Table Of ContentA LARGEDEVIATIONSAPPROACHTO SENSOR SCHEDULING FOR
DETECTION OF CORRELATED RANDOMFIELDS
YoungchulSung,Lang Tong,andH. VincentPoor
ABSTRACT Weassumethateachsensorinthefieldtakesanoisymeasure-
5 ment. Atlocationx ,thesensormeasurementy comesfromthe
0 Theproblemofschedulingsensortransmissionsforthedetectionofcorre- followingbinaryhypiotheses i
latedrandomfieldsusingspatiallydeployedsensorsisconsidered. Using
0
thelargedeviations principle, aclosed-form expressionfortheerrorex-
2 ponentofthemissprobabilityisgivenasafunctionofthesensorspacing H0 : yi=wi, (2)
n andsignal-to-noiseratio(SNR).Itisshownthattheerrorexponenthasa H1 : yi=si+wi, i=1,2,··· ,n,
a distinctcharacteristic: athighSNR,theerrorexponentismonotonically
J increasing withrespect tosensorspacing, while atlow SNRthereis an where si =∆ s(xi), and wi are i.i.d. sensor measurement noises
optimalspacingforscheduledsensors. from (0,σ2)withaknownvarianceσ2. Wewillnotfocushere
1 N
on minimizingthenumber of bitsfor thequantization ofy , nor
2 i
1. INTRODUCTION
willwetackletheproblem of how yi aresent tothefusioncen-
] terfordetection. Theseareimportantdesignissuesthatmustbe
Oneof themost critical designconstraints for largescale sensor
T treatedseparately. Ourfocusinthispaperisthat,ifdataaretobe
networksisenergyefficiency. Inthecontextofdetectingspatially
I collectedfromthesensorfield,wheretheyshouldbecollected.For
s. correlatedsignals,thismeansthat,givenadesiredlevelofdetector energyefficiency,weaimtofindaschedulingschemethatrequire
performance, one should minimize the amount of data required,
c asfewsamplesaspossible.
[ andcollectdatafromjudiciouslychosenareas.
Weconsider inthispaperthedetectionofaonedimensional 1.1. SummaryofResults
1 diffusion process that has a spatial state space structure as illus- WeadopttheNeyman-Pearsonformulationbyfixingthedetector
v tratedinFig.1.Specifically,weassumethattheunderlyingsignal size α and minimizing the miss probability when the number n
6 s(x)isthestationarysolution1ofthediffusionequation ofsamplesislarge. Specifically,weareinterestedintwoclosely
5 connected designproblems: thelocations X = x wheredata
i
0 ds(x) shouldbecollected,andtherateofdecayofmis{spr}obabilityfor
1 = As(x)+Bu(x), x 0, (1)
dx − ≥ detectors of size α. We will make the assumption that samples
0
are collected uniformly with equal spacing ∆ = ∆. Such an
5 whereA 0andBareknown,andtheinitialconditionisgiven i
≥ idealized assumption can only be approximated in practice, but
0 bys(0)∼N(0,Π0). Theinputprocessu(x)iszero-meanwhite itdoesprovideananalyticallytractableformulationthatleadsto
/ Gaussian,independentofbothsensornoiseands(0).
s insightsintoenergyefficientdatacollection.
c ThemissprobabilityP (∆,n;α,SNR)isafunctionofsen-
M
: ∆
v s(x) i sor spacing ∆, the sample size n as well as detector size α and
i SNRΓ =∆ Π0. Theenergyefficientschedulingproblemcanthen
X σ2
beformulatedasoneofoptimizingtheerrorexponentK (∆,SNR)
α
r underthelargedeviationprinciplewhere
a
1
0 xi xi+1 xn Kα(∆,Γ)=nlim nlogPM(∆,n;α,Γ). (3)
→∞
Fig.1.Signalandsensorlocation
Theconnectionoftheaboveformulationwithenergyefficientsen-
sorschedulingisnaturalwhenthesamplesizenisdirectlyrelated
Y.SungandL.TongarewiththeSchoolofElectricalandComputerEngineering,
withthenumberoftransmissions(suchasthecaseinSensorNet-
CornellUniversity,Ithaca,NY14853. Email:{ys87,ltong}@ece.cornell.edu. H.V.
PooriswithDept. ofElectricalEngineering,PrincetonUniversity,Princeton,NJ workswithMobileAccess(SENMA)).
08544.Email:[email protected]. We derive a closed-form expression for the error exponent
ThisworkwassupportedinpartbytheMultidisciplinary University
K (∆,Γ)ofmissprobability(whichisindependentofα)byex-
ResearchInitiative (MURI)undertheOfficeofNavalResearchContract α
ploitingthestate-spacestructureofalternativehypothesesandmak-
N00014-00-1-0564. Prepared through collaborative participation in the
ingaconnectionwithKalmanfiltering.WeshownextthatK (∆,Γ)
CommunicationsandNetworksConsortiumsponsoredbytheU.S.Army α
Research Laboratory under the Collaborative Technology Alliance Pro- has a distinct phase transition: when SNRΓ > 1, Kα(∆,Γ) is
gram, Cooperative Agreement DAAD19-01-2-0011. The work of H.V. a monotonically increasing function of ∆, indicating that sensor
PoorwassupportedinpartbytheOfficeofNavalResearchGrantN00014- spacing∆shouldbemadeaslargeaspossible(forfixedbutlarge
03-1-0102. sample size). When SNRΓ < 1, on theother hand, K (∆,Γ)
α
1Thestationarysolutionof(1)requiresaconditiononA,B,Π0given achievesthemaximumatsome∆∗,whichmeansthatthereisan
byΠ0= B2A2. optimal spacing among collected samples. We also provide an
implicit equation for ∆∗. We also present simulation results to cally,ReandR˜earegivenby
demonstratethepredictedbehavior.
1.2. RelatedWork Re = P+σ2, (7)
The detection of Gauss-Markov process in Gaussian noise is a R˜e = σ2 1+ P2+2σ2Pa2+P2(1−a2)σ4!, (8)
classical problem. See [5] and references therein. Our work re-
[σ2(1−a2)−Q]2+4σ2Q−σ2(1−a2)+Q
liesontheconnectionbetweenlikelihoodratioandtheinnovation P = , (9)
processthroughKalmanfilteringbySchweppe[6].Whilethecon- p 2
nectionbetweenKalmanfilteranderrorexponentisacontribution Infrequencydomain,
of this paper, there is an extensive literature on the large devia-
tionapproach tothe detectionGauss-Markov process [9, 10, 11, 1 2π
K (∆,Γ)= D( (0,σ2) (0,S (ω)))dω, (10)
12,13,14,15]. Theseresultsdonotprovideexplicitexpressions α 2π N ||N y
(withanexceptionfornoiselessARprocesses) fromwhichopti- Z0
malschedulingcanbeobtained. whereD( ) istheKullback-Leiblerdistance, andthespectrum
·||·
Thesensorschedulingproblemcanalsobeviewedasasam- Sy(ω)of yi underH1isgivenby
{ }
plingproblem. Tothisend,BahrandBucklew[11]optimizedthe
esixgpnoanlemntondueml(elroiwcalplaysusnsdigenraalBinayceoslioarnedfonrmoiusela)t,iothne.yFsohroawsepdectihfiact Sy(ω)=σ2+ 1 Π20a(1co−sωa2+)a2. (11)
theoptimalsamplingdependsonSNR,whichwealsoshowinthis −
paperinadifferentsetting. Proof:see[17].(cid:4)
Fornotationalconvenience,weuseKfortheerrorexponent.
2. SYSTEMMODEL
3.1. PropertiesofErrorExponent
We derive the discrete model for the sampled data of the signal
s(x)similarlyto[4]. Forequallyspacedsensorswithspacing∆, First,itiseasilyseenfromTheorem1thatKisacontinuousfunc-
thedynamicsofsampledsignalaredescribedby tionof thecorrelationcoefficienta(0 a 1)foragivenΠ0
andσ2sinceR σ2 >0. ≤ ≤
e
≥
si+1 = asi+ui, (4)
∆ Theorem2 TheerrorexponentispositiveforanySNRΓand0
a = e−A∆, ui = e−AxBu(xi+1−x)dx, a<1.Furthermore, ≤
Z0
(i) fori.i.d. observations(a = 0),theerrorexponentreduces
Themeanandvarianceofprocessuiaregivenby totheKullack-LeiblerdistanceD(p0 p1)wherep0 (0,σ2)
Eui=0, Q=∆ Var(ui)=Π0(1−a2). (5) (ii) afonrdPpe1rf∼ectNly(c0o,rΠre0la+tedσ2s)ig;nal(a =|1|),theerrorex∼poNnent
iszeroforanySNRΓ,andthemissprobabilityisbounded
The signal samples s form an autoregressive sequence under
H1. NoticethatEs2i{=i}Π0foralliandthevalueofadetermines by
theamountofcorrelationbetweensignalsamples.Fori.i.d.signal
1 1
samples we have a = 0 and a = 1 for the perfectly correlated ( D)cn−1/2 PM cn−1/2 (12)
√2π − ≤ ≤ √2π
signal. Noticealsothatthenoisyobservation y arenotautore-
i
{ }
gressive;theyfollowthehiddenMarkovmodel.
for sufficiently large n, where c and D (0, 1 ) are
∈ √2π
3. ERROREXPONENT positiveconstants.
Inthissection,weinvestigatetheperformanceofNeyman-Pearson
ThepositivityofK isimmediatefromTheorem1. Thecase
detectorwithalevelα (0,1).Wepresentaclosed-formexpres-
∈ whena = 0correspondstotheStein’slemmaforthei.i.d. case.
sionoftheerrorexponentofmissprobabilityK (∆,Γ)defined
α Unlessa=1,themissdetectionprobabilityalwaysdecaysexpo-
in(3).Thetheorembelowcomesfromthefactthatthelimitofthe
nentially.
normalizedlog-likelihoodratiounderH0isthebesterrorexponent Fortheperfectlycorrelatedcase(a=1),themissprobability
for general ergodic cases[16]. By expressing the log-likelihood doesnotdecayexponentially; itdecayswithΘ( 1 )asshownin
ratiothroughtheinnovation representation, wemake theclosed- √n
(12). Thisisexplainedbytheformoftheoptimaldetector. Itcan
formcalculationoferrorexponenttractable.
beshownthatthesufficientstatisticisgivenby
Theorem1(Errorexponent) FortheNeyman-Pearson detector n
aonfdth0e hypaothese1s,(t2h,e4b)eswtitehrrloerveelxpαon∈ent(0o,f1m)is(is.ep.roPbaFbi≤lityαis) T =| yi|2. (13)
indepen≤dento≤fαandisgivenby Xi=1
Kα(∆,Γ) = −1log σ2 + 1R˜e − 1, (6) UsynmdmereHtri0c,awbeouhtavzeero,nit=he1yNiey∼mNan(-P0,eanrsσo2n).deStienccteorfw(xit)h=levxe2liαs
2 Re 2Re 2 issimplygivenby P
whereR andR˜ arethesteady-statevariancesoftheinnovation
e e
processofyi calculatedunderH1 andH0,respectively. Specifi- δncor =1{| ni=1yi|≥zn}, (14)
P
wherezn = √nσQ−1(α2). (Q(·)isthetailprobability ofstan- AtlowSNR,noiseintheobservationdominates.Thusmaking
dardnormaldistribution).SinceP1,n = (0, n2Π0+nσ2),the signalmorecorrelatedprovidesthebenefitofnoiseaveraging.The
probabilityofmissingisgivenby N lowertheSNR,thestrongerthecorrelationisdesiredasshownin
Fig.2(right).Notehoweverthatexcessivecorrelationinthesignal
P =1 2Q √nσQ−1(α2) . (15) dthoaetstnh’etpmraoxviimdeumnewvaliunefoorfmeartrioornebxypothneenotbissemrvuactihonla.rgNeorttihcaenatlhsoe
M − √n2Π0+nσ2! valuefori.i.d. caseatlowerSNR.Hence,theoptimalcorrelation
gives much better performance with the same number of sensor
Forlargen, PM behavesas1−2Q(√cn)wherecisaconstant, observationsforlowSNRcases.
whichleadsto(12). A significance of the above theorem is the determination of
Havingobtainedthebehavioroferrorexponentattwoextreme optimalsensorspacing.Inparticular,theoptimaldistanceisgiven
correlationcases,wenowshowthattheerrorexponenthasdistinct by
characteristicsatdifferentSNRregimes. ∆∗ = log(am)/A, (17)
−
forthesameunderlyingphysicalphenomenondescribedby(1).
Theorem3(Kvs.correlation-highSNR) K ismonotonede-
creasing as the correlation strength increases (i.e. a 1) if the 1 2
↑ 0.9 1.8
SNRΓ>1.
0.8 1.6
0.7 1.4
0.6 1.2
Theabovetheoremshowsthatthei.i.d. observationsgivethe am0.5 ∆o 1
besterrorperformanceforthesameSNRwiththemaximumerror 0.4 0.8
exponent D(N(0,1)||N(0,1+Γ)) whenSNRislarger thanunity. 0.3 0.6
0.2 0.4
Fig. 2(left)showstheerrorexponent asafunctionofthecorre- 0.1 0.2
lationcoefficient a. Inthecontext sensor scheduling, ifntrans- 100−1 SN1R00, Γ 101 100−1 SNR, Γ 100
missionsarerequiredforthedesiredmissprobability,itisbestto
maximizethedistancesbetweenthescheduledsensors. Theintu- Fig.3.Left:amvsΓ, right:∆∗vsΓ(<1)
itionisthat,athighSNR,thesignalcomponentintheobservation
is strong, and the innovation process contains more information Fig. 3showsthevalueofa thatmaximizestheerrorexpo-
m
aboutthehypotheses. Thusthereisbenefittomakethesignalless nentasafunctionofSNR.AsshowninthefiguretheSNRofunity
correlated. givessharptransitionbetweentwodifferentbehaviorsoferrorex-
ponentw.r.t.correlationstrength.
0.19 SNR= 10dB 0.00.4055 SSSNNNRRR=== −−−369dddBBB Finally,weinvestigatethebehavioroftheerrorexponentw.r.t.
0.8 D(p0||p1) 0.04 SNR.
0.7 0.035
Error exponent, K000...456 Error exponent, K0.000..200523 caTrhe<aeos1irn.egMmao5sreS(oKNvRevr,sinaSctNrheiRags)hesSTNfhoeRreKarrgoiinvrceernexapcsooernsreenllitanteKiaornliyscowmeioftfihncolioteognntSe0NiRn≤-.
0.3 0.015
0.2 0.01
0.1 0.005
00 0.1 0.2 0.3Corr0e.4lation0 c.5oeffic0ie.6nt, a0.7 0.8 0.9 1 00 0.1 0.2 0.3Corr0e.4lation0 c.5oeffic0ie.6nt, a0.7 0.8 0.9 1 3
2.5
Fig.2.Kvs.a(left:SNR=10dB, right:SNR=-3,-6,-9dB)
thereInexciosntstraasnt,oeprtriomraelxcpoornreelnattiaotnl.owFigS.N2R(irsignhot)tmshoonwostotnhiec,erarnodr Error exponent, K1.125
exponent forthecasethattheSNRislessthanunity. Asshown,
0.5
thei.i.d.caseisnotthebestfortheerrorperformanceforthesame
SNR. The error exponent initially increases as a increases, and −010 −5 0 Sign5al−to−No10ise Ratio1 5[dB] 20 25 30
thendecreasestozeroasaapproachesone. ForSNRof-6dBthe
errorexponentislessthanthecaseof-3dBanditisseenthatam Fig.4.KversusSNR(a=e−1)
isshiftedclosertoone. Thefollowingtheoremcharacterizesthe
optimalcorrelation. The log SNR increase of K w.r.t. SNRissimilar tothe case of
diversitycombiningofRayleigh-fadedmultipathsinadditivenoise
Theorem4(Kvs.correlation-lowSNR) Thereexistsanon-zero sinceinbothcasesthesignal component israndom. Comparing
valuea ofthecorrelationcoefficientthatachievesthemaximum withthedetectionofadeterministicsignalinnoisewheretheerror
m
K forSNR < 1,anda isgivenbysolvingthefollowingequa- exponent is proportional to SNR, the increase of error exponent
m
tion. w.r.t.SNRismuchslowerforthecaseofstochasticsignalinnoise.
a4
[1+a2+Γ(1 a2)]2 2(r + )=0, (16)
− − e re 3.2. SensorPlacement
wherer = R /σ2. Futhermore,a convergestooneasSNRΓ So far we have assumed that sensors have been placed and the
e e m
goestozero. problem is to schedule the transmission of selected sensors. A
relatedproblemissensorplacement. Shouldnsensorsbeplaced 200 sensors whereas it is 10−4 with 20 sensors for 10 dB SNR
tocoveraslargeanareaaspossible,orshouldtheybeclusteredin case.
asubregionofthesignalfield? Theorem3suggeststhat,athigh
SNR, these sensors should be placed with maximum separation
5. CONCLUSIONS
subject to the size of the field. At low SNR, however, sensors
shouldbeplacedwiththeoptimalseparation. Thisimpliesthat,if Wehaveconsideredthedetectionofcorrelatedsignalusingnoisy
thesensor fieldislarge, itisbetter tocluster thesensor insome sensors.WehavederivedthebesterrorexponentfortheNeyman-
region,leavingotherareaswithoutsensors. Pearsondetectorsatisfyingafixedsizeconstraintusingtheinno-
vationsandthespectraldomainapproaches. Wehavealsoinves-
4. SIMULATIONRESULTS
tigatedthepropertiesoftheerrorexponent. Theerrorexponentis
WeconsideredtheNeyman-Pearsondetectoroffirst-orderautore-
a function of SNR and correlation strength. Thebehavior of er-
gressivesignaldescribedby(4).WeconsideredSNRof10dBand
ror exponent w.r.t. correlation strengthissharply divided totwo
-3dB,andseveralaforeachSNR.Theprobabilityoffalsealarm
regions depending on SNR. For SNR larger than unity the error
wasset0.1%forallcases. Fig.5showsthemissprobabilityw.r.t.
exponent ismonotone decreasing as correlation becomes strong.
Ontheotherhandithasanon-i.i.d.correlationstrengththatgives
themaximumslopeforSNRsmallerthanone.Usingtheproperty
100 of errorexponent, theoptimalstrategyforsensor schedulinghas
beenderived.
Theresultspresentedinthispaperhaveageneralizationtothe
10−1
vectorstate-spacemodel. Suchageneralizationisusefultocon-
sidermoregeneralschedulingschemes. Inpractice,thelocations
PM10−2 of sensorsarerandom, andsensorsmayceasetofunction. Thus
randomsamplingmayneedtobeconsidered.
10−3
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