## 5 Proposed Solution
### Multivariate Spatial Correlation  
WSNs consist of multiple nodes spread in a redundant way. Thus, we get a fault tolerant system through dense networks. On the other hand, these networks are usually composed of resource constrained devices. The energy is supplied by batteries and energy consumption can be better managed when the correlations from monitoring applications are taken. Therefore, we can develop solutions which reduce data traffic in the network. The spatial correlation can be exploited to optimize data communication to the sink and between neighbor sensor nodes [2,12,14].  
The spatial correlation happens due to similarities of data being sent to the sink by several sources from high density network [2]. As mentioned in the previous section, [PERSON]'s coefficient [Equation (1)] does not calculate the multivariate spatial correlation. We propose the use of the Euclidean distance to determine the multivariate spatial correlation between two multidimensional vectors, instead of using Pearson's coefficient. The Euclidian distance shows how close a multidimensional vector is to another. The Euclidian distance is defined as follows:  
\[d_{X_{N},X_{V}}=\sqrt{\sum\
olimits_{j=1}^{k}(x_{N_{j}}-x_{V_{j}})^{2}} \tag{5}\]  
where \(X_{N}=x_{N_{1}},...,x_{N_{j}}\) and \(X_{V}=x_{V_{1}},...,x_{V_{j}}\). In our case \(d_{X_{N},X_{V}}\) represents the correlation between two multidimensional vectors of dimension \(k\) with \(j=1,...,k\) to be compared in terms of their correlations. Each vector contains the values of \(\beta\) and \(\alpha\) coefficients of each gathered variable by sensor node \(N\) and its neighbor sensor node \(V\).  
Figure 4: Proposed mechanism diagram.  
The smaller the Euclidean distance is, the greater is the correlation between two vectors. Thus, we can compare coefficients \(\beta\) and \(\alpha\) of the multiple linear regression generated from consecutive readings gathered by a sensor node to \(\beta\) and \(\alpha\) coefficients from its neighbor sensor nodes at a given time. The sensor node checks if there is correlation between itself and its neighbor sensor nodes (Step #3), before sending a packet containing \(\beta\) and \(\alpha\) coefficients of the multiple linear regression function. If the Euclidian distance is close to 0 (zero), then it means that a packet with the same content was previously sent by any other neighbor sensor node (Step #4).  
In our proposed solution, the sensor node detects if there is multivariate spatial correlation between itself and its neighbor node by tree-based routing. This is similar to the compression mechanism adopted by [PERSON] _et al._[8]. The sensor node checks the relationship degree of coefficients \(\beta\) and \(\alpha\) by calculating the value of \(d_{X_{N},X_{V}}\) [Equation (5)].  
The sensor node does not send coefficients \(\beta\) and \(\alpha\) of the current readings to the sink if the Euclidian distance is 0 (zero). It eliminates the overlapping of information between neighbor sensor nodes. Thus, some sensor nodes do not send data packets at a given time. Therefore, it reduces the broadcast between neighbor sensor nodes and also the data forwarded by the relays.