## 3 Grid-Based Event Detection
### Isolating Faulty Nodes Using Confidence Level Evaluation  
Permanent faults, unless the number of faulty nodes is negligibly small, also degrade the event detection performance at the cluster head. Since the number of faulty nodes is expected to increase with time, it is desirable to isolate them as soon as they are detected and identified. In our grid-based event detection, each cluster head receives reports from its member nodes, and makes a decision \(D\) based on a threshold \(\theta\), where \(D\) = 1 indicates an event. Each cluster head maintains confidence levels of its member nodes to isolate nodes with permanent faults when their confidence levels reach the assigned lower bound, resulting in better event detection performance at the cluster head. Depending on the decision made and the reports from its member nodes, the cluster head updates the confidence levels of the member nodes, reflecting the correctness of the reports. These updates need to be careful since fault-free nodes might generate some incorrect reports. Let \(c_{k}\), ranging from 0 to 1 and initialized to 1, represent the confidence level of node \(v_{k}\). At the end of the event or fault detection cycle, the confidence levels of the member nodes are updated to reflect the correctness of their reports as follows. For \(D\) = 0 (_i.e._, the decision is no-event) in Table 3, the confidence level \(c_{k}\) of node \(v_{k}\) is increased by \(\alpha\) to min (\(c_{k}+\alpha\), 1) if the node reported a 0. If it reported a 1 instead, \(c_{k}\) is lowered by \(\beta\) to max (\(c_{k}-\beta\), 0). The values of \(\alpha\) and \(\beta\) need to be assigned depending on the fault behavior, if the best performance is necessary. In our evaluation, \(\alpha\) = \(\beta\) = 0.1 is chosen without loss of generality. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \(w\) & \(q\) = 1 & \(q\) = 2 & \(q\) = 3 & \(q\) = 4 & \(q\) = 5 & \(q\) = 6 \\ \hline
2 & 0.360 & 0.040 & - & - & - & - \\
3 & 0.488 & 0.104 & 0.008 & - & - & - \\
4 & 0.590 & 0.181 & 0.027 & 0.002 & - & - \\
5 & 0.672 & 0.263 & 0.058 & 0.007 & 0.000 & - \\
6 & 0.738 & 0.345 & 0.099 & 0.017 & 0.002 & 0.000 \\ \hline \end{tabular}
\end{table} 
Table 2: The effective transient fault probability \(\tilde{p}_{t}\) for \(p_{t}\) = 0.2. For \(D=\) 1, on the other hand, it is not easy to figure out if \(v_{k}\) reported correctly since the event boundary is unknown. Especially for a sensor node with limited resources and small event regions, it becomes difficult to locally figure out the exact boundary. As far as fault detection/isolation is concerned, however, it would be acceptable not to update the confidence levels when \(D=\) 1 for the following two reasons: (i) A stuck-at-1 node (outside the event region) can be identified and isolated when \(D=\) 0. Hence the last row in the table does not cause a problem; (ii) A stuck-at-0 node in an event region can hardly be identified in the case of no-event. It, however, can be detected if sensor readings of stuck-at-0 nodes do not change or are confined to a extremely small range while those of fault-free nodes vary notably over time. This type of stuck-at-0 can be identified and reflected in the confidence level even when \(D=\) 0, although we do not include this in the subsequent simulation in order to estimate the worst case performance of the proposed scheme. If some existing sophisticated techniques are employed to figure out the exact boundaries, Table 3 can readily be modified. The simulation results in the next section, however, show that high performance can still be obtained even without isolating stuck-at-0 nodes when the permanent fault probability \(p_{p}\) is 0.2. In addition, the performance gain achieved by removing stuck-at-0 nodes in that case will be shown to be marginal. Sensor nodes are logically removed from the network and cannot participate in the event detection process when their confidence levels reach the lower bound (0 in this paper).