# An Adaptive Dempster-Shafer Theory of Evidence Based Trust Model in Multiagent Systems
## 3 Dempster-Shafer Theory of Evidence
[PERSON] distance [45] is one of the rational and popular definitions of distance, which is identified based on [PERSON]'s geometric interpretation of evidence theory [48]. The power set of the frame of discernment space and a distance and vectors are defined with the BPA as a particular case of vectors [45]. [PERSON]'s distance is defined as  
\[d(m_{i},m_{j})=\sqrt{\frac{1}{2}(\vec{m_{i}}-\vec{m_{j}})^{T}D(\vec{m_{i}}- \vec{m_{j}})} \tag{5}\]  
with \(m_{i}\), \(m_{j}\) being two BPAs under the frame of discernment \(\Omega\) matrix. The element in \(D\) is defined as \(D(A,B)=|A\cap B|/|A\cup B|\), \(A,B\in\Omega\), and \(|A|\) represents cardinality. [PERSON]'s distance is an efficient tool to quantify the dissimilarity of two BPAs. **Definition 6**. : _Entropy of Dempster-Shafer theory._  
The concept of entropy comes from physics, which is a measure of uncertainty and disorder [49; 50]. The Shannon entropy has been widely accepted for uncertainty measure ment (or information measurement) in information theory [51]. However, the uncertainty estimation of BPAs of the Dempster-Shafer theory of evidence still needs trial and error. Since the first definition of a BPA in the evidence theory was given by [PERSON] in 1982 [52], many definitions of entropy of the Dempster-Shafer theory have raised, for instance, the definitions proposed by [PERSON] [53], [PERSON] [54], [PERSON] [55], [PERSON] [56], [PERSON] [57], and so on. However, these definitions are still imperfect. For example, Deng entropy has some limitations when the propositions are of the intersection. Thus, several methodologies have been applied for Deng entropy modification [58; 59]. Recently, [PERSON] and [PERSON] proposed a new definition, in which five of the six listed properties of entropy are satisfied [60]. Whether there exists a definition that satisfies the six properties is still an open issue, but the proposed entropy in [60] is acceptable and could be regarded as near-perfect. As a result, we use the definition of entropy of the Dempster-Shafer theory in the literature [60] for uncertainty estimation in this paper. For the frame of discernment defined in \(\Omega\) where \(x\in\Omega\) and \(A\) is a focal element, the entropy of the BPA \(m\) is defined as follows:  
\[H(m)=H_{s}(Pl\_Pm)+H_{d}(m)=\sum_{x\in\Omega}Pl\_Pm(x)log(\frac{1}{Pl\_Pm(x)}) +\sum_{A\in 2^{\Omega}}m(A)log(|A|) \tag{6}\]  
where the first part is employed to measure conflict [60; 61], and \(Pl\_Pm(x)\) is defined as the plausibility transform [60; 62], where  
\[Pl\_Pm(x)=\frac{Pl_{m}(x)}{\sum_{x\in\Omega}(Pl_{m}(x))} \tag{7}\]  
The second part in Equation (6) can be regarded as non-specificity of the BPA \(m\)[56]. That is to say, the uncertainty of the Dempster-Shafer theory of evidence can be captured from two aspects, namely the conflicts between exclusive elements in set \(\Omega\) and the non-specificity uncertainty. Both components are on the scale \([0,log(|\Omega|)]\), thus, the maximum entropy of a BPA is \(2 log(|\Omega|)\). **Example 1**. : _An example of the Dempster-Shafer theory of evidence is given as follows. Suppose an animal stands far away from us, perhaps a cat or a dog. Then the frame of discernment is presented as \(\Omega=\{cat,dog\}\), and three passers-by show their opinions. Their opinions are shown as_  
1. \(m_{1}(Cat)=0.6\)_,_ \(m_{1}(Dog)=0.3\)_,_ \(m_{1}(Cat,Dog)=0.1\)__
2. \(m_{2}(Cat)=0.4\)_,_ \(m_{2}(Dog)=0.4\)_,_ \(m_{2}(Cat,Dog)=0.2\)__
3. \(m_{3}(Cat)=0.1\)_,_ \(m_{3}(Dog)=0.4\)_,_ \(m_{3}(Cat,Dog)=0.5\)__  
In terms of evidence distance, we have the following calculation process:  
1. \(\overrightarrow{m_{1}}=\{Cat,Dog,(Cat,Dog)\}=(0.6,0.3,0.1)\). Similarly, \(\overrightarrow{m_{2}}=(0.4,0.4,0.2)\), \(\overrightarrow{m_{3}}=(0.1,0.4,0.5)\); \(\begin{array}{c}Cat\quad\quad Dog\quad(Cat,Dog)\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\1. \(Pl\_P_{m}(Cat)=\frac{Pl_{m}(Cat)}{\sum_{x\in\Omega}(Pl_{m}(x))}=\frac{0.6\ +\ 0.1}{0.6\ +\ 0.1\ +\ 0.3\ +\ 0.1}=\frac{7}{11}\). In the same manner, \(Pl\_P_{m}(Dog)=\frac{4}{11}\);
2.