SENSOR SELECTION FOR RANDOM FIELD ESTIMATION IN WIRELESS SENSOR NETWORKS

Transcription

1 J Syst Sci Complex (2012) 25: SENSOR SELECTION FOR RNDOM FIELD ESTIMTION IN WIRELESS SENSOR NETWORKS Yang WENG Lihua XIE Wendong XIO DOI: /s Received: 10 May 2010 / Revised: 30 January 2011 c The Editorial Office of JSSC & Springer-Verlag Berlin Heidelberg 2012 bstract This paper studies the sensor selection problem for random field estimation in wireless sensor networs. The authors first prove that selecting a set of l sensors that minimize the estimation error under the D-optimal criterion is NP-complete. The authors propose an iterative algorithm to pursue a suboptimal solution. Furthermore, in order to improve the bandwidth and energy efficiency of the wireless sensor networs, the authors propose a best linear unbiased estimator for a Gaussian random field with quantized measurements and study the corresponding sensor selection problem. In the case of unnown covariance matrix, the authors propose an estimator for the covariance matrix using measurements and also analyze the sensitivity of this estimator. Simulation results show the good performance of the proposed algorithms. Key words BLUE, covariance matrix, exchange algorithm, NP-completeness, quantization, random field, sensor selection. 1 Introduction The developments in micro-electro-mechanical systems technology, wireless communications and digital electronics have made the deployment of low-cost wireless sensor networs (WSNs) in large scale with small size sensor nodes possible [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 salinity of sea water, etc. 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 a underlying parameter vector. In most applications, the WSN nodes are powered by small batteries which limit the lifetime of a WSN. Therefore, energy-efficient algorithms in WSNs are important. Yang WENG College of Mathematics, Sichuan University, Chengdu , China. wengyang@scu.edu.cn. Lihua XIE School of Electrical and Electronic Engineering, Nanyang Technological University, , Singapore. elhxie@ntu.edu.sg. Wendong XIO School of utomation and Electrical Engineering, University of Science and Technology Beijing, Beijing , China. wendongxiao68@gmail.com. This wor was supported by the National Natural Science Foundation of China-Key Program under Grant No and the National Natural Science Foundation of China under Grant No Part of this wor was presented at the 29th China Control Conference, Beijing, China, July This paper was recommended for publication by Editor Yiguang HONG.

2 SENSOR SELECTION FOR RNDOM FIELD ESTIMTION IN WSNs 47 It is desirable that only part of the sensor nodes are tased at any time without compromising the networ performance. Recently, many approaches for sensor selection have been proposed for estimation and detection. In [3], a utility is defined for each set of sensors, and the sensors are selected to maximize the utility while satisfying a given energy constraint. The sensor selection problem for estimation of a deterministic parameter has been studied in [4] via convex optimization using convex relaxation. heuristic method has been given for a subset selection. This method pursues not only a suboptimal choice of measurements, but also a bound on how well the global optimal choice performs. The sensor selection problem for event detection in WSN has been addressed in [5]. dimension reduction method has been introduced by selecting a subset of sensors that maximize the Kullbac-Leibler distance between the selected measurement distributions. However, the aforementioned results cannot be applied to the random field estimation problem in WSNs. So far, few wor has been done for the random field estimation in WSNs. In [6], the optimal sensor placement for the Gaussian process estimation problem is formulated as a problem of maximizing the mutual information between the chosen locations and other locations. This combinatorial problem is proved to be NP-complete, and a polynomial-time approximation greedy algorithm has been proposed. lthough the sensor placement for random field estimation is addressed, the Gaussian assumption limits the applications of this wor. In application scenarios of WSN, there are two issues that should be considered. The first is the wireless bandwidth shared among sensor nodes and the fusion center. Quantization has been viewed as a fundamental element to save bandwidth by reducing the amount of data to represent a signal [7 8]. Quantization has been well studied in digital signal processing and control where a signal with continuous values is quantized due to a finite word-length of microprocessor [9]. In WSNs, quantization is also necessary to reduce the energy consumption as communications cost most of the energy and the amount of energy consumed is related to the amount of data transmitted. The second issue is that the covariance structure of the random field is not nown a priori in many situations since sensor nodes are deployed in an unnown environment or deployed in a hostile region. Estimation of the covariance structure of a spatial process is a fundamental prerequisite for the design of a monitoring networ since an inaccurate estimate of the covariance matrix leads to poor estimation accuracy [10]. In this paper, the sensor selection problem for random field estimation is considered. The main contributions of this paper are in three aspects. First, we formulate the sensor selection for random field estimation as an optimization problem and proved the NP-completeness of this problem. Unlie other results on sensor selection for random field estimation (e.g., [6]), we do not assume the distribution of the random field. Second, we propose a heuristic algorithm to pursue the optimal solution of our proposed optimization problem. lthough those exchangetype algorithms were already used in statistics for regression problem, it is still not applied to the sensor selection problem. Third, we have considered the application scenarios for our sensor selection. We propose a best linear unbiased estimator (BLUE) for a Gaussian random field with quantized measurements and study the corresponding sensor selection problem. In the case of unnown covariance matrix, we propose an estimator for the covariance matrix using measurements and also analyze the sensitivity of this estimator. The rest of this paper is organized as follows. In Section 2, the formulation of the sensor selection problem in random field estimation in WSNs is derived. The complexity of the sensor selection problem for minimizing the determinant of the estimation error covariance matrix with a given number of sensors is analyzed in Section 3. heuristic algorithm for this sensor selection problem is proposed to pursue a suboptimal solution in Section 4. pplication scenarios of sensor selection for random field estimation in WSN with quantization and unnown covariance matrix are discussed in Section 5. Simulation results are reported in Section 6 to show the

3 48 YNG WENG LIHU XIE WENDONG XIO performance of our methods. Concluding remars are given in Section 7. 2 Problem Formulation 2.1 Random Field Model Denote the random field under discussion as {Z(s),s D}, whered 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 = {s 1,s 2,,s n }. Each sensor can tae 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 and the covariance function as M(s) =E(Z(s)), s D (1) K(s, t) =cov(z(s),z(t)), s, t D. (2) We study a single-snapshot scenario in a WSN, in which each sensor node can tae only one measurement. For a given sensor node i at location s i, we denote its observation as Z(s i ). We can reconstruct the random field using the observations form n sensor nodes, namely, Z Pn =(Z(s 1 ),Z(s 2 ),,Z(s n )) T. We can use a simple riging predictor to reconstruct the value of Z(s) where this predictor at s Dcorresponds to the BLUE [11] Ẑ(s) =E(Z(s)) + Σ s,pn Σ 1 P n Z Pn with error variance: ε(s, P n )=Var(Z(s)) Σ s,pn Σ 1 P n Σ Pn,s, where Σ s,pn = cov(z(s),z Pn ), Σ Pn = cov(z Pn,Z Pn ), and E(Z(s)) and Var(Z(s)) are the expectation and variance of Z(s), respectively. Here, we do not restrict the field and its distribution to be stationary and isotropic. 2.2 Random Field Estimation in WSNs We want to choose the optimal sensor locations for a number of sensors to reconstruct the whole field under certain criteria. natural objective is to choose sensor locations to tae observations to minimize the distortion in the estimate of the whole field. However, as a random field is a multi-dimensional 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. ssume that, by using n sensors, we can discretize the random field Z(s) into a sequence {Z(s 1 ),Z(s 2 ),,Z(s n )}, which is denoted as Z =(Z 1,Z 2,,Z n ) T (3) with an index set I = {1, 2,,n} and the corresponding covariance matrix Σ. 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, we want to select l random variables from the vector to estimate the rest of the r (r = n l) variables in the sense of minimum error variance. We denote the selected vector by Z =(Z i1,z i2,,z il ) T,

4 SENSOR SELECTION FOR RNDOM FIELD ESTIMTION IN WSNs 49 and denote the vector to be estimated by Z Ā = Z\Z. (4) Without loss of generality, we further assume that M(s) =0, s D. The BLUE for the rest of the variables can be presented as with error covariance matrix Ẑ Ā = Σ Ā Σ 1 Z, (5) D Ā =cov(ẑā Z Ā )=Σ Ā Σ Ā Σ 1 Σ Ā, (6) where Σ =cov(z ), Σ Ā =cov(z Ā ), Σ Ā =cov(z Ā,Z ). In this paper, we always assume that covariance matrix Σ is invertible for an arbitrary random vector Z, 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, which is closely related to the optimal experiment design [4,12] originally proposed by Wald [13] and extended by Kiefer [14]. Since Kiefer s seminal wor, a large amount of literature can be found for dealing with theoretical aspects of optimal design. Similar to the traditional experimental design theory, a monotonic function can be used to compare the efficiency of various variables of Z i1,z i2,,z il. The most widely used criterion for the experiment design problem is nown as the D-optimal [15]. Definition 1 The criterion is called D-optimal if the determinant of the error covariance matrix D Ā is minimized, i.e., Z =argmin D Ā, (7) I where is the determinant function. While there might not always exist a direct relation between energy efficiency and the number of active sensors, reducing the number of active sensors generally leads to less energy consumption. In WSNs, in order to conserve energy and prolong networ lifetime, it is necessary to select a group of sensor nodes for collecting observation data to estimate the field, while with other nodes inactive (sleeping). The goal of this paper is to propose an approach to obtain optimal estimation performance under the energy consumption constraint of WSN which is formulated as follows, where # is the cardinality of set. 3 NP-Completeness argmin Σ Ā Σ Ā Σ 1 Σ Ā, (8) :#=l Here we will show that the decision version of the optimization problem (8) is NP-complete. The decision version of our optimization problem is as follows. Definition 2 The decision version of the sensor selection for random field estimation problem: Instance: random variable set {Z 1,Z 2,,Z n } with an index set I = {1, 2,,n} and the corresponding covariance matrix Σ, positive integer l and given number M.

5 50 YNG WENG LIHU XIE WENDONG XIO Question: Is there a subset Isuch that, when # l, the determinant of estimation error covariance matrix given in (8) is at most M? Theorem 1 There is no polynomial time algorithm that solves the sensor selection for random field estimation problem (8), unlessp = NP. Proof First, the decision version of our problem is in NP: nondeterministic algorithm needs to guess a subset of I with less cardinality than l and chec in polynomial time that the determinant of estimation error covariance matrix D Ā is less than the given value M. Next, to prove that our problem (8) is NP-complete, we show that this problem contains an NP-complete problem as a special case. Using the determinant identity, B B T C = C B 1 B T, where the matrix is invertible, we can obtain Σ = Σ Σ Ā = Σ Σ Ā Σ Ā Σ 1 Σ Ā. (9) Σ Ā Σ Ā Since the covariance matrix D is invertible for arbitrary random vector Z, I. Recalling the error covariance matrix of Z Ā, according to Z in (6) and the determinant identity in (9), we can get Σ = Σ D Ā. s a result of the determinant of Σ is constant, the problem (8) is equivalent to argmax Σ. (10) :#=l Under the Gaussian assumption, the differential entropy of the multivariate normal distribution is [16] H() = 1 2 (l + l ln(2π)+ln Σ ), (11) where H( ) is the Shannon entropy. Thus, problem (10) is restricted to argmax H(). (12) :#=l The decision version of problem (12) is as follows. Under the Gaussian assumption, with a given number M and covariance matrix Σ of all sites, is there a subset Iof cardinality l such that H() M? This problem is proved to be NP-complete in [17]. Restrict to our problem (8) by allowing only instances having Gaussian distribution for all sites. Therefore, our problem contains an NP-complete problem as a special case. 4 Exchange lgorithm Since the formulated optimization problem (8) is proved to be NP-complete, the standard method of dealing with such combinatorial optimization problems is to propose a heuristic algorithm to approximate the optimal solution [18 19]. Our proposed iterative algorithm is similar to the Fedorov s exchange algorithm, which constructs a discrete D-optimal design for a regression problem [15,20]. Intuitively, in order to minimize the estimation error, we should select a most informative subset with pre-specified size from a set of correlated random variables. The

6 SENSOR SELECTION FOR RNDOM FIELD ESTIMTION IN WSNs 51 basic strategy for the exchange algorithm is a swap operation at each iteration, where one sensor node with the least contribution to the estimation performance will be deleted from the chosen sensors {i 1,i 2,,i l }, and another sensor with the largest improvement to the estimation performance will be added at each step. The iterations will continue until no improvement can be made by swapping. t the -th iteration of our algorithm, the current set of active sensor nodes is denoted as = {i 1,i 2,,i l }, and the sleeping nodes which need to be estimated is denoted as Ā = I\. Theorem 2 Given the current set of active sensor nodes = {i 1,i 2,,i l } at the -th step, a swap consists of two stages, i.e., a deletion stage and an addition stage. t the deletion stage, node i with the least contribution will be deleted from the set of active nodes. Node i is the one with minimum error variance estimated by other active sensor nodes, i.e., i =argmin j D j ( j ), (13) where D j ( j ) denotes the error variance of Z j estimated by the set j = \{j}. The candidate nodes for the next stage can be denoted as Ā+ = Ā {i }. t the addition stage, node i + with most contribution will be activated to form the set of active nodes +1.Nodei + is the one with maximum error variance estimated by i, i.e., i + =argmax j Ā+ D j ( i ). (14) Proof t the deletion stage, following the proof of Theorem 1, we now that Z =argmin I D Ā =argmax I ccording to the determinant identity (9), we have Σ Σ = j Σ j,j Σ j, j Σ jj = Σ j Σ jj Σ j, j where Σ j, j Σ 1 j Σ., Σ j,j =cov(z j,z j ), Σ jj =cov(z j ). Therefore, for the given, the node with the minimum error variance estimated by other active sensor nodes is the least contribution one to the estimation performance, i.e., i = Z =argmind j ( j ). j t the addition stage, we assume that node j will be added to the set of current active nodes i, Σ j + = = Σ i Σ j, i Σ i Σ i,j Σ jj Σ jj Σ j, i Σ 1 i. Σ i,j

7 52 YNG WENG LIHU XIE WENDONG XIO fter the deletion stage at this step, the set i j Ā+,wehave is given. Therefore, for an arbitrary node i + =argmax j Ā+ =argmax j Ā+ =argmax j Ā+ =argmax j Ā+ Σ j + Σ i Σ jj Σ j, i Σ jj Σ j, i D j ( i ), Σ 1 i Σ 1 i Σ i,j Σ i,j which ends the proof of this theorem. Remar 1 t the deletion stage of each iteration, the node best explained by the other active nodes will be deleted from the set of active nodes; at the addition stage, the activated node is the worst explained node by the observations made at the activated nodes in the previous steps, i.e., the node with worst estimation performance estimated by the active set. The iterative algorithm can give better results if multiple nodes can be deleted and added in both stages, while with a more expensive computation cost. ccording to Theorem 2, we present a centralized framewor for sampling and estimation under a certain distortion constraint. Suppose that a fusion center (or a sin) is deployed in the networs, all the sensor nodes can transmit the observations to the fusion center. The fusion center nows the exact position of each sensor node and the covariance matrix for all sensor nodes. Starting with l arbitrary nodes active, the fusion center will employ sensor selection by swapping one of the current active sensors with one of the sleeping sensors iteratively. The iterative algorithm will be terminated until no improvement can be made by swapping. The centralized approximation algorithm is summarized in lgorithm 1. lgorithm 1: Centralized sensor selection scheme 1) Start with =0andl arbitrary nodes 0 active; 2) While Σ > Σ 1 do a) Delete the node i from the set of active sensor nodes with the least contribution to the estimation performance according to (13), i.e., i =argmin j D j ( j ); b) dd node i + to the set of active nodes with the most contribution to the estimation performance according to (14), i.e., i + =argmax j Ā+ D j ( i ); c) Fusion center updates the set of active sensor node to +1 and = +1; 3) end while.

8 SENSOR SELECTION FOR RNDOM FIELD ESTIMTION IN WSNs 53 5 pplication Scenarios Two application scenarios are investigated in this section. First, the sensor nodes can transmit the measurements to fusion center with infinite bandwidth of communication. However, the communication bandwidth is always limited for each node with small size and low energy in WSNs. We should propose an estimation approach at fusion center to get the BLUE for the unselected sites according to the quantized measurements of the selected sites. Second, in the case of a WSN is deployed in an unnown environment, the priori covariance matrix is not available in many situations. We can get an approximated BLUE for the unselected sites resort to the estimated covariance matrix. The sensitivity property of this approximated BLUE should be considered. 5.1 Estimation with Quantized Measurements In this subsection, we consider the random field estimation with quantized measurements in WSNs. Suppose the sensor nodes can quantize the measurements before transmitting them to the fusion center. The estimation at the fusion center can only be based on the quantized measurements which can be treated as the compressed measurements from the local sensors. In this section, we consider a fixed quantizer and the channel can transmit messages of log 2 N bits without error, where N 2 is the number of quantization levels. Without loss of generality, we assume that the random field Z(s) has zero mean. The selected sensor node quantizes its measurements Z i to obtain an output Y i, which does not depend on the information from other nodes. Each Y i is then transmitted to the fusion center. Following the previous notation in (4), the measurements taen by the selected sites are denoted as Z =(Z i1,z i2,,z il ) T. Denote the quantizer outputs of Z as Y = (Y 1,Y 2,,Y l ) T and the sites to be estimated are denoted as Z Ā = Z\Z.TheBLUEforZ Ā according to measurements Y is with error covariance matrix Ž Ā = K Y = E(Z Ā Y T )[E(Y Y T )] 1 Y (15) Ď Ā =cov ( Ž Ā Z Ā ) = ΣĀ K E(Y Y T ) KT. (16) Denote the given quantizer as Q( ). Whenever the input sample falls into the quantization interval B =[d,d +1 ], the quantizer output is y. The ij-th element of Y s covariance matrix can be calculated as follows where [E(Y Y T )] ij = E(Y i Y j ) N N = y i y j P {Z i B i,z j B j }, (17) i=1 P {Z i B i,z j B j } = B i B j p(x 1,x 2 )dx 1 dx 2 and p(x 1,x 2 ) is the joint probability density function of Y i and Y j. For a Gaussian random field, the optimal quantization level which means the minimum distortion with quantization can be chosen according to [21].

9 54 YNG WENG LIHU XIE WENDONG XIO Furthermore, the ij-th element of covariance matrix E(Z Ā Y T ) can be calculated as [E(Z Ā Y T )] ij = E(Z i Y j ) N = E(Z i Z j B j )y j P {Z j B j }. (18) i=1 For a stationary Gaussian process the expectation of Z i conditioned on Z j B j can be calculated as E(Z i Z j B j )= E(Z iz j )E(Z j Z j B j ) ΣZ 2, where ΣZ 2 is the variance of each site in the field. ccording to (17) and (18), the fusion center can calculate the error covariance when it receives the quantized measurements from the sensor nodes. Therefore, the exchange algorithm proposed in Section 4 can be implemented for the quantization case. 5.2 Estimation with Unnown Covariance Matrix ll results proposed in the previous sections have essentially utilized the fact that the covariance matrix Σ is nown. However, when a WSN is composed of a large number of sensor nodes which are densely deployed in an unnown environment, the priori covariance matrix is probably not available. Having repeated observations at every deployed node site of {s 1,s 2,,s n } one can directly estimate Σ. Suppose each sensor can tae m measurements, and the measurement taen of site s i at time j is denoted as z ij. The traditional estimator for covariance matrix is the sample covariance matrix (SCM), which is an unbiased and efficient estimator given by Σ = 1 m m Z j Z T j, (19) where Z j =(z 1j,z 2j,,z nj ) T is the measurements vector of all sites at time j, and{z ij },j = 1, 2,,m are i.i.d. from random variable Z i. Without loss of generality, we still assume random field with zero mean. Intuitively, we should replace the covariance matrix in (5) and (6) with the corresponding SCM. In this subsection, we first analyze why such a substitution maes sense, then we will analyze the sensitivity property of the estimator with estimated covariance matrix. The BLUE Z Ā by the selected sites set Z is obtained by minimizing the following square ris matrix V (C) =E[(Z Ā CZ )(Z Ā CZ ) T ]. Direct minimization of V (C) gives the optimal C as follows in BLUE sense C = Σ Ā Σ 1. If m measurements of Z are available, a reasonable approximate of V (C) is Ṽ (C) = 1 m m (Z Āj CZ j )(Z Āj CZ j ) T, where Z j and Z Āj are the measurements for selected nodes set and unselected nodes set Ā at time j, respectively. Since matrix Ṽ (C) is not a definite matrix, we minimize its trace for

10 SENSOR SELECTION FOR RNDOM FIELD ESTIMTION IN WSNs 55 instead, C =argmin C =argmin C =argmin C =argmin C tr(ṽ (C)) [ 1 m ] tr (Z m Āj CZ j )(Z Āj CZ j ) T m (Z Āj CZ j ) T (Z Āj CZ j ) m (Z T Āj ZĀj 2Zj T CT Z Āj + Zj T CT CZ j ), where tr( ) is the trace function. Differentiating Ṽ (C) with respect to C, wehave Ṽ (C) ( m C =2C ) m Z j Zj T 2 Z Āj Zj. T Hence, ( m C =(Z Āj Zj) T Z j Zj T = ( )( 1 1 m ZĀj Zj T m = Σ 1 Ā Σ, ) 1 m Z j Zj T where Σ Ā and Σ denote the corresponding SCM. The substitution of real covariance matrix to SCM leads to an approximated BLUE estimator with the approximated error covariance matrix: ) 1 Z Ā = Σ 1 Ā Σ Z (20) D Ā =cov(z Ā Z Ā )= Σ Ā Σ 1 Ā Σ Σ Ā. (21) The exchange algorithm proposed in Section 4 can be implemented by exploiting this estimated covariance matrix. lthough the SCM is an unbiased and efficient estimator, deviation always exists when only finite observations are available. The sensitivity problem of the approximated BLUE of (20) arises. We formulate the deviation between the real covariance matrix and SCM as a perturbation Δ in the following ( Σ Σ Ā Σ = Σ Ā Σ Ā ) ( ) Σ +Δ = 1 Σ Ā +Δ 12 = Σ +Δ, Σ Ā +Δ 21 Σ Ā +Δ 2 where Δ 12 =Δ T 21. The deviation of BLUE and approximated BLUE is investigated in the following theorem. Theorem 3 ssume Σ and Σ +Δ 1 are nonsingular. In addition, the perturbations for each bloc in Σ satisfy Δ 1 ε 1 Σ, Δ 2 ε 2 Σ Ā, Δ 12 = Δ 12 ε 3 Σ Ā =

11 56 YNG WENG LIHU XIE WENDONG XIO ε 3 Σ Ā and Σ 1 Δ 1 < 1. The deviation of BLUE and approximated BLUE can be bounded as D Ā D Ā ε 2 L 2 + ε2 3 +2ε 3 κε 1 1 κε 1 L 1, where is the Euclidean norm, L 1 = Σ Ā Σ 1 Σ Ā, L 2 = Σ Ā and κ = Σ 1 Σ. Proof We consider the deviation of the corresponding error covariance matrices D Ā and D Ā, D Ā D Ā =( Σ Ā Σ Ā Σ 1 Σ Ā) (Σ Ā Σ Ā Σ 1 Σ Ā) =Δ 2 (Σ Ā +Δ 21 )(Σ +Δ 1 ) 1 (Σ Ā +Δ 12 )+Σ Ā Σ 1 =Δ 2 (Σ Ā +Δ 21 )[(Σ +Δ 1 ) 1 Σ 1 ](Σ Ā +Δ 12 ) +Δ 21 Σ 1 Σ Ā +Δ 21 Σ 1 Δ 12 + Σ Ā Σ 1 Δ 12. Σ Ā ccording to the results on the inverses of perturbations for nonsingular matrices [22], Thus, Σ +Δ 1 ) 1 Σ 1 Σ Σ 1 Δ 1 (1 Σ 1 ΔΣ ) Σ. (22) D Ā D Ā Δ 2 + (Σ Ā +Δ 21 )[(Σ +Δ 1 ) 1 Σ 1 ](Σ Ā +Δ 12 ) + Δ 21 Σ 1 Σ Ā + Δ 21 Σ 1 Δ 12 + Σ Ā Σ 1 Δ 12 ε 2 Σ Ā +(1+ε 3 ) 2 Σ Σ 1 Σ Ā Δ 1 (1 Σ 1 ΔΣ ) Σ Σ Ā +2ε 3 Σ Ā Σ 1 Σ Ā + ε 2 3 ΣĀ Σ 1 Σ Ā ε 2 L 2 + ε2 3 +2ε 3 κε 1 L 1. 1 κε 1 Remar 2 This theorem shows that when ε i,i=1, 2, 3 are sufficiently small, the approximated BLUE using SCM is close to the one using real covariance matrix. 6 Simulations In this section, we present simulation experiments to illustrate the effectiveness of our proposed algorithm. We randomly generate n = 100 sensor nodes in a square unit. Figure 1 shows an example distribution of sensor nodes. The covariance matrix Σ = {Σ ij } n n for the random field is randomly generated accordingtothespatialmodel { Σ 2 Σ ij = i, i = j, Σ i Σ j exp( αd 2 ij ), i j, (23) where the scaling constant α =0.5 measures the intensity of correlation between two nodes [11]. {Σ i } are generated independently from the normal distribution with mean zero and variance 10. In the first simulation, we select 10 sensors to estimate the rest sites. Under the D-optimal criterion, the distortion for the estimation of the random field will be minimized while the

12 SENSOR SELECTION FOR RNDOM FIELD ESTIMTION IN WSNs Figure 1 n example node distribution in a square area determinant of covariance matrix for selected nodes is maximized. The estimation performance during the iterative process is shown in Figure 2. The set of active nodes is initialized by starting with randomly chosen 10 nodes. For comparison, the initial set of active nodes in both raw and quantized measurements cases are the same. The raw measurements are quantized into 4-bits before transmitting to the fusion center. Here, we use the optimal quantizer for Gaussian distribution designed in [21]. 3 x 104 determinant of covariance matrix for selected sites Raw measurments Quantized measurements iterative step Figure 2 Exchange algorithm for both raw and quantized measurements In the second simulation we show the estimation performance improves with increase of the active sensor nodes. Figure 3 shows the increase of Σ which means the estimation error for the random field decreases. The estimation performance for quantized measurements is quite satisfactory with a moderate number of bits transmitted to the fusion center in both the simulations.

13 58 YNG WENG LIHU XIE WENDONG XIO 4 x 106 determinant of covariance matrix for selected sensors Raw measurements Quantized measurements number of sensors selected Figure 3 Estimation performance with different number of active nodes In the third simulation, we show that the sensitivity property of approximated BLUE which uses the SCM instead of the real covariance matrices. Figure 4 gives a comparison of iterations for BLUE and approximated BLUE with different number of measurements. The iterations of the exchange algorithms have a similar tendency for the three estimators. Both the first and This simulation show that the proposed exchange algorithm will be terminated in about 10 steps. ctually, we investigate the number of iteration steps for our proposed exchange algorithm by 500 Monte Carlo simulations. The result shows that the number of iteration steps will be terminated in dozens of steps. determinant of covariance matrix for selected sites estimated covariance with 20 measurements estimated covariance with 30 measurements nown covariance matrix iterative step Figure 4 Comparison of exchange algorithm for BLUE and approximated BLUE 7 Conclusion In this paper, we have formulated the random field estimation problem in WSN as a sensor selection problem. The combinatorial optimization problem of choosing sensor nodes to min-

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