## 3 Proposed Real-Time Data Fusion Method Based on a New Distance Metric
### Data Fusion for Object Information
#### 3.1.4 Proposed Data Fusion for Determining Existence  
Based on the proposed distance, we leverage technique of the weighted DS combination rule for data fusion [15; 23; 26; 35]. Let us define \(\mathcal{M}_{i}\) as the set of vehicles who can communicate with vehicle \(i\) and \(i\) itself, and \(\mathcal{I}_{i}\) as the set of detected objects which are located in vehicle \(i\)'s sensing area. Denote the set of vehicles associated with vehicle \(i\) who can detect object \(k\) by \(\mathcal{M}_{i(k)}=\{j\in\mathcal{M}_{i}|k\in\mathcal{I}_{j}\}\). Figure 3: Example of the proposed distance given \(w(\mathrm{e})=2,w(\mathrm{n})=1\), and \(w(\mathrm{u})=3\). Suppose the vehicle \(i\) currently receives \(n\)-th data point. Define the set of vehicles from which vehicle \(i\) has received sensing data on object \(k\) including \(i\) as \(\mathcal{M}_{i(k)}(n)\). We have that \(|\mathcal{M}_{i(k)}(n)|=n+1\). Without loss of generality (WLOG), we assume that \(j\)-th data is transmitted from vehicle \(j\in\mathcal{M}_{i(k)}\setminus\{i\}\). Vehicle \(i\) calculates distances by (10) between \(m_{k}^{n}\) and \(m_{k^{\prime}}^{l}\forall l\in\mathcal{M}_{i(k)}(n-1)\), and we define \(n\times 1\) similarity matrix [23; 25; 26], which is given by  
\[SM_{k}^{ni}=\mathbf{1}-[d_{k}^{ni}\ d_{k}^{n1}\ d_{k}^{n2}\ \cdots\ d_{k}^{nn-1}]^{\mathrm{T}}. \tag{15}\]  
where \(\mathbf{1}\) is all-ones matrix. Due to symmetry property from Proposition 1, we have \(d_{k}^{ij}=d_{k}^{ii},\forall j=1,\cdots,n\), and thus \(S^{i}(n)\) is a symmetric matrix. Based on (15), we can construct \(n\)-th block matrix \(S^{i}(n)\) such that  
\[S^{i}(n)= \begin{bmatrix}S^{i}(n-1)&SM_{k}^{ni}\\ (SM_{k}^{ni})^{\mathrm{T}}&1\end{bmatrix} . \tag{16}\]  
Let us denote \((p,q)\) element of \(S^{i}(n)\) as \(S^{i}_{(p,q)}(n)\). We define \((n+1)\times 1\) supplementary matrix \(SU\) and credibility matrix \(CR\)[23; 26], respectively, where \(p\)-th elements are defined by  
\[SU_{p}=\sum_{q=1,q\
eq p}^{n+1}S^{i}_{(p,q)}(n),\quad CR_{p}=\frac{SU_{p}}{\sum _{p=1}^{n+1}SU_{p}},\forall p=1,\cdots,n+1. \tag{17}\]  
Note that \(p=1\) represents the values from vehicle \(i\), and the derived \(CR_{p}\) can be regarded as the weights on \((p-1)\)-th received confidence for all \(p\in[2,n+1]\). Let us define \(3\times 1\) weighted confidence vector about object \(k\) at vehicle \(i\) as \(\mathbf{\hat{m}}_{k}^{i}\). We calculate \(\mathbf{\hat{m}}_{k}^{i}\) based on (17) such that  
\[\mathbf{\hat{m}}_{k}^{i}=(CR_{1}\times\mathbf{m}_{k}^{i})+\sum_{l=1}^{n}(CR_{l+1} \times\mathbf{m}_{k}^{l}). \tag{18}\]  
The elements in (18) are the weighted confidences of E, N, and U. We define the set of elements of \(\mathbf{\hat{m}}_{k}^{i}\) as \(\hat{M}_{k}^{i}=\{\hat{m}_{k}^{i}(\mathrm{E}),\hat{m}_{k}^{i}(\mathrm{N}), \hat{m}_{k}^{i}(\mathrm{U})\}\). Next, we can apply the classical DS procedure in (3) to fuse \(\tilde{M}_{k}^{i}\) for \(n-1\) times [25; 26; 35; 47]. The algorithm for existence calculation is presented in Algorithm 1. ```
1:function\([\mathcal{E},\mathcal{N},\mathcal{U}]=\textsc{Existence}\ (S^{i}(n-1),M_{k}^{j},\forall j\in\mathcal{M}_{i(k)}(n))\)
2:Initialize:\(\mathcal{E},\mathcal{N},\mathcal{U}\gets 0\)
3:\(d_{k}^{nj}\leftarrow\) Calculate conflict distance between \(\mathbf{m}_{k}^{n}\) and \(\mathbf{m}_{k}^{j}\), \(\forall j\in\mathcal{M}_{i(k)}(n)\setminus\{n\}\) from (10)
4:\(SM_{k}^{ni}\leftarrow\mathbf{1}-[d_{k}^{ni}\ d_{k}^{ni}\ d_{k}^{n2}\ \cdots\ d_{k}^{nn-1}]^{\mathrm{T}}\)
5:\(S^{i}(n)\leftarrow \begin{bmatrix}S^{i}(n-1)&SM_{k}^{ni}\\ (SM_{k}^{ni})^{\mathrm{T}}&1\end{bmatrix}