## 3 Improved Multi Mobile Agent Data Fusion Scheme
### Data Fusion Algorithm  
We assume that the \(N\) sensor nodes follow a random distribution in the region of \(b^{2}\), and that the position of each sensor is independent of each other. This satisfies the uniform distribution:  
\[\mathrm{f}(x_{i},y_{i})=\left\{ \begin{array}{rl}\frac{1}{b^{2}},&-\frac{b}{2} \leq x_{i},y_{i}\leq\frac{b}{2}\\ 0,&\textit{otherwise}\end{array} \right. \tag{2}\]  
where \(i=1,...,N\), \((x_{i},y_{i})\) is the coordinate value of sensor node \(i\). We assume that the noise of local sensor nodes follows the standard Gauss distribution:  
\[n_{i}\sim N(0,1) \tag{3}\]  
For sensor node \(i\), the two-element hypothesis detection problem is:  
\[H_{1}:s_{i}=a_{i}+n_{i}\quad H_{0}:s_{i}=n_{i} \tag{4}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Identification** & **Data Space** & **Service Know-How** & **Itinery** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Application proxy data structure. where \(s_{i}\) is the received signal, \(a_{i}\) is the signal amplitude. The signal energy emitted by the target decreases with the increase of distance. We employ an isotropic signal power decay model as follows:  
\[a_{i}=\sqrt{\frac{P_{0}}{1+ad_{i}^{n}}} \tag{5}\]  
\(P_{0}\) is the signal power of the target when the distance is 0, \(d_{i}\) is the distance between the target and the local sensor node \(i\):  
\[d_{i}=\sqrt{(x_{i}-x_{t})^{2}+(y_{i}-y_{t})^{2}} \tag{6}\]  
(\(x_{t}\), \(y_{t}\)) are the coordinates of the target, \(n\) is the signal attenuation index. Its range is (2, 3), \(a\) is an adjustable constant. The greater the value, the greater the signal attenuation. We can define a parameter at first:  
\[\text{SNR}_{i}=a_{i}^{2}=\frac{P_{0}}{1+ad_{i}^{n}} \tag{7}\]  
When the distance is 0, SNR value is:  
\[\text{SNR}_{0}=10 log_{10}^{(P_{0})} \tag{8}\]  
We assume that all sensor nodes make decisions with the same threshold \(\tau\), threshold \(\tau\) and false alarm rate \(P_{fa}\) satisfies the following relations:  
\[P_{fa}=\int_{\tau}^{\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{t^{2}}{2}}dt=Q(\tau) \tag{9}\]  
\[\text{Or}\ \tau=Q^{-1}\Big{(}P_{fa}\Big{)} \tag{10}\]  
\(Q(\cdot)\) is a complementary function of the standard Gauss distribution function, that is:  
\[\text{Q(x)}=\int_{\tau}^{\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{t^{2}}{2}}dt \tag{11}\]  
The detection probability of the local sensor \(i\) is:  
\[ \begin{split} P_{di}&=\int_{\tau}^{\infty}\frac{1 }{\sqrt{2\pi}}e^{-\frac{(t-a_{i})^{2}}{2}}dt\\ &=Q(\tau-\sqrt{\frac{P_{0}}{1+ad_{i}^{n}}})\end{split}  \tag{12}\]  
Using binary data \(m_{i}\) = {0,1} to represent sensor nodes, when the value of \(m_{i}\) is 1, it indicates that sensor node \(i\) has probe event, otherwise the value is 0. All sensor nodes within a cluster are represented as \(M\) = {\(m_{i}\): \(i\) = 1,..., \(N\)}, When the mobile agent assigned by AMS collects the data \(M\), the fusion center DM makes the final decision with the integration algorithm. We know that the best decision fusion rule is the Chair-Varshney fusion rule. Its statistical function is:  
\[\Lambda_{1}=\sum_{i=1}^{N}\left[m_{i}ln\frac{P_{di}}{P_{di}}+(1-m_{i})ln\frac {1-P_{di}}{1-P_{di}}\right] \tag{13}\]  
Each sensor node of the false alarm rate can be (\(P_{tai}\) = \(P_{fu}\)) by Formula, we only need to know the threshold can be \(\tau\). However, the detection rate of \(P_{di}\) for each sensor node is very difficult, because the distance between the sensor node and the target and the signal amplitude of the target determine the detection rate. All sensor nodes may directly send collected data to the fusion center DM, and the fusion center makes decisions based on original testing. However, the direct transmission of raw data needs large overhead due to low energy and bandwidth, so the transfer of binary data to the processing center is a very good way to deal with this.