Distributed parameter estimation in wireless sensor networks using fused local observations

Transcription

1 Dstrbuted parameter estmaton n wreless sensor networks usng fused local observatons Mohammad Fanae, Matthew C. Valent, Natala A. Schmd, and Marwan M. Alkhweld Lane Department of Computer Scence and Electrcal Engneerng West Vrgna Unversty, Morgantown, WV, USA ABSTRACT The goal of ths paper s to relably estmate a vector of unknown determnstc parameters assocated wth an underlyng functon at a fuson center of a wreless sensor network based on ts nosy samples made at dstrbuted local sensors. A set of nosy samples of a determnstc functon characterzed by a fnte set of unknown parameters to be estmated s observed by dstrbuted sensors. The parameters to be estmated can be some attrbutes assocated wth the underlyng functon, such as ts heght, ts center, ts varances n dfferent drectons, or even the weghts of ts specfc components over a prened bass set. Each local sensor processes ts observaton and sends ts processed sample to a fuson center through parallel mpared communcaton channels. Two local processng schemes, namely analog and dgtal, are consdered. In the analog local processng scheme, each sensor transmts an amplfed verson of ts local analog nosy observaton to the fuson center, actng lke a relay n a wreless network. In the dgtal local processng scheme, each sensor quantzes ts nosy observaton before transmttng t to the fuson center. A flat-fadng channel model s consdered between the local sensors and fuson center. The fuson center combnes all of the receved locally-processed observatons and estmates the vector of unknown parameters of the underlyng functon. Two dfferent well-known estmaton technques, namely maxmum-lkelhood ML, for both analog and dgtal local processng schemes, and expectaton maxmzaton EM, for dgtal local processng scheme, are consdered at the fuson centre. The performance of the proposed dstrbuted parameter estmaton system s nvestgated through smulaton of practcal scenaros for a sample underlyng functon. Keywords: Dstrbuted parameter estmaton, maxmum lkelhood ML, expectaton maxmzaton EM, analog/dgtal local processng, local quantzaton, ntegrated mean-squared error, flat-fadng channel, fuson center, wreless sensor networks.. INTRODUCTION Wreless sensor networks WSNs are typcally formed by a large number of densely-deployed geographcallydstrbuted sensors wth lmted capabltes that cooperate wth each other to acheve a common goal. One of the most mportant applcatons of such networks s dstrbuted parameter estmaton, whch s the frst step n a wder range of applcatons such as event detecton and classfcaton. In a WSN performng dstrbuted parameter estmaton, dstrbuted local sensors observe the condtons of ther surroundng envronment, process ther local observatons, and send ther processed data to a fuson center, whch then combnes all of the locallyprocessed samples to perform the ultmate global estmaton. Ths paper studes the problem of estmatng a vector of unknown parameters assocated wth a determnstc functon at the fuson center of a WSN from ts dstrbuted nosy samples observed by local sensors and communcated through parallel flat-fadng channels. Two local processng schemes, namely analog and dgtal, wll be consdered. In the analog local processng scheme, each sensor acts as a pure relay and transmts an amplfed verson of ts raw analog local observaton to the fuson center. In the dgtal local processng method, each sensor quantzes ts local nosy observaton and sends ts quantzed sample to the fuson center usng a dgtal modulaton format. A major applcaton of ths work could be n cases, where the estmated parameters can then be used n detecton and classfcaton of the underlyng object that has created the observed nfluence-feld ntensty functon. The man contrbuton of ths paper s a generalzed formulaton of dstrbuted parameter estmaton n the context of wreless sensor E-mal: mfanae@mx.wvu.edu, {matthew.valent and natala.schmd}@mal.wvu.edu, and malkhwel@mx.wvu.edu.

2 networks, where local observatons are not lnearly dependent on the underlyng parameters to be estmated and no specfc observaton model has been consdered n the analyss. Xao et al. have consdered the lnear coherent decentralzed mean-squared error MSE estmaton of an unknown vector under strngent bandwdth and power constrants, where the local observaton models, compresson functons at local sensors, and the fuson rule at the fuson center are all lnear. As a result of the bandwdth constrant, each sensor n ther proposed framework transmts to the fuson center a fxed number of real-valued messages per observaton. The power constrant n ther model lmts the strength of the transmtted sgnals. Based on ther proposed algorthm, each sensor lnearly encodes ts observatons usng a lnear amplfyand-forward codng strategy, and the fuson center apples a lnear mappng to estmate the unknown vector based on the receved messages from local sensors through a coherent multple-access channel MAC assumng a perfect synchronzaton between sensors and the fuson center so that the transmtted messages from local sensors can be coherently combned at the fuson center. It s shown n 2 that f the sensor statstcs are Gaussan,.e., the parameter to be estmated and the ndependent observaton noses are Gaussan, and the communcaton channels between local sensors and the fuson center are standard Gaussan multple-access channels, a smple uncoded analog-and-forward scheme, n whch each sensor s channel nput s merely a scaled verson of ts nosy observaton, drastcally outperforms the separate source-channel codng approach n the sense of mean-squared error. Therefore, the proposed decentralzed estmaton algorthm n can perform optmally n applcatons that satsfy the requrements summarzed n 2. In ths paper, a general non-lnear model for the dstrbuted local observatons s consdered and analyzed. Ishvar et al. 3 have consdered the problem of estmaton wth unrelable communcaton lnks and have derved an nformaton-theoretc achevable rate-dstorton regon characterzng the per-sample sensor bt-rate R versus the estmaton error D. In partcular, they have consdered two cases of fully-dstrbuted estmaton schemes wth no nter-sensor collaboraton and localzed or collaboratve estmaton schemes n whch t s assumed that the network s dvded nto a number of sensor clusters, where collaboraton s allowed among sensors wthn the same cluster but not across clusters. Nowak et al. 4 have studed the theoretcal upper and lower bounds and tradeoffs between the estmaton error and energy consumpton n WSNs as functons of local sensor densty. They have proposed practcal n-network processng approaches based on herarchcal data handlng and multscale parttonng methods for the estmaton of two-dmensonal nhomogeneous felds composed of two or more homogeneous smoothly-varyng regons separated by smooth boundares. Rbero and Gannaks 5 have proposed a dstrbuted bandwdth-constraned maxmum-lkelhood ML estmaton method for estmatng a scaler determnstc parameter n the presence of zero-mean addtve whte Gaussan observaton nose usng only quantzed versons of the orgnal local observatons perfectly avalable at the fuson center. In a sequel work, they have consdered more realstc sgnal models such as known unvarate but generally non-gaussan nose probablty densty functons pdfs, known nose pdfs wth a fnte number of unknown parameters, completely unknown nose pdfs, and practcal generalzatons to vector parameters and multvarate and possbly correlated nose pdfs 6. The observaton model n ths work could n general be a non-lnear functon of the vector parameter to be estmated, but t stll gnores the effects of mperfect wreless channels between local sensors and the fuson center. It s shown n these works that transmttng a few bts or even a sngle bt per sensor can approach the performance of the estmator based on unquantzed data under realstc condtons. In ths paper, a general observaton model wll be consdered n whch local observatons are not necessarly lnear functons of the vector of parameters to be estmated. Furthermore, mpared flat-fadng channels between local sensors and the fuson center wll be consdered n the analyss. Nu and Varshney 7 have proposed a sgnal ntensty- or energy- based ML target locaton estmaton scheme to estmate the coordnates of an energy emttng source usng quantzed local observatons, whch are assumed to be perfectly receved by the fuson center. They have consdered an sotropc sgnal ntensty attenuaton model n whch the ntensty of the sgnal s assumed to be nversely proportonal to the square of the dstance from ts source. An optmal nfeasble as well as two heurstc practcal desgn methods for optmal local quantzaton thresholds are proposed by them that mnmze the summaton of estmaton error varances for the target s two coordnates. In a sequel work, Ozdemr et al. 8 have added the effects of mperfect fadng and nosy wreless communcaton channels between local sensors and the fuson center to ther problem formulaton and have developed smlar results. Maşazade et al. 9 have consdered a smlar problem n whch the quantzed verson

3 of sensor measurements are transmtted to the fuson center over error-free channels. They have proposed an energy-effcent teratve source localzaton scheme n whch the algorthm begns wth a coarse locaton estmate obtaned from measurement data from a set of anchor sensors. Based on the accumulated nformaton at each teraton, the posteror pdf of the source locaton s approxmated usng an mportance-samplng-based Monte Carlo method, whch s then used to actvate a number of non-anchor sensors selected to mprove the accuracy of the source locaton estmate the most. Dstrbuted compresson of measurement data pror to transmsson s also employed at the non-anchor sensors to further reduce the energy consumpton. It s shown that ths teratve scheme reduces the communcaton requrements by selectng only the most nformatve sensors and compressng ther data pror to transmsson to the fuson center. An teratve non-lnear least-square receved sgnal strength RSS-based locaton estmaton technque s also proposed n 0 for jont estmaton of unknown locaton coordnates and dstance-power gradent, whch s a parameter of rado propagaton path-loss model. Salman et al. have proposed a low-complexty verson of ths approach. Although the llustratve case study for our numercal smulaton results n ths paper consders estmatng the locaton of the center of a generc Gaussan functon, among ts other parameters, ts scope s not lmted to estmatng only the locaton. In other words, ths paper consders a more general framework than the ones consdered n the aforementoned works and s not based on the sgnal propagaton model n the observaton envronment. The rest of ths paper s organzed as follows: Secton 2 descrbes the model of the dstrbuted parallel fuson WSN that wll be consdered n our analyss and nes the precse problem that we are consderng n ths paper. In Secton 3, the ML estmate of a vector of unknown parameters assocated wth a determnstc twodmensonal functon s derved for the case of analog local processng scheme. Secton 4 consders the same problem for the case of dgtal sgnal processng method. As t s shown n ths secton, the ML estmate n ths case s nfeasble to be found effcently n a dstrbuted manner. Therefore, a lnearzed verson of the expectaton maxmzaton EM algorthm wll be proposed n Secton 5 to numercally fnd the ML estmate of the vector of unknown parameters n the case of dgtal local processng scheme. Secton 6 presents the numercal results of our smulatons to study the performance of the proposed dstrbuted estmaton framework for a specal twodmensonal Gaussan-shaped functon, whose assocated parameters to be estmated are ts heght and center. The effects of dfferent parameters of the WSN on the performance of the proposed system wll be studed n ths secton. Fnally, n Secton 7 we conclude our dscussons and summarze the man achevements of ths work. Notatons. Throughout ths paper, the followng notatons are used. Boldface upper-case letters denote matrces, boldface lower-case letters denote column vectors, and standard lower-case letters denote scalars. The superscrpts T and denote the matrx transpose and matrx nverse operators, respectvely. The operator dag manpulates the dagonal elements of a matrx or puts a column vector nto a dagonal matrx. s the absolute value of a complex number or the determnant of a square matrx. 2. SYSTEM MODEL AND PROBLEM STATEMENT Consder a set of K spatally-dstrbuted sensors formng a WSN as shown n Fg.. Suppose that the sensors are located wthn the doman of a two-dmensonal functon gx, y that s completely known except for a set of unknown determnstc parameters denoted by θ θ, θ 2,..., θ p T. The ultmate goal of the underlyng WSN s to relably estmate the vector of unknown parameters θ usng dstrbuted nosy samples of functon gx, y provded by local sensors to a fuson center through parallel mpared flat-fadng channels. Assume that the th sensor,, 2,..., K, observes a nosy verson of a sample of functon gx, y at ts locaton as r gx, y + w, where r s the local sensor s nosy observaton, x, y s the locaton of the th sensor n the network, g gx, y s the sample of functon gx, y at locaton x, y, and w s zero-mean spatally-uncorrelated addtve whte Gaussan nose wth varance σ 2 O,.e. w N 0, σ 2 O. Sensors can be placed on a unform lattce or dstrbuted at random over the observaton area covered by the WSN. Throughout our dscussons n ths paper, we assume that the locaton of dstrbuted sensors s known at the fuson center.

4 ..... H M- x K u Channel K Sensor K K PY K U K y K h n u r z Sensor h 2 n 2 gx,y:θ r 2 u 2 Sensor h K n K z 2... Fuson Center θ ML r K Sensor K u K z K Fgure : System model of a typcal WSN for dstrbuted estmaton of a determnstc parameter vector θ. Note that the observaton model ntroduced n Equaton covers a number of varatons n dfferent applcatons. For example, t s a generalzaton of the well-studed lnear observaton model n whch r a T θ + w, where a a, a 2,..., a p T s the local observaton gan vector for sensor consdered for example by Rbero and Gannaks 5. Furthermore, t s also a generalzed verson of the observaton model consdered by Nu and Varshney 7 and Ozdemr et al. 8 Each sensor processes ts local observaton based on ts local processng rule γ and sends ts result, denoted by u γ r, to the fuson center through an mpared communcaton channel. In ths paper, two local processng schemes, namely analog and dgtal, are consdered. In the analog local processng scheme, each sensor acts as a pure relay and transmts an amplfed verson of ts raw analog local observaton to the fuson center. In the dgtal local processng method, each sensor quantzes ts local observaton and sends ts quantzed sample to the fuson center usng a dgtal modulaton format. We wll formulate detaled analyss of the local analog processng scheme n Secton 3 and that of the local dgtal processng scheme n Sectons 4 and 5. Suppose that locally-processed sample s transmtted to the fuson center over parallel ndependent flat fadng channels. In other words, assume that the receved sgnal from sensor at the fuson center s z h u + n,, 2,..., K, 2 where h s the fadng gan of the channel between sensor and the fuson center and n s zero-mean spatallyuncorrelated addtve whte Gaussan nose wth varance σ 2 C,.e. n N 0, σ 2 C. It s assumed that channel fadng gans are uncorrelated and completely known at the fuson center. Upon recevng the vector of locally-processed observatons from dstrbuted sensors z z, z 2,..., z K T, the fuson center combnes them to estmate the vector of unknown determnstc parameters θ. In the followng three sectons, we wll formulate the ML estmaton of θ at the fuson center for the two cases of analog and dgtal local processng schemes and the EM algorthm to numercally fnd the ML estmate of θ at the fuson center for the dgtal local processng scheme. 3. ML ESTIMATION OF θ: ANALOG LOCAL PROCESSING CASE Suppose that each sensor acts as a pure relay and transmts an amplfed verson of ts local raw analog observaton to the fuson center. In other words, u α r,, 2,..., K, 3 where α s the local amplfcaton gan at sensor, assumed to be known at the fuson center. Note that n ths paper, we do not try to optmze the local amplfcaton gans as t s consdered by Banavar et al. 2 for the problem of dstrbuted scalar random parameter estmaton. The receved sgnal from sensor at the fuson center can then be represented as z h α r + n h α g + h α w + n,, 2,..., K. 4

5 Assumng complete knowledge of the local observaton amplfcaton gans and channel fadng gans, the fuson center can make a lnear transformaton on the receved sgnal from each local sensor to fnd z h α z g + w + h α n g + v, 2,..., K, 5 where v w + h α n s a zero-mean whte Gaussan random varable wth varance σ 2 σo 2 + σ2 C. The h α 2 estmaton process at the fuson center can then be performed based on vector z z, z 2,..., z K T. The probablty densty functon of the lnearly-processed receved vector of local observatons from dstrbuted sensors at the fuson center s gven by f z z : θ 2πK Σ exp 2 z g T Σ z g, 6 where Σ dag σ, 2 σ2, 2..., σk 2 s the dagonal matrx of cumulatve nose varances and g g, g 2,..., g K T s the vector of samples of functon gx, y at sensor locatons. The jont log-lkelhood functon of the vector of unknown parameters θ can then be found as l θ ln f z z : θ ln Σ + z g T Σ z g a z g T Σ z g K z g 2, 7 σ 2 where a s based on the fact that Σ s ndependent of θ. The maxmum-lkelhood estmate of the vector of unknown parameters θ s the maxmzer of ts log-lkelhood functon gven n Equaton 7. In other words, θ ML arg max θ arg mn θ arg mn θ l θ z g T Σ z g K σ 2 z g 2. 8 Based on Equaton 8, the ML estmate of the vector of unknown parameters θ ML can be found as the soluton of the followng system of equatons: θ l θ θml 0, 9 T where θ θ, θ 2,..., θ p s the gradent operator wth respect to θ. Ths system of equatons can be smplfed n the vector form by substtutng l θ from Equaton 7 as g θ Σ z g 0, 0

6 where g θ s a p K matrx of partal dervatves of the components of vector g wth respect to the components of θ ned as g g 2 g θ θ K θ g g g 2 g θ 2 θ 2 K θ 2 θ g g 2 θ p θ p The system of equatons derved n Equaton 0 can be rewrtten n the scalar format as K σ 2 g θ j z g g K θ p 0, j, 2,..., p. 2 Ths system of equatons s hghly non-lnear n unknown parameters for most practcal applcatons and does not have a closed-form soluton. In Secton 6, ths system wll be solved usng numercal methods to estmate a vector of unknown parameters assocated wth a specfc two-dmensonal Gaussan-shaped functon of nterest gx, y. 4. ML ESTIMATION OF θ: DIGITAL LOCAL PROCESSING CASE Suppose that sensor quantzes ts local observaton r to b log 2 M bts and sends the ndex of ts quantzed mult-bt sample to the fuson center, where M s the number of quantzaton levels for sensor. Let L {β 0, β,..., β M } be the set of quantzaton thresholds for sensor, where β l s the lth quantzaton threshold of the th sensor, β 0, and β M,, 2,..., K. The local processng rule of sensor s then ned as a functon γ : R {0,,..., M }, whose values are determned as u l β l r < β l +, l 0,,..., M, and, 2,..., K. 3 Based on the local observaton model ntroduced n Equaton, the probablty densty functon of each sensor s quantzed output sample can be found as f U u : θ M l0 Q l δ u l, u 0,,..., M, and, 2,..., K, 4 where δ denotes the dscrete Drac delta functon, Q l s ned as β l g β l + g Q l Q Q, l 0,,..., M, and, 2,..., K, 5 σ O σ O and Q s the complementary dstrbuton functon of the standard Gaussan random varable ned as Qx exp 2 2π t2 dt. 6 x Lke n the case of analog local processng, assumng complete knowledge of the channel fadng gans, the fuson center can make a lnear transformaton on the receved sgnal from each local sensor to fnd z h z u + h n u + v, 2,..., K, 7

7 where v h n s a zero-mean Gaussan random varable wth varance σ 2 σ2 C h. The estmaton process at 2 the fuson center can then be performed based on vector z z, z 2,..., z K T. The probablty densty functon of the lnearly-processed receved vector of local observatons from dstrbuted sensors at the fuson center s gven by f z z : θ a b c K f Z z : θ K M u 0 K M u 0 K M l0 f Z U z u f U u : θ { exp 2πσ 2 M u 0 z u 2 2σ 2 exp z u 2 2πσ 2 l0 2σ 2 } M l0 Q l δ u l Q l δ u l K M Q 2πσ 2 l exp z l2 2σ 2, 8 where a s based on the fact that the receved local observatons from dstrbuted sensors are spatally uncorrelated, the summaton n b s based on the theorem of total probablty and over all M possble realzatons of each u, and c s based on the fact that the nner summaton over u can be smplfed usng the propertes of dscrete Drac delta functon as M u 0 exp z u 2 2σ 2 Q l δ u l exp z l2 2σ 2 Q l. 9 Therefore, the jont log-lkelhood functon of the vector of unknown parameters θ can be found as l θ ln f z z : θ K M ln Q l exp z l2 2σ l0 The maxmum-lkelhood estmate of the vector of unknown parameters θ s the maxmzer of ts log-lkelhood functon gven n Equaton 20. In other words, θ ML arg max θ Therefore, θ ML can be found as the soluton of the followng system of equatons: l θ. 2 θ l θ θml 0, 22 whch can be smplfed by the substtuton of l θ from Equaton 20 as M K l0 A,j l exp z l2 2σ 2 0, j, 2,..., p, 23 M l0 Q l exp z l2 2σ 2

8 where A,j l s the partal dervatve of Q l wth respect to θ j ned as A,j l { Q l} θ j g 2πσO 2 θ j exp g2 2σO 2 B l, 24 where B l s ned as B l 2g β l β 2 exp 2g β l + β 2 2σO 2 exp 2σO 2 l Ths system of equatons gven n Equaton 23 s hghly non-lnear n unknown parameters for most practcal applcatons and does not have a closed-form soluton. In practce, an effcent numercal approach has to be devsed to fnd the ML estmate formulated n ths secton. In the next secton, the expectaton-maxmzaton algorthm 3 wll be developed as an effcent teratve approach to numercally fnd the ML estmate of a vector of unknown parameters for the case of dgtal local processng scheme. 5. LINEARIZED EM SOLUTION: DIGITAL LOCAL PROCESSING CASE Based on the expectaton-maxmzaton algorthm ntroduced by Dempster et al. 3, we consder lnearly-processed receved vector of local observatons from dstrbuted sensors at the fuson center z ned n Equaton 7 as ncomplete data set, and the par of the vector of local observatons r r, r 2,..., r K T ned n Equaton and the vector of lnearly-transformed channel addtve whte Gaussan nose varables v v, v 2,..., v K T,.e. {r, v}, as complete data set. The mappng z u + v relates the ncomplete and complete data spaces, where u u, u 2,..., u K T s the vector of locally-quantzed observatons of dstrbuted sensors based on u γ r as the known local quantzaton rule of sensor ned n Equaton 3. The jont probablty densty functon of the complete data set parametrzed by the vector of unknown parameters θ s found as f CD r, v : θ a f r r : θ f v v 2πK Σ O exp 2 r gt Σ O r g 2πK Σ C exp 2 vt Σ C v 26 where a s based on the fact that r and v are ndependent Gaussan random vectors, Σ O dag σo 2, σo 2 2,..., σo 2 K and ΣC dag σ, 2 σ2, 2..., σk 2 are the dagonal matrces of observaton nose varances and transformed channel nose varances σ 2 complete data set s ned as σ2 C h, respectvely. The jont log-lkelhood functon of the 2 l CD r, v : θ ln f CD r, v : θ r g T Σ O r g K σo 2 r g 2, 27 where the terms ndependent of the vector of unknown parameters θ are omtted. k Let θ be the estmate of the unknown vector of parameters at the kth teraton of the expectatonmaxmzaton algorthm. To further refne and update the estmates of unknown parameters, we alternate the expectaton and maxmzaton steps ned as follows:

9 Expectaton Step E-Step: Durng the expectaton step, the condtonal expectaton of the jont loglkelhood functon of the complete data set, gven the ncomplete data set and θ k, s found as k F θ θ E l CD r, v : θ z k, θ K E r g 2 z k, θ, 28 σ 2 O where E denotes the expectaton operaton wth respect to the condtonal pdf of the complete data set, gven the ncomplete data set and the estmate of the vector of parameters at the kth teraton as f CD ID r, v : θ k z f r z r : θ k z f v z v : θ k z, 29 where r and v are two ndependent Gaussan random vectors, f v z nsde the expectaton operaton, and based on the Bayes rule, v : θ k z s ndependent of the argument f r z r : θ k z z : θ k k r f r r : θ. 30 k z : θ f z r f z Based on the theorem of total probablty, f z z : θ k Note that f z } {{ } K f z r z : θ k r f r r : θ k dr. 3 z : θ k s ndependent of the argument nsde the expectaton operaton n Equaton 28. Maxmzaton Step M-Step: Durng the maxmzaton step, the next estmate of the vector of unknown parameters s found as the maxmzer of the result of the expectaton step. In other words, θ k+ arg max F θ θ k, 32 θ whch can be rewrtten as θ j { F θ θ k} θk+ Ths system of equatons can be specfed more precsely by substtuton of F 0, j, 2,..., p. 33 θ θ k from Equaton 28 as E K σ 2 O r g g θ j z, k θ θk+ 0, j, 2,..., p. 34 The above condtonal expectaton can be found usng the probablty densty functons ned n Equa-

10 tons 29 3 as K E r g g θ j z, σ 2 O k θ a K K σo 2 E r g g z k θ, θ j σ 2 O K σ 2 O K σ 2 O r g g f CD ID r, v : θ θ k z j r g g θ j f Z z f R Z r : θ k z dr dr dv k T k,j z, 35 : θ where a s based on Equaton 29 and the ndependence of v from the argument of the expectaton operaton and T k,j z s ned as T k,j z r g g f Z z θ R : θ k r f R r : θ k dr. 36 j Note that based on Equaton 7, the condtonal pdf of Z, gven R, s Gaussan wth mean u and varance σ 2 g k σ2 C h,.e. Z 2 R N u, σ 2. Moreover, based on Equaton, the pdf of R s Gaussan wth mean and varance σo 2,.e. R N g k, σo 2, where g k g x, y : θ k s the estmate of the underlyng functon gx, y at locaton x, y wth the vector of unknown parameters θ replaced by ts estmate at the kth teraton of the EM algorthm. Usng these two probablty densty functons, T k,j z can be found as T k,j z r g g 2 exp z u 2 r g k θ j 2πσ 2 2σ 2 exp 2πσ 2 2σO 2 dr O M βl+ r g g 2 exp z l2 r g k 2π σ 2 σ2 O l0 β l θ j 2σ 2 exp 2σO 2 dr M exp z l2 g 2πσ 2 2σ 2 Γ k l, θ j 37 where Γ k l s ned as where Q k l0 Γ k l l Q l gg k Λ k l exp βl+ r g exp 2πσO 2 β l 2 σo 2 2π exp g k 2σO 2 Λ k l + and Λ k l s ned as 2g k β l β 2 l 2g k 2σO 2 exp r g k 2σ 2 O 2 dr g k g Q k l, 38 β l + β 2 l + 2σO 2. 39

11 Combng Equatons 35 and and replacng them n Equaton 34 wll result n a non-lnear system of equatons n terms of θ k+. In other words, Equaton 34 can be rewrtten as follows to gve a new update of the vector of unknown parameters at the k + th step of the EM algorthm usng ts values derved n the kth step: K σ 2 O 2πσ 2 f Z z k : θ g k+ θ j M k z g k+ g k+ N k z θ j 0, j, 2,..., p, 40 where M k z and N k M k z N k z z are ned as M l0 M l0 exp z l2 2σ 2 Q k l exp σ 2 O z l2 2σ 2 2π exp g k 2σ 2 O 2 Λ k l + g k Q k l 4a. 4b It can easly been seen that the system of equatons gven n Equaton 40 s hghly non-lnear wth respect to the components of the vector of unknown parameters to be estmated θ k+. Furthermore, the soluton set for ths system of equatons s non-convex. Therefore, a numercal soluton should be found for ths system of equatons at each teraton of the EM algorthm. In Secton 6, ths approach wll be utlzed to numercally fnd the ML estmate of a vector of unknown parameters assocated wth a specfc two-dmensonal Gaussan-shaped functon of nterest gx, y. Newton s method s appled to lnearze the system of equatons gven n 40 as brefly descrbed n the followng. Let f θ k+ 0 be the vector form of the system of equatons gven n 40, where each component of the vector f s a non-lnear functon of the components of the vector of unknown parameters θ k+. At each step n of the Newton s lnearzaton method, the followng system of lnear equatons s solved to fnd a new update for θ k+ n+ based on the prevous estmate of θ k+ n : where J θ k+ n J θ k+ n θ k+ n+ θ k+ n f θ k+ n, n 0,, 2,..., 42 s the p p Jacoban matrx of vector functon f evaluated at θ k+ n. The element at the rth row and cth column of the Jacoban matrx J evaluated at θ k+ n J r,c θ k+ n f r θ θ c θθ k+ n where f r s the rth component of the vector functon f. s ned as, r, c, 2,..., p, CASE STUDY AND NUMERICAL ANALYSIS In ths secton, we wll numercally evaluate the performance of the ML estmaton framework developed n prevous sectons for both analog and dgtal local processng schemes for a specfc functon of nterest gx, y.

12 6. Smulaton Setup, Parameter Specfcaton, and Performance Measure Defnton Suppose that a two-dmensonal Gaussan-shaped functon ned as gx, y h exp x xc 2 2 σx 2 + y y c 2 σy 2 s beng sampled by the WSN under study, where h s the maxmum ntensty or heght of the functon, x c, y c s the locaton of ts center, and σ 2 x and σ 2 y are known spread of the functon n the x and y drectons, respectvely. In our smulatons, we fx these values to be σx 2 4 and σy 2. Let θ h, x c, y c T be the vector of unknown determnstc parameters assocated wth functon gx, y. The goal s to estmate these p 3 determnstc parameters usng dstrbuted possbly sparse nosy samples of the functon acqured by a WSN and transmtted to a fuson center through mpared parallel flat-fadng channels. Suppose that K sensors are randomly dstrbuted n the observaton envronment, whch s assumed to at least cover the area A x c 3σ x, x c + 3σ x y c 3σ y, y c + 3σ y, where X Y denotes the Cartesan product between sets X and Y. Ths choce of the observaton envronment guarantees that all of the sensors are wthn the doman of the underlyng functon gx, y, and that almost all of the doman of ths functon s covered by dstrbuted sensors. In our smulatons, we assume that the observatons made by dstrbuted sensors are homogeneous. In other words, t s assumed that the varances of addtve whte Gaussan observaton noses for all sensors are the same,.e. In our smulatons, the observaton sgnal-to-nose rato SNR s ned as σ 2 O 44 σo 2 σo 2 2 σo 2 K. 45 SNR O 2σ 2 O. 46 Note that ths nton of SNR O s the SNR at a reference dstance from the center of the functon x c, y c at whch the strength of the functon samples s unt. As t was mentoned n Secton 2, there are two man classes of local processng schemes, namely analog and dgtal. Both of these schemes are consdered n our smulatons. For the analog local processng scheme, t s assumed that the observaton amplfcaton gan s absorbed n the channel fadng gan and normalzed to one as descrbed n the followng paragraph. For the dgtal local processng scheme, t s assumed that the quantzaton rule at all of the sensors s the same. Therefore, all sensors quantze ther local analog observatons to b b b 2 b K bts and use the same number of quantzaton levels M M M 2 M K. Each sensor uses a determnstc unform scalar quantzer, whose set of quantzaton thresholds s known and s the same for all sensors. In other words, βl β l,, and l 0,,..., M. In our smulatons, the set of quantzaton thresholds for all sensors s chosen as βl lh M, l, 2,..., M. The parallel ndependent channels between dstrbuted sensors and the fuson center are assumed to be flat Raylegh fadng channels. The observaton and channel SNRs and the channel fadng gans are normalzed to ensure that E h 2 or more generally, E α h 2, when applcable,, 2,..., K. When the sensors are located close to each other and the fuson center s far away from them, the dstance between the sensors and the fuson center s approxmately the same for all sensors, and ths assumpton s vald. In our smulatons, t s assumed that the parallel channels between dstrbuted sensors and the fuson center have homogeneous noses. In other words, t s assumed that the varances of addtve whte Gaussan channel noses n the channels between all sensors and the fuson center are the same,.e. In our smulatons, the channel sgnal-to-nose rato SNR s ned as σ 2 C σc 2 σc 2 2 σc 2 K. 47 SNR C 2σ 2 C. 48

13 Agan, ths nton of SNR C s the SNR at a reference dstance from each sensor at whch the strength of the faded transmtted sgnal s unt. For the analog local processng scheme, the ML estmate of the vector of unknown parameters θ s found based on the system of non-lnear equatons gven n Equaton 2. Ths system of equatons s lnearzed and solved teratvely usng Newton s method as brefly descrbed at the end of Secton 5. For the dgtal local processng scheme, the lnearzed EM algorthm gven n Equatons 40 4 s used as an teratve, effcent approach to numercally fnd the ML estmate of θ. To solve ths system of non-lnear equatons, Newton s lnearzaton method s appled, as brefly descrbed at the end of Secton 5, based on Equaton 42. We have chosen the estmaton ntegrated mean-squared error IMSE ned as IMSE MSE x, y dxdy 49 A as the performance measure to evaluate the ML estmaton technques developed n ths paper, where A s the two-dmensonal observaton envronment and MSE x, y s the locaton-dependent estmaton mean-squared error at locaton x, y A ned as MSE x, y E h,w,n gx, y ĝx, y 2, 50 where E denotes the expectaton wth respect to observaton nose, channel nose, and channel fadng gan realzatons, gx, y s the sample of the underlyng functon at locaton x, y, and ĝx, y s ts estmated value based on the estmate of the vector of unknown parameters. In our smulatons, the result of MSE x, y s also averaged wth respect to the random locaton of dstrbuted sensors n the observaton envronment. 6.2 Effects of K, SNR O, and SNR C on Dstrbuted Parameter Estmaton Performance The performance of the proposed dstrbuted parameter estmaton framework s a functon of, n part, the number of dstrbuted sensors n the observaton envronment, observaton SNR, and channel SNR. In Fg. 2, the ntegrated mean-squared error at the fuson center versus the number of dstrbuted sensors n the observaton envronment K s shown for dfferent values of observaton SNR SNR O and channel SNR SNR C. Fgure 2a shows ths performance measure for applyng ML estmaton technque at the fuson center, when the analog local processng scheme s used. Fgure 2b shows the IMSE as the estmaton performance measure for applyng the lnearzed EM estmaton algorthm at the fuson center, when the dgtal local processng scheme s used wth M 8 quantzaton levels. As t can be seen n ths fgure, as the number of local sensors n the observaton envronment ncreases, the IMSE decreases monotoncally. Ths concluson s vald for both analog and dgtal local processng schemes. Furthermore, n can be observed from Fg. 2 that the performance mprovement due to the ncrease n the densty of the local sensors n the observaton envronment s more consderable when there are small number of sensors. As the number of sensors ncreases, the percentage of ths performance mprovement decreases, and the dstrbuted parameter estmaton system acheves an acceptable performance n terms of the IMSE at a moderate number of local sensors n the observaton envronment. Fgure 3 shows the ntegrated mean-squared error at the fuson center versus the observaton SNR SNR O for dfferent values of the number of dstrbuted sensors n the observaton envronment K and channel SNR SNR C. Fgure 3a shows the IMSE as the performance measure for applyng ML estmaton technque at the fuson center, when the analog local processng scheme s used. Fgure 3b shows the IMSE as the estmaton performance measure for applyng the lnearzed EM estmaton algorthm at the fuson center, when the dgtal local processng scheme s used wth M 8 quantzaton levels. Smlar to detaled dscussons provded n analyzng Fg. 2, as the observaton SNR ncreases, the IMSE decreases monotoncally for both cases of analog and dgtal local processng schemes. Furthermore, t can be observed from Fg. 3 that the performance mprovement due to the ncrease n the observaton SNR s more consderable at low SNRs. As the observaton SNR ncreases, the percentage of ths performance mprovement decreases, and the dstrbuted parameter estmaton system acheves an acceptable performance n terms of the IMSE at a moderate observaton SNR. Smlar results and dscussons could have been provded for analyzng the effects of channel SNR on the performance of the dstrbuted parameter estmaton framework, whch s omtted because of the lmted space.

14 One of the most mportant ponts to be notced n Fgs. 2 and 3 s that for the same values of K, SNR O, and SNR C, the ML estmaton at the fuson center based on the analog local processng scheme outperforms the lnearzed EM estmaton at the fuson center based on the dgtal local processng schemes wth M 8 quantzaton levels. Ths observaton s expected for two man reasons. Frst, the lnearzed EM algorthm s an effcent teratve algorthm for numerc calculaton of the ML estmate and therefore, t shows a degraded performance compared to the exact ML estmate. Second, by performng dgtal sgnal processng and quantzng ther nosy observaton, local sensors are ntroducng some form of quantzaton nose n the processed samples to become avalable at the fuson center. Therefore, the results of estmaton at the fuson center based on the receved analog samples s more relable than those based on the quantzed versons of local samples. 6.3 Effects of M on Dstrbuted Lnearzed EM Parameter Estmaton Performance Besdes K, SNR O, and SNR C, one of the major parameters that affects the performance of the lnearzed EM estmaton at the fuson center, when the dgtal local processng scheme s used, s the number of quantzaton levels at local sensors,.e. M. Fgure 4 shows the ntegrated mean-squared error at the fuson center versus the number of quantzaton levels at local sensors M for dfferent values of the number of dstrbuted sensors r IMSE ted Mean Sq quared Error Integrat SNR C 0 0dB and dsnr O 0 0dB SNR C 5 db and SNR O 0 db SNR C 0 db and SNR O 5 db SNR C 5 db and SNR O 5 db K Number of Sensors a ML estmaton at the fuson center based on analog local processng. red Error IM MSE Integrated Mean Squar SNR C 0 db and SNR O 0 db SNR C 5 db and SNR O 0 db SNR C 0 db and SNR O 5 db SNR C 5 db and SNR O 5 db K Number of Sensors b Lnearzed EM estmaton at the fuson center based on dgtal local processng wth M 8 quantzaton levels. Fgure 2: Integrated mean-squared error IMSE versus the number of dstrbuted sensors n the observaton envronment K for dfferent values of observaton SNR SNR O and channel SNR SNR C.

15 IMSE Integrated Mean Squ uared Error K50 and SNR C 0 db K50 and SNR C 5 db K00 and SNR C 0 db K00 and SNR C 5 5dB Observaton SNR SNR O a ML estmaton at the fuson center based on analog local processng. r IMSE ated Mean Sq quared Error Integra K50 and SNR C 0 0dB K50 and SNR C 5 db K00 and SNR C 0 db K00 and SNR C 5 db Observaton SNR SNR O b Lnearzed EM estmaton at the fuson center based on dgtal local processng wth M 8 quantzaton levels. Fgure 3: Integrated mean-squared error IMSE versus observaton SNR SNR O for dfferent values of the number of dstrbuted sensors n the observaton envronment K and channel SNR SNR C. n the observaton envronment K, observaton SNR SNR O, and channel SNR SNR C, when the lnearzed EM estmaton algorthm s appled at the fuson center and the dgtal local processng scheme s used at local sensors. As t can be seen n ths fgure, as the number of quantzaton levels at local sensors ncreases, the IMSE decreases monotoncally. Agan, t can be observed from Fg. 4 that the performance mprovement due to the ncrease n the umber of quantzaton levels at local sensors s more consderable for small values of M. As the umber of quantzaton levels at local sensors ncreases, the percentage of ths performance mprovement decreases, and the dstrbuted parameter estmaton system acheves an acceptable performance n terms of the IMSE at a reasonably low number of quantzaton levels. It s worth mentonng that Fg. 4 shows that even M 4 quantzaton levels at local sensors can acheve a very good performance n terms of the IMSE. In other words, even f the local sensors quantze ther nosy observatons just to two bts, the system shows an acceptable performance. Ths concluson emphaszes on the energy effcency of the proposed parameter estmaton framework that could lead to a hgher lft-tme of dstrbuted sensors n the network. In other words, local sensors do not need to waste a lot of energy to send very hgh-resoluton observatons quantzed to a large number of quantzaton levels to acheve an acceptable estmaton error performance at the fuson center n terms of the IMSE.

16 r IMSE Integra ated Mean Square Error K50 50, SNR C SNR O 0 db K50, SNR C SNR O 5 db K00,SNR C SNR O 5 db M Number of Quantzaton Levels Fgure 4: Integrated mean-squared error IMSE versus the number of quantzaton levels at local sensors for lnearzed EM estmaton at the fuson center based on dgtal local processng for dfferent values of the number of dstrbuted sensors n the observaton envronment K, observaton SNR SNR O, and channel SNR SNR C. 7. CONCLUSIONS In ths paper, the problem of dstrbuted estmaton of a vector of unknown determnstc parameters assocated wth a two-dmensonal functon was consdered n the context of WSNs. Each sensor observes a sample of the underlyng functon at ts locaton corrupted by an addtve whte Gaussan observaton nose, whose samples are spatally uncorrelated across sensors. After local processng, each sensor transmts ts locally-processed sample to the fuson center of WSN through parallel flat-fadng channels. Two local processng schemes were consdered, namely analog and dgtal. In the analog local processng scheme, each sensor transmts an amplfed verson of ts local analog nosy observaton to the fuson center, actng lke a relay n a wreless network. In the dgtal local processng scheme, each sensor quantzes ts nosy observaton usng a determnstc unform scaler quantzer before transmttng ts dgtally-modulated verson to the fuson center. The ML estmate of the vector of unknown parameters at the fuson center was derved for both analog and dgtal local processng schemes. Snce the ML estmate for the case of dgtal local processng scheme was too complcated to effcently be mplemented, an effcent teratve EM algorthm was proposed to numercally fnd the ML estmate n ths case. Numercal smulaton results were provded to evaluate the performance of the proposed dstrbuted parameter estmaton framework n a typcal WSN applcaton scenaro. As shown n the results of these smulaton, the proposed dstrbuted estmaton framework can acheve a very good performance n terms of the ntegrated mean-squared error for reasonable values of the parameters of the system ncludng the number of dstrbuted local sensors n the observaton envronment, observaton SNR, channel SNR, and the number of quantzaton levels for dgtal local processng scheme. In partcular, numercal performance analyss showed that even wth a low number of quantzaton levels at dstrbuted sensors,.e. hgh energy-effcency, the estmaton framework could provde a very good performance n terms of the ntegrated mean-squared error. REFERENCES. J.-J. Xao, S. Cu, Z.-Q. Luo, and A. J. Goldsmth, Lnear coherent decentralzed estmaton, IEEE Transactons on Sgnal Processng 56, pp , February M. Gastpar, Uncoded transmsson s exactly optmal for a smple gaussan sensor network, IEEE Transactons on Informaton Theory 54, pp , November P. Ishwar, R. Pur, K. Ramchandran, and S. Pradhan, On rate-constraned dstrbuted estmaton n unrelable sensor networks, IEEE Journal on Selected Areas n Communcatons 23, pp , Aprl R. Nowak, U. Mtra, and R. Wllett, Estmatng nhomogeneous felds usng wreless sensor networks, IEEE Journal on Selected Areas n Communcatons 22, pp , August 2004.

17 5. A. Rbero and G. B. Gannaks, Bandwdth-constraned dstrbuted estmaton for wreless sensor networks part I: Gaussan case, IEEE Transactons on Sgnal Processng 54, pp. 3 43, March A. Rbero and G. B. Gannaks, Bandwdth-constraned dstrbuted estmaton for wreless sensor networks part II: unknown probablty densty functon, IEEE Transactons on Sgnal Processng 54, pp , July R. Nu and P. K. Varshney, Target locaton estmaton n sensor networks wth quantzed data, IEEE Transactons on Sgnal Processng 54, pp , December O. Ozdemr, R. Nu, and P. K. Varshney, Channel aware target localzaton wth quantzed data n wreless sensor networks, IEEE Transactons on Sgnal Processng 57, pp , March E. Maşazade, R. Nu, P. K. Varshney, and M. Kesknoz, Energy aware teratve source localzaton for wreless sensor networks, IEEE Transactons on Sgnal Processng 58, pp , September X. L, RSS-based locaton estmaton wth unknown path-loss model, IEEE Transactons on Wreless Communcatons 5, pp , December N. Salman, M. Ghogho, and A. H. Kemp, On the jont estmaton of the RSS-based locaton and path-loss exponent, IEEE Wreless Communcatons Letters, pp , February M. K. Banavar, C. Tepedelenloglu, and A. Spanas, Estmaton over fadng channels wth lmted feedback usng dstrbuted sensng, IEEE Transactons on Sgnal Processng 58, pp , January A. P. Dempster, N. M. Lard, and D. B. Rubn, Maxmum lkelhood from ncomplete data va the EM algorthm, Journal of the Royal Statstcal Socety, Seres B Methodologcal 39, pp. 38, December 977.