J Control Theory Appl 2011 9 (1) 44–50 DOI 10.1007/s11768-011-0240-y
Sensor selection for parameterized random field estimation in wireless sensor networks Yang WENG 1,3 , Wendong XIAO 2 , Lihua XIE 1 1.School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798; 2.Institute for Infocomm Research, Singapore 138632; 3.School of Mathematics, Sichuan University, Chengdu Sichuan 610064, China
Abstract: We consider the random field estimation problem with parametric trend in wireless sensor networks where the field can be described by unknown parameters to be estimated. Due to the limited resources, the network selects only a subset of the sensors to perform the estimation task with a desired performance under the D-optimal criterion. We propose a greedy sampling scheme to select the sensor nodes according to the information gain of the sensors. A distributed algorithm is also developed by consensus-based incremental sensor node selection through information quality computation for and message exchange among neighboring sensors. Simulation results show the good performance of the proposed algorithms. Keywords: Random field estimation; Parametric trend; Wireless sensor network; Sensor selection; NP-completeness; Distributed processing
1
Introduction
The developments in microelectromechanical system technology, wireless communications, and digital electronics have enabled the deployment of low-cost wireless sensor networks (WSNs) in large-scale using small-sized sensor nodes [1]. WSNs have been used to monitor various phenomena, such as the moisture content in an agricultural field, the temperature distribution in a building, the pH value in a river, and the salt concentration in the sea. Usually, the observed data in a WSN are spatially correlated with a covariance structure that may be modeled as a random field [2], which is a generalized stochastic process with the underlying parameter being a multidimensional vector instead of a simple real. Due to the high density of the network, sensor observations are highly correlated in the space domain [3]. The spatial correlated data from the sensor nodes have been widely used for estimation, detection, classification, etc. [4∼6]. In most applications, the WSN nodes are powered by small batteries, which restrains the lifetime of a WSN. Therefore, energy-efficient algorithms in WSNs are especially important. It is desirable that only part of the sensor nodes are tasked at any time without compromising the network performance. An adaptive energy-efficient multisensor scheduling scheme has been proposed for collaborative target tracking in WSNs [7]. It calculates the optimal sampling interval to satisfy a specification on predicted tracking accuracy, selects the cluster of tasking sensors according to their joint detection probability, and designates one of the tasking sensors as a cluster head for estimation update and sensor scheduling according to a cluster head energy measure function. A novel energy-efficient adaptive sensor scheduling approach has been proposed in [8]. This approach jointly selects tasking sensors and determines their associated sam-
pling intervals according to the predicted tracking accuracy and tracking energy cost. In [9], a framework has been proposed for specifying the information gain of observations at each set of sensors, and an algorithm has been presented to select a sequence of sets to observation that total information gain is maximized while not exceeding the available energy. The optimal sensor selection problem has been formulated as maximizing the mutual information between the chosen locations and the locations that are not selected [10]. This combinatorial problem has been proven to be NPcomplete, and a polynomial-time greedy algorithm with a given approximation has been proposed. In [11], the sensor selection problem has been solved via convex optimization. The convex relaxation followed by a local optimization method has been proposed. This method pursues not only a suboptimal choice of measurements, but also a bound on how well the global optimal choice does. However, a fusion center is needed in the aforementioned works for sensor selection. Distributed algorithms have many advantages on robustness and scalability in large-scale WSNs against the centralized scheme with a fusion center. The WSNs without any fusion center are highly robust due to their distributed nature, node redundancy, and avoiding the dangers caused by the failure of the fusion center. Robust, asynchronous, and distributed algorithms have seen more and more opportunities as the intelligent and autonomous sensors play a more important role in the networks. A lot of distributed solutions have been proposed for estimation, detection, and control [12∼14]. In [15], a distributed sampling scheme has been proposed to estimate an unknown parameter with correlated sensor data. The proposed algorithm selects a nearly minimum number of active sensors to ensure the estimation performance, which leads to energy efficiency.
Received 16 October 2010. This work was partly supported by the National Natural Science Foundation of China-Key Program (No. 61032001) and the National Natural Science Foundation of China (No. 60828006). c South China University of Technology and Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2011
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Motivated by these observations, we focus on designing both centralized and distributed sensor selection schemes for estimating the parametric trend of a random field in this paper. We consider a WSN with many redundant nodes, each of which can observe the physical phenomenon in the field. For each work period, the network selects a subset of sensor nodes to be active for observations to achieve a desired estimation performance. Intuitively, more active sensors can achieve higher estimation performance but at high energy cost. We formulate the random field estimation problem in WSN as a sensor selection problem that select a set of sensor nodes whose total information quality is maximized while not exceeding the available energy. We propose a framework for specifying the information gain of each sensor node and also propose a greedy algorithm to select as small number of active nodes as possible to achieve the desired performance. The rest of this paper is organized as follows. In Section 2, we formulate the random field estimation problem in WSN as a sensor selection problem. The estimation for random field with parametric trend is discussed in Section 3. The distributed sampling scheme is presented in Section 4. Simulation results are reported in Section 5 to show the performance of our method. Concluding remarks are given in Section 6.
2
Problem formulation
2.1 Random field model Denote the random field under discussion as {Z(s), s ∈ D}, where D is a compact set of Euclidean space and Z(s) is a random variable at s. There are n sensor nodes that are deployed in D with locations P = {s1 , · · · , sn }. Each sensor can take measurement of this field without noise. We wish to reconstruct this field by the observations from the distributed sensor nodes. In order to specify a random field, we denote the mean function for the field as M(s) = E(Z(s)), ∀s ∈ D, and the covariance function as K(s, t) = cov (Z(s), Z(t)) , ∀s, t ∈ D. We study a single-snapshot scenario in a WSN, in which each sensor node can take only one measurement. For a given sensor node i at location si , denote its observation as Z(si ). We can reconstruct the random field using the observations form n sensor nodes, namely, ZPn = (Z(s1 ), · · · , Z(sn ))T . We can use simple kriging predictor to reconstruct the value of Z(s) where this predictor at s ∈ D corresponds to the best linear unbiased estimator (BLUE) [16] −1 ˆ ZPn Z(s) = E(Z(s)) + Σs,Pn ΣP n
with error variance: −1 ΣPn ,s , ε(s, Pn ) = Var(Z(s)) − Σs,Pn ΣP n
where Σs,Pn = cov(Z(s), ZPn ), ΣPn = cov(ZPn , ZPn ), and E(Z(s)) and Var(Z(s)) are the expectation and variˆ ance of Z(s), respectively. Z(s) is the minimum variance estimator of Z(s). Here, we do not restrict the field to be stationary and isotropic as well as the distribution of the field.
2.2 Discretization and optimality criteria We want to choose the optimal sensor locations for a number of sensors to reconstruct the whole field under certain criteria. A natural objective is to choose sensor locations to take observations to minimize the distortion in the estimate of the whole field. However, as a random field is a multidimensional vector space or even a manifold, utilizing several observations to estimate random field is complicated. One convenient way is to consider the discrete case with finite random variables by mapping the space to a list of random variables. Assume that, by using n sensors, we can discretize the random field Z(s) into a sequence {Z(s1 ), · · · , Z(sn )}, T which is denoted as Z = (Z1 , · · · , Zn ) with an index set I = {1, 2, · · · , n}. We want to estimate Z with several observations from the locations where we can deploy the sensor nodes. Mathematically, the problem is that we have a random vector Z with dimension n to be estimated, and we want to select l random variables to estimate the rest r ones (r = n − l) in the sense of minimum error variance. We denote the selected vector by T
ZA = (Zi1 , · · · , Zil ) , and denote the vector to be estimated by ZA¯ = Z\ZA . Without loss of generality, we further assume that M(s) = 0, ∀s ∈ D. The best linear unbiased estimator is well known and can be presented as follows: −1 ZA ZˆA¯ = ΣAA ¯ Σ A
with error covariance matrix: −1 DA¯ = cov(ZˆA¯ − ZA¯) = ΣA¯ − ΣAA ¯, ¯ ΣA ΣAA where ΣA = cov(ZA ), ΣA¯ = cov(ZA¯), ΣAA ¯ = cov(ZA ¯ , ZA ). In this paper, we always assume that covariance matrix DA is invertible for an arbitrary random vector ZA , A ⊂ I. For the random field estimation in WSNs, we can consider that each discretized location for the random field can be reached by a sensor node since a highly dense deployment is possible [1]. Therefore, our problem can be regarded as the sensor selection problem that is closely related to the optimal experiment design [10,11,17], which was originally proposed by Wald [18] and extended by Kiefer [19]. Since Kiefer’s seminal work, a large amount of literature can be found for dealing with theoretical aspects of optimal design. Similarly to the traditional experimental design theory, some monotonic function Q( · ) can be used to compare the efficiency of various variables of Zi1 , · · · , Zil . In this paper, we consider the D-optimal criteria [20], in which the determinant of the error covariance matrix DA¯ is minimized, i.e., ZA = arg min |DA¯|. A⊂I
where | · | is the determinant function. 2.3 Random field estimation in sensor networks In WSNs, in order to conserve energy and prolong network lifetime, very often, it is necessary to select a group of sensor nodes for collecting observation data to estimate the field in a cooperative way, while other nodes are inactive (sleeping). The goal of this paper is to propose a method to obtain
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optimal estimation performance under the energy consumption constraint of WSN, which is formulated as follows: −1 arg min |ΣA¯ − ΣAA (1) ¯ ΣA ΣAA ¯| A:C(A)=l
where C(A) is the cardinality of set A. In general, selecting a set of l sensors that minimize the estimation error under the D-optimal criterion belongs to a class of combinatorial optimization problems and is typically NP-hard. We have proven that problem (1) is NP-complete in [21].
3
Parameterized random field estimation
For statistical model, a random response variable is usually decomposed into a mathematical structure describing the mean and an additive stochastic structure describing variation and covariation among the responses. Therefore, we can describe the random field as the following model: (2) Z(s) = x (s)β + e(s), where x(s) is a m-dimension vector of known functions, which is also called regression coefficient, and θ is a corresponding vector of parameters to be estimated. The first term of right side in (2) is the trend for the random field. The stochastic part of the random field is model as an additive error with zero mean and a covariance function K(s, t), which is the same as the covariance in (1). Considering the discretization of parametric model for random field estimation in WSNs, n sites are deployed each with a sensor node for taking observations. The number of sensor nodes n can be very large when the network is deployed densely or covers a large area. We want to select some of the sensor to work such that we can reconstruct the random field under some performance considerations, i.e., Z(si ) = x (si )β + e(si ), i = 1, · · · , n. We can stack all sensor measurements into a vector form and have Z = Xβ + e, (3) where Z = (Z(s1 ), · · · , Z(sn )) , X = (x(s1 ), · · · , x(sn )) , e = (e(s1 ), · · · , e(sn )) . The covariance matrix for e is denoted as Σ. The set of selected nodes and the set of rest nodes needed to be estimated ¯ respectively. The BLUE for the paare denoted as A and A, rameter β in the least-squares sense [22] is βˆA = (X Σ −1 XA )−1 X Σ −1 ZA A
A
A
A
with error covariance matrix −1 VA = (XA ΣA XA )−1 , where A = {i1 , · · · , il } denotes the set of selected nodes, and ZA , XA , ΣA denote the corresponding selected observations, regression coefficients, and covariance matrix, respectively. The computation of the BLUE for β can be facilitated if the network has a priori knowledge of the covariance structure Σ. In practice, the matrix Σ can be estimated from the measurements at all sensors. Lemma 1 According to the estimated parameter βˆA by the selected sites A, the estimation of random vector ZA¯, which corresponds to the rest sites A¯ in the random field, can be written as −1 ˆA ) ZˆA¯ = XA¯βˆA + ΣAA ¯ ΣA (ZA − XA β with error covariance matrix −1 ˜ A¯ = DA¯ + TA¯ = (ΣA¯ − ΣAA D ¯ ) + N VA N , ¯ ΣA ΣAA
−1 where N = XA¯ − XA ΣA ΣAA¯. The proof of this lemma is given in the appendix. From Lemma 1, we can see that the error covariance matrix for ZA¯ consists of two terms. The first term DA¯ is the error covariance for the stochastic part in the parametric model (3), while the second term TA¯ is related to the trend that is the deterministic part for the model. Again, we consider the D-optimal criterion problem under the constraint of the observation costs of each sensor node in the field, i.e., (4) arg min |DA¯ + TA¯|, A:C(A)=l
which has an additional term corresponding to the parametric trend in the objective function. As the optimization problem (1) is a special case of (4), which is at least NPcomplete, the exchange algorithm can be implemented to pursue the suboptimal approximation with polynomial complexity according to reference [21]. However, the distortion term TA¯ corresponding to the parametric trend leads to high-dimensional matrix operation, which makes the exchange algorithm inefficiency. Therefore, we propose an alternative criterion that focuses on the estimation performance of the parametric trend. We also consider the D-optimal criterion for estimation of parameter β, ZA = arg min |VA |. The problem that miniA⊂I
mizes the determinant of estimation error covariance matrix of parameter with a given number of sensor nodes is arg min |VA |. (5) A:C(A)=l
Intuitively, in order to minimize the estimation error, we should select a most informative subset with prespecified size from a set of correlated random variables. In the Gaussian case, this selection problem is proven to be NPcomplete, as given in the following theorem [23]. Theorem 1 Assume the random field is Gaussian. Given rational M and covariance matrix Σ of all sites, deciding whether there exists a subset A ⊆ I of cardinality l such that H(A) M is NP-complete, where H( · ) is the entropy of ZA . The differential entropy of the multivariate normal distribution is [24] 1 H(A) = (l + l ln(2π) + ln |ΣA |). 2 It is easy to show that the following problem is NP-complete even under Gaussian assumption (6) arg min |ΣA |. A:C(A)=l
When XA is an identity matrix, VA = ΣA . Therefore, problem (6) is a special case of problem (5), which is also NP-complete. We can switch the objective and constraint of problem (5) (7) arg min l, |VA |C0
where l denotes the selected nodes and C0 is the given estimation accuracy requirement. Problem (5) and problem (7) are equivalent from optimization point of view. Due to the NP-completeness of our sensor selection problems, we shall develop a heuristic method to solve the optimization problem (5). Instead of minimizing the determinant of error covariance with given number of sensors, we propose a greedy algorithm to select a nearly minimum number of active sen-
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sors to achieve a desired estimation performance. We give the following lemma to show that the D-optimal criterion for parameter estimation is monotonic to the number of selected nodes. Lemma 2 Denote the current set of active sensor nodes as Ak = {i1 , · · · , ik }. For arbitrary node j ∈ A¯k that is selected in the next step, the estimation error can be decreased as (8) |Vk | − |Vk+1 | = |Vk+1 | αj−1 ϕj Vk ϕj , where Vk is the covariance corresponding to Ak and −1 ϕj = x(sj ) − XA ΣA ΣAk j , k k −1 αj = Dj (Ak ) = Σjj − ΣjAk ΣA ΣAk j . k
The proof of this lemma is given in the appendix. When the current active node set Ak = {i1 , · · · , ik } is given, we can define the information gain for arbitrary node j ∈ A¯k as Uj|Ak = |Vk | − |Vk+1 | = |Vk+1 | αj−1 ϕj Vk ϕj . From (8), we can easily obtain −1 |Vk+1 | = |Vk−1 |(1 + αj−1 ϕj Vk ϕj ). Therefore, the node to be added is the one with the maximal information gain according to the current selected node ϕj Vk ϕj , set Ak , i.e., ik+1 = arg max Uj|Ak = arg max ¯k ¯k αj j∈A j∈A which only depends on the current active nodes Ak . To meet the desired estimation performance with a nearly minimum number of measurements, the sampling algorithm should choose the sensor that is most informative with respect to the previous sensors. The basic strategy of centralized sampling scheme is that initializing with one node active, i.e., A1 = {i1 }, successively choose one sensor with maximal information gain from the sleeping nodes. The procedure continues until the desired performance is achieved. The details of sampling algorithm is summarized in Algorithm 1. Algorithm 1 (Centralized sampling scheme) 1) Start with k = 1 and A1 = {i1 }, with i1 being randomly chosen; 2) While |VAk | > D0 do a) For each node j ∈ A¯k , compute ik+1 = arg max Uj|Ak ; ¯k j∈A
b) Fusion center updates the set of active sensor node to Ak+1 and k = k + 1; 3) end while. Remark 1 From Algorithm 1, we can see that select one sleeping node to be activated does not need highdimensional matrix operation for the parametric model, since the number of active nodes is small compared with the number of sites need to be estimated. The computational complexity of matrix inversion can be reduced by using the recursive algorithm for inversion of covariance matrix [15].
4
Distributed sampling scheme
In this section, a distributed scheme for random field estimation will be proposed. Consider a WSN with sensor nodes spatially distributed in the field. The network needs to select a subset of sensor measurements to estimate the unknown parameters in a distributed manner by relying on local computations and neighborhood information exchanges. We model the WSN using a graph with the vertex
set V = {1, · · · , n} and the edge set E = {(i, j) ∈ V × V}. In this paper, we assume that each sensor node has a constant communication range r. Each sensor node is a vertex of the graph. Two sensor nodes i and j form an edge of the graph if they are connected with a distance less than or equal to r, i.e., E = {(i, j)|dij r}. We assume the graph is connected, i.e., there exists a path in E for any two vertices. The set of neighbors of node i is defined as the nodes that can communicate to node i directly, i.e., N (i) = {j|(i, j) ∈ E}. The sleeping node set and active node set of N (i) are denoted as Ns (i) and Na (i), respectively. Assuming A is a ˜ which is subset of I, the coverage of A is denoted as A, definedas the union of neighbor sets for all node in A, i.e., A˜ = N (i). i∈A
We propose a distributed scheme for sampling and estimation the parameter β to achieve a desired estimation performance. The basic strategy of distributed sampling scheme is to successively choose one sensor with maximal information gain from the sleeping nodes within the coverage of the current active node set by local computation and message exchange between neighboring sensors. The iterative procedure continues until the estimation performance requirement is met. The distributed iterative algorithm is described in Algorithm 2. Algorithm 2 (Distributed sampling scheme) 1) Start with k = 1 and A1 = {i1 }, i1 is a randomly chosen sensor node; 2) While |VAk | > D0 do a) For each node il ∈ Ak , compute imax = arg max Uj|Ak ; l j∈Ns (il )
b) Each active node il sends a message Mil includand the associated information gain ing the index imax l to its active neighboring nodes Na (il ). Message Uimax |A k l max , U Mil = {imax il |Ak } will be sent to its neighboring l nodes only if it has been updated; c) For each node il ∈ Ak , according to the receiving messages {Mim , im ∈ Na (il )} from active neighboring by nodes, node il updates its index imax l imax = arg l
max
im ∈Ns (il )
Uim |Ak
and updates the associated efficiency; d) Until all the active sensor nodes achieve consensus with the index imax , let ik+1 = imax . An awakening mesl l sage is sent to node ik+1 by its nearest neighbor node as well as the active index information Ak ; e) Each active node adds ik+1 into the active nodes set Ak+1 = {i1 , i2 , · · · , ik , ik+1 }, and k = k + 1; 3) end while. In this proposed distributed algorithm, the iterative procedure does not need a fusion center deployed in the networks; instead, the sensor nodes accomplish the estimation task in a collaborative way. Comparing with the number of the sites need to be estimated, the number of the nodes that have been activated is much smaller. The algorithm requires only the correlations with the active sites are needed when compute the performance improvement of one node during the swap, which means the energy efficiency for the distributed algorithm. During the iterative procedure, each node in the active
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set examines its neighbors and finds the best local candidate that has the most information gain. The computation of information gain is only performed locally without invoking any message exchange. Each node then exchanges the information of its candidate such that the most informative node can be identified in a distributed way. Meanwhile, each sensor does not have to transmit the information of its candidate every time unless its candidate has been updated by its active neighbors. This simple mechanism can avoid unnecessary transmission and save a significant amount of energy. Since only the information of the newly selected node is shared among the current set of active nodes, the iterative procedure does not cause much overhead. Actually, a maximum consensus filter is implemented at each iteration to make all the active nodes to achieve agreement on added node at the current step. Furthermore, exchange of up to k − 1 times to achieve consensus for the worst case at kth iteration. Therefore, when the iteration stop at L step, the toL(L − 1) = O(L2 ). tal communication steps are less than 2
5
In the parametric model of random field, we consider a simple linear model Zi = β + ei , i = 1, 2, · · · , n. The covariance matrix Σ for the stochastic part in the parametric model is generated according to the spatial model 2 i = j, σi , Σ= 2 σi σj exp(−αdij ), i = j with α the scaling constant that measures the intensity of correlation between two nodes [16]. Starting with one node activated, both centralized and distributed sampling algorithms are implemented until the desired estimation performance is met. For comparison, the node with least variance is selected as the initial node for both centralized and distributed. The estimation performance for parameter β is illustrated in Fig. 2 for both centralized and distributed cases. The plot is averaged over 1000 Monte Carlo simulations.
Simulation
In this section, we will present simulation results to illustrate the effectiveness of our proposed algorithm. We randomly generate n = 100 sensor nodes in a 10 × 10 m2 area. Two nodes will be connected with an edge while the distance between them is less than the communication range r of each node. Fig. 1 shows two case of communication range that r = 2.5 and r = 3. Fig. 2 The estimation error of β averaged over 1000 simulations.
In the distributed sampling scheme, each node exchanges the information of its candidate such that the most informative node to the neighbors only when its candidate has been updated by its active neighbors. The algorithm with small communication range may consume fewer communication steps since the number of candidate nodes for all active nodes is fewer than the one with larger communication range. The total communication steps of the proposed distributed sampling algorithm is demonstrated in Fig. 3. The plot is averaged over 1000 Monte Carlo simulations.
Fig. 3 Total communication steps of the distributed sampling scheme averaged over 1000 simulations.
6 Conclusions
Fig. 1 Network connection with different communication range.
In this paper, we have formulated the estimation problem of the random field with parametric trend as a sensor selection problem that selects a minimum number of active sensor nodes while achieving the desired estimation performance. We has proposed a framework for specifying the information gain of each node according to the previous active nodes as well as greedy algorithms to successively choose
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one sensor with maximal information gain from among the sleeping nodes to achieve the desired performance for both centralized and distributed sampling schemes. The simulation results have shown good performance of our proposed algorithms.
[21] Y. Weng, L. Xie, W. Xiao. Random field estimation with quantized measurements in sensor networks[C]//Proceedings of the 29th Chinese Control Conference. Beijing, 2010: 6203 – 6208.
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Appendix Proof of Lemma 1 Recall the estimation of ZA¯ in (4) −1 ZˆA¯ = XA¯ βˆA + ΣAA (ZA − XA βˆA ) ¯ Σ A
with error covariance matrix to be verified −1 ˜ A¯ = (ΣA¯ − ΣAA D ¯) ¯ ΣA ΣAA −1 −1 +(XA¯ − XA ΣA ΣAA¯ )VA (XA¯ − XA ΣA ΣAA¯ ) . The error covariance matrix can be calculated ˜ A¯ = E(ZA¯ − ZˆA¯ )(ZA¯ − ZˆA¯ ) D −1 = E[XA¯ β + eA¯ − XA¯ βˆA − ΣAA ¯ Σ A
· (XA β + eA − XA βˆA )] · [XA¯ β + eA¯ − XA¯ βˆA −1 −ΣAA (XA β + eA − XA βˆA )] ¯ Σ A
−1 ˆA ) = E[XA¯ (β − βˆA ) − ΣAA ¯ ΣA XA (β − β −1 ˆ +(eA¯ − ΣAA eA )] · [XA¯ (β − βA ) ¯ Σ
−1 −1 ˆA ) + (eA¯ − ΣAA −ΣAA ¯ ΣA XA (β − β ¯ ΣA eA )] −1 = XA¯ VA XA¯ − XA¯ VA XA ΣA ΣAA¯ −1 −1 −1 −ΣAA ¯ ¯ ΣA XA VA XA ¯ ΣA VA ΣA ΣAA ¯ + ΣAA −1 +(ΣA¯ − ΣAA ¯) ¯ ΣA ΣAA −1 = (ΣA¯ − ΣAA ¯) ¯ ΣA ΣAA −1 −1 +(XA¯ − XA ΣA ΣAA¯ )VA (XA¯ − XA ΣA ΣAA¯ ) . One more statement is needed during the deduction that XA¯ (β − βˆA ) is orthogonal with (eA¯ − eˆA¯ ), where −1 eˆA¯ = ΣAA ¯ ΣA eA .
It can be proven E[XA¯ (β − βˆA )][eA¯ − eˆA¯ ]
−1 −1 = E[XA¯ (β − (XA ΣA XA )−1 XA ΣA (XA β + eA ))] −1 · [eA¯ − ΣAA Σ e ] ¯ A A −1 −1 −1 = E[XA¯ (XA ΣA XA )−1 XA ΣA eA ][eA ΣA ΣAA¯ − eA¯ ] = 0. Proof of Lemma 2 Denote Ak as the selected node set. For notation simplicity, we denote XAk as Xk , the notation with subscript j denotes the sensor could be added at the next step. The error covariance matrix for β can be written as !−1 ! Σk Σkj Xk −1 Vk+1 = ( Xk xj ) Σjk Σjj xj ! Xk = ( Xk xj ) · Φ · xj
= Vk−1 + αj−1 ϕj ϕj , where Φ=
Σk−1 + Σk−1 Σkj αj−1 Σjk Σk−1 − Σk−1 Σkj αj−1 −αj−1 Σjk Σk−1 αj−1
! ,
ϕj = xj − Xk Σk−1 Σkj , αj = Σjj − Σjk Σk−1 Σkj , and the second equation follows the block-wise inversion for matrix. Considering the D-optimal criteria and the recursion formula
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for determinant −1 |Vk+1 | = |Vk−1 |(1 + αj−1 ϕj Vk ϕj ). Therefore, we have |Vk | = |Vk+1 |(1 + αj−1 ϕj Vk ϕj ) and |Vk | − |Vk+1 | = |Vk+1 | · αj−1 ϕj Vk ϕj . Yang WENG received his B.S. and Ph.D. degrees from the Mathematics Department, Sichuan University, Chengdu, China, in 2001 and 2006, respectively. He has been a lecturer at the School of Mathematics, Sichuan University, Chengdu, China, since 2006. He was a research fellow at the Nanyang Technological University, Singapore, from August 2008 to July 2010. His current research interests include statistic signal processing, signal detection, and estimation. E-mail:
[email protected]. Wendong XIAO received his B.S. degree in Mathematics and Ph.D. degree in Automatic Control from the Northeastern University, China, in 1990 and 1995, respectively. Currently, he is a research scientist at the Institute for Infocomm Research, Agency for Science, Technology and Research (A∗Star), Singapore. Previously, he held research and academic positions at the POSCO Technical Research Laboratories (South Korea), Northeastern University (China), and Nanyang Technological University (Singapore). His current research interests include collaborative signal processing, localization and tracking, communication protocols, and information-driven resource management in wireless ad hoc, sensor, and mesh networks. He is a senior
member of IEEE and a member of ACM. E-mail:
[email protected]. Lihua XIE received his B.E. and M.E. degrees in Electrical Engineering from Nanjing University of Science and Technology in 1983 and 1986, respectively, and Ph.D. degree in Electrical Engineering from the University of Newcastle, Australia, in 1992. Since 1992, he has been with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, where he is currently a professor and the Director, Centre for Intelligent Machines. He held teaching appointments at the Department of Automatic Control, Nanjing University of Science and Technology from 1986 to 1989. He has also held visiting appointments with the University of Newcastle, the University of Melbourne, Hong Kong Polytechnic University, and South China University of Technology. Dr. Xie’s research interests include robust control and estimation, networked control systems, time delay systems, and control of disk drive systems, and sensor networks. In these areas, he has published over 180 journal papers and co-authored two patents and the books H-infinity Control and Filtering of Two-dimensional Systems (with C. Du); Optimal and Robust Estimation (with F. L. Lewis and D. Popa) and Control and Estimation of Systems with Input/Output Delays (with H. Zhang). He is an associate editor of Automatica, IEEE Transactions on Control Systems Technology, the Transactions of the Institute of Measurement and Control, and Journal of Control Theory and Applications, and is also a member of the Editorial Board of IET Proceedings on Control Theory and Applications. He served as an associate editor of IEEE Transactions on Automatic Control from 2005 to 2007, IEEE Transactions on Circuits and Systems-II from 2006 to 2007, International Journal of Control, Automation and Systems from 2004 to 2006, and the Conference Editorial Board, IEEE Control Systems Society from 2000 to 2005. He was the General Chairman of the 9th International Conference on Control, Automation, Robotics and Vision. Dr. Xie is a fellow of IEEE, and a fellow of Institution of Engineers, Singapore. E-mail:
[email protected].
