## 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
\)
6: Calculate \(\mathbf{\hat{m}}_{k}^{i}\) based on \(S^{i}(n)\) by (17) and (18)
7:\(\{\mathcal{E},\mathcal{N},\mathcal{U}\}\leftarrow\{\hat{m}_{k}^{i}(\mathrm{E}),\hat{m}_{k}^{i}(\mathrm{N}),\hat{m}_{k}^{i}(\mathrm{U})\}\)
8:for\(t=1,\cdots,n\)do
9:\(\mathcal{E}\leftarrow\mathcal{E}\oplus\hat{m}_{k}^{j}(\mathrm{E}),\mathcal{N} \leftarrow\mathcal{N}\oplus\hat{m}_{k}^{j}(\mathrm{N}),\mathcal{U}\leftarrow \mathcal{U}\oplus\hat{m}_{k}^{j}(\mathrm{U})\)
10:endfor
11:return\(\{\mathcal{E},\mathcal{N},\mathcal{U}\}\)
12:endfunction
```  
**Algorithm 1** Existence calculation for \(i\) about object \(k\)  
Note that the proposed scheme has polynomial time complexity, which enables its real-time data fusion. The DS rule in general has high computation complexity for data fusion, \(O(2^{2|\Theta|})\), which is a main drawback. However, we have 3 elements in \(2^{\Theta_{E}}\), and thus the complexity of calculating the DS rule is \(O(1)\). Specifically, the vehicles which detect object \(k\) participate in the data fusion, thus vehicle can obtain the fused output with \(O(|\mathcal{M}_{i(k)}|)\) complexity for each time instant, which in turn we have \(O(N)\) complexity in the worst case due to \(|\mathcal{M}_{i(k)}|\leq N\). Furthermore, the considered data fusion scheme is valid irrespective of the order of data arrivals as well. **Proposition 2**. : _Given a set of data points, let \(\Pi_{u}\) and \(\Pi_{v}\) be the arbitrary order of data points where \(\Pi_{u}\
eq\Pi_{u}\) for \(u\
eq v\), and we denote corresponding outputs as \(\tilde{M}_{k}^{i}(\Pi_{u})\) and \(\tilde{M}_{k}^{i}(\Pi_{v})\), respectively. Then, we have \(\tilde{M}_{k}^{i}(\Pi_{u})=\tilde{M}_{k}^{i}(\Pi_{v})\)._  
Proof of Proposition 2.: Suppose that vehicle \(i\) receives \(n\)-th data. We suppose that there are different data arrival orders which are denoted by \(\Pi_{u}\) and \(\Pi_{v}\), \(|\Pi_{u}|=|\Pi_{v}|=n\). The hat and check symbols denote the notations related to \(\Pi_{u}\) and \(\Pi_{v}\), respectively, e.g., \(\hat{S}^{i}(n)\) and \(\tilde{S}^{i}(n)\) are similarity block matrices in (16) corresponding to \(\Pi_{u}\) and \(\Pi_{v}\). WLOG, we focus on the data comes from vehicle \(j\). Let us define the order of this data in \(\Pi_{u}\) and \(\Pi_{v}\) as \(\Pi_{u}^{j}\) and \(\Pi_{u}^{j}\), respectively. For \(\Pi_{u}^{j}\)-th data in \(\Pi_{u}\), we can express the associated element in supplementary matrix such that  
\[\hat{S}\hat{U}_{\Pi_{u}^{j}+1} =\sum_{q=1,q\
eq\Pi_{u}^{j}+1}^{n+1}\hat{S}_{(\Pi_{u}^{j}+1,q)}^{ i}(n) \tag{19}\] \[=(1-\tilde{d}_{k}^{\Pi_{u}^{j},i})+\sum_{q<\Pi_{u}^{j}}(1-\tilde{ d}_{k}^{\Pi_{u}^{j},q})+\sum_{q>\Pi_{u}^{j}}(1-\tilde{d}_{k}^{\Pi_{u}^{j}})\] (20) \[=(1-\tilde{d}_{k}^{\Pi_{u}^{j},i})+\sum_{q=1,q\
eq\Pi_{u}^{j}}^{n }(1-\tilde{d}^{\Pi_{u}^{j},q})\] (21) \[=(1-\tilde{d}_{k}^{\Pi_{u}^{j},i})+\sum_{q=1,q\
eq\Pi_{v}^{j}}^{n }(1-\tilde{d}^{\Pi_{v}^{j},q})=\hat{S}\hat{U}_{\Pi_{v}^{j}+1} \tag{22}\]  
The equalities in (19) and (20) come from the definitions in (15) and (17), respectively. The third equality in (21) holds due to the _symmetry_ of the proposed distance. Finally, we can derive the last equality, as the both \(\Pi_{u}^{j}\)-th and \(\Pi_{v}^{j}\)-th data refer to the same information from vehicle \(j\). Therefore, we have that  
\[\hat{C}\hat{R}_{\Pi_{u}^{j}+1}\times\hat{m}_{k}^{\Pi_{u}^{j}}=\hat{C}\hat{R}_{ \Pi_{v}^{j}+1}\times\hat{m}_{k}^{\Pi_{v}^{j}}. \tag{23}\]  
We can apply this process to all the data points, which results in the same weighted output irrespective of any arrival order. We can extend Proposition 2 for any order of data points, which results in that given a set of data points, we have the same final output irrespective of the order of fusion. This property guarantees an independence of the data fusion scheme. In other words, as it is a data-driven fusion method, we can focus on the data synthesis rather than external influence such as channel condition, etc.