## 3 Fuzzy-BLS Model for Slope Risk Assessment
### Fuzzy-BLS Risk Assessment Model Structure
Assuming that the number of enhanced nodes is \(m\), the output matrix of the enhanced layer can be expressed as follows:  
\[H^{m}=(H_{1},H_{2},\cdots,H_{m}) \tag{14}\]  
where \(H_{j}=\tilde{\varsigma}_{j}(Z^{n}W_{h_{j}}+\beta_{h_{j}})\); \(W_{hj}\) and \(\beta_{hj}\) are the weights and biases, respectively, and their values are randomly generated from [0, 1]. Thus, the defuzzified output vector of the \(i\)-th fuzzy system can be expressed as follows:  
\[F_{pi}=\left(\sum_{k=1}^{K_{i}}\omega_{pk}^{i}\left(\sum_{i=1}^{M}\delta_{k1}^ {i}\alpha_{kk}^{i}x_{pt}\right)\cdots\sum_{k=1}^{K_{i}}\omega_{pk}^{i}\left( \sum_{t=1}^{M}\delta_{kC}^{i}\alpha_{kk}^{i}x_{pt}\right)\right) \tag{15}\]  
where \(C\) is the dimension of the output vector, which is equal to four in the risk assessment experiments in this study. Thus, for all training samples, \(F\), the output of all fuzzy subsystems is shown in Equation (16). \[F^{n}=\sum_{i=1}^{n}diag\Big{\{}\sum_{t=1}^{M}\alpha_{kt}^{i}x_{1t},\cdots, \sum_{t=1}^{M}\alpha_{kt}^{i}x_{Nt}\Big{\}} \begin{pmatrix}\omega_{11}^{i}& \cdots&\omega_{1K_{i}}^{i}\\ \vdots&\ddots&\vdots\\ \omega_{N1}^{i}&\cdots&\omega_{1 NK_{i}}^{i}\end{pmatrix}\begin{pmatrix}\delta_ {11}^{i}&\cdots&\delta_{1C}^{i}\\ \vdots&\ddots&\vdots\\ \delta_{K1}^{i}&\cdots&\delta_{K_{i}C}^{i}\end{pmatrix}  \tag{16}\]  
Therefore, the final output of the Fuzzy-BLS is determined as follows:  
\[\Theta=F^{n}+H^{m}W_{e} \tag{17}\]  
where \(W_{e}\) is the parameter matrix of the Fuzzy-BLS network. Based on the output, \(\Theta\), corresponding to Equation (9), the slopes are classified into the corresponding classes according to the following equation:  
\[L=\max\{\theta_{i-X}\}=\begin{cases}\theta_{i-A}\,,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,  
The figure shows that when the number of fuzzy subsystems reaches six, the accuracy rate reaches 90% and above and remains stable with the increased number, and the accuracy rate reaches a maximum of 92.08% when \(Nt=9\). Therefore, we chose to set the number of fuzzy subsystems of the final risk assessment model to nine. We set the search range of fuzzy rules for each fuzzy subsystem of the Fuzzy-BLS model \(Nr=2\), the number of augmentation nodes \(Ne=20\), and the number of fuzzy subsystems \(Nt=9\). During the experimental process, we found that the number of training samples has an impact on the performance of the Fuzzy-BLS model, so the training set and the test set (the same situation for each risk level) were divided into 6:4, 7:3, 8:2, 9:1, and 19:1, five ratios used to train the model, and the test performance is shown in Table 2.  
Given the data analysis in the table above, the test accuracy of the Fuzzy-BLS model is low when the training samples are lower than 80%, and it increases significantly with an increase in the training samples, until the training samples are more than 80%, reaching more than 90%. Training accuracy is the highest when the number of training samples is 8:2. When the number of samples increases, the training accuracy begins to decrease, and the test accuracy shows a gradual improvement trend, but this improvement is small. At the same time, the training time becomes longer and longer, and there are large fluctuations in the accuracy of different risk levels. Therefore, the 8:2 dataset ratio is finally selected as the optimal scheme for network training. This ratio ensures that the model maintains stable and high accuracy in all levels of risk assessment, while also controlling the training time to a certain extent. With a dataset ratio of 8:2, this study selected BLS, eXtreme Gradient Boosting (XG-Boost), and Support Vector Machine (SVM) to conduct comparative experiments on the effect of introducing fuzzy subsystems. The aim was to verify the superiority of the proposed FBLS model in landslide risk assessment. The experimental results of different models are presented in Table 3.