## 3 Results and Discussion
### Simulations for Chain-Based Network
To achieve a very low MSE, for instance consider 0.06, as it is evident from Figure 7, we need about 35 additional nodes for AVG and 31 additional nodes for the proposed method than EBDKF. However, to achieve MSE of 0.228, the proposed method will require 10 additional nodes while AVG will require around 21 more additional nodes than required by EBDKF method. When the MSE requirement goes above 0.25, the proposed method outperforms both AVG and EBDKF. For instance, to achieve MSE of 0.30, EBDKF will require 35 more nodes and AVG will require 42 more nodes than the proposed method. These results verify the network MSE obtained in Figure 4.0 Therefore, using less resources (nodes), the proposed method can achieve an MSE as low as 0.3 than other data fusion techniques considered. The data retransmission is another technique used to mitigate the impacts of transmission errors. In particular, we analyze the network MSE vs. maximum number of additional retransmissions allowed per hop per data frame/packet required for AVG, EBDKF, and the proposed statistical method, as shown in Figure 4.1 Here, we consider an equally spaced 50 nodes in a chain-based network. The BER of a link is equal to 0.05. Here we have made the assumption that communication channel errors are not affecting on acknowledgment frames/packets so that an acknowledgment sent is always received. We vary the maximum number of retransmission attempts for each hop per data frame/packet between 0 and 4. Figure 4.2 The number of nodes required to achieve a given percentage error using the proposed statistical method, average, and EBDKF data fusion for IoT platform. It is observed that the proposed method maintains the network MSE less than 0.3 under all conditions having a very low gradient suggesting that its network MSE is very slightly decreased in a retransmission allowed network. The data retransmission does not provide any additional benefit for the proposed method to improve the accuracy. On the other hand, for AVG method, it is required to set the maximum retransmission attempts to 3 to reduce the MSE less than 0.30. Further, the EBDKF method shows a very high network MSE when no retransmissions are allowed having least accuracy out of the three fusion methods. But it can improve drastically its network MSE to 0.29 getting almost equal MSE performance to the proposed method and outperforming Average fusion when a maximum of one retransmission is allowed as evident from Figure 4.3  
Even though retransmissions can reduce network MSE, excessive energy for both information and acknowledgment retransmissions, representing excessive use of communication resources, are the major drawbacks. Therefore, it is evident that the proposed method can outperform both AVG and EBDKF at high BER value like 0.05, as it can achieve least network MSE with least usage of energy and communication resources. The usage of communication resource for frame/packet transmission in three schemes for chain-based network can be generalized as follows. Assume that there are \(k\) number of nodes and n is the number of information frames/packets that are transmitted within a single link per data gathering cycle in a chain-based WSN in which retransmissions are not allowed. Also, let \(L\) be the maximum number of retransmissions allowed for a node per information frame/packet. Depending on the channel conditions, the maximum retransmission attempts might not be used in each link. The number of retransmissions will depend on the BER(\(P_{e}\)). All nodes will transmit at least one time. Then, when \(L=1\), the number of nodes retransmitting one time will directly depend on the value of \(P_{e}\). When \(L=2\), it can be assumed to be proportional to \({P_{e}}^{2}\) and so on and so forth. Therefore,  
Figure 4.4 The performance comparison among the proposed statistical, AVG, and EBDKF information fusion methods in a retransmission allowed chain IoT platform.