# A Novel Ensemble Belief Rule-Based Model for Online Payment Fraud Detection
## 4 Constructions of Ensemble BRB  
In this section, the main process of constructing ensemble belief rule base model is shown. In Figure 5, the modeling process of the ensemble BRB has been divided into five steps. Each modeling step can be described as follows. Step 1. Data sampling. This process involved two rounds of data sampling. The first round was used to split the original dataset into the training dataset and test dataset. After that, a novel data sampling strategy was used to generate the sub-datasets used for generating the weak classifiers (i.e., the weak BRB model). It is important to note that the dataset used in the experiment was large and extremely imbalanced. Therefore, the sampling needed to be carefully balanced to ensure sample diversity while retaining as much information as possible. In this study, for the non-fraudulent data, approximately 60% to 70% of the total samples were extracted, while all the fraudulent data were included in the training process. The sampling process can be described as follows. Figure 5: Construction process of ensemble BRB. In a given dataset, \(DS=\{P,N\}\), \(P=\{x_{1}^{+},x_{2}^{+},\ldots,x_{n+^{\prime}}^{+}\}\) is the set of positive samples and \(N=\{x_{1}^{-},x_{2}^{-},\ldots,x_{n-}^{-},\}\) is the set of negative samples. \[N^{\prime}{}_{i}=Sampling(N,\gamma),\gamma\in(0,1] \tag{14}\]  
\[DS^{\prime}{}_{i}=P\cup N^{\prime} \tag{15}\]  
where \(\gamma\) denotes the ratio of data sampling, \(N^{\prime}{}_{i}\) is the \(i\)-th set of negative samples after data sampling and \(DS^{\prime}{}_{i}\) is the \(i\)-th dataset after data sampling. Step 2. Feature selection. In this step, feature selection was performed on the data subsets generated in the previous step. Here, we used the XGBoost feature selection method to select features from the generated data subsets. By applying feature selection in the process of the weak BRBs' construction, the combinatorial explosion problem can be avoided and each BRB's performance can be ensured. The feature selection process can be described as in Equation (16). \[\{x_{a,i},x_{\beta,i}\}=XGBoostFS(DS^{\prime}{}_{i}) \tag{16}\]  
where \(x_{a,i},x_{\beta,i}\) denotes the selected two features of \(DS^{\prime}{}_{i}\) and \(XGBoostFS\) means the method of feature selection based on XGBoost feature importance analysis. Step 3. BRB construction. In this step, based on the generated dataset and the selected attributes, the construction process of the weak BRBs can begin. This process involved steps such as referential value setting, initial confidence rule construction, parameter optimization and so on. The process of BRB construction can be described as in Equation (17). \[ \begin{array}{l}wBRB_{i}=model(x_{a,i},x_{\beta,i},Param_{i},CMA-ES)\\ i=1,2,\ldots,num\end{array}  \tag{17}\]  
where \(wBRB_{i}\) denotes the \(i\)-th weak BRB model, \(model\) is the construction process of BRB and \(num\) is the number of weak BRB models. \(Param_{i}\) means the parameters used for \(i\)-th BRB construction such as referential values of attributes, referential values of output, weight of attributes and weights of rules. \(CMA-ES\) denotes the CMA-ES algorithm (Covariance Matrix Adaptation-Evolution Strategy) used for the parameter optimization. Step 4. Model selection. In the previous step, each weak BRB model will produce a prediction result. In this phase, the trained models were tested and filtered because the sampling was random, so it could not guarantee that every generated model would be useful. Since ensemble learning frameworks can only achieve the best performance when each model has a certain level of accuracy and is independent, it was crucial to filter out and select the models with adequate accuracy. Step 5. Combining weak BRBs. Finally, all the weak BRB classifiers were added to the ensemble by adopting the majority voting strategy. After the ensemble, the entire modeling process for the integrated BRB was completed. The process of voting strategy can be described as in Equation (18).