# Link Investigation of IEEE 802.15.4 Wireless Sensor Networks in Forests
## 4 Performance of In-Network Links
### Exploring Link Correlation  
In this section, we propose a new performance metric for in-network links, the link correlation, which is potentially helpful to estimate or forecast the link performance and then can serve as a metric in network protocol designs. In general, the link correlation weighs the homogeneity of two links with a common sender in terms of the temporal variation of link quality. The left part of Figure 17 shows that sender \(u\) has five neighbors and then five corresponding directional links, \(\ell_{u,a}\), \(\ell_{u,b}\),..., and \(\ell_{u,e}\); the right part shows the PRR variation as the time increases. It is clear that in Figure 17, the two PRRs of links \(\ell_{u,b}\) and \(\ell_{u,e}\) vary with a similar pattern, while neither of them is congenial with the PRR of link \(\ell_{u,a}\). In such a case, the two links, \(\ell_{u,b}\) and \(\ell_{u,e}\), are regarded to be correlated; we can statistically estimate the PRR of \(\ell_{u,e}\) by the PRR of \(\ell_{u,b}\) if the PRR variation pattern of \(\ell_{u,b}\) can be known a priori. We measure the correlation between two links by calculating the Pearson correlation coefficient of the PRRs of both links and define the correlation by Equation (3), where \(\ell_{1}\) and \(\ell_{2}\) are two directional links with a common sender, and \(Prr_{1}\) and \(Prr_{2}\) are two variables representing the PRR measurements for the two links, respectively. \[Corr(\ell_{1},\ell_{2})=\left\{ \begin{array}{ll}0,&\text{if }\sigma_{Pr_{1}} \cdot\sigma_{Pr_{2}}=0;&\text{otherwise}\\ &\\ \frac{E[Prr_{1}\cdot Prr_{2}]-E[Prr_{1}]\cdot E[Prr_{2}]}{\sigma_{Prr_{1}} \cdot\sigma_{Pr_{2}}}\end{array} \right. \tag{3}\]  
Noticeably, the link correlation we emphasize here is different from the packet correlation considered in prior works [24; 25; 26]. The packet correlation is to profile the temporal correlation between two successive packets transmitted through a single link, and it is often calculated by a conditional probability. Specifically, the packet correlation could be employed in designing more efficient MAC schemes, e.g., adaptively deciding the back-off duration of retransmissions [24]. The link correlation, defined on two links with a common sender, is to quantify to what extent two such incident outgoing links perform similarly, i.e., to measure the possibility of deducing the performance of a link with the known performance of another sender-shared link. The link correlation metric can facilitate the decision making in the resource-constrained networking controls for wireless sensor networks. In  
Figure 16: Symmetry variation of links with different link quality. (**a**) Good link; (**b**) Middle link; (**c**) Poor link. Figure 17, for instance, if there exists significant correlation between the links \(\ell_{u,a}\) and \(\ell_{u,b}\) and the current link quality of \(\ell_{u,a}\) is known, then we do not have to probe the link \(\ell_{u,b}\) within at least a short window, say a few or tens of seconds; we can estimate the PRR of \(\ell_{u,b}\) according to the PRR variation of \(\ell_{u,a}\), thereby saving the precious system resources. Figure 18 shows four typically different link correlations. Figure 18a shows an obvious _good_ correlation between links S6-S2 and S6-S7, while Figure 18d shows a _bad_ correlation between links S11-S2 and S11-S4; there is no significant correlation between links S8-S5 and S8-S12 in Figure 18b, and there is no correlation between links S4-S6 and S4-S11 in Figure 18c. Figure 19a shows the correlation distribution of all of the pairs of incident links, and Figure 19b plots the cumulative distribution of the absolute value of link correlation. We find that among all of the pairs of links having a common sender, 91% of them are less than 0.6, and 80% of them are less than 0.4, in terms of the absolute correlation coefficient. For each sensor node S\({}_{i}\), Figure 20 plots the distribution of the \(Corr(\ell_{m},\ell_{n})\) where links \(\ell_{m}\) and \(\ell_{n}\) both start from sender S\({}_{i}\).