}(x_{pt}) \tag{11}\]  
where \(\mu^{i}_{kt}(x_{pt})\) is the Gaussian affiliation function corresponding to \(x_{pt}\). \[u^{i}_{kt}(x_{pt})=e^{-\frac{(x_{pt}-\frac{i^{i}_{kt}}{\delta^{i}_{kt}})^{2}}{ \delta^{i}_{kt}}} \tag{12}\]  
where \(c^{i}_{kt}\) and \(\delta^{i}_{kt}\) denote the width and center of the Gaussian affiliation function, respectively, and are obtained by initializing the clustering center. Therefore, the weighted activation strength of each fuzzy rule is calculated as shown in Equation (13). \[\omega^{i}_{sk}=\frac{\tau^{i}_{sk}}{\sum_{k=1}^{K_{k}}\tau^{i}_{sk}} \tag{13}\]  
Figure 4: Fuzzy-BLS structure diagram. Thus, the output vector of the \(i\)-th fuzzy subsystem is \(Z_{pi}=(\omega_{pi1}^{i}z_{pi1}^{i},\omega_{p2}^{i}z_{i2}^{i},\cdots,\omega_{pK_{ i}}^{i}z_{pK_{i}}^{i})\). Then, the intermediate output matrix of all fuzzy subsystems is \(Z^{n}=(Z_{1},Z_{2},\cdots,Z_{n})\). The intermediate output matrix, \(Z^{n}\), is input in the augmentation node for nonlinear transformation, which can effectively deal with nonlinear characteristics in the data.