# RPC-EAU: Radar Plot Classification Algorithm Based on Evidence Adaptive Updating
## 3 Proposed Method
### Construction of Belief Function  
In belief function classification, the most important thing is to construct a reasonable mass function for each decision evidence. Consider a radar plot classification problem in which the \(N\) objects \(\{O_{1},\cdots,O_{N}\}\) have to be classified into a certain class from \(\Omega=\{\omega_{T},\omega_{c},\Theta\}\). Here, \(\omega_{T}\) and \(\omega_{c}\) represent the target and clutter, respectively. While \(\Theta\) represents the uncertain class, the samples within it need to be further divided into \(\omega_{T}\) and \(\omega_{c}\) by additional information. Firstly, we need to use the feature extraction method in Section 3.1 to obtain the features of these radar plots. And the feature set \(\{F_{1},\cdots,F_{N}\}\) of these \(N\) radar plots can be obtained. Then, we calculate the feature distance between each radar plot using the following formula. \[d_{ij}=\sqrt{\sum_{q=1}^{10}\left(O_{i}^{q}-O_{j}^{q}\right)^{2}}, \tag{12}\]  
where \(q\) represents the \(q\)th dimensional feature of targets \(O_{i}\) and \(O_{j}\), with values ranging from 1 to 10. Here, \(d_{ij}\) is the Euclidean distance between \(O_{i}\) and \(O_{j}\). After pairwise calculations of all radar plots, the following \(N\times N\) symmetric feature distance matrix \(D\) can be obtained. \[D=\left[ \begin{array}{ccccc}0&d_{1,2}&\cdots&d_{1,N-1}&d_{1,N}\\ d_{2,1}&0&\cdots&d_{2,N-1}&d_{2,N}\\ &&\ddots&&\\ d_{N-1,1}&d_{N-1,2}&\cdots&0&d_{N-1,N}\\ d_{N,1}&d_{N,2}&\cdots&d_{N,N-1}&0\end{array} \right] \tag{13}\]  
The reason why the elements of the main diagonal of matrix \(D\) have values of 0 is that they represent the distance between the sample and itself. The information contained in this feature distance matrix \(D\) can be used to guide us in constructing decision evidences. A very effective strategy is to select other neighbors near a certain radar plot to be achieved. Assuming we construct the mass function of \(O_{i}\), and \(\{O_{1},\cdots,O_{K}\}\) is the set of \(K\) nearest neighbor samples obtained from the feature distance matrix \(D\). Support that the class label of neighboring sample \(O_{j}\) is \(\omega_{j}\in\{\omega_{T},\omega_{c},\Theta\}\). Then, the mass function constructed by \(O_{j}\) on \(O_{i}\) can be defined as the following formula. \[\left\{ \begin{array}{l}m_{ij}(\omega_{j})=\alpha_{ij}\\ m_{ij}(\Theta)=1-\alpha_{ij}\end{array} \right. \tag{14}\]  
Here, \(\alpha_{ij}\) is the confidence level that \(O_{j}\) considers \(O_{i}\) to belong to class \(\omega_{j}\); therefore, the value of \(\alpha_{ij}\) ranges from 0 to 1. Then, the key is to convert \(d_{ij}\) to the value interval through a certain mapping function. We find the maximum feature distance \(d_{\text{max}}\) and the minimum feature distance \(d_{\text{min}}\) of \(K\)-nearest neighbor samples as follows:  
\[d_{\text{max}}=\max_{j=1}^{K}\{d_{ij}\} \tag{15}\]\[d_{\min}=\min_{j=1}^{K}\{d_{ij}\} \tag{16}\]  
We then modify the feature distance of each neighboring sample based on the maximum and minimum values of these feature distances. \[d_{ij}^{\prime}=(d_{ij}-d_{\min})/(d_{\max}-d_{\min}) \tag{17}\]  
Many attempts have been made in the design of mapping functions such that \(d_{ij}^{\prime}\rightarrow\alpha_{ij}\), and exponential functions have been proven to be the most suitable. Therefore, the mapping function in this paper is defined as:  
\[\alpha_{ij}=\alpha_{0}\alpha_{i}\exp(-(d_{ij}^{\prime})^{\beta}) \tag{18}\]  
Here, parameter \(\alpha_{0}\) is a fixed value used to adjust the mapping, which can be heuristically set to 2. The parameter \(\alpha_{i}\) is related to the distribution of \(K\)-nearest neighbor samples in the feature space. And the value of the parameter \(\beta\) is usually selected as 1 or 2. \[\alpha_{i}=\sum_{j=1}^{K}d_{ij}^{\prime}/K \tag{19}\]  
Through the above methods, the mass functions of \(K\) pieces of evidence \(\{m_{i1},m_{i2},\cdots,m_{iK}\}\) for \(O_{i}\) can be obtained sequentially. In the same way, decision evidence for all radar plots can also be constructed.