## 2 Information Fusion Process Based on D-S Evidence Theory
### Calculation Process for MIFMS
First, in order to reduce the information amount and lower the burden on the system as much as possible, each signal (light, smoke, temperature, gas and moisture) is only classified into six grades. Thus, the number of fire cases is \(6^{5}=7776\) at most. The system will work more efficiently when dealing with less than 7776 species; however, here, 7776 conditions will be discussed. Because there are many fire cases, one microprocessor is set at each detected spot. In this way, each subsystem is an information fusion system. The following is the decision-making process of the microprocessor. The 7776 species constitute a fire alarm framework \(\Theta=\{H_{1}\), \(H_{2}\), \(\ldots\), \(H_{7776}\}\). According to D-S combination rules, the reports are integrated from sensors that support the same supposition. Suppose there are \(S_{j}\) reports from five sensors supporting \(H_{j}\), \(j=1\), \(2\), \(\ldots\), 7776. Then, the support degree after integrating \(S_{j}\) reports is: \[M^{S_{j}}\big{(}H_{j}\big{)}=1-\prod_{i=1}^{S_{j}}M_{i}\big{(} \Theta\big{)}=1-\prod_{i=1}^{S_{j}}\big{[}1-M_{i}\big{(}H_{j}\big{)}\big{]}\] (5) \[M^{S_{j}}\big{(}\Theta\big{)}=\prod_{i=1}^{S_{j}}\Big{[}1-M_{i} \big{(}H_{j}\big{)}\Big{]}\] (6) Where \(M^{S_{j}}(H_{j})\) is the general degree to which \(S_{j}\) sensors support \(H_{j}\). By D-S combination rules, the denominators of the support degree are the same and can be ignored. Suppose \(W\) is the total number of support degrees. \(Q^{S_{i}}\left(H_{i}\right)\) is the numerator of the support degrees to which integrated \(W\) support \(H_{i}\), then:  
\[Q^{S_{i}}\left(H_{i}\right)-Q^{S_{i}}\left(H_{j}\right)=\frac{M^{S_ {i}}\left(H_{i}\right)}{M^{S_{i}}\left(\[PERSON]\right)}\prod_{t=1}^{W}M^{S_{i}} \left(\[PERSON]\right)-\frac{M^{S_{j}}\left(H_{j}\right)}{M^{S_{j}}\left(\[PERSON] \right)}\prod_{t=1}^{W}M^{S_{i}}\left(\[PERSON]\right)\] \[=\] \[= \left[M^{S_{i}}\left(H_{i}\right)-M^{S_{j}}\left(H_{j}\right) \right]\times\prod_{t=1,j}^{W}M^{S_{i}}\left(\[PERSON]\right) \tag{7}\] \[= \left[\left[1-\prod_{t=1}^{S_{i}}\left(1-M_{t}\left(H_{i}\right) \right)\right]-\left[1-\prod_{t=1}^{S_{i}}\left(1-M_{t}\left(H_{j}\right) \right)\right]\right]\times\prod_{t=1,j}^{W}M^{S}\left(\[PERSON]\right)\] \[= \left[\prod_{t=1}^{S_{j}}\left(1-M_{t}\left(H_{j}\right)\right)- \prod_{t=1}^{S_{i}}\left(1-M_{t}\left(H_{i}\right)\right)\right]\times\prod_{ t=1\atop t=i,j}^{W}M^{S}\left(\[PERSON]\right)\]  
Suppose the above polynomial has a value greater than zero, then:  
\[\prod_{t=1}^{S_{j}}\left(1-M_{t}\left(H_{j}\right)\right)>\prod_{t=1}^{S_{j}} \left(1-M_{t}\left(H_{i}\right)\right) \tag{8}\]  
Taking the logarithm for both sides,  
\[\sum_{t=1}^{S_{j}}\ln\left(1-M_{t}\left(H_{j}\right)\right)-\sum_{t=1}^{S_{j} }\ln\left(1-M_{t}\left(H_{i}\right)\right)>0 \tag{9}\]  
The above equation can be written in simple form as \(\sum\alpha_{t}\mu_{t}>0\), where:  
\[\alpha_{t}= \begin{vmatrix}\left(1/\,j\right)\times\ln\left[1-M_{t}\left(H_{j }\right),\text{when}\mu_{t}=j\right]\\ \left(-1/\,j\right)\times\ln\left[1-M_{t}\left(H_{i}\right),\text{when}\mu_{t }=i\right]\end{vmatrix}  \tag{10}\]  
In summary, we can get the decision-making rule in the following,  
\[f\left(S_{i},S_{j}\right)= \begin{vmatrix}H_{i},\text{if}\sum\alpha_{t}\mu_{t }>0\\ H_{j},\text{otherwise}\end{vmatrix}  \tag{11}\]  
using the above method repeatedly, the microprocessor makes a decision as to whether the fire alarm grade is \(H_{m}\). It satisfies \(f\left(S_{m}\,,S_{j}\right)=H_{m}\,,\,i=1,\,2,\,\ldots,\,7776\). So far, the information fusion process in the space domain has been calculated. However, in the fire detection system, the sensor is periodically or continuously monitoring the target. Fire, in terms of the measurement of light, smoke, temperature, gas and moisture, should be put in sequence, noting that the values are also gradually changing. Some fires start with smoke, then flames appear; others start with flames, and smoke appears only later.