Adaptive Link Selection Strategies for Distributed Estimation in Wireless Sensor Networks

0.22 MB·

by  Songcen Xu

ADAPTIVELINK SELECTION STRATEGIES FORDISTRIBUTED ESTIMATION IN
WIRELESS SENSOR NETWORKS
SongcenXu, RodrigoC. deLamareandH. VincentPoor
CommunicationsResearch Group,Department ofElectronics
UniversityofYork, U.K.
Princeton University
Emails: [email protected],[email protected],[email protected]
4
1
0
2
ABSTRACT network in order to select a subset of links that result in an im-

n proved estimation performance. The first approach consists of an
In this work, we propose adaptive link selection strategies for
a exhaustive searchbased link selection (ESLS) algorithm, whereas
distributedestimationindiffusion-type wirelessnetworks. Wede-
J the second technique is based on a sparsity-inspired link selection
velop an exhaustive search-based link selection algorithm and a
5 (SILS) algorithm. For the ESLS algorithm, we consider all pos-
sparsity-inspiredlinkselectionalgorithmthatcanexploitthetopol-
1 sible combinations for each node with its neighbors and choose
ogyofnetworks withpoor-qualitylinks. Intheexhaustivesearch-
thecombinationassociatedwiththesmallestEMSEvalue. Forthe
basedalgorithm, wechoose theset ofneighbors that resultsinthe
] SILS algorithm, we introduce the RZA strategy into the adaptive
T smallestexcessmeansquareerror(EMSE)foraspecificnode.Inthe
linkselectionalgorithm. TheRZAstrategyisusuallyemployedin
sparsity-inspired link selection algorithm, a convex regularization
I applicationstodealwithsparsesystemsinsuchawaythatitshrinks
. isintroduced todeviseasparsity-inspiredlinkselectionalgorithm.
s thesmallvaluesintheweightvectortozero,whichresultsinabetter
c The proposed algorithms have the ability to equip diffusion-type convergence rateand steady-stateperformance. Unlikeprior work
[ wireless networks and to significantly improve their performance.
withsparsity-awarealgorithms[3,6,7,8,9,10],theproposedSILS
Simulationresultsillustratethattheproposedalgorithmshavelower
1 algorithm exploits the possible sparsity of the EMSE associated
EMSE values, a better convergence rate and significantly improve
v witheach of the linksin anopposite way. Weintroduce a convex
thenetworkperformancewhencomparedwithexistingmethods.
5 penalty, i.e., an ℓ1–norm term to adjust the combined coefficients
8 Index Terms— Adaptive link selection, diffusion networks, for each node with its neighbors in order to select the neighbor
7 wirelesssensornetworks,distributedprocessing. nodesthatyieldthesmallestEMSEvalues.Foraspecifiednode,we
3 calculate the EMSE all its neighbor nodes including the specified
. node itselfthrough theprevious estimation. Forthenode withthe
1 1. INTRODUCTION
maximum EMSE,weimpose a penalty and give thesame amount
0
ofawardtothenodewiththeminimumEMSE.TheproposedSILS
4 Distributed strategies have become very popular for parameter es-
algorithm performs thisprocess automatically. By using the SILS
1 timationin wireless networks and applications such as sensor net-
: works and smart grids [1, 2, 3, 4]. In this context, for each spe- algorithm some nodes with bad performance will be shrunk and
v some poor nodes will taken into account when their performance
cific node a set of neighbor nodes collect their local information
i improves,whichmeansthenetworktopologywillchangeautomat-
X andtransmittheestimatesbacktoaspecificnode. Manyworksin
ically as well. Simulation results for an application to distributed
theliteraturehaveproposedstrategiesfordistributedprocessing:in-
r
a cremental[1],diffusion[2],sparsity-aware[3]andconsensus-based system identification illustrate that, the proposed ESLS and SILS
algorithms have a better convergence rate and lower EMSE value
strategies[4]. Withdiffusionstrategies[2]andaleast-meansquare
whencomparedwiththeexistingdiffusionLMSstrategies[2].
(LMS)algorithm,theneighborsforeachnodearefixedandthecom-
binedcoefficientsarepre-calculatedafterthenetworkisgenerated. Thispaperisorganizedasfollows. Section2describesthedis-
Thisapproachmaynotprovideanoptimizedestimationperformance tributedprocessinginthenetworksandproblemstatement. Insec-
foreachspecifiednodebecausetherearelinksthataremoreseverely tion3, the proposed link selectionalgorithms are introduced. The
affectedbynoiseorfading.Moreover,whenthenumberofneighbor numerical simulation results are provide in section 4. Finally, we
nodes is large, each node requires a large network bandwidth and concludethepaperinsection5.
transmit power. In order to reduce this bandwidth requirement, a Notation: Weuseboldfaceuppercaseletterstodenotematrices
single-linkdiffusionstrategyhasbeenreportedrecentlyin[5],where andboldfacelowercaseletterstodenotevectors. Weuse(·)T and
each node selects itsbest neighbor node. A key problem withthe (·)−1 denote the transpose and inverse operators respectively, and
strategiesreportedsofarintheliteratureisthattheydonotexploit (·)∗forconjugatetransposition.
thetopologyofwirelessnetworksandtheknowledgeaboutthepoor
linkstoimprovetheperformanceofestimationalgorithms.
In order to optimize the performance of distributed estimation 2. DISTRIBUTEDPROCESSINGINWIRELESS
techniques in wireless networks and minimize the excess mean- NETWORKSANDPROBLEMSTATEMENT
square error (EMSE) associated with the estimates, we propose
two adaptive link selection algorithms. The proposed techniques Weconsideradiffusion–typewirelessnetworkwithNnodeswhich
exploittheknowledgeaboutthepoorlinksandthetopologyofthe have limitedprocessing capabilities. At every timeinstant i, each
nodektakesascalarmeasurementd(i)accordingto:
k
20
d(ki) =ωH0 xk(i)+nk(i), i=1,2,...,N, (1) 18 14 12 11
wherex(ki)istheM×1inputsignalvector,nk(i)isthenoisesample 16 10
ateachnodewithzeromeanandvarianceσ2 .Through(1),wecan
v,k 14 4
seethat themeasurements forall nodes arerelatedtoanunknown 20 18
vectorω . Fig.1showsanexampleforadiffusion–typewireless 12
0
network with 20 nodes. The aim for the diffusion – type network
istocomputeanestimateofω0 inadistributedfashion,whichcan 10 15 7 9
minimizethecostfunction: 8
19 8
J (ω)=E|d(i)−ωHx(i)|2, (2) 6
ω k k 13 3
4 6
whereEdenotestheexpectationoperator.Tosolvethisproblem,one 17
possiblebasic diffusion strategyisthe adapt–then–combine (ATC) 2 2
diffusionstrategy[2]: 00 2 4 166 8 101 12 14 16 518 20
ψ(i) =ω(i−1)+µ x(i)[d(i)−ω(i−1)∗x(i)]∗,
k k k k k k k
 ω(ki) = cklψl(i), (3) Fig.1.Networktopologywith20nodes
l∈PNk

wherec isthecombinecoefficient, whichiscalculatedunderthe
kl
Metropolisrules: where{nk}isthetotalnumberofnodeslinkedtonodekincluding
nodekitself.Thiscombinatorialstrategywillcoverallcombination
ckl = max(n1k,nl), ifk6=l arelinked choicesforeachnodekwithitsneighbors.
 ckl =0, forkandlnotlinked AfterthetentativesetΩs isdefined,weredefinethecostfunc-
 ckk =1− ckl, fork=l tionas
 l∈NPk/k (4) Jψ(ψ),E|d(ki)−x(ki)∗ψ|2, (7)
andshouldsatisfy: where
ψ, c ψ(i), (8)
kl l
ckl =1,l∈Nk∀k (5) lX∈Ωs
Xl andtheerrorpatternisintroducedas:
Forthiskindofstrategy,thechoiceoftheneighborsforeachnode e(i) ,d(i)−x(i)∗ c ψ(i). (9)
isfixed,thissituationwillcauseaproblemwhensomeoftheneigh- Ωs k k kl l
bornodeshaveapoorperformance, andthereisnochanceforthe lX∈Ωs
nodetodiscardthebadperformanceneighborsinsteadofcontinue Foreachnodek,thestrategythatfindsthebestsetΩ shouldsolve
s
tousetheirinformation. Tooptimizethedistributedprocessingand thefollowingoptimization:
improve the performance, we need to provide each node some re-
finementsandtheabilitytoselectitslinks. Ω =argminJ (ψ) (10)
s ψ
Ωs
c
3. PROPOSEDADAPTIVELINKSELECTION whichisequivalenttominimizingtheerrore(i).Afterallstepshave
ALGORITHMS Ωs
beencompleted,thediffusionstepin(3)canbemodifiedas:
Inordertooptimizethedistributedprocessingandimprovetheper- ω(i) = c ψ(i). (11)
formanceofthenetwork,weproposetheESLSandtheSILSalgo- k kl l
Xc
rithms.Thesetwoalgorithmicstrategiesgivethenodestheabilityto l∈Ωs
choosetheirneighborsbasedontheirEMSEperformance.
Atthispoint,themainstepsoftheESLSalgorithmhavebeencom-
pleted. Theproposed ESLSalgorithmfindsthebest set Ω which
s
3.1. Exhaustivesearch-basedlinkselection(ESLS) minimizes the cost function in (7) and then uses this set of nodes
c
TheESLSemploysanexhaustivesearchtoselectthelinksthatyield toobtainω(ki) through(11). TheESLSalgorithmissummarizedin
the best performance in terms of EMSE. For our proposed ESLS Table1.WhentheESLSisimplementedinnetworkswithsmalland
low-powersensors,thecostmaybecomeexpensive,asthealgorithm
algorithm,weemploytheadaptationstrategygivenby
in(6)requiresanexhaustivesearchandneedsmorecommunication
ψ(ki) =ωk(i−1)+µkxk(i)[dk(i)−ωk(i−1)∗xk(i)]∗ resourcestoexamineallthepossiblesetsΩs.
andredefinethediffusionstep. 3.2. Sparsity-inspiredlinkselection(SILS)
First,weintroduceatentativesetΩ usingacombinatorialap-
s
proachdescribedby The ESLS algorithm outlined above mentioned problem needs to
examineallpossiblesetstofindasolution,whichmightresultinan
Ω ,Cj , j =1,2,...,nk, (6) unacceptablecomputationalcomplexityforlargenetworksoperating
s nk
betransformedintothelinkselectionmethodas:
Table1.TheESLSAlgorithm
FInoirtieaaliczhe:tiωm(ke−i1n)s=ta0nti=1,2,...,n ω(ki) =lX∈Nk[ckl−ρ∂∂fe1l(e(li))]ψ(li) (14)
Foreachnodek=1,2,...,N
ψ(i) =ω(i−1)+µ x(i)[d(i)−ω(i−1)∗x(i)]∗ where ρ is used to control the strength of the algorithm and it is
k k k k k k k updatedbyusing
end
Foreachnodek=1,2,...,N ∂f (e(i)) sign(e(i))
findallpossiblesetsofΩ 1 l =ε l (15)
e(i) =d(i)−x(i)∗ cs ψ(i) ∂el 1+ε|ξmin|
Ωs k k kl l
l∈PΩs In(15),theparameterξ standsfortheminimumvalueofe(i) in
Ω =argmine(i) min l
s Ωs Ωs eachgroupofnodesincludingeachnodek anditsneighbors. The
ωc(i) = c ψ(i) functionsign(x)isdefinedas
k kl l
c
l∈PΩs x/|x| x6=0
end sign(x)= (16)
end (cid:26) 0 x=0
Tofurther simplifytheexpression in(14), weintroduce thevector
andmatrixquantitiesrequiredtodescribetheadaptationprocess.We
intime-varyingscenarios. Tosolvethecombinatorialproblemwith firstdefineavectorcthatcontainsthecombinationcoefficientsfor
alowcomplexity,weproposetheSILSalgorithmwhichisassimple eachgroupofnodesincludingnodekanditsneighborsasdescribed
asastandarddiffusionLMSalgorithmandissuitablefor adaptive by
implementationsandscenarioswheretheparameterstobeestimated c,[c ] (17)
(k,l) l∈Nk
areslowlytime-varying. Then,weintroduceamatrixΨthatincludesalltheestimatedvectors
whicharegeneratedaftertheadaptationstepin(3)foreachgroupas
2#-3,) 2#-3,) ! givenby
!"#$%&’()*+#$,)) Ψ,[ψl(i)]l∈Nk (18)
*-./$&’01) Anerrorvectorethatcontainsalltheerrorvaluescalculatedthrough
(13)foreachgroupisexpressedby
4/5,%) 4/5,%)
#6)!"#$%&’()*+#$,)*-./$&’01)
e,[e(i)] (19)
2#-3,) 2#-3,) l l∈Nk
To employ the sparsity –inspired approach, we have modified the
!78!))
*-./$&’01) vectoreinthefollowingway.Themaximumvaluee(i)inewillbe
l
4/5,%) 4/5,%) setto|e(li)|,whiletheminimumvaluee(li)willbesetto−|e(li)|.For
96)!78!)*-./$&’01) theremainingentries,theywillbesettozero. Finally,byinserting
(15)-(19)into(14),thediffusionstepwillbechangedto
Fig.2.SparsityAwareTechnology
Nk ∂f
ω(i) = [c −ρ 1(e )]Ψ
k j ∂e j j
Fig. 2showsthedifferencebetweentheexistingsparsity-aware Xj=1 j
(20)
methodsandtheproposedsparsity-inspiredstrategy. InFig. 2(a),
wecanseethat,afterbeingprocessedbyasparsity-awarealgorithm, = Nk[c −ρε sign(ej) ]Ψ
thenodeswithsmallerrorvalueswillbeshrunktozero.Incontrast, Xj=1 j 1+ε|ξmin| j
thesparsity-inspiredalgorithmwillshrinkthenodeswithlargeerror
valuestozeroasillustratedinFig.2. TheproposedSILSalgorithmperformslinkselectionbytheadjust-
In the proposed SILS algorithm, we introduce the convex ment of the combination coefficients through c in (20). For the
penalty term ℓ –norm into the diffusion step in (3) to perform neighbornodewiththelargestEMSEvalue,afterourmodifications
1
thelinkselection. Differentpenaltytermshavebeenconsideredfor fore,itse(i)valueinewillbeapositivenumberwhichwillleadto
l
thistask. Wehaveadoptedthereweightedzero–attracting strategy thetermρε sign(ej) in(20)beingpositivetoo.Thismeansthatthe
[3]intothediffusionstepin(3)becausethisstrategyhasshownan 1+ε|ξmin|
combiningcoefficientforthisnodewillbereducedandtheweight
excellentperformanceandissimpletoimplement.
for this node to build the ω(i) is reduced too. In contrast, for the
First,weconsiderthefollowingregularizationfunction: k
neighbornodewiththeminimumEMSE,asitse(i) valueinewill
l
f (e(i)))= log(1+ε|e(i)|) (12) be a negative number, the term ρε sign(ej) in (20) will be nega-
1 l l 1+ε|ξmin|
lX∈Nk tive too. As a result, the weight for this node associated with the
minimumEMSEtobuildtheω(i) isincreased. For theremaining
wheretheerrorpatternel(i)isdefinedas: neighbor nodes, the e(i) value ink e is zero, which means the term
l
e(li) ,dk(i)−xk(i)∗ψl(i) (l∈Nk) (13) tρoεb1us+iigεlnd|ξ(methjien)|ωi(ni)(.2T0)heispzreorcoesasnfdorththereeciosmnboincahtainognecofoerffithcieeinrtwsiesigshtitlsl
k
andεistheshrinkagemagnitude.Then,thediffusionstepin(3)can satisfied(5).TheSILSalgorithmissummarizedinTable2.
Table2.TheSILSAlgorithm
0
Initialize:ω(−1)=0 Diffusion ATC LMS without link selection
k
Foreachtimeinstanti=1,2,...,n −5 Diffusion ESLS link selection
Foreachnodek=1,2,...,N Diffusion SILS link selection
ψ(i) =ω(i−1)+µ x(i)[d(i)−`ω(i−1)∗x(i)]∗ −10 EMSE Bound
k k k k k k k
end
−15
Foreachnodek=1,2,...,N B)
e(li) =d(ki)−xk(i)∗ψl(i) (l∈Nk) SE (d−20
c=[c ] M
(k,l) l∈Nk E
Ψ=[ψ(i)] −25
l l∈Nk
e=[e(i)]
l l∈Nk −30
Findthemaximumandminimumtermsine
Modifiedease=[0···0,|el(i)|,0···0,−|el(i)|,0···0] −35
max min
ξmax=min(el(i)) |{z} | {z } −400 80 160 240 320itera40ti0ons 480 560 640 720 800
ω(i) = Nk[c −ρε sign(ej) ]Ψ
k j 1+ε|ξmin| j
jP=1 Fig.3.NetworkEMSEcurvesinastaticscenario
end
end
5
For the ESLS and SILS algorithms, we redesign the diffusion Diffusion ATC LMS without link selection
stepandemploythesameadaptationprocedure,whichmeansthese Diffusion ESLS link selection
0 Diffusion SILS link selection
twoalgorithmshavetheabilitytoequipanydiffusion–typewireless
EMSE Bound
networksbesidestheLMSstrategy.ThisincludesthediffusionRLS
−5
strategy[11]andthediffusionconjugategradientstrategy[12].
−10
B)
4. SIMULATIONRESULTS E (d−15
S
M
E
In this section, we compare our proposed diffusion link selection −20
algorithms ESLS and SILS with the traditional diffusion ATC al-
gorithm[2]basedontheperformanceofEMSE.Withthenetwork −25
topologystructureinFig. 1,weintroduceN=20nodesinthissys-
tem. The length for the unknown parameter ω is M=10 and it −30
0
is generated randomly. The input signal is generated as u(i) =
k −35
[u(ki) uk(i−1) ... u(ki−M+1)]anduk(i) = xk(i)+ρkuk(i−1),where 0 80 160 240 320 itera40ti0ons 480 560 640 720 800
x(i)isawhitenoiseprocesstoensurethevarianceofu(i)σ2 =1.
k k u,k
ThenoisesamplesaremodeledascomplexGaussiannoisewithvari- Fig.4.NetworkEMSEcurvesinatime–varyingscenario
anceofσ2 = 0.001. Thestepsizeforallthesethreealgorithmsis
k
µ= 0.045. Forthestaticscenario,thesparsityparametersofSILS
algorithmaresettoρ = 4∗10−3 andε = 10. Theresultsareav-
eragedover100independentruns.FromFig.3,wecanseethat,the
ESLShasthebestperformanceonboththeEMSEandconvergence 5. CONCLUSION
rate, it has around a 5 dB gain over the traditional diffusion ATC
algorithm. SILSisabitworsethantheESLS,butstillsignificantly
Inthispaper, twoadaptive link selectionstrategieshave been pro-
betterthanthestandarddiffusionATCalgorithmbyabout4dB.For
posedfordistributedestimationindiffusion–typewirelessnetworks.
thecomplexityandprocessingtime, SILSisassimpleasthestan-
TheESLSalgorithmusesanexhaustive searchtoperformthelink
darddiffusionATCalgorithm,whileESLSismorecomplex.Forthe
selection, and the SILSemploys a sparsity-inspired approach with
time–varyingscenario,thesparsityparametersofSILSalgorithm
aresettoρ = 6∗10−3 andε = 10. Theunknownvectorω isis theℓ1 –norm penalization. TheESLSalgorithmchooses thebest
0 set of nodes, while the SILS algorithm shrinks the node with the
definedbythefirst–orderMarkovvectorprocess:
highesterrorvaluesandawardsthenodewiththesmallesterrorsin
eachgroup. Numericalresultshaveshownthatthetwoproposedal-
ω0(i+1) =ω0(i)+z(i) (21) gorithmsachieveabetterconvergencerateandlowerEMSEvalues
thanthealgorithmsin[2]. Theseresultsholdfor other algorithms
wherez(i) isanindependent zero–meangaussianvectorprocess. includingRLSandCGtechniques. TheESLSandSILSalgorithms
Fig. 4 shows that, for the time – varying scenario, the ESLS still can beused inany kind of diffusion – typewirelessnetworks and
performsbest,whiletheSILShasthesecondbestperformance. appliedtoproblemsofstatisticalinferenceinsmartgrids.
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0 80 160 240 320 400 480 560 640 720 800
iterations