## 2 Materials and Methods
### Contrast
#### 2.2.3 Fuzzy Broad Learning System Architecture  
The fuzzy broad learning system (FBLS) [32] is architecture based on the random vector functional link neural network and pseudo-inverse theory. This design enables the system to learn quickly and incrementally, as well as to remodel the system without the need for retraining. The FBLS model, depicted in Figure 3, consists of an input layer, a fuzzy subsystem layer, an enhancement node composite layer, and an output layer. In the composite layer, multiple groups of fuzzy subsystems are generated from the input data using fuzzy rules. These fuzzy subsystems are then enhanced as enhancement nodes, each with different random weights. The output is computed by connecting all the features and enhancement nodes to the output layer. The main concept behind the fuzzy BLS is to map the input data into a set of fuzzy rules and then utilize the BLS algorithm to learn and optimize the weights and parameters of these rules. These fuzzy rules are composed of fuzzy sets and their corresponding membership functions, which describe the fuzzy relationship between the inputs and outputs. By employing fuzzy logic's inference, the system can make predictions and classifications based on the fuzzy representation of the input data. The FBLS is a neural network model that incorporates the random vector functional link neural network and pseudo-inverse theory. This design allows the system to learn and adapt quickly, without the need for retraining. As shown in Figure 3, it consists of an input layer, a fuzzy subsystem layer, an enhancement layer, and an output layer. In the fuzzy subsystem layer, multiple groups of fuzzy subsystems are generated from the input data using fuzzy rules. These fuzzy subsystems are then transformed as enhancement nodes using random weights. The output is computed by connecting all the fuzzy subsystems and enhancement nodes to the output layer. The fuzzy rules consist of fuzzy sets and their corresponding membership functions, which describe the fuzzy relationship between the inputs and outputs. By employing fuzzy logic's inference, the system can make predictions and classifications based on the fuzzy representation of the input data. The specific implementation steps are as follows:In a fuzzy BLS, suppose there are n fuzzy systems and m groups of enhancement nodes. The input features are denoted by \(X\in\mathbb{R}^{N\times M}\), where M is the dimension of each feature vector. Assume that the i th fuzzy subsystem has \(K_{i}\) fuzzy rules of the following form. \[\text{if}\ x_{1}\text{is}\ A_{k1},x_{2}\text{is}\ A_{k2},...,\text{and}\ x_{M} \text{is}\ A_{kM}\text{then}z_{\text{sk}}^{i}=\hat{\text{f}}_{k}^{i}(x_{\text{s1 }},x_{\text{s2}},...,x_{\text{sM}}),\ k=1,\ 2,\ldots,\text{K}_{i} \tag{10}\]  
where \(\text{A}_{\text{kj}}\) is a fuzzy set, \(x_{\text{j}}\) is the system input (j = 1, 2, \(\ldots\), M), and K is the number of rules. Then, we adopt the first-order Takagi-Sugeno (TS) fuzzy system and we let  
\[z_{\text{sk}}^{i}=\hat{\text{f}}_{k}^{i}(x_{\text{s1}},x_{\text{s2}},...,x_{ \text{sM}})=\sum_{t=1}^{M}\alpha_{\text{kt}}^{i}x_{\text{st}} \tag{11}\]  
where \(\alpha_{\text{kt}}^{i}\) is the coefficient. The fire strength of the k th fuzzy rule in the i th fuzzy subsystem is  
\[\tau_{\text{sk}}^{i}=\prod_{t=1}^{M}\mu_{\text{kt}}^{i}(x_{\text{st}}) \tag{12}\]  
Then, we denote the weighted fire strength for each fuzzy rule as  
\[\omega_{\text{sk}}^{i}\frac{\tau_{\text{sk}}^{i}}{\sum_{k=1}^{K_{i}}\tau_{ \text{sk}}^{i}} \tag{13}\]  
The Gaussian membership function is chosen for \(\mu^{i}_{\text{kt}}(x)\) that corresponds to fuzzy set \(\text{A}^{i}_{\text{kt}}\), which is defined as follows:  
\[\mu_{\text{kt}}^{i}(x)=\text{e}^{-(\frac{x-x_{\text{kt}}^{i}}{\sigma_{\text{kt }}^{i}})^{2}} \tag{14}\]  
where \(\text{c}_{\text{kt}}^{i}\) and \(\sigma_{\text{kt}}^{i}\) are width and center, respectively.