## 4 Confidence Determination of Multi-Sensor Data
### Judement of Confidence Distance  
When the location information of the unmanned ship is positioned based on the ZigBee-GPS/BDS multi-sensor positioning system, its sensor characteristic function can be described by the Gaussian distribution curve \(p\) (\(x_{i}\)) [20], and the positioning data \(x_{i}\) follows the normal distribution \(N(\mu,\sigma)\), namely  
\[p(x_{i})=\frac{1}{\sqrt{2\pi\sigma}}e^{\frac{(x_{i}-p)^{2}}{2\beta^{2}}} \tag{21}\]  
where \(x_{i}\) is the \(i\)th positioning data of the sensor, \(\mu\) is the true value of the positioning data, \(\theta\) is the measurement accuracy of the sensor information, and \(\sigma\) is the measurement error of the sensor information, \(i=1,2,\,\ldots,\,n\).  
Let \(\mathbf{X}=\{\mathbf{H}^{\sigma},\mathbf{H}^{\beta},\mathbf{H}^{\text{b}}\}\), \(x_{i}^{\text{ZGB}}=\{\,h_{i}^{z},h_{i}^{\beta},\,h_{i}^{\eta}\}\), \(\forall\)\(i\in[1,\,n]\), \(x_{i}^{\text{ZGB}}\in\mathbf{X}\): the confidence distance of the sensor positioning data is determined using the defined \(p(x_{i})\). The specific steps are as follows:  
Step 1 calculate the confidence distance among the positioning data of the ZigBee-GPS/BDS unmanned-ship positioning system at different times.  
\[d_{i}(k)=p_{r}\left\{\frac{\left|x_{i}^{\text{ZGB}}(k)-x_{i}^{\text{ZGB}}(k+1 )\right|}{\sqrt{\widehat{\tau}}\min(\tau_{i}^{\text{ZGB}}(k),\tau_{i}^{\text{ ZGB}}(k+1))}\right\} \tag{22}\]  
where \(p_{r}\{\cdot\}\) is a probability function, \(x_{i}^{\text{ZGB}}(k)\) and \(x_{i}^{\text{ZGB}}(k+1)\) are the observation values of the \(i\)th positioning data at time \(k\) and \(k+1\), respectively, \(\tau_{i}^{\text{ZGB}}(k)\) and \(\tau_{i}^{\text{ZGB}}(k+1)\) are the measurement variances of the \(i\)th positioning data at time \(k\) and \(k+1\), and \(\widetilde{\tau}\) is the mean of the measurement variance, \(k=1,2,\,\ldots,\,T\).  
Step 2 determine the credibility of the sensor information at time \(k\) according to the confidence level \(\varepsilon\) of the preset sensor support.  
\[\left\{\begin{array}{l}\text{High data credibility},\,\,if\,d_{i}(k)\leq 1-\varepsilon \\ \text{Low data credibility},\,\,if\,d_{i}(k)>1-\varepsilon\end{array}\right. \tag{23}\]Step 3 Divide the positioning data into corresponding confidence intervals according to the confidence distance of the positioning data [21].  
\[\left\{\begin{array}{l}d_{i}(k)>1-\varepsilon,\ x_{i}^{\text{ZGB}}(k)\in[\frac{ \varepsilon}{\tau}-\iota,\frac{\varepsilon}{\tau}]\\ d_{i}(k)\leq 1-\varepsilon,\ x_{i}^{\text{ZGB}}(k)\in[\frac{\varepsilon}{\tau}, \frac{\varepsilon}{\tau}+\iota]\end{array}\right. \tag{24}\]  
where \(\iota\) is the variable coefficient of the sampling sample in the probability interval.  
In this paper, the credibility between sensor information at different times was determined by calculating the ZigBee-GPS/BDS confidence distance, and the determined data was divided into corresponding confidence intervals according to the probabilistic measurement value between the positioning data, which caused the credible data to be close to the region with higher confidence. Giving priority to the confidence factors of the sensor positioning data is helpful in improving the efficiency of data fusion. The set confidence level \(\varepsilon\) corresponded to the mean value \(\widehat{\tau}\) of the sensor measurement variance, which was used to represent the change in the confidence degree of the multi-sensor positioning data at different times. The classification of the corresponding confidence interval also changed when the mean value \(\widehat{\tau}\) of the measurement variance was different, which improved the assignment accuracy of the data fusion algorithm to the sensor positioning data.