## 3 Proposed Real-Time Data Fusion Method Based on a New Distance Metric
### Data Fusion for Object Information
#### 3.1.3 Proposed Metric for Evidence Distance  
Based on the confidence assignment, in [PERSON]'s approach [24], the similarity between two subsets is measured by their cardinality using the Jaccard index, \(\frac{|A\cap B|}{|A\cup B|}\), and the distance increases with the degree of conflict between two sets of confidences. Specifically, given two confidences \(m_{1}\) and \(m_{2}\) on the same frame of discernment \(\Theta\), the Jousselme distance between them is defined as follows [24],  
\[\hat{d}_{1,2}=\sqrt{\frac{1}{2}(\mathbf{m}_{1}-\mathbf{m}_{2})^{\mathrm{T}}P(\mathbf{m}_{1 }-\mathbf{m}_{2})}, \tag{6}\]  
where \(\mathbf{m}_{1}\) and \(\mathbf{m}_{2}\) are \(|2^{\Theta}|\times 1\) vectors associated with \(m_{1}\) and \(m_{2}\), respectively, and \(P\) is \(|2^{\Theta}|\times|2^{\Theta}|\) matrix whose elements are defined by  
\[P(A,B)=\frac{|A\cap B|}{|A\cup B|},\ \forall A,B\in 2^{\Theta}. \tag{7}\]  
The underlying premise of this approach is that elements of the frame of discernment are equally important. However, in navigational safety, false negative is significantly dangerous [44; 45]. That is, the sensor did not report the existence of an object although the object existed. Therefore, we would like to put more emphasis on the confidence of E than the other elements. This results in an asymmetrical importance among the elements in the frame of discernment. In this paper, we propose a novel distance metric to measure conflict by applying asymmetric weights to each element in the frame of discernment, and the weighted DS combination [15; 25; 26; 35] is considered to fuse the confidences according to the proposed conflict distance. **Definition 1**. : _For the proposed distance metric, we define the weighted cardinality for \(A\in 2^{\Theta}\) as follows,_  
\[|A|_{w}:=\sum_{x\in A}w(x), \tag{8}\]  
_where \(w(\cdot)\) is a weight function \(w:\Theta\rightarrow\mathbb{R}_{++}\)._  
Through \(w(\cdot)\), we can flexibly adjust the weights depending on the importance of the associated elements, i.e., a larger weight indicates a higher importance. By doing so, we can put asymmetric emphasis on the elements in the frame of discernment. Our approach generalizes the similarity measure used by the existing methods in which the volume of similarity is determined by the number of elements in subsets having identical weight [24; 27]. We compute the weighted similarity matrix \(Q\) as follows,  
\[Q(A,B):=\frac{\left|A\cap B\right|_{w}}{\left|A\cup B\right|_{w}}=\frac{\sum_ {x\in A\cap B}w(x)}{\sum_{x\in A\cup B}w(x)},\ \forall A,B\in 2^{\Theta}. \tag{9}\]  
For \(\Theta_{E}\), given the confidences of vehicle \(i\) and \(j\) about object \(k\), the distance between them is defined by  
\[d_{k}^{ij}=\sqrt{\frac{1}{2}(\mathbf{m}_{k}^{i}-\mathbf{m}_{k}^{j})^{ \mathrm{T}}Q(\mathbf{m}_{k}^{i}-\mathbf{m}_{k}^{j})}, \tag{10}\]  
where \(\mathbf{m}_{k}^{i}\) is \(3\times 1\) vector of \(\left(m_{k}^{i}(\mathrm{E})\ m_{k}^{i}(\mathrm{N})\ m_{k}^{i}(\mathrm{U}) \right)^{\mathrm{T}}\). We show that the proposed distance in (10) satisfies all four conditions on distance metrics, which are _non-negativity_, _nondegeneracy_, _symmetry_, and _triangle inequality_[24]. **Proposition 1**. : _The proposed distance \(d_{k}^{ij}\) is a distance metric on \(2^{\Theta_{E}}\)._Proof of Proposition 1.: For \(2^{\Theta_{E}}\), we have 3 elements given by \(\{\mathrm{E},\mathrm{N},\mathrm{U}\}\). Let us denote the weighted cardinalities corresponding to each element as \(w(\mathrm{e})\), \(w(\mathrm{n})\), and \(w(\mathrm{u})\), respectively, i.e., \(w(\mathrm{e}):=\sum_{x\in\mathrm{E}}w(x)\). We can compute the weighted similarity matrix \(Q\) by (9), and suppose that the matrix \(Q\) is partitioned into the form such that  
\[Q= \begin{bmatrix}1&0&Q(\mathrm{E},\mathrm{U})\\ 0&1&Q(\mathrm{N},\mathrm{U})\\ Q(\mathrm{E},\mathrm{U})&Q(\mathrm{N},\mathrm{U})&1\end{bmatrix} = \begin{bmatrix}A& B\\ B^{\mathrm{T}}&C\end{bmatrix} ,\text{ where }A= \begin{bmatrix}1&0\\ 0&1\end{bmatrix} ,B=