# A Matrix-Based Proactive Data Relay Algorithm for Large Distributed Sensor Networks
## 3 Problem Description
### Information Quality  
We use \(L_{i}^{h}\subseteq L_{i}\) to express the subset of data that is relevant, and indicates information by \(I_{h}\). The quality \(Q(L_{i}^{h})\) of fused data in subset \(L_{i}^{h}\) can be calculated based on the fusion rule of sensors. Let us take Dempster-Shafer rule [28] as an example,  
\[Q(L_{i}^{h})=\max_{T_{k}\in T}c_{L_{i}^{h}}(T_{k})+c_{L_{i}^{h}}(CLUTTER) \tag{2}\]where \(c_{L_{i}^{h}}(T_{k})\) and \(c_{L_{i}^{h}}(CLUTTER)\) are elements of \(c_{L_{i}^{h}}=\{c_{L_{i}^{h}}(T_{1}),..,c_{L_{i}^{h}}(T_{M}),c_{L_{i}^{h}}(CLUTTER )\}\), representing the basic probability assignment for the fused data indicating information \(l_{h}\). This can be calculated as \(c_{L_{i}^{h}}=\oplus_{d_{i}\in L_{i}^{h}}c_{d_{i}}\), where \" \(\oplus\) \" is the operator in D-S rule of combination. If the number of these data reaches the minimum number threshold \(\omega_{n}\) and the quality of fused data is more than the minimum value threshold \(\omega_{Q}\): \(Q(L_{i}^{h})>\omega_{Q}\) &\(|L_{i}^{h}|>\omega_{n}\), these data will be fused into a piece of information \(l_{h}\). The quality of the whole local data set is the sum of the quality of data sets:  
\[Q(L_{i})=\sum_{L_{i}^{h}\subseteq L_{i}}Q(L_{i}^{h}) \tag{3}\]  
As data is only fused with relevant data, adding one piece of data to a local dataset can only affect the quality of its related data set. The incremental quality of a database after receiving \(d_{j}\) can be calculated as  
\[\Delta Q(d_{j},L_{i})=Q(L_{i}\cup\{d_{j}\})-Q(L_{i})=Q(L_{i}^{h}\cup\{d_{j}\}) -Q(L_{i}^{h}) \tag{4}\]  
Maintaining a precise value of the incremental quality based on Equation (4) for each neighbors is computational high. However, it is unnecessary to keep it so precise and an estimation of this value is enough for the relay decision. What we need is to obtain those data that have higher confidence and have more relevant data in the local data set. To simplify, we use the utility of a data set to approximately estimate the quality of same. The utility can easily be calculated as the sum of utility between data:  
\[U(L_{i})=1/2\sum_{d_{j}\in L_{i}}\sum_{d_{i}\in L_{i}}U(d_{j},d_{l})\propto Q (L_{i}) \tag{5}\]  
where \(U(d_{j},d_{l})\) is the utility between data. The value can be computed either according to the fusion rule of sensors, or given by an expert knowledge system. For example, utility can be looked up in a utility table stored in sensors as their domain knowledge. Particularly, \(U(d_{j},d_{j})=0\), and if \(d_{j}\) and \(d_{l}\) are not relative, \(U(d_{j},d_{l})=0\). Therefore the incremental quality can be approximately represented by the incremental utility:  
\[\Delta Q(d_{j},L_{i}) \propto \tag{6}\] \[\Delta U(d_{j},L_{i}) =U(L_{i}\cup\{d_{j}\})-U(L_{i}) =\begin{cases}\sum_{d_{l}\in L_{i}}U(d_{j},d_{l})&\text{if }d_{j}\
ot\in L_{k};\\ 0&\text{if }d_{j}\in L_{i}\end{cases}\]  
If \(v_{i}\) already has observation about \(d_{hr}\), this data becomes redundant and makes no contribution to fusion, and as such, the benefit of broadcasting is 0. If \(v_{i}\) does not have this data, it may be helpful to sensor \(v_{i}\) to increase the fusion probability. The benefit in this case can be represented by the sum of utility between this data and other data in local data set.