## 2 Materials and Methods
### Contrast
#### 2.2.3 Fuzzy Broad Learning System Architecture
In the FBLS, fuzzy subsystems deliver the intermediate vectors to the enhancement layer, which performs a further non-linear modification while retraining the input features. The intermediate vector \(x_{\text{s}}\), without aggregation in the i th fuzzy subsystem, is  
\[Z_{\text{si}}=\left(\omega_{\text{s1}}^{i}z_{\text{s1}}^{i},\omega_{\text{s2} }^{i}z_{\text{s2}}^{i},...,\omega_{\text{sK}}^{i}z_{\text{sK}_{i}}^{i}\right) \tag{15}\]  
For all training samples X, the i th fuzzy subsystem's output matrix is  
\[Z_{i}=(Z_{1},Z_{2},...,Z_{n})\epsilon\mathbb{R}^{N\times K_{i}},i=1,\ 2,\ldots,n \tag{16}\]  
Then, we designate an intermediate output matrix for n fuzzy subsystems by  
\[Z^{n}=(Z_{1},Z_{2},...,Z_{n})\epsilon\mathbb{R}^{N\times(K_{1}+K_{1}+...+K_{n })} \tag{17}\]  
Then, \(Z^{n}\) is passed to the enhancement layer for further non-linear processing. Consider the j th enhancement node group to be made up of Lj neurons represented by  
\[H^{j}=\delta_{j}\Big{(}Z^{n}W_{h}+b_{h}\Big{)}\epsilon\mathbb{R}^{N\times L_{ j}} \tag{18}\]  
where \(W_{hj}\) and \(b_{hj}\) are the j th enhancement node group's random weights and bias terms, respectively. The enhancement layer's output matrix is  
\[H^{j}=(H_{1},H_{2},...,H_{m}) \tag{19}\]  
We now consider the output of each fuzzy subsystem, a process known as defuzzification. The defuzzification output \(F^{n}\) and the enhancement layer output matrix \(H^{m}\) will then be combined and sent to the top layer. First, consider the j th fuzzy subsystem's output vector. \[z_{\text{sk}}^{i}=\hat{\text{f}}_{k}^{i}(x_{\text{s1}},x_{\text{s2}},...,x_{ \text{sM}})=\sum_{t=1}^{M}\alpha_{\text{kt}}^{i}x_{\text{st}} \tag{11}\]0 \[=\sum\limits_{t=1}^{M}\alpha_{kt}^{i}x_{st}\left(\omega_{s1}^{i},...,\omega_{sK_{i}}^{i}\right)\left( \begin{matrix}\varphi_{11}^{i}&\cdots& \varphi_{1C}^{i}\\ \vdots&\ddots&\vdots\\ \varphi_{K_{i}1}^{i}&\cdots&\varphi_{K_{i}C}^{i}\end{matrix} \right)\]  
where \(\varphi_{kC}^{i}\) is a new parameter in the FBLS. To reduce the number of parameters, we convert \(\alpha_{kt}^{i}\) to \(\varphi_{kC}^{i}\alpha_{kt}^{i}\) and calculate the pseudo-inverse. For all fuzzy subsystems, we set  
\[D=diag\left\{\sum\limits_{t=1}^{M}\alpha_{kt}^{i}x_{1t},...,\sum\limits_{t=1}^ {M}\alpha_{kt}^{i}x_{Nt}\right\} \tag{21}\]  
\[\pi^{i}=\left( \begin{matrix}\omega_{11}^{i}&\cdots&\omega_{1K_{i}}^{i}\\ \vdots&\ddots&\vdots\\ \omega_{N1}^{i}&\cdots&\omega_{NK_{i}}^{i}\end{matrix} \right) \tag{22}\]  
\[\varphi^{i}=\left( \begin{matrix}\varphi_{11}^{i}&\cdots&\varphi_{1C}^{i}\\ \vdots&\ddots&\vdots\\ \varphi_{K_{i}1}^{i}&\cdots&\varphi_{K_{i}C}^{i}\end{matrix} \right) \tag{23}\]  
The three terms are then combined and rewritten as F. The output of i th fuzzy subsystem for the input data \(X\) is  
\[F_{i}=\left(F_{1i},F_{2i},...,F_{Ni}\right)^{T}\triangleq D\pi^{i}\varphi^{i} eR^{N\times C} \tag{24}\]  
The top layer's aggregate output of n fuzzy subsystems can then be obtained as follows:  
\[F^{n}=\sum\limits_{i=1}^{n}F_{i}=\sum\limits_{i=1}^{n}D\pi^{i}\varphi^{i} \tag{25}\]  
Now consider the top layer. We send the value of defuzzification \(F^{n}\), which is related to the output of the enhancement layer \(H^{m}\) and is connected to the weight matrix of the enhancement layer, denoted as \(W_{e}\in R^{(L+1+2+Lm)\times C}\), to the top layer of the FBLS. As a result, the FBLS final output is equal to 2. \[\hat{Y}=F^{n}+H^{m}W_{e}=D\pi\varphi+H^{m}W_{e}=(D\pi,H^{m})W \tag{26}\]  
\(W\) denotes the parameter matrix of a fuzzy BLS. \(W\) can be quickly calculated via the pseudo-inverse given training targets \(Y\). \[W=(D\pi,H^{m})^{+}Y=\left((D\pi,H^{m})^{T}((D\pi,H^{m}))^{-1}\right)(D\pi,H^{m} )^{T} \tag{27}\]  
A certain number of features and enhancement nodes are selected in advance to train the BLS model. The parameters' range for selections 10-20 for the number of enhancement node groups and 10-50 for the number of feature node groups. Suitable parameters are selected from these ranges to enhance the robustness of the model.