Research Article Data Reduction with Quantization Constraints for Decentralized Estimation in Wireless Sensor Networks

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1 Matheatical Probles in Engineering Volue 014, Article ID 93358, 8 pages Research Article Data Reduction with Quantization Constraints for Decentralized Estiation in Wireless Sensor Networks Yang Weng College of Matheatics, Sichuan University, Chengdu, Sichuan , China Correspondence should be addressed to Yang Weng; wengyang@scueducn Received August 013; Revised 15 Deceber 013; Accepted 19 Deceber 013; Published 9 January 014 Acadeic Editor: Shuli Sun Copyright 014 Yang Weng his is an open access article distributed under the Creative Coons Attribution icense, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original work is properly cited he unknown vector estiation proble with bandwidth constrained wireless sensor network is considered In such networks, sensor nodes ake distributed observations on the unknown vector and collaborate with a fusion center to generate a final estiate Due to power and counication bandwidth liitations, each sensor node ust copress its data and transit to the fusion center In this paper, both centralized and decentralized estiation fraeworks are developed he closed-for solution for the centralized estiation fraework is proposed he coputational coplexity of decentralized estiation proble is proven to be NP-hard and a Gauss-Seidel algorith to search for an optial solution is also proposed Siulation results show the good perforance of the proposed algoriths 1 Introduction he developents in icroelectroechanical systes technology, wireless counications, and digital electronics have enabled the deployent of low-cost wireless sensor networks (WSNs) in large scale using sall size sensor nodes [1] In such networks, the distributed sensors collaborate with a fusion center to jointly estiate the unknown paraeter If fusion center receives all easureent data fro all sensors directly and processes the in real tie, the corresponding processing of sensor data is known as the centralized estiation, which has several serious drawbacks, including poor survivability and reliability, heavy counications, and coputational burdens Since all sensors have liited battery power, their coputation and counication capability are severely liited; the decentralized estiation ethods are widely discussed in recent years [ 6] In the decentralized estiation fraework, every sensos also a subprocessor It first preprocesses the easureents in ters of a criterion and then transits its local copression data to the fusion center Upon receiving the sensor essages, the fusion center cobines the according to a fusion rule to generate the final result In such networks, less inforation is transitted leading to a significant power-saving advantage which is very iportant in the case of WSNs o iniize the counication cost, only liited aount of inforation is allowed to be transitted through networks; diensionality reduction estiation ethods have attracted considerable attentions [7 9] he basic idea of the diensionality reduction estiation strategy is to prefilter the high-diensional observation vector by a linear transforation (atrix) to project the observation onto the subspace spanned by basis vectors and filter the result with a low-rank estiation Indeed, diensionality reduction estiation and filtering are iportant for a wide range of signal processing applications where data reduction, robustness against noise, and high coputational efficiency are desired Quantization has been viewed as a fundaental eleent in saving bandwidth by reducing the aount of data to represent a signal and well studied in digital signal processing andcontrolwhereasignalwithcontinuousvaluesisquantized due to a finite word-length of icroprocessor [10] In WSNs, quantization is also necessary to reduce the energy consuption as counications consue the ost energy as the aount of energy consued is related to the aount of data transitted An interesting distributed estiation approach based on the sign of innovation (SOI) has been developed for dynaic stochastic systes in [11] whereonly transission of innovation of a single bit is required A general ultiple-level quantized innovation Kalan filter for

2 Matheatical Probles in Engineering estiation of linear dynaic stochastic systes has been presented in [1] he solution to the optial filtes given in ters of a siple Riccati recursion as in the standard Kalan filter A rando field estiation proble with quantized easureents in sensor networks has been considered in [13] In the early work [14], the trade-off between diension reduction and quantization in iniu ean squared error estiation proble is investigated In this paper, different fro the existing work, the diensionality reduction and quantization for local data copression are considered in an integrated way Data reduction with quantization constraints estiation for an unknown vectos forulated as an optiization proble Both centralized and decentralized estiation fraeworks are developed he closed-for solution for the centralized estiation fraework is proposed By using coputational coplexity theory, the intractability of decentralized estiation proble is established A Gauss-Seidel type iteration algorithtosearchforanoptialsolutionisalsoproposed for the decentralized estiation proble he rest of this papes organized as follows With given counication bandwidth, the bits allocation proble has been forulated as an optiization proble in Section he closed-for solution for the centralized estiation fraework is proposed in Section 3 he coputational coplexity of decentralized estiation proble is proved to be NPhard and a Gauss-Seidel algorith to search for an optial solution is also proposed in Section 4 Siulation results arereportedinsection 5 to show the perforance of our ethods Concluding rearks are given in Section 6 Proble Forulation Consider a sensor network deployed with sensor nodes Each sensor, say the ith sensor, can take observation y i R n i which is correlated with an unknown rando paraeter x R he observations will be transitted to a fusion center to estiate the unknown paraeter x under soe certain criterion In this paper, we consider the iniu ean squared error (MMSE) criterion [15, 16] hrough a transfor atrix K i R n i, n i, each sensor transfors the observation into a 1vector K i y i, whereafter the transfored vector will be quantized into several bits and transited to the fusion center In this paper, we assue that there is no inforation exchange aong sensors We also assue without loss of generality that the unknown paraeter x and observations y i are zeroean he auto- and cross-covariance atrices R xx,r xyi,r yi y j, i, j {1,,}are available at the FC he role of FC is to cobine thereceivedquantizationinforation according to {Q (K 1 y 1 ),Q(K y ),,Q(K y )}, (1) x =f(q(k 1 y 1 ),Q(K y ),,Q(K y )), () where f( ) is the fusion function and Q( ) is a given quantizer Our goal is to design the linear transfors {K 1,,K } and the fusion function f( ) such that the ean squared error (MSE) is as sall as possible under the constraint that the totalnuberofbitscanbetransittedtofchroughout this work, we will focus only on the linear design of fusion function f( ) which can be represented in the for x =f(q(k 1 y 1 ),Q(K y ),,Q(K y )) = C i Q(K i y i ) he given quantizer Q( ) is considered as a iniu squared error distortion quantizer [17] We assue that the quantizenput vector Z = (z 1,,z r ) is a rando vector with uncorrelated coponents Each coponent has zero ean and variance σ i Under the Gaussian assuption, the quantizer output Q(Z) is treated as a noise source that introduces independent white noise V as whoseeaniszeroandcovarianceis[17] Σ VV =E(VV ) (3) Z=Q(Z) V, (4) = diag {γ 1,,γ r } = diag {σ 1 B 1,,σ r B r }, where γ i =σ i B i is the squared error distortion for the ith coponent and B i is the bits used by the the ith coponent z i herefore, the ean squared error at the fusion center can be calculated as follows: E x =E =E C i Q(K i y i ) C i (K i y i V i ) =E i K i y i C s=1 t=1 E C iv i c i st γ it =E i K i y i c C i st σ it B it, s=1 t=1 where K i is the local linear transfor operator, C i ={c i st } isthefusionoperatoratthefusioncenterandb it is the quantization bits for the tth eleents of vector K i y i he optialestiationof rando vectorx undendividual sensor bandwidth constraint can be forulated as follows: iniize E( x ) (5) (6) (7) B it B, C i R, K i R n i t=1

3 Matheatical Probles in Engineering 3 By appropriate pre- and postwhitening process if necessary, we assue without loss of generality that the auto- and cross-covariance atrices R xx,r xyi,r yi y j, i,j {1,,} have full rank and the eleents of observation vector taken by each sensor are uncorrelated [18] 3 Centralized Data Reduction with Quantization Constraints In this section, we consider a siple centralized fraework where the entire data is available at a single sensor node and the centralized case of optiization proble (7) can be siplified as follows: iniize E x =E K(r) y r B i B, K(r) R n, r σ i B i where K(r) is the approxiation atrix and B is the total bits to be transitted he optial estiation without observation copression in the MMSE sense is as follows: x =R xy R 1 yyy=ky, (9) with estiation error covariance atrix P=R xx R xy R 1 yy R yx, (10) where K is called optial estiation atrix We write forula (9) as a linear odel by introducing an estiation error e as x= xe=r xy R 1 yyye (11) We consider the proble that the optial estiation atrix K is replaced by an approxiating atrix K(r) with lower rank r<n With a given copressed diension r, we want to find an optial K(r) such that the MMSE is as sall as possible he linear odel (11)isodifiedas (8) x= x (r) e(r) =K(r) ye(r) (1) he estiation error covariance atrix can be calculated as follows: P (r) =E(e(r) e(r) ) =E((eKy K(r) y) (e Ky K (r) y) ) =P(K K(r)) R yy (K K(r)) (13) herefore, the approxiation atrix K(r) introduces an extra variance ter ε = tr [(K K(r)) R yy (K K(r)) ] = tr [(KR 1/ yy K(r) R1/ yy )(KR1/ yy K(r) R1/ yy ) ] (14) he atrix KR 1/ yy =R xyr 1/ yy hasansvdofthefor KR 1/ yy =UΣV (15) By iniizing ε, it is not hard to show that the best approxiation atrix is he extra variance is then K (r) =UΣ(r) V R 1/ yy (16) ε = λ i, (17) i=r1 where λ r1,,λ are the sallest rsingular values of KR 1/ yy herefore the optiization proble (18) is as follows: iniize i=r1 r λ i r B i B,r σ i B i (18) If s given, we can solve this optiization proble by a aplacian ultiplier [19] 4 Decentralized Data Reduction with Quantization Constraints et us now consider the estiation fraework in a ultisensor setup, under a total available rate B which has to be shared aong all sensors In the decentralized anner, the ith sensor transfors the observation y i R n i into a 1vector K i y i through a transfor atrix K i R n i, n i,whereafter the transfored vector will be quantized into several bits and transitedtothefusioncenterbythelinearfusionruleat the fusion center, the ean squared error at the fusion center can be calculated as follows: E x =E C i Q(K i y i ) =E i K i y i c C i st σ it B it, s=1 t=1 (19) where K i is the local linear transfor operator, C i ={c i st } isthefusionoperatoratthefusioncenterandb it is the quantization bits for the tth eleents of vector K i y i herefore, the decentralized estiation of rando vector x under

4 4 Matheatical Probles in Engineering individual sensor bandwidth constraint can be forulated as follows: iniize E x =E i K i y i C c i st σ it B it s=1t=1 B it B, C i R, K i R n i t=1 (0) heore 1 he coputational coplexity of solving proble (0) is NP-hard even in the case with absence of channel distortions for quantization of each sensor Proof We present the siplified forulations to analyze the coputation coplexity of proble (7) et the distributed sensor nodes ake observations on a coon rando paraeter vector x R according to y i =H i xv i,,,,, (1) where H i R i is the observation atrix and V i R i is the additive noise which is zero ean and spatially uncorrelated According to [18], we can assue that the sensor noises are uncorrelated with the input signal x Without loss of generality, we can assue that the unknown paraeter vector x has an autocovariance atrix R xx =I and the noise covariance atrix is R Vi =I i hemseatthefccanbecalculatedasfollows: E x =E C i Q(K i y i ) =E i K i y i c C i st σ it B it s=1 t=1 () In the absence of channel distortions, the MSE at the FC can be siplified as E( x )=E =E C i Q(K i y i ) C i (K i y i V i ) =E i K i y i C s=1 t=1 =E =E i K i y i C C i K i (H i xv i ) E C iv i c i st γ it =E( ( = r (E (x ( = r [(I C i K i (H i xv i )) C i K i (H i xv i )) C i K i (H i xv i )) C i K i (H i xv i )) (I = r [(I [ ) C i K i H i )E(xx ) C i K i H i ) (C i K i E(V i V i )K i C i )] C i K i H i )(I (C i K i K i C i ) ] ] = I i K i H i C F C i K i H i ) r (C i K i K i C i ), (3) where the last step follows fro the independence assuptions and the fact that the autocovariance of x and V i is noralized to I and I i,respectively;thenotationr( ) denotes the trace of a atrix, and the subscript F denotes the usual Frobenius nor of a atrix herefore, the optial linear DES design proble undendividual sensor power constraint can be forulated as follows: iniize E x = I i K i H i C F r (C i K i K i C i ) r (K i K i ) k, C i R, K i R n i, (4) where k is the total rate constraint since the transission power for senso to send K i y i to fusion centes linearly proportional to r(k i K i )

5 Matheatical Probles in Engineering 5 Fro (3), the MSE at the FC can be written as E( x ) = r [(I [ C i K i H i )(I (C i K i K ic i ) ] ] C i K i H i ) Following the fact of atrix derivatives of traces [0], (5) be thought of as being erged into one node are denoted by y In line with this, we can partition the covariance atrix of the entire vectonto four parts, according to R y =( R y 1 R y1 y R y y 1 R y ) (8) Denoting ξ=c 1 K 1 y 1, η=k y, the distortion by diension reduction is E x =E (C 1K 1 y 1 C K y ) =E ξ C η (9) X r (XA) =A, X r (X BX) = BX B X, (6) we can eliinate variables {C i } by iniizing E x x with respect to {C i } As a result, the optiization proble (4) is equivalent to he optial estiation atrix is C =R ξη R 1 η = E [(x C 1 K 1 y 1 )(K y ) ](E(K y )(K y ) ) 1 =(R xy K C 1K 1 R y1 y K )(K R y K ) 1 K y (30) iniize r (I H i K i (K i K i) 1 K i H i ) r (K i K i ) k, K i R n i 1 (7) When K i is a vector, proble (7)isequivalentto iniu su of squares problewhichisnp-coplete[1] herefore, the coputational coplexity of solving proble (0) is NP-hard even in the case with absence of channel distortions for quantization of each sensor ake (30)into(9)as E x = C 1K 1 y 1 (R xy K C 1K 1 R y1 y K ) (K R y K ) 1 K y = ζ C 1K 1 ] Denote (31) Reark he NP-copleteness of optiization proble (0) leads to the intractability of finding the globally optial solution in polynoial tie Instead of finding a globally optial solution, a locally optial solution ay be sufficient in any applications An effective heuristic algorith should be proposed to search for the optial solution of optiization proble (0) An algorith that could be used to search for the optial solution is the Gauss-Seidel type iteration algorith which ayconvergetoalocallyoptialsolutionandwidelyusedin estiation, detection, and classification with sensor networks [ 7] In this paper, a Gauss-Seidel type iteration algorith is proposed to search the optial solution for that proble sensor by sensor Suppose that all sensor nodes except node j have fixed transforation atrix K l, l =j, l = 1,,hegoalis to deterine the optial K j Fro the perspective of a selected node j, suppose that all other nodes have decided on (arbitrary) suitable approxiations of their observations, and thequestionbecoestooptiallychoosetheapproxiation to be provided by terinal j, wherewewithoutlossof generality set j = 1 Observations taken by sensor node 1 are denoted by y 1 he reaining observations which ay ζ=r xy K (K R y K ) 1 K y, ] =y 1 R y1 y K (K R y K ) 1 K y Equation (31) is siplified as (3) E x = ζ C 1K 1 ] (33) Obviously, the optial solution of sensor by sensor optiization proble can be solved because the question has been reducedtothatinthecentralizedcasebasedontheprevious analysis, it is easy to construct a Gauss-Seidel type iteration algorith to search for an optial solution of optiization proble (0) We oit it here 5 Siulations In this section, we ipleent several siulations to show the perforance of our proposed ethod Both centralized and decentralized estiation fraeworks are considered 51 Centralized Estiation Fraework In centralized estiation fraework, entire data is available at a single sensor

6 6 Matheatical Probles in Engineering MSE 15 MSE CRB Centralized estiation (bits) Figure 1: Estiation perforance for centralized estiation with different bandwidth constraints J-0 bits J-5 bits J-30 bits Reduced diension (r) J-di CRB Figure : Coparison of estiation perforance with different reduced diensions and the observation data copression is needed in order to reduce counication requireent Consider a linear odel y=hxε, H R n, (34) where H R n and ε is a white noise with covariance atrix Σ ε = σ I In addition, x and ε are uncorrelated In siulation, we set n = 50, = 10, σ = 1,andΣ x = RR,whereR is drawn fro a standard noral distribution he estiation perforance for centralized estiation fraework with different bandwidth constraints is shown in Figure 1 he botto solid line is the Craer-Rao lower bound(crb)hediensionreductionandquantization lead to the gap between the centralized estiation curve and CRB We plot the estiation perforance for different reduced diensions in Figure hree cases of bandwidth constraint are considered (B = 0, 5, 30)he bottosolidlineis the CRB he blue line with circle is the MSE for the data only with diension reduction When quantization is ipleented after diension reduction, the optial strategy allocates the bandwidth to the ost iportant diension Do not waste the bandwidth on the less iportant diension which would leads to bad perforance he coparison of estiation perforance for different signal-to-noise ratios (SNR) is shown in Figure 4heSNRis defined as SNR = r (HΣ yh ) nσ (35) 5 Decentralized Estiation Fraework In decentralized estiation fraework, the distributed sensors collaborate with a fusion center to jointly estiate the paraeter ssince all sensors have liited battery power, their coputation and counication capability are severely liited As a result, local data copression is needed in order to reduce counication requireent et the 3 distributed sensor nodes ake observations on a coon rando paraeter vector x R according to y i =H i xv i,,,3, (36) where H i R i is the observation atrix and V i R i is the additive noise which is zero ean and spatially uncorrelated In addition, x and V i are uncorrelated In siulation, we set i =15, =10, σ =1,andΣ x =RR,whereR is drawn fro a standard noral distribution he coparison of centralized and decentralized estiation perforance is shown in Figure 3hebottosolidline is the CRB he estiation for centralized and decentralized fraeworkareplottedinreddashlinewithcircleandblue dot line with square for different bandwidth constraints, respectively he decentralized estiation perforance is slightly worse than the centralized estiation since the Gauss-Seidel ethod cannot guarantee the optial solution [] 6 Conclusion In this paper, we have considered a bandwidth constrained sensor network in which a set of distributed sensors and

7 Matheatical Probles in Engineering 7 MSE Reduced diension (r) he coputational coplexity of decentralized estiation proble has been proved to be NP-hard and a Gauss-Seidel type iteration algorith to search for an optial solution has been also proposed Siulation results show the good perforance of the proposed algoriths Conflict of Interests he authors declares that there is no conflict of interests regarding the publication of this paper Acknowledgents hisworkissupportedbythenationalnaturalsciencefoundation of China under Grant no and the National Natural Science Foundation of China-Key Progra under Grant no References J-di-SNR J-di-SNR6 J-30 bits-snr J-30 bits-snr6 Figure 3: Coparison of estiation perforance with different SNR MSE CRB Centralized estiation Decentralized estiation (bits) Figure 4: Coparison of estiation perforance for centralized and decentralized fraework a fusion center collaborate to estiate an unknown vector With given counication bandwidth, the bits allocation proble has been forulated as an optiization proble Both centralized and decentralized estiation fraeworks have been developed he closed-for solution for the centralized estiation fraework has been proposed [1] I F Akyildiz, W Su, Y Sankarasubraania, and E Cayirci, Wireless sensor networks: a survey, Coputer Networks, vol 38,no4,pp393 4,00 [] W-M a and A R Reiban, Design of quantizers for decentralized estiation systes, IEEE ransactions on Counications, vol 41, no 11, pp , 1993 [3] M Mallick, S Coraluppi, and C Carthel, Advances in asynchronous and decentralized estiation, in Proceedings of the IEEE Aerospace Conference, vol 4, pp , IEEE, March 001 [4] Z-Q uo, Universal decentralized estiation in a bandwidth constrained sensor network, IEEE ransactions on Inforation heory,vol51,no6,pp10 19,005 [5] Z-Q uo, An isotropic universal decentralized estiation schee for a bandwidth constrained ad hoc sensor network, IEEE Journal on Selected Areas in Counications, vol3,no 4, pp , 005 [6] CAysalandKEBarner, Constraineddecentralizedestiation over noisy channels for sensor networks, IEEE ransactionsonsignalprocessing, vol 56, no 4, pp , 008 [7] YHua,MNikpour,andPStoica, Optialreduced-rankestiation and filtering, IEEE ransactions on Signal Processing, vol49,no3,pp ,001 [8] YZhu,ESong,JZhou,andZYou, Optialdiensionality reduction of sensor data in ultisensor estiation fusion, IEEE ransactions on Signal Processing, vol53,no5,pp , 005 [9]HZhu,IDSchizas,andGBGiannakis, Power-efficient diensionality reduction for distributed channel-aware kalan tracking using WSNs, IEEE ransactions on Signal Processing, vol 57, no 8, pp , 009 [10] D Williason, Digital Control and Ipleentation, Prentice- Hall, 1991 [11] A Ribeiro, G B Giannakis, and S I Roueliotis, SOI-KF: distributed Kalan filtering with low-cost counications using the sign of innovations, IEEE ransactions on Signal Processing,vol54,no1,pp ,006 [1] K You, Xie, S Sun, and W Xiao, Multiple-level quantized innovation Kalan filter, in Proceedings of the 17th IFAC World Congress, pp , July 008

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