## 4 Simulation Results  
Computer simulation is performed in a sensor network where 1,024 sensor nodes are randomly deployed in a 256 \(\times\) 256 square area. The network is divided into 8 \(\times\) 8 grids. Each grid is further divided into 2 \(\times\) 2 sub-grids. Hence each grid has 16 nodes on average. Detection accuracy (DA) and false alarm rate (FAR) are employed as the performance metrics. Experiments are conducted in the following order. First, we estimated the performance improvement due to the smoothing filter. We then evaluated the proposed grid-based event detection to show its effectiveness in achieving high performance even for small events. Finally, the effect of flattening \(\theta\) for randomly deployed sensor networks is estimated. In the first experiment, only transient faults are assumed to evaluate the effectiveness of filtering. Four different values of \(p_{t}\) (0.0 \(\leq p_{t}\leq\) 0.2) are chosen for \(m=2\) (_i.e._, \(l=2r\)), \(w=4\), \(q=3\), and \(\theta=\text{ min}(\frac{d}{4},3)\). The resulting DA and FAR are shown in Figure 3, where our CDF (cooperative decision with filtering) and CD (cooperative decision without filtering), and the well-known MV(majority voting) are compared. Figure 2: An illustration of the proposed grid-based event detection. For both CDF and CD almost perfect detection performance has been achieved, as expected. A significant difference, however, is noticed when FAR is compared. When \(p_{t}\) = 0.2, FAR for CD reaches 0.4, while that for CDF remains very close to 0. MV, although inadequate for a small event region, shows relatively poor DA performance due to the insufficient number of sensor nodes to pass the threshold. We then conducted simulation to evaluate the proposed grid-based event detection scheme. The performance is evaluated for four different values of \(p_{p}\) when \(p_{t}\) = 0.2. For comparison purposes, event detection without inter-cluster communications, named LDF (local decision with filtering), is also included. The performance of CDF is compared with LDF and MV in Figure 4, where effectiveness of the proposed scheme is demonstrated. Both DA and FAR for CDF are very close to 1 and 0, respectively, whereas LDF slowly loses its DA performance. For \(p_{p}\) = 0.2, the difference is approximately 0.05. MV does not perform well as expected. Figure 4: Comparison of CDF and LDF. Figure 3: Improving DA and FAR using a smoothing filter. Event detection accuracy may change with the event size. In a relatively large event region, the simple majority voting will achieve high performance. As the event size becomes smaller, increasing \(q_{d}\) is necessary to maintain high overall performance. DA for three different event region sizes are shown in Figure 5, where \(\theta=\text{min}\ (\frac{d}{4},\ 3)\) is shown to be adequate for \(l=2r\). For \(l=2.5r\) and \(l=3r\), however, some improvements are desirable. Lowering \(q_{c}\) will help improve DA with a negligibly small increase in FAR as shown in Figure 6, where \(\theta=\text{min}\ (\frac{d}{4},\ 2)\) is chosen. As addressed in the previous section, stuck-at-0 nodes can be detected when they are in an event region and the event boundary is identified. They can also be detected if sensor readings of a stuck-at-0 node are confined to a relatively small range over a long period of time compared to the readings of a normal node. If stuck-at-0 nodes are isolated, some additional gain can be expected. The improvements are shown as shown in Figure 7, where CDFS (cooperative decision with filtering and stuck-at-0-removal) and CDF (cooperative decision with filtering) are compared. As noted in the figure, the difference in DA is negligibly small for a relatively small \(p_{p}\). Finally, we conducted simulation to see the performance changes due to flattening \(\theta\). DA and FAR for two different values of \(\theta\), \(\theta_{1}=\frac{d}{4}\) and \(\theta_{2}=\text{min }(\frac{d}{4},\ 3)\), are shown in Table 4 Some notable improvements in DA are observed with a negligibly small increase in FAR.