## 3 Improved Multi Mobile Agent Data Fusion Scheme
### Data Fusion Algorithm
We assume that \(P_{di}=P_{d}\), \(P_{di}=P_{fi}\), and \(P_{d}>P_{fi}\), the statistical function is simplified as follows:  
\[\Lambda_{2}=\sum\limits_{i=1}^{N}m_{i} \tag{14}\]  
Therefore, the fusion rules of fusion centers are:  
\[\Lambda_{2}=\sum\limits_{i=1}^{N}m_{i}\simeq\frac{H_{1}}{H_{0}}T \tag{15}\]  
We want to detect the number of mobile agents to collect, and use the total \(m_{i}\) value and the threshold value \(\tau\). The main task of Processing center, including data management for the DM, is data mining and fusion. For data fusion level, we calculate the false alarm rate \(P_{fi}\) as:  
\[P_{fa}=Pr\{\Lambda_{2}=\sum\limits_{i=1}^{N}m_{i}\geq T|H_{0}\} \tag{16}\]  
Obviously, based on the assumption of \(H_{0}\), the total number of all detected events \(\Lambda=\sum\limits_{i=1}^{N}m_{i}\) follows the binomial distribution Binomial (\(N\), \(P_{fi}\)). Therefore, for a given threshold \(T\), the rate of false alarm is:  
\[P_{fa}=\sum\limits_{i=\lceil\Upsilon\rceil}^{N}\big{(} \begin{array}{c}N\\ 1\end{array} \big{)}P_{fa}^{i}\big{(}1-P_{fa}\big{)}^{N-i} \tag{17}\]  
This article focuses on the large sensor networks, so when the number of network nodes is large enough, Equation (17) approximate expression is:  
\[P_{fa}\approx Q(\frac{T-NP_{fa}}{\sqrt{NP_{fa}\Big{(}1-P_{fa}\Big{)}}}) \tag{18}\]  
For a given threshold \(T\), to get the Receiver Operating Characteristic curve (ROC), false probability as the horizontal axis and the hit probability as the vertical axis form the coordinate chart. The curves of the different results obtained by the different judgment criteria under the specific stimulus conditions need the corresponding \(P_{d}\) value. Due to the different detection rate of different local sensor nodes, the total number of \(\Lambda_{2}\) detection does not follow the binomial distribution under the \(H_{1}\) assumption, so it is very difficult to derive the analytical expression from the \(\Lambda_{2}\) distribution. We know that \(m_{i}\) follows the Bernoulli distribution, and the probability of success is \(P_{di}\). When the sensor node \(N\) is large enough, \(\{m_{1},...,m_{N}\}\) is independent of each other under the H\({}_{1}\) assumption. According to CLT and the distribution of \(\Lambda_{2}\) are close to the Gauss distribution, we calculate the mean \(\eta\) and variance \(\sigma^{2}\) respectively as:  
\[\eta\approx E\{\Lambda_{2}|H\}=\sum\limits_{i=1}^{N}E\{I_{i}|H_{1}\}=\sum \limits_{i=1}^{N}P_{di} \tag{19}\]  
\[\sigma^{2}=var\{\Lambda_{2}|H_{1}\}=\sum\limits_{i=1}^{N}var\{I_{i}|H_{1}\} =\sum\limits_{i=1}^{N}P_{di}(1-P_{di}) \tag{20}\]\(P_{di}\) is a function of \([x_{i},y_{i},x_{t},y_{t}]\), in this paper, when \(N\) is large enough, we can approximate Equation (19) as follows:  
\[\eta(x_{t},y_{t})=\sum_{i=1}^{N}P_{di}(x_{i},y_{i},x_{t},y_{t}){\approx}\sum_{i= 1}^{N}E\{P_{di}(x_{i},y_{i},x_{t},y_{t})\}=N\overline{P_{d}}(x_{t},y_{t}) \tag{21}\]  
Among them:  
\[P_{d}(x,y,x_{t},y_{t})=Q(\tau-\sqrt{\frac{P_{0}}{1+\alpha((x-x_{t})+(y-y_{t}))^ {\frac{\alpha}{2}}}}) \tag{22}\]  
Similarly:  
\[\sigma^{2}(x_{t},y_{t})=N\sigma^{-2}(x_{t},y_{t}){=}\frac{N}{b^{2}}\int_{-\frac {b}{2}}^{\frac{b}{2}}\int_{-\frac{b}{2}}^{\frac{b}{2}}(1-P_{d}(x,y,x_{t},y_{t} ))P_{d}(x,y,x_{t},y_{t})dxdy \tag{23}\]  
Therefore:  
\[P_{d}(x_{t},y_{t})=Pr\{\Lambda_{2}\geq T\}{\approx}\ Q(\frac{T-\eta(x_{t},y_{ t})}{\sigma(x_{t},y_{t})}) \tag{24}\]  
We assume that the target has an even distribution in the ROI, and that the average \(P_{d}\) is:  
\[P_{d}=\frac{1}{b^{2}}\int_{-\frac{b}{2}}^{\frac{b}{2}}\int_{-\frac{b}{2}}^{ \frac{b}{2}}P_{d}(x_{t},y_{t})dx_{t}dy_{t} \tag{25}\]  
We make some reasonable assumptions to simplify the calculation, assuming that ROI is very large, the signal energy of the target decreases with the increase of distance (when \(\alpha\) takes a large value).