## 2 Information Fusion Process Based on D-S Evidence Theory
### Calculation Process for MIFMS  
The basic definitions of D-S evidence theory are described as follows. 1. Basic probability assignment function: \(A_{i}\) is a grade of the information detected by a sensor in alarm framework \(\Theta\), if function \(M_{j}\) satisfies \(M_{j}\) (\(\omega\)) = 0 and \(\sum\limits_{i}M_{j}(A_{i})\)=1 under the map \(2^{\Theta}\rightarrow[0,\,1]\); \(M_{j}\) is called a basic probability assignment function based on \(2^{\Theta}\). \(M_{j}\) (\(A_{i}\)) is the basic probability assignment for proposition \(A_{i}\) and denotes the degree of trust in the assignment. \(A_{i}\) is called the focus element. 2. Belief function: The belief function denotes the total degree to which a grading of the information is supported by the obtained evidence. For Grades A and B satisfying \(B\subseteq A\), \(A\subseteq\Theta\), \(B\subseteq\Theta\). Define the following function, \[\textit{Bel: }2^{\Theta}\rightarrow[0,\,1]\] (1) \[\textit{Bel}\big{(}A\big{)}=\sum\limits_{B\subseteq A}M\big{(}B \big{)}\] (2) \[\textit{Bel}\ \text{is the belief function of }\Theta.\]
3. Likelihood function: The likelihood function denotes the degree to which the grading cannot be rejected by the obtained evidence. Given a map \(\textit{Pl: }2^{\Theta}\rightarrow[0,\,1]\), it is defined as: \[\textit{Pl}\big{(}A\big{)}=1-\textit{Bel}\big{(}\vec{A}\big{)}=\sum\limits_{B \cap A\
eq\phi}M\big{(}B\big{)}\] (3) Equation (3) can be called the likelihood function or upper limit function. 4. Dempster-Shafer combination rules: Let us consider the belief functions of the first two sensors \(\textit{Bel}_{1}\) and \(\textit{Bel}_{2}\). \(M_{1}\) and \(M_{2}\) are the corresponding basic probability assignment functions. The focus elements are, respectively, \(A_{1}\), \(A_{2}\), \(\ldots\), \(A_{n}\) and \(B_{1}\), \(B_{2}\), \(\ldots\), \(B_{m}\). Then, the new probability \(M\) (\(C\)), \(C=A_{i}\cap B_{j}\) can be derived by combining these two pieces of evidence according to Dempster-Shafer combination rules in the following way. Figure 1: Basic system structure of a multi-sensor intelligent fire monitoring system (MIFMS) based on D-S theory. \[M\left(C\right)= \begin{bmatrix}0,C=\phi\\ \frac{1}{1-K}\sum_{A\cap B_{j}=C}M_{1}\big{(}A_{i}\big{)}M_{2}\big{(}B_{j}\big{)},C\
eq\phi\end{bmatrix}  \tag{4}\]  
Where \(K=\sum_{A\cap B_{j}=\phi}M_{1}\big{(}A_{i}\big{)}M_{2}\big{(}B_{j}\big{)}\). The rule gives the probability of the cross set of two original sets. The coefficient \(K\) guarantees \(M\left(C\right)\leq 1\). In Equation (4), if \(K\
eq 1\), \(M\) makes a certain basic probability assignment. On the other hand, if \(K=1\), we regard that \(M_{1}\) and \(M_{2}\) are contradictory, and it is not possible to combine the two basic probability assignments. For multiple pieces of evidence, we can combine the pieces of evidence one by one. Each subsystem of MIFMS has five pieces of evidence; they are combined step by step. There are five basic properties of D-S evidence theory. 1. Commutability: \(m_{1}\in m_{2}=m_{2}\in m_{1}\)
2. Associability: \(m_{1}\in(m_{2}\in m_{3})=(m_{1}\circ m_{2})\in m_{3}\)
3. Monotonic: Assume that \(m_{1}\) and \(m_{2}\) are monotonous, when applying the rules to generate a new synthetic \(m_{3}\); \(m_{3}\) is also monotonous. 4. Identity: Let the basic distribution function \(m_{0}\): \(2^{\Theta}\rightarrow[0\), \(1]\) be empty; there is, \(m_{0}\) (\(\Theta\)) = 1, \(m\) (other) = 0; for all of the basic probability functions \(m\), we have \(m\circ m_{0}=m\); \(m_{0}\) is the only basic probability assignment (BPA) satisfying the formula. 5. Polarization resistance: The evidence theory of polarization means that for one proposition, when there is an expert opinion, more persuasive synthetic effects are generated than just the one. 5. Final decision of the microprocessor: The number of fire styles is confirmed by the following method.