# RPC-EAU: Radar Plot Classification Algorithm Based on Evidence Adaptive Updating
## 2 Related Work  
### Belief Functions  
The belief function theory, also known as evidence theory, has been proven to be highly effective in handling uncertain and imprecise information. In belief functions, a problem domain is represented by a finite set \(\Omega=\{\omega_{1},\cdots,\omega_{c}\}\) called the frame of discernment. The set of all subsets of \(\Omega\) is called the power set of \(\Omega\) and is denoted \(2^{\Omega}\). \(\omega\) is an independent event under the discernment frame \(\Omega\). The combination results of all subsets of \(\Omega\) are included in \(2^{\Omega}\). For example, if \(\Omega=\{\omega_{1},\omega_{2},\omega_{3}\}\), then \(2^{\Omega}=\{\varnothing,\{\omega_{1}\},\{\omega_{2}\},\{\omega_{3}\},\{\omega_{ 1},\omega_{2}\},\{\omega_{1},\omega_{3}\},\{\omega_{2},\omega_{3}\},\Theta\}\). \(\varnothing\) is an empty set, representing an impossible event with a confidence assignment of \(0\). \(\Theta\) represents complete lack of knowledge and uncertainty as to which independent event occurred. The mass function \(m(\cdot)\) is used to represent the degree support for each element \(A\) from \(2^{\Omega}\). It satisfies a mapping relationship from \(2^{\Omega}\) to \([0,1]\), such that:  
\[\left\{ \begin{array}{l}m(\varnothing)=0\\ \sum\limits_{A\subseteq 2^{\Omega},A\
eq\varnothing}m(A)=1\end{array} \right. \tag{1}\]  
The subsets \(A\subseteq 2^{\Omega}\) are called the focal elements of \(m(\cdot)\). If \(A\
eq\varnothing\), then \(m(A)>0\). And the sum of the basic assignments of all nonempty subsets is 1. For the subset \(A\subseteq 2^{\Omega}\), the belief function \(Bel\) represents the probability that the evidence supports its validity. The belief function can be defined as the following equation. \[Bel(A)=\sum\limits_{B\subseteq A}m(B) \tag{2}\]  
The plausibility function \(Pl\) represents the probability that the evidence does not contradict \(A\), and can be defined by the following equation. \[Pl(A)=\sum\limits_{B\cap A\
eq\varnothing}m(B). \tag{3}\]  
In order to provide readers with a clear understanding of the meanings of \(Bel\) and \(Pl\), Figure 1 is presented as follows. For example, we are certain that the probability of event \(A\) occurring is 0.5 and the probability of its not occurring is 0.2. Then, we will obtain the results that \(Bel=0.5\), \(Pl=0.8\), \(uncertain=0.3\), \(negative=0.2\). The combination of different pieces of evidence plays an important role in belief functions. Assuming that \(m_{1}(B)\) and \(m_{2}(C)\) are two independent pieces of evidence to be fused, \(B\) represents the various propositions provided by Evidence 1, and \(C\) represents the various propositions provided by Evidence 2. \(A\) is obtained through the intersection operation of \(B\) and \(C\), which represents the various propositions of the combined evidence. The combination of \(m_{1}(B)\) and \(m_{2}(C)\) with Dempster's (D-S) rule is defined by:  
\[m(A)=\left\{ \begin{array}{ll}\frac{\sum\limits_{B\cap C=A}m_{1}(B)m_{2}(C)}{1 -k},&\forall A\in 2^{\Omega},&A\
eq\varnothing\\ 0,&A\
eq\varnothing\end{array} \right. \tag{4}\]  
where  
\[k=\sum\limits_{B\cap C=\varnothing}m_{1}(B)m_{2}(C) \tag{5}\]  
For the nonempty \(A\subseteq 2^{\Omega}\), where \(k\) represents the degree of conflict between \(m_{1}\) and \(m_{2}\), such that \(k<1\), if the value of \(k\) is close to 1, indicating a high conflict between the two pieces of evidences, the D-S rule will have certain shortcomings. To deal with this problem, some improvement rules have been studied. The most representative one among them is the Yager rule, which can assign the degree of conflict set as uncertain by the following combined formula. \[\left\{ \begin{array}{ll}m(\varnothing)=0\\ m(A)=\sum\limits_{B\cap C=A}m_{1}(B)m_{2}(C)&\forall A\in 2^{\Theta},A\
eq \varnothing,A\
eq\Theta\\ m(\Theta)=m_{1}(\Theta)m_{2}(\Theta)+\sum\limits_{B\cap C=\varnothing}m_{1}(B)m_ {2}(C)\end{array} \right. \tag{6}\]  
In addition, there are many other improvement rules, such as Smets, Dubois and Prade, PCR5, DSmT, and so on.