3.1.1. Simulations for Data Accuracy of a Deterministic
Constant (Slowly Varying) Parameter Having Bounded Sensor Noise and Measurements
In this section, we compare the
and the
when the proposed statistical information fusion method; the EBDKF and AVG method are employed in the presence of measurement errors, quantization errors, and transmission errors. We run 600 iterations for a single run in order to get enough steady state measurements while there are multiple number of runs for a simulation. This algorithm is used to get
and then as seen in Section 3 in [
44]; it is used to obtain the estimate of deterministic parameter
, having an expectation of 1.8. One of the reasons for selecting 1.8 as the system parameter for analysis is since it is close to zero such that sensor measurements will rarely exceed sensor measurement limit. Also, it is not one of the quantized output values in the implemented quantized scheme of 8 bits as the nearest quantized output values to it are 1.79688 and 1.875. Therefore, even the highest accurate estimation of 1.80 will cause a quantization error of 0.00312 so that effect of quantization error will be reflected on the generated results. In these simulations, we generate different network conditions for different runs and in each network condition, data is gathered 600 times. The final results are obtained by averaging these simulation results. We first analyze the impact of transmission errors. A network that consists of 50 nodes is considered and the BER of a link is increased by reducing the distances between nodes. These values are varied between
and 0.05 by changing the distance between two consecutive nodes from 24.45 m to 41.5 m. We generate heterogeneous measurement noise variances by setting
and
with variance upper bounded by 0.7.
Figure 4 compares the Network MSE and MSE at sink nodes of the proposed statistical method, EBDKF, and AVG method.
We can observe that the proposed statistical method outperforms AVG method for the considered BER range. The performance gap increases significantly with the BER. The proposed method also outperforms the EBDKF when the BER, due to communication errors, is greater than 0.015. For very low BER values (<10
), the accuracy of proposed method is slightly higher than Diffusion Kalman Filtering. It can be observed from
Figure 4 that for higher BER (>0.01), the EBDKF algorithm’s network mean square error increases with a rising gradient. That is due to the fact that EBDKF heavily relies on multiple transmissions among neighbors for a single diffusion estimate at an iteration step which can get highly inaccurate under transmission errors. Due to that, as it is evident from
Figure 4 EBDKF is superior than the proposed method in terms of information accuracy for moderate bit error rates in the range of
to 0.0115. However, EBDKF shows large MSE under high communication errors which can occur in a Smart Grid environment. EBDKF’s accuracy degrades even below average fusion method when BER is greater than 0.0175. However, on the other hand, the proposed method is aware of the communication errors and adjust the weighting coefficient automatically which is
in Equation (
24). When communication errors increase,
also increases, so that it having a smaller weight in the fusing equation. Due to this as evident from
Figure 4, MSE in proposed method increase with a very low gradient with BER.
As the proposed statistical method accurately models the received information, we are able to eliminate any bias that can occur in the final result by dividing each received data with corresponding coefficient. Furthermore, the use of the weighting property in the conventional BLUE estimator helps to achieve the minimum variance for the received information. Thus, the proposed statistical method is able to maintain the network mean square error less than 0.29 for the considered BER range.
Diffusion techniques do not differentiate between the parent and child nodes; they identify both child nodes and the parent as neighbors. For chain structure employing diffusion techniques, the number of neighbors of the intermediate nodes is always equal to two, whereas the number of neighbors of the first and sink node is equal to one. The number of child nodes per network and the sink node for a WSN employing the proposed statistical method and the AVG fusion techniques are always equal to one. Therefore, the accuracy of the information fusion for network and the sink node will be similar for networks employing AVG and the proposed method, as is evident from the results obtained in
Figure 4. When the number of nodes is high, so is the number of intermediate nodes. Therefore, the average number of neighbors for a network employing EBDKF is equivalent to two. But, the sink node will definitely have only one neighbor. The performance gap of EBDKF for network MSE error and MSE of sink node observed in
Figure 4 must be due to this difference in number of neighbors of the sink node and the network. For low BER values, the network MSE has been lower than sink node’s MSE. That is because an estimate produced by a diffusion technique becomes more accurate when the number of neighbors is higher under low communication errors. However, the higher number of neighbors will greatly reduce the accuracy under high communication errors for diffusion techniques. That is why the sink having only one neighbor outperforms its network having two neighbors for high BER values. However, still, we can observe that the proposed method’s sink node’s MSE is lesser than MSE of sink node employing EBDKF for BERs higher than 0.025. However, the accuracy of the sink node of EBDKF fusion technique is always lesser than accuracy of a sink node with Average data fusion for all BER s.
To have a very good understanding about the network’s estimate of the system parameter under different BER for different information fusion schemes, we plot the mean value of the estimate at each node and the mean value of each node’s 95% confidence interval limits as illustrated in
Figure 5.
When analyzing the results obtained in
Figure 5, it can be seen that for low BER values, the network mean estimate is very close to the system state of 1.8 for all fusion techniques. However, the performance in terms of accuracy is different due to difference in the variance of the estimates of different fusion techniques. This variance can be identified by looking at the size of the 95% confidence interval in
Figure 5. In very low BER conditions (<10
), the 95% confidence interval is also narrow and its width is in decreasing order for Average, EBDKF, and proposed method proving the previous result obtained for network MSE for very low probability of communication errors. This is because we can say that when the confidence interval for the results is higher, so is the network mean square error. The mean estimate of the proposed method decreases slightly and remains very close to the real value of 1.8 when BER increases outperforming the accuracy of mean estimate of other two fusion techniques. However, the mean value tends to deviate more from real value of 1.8 when BER increases for other two data fusion techniques where the AVG method having a moderate deviation and EBDKF having the highest deviation. The reason for observing high network MSE for high BER for average and EBDKF can be explained by looking at the increasing size of the 95% confidence interval for those two methods as evident from
Figure 5. In contrast, the proposed method’s confidence interval is only slightly increased with increasing BER, so that it is showing least network MSE for high BER. Further, the average method’s accuracy approaches that of proposed method when
as seen from the confidence interval and observable in
Figure 4 as well.
We further analyze the impact of the transmission errors on each individual bit in a data frame/packet.
Figure 6 depicts how each individual bit in a data frame/packet experience two different BER conditions. Here, we compare the mean value of bit wise accuracy of the quantized output of fused result of each node with the quantized value of the system parameter. We set 100% error if a bit is altered and 0% error if the particular bit of the quantized fused output is same as the corresponding bit of the quantized value of system parameter. Here, we get the mean of the results for particular bit considering all the nodes to obtain the network bit level performance for first five most significant bits.
The significant bits can drastically change the final value from the true value. For example, the impact of MSB is higher than the total combination of next four significant bits. When BER is 0.005, the average error probabilities of first five significant bits of EBDKF is lesser than both AVG and proposed method. The percentage error of first two Significant bits for proposed and AVG methods remain at very low values but higher than EBDKF. The decreasing order of magnitude of percentage error in first two most significant bits are in the order of AVG, proposed method, and EBDKF. Therefore, this proves why it is seen a high network MSE and high width of 95% confidence interval in
Figure 4 and
Figure 5, respectively, in decreasing order as AVG, proposed method and EBDKF when BER is 0.005. However, when BER is 0.05, the preceding order changes to EBDKF, AVG, and the proposed method proving that proposed method shows lower error probabilities for the first two most significant bits compared to AVG and EBDKF as seen in
Figure 6. When observing the size of the confidence intervals at a BER of 0.05, only the proposed method’s interval is small as seen in
Figure 5. The size of the confidence interval is moderately large for Average and very large for EBDKF. Same order is observed for first two MSB s for BER of 0.05 in
Figure 6. Therefore, we can use the proposed method to protect these most significant bits successfully under high communication error environments. This explains why we observe a significant performance gap among the three different fusion schemes at BER 0.05 in
Figure 4 and
Figure 5. It can also be observed from
Figure 6 that the percentage error of both third and fourth significant bits of all considered fusion schemes have increased when BER is 0.05 compared to BER of 0.005. However, the percentage error of first two significant bits for proposed method is reduced by a very small amount and that for Average and EBDKF have been significantly increased as the BER is increased.
Redundant node deployment can be used to improve the accuracy of fusion schemes. We simulate the information fusion process in chain-based network by varying the number of nodes. Here, the distance between the sink node and the furthest node is kept constant at 2500 m.
Figure 7 shows the number of nodes required for AVG, EBDKF, and proposed statistical methods to maintain a given network mean square error.
It is evident from
Figure 7 that when more sensor nodes are added, the network MSE of all the aggregation schemes decrease. However, the rate of decrement of MSE depends on the aggregation method. 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. 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 8. 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.
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 8.
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(
). All nodes will transmit at least one time. Then, when
, the number of nodes retransmitting one time will directly depend on the value of
. When
, it can be assumed to be proportional to
and so on and so forth. Therefore, total Transmissions (
N) considering only information frame/packet (without acknowledgments) transfer will be as shown in Equation (
38),
where
is a constant. Using Equation (
38), an upper bound on number of total transmissions can be derived as depicted in Equation (
39) for the cases when
.
As proved in the preceding section,
in the proposed method, EBDKF, and Average fusion methods, respectively, to achieve a network MSE less than 0.3. It is evident that
in proposed method and average method as a node will send only one fused data frame/packet in a link. As there are two
,
,
for each link in EBDKF per data gathering cycle,
. Therefore, by substituting in Equation (
39) for upper bound,
,
,
. Therefore, when the network MSE is less than 0.3, total retransmissions will be lesser than the upper bound. In order to obtain only retransmission upper bound,
in Equation (
38) will have to be replaced by
L. When acknowledgment frames/packets are also considered, the total transmissions will be even more than the values calculated in preceding sections.
Although, the information retransmissions have been shown as a solution to improve the accuracy of the information fusion with AVG and EBDKF methods as it was stated in preceding section, they consume additional energy which is a major disadvantage in WSN s. To get the exact number of total retransmissions when BER is 0.05,
Figure 9 was obtained to visualize the total retransmission attempts for data frames/packets utilized in the network when the maximum retransmission attempts are limited to one, two, three, and four among EBDKF and AVERAGE fusion techniques. We do not observe the total retransmissions of proposed method, as we came to conclusion that frame/packet retransmissions very slightly improve the accuracy of the estimate generated by proposed method.
It can be clearly observed from
Figure 9 that all values are within the upper bound for retransmission of 157 and 294 for AVG and EBDKF when
and
, respectively. As observed in
Figure 9, approximately total of 24 retransmission attempts are used in AVG method and 93 in EBDKF to achieve a network MSE less than 0.3. These additional retransmissions consume excessive energy for information retransmissions. Therefore proposed method can effectively save transmission energy compared to other fusion algorithms.