Progress In Electromagnetcs Research M, Vol. 70, 135 143, 2018 An Alternaton Dffuson LMS Estmaton Strategy over Wreless Sensor Network Ln L * and Donghu L Abstract Ths paper presents a dstrbuted estmaton strategy called alternaton dffuson LMS estmaton (AD-LMS) to estmate an unknown parameter of nterests from nosy measurement over wreless sensor network. It s useful n the wreless sensor networks where robustness and low consumpton are desred features. Dffuson LMS s ntroduced n ths estmaton strategy to mprove the performance and reduce the communcaton burden. Wth the proposed strategy, whether each node dstrbutes ts estmaton depends on an alternatve parameter. The node only exchanges ts estmaton when the nstant tme meets some condtons. Next, each node combnes the estmatons of neghbors wth ts own estmaton usng combnaton coeffcents upon the topology of the network. At last, the nodes update ther estmatons wth a normalzed LMS algorthm. The proposed AD-LMS strategy s compared to standard dffuson strategy. The results show that they acheve exactly the same coverage rate and nearly the network performance (network MSD and steady-state MSD) of standard dffuson strategy whle reducng the communcaton burden sgnfcantly. 1. INTRODUCTION In a wreless sensor network, the nodes collect data n a dstrbuted way n some applcatons such as target localzaton and trackng, envronment montorng, spectrum sensng, and automotve radars [1]. An unknown common parameter of nterest s the dstorton of the collected regresson data by nose, whch occurs when the local copy of the underlyng system nput sgnal at each node s corrupted by varous sources of mparment such as measurement or quantzaton nose [2]. A bg problem s how to estmate the unknown parameter from the obtaned data from each node n a WSN [3]. To solve the problem of the parameter estmaton n a WSN, there have been two man strateges n recent years: one s centralzed strategy, and the other s dstrbuted strategy [4]. In a centralzed strategy, all the nodes need to send ther estmatons to a central node to process and estmate the unknown parameter. The central node can offer an estmaton after obtanng the whole nformaton of the network. However, a network wth ths strategy ncreases the cost greatly. The power of sensor node whch s usually suppled by battery runs out quckly by usng the centralzed strategy, and ths s unacceptable. Snce the WSNs are lmted wth energy, and the connecton between nodes are multhop, dstrbuted strateges have attracted more and more attenton. In a dstrbuted strategy, each node estmates the parameter based on ts own local computaton and the estmaton nformaton receved from ts neghbors wthout the help of the central node [5]. The exstng dstrbuted strateges can be classfed nto ncremental [6, 7], dffuson [2, 8 11] and herarchcal strateges [12, 13]. The dffuson LMS strategy s the most popular strategy, and we focus on t n ths paper. Each node performs an LMS update after exchangng the estmaton wth ts neghbors n a dffuson strategy [9]. Compared wth the centralzed strategy, t can acheve scalablty, robustness and low communcaton burden. There are many dstrbuted dffuson strateges proposed n the past papers. In work [8], a smple Receved 23 Aprl 2018, Accepted 29 June 2018, Scheduled 13 July 2018 * Correspondng author: Ln L (lln0217@tju.edu.cn). The authors are wth the School of Electrcal and Informaton Engneerng, Tanjn Unversty, Tanjn 300072, Chna.
136 L and L adaptve dffuson LMS strategy s llustrated. [9] analyses the performance of CTA, and ATC dffuson strategy n a dstrbuted network [11] uses the normalze step-sze n the adaptve stage to adapt the nput sgnal. Shao et al. [14] propose a robust dffuson estmaton algorthm based on a mnmum error entropy crteron wth a self-adjustng step-sze to gan a fast speed of convergence. As most networks contan a large number of nodes and a complex topology, the communcaton burden of estmaton s stll consderable n a dstrbuted dffuson strategy. The broken-motfs dffuson LMS (BM-LMS) algorthm [15] reduces the communcaton burden wth only a subset of edges whch are partcpated n communcatons. Consderng the communcaton burden n a dstrbuted network, we propose a new dstrbuted estmaton strategy, called alternaton dffuson LMS estmaton (AD-LMS). In ths paper, each node dstrbutes ts estmaton dependng on an alternatve parameter. The node only exchanges ts estmaton when t s chosen. Next, each node combnes the estmatons from other nodes wth ts own estmaton usng combnaton coeffcents upon the topology of the network. At last, each node performs an LMS update of estmaton wth a normalzed step-sze. Moreover, by usng the proposed AD-LMS strategy, the communcaton burden n the whole network has been sgnfcantly reduced wth a lttle nfluence on the network performance. The paper s organzed as follows. In Secton 2, we state the estmaton problem and defne the cost functons. Then, the dervaton of the dffuson soluton strategy s presented. In Secton 3, we descrbe our AD-LMS strategy. In Secton 4, we provde detaled smulaton results of a dstrbuted network wth 50 nodes to llustrate the performance of our strategy compared wth the exstng dffuson strateges. In Secton 5 we have a concluson of ths paper. 2. PROBLEM STATEMENT 2.1. Network Model In ths paper, we consder a WSN wth N nodes. A typcal topology of the WSN s llustrated n Fg. 1. The nodes are denoted by neghbors as they can exchange ther nformaton drectly wthout transferrng. A usual lnear regresson model [16] s shown as follows: d (t) = wo T u (t) + v (t) (1) The node outputs a scalar measurement d (t) at nstant tme t whch relates to the nput regresson vector u (t) andthetrueparameterw o,whered (t) s a scalar value. u (t) sanm 1 vector so s w o. v (t) denotes the observaton nose or dsturbance of each node I, and v (t) s ndependent and unrelated. We assume that v (t) of each node I at nstant tme t s a random sgnal wth zero mean and varance σv, 2. 7 7 [ 1(t), 1 ] d u(t) 1 2 2 2 Ω 7 8 6 6 6 8 8 7 3 3 5 5 Fgure 1. Model of a typcal wreless sensor network. 5 3 4 4 9 4 [ d(t), u (t) ] 9 9
Progress In Electromagnetcs Research M, Vol. 70, 2018 137 2.2. Cost Functon To acheve an estmaton vector w for w o, the global cost functon [16] of the whole network should be mnmzed gven by N J global (w) = E d (t) u T (t)w 2 (2) =1 where E denotes the expectaton operator. Assume that the process u (t) s jontly wde sense statonary. A centralzed least mean square (LMS) algorthm update [17] s shown as w(t + 1) = w(t) + μ where μ>0, μ s a step sze, and w(t) s the estmaton of w o n tme t. N u (t)(d (t) u T (t)w(t)) (3) =1 2.3. Dstrbuted Dffuson Strategy By the centralzed LMS algorthm, the whole network nformaton should be collected and processed n a central node. To send and transmt the nformaton to central node, the communcaton burden s greatly ncreased [18]. It s mpractcal n a WSN due to the lmted resources of nodes. Moreover, f there are some lnk falures and changes n the network, the centralzed algorthm wll not have a good performance [19]. On the contrary, we ntroduce the dstrbuted dffuson strategy to overcome these drawbacks. In a dstrbuted estmaton strategy, each node only needs to exchange the nformaton wth ts neghbors to acheve the estmaton. It s assumed that two nodes are connected f they can communcate wth each other drectly []. The neghbor denoted by Ω of node s a set of nodes (nclude node tself) whch are connected wth node. Each node can process ts local estmaton and get the dffuson estmatons from ts neghbors. In Fg. 1, there s an example of a network consstng of ten nodes. The arrows ndcate the connectons of the nodes whle the nodes at the end of an arrow can exchange nformaton wth each other. The neghbor of node 7 denoted by Ω 7 ncludes nodes 5, 6, 7, 8. In ths case, the dstrbuted estmaton does not collapse even f some nodes fal. The dstrbuted strategy s commonly performed n two stages: adapton and combnaton. Based on the topology of the network, the estmatons are combned wth combnaton coeffcents. γ + j Ω γ j =1 j (4) where γ s the combnaton coeffcent of tself, and the γ j s the combnaton coeffcent of node j n ts neghbors, satsfyng Eq. (4). In ths paper we use the Metropols rules [20] to get the combnaton coeffcents wth Eq. (5). 1 γ j = f j Ω j max( Ω, Ω j ) γ j =0 fj/ Ω j γ =1 (5) γ j f j Ω j j Ω where Ω denotes the cardnalty of the set Ω. In ths paper, we seek to estmate the parameter of w o only by processng the nformaton of the neghbors n a dstrbuted dffuson strategy. Node has a pror estmate w (t) of parameter w o n the nstant tme t. The update functon s generated n Eq. (6). { w (t + 1) = arg mn γ w w (t) 2 + } w j Ω, j γ j w w j (t) 2 + μ (d (t) u T (t)w )2 (6) where μ s the step sze of node.
138 L and L To smplfy the update functon, we expand the last tem (d (t) u T (t)w )2 of the unknown w around w j (t) n Taylor formula. (d (t) u T (t)w )2 = e 2 j(t) 2e j (t)u T (t)(w w j (t)) + o w 2 (7) where e j (t) = d (t) u T (t)w j (t). In the same way, the expanson of the last term around w (t) s Eq. (8). (d (t) u T (t)w )2 = e 2 (t) 2e (t)u T (t)(w w j (t)) + o w 2 (8) where e (t) = d (t) u T (t)w (t). Then, we put Eqs. (7) and (8) nto Eq. (6). Snce the combnaton coeffcents satsfy Eq. (4), we have the functon n Eq. (9). γ w w (t) 2 + γ j w w j (t) 2 j Ω, j w (t + 1) = arg mn +μ γ [e 2 w (t) 2e (t)u T (t)(w w (t))] (9) [e 2 j (t) +μ γ j 2e j (t)w j (t)u T (t)(w w j (t))] j Ω, j The term wthn the large braces s a functon of w.togetw (t + 1), we dfferentate the functon of w and let t equal to 0. Then the dstrbuted update estmaton w (t + 1) s shown n Eq. (). where w (t + 1) = ϕ (t + 1) + μ u (t)(d (t) u T (t)ϕ (t + 1)) () ϕ (t + 1) = γ w (t) + j Ω, j γ jw j (t) (11) Eq. (11) s regarded as the combnaton stage, and Eq. () s the adaptve stage. dffuson LMS s shown n Fg. 2. The dstrbuted j1 w j1 (t) w (t) j3 w j3 (t) w (t) w (t) j Σ ϕ (t + 1) u(t) ϕ (t + 1) w (t + 1) Informaton exchange j2 w j2 (t) Combnaton stage γ γ j Adaptve stage d() t Fgure 2. Dstrbuted dffuson strategy. 3. THE PROPOSED AD-LMS STRATEGY By the strategy n Secton 2, the amount of nformaton exchange s reduced. However, the communcaton burden s stll consderable n WSN. If a node updates ts estmaton all by tself wthout cooperaton, the network performance s bad whch cannot meet the requrement of estmaton. Then, we propose our AD-LMS to balance the network performance and communcaton burden. To reduce the communcaton burden and energy consumpton, the proposed AD-LMS strategy uses an alternaton way whle the node does not need exchange ts own estmaton n every tme t. Wth the proposed AD-LMS strategy, we set a alternatve parameter P whch decdes the dffuson. The whole workng tme s dvded nto epochs wth P slots. That s to say, a slot s the nstant tme t. In the slots except the last one, the nodes enter the adaptve stage drectly by usng ts own estmaton to update. Then n the last slot, the node exchanges the nformaton wth ts neghbors to combne the estmatons.
Progress In Electromagnetcs Research M, Vol. 70, 2018 139 In other words, the node only needs to exchange ts estmaton when the nstant tme t modp = 0 whch means that t s chosen. Wth the AD-LMS strategy, the communcaton burden s 1/P of that n the standard dffuson strategy. Snce the performance of the LMS algorthm strongly depends on the step sze parameter μ, the normalzed algorthm s used n ths paper. We use μ = μ /u T (t)u (t) nstead of μ. The AD-LMS strategy s llustrated n detal n Table 1. Table 1. AD-LMS strategy. AD-LMS Intalze: Set the alternatve parameter P; For each node w (0) = 0 for where w s M 1 estmaton vector end Runnng: For eachtmenstantt=1, 2,...,T For each node = 1, 2,...,N If t mod P =0 Combnaton: ϕ (t + 1) = γ w (t) + j Ω, j γ jw j (t) else ϕ (t + 1) = w (t) end Adaptaton: w (t + 1) = ϕ (t + 1) +μ u (t)(d (t) u T (t)ϕ (t + 1)) end end 80 Network Toplogy Fgure 3. WSN topology. 70 50 48 60 43 46 49 47 44 45 38 40 41 37 36 39 42 50 29 30 31 32 40 24 25 33 34 35 22 26 27 30 23 28 18 20 15 17 19 21 20 9 16 11 8 13 14 12 3 5 1 0 2 4 6 7 0 20 30 40 50 60 70 4. SIMULATION RESULTS To llustrate the performance of the proposed strategy n ths paper, we compare our AD-LMS wth other LMS strateges. In ths smulaton, the consdered network topology n Fg. 3 s a WSN wth
140 L and L 15.8-4 14.5.6 Tr(R ) 14 13.5 13 12.5 12 Nose varance v, 2.4.2 9.8 9.6 9.4 9.2 11.5 0 5 15 20 25 30 35 40 45 50 Nodes number 9 0 5 15 20 25 30 35 40 45 50 Nodes number Fgure 4. Trace of regressors. Fgure 5. Nose varance. N = 50 nodes. Communcaton burden covers number of transmtted packets, packet delvery rato, data delay, or processng load. Snce the packet delvery raton and data delay are the same as other LMS strateges. There should be a postve correlaton between the number of transmtted packets and the processng load. We use the number of transmtted packets to evaluate the communcaton burden compared wth other strateges as n [9, 15]. The red astersks represent the sensor node, and the blue lnes represent the communcaton lnk wthn the network. In our smulaton, we use the nput regressors of each node whch are generated as sample vectors u,t =[u (t) u (t 1)...u (t M + 1)] T of an AR-1 [21] process of the form u (t) =x (t)+ρ u (t 1) whereρ =0.5 s a correlaton coeffcent, and x (t) s a whte nose process wth σ x, = 1. The parameter M s set to, and the nput regresson vector u,t s wth dmensons. The trace of each node s regresson matrx R =E(u (t)u T (t)) s shown n Fg. 4. The nose nput v (t) at each node s zero-mean Gaussan, and we show the varant of each node s nose n Fg. 5. The nput regressors and nose are temporary and spatally ndependent of each other. The step sze of LMS wthout cooperaton, standard dffuson LMS and our ADLMS s set μ =0.4/u T (t)u (t). We can set the alternatve parameter from 1 to I + 1. AD-LMS strategy s the same as the LMS wthout communcaton when P = I + 1, and t s the same as standard DLMS when P = 1. In our smulaton, the alternatve parameter of AD-LMS strategy s set to 2, 5, 8 to compare wth other strateges. All the curves shown n the fgures are the average results of 50 ndependent runs. To evaluate each strategy, we use mean-square devaton (MSD) of the whole network defned as MSD(dB)= 20log( 1 N E W(t) W o 2 ), shown n Fg. 6. At nstant tme 300, the network MSD of the proposed AD-LMS and standard dffuson s below 40 db whle the LMS wthout cooperaton strategy s about 35 db. The standard DLMS has the best MSD performance, and the MSD of AD-LMS wth P = 2 s near the standard one. Wth P ncreasng, the network MSD performance s worse. The convergence rates of all the strateges are exactly the same. To evaluate the performance n the steady state, we average the data of the last 500 nstant tmes as a steady state. In ths paper, we defne the steady-state MSD of node as MSD (db) = 20log(E w w (t) 2 ). In Fg. 7, the steady-state MSD of AD-LMS wth P = 2, 5, 8 s about 50 db, 46 db, 44 db. The MSD of standard DLMS s about-53 db, and the no-dffuson strategy s about 36 db. Table 2 llustrates the comparson of average steady-state MSD per node. When P = 2, 5, 8, respectvely, the AD-LMS gan 94.3%, 84.4%, and 82% MSD performance of standard DLMS. From the results n Fg. 6, Fg. 7 and Table 2, the proposed AD-LMS has exactly the same convergence rate and a good MSD performance almost as the standard DLMS wth a small alternatve
Progress In Electromagnetcs Research M, Vol. 70, 2018 141 MSD (db) 0 - -20-30 -40-50 LMS wthout cooperaton standard DLMS AD-LMS P=2 AD-LMS P=5 AD-LMS P=8 Steady state MSD (db) -36-38 -40-42 -44-46 -48-50 LMS wthout cooperaton standard DLMS AD-LMS P=2 AD-LMS P=5 AD-LMS P=8-60 -52-70 0 200 400 600 800 00 1200 1400 1600 1800 2000 tme Fgure 6. Network MSD. 5 15 20 25 30 35 40 45 50 Nodes number Fgure 7. Steady-state MSD. Fgure 8. Average number of transmtted packets per tme. parameter. Fg. 8 shows the average number of transmtted packets per tme. The average number n standard DLMS s 25000. As the node does not exchange the estmaton all the tme, the number of AD-LMS wth P = 2, 5, 8 s respectvely 12500, 5000 and 3125. Snce the standard dffuson strategy has a heavy communcaton burden n the WSN, and the network performance by usng the LMS strategy wthout cooperaton cannot meet the requrement of estmaton, our AD-LMS strategy reduces the communcaton burden sgnfcantly. Table 3 shows the comparson of communcaton burden wth standard dffuson strategy and AD-LMS wth the alternatve parameters n ths smulaton. By the AD-LMS, the network performance and communcaton are balanced. We can set the alternatve parameter dependng on whch one s more concerned n a specfc network. Table 2. Average steady-state MSD comparson. LMS wthout cooperaton Average Steady-state MSD (db) Standard Dffuson LMS AD-LMS P = 2 AD-LMS P = 5 AD-LMS P = 8 35.9 53.31 50.04 45.82 43.78
142 L and L Table 3. Communcaton burden comparson. Communcaton burden Standard Dffuson LMS AD-LMS P = 2 AD-LMS P = 5 AD-LMS P = 8 0% 50% 20% 12.5% 5. CONCLUSION In ths paper, a dstrbuted estmaton strategy denoted by alternaton dffuson LMS estmaton (AD- LMS) for WSN s proposed to estmate an unknown parameter wth less communcaton burden. We descrbe the dffuson LMS n a WSN and the dervaton of the algorthm. Snce the communcaton burden s stll hgh n the standard dffuson way, we propose our AD-LMS. Wth an alternatve parameter, each node only needs to exchange ts estmaton n some specfc nstant tmes. Hence the communcaton burden decreases consderably. Compared wth the standard dffuson strategy, the same coverage rate s acheved wth a lttle nfluence on MSD performance. Through settng the alternatve parameter of the AD-LMS, we can balance the network performance and network communcaton burden. REFERENCES 1. Rahman, M. U., Performance analyss of MUSIC DOA algorthm estmaton n multpath envronment for automotve radars, Internatonal Journal of Appled Scence & Engneerng, Vol. 14, 125 132, 2016. 2. Abdolee, R. and B. Champagne, Dffuson LMS strateges n sensor networks wth nosy nput data, IEEE/ACM Transactons on Networkng, Vol. 24, 3 14, 2015. 3. Lopes, C. G. and A. H. Sayed, Incremental adaptve strateges over dstrbuted networks, IEEE Transactons on Sgnal Processng, Vol. 55, 4064 4077, 2007. 4. Lu, Y., C. L, W. K. S. Tang, and Z. Zhang, Dstrbuted estmaton over complex networks, Informaton Scences, Vol. 197, 91 4, 2012. 5. Arabloue, R., Y. F. Huang, S. Werner, and K. Doğançay, Reduced-communcaton dffuson LMS strategy for adaptve dstrbuted estmaton, Sgnal Processng, Vol. 117, 355 361, 2014. 6. Sahoo, U. K., G. Panda, B. Mulgrew, and B. Majh, Robust ncremental adaptve strateges for dstrbuted networks to handle outlers n both nput and desred data, Sgnal Processng, Vol. 96, 300 309, 2014. 7. Cattvell, F. S. and A. H. Sayed, Analyss of spatal and ncremental LMS processng for dstrbuted estmaton, IEEE Transactons on Sgnal Processng, Vol. 59, 1465 1480, 2011. 8. Lopes, C. G. and A. H. Sayed, Dffuson least-mean squares over adaptve networks: Formulaton and performance analyss, IEEE Transactons on Sgnal Processng, Vol. 56, 3122 3136, 2008. 9. Cattvell, F. S. and A. H. Sayed, Dffuson LMS strateges for dstrbuted estmaton, IEEE Transactons on Sgnal Processng, Vol. 58, 35 48, 20.. Chen, J. and A. H. Sayed, Dffuson Adaptaton Strateges for Dstrbuted Optmzaton and Learnng over Networks, IEEE Press, 2012. 11. Fernandezbes, J., J. A. Azpcuetaruz, M. T. M. Slva, and J. Arenasgarca, A novel scheme for dffuson networks wth least-squares adaptve combners, Vol. 248, 1 6, 2012. 12. Tewar, M. and K. S. Vasla, Performance study of SEP and DEC herarchcal clusterng algorthm for heterogeneous WSN, 2014 6th Internatonal Conference on Computatonal Intellgence and Communcaton Networks, 385 389, Bhopal, Inda, 2014. 13. Senouc, M. R., A. Mellouk, H. Senouc, and A. Assan, Performance evaluaton of network lfetme spatal-temporal dstrbuton for WSN routng protocols, Journal of Network and Computer Applcatons, Vol. 35, 1317 1328, Jul. 2012.
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