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International Journal of Distributed Sensor Networks
Volume 2014 (2014), Article ID 148234, 12 pages
Model Selection Approach for Distributed Fault Detection in Wireless Sensor Networks
1Department of Statistics, West Bengal State University, Barasat, India
2ASD, Indian Statistical Institute, 203 B. T. Road, Kolkata 700 108, India
3Chennai Mathematical Institute, Chennai, India
Received 2 April 2013; Revised 13 November 2013; Accepted 14 November 2013; Published 20 January 2014
Academic Editor: Shuai Li
Copyright © 2014 Mrinal Nandi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Sensor networks aim at monitoring their surroundings for event detection and object tracking. But, due to failure or death of sensors, false signal can be transmitted. In this paper, we consider the problems of distributed fault detection in wireless sensor network (WSN). In particular, we consider how to take decision regarding fault detection in a noisy environment as a result of false detection or false response of event by some sensors, where the sensors are placed at the center of regular hexagons and the event can occur at only one hexagon. We propose fault detection schemes that explicitly introduce the error probabilities into the optimal event detection process. We introduce two types of detection probabilities, one for the center node, where the event occurs, and the other one for the adjacent nodes. This second type of detection probability is new in sensor network literature. We develop schemes under the model selection procedure and multiple model selection procedure and use the concept of Bayesian model averaging to identify a set of likely fault sensors and obtain an average predictive error.
Traditional and existing sensor-actuator networks use wired communication, whereas wireless sensor networks (WSN) provide radically new communication and networking paradigms and myriad new applications. The wireless sensors have small size, low battery capacity, nonrenewable power supply, small processing power, limited buffer capacity, and low-power radio. They may measure distance, direction, speed, humidity, wind speed, soil makeup, temperature, chemicals, light, and various other parameters.
Recent advancements in wireless communications and electronics have enabled the development of low-cost WSN. A WSN usually consists of a large number of small sensor nodes, which are equipped with one or more sensors, some processing circuit, and a wireless transceiver. One of the unique features of a WSN is random deployment in inaccessible terrains and cooperative effort that offers unprecedented opportunities for a broad spectrum of civilian and military applications, such as industrial automation, military surveillance, national security, and emergency health care [1–3]. Sensor networks are also useful in detecting topological events such as forest fires .
Sensor networks aim at monitoring their surroundings for event detection and object tracking [1, 5]. Because of this surveillance goal, coverage is the functional basis of any sensor network. In order to fulfill its designated tasks, a sensor network must fully cover the Region of Interest (ROI) without leaving any internal sensing hole [6–9]. So far, a number of movement-assisted sensor placement algorithms have been proposed. An exclusive survey on these topics is presented by Li et al. . On the other hand sensor could die or fail at runtime for various reasons such as power depletion and hardware defects. So, even after the ROI is fully covered by the sensors, wrong information can be communicated by some sensors or sensors may fail to detect the event due to noise or obstructions. Chen et al.  have proposed a distributed localized fault detection algorithm for WSN, where each sensor identifies its own status to be either good or faulty and the claim is then supported or reverted by its neighbors. The proposed algorithm is analyzed using a probabilistic approach. Sharma et al.  have characterized the different types of fault and proposed a different algorithm for fault detection considering different types of fault. Some of the methods are statistical, like using histogram and so forth. Both works can only detect the faulty sensors, but not the event.
One of the important sensor network applications is monitoring inaccessible environments. Sensor networks are used to determine event regions and boundaries in the environment with a distinguishable characteristic [13–15]. The basic idea of distributed detection  is to have each of the independent sensors make a local decision (typically, a binary one; i.e., an event occurs or not) and then combine these decisions at a fusion sensor (the sensor which collects the local information and takes the decision) or at a base station to generate a global decision.
A closely related area is neural network. Several works are there in the literature on this area. Li and Qin  find a feasible solution to a class of nonlinear inequalities defined on a graph proposing a recurrent neural network. The convergence of the neural network and the solution feasibility to the defined problem are both theoretically proven. They proposed neural network features as a parallel computing mechanism and a distributed topology isomorphic to the corresponding graph which is suitable for distributed real-time computation. The proposed neural network is applied to range-free localization of WSNs. Li et al.  show that feasible solution set to the same problem is often infinity and Laplacian eigenmap is used as heuristic information to gain better performance in the solution. A continuous-time projected neural network and the corresponding discrete-time projected neural network are both given to tackle this problem iteratively. The effectiveness of the proposed neural networks is tested and compared with others via its applications in the range-free localization of WSNs. Location information is useful for mobile phones. There exists a dilemma between the relatively high price of GPS devices and the dependence of location information acquisition on GPS for most phones in current stage. Li et al.  formulate the problem as an optimization problem defined on the Bluetooth network. The solution to this optimization problem is not unique. Heuristic information is employed to improve the performance of the result in the feasible set. They used recurrent neural networks to solve the problem distributively in real time. The convergence of the neural network and the solution feasibility to the defined problem are both theoretically proven. The hardware implementation of the proposed neural network is also explored in this paper.
Distributed algorithms are also used for a network dynamic system. Li et al. [20, 21] studied the decentralized control and kinematic control of multiple redundant manipulators for the cooperative task execution problem. The problem is formulated as a constrained quadratic programming problem, and then a recurrent neural network with independent modules is proposed to solve the problem in a distributed manner. They proposed a novel strategy capable of solving the problem, even though there exist some manipulators unable to access the command signal directly.
Another related area is the winner-take-all (WTA) competition, which is widely observed in both inanimate and biological media and society. Many mathematical models are proposed to describe the phenomena discovered in different fields. These models are capable of demonstrating the WTA competition. Li et al. [22, 23] make steps in that direction and present a simple model, which produces the WTA competition by taking advantage of selective positive-negative feedback through the interaction of neurons via -norm. They also present a class of recurrent neural networks to solve quadratic programming problems. Different from most existing recurrent neural networks for solving quadratic programming problems, the proposed neural network model converges in finite time and the activation function is not required to be a hard-limiting function for finite convergence time. The stability, finite-time convergence property, and the optimality of the proposed neural network for solving the original quadratic programming problem are proven in theory. Extensive simulations are performed to evaluate the performance of the neural network with different parameters. In addition, the proposed neural network is applied to solving the k-winner-take-all (k-WTA) problem.
1.1. Our Motivation
In this paper, we are interested in one particular query: determining event in the environment (i.e., ROI) with a distinguishable characteristic. We assume the ROI to be partitioned into suitable number of congruent regular hexagonal cells (i.e., we can think of ROI as a regular hexagonal grid). This physical structure of ROI is not a requirement for the theoretical analysis, and we can do the similar analysis with another structure also. Suppose that sensors are placed a priori at the center (which are known as nodes) of every hexagon of the grid. We assume that the sensors are connected to its adjacent sensor nodes in the sense that a hexagon will be strongly covered by its center node and weakly covered by the adjacent nodes. If event occurs in the hexagon where a particular sensor lies, then that particular sensor can detect the event with a greater probability, whereas if event occurs in any adjacent hexagon, then the particular sensor can detect the event with a lesser probability. Hence, only one node (center node of the event hexagon) can detect the event hexagon with greater probability, say , and adjacent nodes (six for interior nodes and less for boundary nodes) can detect the event hexagon with lesser probability, say , with . We assume that no other sensor can detect the event hexagon. In this paper, unlike the previous works, we assume that if the event occurs, then it occurs at only one hexagon of the grid which will be known as event hexagon, and there is no fusion sensor. All sensors can communicate with the base station and the base station takes the decision about the query. As an example, consider a network of devices that are capable of sensing mines or bombs, if we assume that a few mines or bombs may be placed on a particular area of ROI. Information from these devices can be sent to a nearby police station or a central facility. Then, an important query in this situation could be whether a particular hexagon is the event hexagon or not (i.e., mines or bombs are placed or not).
One fundamental challenge in the event detection problem for a sensor network is the detection accuracy which is disturbed by the noise associated with the detection and the reliability of sensor nodes. A sensor may fail to detect the event due to natural obstruction or any other causes. After detecting the event, a sensor can send false message to the base station due to some technical reasons. The sensors are usually low-end inexpensive devices and sometimes exhibit unreliable behavior. For example, a faulty sensor node may issue an alarm, even though it has not received any signal for event or it cannot detect any event and vice versa. Moreover, a sensor may be dead, in which case the sensor cannot send any alarm.
1.2. Previous Work and Our Contribution
Lou et al.  consider two important problems for distributed fault detection in WSN: how to address both the noise-related measurement error and sensor fault simultaneously in fault detection and how to choose a proper neighborhood size for a sensor node in fault correction such that the energy could be conserved. They propose a fault detection scheme that explicitly introduces the sensor fault probability into the optimal event detection process. They show that the optimal detection error decreases exponentially with the increase of the neighborhood size.
Krishnamachari and Iyengar  propose a distributed solution for canonical task in WSN (i.e., the binary detection of interesting environmental events). They explicitly take into account the possibility of sensor measurement faults and develop a distributed Bayesian algorithm for detecting and correcting such faults.
Nandi et al.  consider the problem of distributed fault detection in wireless sensor network (WSN), where the sensors are placed at the center of a particular square (or hexagon) of the grid covering the ROI. They proposed fault detection schemes that explicitly introduce the error probabilities into the optimal event detection process. They developed the schemes under the consideration of Neyman-Pearson hypothesis test and Bayes test. They also calculate type I and type II errors for different values of the parameters.
In almost all the previous works, except , authors assume that event occurs over a region and there are fusion sensors that collect the information locally and take a decision. Since they do not introduce the concept of base station, there is no concept of response probability. Also, they assume that information is spatially correlated. Unlike the previous work, in this paper, we assume that if event occurs, then it occurs at only one cell of the ROI and there is no fusion sensor. All the sensors send information to the base station. We introduce the probability model in two different stages: firstly, when a sensor detects the event and, secondly, when a sensor sends the message to the base station. In the previous works, only one type of detection probability has been introduced to simulate the different error probabilities for some specific values of the parameters. In this paper, we introduce two different detection probabilities and obtain analytically the exact test and estimate the error probabilities by simulation. In almost all the previous works, authors assume the ROI to be a square grid. The hexagonal grid is better in the sense that a minimum number of sensors are required to cover the entire ROI .
In our theoretical analysis, the sensor fault probabilities are introduced into the optimal event detection process. We apply model selection approach, multiple model selection approach, and Bayesian model averaging methods [27, 28] to find a solution of the problem. We develop the schemes using the model selection technique. We calculate different error probabilities and find some theoretical results.
In all previous works, the authors assume only one detection probability. We introduce two detection probabilities, and , one for the center node and other for the adjacent nodes. Even if the center node may fail to detect the event, the adjacent nodes may detect the event, and vice versa. We consider these probabilities and show that, in various situations, the adjacent nodes play key role to detect the event. One can introduce more detection probabilities and analyze the situation in similar manner.
The parameters and , the detection probabilities of a sensor, and error probabilities (see Section 3) cannot be estimated from the real life situations but need to be estimated beforehand by some experimentation. The prior probabilities of various events also cannot be estimated but may be known in some cases. Finally, we calculate the error probabilities numerically for some values of the parameters of our model and make some concluding remarks analyzing the results.
2. Statement of the Problem and Assumptions
In this section, we describe the problem in more specific terms and state the assumptions that we make.
Sensors are deployed or manually placed over ROI to perform event detection (i.e., to detect whether an event of interest has happened or not) in ROI. If sensors are deployed from air then, using actuator-assisted sensor placement or by movement-assisted sensor placement, sensors are so placed that sensor network covers the entire ROI. This ROI is partitioned into suitable number of regular hexagons (i.e., we can think of the ROI as a regular hexagonal grid), as shown in Figure 1. Sensors are placed a priori at every center (which are known as nodes) of the regular hexagons. Sensors have two detection probabilities. The sensor network covers the entire ROI and there is only one event hexagon, as discussed before.
Each sensor node determines its location through beacon positioning mechanisms  or by exploiting the Global Positioning System (GPS). Through a broadcast or acknowledge protocol, each sensor node is also able to locate the neighbors within its communication radius. Sensors are also able to communicate with the base station. Base station will take the decision. In this paper, we assume that event occurs at one particular hexagon of the grid which will be known as event hexagon or event does not occur (in that case we say that ROI is normal). All sensors can communicate with the base station and base station takes the decision by combining the information received from all the sensors.
There are two phases in the whole process. The first one is detection phase, when the sensor at the center of a regular hexagon tries to detect the event. The sensor at the center of the event hexagon can detect the event hexagon with greater probability and the sensors at the adjacent nodes (see Figure 1) can detect the event hexagon with lesser probability . We also assume that there is a prior probability that a particular hexagon is an event hexagon. The next phase is response phase, in which sensors send message to the base station. Even if the event hexagon is detected by a sensor, it may not respond (i.e., send message to the base station that no event occurred in that cell and the neighboring cells due to some technical fault) with some probability; then we say that the sensor is a faulty sensor. Conversely, if event hexagon is not detected or there is no event hexagon at all (i.e., ROI is normal), then also a faulty sensor can send the wrong information to the base station with some probability. A sensor is said to be a dead sensor if the sensor does not work. A dead sensor sends no response in either cases.
Each sensor sends information to the base station. As the sensors may send wrong information, the base station takes the important role in identifying the event hexagon. Base station will collect all information and take a decision about the event hexagon according to a rule which we have to find out. Our job is to find a rule for the base station such that base station works most efficiently.
2.1. Notations and Assumption
Our problem is to develop a strategy for the base station to take decision about event hexagon (i.e., which hexagon of the ROI is the event hexagon, if at all). Let be the set of all nodes. For , define as the set of adjacent node(s) of , and let be the number of adjacent node(s) of . Hence, . Call a node interior if . Let be the sensor which is placed at the node , and let be the hexagon where the node is placed (i.e., is the center of ). For , let denote the true status of the node . That is, if event occurs at , and otherwise. Also define if detects no event, and if detects the event in or , for . Finally define if does not respond; that is, the sensor informs the base station that event does not occur at or for , and if responds; that is, the sensor informs the base station that the event has occurred in or , for .
Now we make one natural assumption that once detection phase is completed, response of a sensor depends only on what it detects but not on whether the event has actually occurred or not; that is, . We also assume that the sensors work independently and identically.
Since we assume that there is at most one event hexagon, or .
The possible true scenarios are, therefore, represented by the following different models: : ( for all ), and, for each , : ( and for all ).
and, for all , .
In particular, we may assume 's to be the same for all . We denote any probability under the model as and under the model as .
We also make the followings assumptions.(i)For all , and .(ii)For all , and .(iii)For all and .(iv) and are independent for .(v)The responses from different nodes are independent under a particular model; that is, 's are independent under for a fixed .
3. Theoretical Analysis of Fault Detection
In this section we discuss some theoretical results. In real situations, may be very large. Given the network of the sensor nodes and some prior knowledge about the nature of event, one may have fairly good idea about the set of feasible regions for the event. Formally, instead of all possible models, one may be able to restrict to a set containing all the feasible models. For example, if the event is known to take place in a particular region, we can restrict our models accordingly.
3.1. Model Selection Approach
Consider Hence, under the model follows , for all , and the likelihood of the data , under the model , is So . Hence, for all , under follows . Similarly, for all , under follows , where and, under follows for all . Note that since . Hence the likelihood for the model , given , is Let , so that with the corresponding observed values denoted by Therefore, where are independent of .
In model selection approach, the model resulting in the maximum value of the likelihood is selected. Note that, since there is no parameter being estimated, this is equivalent to the well-known Akaike Information Criterion (AIC) . Therefore, the base station will accept the model if Otherwise, as is positive, accept the model for which is maximum among all . If values of are equal for more than one , then we can select one of the corresponding models with equal probability. If we want to maximize the likelihood for the models corresponding to the interior nodes only, so that is fixed, then we need to maximize among all .
3.2. Multiple Model Selection
Instead of selecting one particular model, one may want to select more than one model with approximately similar log likelihood values to the maximum one. We can consider the set of models where is a suitable constant close to . This is usually chosen according to the resource available. This is similar to the idea of Occam's window in the context of Bayesian model selection . This may be interpreted as the interval estimation for the true model.
Note that is an increasing function of , as is positive. We consider only the following set of models where , for all , with . In particular, if we consider the interior nodes only, then we consider the set of models given by We can select multiple models using some other criteria. One such may be to select all the models (one or more) for which the maximum value of the likelihood is attained. Let be the set of nodes corresponding to all these models, including “” corresponding to if it has the maximum value of the likelihood. Then this method selects all the models with . By another criterion, one may select the models , for ; that is, is a node in or any of the neighboring nodes of a node in . Note that for is the empty set. One can combine these two types of criteria and come up with many others.
3.3. Bayesian Model Averaging
Bayesian model averaging is an effective method to solve a decision problem when there are many alternative hypotheses or models, which are complicated . Suppose that are the models considered and denotes the given data. The posterior probability for model is given by where denotes the probability of observing data under the model (which is essentially the likelihood under ) and is the prior probability that is the true model (assuming that one of the models is true).
In this work, the data is and the models are as defined in Section 3.2. Hence, the posterior probability for model is
We select the model if is greater than , for all ; otherwise, select for which is maximum among all . Hence, if 's are all equal, then Bayesian approach is the same as the likelihood approach.
4. Some Important Considerations and Error Probabilities
In this section, we consider some important issues related to the problem of fault detection and the proposed methodology including calculation of errors (e.g., false detection, etc.) and detection probabilities.
The following probabilities give some idea about the role of neighboring nodes, along with the center node, in detection or false detection, of event. For example, gives the probability of a false detection by the th node and not by the neighboring nodes, while gives the probability of a false negative by the th node, with all the neighboring nodes detecting the event. Since, given a particular model, and are independent, calculation of such probabilities is simple as given in the following. For any and ,(1), (2), (3), (4). which can be numerically obtained using the joint distribution of and under the model . The maximum of these probabilities over all gives a lower bound for the probability that a node is considered to be an event node when the ROI is normal. On the other hand, the sum over all gives an upper bound for the same. Similarly, for , which can be again numerically obtained using the joint distribution of and under the model . This probability gives some idea about the error that when th node is the event node and it is not detected.
As noted in Section 3.1, we select the model for which is the maximum, for . The random variable is, therefore, of some interest, the distribution of which under different models is useful in calculating many error probabilities. We first find the distribution of under the model . Note that takes values , corresponding to and , for , and . Assume that, for convenience, the values of for different and are all distinct. Therefore, for and , For or , one can find in similar manner, although the calculation is very tedious as there are many subcases. Ideally, one is interested in probability of errors occurring at the level of base station. For example, the two important errors are not selecting when is true (false positive) and selecting when is true for some (false negative). Theoretical calculation of these error probabilities is complicated. We, therefore, use simulation technique to estimate these and similar error probabilities.
5. Simulation Study
We consider a hexagonal grid and we run the programme 10000 times. The simulation is performed using the -code, and required random numbers are generated using the standard -library.
In our simulation study, we consider different criteria, as discussed in Sections 3.1 and 3.2, for estimating the error probabilities or, equivalently, the success rate. First consider the probability of selecting , when it is true. Let denote the proportion of correct detection of normal situation, when model is true, using the model selection method of Section 3.1. That is, gives an estimate of . Then gives an estimate of the false positive rate.
When is true for some , let denote the proportion of correct decision for the event node using the model selection method of Section 3.1, so that it estimates . Note that, for each simulation run, the event hexagon is chosen randomly, so that gives an average value over all . In this context, this probability is the same for all the interior nodes. Then, gives an estimate of the corresponding error probability of not selecting , when it is true.
Note that, in this problem of fault detection with a single event node, the likelihood value, for a given observed data configuration, may be equal for more than one models. Therefore, quite often, the maximum value of the likelihood may be attained by more than one model. The model selection method of Section 3.1, which selects one of these models randomly in such cases, may often not select the correct model. Therefore, the method of Section 3.2, which selects more than one model having similar likelihood value, may be preferred and will have better chance of selecting the correct model. We now consider some of those methods in the following.
Let us first consider the method in which all the models corresponding to the maximum value of the likelihood are selected. Let denote the proportion of correct selection of the model , when it is true, by this method. Then estimates the probability , which is always more than or equal to the quantity estimated by , as remarked before. We also consider the method in which all the models having maximum likelihood along with their neighborhood models are selected. A model is a neighborhood model of the model if is a neighboring node of . If denotes the proportion of correct selection of the model , when it is true, by this method, then estimates . Clearly, . Similarly, if denotes the proportion of correct selection of the model , when it is true, by selecting all those models with likelihood value being more than of the maximum likelihood (i.e., the method of Section 3.2 with ), then estimates the probability with denoting the maximum value of the likelihood.
Suppose that denotes the average number of selected nodes to be searched corresponding to . Clearly, because we need no search when is selected. When event occurs and we consider only one from , we need at most one search (since no search is needed if is selected) and we have . In our simulation, we find in all the cases; which means that, in simulation, has not been selected when event occurred. Note that since we consider all 's in for searching. Again, as before, . Also, by definition, . Table 1 presents the different 's and 's based on simulation for different values of , and with and taking values 0.9 and 0.99, taking values 0.01 and 0.001, and taking values 0.0, 0.3, 0.4, 0.5, and 0.6. The choice of and reflects the corresponding high probability, whereas that of reflects small probability, which is desirable in a good sensor. Since the primary interest is to study the effect of detection by neighboring nodes, we consider as 0 (which means that there is no effect of neighboring nodes) and some positive values less than .
Note that the probability of correct detection under depends only on . This is also evident in Table 1. Intuitively, if is high, then the proportion of correct detection in normal situation is low. In Table 1, we see that is 0 for , varies from 0.35 to 0.37 for , and varies from 0.90 to 0.91 for (not shown in Table 1). If we consider smaller value of , then the success probability will be higher. Hence must be low as the number of hexagons is high to get better results in normal situation.
We see that the estimated false negative rate, that is, an estimate of , is often in our simulation (not shown in Table 1). This is because, if the event occurs at , then detection of the event by at least one of the nodes belonging to is highly probable. Furthermore, since the grid size is large, one of the nodes belonging to may respond wrongly, though it cannot detect the event. So, under , there is a small probability to select ROI as normal. If we take and the detection probabilities and to be very small, then we may get some positive false negative rate, but this is not a desired condition for a good sensor.
From simulation, we see that, as increases (for positive ), values increase, whereas decrease. As increases, it helps to differentiate between the likelihood values resulting in lower cardinality of the set and lower values of 's. However, since the neighboring nodes help to detect the event, the success probability increases. From simulation, we find that, as increases, success probabilities also increase, but the effect of is more prominent than that of . On the other hand, success probabilities also change with and . Since means , so there is little variability in the likelihood values leading to larger size of .
When , effect of on , and and , and seems to be significant, whereas the same cannot be said for . There is sudden change in 's and 's, when we shift from to , for , but not . So, when is small, the effect of the neighborhood seems to be less.
The values of and are very similar for different values of the parameters, but larger increment in than suggests that the idea of neighboring search is not effective. But is much higher than , so the method of searching all the nodes in is a better idea than that of searching a random node from .
We estimate the success probability by simulation for different values of the threshold ranging from 0.5 to 0.9 (see Table 2). Note that corresponds to the threshold value . We consider , and four values of . From Table 2, we see that the success probability increases as the threshold value decreases and increases. Similarly, the number of search decreases with both and .
One prime object of this paper is to show the effect of the neighboring nodes in detection of an event. In this section, we discuss the role of the neighboring nodes and some other related issues and make remarks.
6.1. Role of the Neighboring Nodes
Since , where , and are as defined in Section 3.1, denotes the weight of the central node compared to the neighboring nodes in the corresponding likelihood. Note that, since , we have , and if is close to 1, then the six neighboring nodes are as important as the event node. So, as the value of increases, the importance of the neighboring nodes decreases. Also, gives some idea about the role of the number of adjacent nodes, that is, . Recall that and are the probabilities of responding (i.e., reporting the node as the event hexagon) by the sensors and , respectively, when is the event hexagon and is a neighboring node of . So, we numerically calculate the quantities , and for some values of the parameters (see Table 3).
From the theoretical results in Section 3.1, we see that and increase as increases, while and do not depend on . On the other hand, while increases with , and decrease and is independent of . Therefore, the importance of the neighboring nodes decreases with and increases with , as expected and observed in Table 3.
6.2. Estimation of the Parameters
In practice, the parameters , and may be unknown. We can, however, estimate the parameters by some experimentation.
Note that, under follows for all . Hence, is the expected value of given . So we perform the experiment by keeping the ROI normal. The proportion of 's having value gives an estimate of . Repeat this experiment several times, so that the average of the proportions over the repeated experiments can be taken as an estimate of .
Note that is the expected value of under . So, we perform the experiment by keeping an event in some node of the ROI. The proportion of 's having value gives an estimate of . Repeat this experiment for several times, so that the average of the proportions over the repeated experiments can be taken as an estimate of . Similar experiments will give estimates of and as well.
6.3. Incorporation of Heterogeneity and Uncertainty in Parameters
Let denote the set of parameters, which has been assumed to be the same for all the nodes. While, in practice there is no reason why the parameters should be same for all the nodes, it is also not clear how these would be different across . This unexplained heterogeneity can be incorporated by assuming the 's, for different , to be independent realizations from a common distribution.
Let denote the set of parameters for node . We assume that , are i.i.d. from some distribution, say . Also assume that, given 's are independent. Note that denotes the joint distribution of the four parameters. For simplicity, we may assume them to be independent, so that can be written as . In this situation, the likelihood for the model is where the integration is over the range of . Similarly, the likelihood for the model can be written as where the integral is over the four-dimensional space given by the range of and is the contribution of the th node to the likelihood , given the value , as described in Section 3.1.
Similar technique can also be used to incorporate parameter uncertainty. Even though the parameters can be assumed to be same for all the nodes, there may be reasonable uncertainty about the constancy of the parameter values. As in the Bayesian paradigm, the set of parameters may be assumed to be a realization from a distribution, say . Then, the likelihoods for the models and are The choice of may be a difficult one. However, sometimes there may be specific information available regarding the distribution of , which can be incorporated in the model.
6.4. When More Sensors Can Detect the Event Square
We may consider the situation when sensing radii are larger and more sensors can detect the event hexagon but with different probabilities. With respect to a particular node, classify the remaining nodes with respect to the probability of detecting the event at that node, which may as well depend on the distance from the particular node. Suppose that the sensors in the th class detect the event hexagon with probability . The theoretical analysis is similar to that of Section 3, but having more probability terms.
6.5. Concluding Remarks
In this paper, we consider the problem of fault detection in wireless sensor network (WSN). We discuss how to address both the noise-related measurement errors ( and ) and sensor fault ( and ) simultaneously in fault detection, where the ROI is partitioned into regular hexagons with the event occurring at only one hexagon. We propose fault detection schemes that explicitly introduce the error probabilities into the optimal event detection process. We develop the schemes under the consideration of model selection technique, multiple model selection technique, and Bayesian model averaging method. The different error probabilities are calculated by means of simulation. Note that the same analysis can be carried out when ROI is partitioned into squares and sensors are placed at the centers.
Nandi et al.  consider similar problem in wireless sensor network (WSN), in which the event can take place at the center of one particular square (or hexagon) of the grid covering the ROI. In our paper, we allow the event square to be any one in the grid. Our approach can also be used for the problem of  with only two models to be considered for selection. In , the authors develop the scheme under the consideration of Neyman-Pearson hypothesis test, where the null and alternative hypotheses correspond to the two models. In model selection approach, we select the model with higher likelihood. In classical Neyman-Pearson hypothesis test, a model is selected if its likelihood is greater than some constant times the likelihood of the other. This constant is fixed before the test depending on the size of the test. In model selection approach, the constant is , leaving no choice for the size of the test. On the other hand, we cannot apply the classical Neyman-Pearson test with more than two models to be considered for selection.
The principle of hypothesis testing places a large confidence in the null hypothesis and does not reject it unless there is strong evidence against it. This safeguard of null hypothesis cannot be ensured in the model selection approach of Section 3.1. However, the multiple model selection approach of Section 3.2 provides some safeguard in this regard.
This principle of model selection can be extended to the situation when there are more than one event hexagon and the objective is to detect the event hexagons. We may also assume that the sensors can detect different types of events. That is, response of sensors may not be only binary; sensors can measure distance, direction, speed, humidity, wind speed, soil makeup, temperature, and so forth and send the measurement of continuous type variables to the base station. One needs a different formulation of the problem in such case which will be taken up in future.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are sincerely thankful to the anonymous reviewers for their detailed comments that helped improve the overall quality of this paper. The authors are also thankful to Sourav Sengupta, Indian Statistical Institute, Kolkata, for his valuable suggestions.
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