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IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 38, NO. 1, FEBRUARY 2008
Adaptive Sensor Placement and Boundary Estimation for Monitoring Mass Objects Zhen Guo, MengChu Zhou, Fellow, IEEE, and Guofei Jiang
Abstract—Sensor networks are widely used in monitoring and tracking a large number of objects. Without prior knowledge on the dynamics of object distribution, their density estimation could be learned in an adaptive manner to support effective sensor placement. After sensors observe the “current” locations of objects, the estimates of object distribution are updated with these new observations through a recursive distributed expectation–maximization algorithm. Based on the real-time estimates of object distribution, an adaptive sensor placement algorithm could be designed to achieve stable and high accuracy in tracking mass objects. This paper constructs a Gaussian mixture model to characterize the mixture distribution of object locations and proposes a novel methodology to adaptively update sensor placement. Our simulation results demonstrate the effectiveness of the proposed algorithm for adaptive sensor placement and boundary estimation of mass objects. Index Terms—Expectation–maximization (EM), Gaussian mixture model (GMM), learning, maximum likelihood (ML), sensor networks, sensor placement, wireless sensor network.
I. I NTRODUCTION
I
N RECENT years, research on wireless sensor networks is burgeoning. Sensor networks become a bridge between the physical world and modern information systems. Among many research topics on sensor networks, one challenge is how to optimally place physical sensors in a field to form an effective sensor network to track a large number of objects. In practice, many sensor placement problems can be mathematically formulated as a set-covering problem. Since the set-covering problem is known to be NP-hard, many sensor placement problems that are equivalent to this problem are also NP-hard in computation. Therefore, it is necessary to develop computationally efficient sensor placement algorithms that can also achieve high detection accuracy in object tracking.
Manuscript received October 3, 2006; revised February 28, 2007 and July 23, 2007. This work was supported in part by Nippon Electric Company, Limited (NEC) Laboratories America, Chang Jiang Scholars Program, PRC Ministry of Education, and the Department of Defense under Subcontract 86190NBS21. This paper was recommended by Associate Editor E. Santos. Z. Guo is with the Countrywide Securities Corporation of Countrywide Financial, Calabasas, CA 91302 USA, and also with the Department of Electrical and Computer Engineering, New Jersey Institute of Technology, Newark, NJ 07102-1982 USA (e-mail:
[email protected]). M. C. Zhou is with the Department of Electrical and Computer Engineering, New Jersey Institute of Technology, Newark, NJ 07102-1982 USA and also with the School of Electro-Mechanical Engineering, Xidian University, Xi’an 710071, China (e-mail:
[email protected]). G. Jiang is with the Robust and Secure Systems Group, NEC Laboratories America, Princeton, NJ 08540 USA (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSMCB.2007.910531
Fig. 1.
Sensor placement example for mass objects monitoring.
Object tracking with sensor networks has received much research attention recently. Most of the work focuses on identifying and tracking one or several individual objects [1]–[3]. However, sufficient research effort has yet to be conducted on the issues of monitoring and tracking a large number of objects with sensor networks. In practice, there are some needs to monitor and track a large number of objects. For example, a surveillance system is needed to monitor crowds in museums or airports, soldiers in battle fields, and wild animals [23] in grasslands. These objects with large population, which are referred to as mass objects in this paper, are usually distributed in a certain way. This is because several points are most attractive to them, and the object distribution around these points is usually dense. In many real-world applications, it is necessary to dynamically and effectively place sensors to track mass objects with high accuracy while deploying only a limited number of sensors. Fig. 1 illustrates an example of effective sensor placement layout for monitoring mass objects. In such cases, we track the locations of mass objects instead of individual objects. Although the individual objects are frequently moving, the topology of mass objects is less likely to change quickly. With recent advances in sensor technologies, we are able to use mobile sensors to track objects. Mobile sensors can move to different places on demand to provide the required coverage. They are used to collaboratively detect targets and monitor environments where they are deployed. As the mass objects’ topology slowly changes, they become capable of acknowledging and moving to the new desired locations. Several studies [4]–[6] propose the use of the signal strength received from sensors to estimate the locations of mass objects. The sensor positions are calculated by modeling signal propagation, which requires adequate signal detection. Therefore, one of the most
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important issues in using sensor networks to track mass objects is to select the right locations for sensor deployment. Effective sensor placement is needed to provide adequate signal coverage while maximizing the tracking accuracy. Sensor placement schemes directly influence the efficiency of resource management and the type of back-end data processing and exploitation in distributed sensor networks. A key challenge in sensor resource management is to determine a sensor field layout that could minimize the sensor deployment cost [15] and provide high detection accuracy and resilience to sensor failures. The estimate of detection accuracy is inherently probabilistic due to the uncertainty that is associated with sensor coverage. Given a limited number of sensors and their limited signal strength, this paper proposes a methodology for improving detection accuracy. To the best of our knowledge, this work is the first effort that adaptively selects the sensor locations, evaluates the detection accuracy, and estimates the dynamical probabilistic boundary for mass objects through an online unsupervised learning method. The rest of the paper is organized as follows. Section II briefly reviews related works. Section III formalizes the problem of effective dynamic sensor placement and analyzes the distribution of mass objects. The surveillance coverage is modeled as a Gaussian mixture model (GMM). Expectation–maximization (EM) and recursive EM solutions are presented to address the coverage problem in Section IV. Section V proposes an approach to implementing distributed EM algorithms in sensor networks. The simulation details and detection performance are discussed in Section VI. Section VII concludes this paper. II. R ELATED W ORK There is much research work on dynamic sensor deployment problems. Wang et al. [12] propose a movement-assisted deployment approach based on Voronoi diagrams to find coverage holes and move sensors from a densely deployed area to a sparsely deployed area to improve the coverage of sensor networks. With uncertainty-aware layout of sensor placement, Zou and Chakrabarty [13] develop a probabilistic model that is targeted at average coverage, as well as at maximizing the coverage of vulnerable areas. Lin and Chiu employ a simulated annealing method to place the sensors in a min–max optimization model [14]. Recently, there has been a trend on developing information-driven algorithm for sensor placement. Cortes et al. proposed an approach [24] for autonomic vehicle network sensing tasks by utilizing a distributed coordination algorithm. A novel theoretical framework for distributed flocking algorithms is designed and analyzed by Olfati-Saber [25] to perform massive distributed sensing tasks by using mobile sensor networks. Another interesting and important issue is network connectivity, which plays a critical role in distributed target tracking with sensor network; Olfati-Saber has recently introduced a framework of distributed Kalman filter, together with a flocking algorithm that further improves the tracking performance [26]. All these methods aim at covering monitored areas with high efficiency and accuracy. In many real scenarios, the essential goal is to precisely locate the moving
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targets. For the problem of detecting thousands of targets, if the mass characteristics of targets can be dynamically learned, an optimal sensor placement strategy can be proposed accordingly. Therefore, we believe that the “target-oriented” approaches are more meaningful than the “area-oriented” ones. The challenge here is how to estimate the mass characteristics of objects. Nowak [11] proposes a distributed version of the EM algorithm to estimate the density of targets for sensor networks. Although the density can be well estimated, this approach is not applicable to dynamic scenarios because the estimates are static and cannot be updated according to the dynamics of mass characteristics. A recursive EM algorithm [9] is proposed to dynamically update the estimates of observations. We believe that a well-designed distributed recursive EM algorithm can provide a feasible solution for effective sensor placement to dynamically track mass objects. III. P ROBLEM F ORMULATION Accurate and computationally efficient sensor detection models are usually required for effective sensor deployment. This paper starts with the same assumption used in [7], i.e., the probability of detecting a target exponentially varies with the distance from the sensor to the target. In other words, the probability that a target can be detected by a sensor that is distance d apart is e−ρd , where ρ is a parameter used to model the rate at which its detection probability diminishes with distance. Obviously, the detection probability is equal to 1 if the target is located exactly at the point where the sensor is. Although a region could be covered by multiple sensors, an object in some regions may be concurrently detected by several sensors. The probability of detecting an individual object in the field should be the mixture probability that sums up the conditional detection probabilities of all sensors. A. Sensor Placement at Most Informative Locations Since only a limited number of sensors can be used for a specific tracking task, it is important to deploy them at locations where the objects can be monitored and tracked with the highest accuracy. In many cases, objects are symmetrically distributed around the points of interests. Such a phenomenon makes it an acceptable assumption that the distribution of objects around these points could be approximated with a Gaussian distribution. To improve the detection accuracy of a target, a sensor has to be placed closer to the target. To monitor mass objects, we propose a method to reduce the sum of distances between sensors and all objects in the field. Based on the above analysis, the sensors should be placed at the position with the local maximal object density to increase the total detection and monitoring performance. Since the objects are frequently moving, and the topology is also changing, the estimates of the new object distribution should be adaptively updated. The historical observations of objects can be used to estimate and learn the centers of object distributions and their boundaries. As an example, consider three groups of mass objects distributed in a 2-D covered area, as illustrated in Fig. 2. Based on such observations, there must be at least three points (regions) of
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Because of these characteristics in monitoring mass objects with wireless sensor networks, we apply a GMM to approximate the exact distribution of the objects. B. Reposition Capability of Sensor Nodes
Fig. 2. (a) Mass objects locations. (b) Sensor placement.
high interests, and each of them must be covered by at least one sensor. Moreover, due to the nature of wireless communication, a poor link quality between sensors that are very far apart could result in a large number of retransmissions, quickly draining out battery power and, hence, decreasing the overall deployment lifetime of the whole sensor network. This suggests that communication distance is a fundamental constraint and should be taken into consideration. To track the statistical properties of sensor data in many applications, approximate estimates are often sufficient and, sometimes, desirable if energy conservation is more important. Commonly, it is desirable to obtain the overall data distribution and the correlation information across the entire network; it is also important to support more complex queries. Given a limited set of sensor readings, we aim at efficiently and accurately answering the queries with a maximum sensing quality and a minimum communication cost.
Traditionally, network coverage is maximized by determining the optimal placement of static sensors in a centralized manner. However, recent investigations on sensor network mobility reveal that mobile sensors can self-organize to provide better coverage than static placement schemes [21]. The effectiveness of static sensor placement or uninformed mobility is usually limited by the following constraints: 1) no prior information about the exact target locations, population density, or motion pattern; 2) limited sensory range; and 3) very large area to be monitored [22]. All these conditions may cause sensor networks that may be unable to cover the entire region of interests. Hence, fixed sensor deployment or uninformed mobility is inadequate in general. Instead, the sensors have to move dynamically in response to the motion and distribution of targets, as well as other sensors, to obtain high coverage. This paper is to address the issues of adaptive sensor coverage and tracking for dynamic target topology. Hence, collaborative microrobots-assisted sensor nodes should be deployed to reach places where tasks would be performed with higher coverage. Therefore, each sensor node is required to have the reposition capability in the proposed approach. There are some popular examples of mobile sensor networks, including RoboMote [18] and Parasitic Mobility [19]. In such networks, mobile nodes can be self-deployed to guarantee the required coverage and self-relocated when network failure happens or when their sensing tasks change. For example, RoboMote is a small wheeled robot that could measure less than 6 cm3 in volume. RoboMote makes itself ideally suitable for mobile sensor networks because of its small size, low production cost, and low power consumption. When all features required for navigation are active, RoboMote consumes 1.5 W and can travel at a speed of 0.27 km/h. Parasitic Mobility is a novel type of mobile sensor network, which realizes mobility without incurring the extra cost of motion. Instead, the node is selectively attached to or embedded with an external mobile host. These hosts include vehicles, fluids, people, and animals. This paper proposes a recursive approach in which sensor nodes are capable of moving at will based on the most updated estimates. Therefore, mobile sensors with their own driven wheels such as RoboMote are ideal for this research. In typical mobile sensor networks, the motion functional unit of a mobile node is composed of the following three main parts [20]: 1) moving parts, e.g., wheels; 2) motors, which provide power to drive the wheels; and 3) microcontrollers, which control the movement of mobile sensors. Although it brings many benefits, augmenting sensor nodes with the reposition capability further emphasizes the challenge of energy efficiency. Power consumption and efficiency is an important research issue for mobile sensor networks. For example, Wang et al. [20] proposed an optimal velocity schedule for minimizing energy consumption. In this paper, we focus on how to effectively place
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sensors for tracking mass objects but do not specifically address the issue of power consumption. C. GMM Scatter points in Fig. 2 are the observations of object locations that were detected by sensors. Note that the observations are the historical positions where individual objects are located. During an interval of information collection, each individual object is moving around and may appear at several different locations. With the objective of monitoring mass objects, we focus on the distributions of clusters rather than any individual object’s movement. In this sense, one individual object may correspond to several observations. The observations, which record all locations where objects have been, are a good clue of building a probabilistic model that describes how likely the objects appear at a specific location. Here, we may take “electron cloud” as an analogy: each point in the Gaussian mixture does not necessarily mean that some objects appear at that location currently. Rather, it describes how likely objects appear at the position. The observations are simply a combination of previous locations where objects have happened to be. For example, there are 1000 observations, which are shown as scatter points in Fig. 2(a), although in fact, there may only be 500 objects that are present in the area. Sensors do not have to identify each individual object and track how and when it moves. Instead, our objective is to analyze the historical observations of object locations, learn the maximum-likelihood (ML) parameters for Gaussian mixtures, and estimate the centers and boundaries of all clusters. In summary, we consider the problem of dynamic sensor placement for locating mass objects as a group in a statistical way rather than identifying and deterministically analyzing the individual objects’ movement. Fig. 2(b) illustrates an example of sensor deployment, where the central points are the positions of sensors, and their sensing regions are enclosed with three solid circles. In this section, we formulate this problem in statistics. Assume that we have a set of N observations {zi } = {(xi , yi )} ∈ S, i ≤ N , which are the previous locations of the objects. Our task is to learn the statistical parameters of each cluster and then move the sensors to the cluster’s center that has the largest probability density in the cluster. The sensor detection probability for mass objects can be increased if the sensor is placed at the center of the cluster, i.e., the position with the maximal probability density in each cluster. In the GMM, we use the following expression to approximate the real observation distribution: p(zi ) =
M
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used later, is the whole set of θ ’s. αj is the mixture weight, which means the probability that observations appear in the jth cluster. p(zi |θj ) is the conditional probability density of the ith observation with respect to cluster j, given that the location of sensor j is known. M is the number of mixture components (i.e., the number of clusters), and N is the number of observations (i.e., the sample size). To model the location distribution of these objects, we can take advantage of the historical observations collected by sensors. However, they are often referred as incomplete data due to the lack of the cluster label information. Based on the limited knowledge, the ML estimate of the parameters of an underlying distribution is desired from a given data set. The incompletedata log-likelihood expression is described as follows: Q(Θ) = log (L(Θ|Z)) = log
N
p(zi |Θ)
i=1
=
N
log
i=1
M
αj pj (zi |θj )
where we define the likelihood function as L(Θ|Z) = p(Z|Θ) =
N
p(zi |Θ).
(1)
j=1
where p(zi ) is the mixture probability density of the ith observation, i.e., the probability density of the observation i in the mixture of Gaussian distributions. The parameter θ = (α, µ, σ) is the combination of mixture weight, mean value, and variance of a cluster, which describe the distribution of a single component of a Gaussian mixture (i.e., cluster). Θ, which will be
(3)
i=1
This function is also called the likelihood of the parameters under given observations. The likelihood function of parameter estimates is, in fact, a measure of coverage likelihood for mass objects. In (3), it is estimated from the product of the mixture probability density function under the assumption of independent and identical distribution (i.i.d.) of individual observations. Now, the optimization problem for coverage and detection becomes the maximization of the likelihood function Θ = arg max Q(Θ).
(4)
Θ
By solving (4), we can find the positions with the local maximal density of each region of interests. They are the optimal locations to place sensors in the sense of detection probability. To dynamically update the estimated sensor placement, we can use the maximum a posteriori (MAP) solution to estimate the dynamic distribution of objects. Similarly, the model selection techniques are based on maximizing the following type of criteria: J (M, θ(M )) = log (L(Θ|Z)) − P (M )
αj pj (zi |θj )
(2)
j=1
(5)
where log(L(Θ|Z)) is the log-likelihood of the available data. This part can be maximized using an ML solution, as mentioned earlier. Increasing the number of sensors (i.e., increasing the mixture components) always increases the log-likelihood, but it may also introduce unnecessary redundant sensors. A penalty function P (M ) is thus introduced to achieve the balance between sensor coverage and the number of deployed sensors. P (M ) is a function of M (the deployed sensor number), and
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it increases as M increases. In this paper, we use the Dirichlet prior proposed in [9] to represent P (M ), which monotonically increases with M and makes the EM algorithm converge quickly. The convergence proof of the EM algorithm with such a penalty function is shown in [9], and its details are beyond the scope of this paper.
We introduce the Lagrange multiplier λ with the constraint αl = 1 and solve the equation M N
∂ g log(αl )p (l|zi , Θ ) + λ αl − 1 = 0. ∂αl l=1 i=1 l (9) Then, we obtain
IV. S TANDARD AND R ECURSIVE EM A LGORITHMS λ = −N αlnew =
A. Standard EM Algorithm The EM algorithm is an iterative procedure that searches for a local maximum of a log-likelihood function. It starts with initial observations and a parameter estimate θ0 . The estimate θk from its kth iteration is obtained using the previous estimate θk−1 as Q(θk |θk−1 ) = E (log p(Zi≤k−1 , zk |θ)|Zi≤k−1 , θk−1 ) .
(6)
This step, which is referred to as the “expectation step (E-step),” finds the expected value of the “complete-data” loglikelihood with respect to the unknown parameters, given the observed positions and current parameter estimates. The E-step equation can be further written as follows: Q(Θ|Θg ) =
M N
log (αl pl (zi |θl )) p (l|zi , Θg )
l=1 i=1
=
M N
log(αl )p (l|zi , Θg )
l=1 i=1
+
(10)
By taking the derivative of the second term of (7) with respect to µl and setting it to zero, we have N
µnew l
=
zi p (l|zi , Θg )
i=1 N
(11) p (l|zi , Θg )
i=1 N
σlnew = i=1
p (l|zi , Θg ) (zi − µnew ) (zi − µnew )T l l N
.
(12)
p (l|zi , Θg )
i=1
log (L(Θ|Z g , z)) p(z|Z g , Θg )
z∈S
=
N 1 p (l|zi , Θg ) . N i=1
M N
log pl (zi |θl )p (l|zi , Θg )
(7)
l=1 i=1
where l is a random variable that labels in which region an individual object belongs, and the superscript g means that the referred parameter is available from the previous iteration [8]. Given Θg , we can compute pj (zi |θjg ) for each observation i and cluster j. In addition, the mixture parameters αj can be thought of as prior probabilities of each mixture component, i.e., αj = p(component j). The mixture parameters are uncorrelated with the observations of zi . Therefore, according to the Bayesian rule, we can compute the following “ownership function” [8]: p (l|zi , Θg ) = =
αlg pl (zi |θlg ) p (zi |θlg ) αlg pl (zi |θlg ) . M αkg pk (zi |θlg )
(8)
k=1
Note that the parameters (αnew , µnew , and σ new ) = θnew calculated in the M-step will be used as Θg in the E-step to compute the new p(l|zi , Θg ). Meanwhile, p(l|zi , Θg ) is also used in the M-step to get the new parameter θnew . The Estep and M-step interactively execute, and the EM algorithm usually converges to a local maximum of the log-likelihood function [17]. Hence, the EM algorithm is a good choice for estimation of mixture distribution, particularly distributed (and unsupervised) applications like the mass objects tracking with distributed sensor networks. Because of the distributed nature of sensor networks, a distributed algorithm is preferred over a centralized algorithm. A distributed EM algorithm is further discussed in Section V. B. Recursive EM Algorithm The recursive EM algorithm is an online discounting version of an EM algorithm. A stochastic discounting approximation procedure is conducted to recursively and adaptively estimate the parameters. In real-time monitoring and tracking of the dynamic objects’ topology, the recursive EM algorithm is better because the parameters are updated with a forgetting factor that degrades the influence of the out-of-date samples. In this section, we briefly introduce the recursive EM algorithm; more details about this algorithm can be found in [9]. As mentioned in Section III, the following type of criteria should be maximized: J (M, θ(M )) = log (L(Θ|Z)) − P (M )
g
To solve the equation Θ = arg maxΘ Q(Θ|Θ ), we can maximize the term that includes parameter αl and the term that includes θl independently since they are not correlated. This step is referred to as the “maximization step (M-step).”
= log p (Z|θ(M )) + log p (θ(M ))
(13)
where we introduce an extra item, namely, log p(θ(M )), which penalizes redundant sensors and complex solutions. For the
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MAP solution, we have ∂ ∂αm
log p(Z|θ) + log p(θ) + λ
M
αm − 1
=0
m=1
M cm where p(θ(M )) ∝ exp M m=1 cm log αm = m=1 αm is a Dirichlet prior [10]. For t data samples, we have (t) αm
1 = K
t
p (l|zi , Θ ) − c g
(14)
i=1
where K = t − M c, and the parameters of the prior are cm = −c = −N/2. If we assume that the parameter estimates do not change much when a new observation is added and the new ownership function p(l|zi, Θt+1 ) can be approximated by p(l|zi , Θt ), we can derive the following recursive updating equations:
otm (z t+1 ) = p lm |zi , Θt+1
t t αm pm z t+1 |θm = p(z t+1 |Θt ) t t+1 om (z ) − cT t+1 t αm = χ + (1 − χ)αm 1 − M cT
(15)
t+1 t t σm = σm + w δδ T − σm
transmission of sufficient statistics from one node to another based on a prescribed sequence through the nodes. Each node computes its local updates for the sufficient statistics. Note that these local updates are computed with local observations and estimates that are available at each sensor node. Assume that the total number of sensors is H. The forward path is prescribed as the cyclic order from 1 to H to 1. It is assumed that the number of available sensors is larger than the number of clusters, i.e., H > M . Each sensor increments the local estimates into the old cumulated estimates and sequentially passes the new cumulated estimates to the downstream sensors, i.e., t θjt = θjt + θh,j
(18)
(16)
where χ = 1/T is a fixed forgetting factor that is used to forget the out-of-date statistics. After the new mixture weight is calculated, the online algorithm should check if there are irrelevant t+1 < 0, we discard the components, If the mixture weight αm component m, set M = M − 1, and renormalize the remaining mixture weights accordingly. As mentioned before, this mechanism of discarding an irrelevant component is achieved by introducing the penalty function p(θ(M )). As discussed earlier, it is straightforward to see that the penalty function always decreases with the number of clusters M . Eventually, the clusters merge to form those necessary ones only. The rest of the parameters are then updated in the following equations: t µt+1 m = µm + wδ
Fig. 3. Communication cycle for message passing.
(17)
t where w = λ(otm (z t+1 )/αm ), and δ = (z t+1 − µtm ). Following these equations, new parameters are updated with new observations. Sensors can slightly move their locations to the updated local maxima and iteratively wait for the next round of parameter estimation.
V. D ISTRIBUTED A LGORITHM I MPLEMENTATION AND C OMPLEXITY A NALYSIS A. Distributed EM Algorithm Assume that, initially, a sufficient number of sensor nodes are evenly distributed over the entire area. All nodes have their local estimates on their observations. The next EM iteration can be computed by performing two message-passing cycles through the sensor nodes. Each message-passing operation involves the
where θjt is the cumulative estimate of cluster j at time t, t is the local estimate of cluster j by sensor node h and θh,j at t. This process can be justified by the fact that the E-step can be separated into H separate expectations, followed by incremental accumulation [11]. At the last sensor node H, the summary of complete sufficient statistics is available and is passed over to node 1. Therefore, the summary of sufficient statistics is incrementally passed in the circle. After two message-passing cycles, the incremental form of cumulative statistics of all observations collected by all nodes could reach every node in the circle. Therefore, all H sensor nodes should have sufficient statistics to compute the M-step and obtain θt+1 . This process is guaranteed to monotonically converge to a local maximum [11]. Fig. 3 illustrates communication cycles in a distributed implementation of the recursive EM algorithm. Small circles indicate sensors that are deployed in the target field. The forwarded messages proceed in a cyclic fashion with a predetermined order, i.e., messages are passed between nodes on the order of 1, 2, . . . , H, 1, 2, . . . , H, 1, . . .. Without loss of generality, we choose H = 8 in Fig. 3. Similarly, the recursive EM algorithm is implemented in a distributed manner to fit the requirements of sensor networks by taking advantage of the accumulation of separate E-steps as well. Each sensor node updates its local parameter estimates in a recursive form by taking all the new local observations as inputs and then passes the local estimates with cumulative estimates to the downstream one. As long as the global statistical parameters are distributed after two messagepassing cycles, each node can tune their parameter estimates in their M-steps. By recursively adding new observations, the distributed recursive EM algorithm can adaptively tune the parameter estimates to capture the changes of the topology
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As long as the current estimates are converged, the distributed EM algorithm continues. At first, node 1 collects new observations. If the difference between new observations (Zhnew ) and old ones (Zhold ) is higher than a given threshold (ε ), the sensor recursively updates cumulative estimates with one new observation at each iteration according to the recursive EM algorithm described in Fig. 4. If this difference is lower than the threshold, the sensor skips the process of updating new estimates to save power and time. Since power consumption is a critical issue for sensor networks, this approach can improve the efficiency of power consumption. The developers of RoboMote sensors discover that moving a sensor with a distance of 1 m consumes an amount of energy approximately similar to transmitting 300 messages on the average [12]. Therefore, our approach can reduce the frequency of sensor repositioning to reduce power consumption. For some energy-critical applications, the threshold (ε ) can be set higher to reduce the power consumption. As long as all new observations collected by sensor node 1 have been used to update the local estimates, the statistics of updated cumulative estimates are then passed to node 2, which will repeat the same process. Each node cyclically updates the cumulative estimates with its new observations and then passes the summary of statistics to its downstream node. Note that at each time, only one sensor node can transmit the message of statistics to the next node. The other critical issue for dynamical sensor placement is to maintain network connectivity. Since all sensors are able to move in the field, we may not be able to keep all sensors always connected. In practice, we may have to integrate some static wireless transmitters as the communication “hubs” for mobile sensor networks. B. Space and Computational Complexity
Fig. 4. Dynamical sensor deployment algorithm.
of mass objects. Our distributed EM algorithm is illustrated in Fig. 4. Initially, H sensors are evenly distributed throughout the covered area; each node collects a set of observations and makes its own local estimates. The cumulative estimates are incremented with the local estimates of each node in a cyclic manner. The message of cumulative statistics is passed through the sensor nodes according to the distributed implementation of the EM algorithm. Note that it takes several message-passing cycles to converge to a local maximum. We can determine the convergence of the estimates by checking whether the increase of the likelihood function at each iteration is greater than the previous likelihood function multiplied with a small coefficient ε (e.g., ε = 10−5 ), i.e., L(Θt |Z) − L(Θt−1 |Z) > ε L(Θt−1 |Z) . Section VI includes some experimental results on the convergence rate.
Necessarily, the space and computational complexity should be taken into consideration for the feasibility of the proposed algorithm. First, consider the analysis of space complexity. Assume that we use a double-precision floating number (8 bytes) to store each parameter. As mentioned before, each cluster contains a three-tuple (α, µ, σ), and the message size would be Nsize = 8 + 24 ∗ M bytes, where M is the number of clusters, and we need 8 bytes extra for the information header to store the cluster information. Since we use the recursive EM algorithm, these messages keep being updated as time goes on, but the space usage is fixed. However, the space usage could still be large if the monitoring area is large and the number of clusters is also large. Fortunately, in our algorithm, the penalty function can reduce the number of clusters. The other concern of clustering algorithms is its computational complexity. Essentially, the calculation of the Mahalanobis distance of each single Gaussian distribution dominates the computational complexity of an EM algorithm. For 2-D Gaussian distributions shown in (19), the Mahalanobis distance is defined as m(z) = ((z − µ) Σ−1 (z − µ))1/2 . For a specific Gaussian distribution p(z), the values of its squared Mahalanobis distance m(z)2 are approximately chi-square distributed. A probability-density-based cluster boundary determined by p(z) can be mapped to a corresponding boundary in
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its Mahalanobis distance. Therefore, we can use the following inequality to determine the sensing coverage in Mahalanobis distance: m(z)2 = (z − µ) Σ−1 (z − µ) = (z − µ) (P ΛP )−1 (z − µ) → − → − = t Λ−1 t =
t21 t2 + 2 λ1 λ2
≤ χ2 (α)
(19)
where λ1 and λ2 are the eigenvalues of the covariance matrix Σ, α is the probability coverage in Gaussian distribution, and − → t = (P )−1 (z − µ) =
t1 t2
Λ=
λ1 0
0 λ2
Fig. 5. Near-optimal sensor deployment simulation.
.
Although the EM algorithm theoretically always converges to the eigenvalues of the covariance matrix, its convergence rate is not very stable [16]. We set the maximum number of iterations to avoid too many communication cycles and reduce the computational complexity. In the worst-case scenario, the computational complexity at each node is then bounded by O(1).
VI. S IMULATION AND P ERFORMANCE A NALYSIS The simulation of our work is implemented with Matlab and C/Numeric recipe on LINUX FEDORDA CORE 4. The object location data set is generated by the known “iris” data [9], and we also add some randomly generated white noise into the data set.
A. Sensor Placement Applications A simulated sensor deployment application is presented in Fig. 5 to demonstrate the effectiveness of the dynamical sensor placement method. We consider the following scenario in our simulations. Assume that a very large number of animals scatter in a wild area, and we hope to cover this area with a sensor network. Assume that there are several points of attractions, and these points are moving or hard to be localized. The probability of animal distribution around these interesting points is assumed to follow a GMM. To maximize the coverage rate, the recursive EM algorithm should be used to find the local maximum first, place critical sensors around those points with the local maximum density while placing other sensors evenly in the sparse area to monitor rare events. Fig. 5 illustrates a simulated scene that mimics this situation with an area of 12 km × 12 km. In this work, 300 objects and 9000 observations of these objects’ locations are generated according to a GMM, whereas the mixture weights are randomly selected. In a timely manner, our proposed method adaptively estimates the “near-optimal” sensor placement for the most recent 300 observations. The last 300 observations are shown in Fig. 5.
Fig. 6. Detection probability coverage.
Shown as triangles in Fig. 5, 14 sensors with a 1.7-km sensing radius are deployed, and three groups of objects are clustered. As mentioned earlier, initially, sensors are uniformly placed in the area. The dynamical sensor placement method calculates the updated location of sensors and then moves the sensors to the desired place. Fig. 5 illustrates the output layout after 9000 data samples are taken into recursive estimation.
B. Sensor Coverage As mentioned in Section III, the detection probability is e−ρd when an object is d units of distance from a sensor. In our simulation, we set ρ = 0.1 (1/km). Fig. 6 demonstrates the better sensor coverage, resulting from the recursive EM algorithm in comparison with that from the evenly placed sensor layout. Clearly, the detection probability is improved with an increase in the number of deployed sensors. As the number of sensors keeps increasing, the improvement of detection probability eventually gets very small. This justifies why we try to deploy as fewest sensors as possible to achieve satisfactory detection probability. Our simulation results also demonstrate that our algorithm adapts to the topological change of mass objects very well so that it can fairly accurately cover most of the mass objects using a small number of sensors.
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vations very well. Note that the fitness score is calculated after the estimates of parameters become stable. D. Adaptive Boundary Estimation
Fig. 7. Histogram of probability. (a) Estimated probability. (b) Empirical probability.
C. Measurement on Model Fitness The estimates of object distribution may not exactly fit with real observations. In this paper, we use the normalized difference between the estimated and empirical probabilities to measure how well the learned model fits with real observations. For convenience, we compare the empirical probability with estimated probability in a discrete manner. The region of a cluster is divided into several equal-sized segments. In practice, the segments of a specific cluster all have elliptic or circular border with a shared center. Therefore, a certain number of segments, e.g., G, in a cluster should include G − 1 “rings” and the innermost circle (or ellipse). We calculate the empirical probability of a certain segment by dividing the number of observations within that segment with the total number of observations in the whole cluster. The estimated probability of a segment is calculated by measuring how likely the points locate in that segment according to the estimated distribution. Fig. 7 shows the histograms of estimated and empirical probabilities, respectively. Each cluster is divided into six segments in the previous simulation. Fig. 7 provides a visualized judgement on how well the model fits the observation. For quantitative analysis on the model’s fitness, we define a “fitness score” as follows: F (θ) = 1 −
G
2
(preal (g) − pest (k)) ∗ 100. G 2 1 preal (g) − G
g=1
(20)
g=1
In (20), preal (g) and pest (g), respectively, represent the empirical probability and estimated probability of data points in the gth segment. 1/G is the mean of the probabilities in all segments under a uniform distribution. A high fitness score indicates that the distribution of mass objects can be well modeled with the learned GMM. Note that the upper bound of the fitness score is 100. The example above collects 5000 observations, and the fitness score in this simulation is 89.31, which demonstrates that the learned GMM fits with real obser-
By estimating the mass characteristics of real observations, we can also estimate the probabilistic boundary of mass objects’ distribution. With the same simulated observations, this section focuses on how to determine the dynamical topology of observed locations. As shown in Fig. 8, the real-time “likely” boundary can be adaptively estimated with certain probability coverage. Since there is no prior knowledge about the clusters, our algorithm starts with 20 clusters. As mentioned in Section IV-B, a penalty function is introduced to discard those unnecessary clusters. This explains why we have several unnecessary clusters in the first couple of estimates, and later, these clusters are eventually merged to three clusters. Note that the “likely” boundary in this paper refers to the ellipses with certain probability coverage, and all points on the border have identical probability densities. In the simulation shown above, we choose the boundaries with 95% probability coverage. This means that the boundary encloses 95% of real observations in a statistical sense. Fig. 8 illustrates the boundary estimates in a timely manner with incoming new observations. E. Convergence Test As mentioned earlier, initially, the sensors are evenly placed throughout the deployment field. Our algorithm starts with original observations and takes certain iterations and communication cycles through all nodes to achieve the initial convergence. The initial convergence should be reached before any dynamic sensor placement. To some extent, the performance and feasibility of our algorithm could be affected by the initial convergence rate. Here, we include an experiment on the convergence to analyze how many communication cycles are needed for different numbers of clusters (i.e., points of interests) and different numbers of deployed sensors. Fig. 9 shows the result of our convergence experiments. Our simulation randomly generates a certain number (e.g., 2, 4, 6, 8, and 10 in our experiment) of points of attractions that are randomly placed in a 12 km × 12 km area. The experiments also show that the number of deployed sensors also imposes impacts on the convergence rate. As illustrated in Fig. 9, we tried 10, 15, and 20 sensors in our experiments. In the figure, it is clear to see that 1) with more points of attractions, more communication cycles are needed for convergence, and 2) with more deployed sensors, more communication cycles are needed for convergence. VII. C ONCLUSION This paper has proposed an adaptive sensor placement method according to the ML estimates of mass object locations. The proposed approach uses a GMM to capture mass characteristics of real observations and searches for the desired locations for sensor placement that could maximize detection probability. To the best of our knowledge, it is the
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Fig. 8. Adaptive boundary estimation at different time. (a) Initial observations. (b) Estimates after 150 samples. (c) Estimates after 300 samples. (d) Estimates after 1500 samples. (e) Estimates after 3000 samples. (f) Estimates after 9000 samples.
estimation. In future works, we need to estimate the observation distributions using models other than a GMM. Additional optimization is also desired to improve the performance of our algorithms. For example, the performance of our algorithm can be improved by a proper initialization methodology that employs hierarchical approaches for distribution estimation [17]. Communication cost and energy efficiency are two other important issues in sensor network research [27]–[29]. More efficient distributed algorithms can be developed to balance detection accuracy with communication cost and power consumption. R EFERENCES Fig. 9.
Communication cycles needed for convergence.
first time that an unsupervised learning method is used to support adaptive sensor placement and monitoring mass objects. We also employ a distributed implementation of an EM algorithm to reduce the communication cost. Our method is able to place sensors at the positions with the local maximal density, which maximizes the detection accuracy of sensor networks. Our method can also be used to estimate mass objects’ boundary. Therefore, it can be applied to anomaly detection tasks in large-scale wireless network surveillance. Our simulation results demonstrate the effectiveness of our methods on adaptive sensor placement and real-time boundary
[1] Y. Xu and W.-C. Lee, “On localized prediction for power efficient object tracking in sensor networks,” in Proc. 1st Int. Workshop MDC, May 2003, pp. 434–439. [2] Z. Guo and M. C. Zhou, “Prediction-based object tracking algorithm with load balance for wireless sensor networks,” in Proc. IEEE Int. Conf. Netw., Sens. Control, Mar. 2005, pp. 756–760. [3] C. Lin and Y. Tseng, “Structures for in-network moving object tracking in wireless sensor networks,” in Proc. 1st Int. Conf. BroadNets, 2004, pp. 718–727. [4] P. Bahl and V. N. Padmanabhan, “RADAR: An in-building RF-based user location and tracking system,” in Proc. IEEE INFOCOM Conf., 2000, pp. 775–784. [5] M. Vossiek and L. Wiebking, “Wireless local positioning—Concepts, solution, application,” IEEE Microw. Mag., vol. 4, no. 4, pp. 77–86, Dec. 2003. [6] S. Ray and R. Ungrangsi, “Robust location detection in emergency sensor networks,” in Proc. IEEE INFOCOM Conf., 2003, pp. 1044–1053. [7] S. Dhillon, “Sensor placement for effective coverage and surveillance in distributed sensor networks,” in Proc. IEEE WCNC, 2003, pp. 1609–1614.
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Zhen Guo received the B.S. degree in material physics from the University of Science and Technology of China, Hefei, in 1999 and the M.S. degree in computer science and Ph.D. degree in computer engineering from the New Jersey Institute of Technology, Newark, in 2001 and 2006, respectively. He is currently a Lead Quantitative Analyst with Countrywide Securities Corporation of Countrywide Financial, Calabasas, CA. He is also with the Department of Electrical and Computer Engineering, New Jersey Institute of Technology.
MengChu Zhou (S’88–M’90–SM’93–F’03) received the B.S. degree from the Nanjing University of Science and Technology, Nanjing, China, in 1983, the M.S. degree from the Beijing Institute of Technology, Beijing, China, in 1986, and the Ph.D. degree in computer and systems engineering from the Rensselaer Polytechnic Institute, Troy, NY, in 1990. In 1990, he joined the New Jersey Institute of Technology, Newark, where he is currently a Professor of electrical and computer engineering and the Director of the Discrete-Event Systems Laboratory. He is the Editor-in-Chief of the International Journal of Intelligent Control and Systems. His research interests are computer-integrated systems, Petri nets, wireless ad hoc and sensor networks, system security, semiconductor manufacturing, and embedded control. He has about 300 publications, including six books and more than 120 journal papers. He ranks top in terms of publications and citations in the areas of Petri nets and automated manufacturing systems based on the data summary and analysis from Scopus—the most extensive database covering engineering and science papers. Dr. Zhou served as an Associate Editor for the IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION from 1997 to 2000, the IEEE TRANSACTIONS ON S YSTEMS , M AN , AND C YBERNETICS —P ART B: C YBERNETICS from 2002 to 2005, and the IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING from 2004 to 2007. He is currently the Managing Editor of the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C and an Associate Editor for the IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS and IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A. He was the recipient of the National Science Foundation’s Research Initiation Award, CIM University-LEAD Award by the Society of Manufacturing Engineers, Perlis Research Award by NJIT, Humboldt Research Award for U.S. Senior Scientists, and Distinguished Lecturer of the IEEE SMC Society. He was the Founding Chair of the Discrete Event Systems Technical Committee of the IEEE SMC Society, and Chair of the Semiconductor Manufacturing Automation Technical Committee of the IEEE Robotics and Automation Society. He is a Life Member of the Chinese Association for Science and Technology-USA and served as its President in 1999.
Guofei Jiang received the B.S. and Ph.D. degrees in electrical and computer engineering from the Beijing Institute of Technology, Beijing, China, in 1993 and 1998, respectively. From 1998 to 2001, he was a Postdoctoral Fellow in computer engineering with Dartmouth College, Hanover, NH. He is currently a Senior Research Staff Member with the Robust and Secure Systems Group, NEC Laboratories America, Princeton, NJ. He leads a group of researchers working on distributed systems, dependable and secure computing, and system and information theory. He is the author of over 60 published technical papers in these areas. Dr. Jiang is an Associate Editor for IEEE SECURITY AND PRIVACY and has served in the program committees of many prestigious conferences.
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