# A Novel Ensemble Belief Rule-Based Model for Online Payment Fraud Detection
## 3 Preliminaries  
### Traditional BRB Model  
BRB is an expert system composed of several rules and the \(k\)-th rule can be described in Equation (1). \[ \begin{array}{l}R_{k}:\text{IF }x_{1}\text{ is }A_{1}^{k}\wedge\ x_{2}\text{ is }A_{2}^{k}\wedge\ldots\wedge\ x_{M}\text{ is }A_{M}^{k}\\ \text{THEN }D=\left\{(D_{1},\beta_{1,k}),(D_{2},\beta_{2,k})\ldots,(D_{N},\beta_{N,k })\right\}\\ \text{WITH rule weight }\theta_{k}\\ \text{AND attribute weight }\delta_{1},\delta_{1},\ldots,\delta_{M}\end{array}  \tag{1}\]  
where \(R_{k}\) denotes the \(k\)-th rule in the BRB. \(x_{1},x_{2},\ldots,x_{M}\) are the antecedent attributes. \(A_{1}^{k},A_{2}^{k},\ldots,A_{M}^{k}\) denotes the referential values of attributes in the \(k\)-th rule. \(D=D_{1},D_{2},\ldots D_{N}\) indicates the consequent ranks. In classification tasks, \(D_{1},D_{2},\ldots D_{N}\) are considered as distinct categories and \(\beta_{1,k},\beta_{2,k},\ldots\beta_{N,k}\) are corresponding belief degrees. \(\theta_{k}\) is the rule weight of the \(k\)-th rule. \(\delta_{1},\delta_{2},\ldots,\delta_{M}\) are the attribute weights. The mechanism of BRB reasoning process can be described as follows. First, given a sample with several attributes, it is necessary to calculate the matching degree of input attributes. The calculating process is shown as follows. \[\alpha_{i,j}^{k}=\left\{ \begin{array}{ll}\frac{A_{i,j+1}^{k}-attr_{i}}{A_{i,j +1}^{k}-A_{i,j}^{k}}&\text{if }j=l\text{ and }A_{i,l}^{k}\leq attr_{i}\leq A_{i,l+1}^{k},\\ \frac{attr_{i}-A_{i,l}^{k}}{A_{i,j+1}^{k}-A_{i,l}^{k}}&\text{if }j=l+1\text{ and }A_{i,l}^{k}\leq attr_{i}\leq A_{i,l+1}^{k},\\ 0&\text{else}.\end{array} \right. \tag{2}\]  
where \(\alpha_{i,j}^{k}\) denotes the matching degrees of the input sample's \(i\)-th attribute belonging to its \(j\)-th referential value \(A_{i,j}^{k}\) in the \(k\)-th rule. \(attr_{i}\) denotes the value of the \(i\)-th attribute \(x_{i}\). After calculating the matching degree of input attributes, the activation weights \(\omega_{k}\) are calculated by Equations (3) and (4). \[\overline{\delta}_{i}=\frac{\delta_{i}}{\max_{j=1,2,\ldots T_{k}}\left\{ \delta_{j}\right\}} \tag{3}\]  
\[\omega_{k}=\frac{\theta_{k}\prod\limits_{i=1}^{T_{k}}\left(\alpha_{i,j}^{k} \right)^{\overline{\delta}_{i}}}{\sum\limits_{l=1}^{K}\left[\theta_{l}\prod \limits_{i=1}^{T_{l}}\left(\alpha_{i,j}^{l}\right)^{\overline{\delta}_{i}} \right]} \tag{4}\]  
where \(\overline{\delta}_{i}\) indicates the normalized attribute weight and \(\omega_{k}\) represents the activation weight of the \(k\)-th rule. Once \(\omega_{k}\) is not 0, the \(k\)-th rule is activated. To better demonstrate the construction of belief rules, Figure 1 shows an example of belief rules composed of hypothetical values. In this example, there are two attributes, \(x_{1}\) and \(x_{2}\), with their corresponding referential values sets set to \(A_{1}^{k}\in\{1,5,7\}\) and \(A_{2}^{k}\in\{12,17,20\}\), respectively. According to the principle of permutation and combination, there are nine rules in total. When a sample data point \((x_{1},x_{2})\) is input into the BRB, according to the figure, the corresponding values \(attr_{1}\) and \(attr_{2}\) are 4 and 19, respectively. Using Equation (2), the matching degrees of \(x_{1}\) and \(x_{2}\) to their respective referential values are calculated. Then, using Equation (3) and (4), the activation weights of the rules are calculated. As a result, only rules \(R_{2}\), \(R_{3}\), \(R_{5}\) and \(R_{6}\) have non-zero activation weights; thus, these four rules are activated. Figure 1: Representation of the belief rules and the activated rules. (3) After calculating the activation weights, the inference of the activated belief rules is aggregated by the analytical ER algorithm via Equations (5)-(7). The inference of the activated belief rules is the analytical ER algorithm described below.