Survey on sensor selection algorithms
By
Nooreddin Naghibolhosseini
Department of Computer Science GRADUATE CENTER OF THE CITY UNIVERSITY OF NEW YORK
A second examination submitted to the Graduate Center of CUNY in accordance with the requirements of the degree of
DOCTOR OF PHILOSOPHY in the Computer Science.
Committee Members:
September 2017
Robert Haralick ........................................
Sven Dietrich ...........................................
Saptarshi Debroy .....................................
.................................................................
Table of Contents
Abstract....................................................................................................................................................1
1.Introduction.......................................................................................................................................2
1.1.Powerrequirementofthesensors........................................................................................................4
1.2.Dependencybetweensensors..............................................................................................................6
1.3.Classificationsofnetworks....................................................................................................................7
1.4.Sensorselection.....................................................................................................................................8
1.4.1.Selectiongoal.................................................................................................................................8
2.4.1.Sensorplacementversussensorselection..................................................................................10
1.5.Sensorprediction................................................................................................................................10
1.6.Motivation...........................................................................................................................................11
2.SensorSelectionforMinimizingWorst-CasePredictionError......................................13
2.1.Introduction.........................................................................................................................................13
2.2.Notations.............................................................................................................................................14
2.3.Adversarybasedselectionandpredictionapproach..........................................................................15
2.3.1.Generalizedresults......................................................................................................................16
2.4.Sensorselection...................................................................................................................................17
2.5.Experimentalresults............................................................................................................................18
2.6.Discussionandcritiques.....................................................................................................................19
3.OnlineDistributedSensorSelection.......................................................................................21
3.1.Introduction.........................................................................................................................................21
3.2.Notations.............................................................................................................................................21
3.3.TheOfflinesensorselection................................................................................................................22
3.4.Theonlinesensorselectionproblem..................................................................................................23
3.5.Centralizedonlinesinglesensorselection...........................................................................................23
3.6.Centralizedonlinemultiplesensorselection......................................................................................25
3.7.Distributedalgorithmforonlinesensorselection...............................................................................26
3.7.1.Onlinedistributedsinglesensorselection...................................................................................27
3.7.2.Thedistributedonlinegreedy(DOG)algorithm...........................................................................30
3.7.3.LazyrenormalizationanddistributedEXP3(Singlesensorselection)..........................................31
3.7.4.LazyDOG.......................................................................................................................................32
3.7.5.ObservationDependent-DistributedOnlineGreedy(OD-DOG)..................................................32
3.8.Discussionandcritiques.....................................................................................................................35
4.OnDynamicData-DrivenSelectionofSensorStreams.....................................................36
4.1.Introduction.........................................................................................................................................36
4.2.Notations.............................................................................................................................................38
4.3.Localregressionclustering..................................................................................................................38
4.4.Sensorselectionprocess.....................................................................................................................39
4.5.SpeedingupwithRankingVariations..................................................................................................41
4.6.LeveragingLocal-RegressionClustersforPrediction...........................................................................41
4.7.Experimentalresults............................................................................................................................41
4.8.Discussionandcritiques......................................................................................................................43
5.Summeryandconclusion............................................................................................................45
6.References........................................................................................................................................46
1
Abstract
Thetopicofsensornetworkshadconsiderableattentioninthepast
decades. Today, the network of devices thatwork as sensors has
beenexpandedtoanevenlargernetworkthantheinternet.Sensors
havefoundtheirpathtothesmartdevicessuchascellphonesand
tablets and by advancement of the technology they can domore
thanasimpledatacollection fromtheenvironment.Forexample,
theycantargetconsumersintherelevantlocationsanddeliversmart
contentstotheirmobilephones.Oneof theproblemsofasensor
network is its cost. This cost can be either a computational cost,
bandwidth,sensors’battery(i.e.lifeofasensor)ordeploymentcost.
Toreducethecostofsensorsitisrecommendedtonotuseallthe
deployedsensorsatthesametime,andinsteadusetheonesthat
givemoreinformationabouttheenvironmentwithlowercost.The
sensor selection algorithms are a group of computer science
algorithmsthataredesignedtosolvetheproblemofselectingthe
mostappropriatesensorsetfromasensornetwork.
2
1. Introduction
Asensornetwork isagroupof sensors,working together toobtain information
abouttheenvironment.Aftertheinternet,thewirelesssensornetwork(WSN)is
thesecondlargestnetworkontheearth.Wirelesssensornetworksarewidelyused
inavarietyofapplications including:emergencyresponse,energymanagement,
medicalmonitoring,logisticsandinventorymanagement,battlefieldmanagement,
industrialapplications,environmentalapplicationsandhomeapplications[5,92].
Themaincomponentofasensornetwork isasensor.Fromthispointwhenwe
refertoanetwork,weimplytomeanasensornetwork.Asensorisadevicecapable
of reading some environmental phenomena and producing the relevant data.
Examples of environmental phenomena are: temperature, humidity, vehicular
movement,lightingconditions,pressure,soilmakeup,noiselevels,thepresenceor
absenceofcertainkindsofobjects,mechanicalstresslevelsonattachedobjects,
andthecurrentcharacteristicssuchasspeed,direction,andsizeofanobjectand
etc.[5].Eachsensorneedsapowersourcetooperate[100].Inaddition,sensors
needsome internalprocessor tocommunicatewitheachotherandprocess the
dataandaninternalmemorythatislargeenoughtoholdthedata.Besidesensor
nodes, theremight be other nodes operating in a network. The server node is
usually responsible for storing the obtained data from the sensors for further
processing.Thesenodesarealsoinvolvedinselectionandsynchronizationofthe
sensors;inthiscase,sometimestheyarecalledthecontrolnodes[34].Thepower
suppliernodesaretheothercriticalnodesandtheirresponsibilityistochargethe
3
sensorspowersource.Thesenodesareusuallymobile(e.g.adroneresponsiblefor
chargingthesensornodesusingwirelesscharging[91,69,41]).
Usually there isonlyonetypeofsensor in thenetwork,however insomecases
theremight bemore than one type of sensor. For example, the data of sonar
sensorscanbeusedwiththedataofmotionsensors[36,96],orinsomecases,like
weather sensors, it is necessary to have multiple sensors in one place work
together,thesetypesofsensornodesarecalledmulti-sensornode[124].
Figure 1: An environmental monitoring multi-sensors for monitoring weather [49].
Thetopologyofthenetworkandthedistancebetweensensorsarerelatedtothe
types of the sensors and the goal of the network. Sometimes the topology is
variable.Anexampleofvariable topology iswhen thesensornodesaremoving
(e.g.Crowdsourcedsensornetworksusingmobilephone[96,136,19,44,35,65,
61]orwearablesensors[60,62]),orwhenthesensornodesarebeingselectedand
selectionchangesovertime[130,134,33,125,101,53,131,132,137,8,72,81].
Theworkson[8,72,81,131,137]havediscussedabouttargettracking;whenever
there is a target trackingproblem, theenvironment indynamicand topology is
4
variablebecausethemovingtargetchangesitslocationbytimeandasensorthat
hadagoodapproximationofthetargetlocationattime𝑇maynothavesuchan
approximationatlatertime.
1.1. Powerrequirementofthesensors
Eachsensorinthenetworkneedsapowersourcetooperate.Thepowersource
can be an internal battery or it might be an external source of energy. The
rechargeable sourceof energy is the third and themost reliable typeof power
sourceinthenetworkwherethebatterycanbechargedusingenergyharvesting
from the environment [55, 6, 113, 107, 104, 90, 13, 12, 38, 115], (e.g. solar
charging)orintheabsenceoftheenvironmentalfeature,itispossibletocharge
thesensorsmanually(e.g.Wirelesschargingusingdrones[82,21,91]).However,
therearesituationswherechargingthesensor’sbatteryusingtheenvironmental
featuresisnotpossibleandthesensorsarehardlyaccessibletobechargedusing
wireless charging methods. Examples of each sensor networks are underwater
sensornetworks[48,18,66,121,47],sincethesenetworksareformedhundreds
ofmetersunderthewater,thereisnosunlighttochargethemandtheyarehardly
accessibleusinginstrumentslikerobotstoreplacetheirbatteries.Inthesesituation
forlong-termsensornetworks,consuminglessenergybyoptimizingthereadingof
thenetworkisimportant(seefigures2and3).
5
Figure 2: An illustration of the mobile under water sensor network (UWSN) architecture for short-term time-critical aquatic exploration applications [25]
Figure 3: An illustration of the mobile UWSN architecture for long-term non-time-critical aquatic monitoring applications [25]
6
Thesensorsinanetworkcaneitherconnectdirectlytogetherbycableortheymay
operateinwirelessmode.Physicalconnectionbetweensensorsisnotflexibleand
itisnotpossibleinvarietyofsituations(e.g.sensorsdroppedfromairplaneinthe
battlefields [16, 68]); on the other hand, wireless sensor networks may have
synchronizationproblems.Forexample,inunderwatersensornetworksthereisa
longlatencybetweenreadingofpackets[46,122,4,117].
1.2.Dependencybetweensensors
Sensorselectionorprediction requirescalculating thedependencybetweenthe
sensornodes.Usuallythereisalagbetweenthereadingvaluesoftwosensors;it
meansthereadingvalueofsensor𝑎attime𝑡maynotbedependentonthereading
valueofsensor𝑏atthesametime,anditmightbedependentonthereadingvalue
of𝑏attime𝑡 − ℎ.Thistypeofdependencyiscalledalagdependency.
Insomecases,thereisnoexactinformationaboutthedependencybetweensensor
nodes and it should be approximated.One of these cases is sensor placement;
sensorplacementistheprocessofselectingthebestsetofsensorplacesforsensor
deployment.Forexample,ifthereare20possiblelocationstodeploythesensors
andthereareonly5availablesensors, then it isnecessary toselectasubsetof
these locations to deploy the sensors. Sensor placement can be converted to
sensorselectionbyassumingthereare20fakesensorsonthefieldandtrytoselect
5 of them. In this case, because there is no real sensor deployed, the relation
betweenthesensorsshouldbeapproximated[27,73,64].
7
1.3.Classificationsofnetworks:
Anetworkcanbeclassifiedusingdifferentapproachesbasedon:1)thenatureof
theobserveddataovertimeand2)thephysicalpropertiesofthenetwork.Inthis
survey,thefirstapproachisused,sinceitismoretheoretical.
1. Dynamicversusstatic:Unlikethestaticnetwork,inthedynamicnetworkthe
dependencybetweensensorschangesbytime.Sometimesintheliterature
dynamicisreferredtothenatureofhavingmotioninthesensors.
2. Centrally controlled or distributed controlled: In a centrally controlled
networkthereisacentralnoderesponsibleforsynchronizationandselection
ofthesensors.Inthedistributedmodel,synchronizationand/orselectionof
thesensorsisperformedbythesensorsthemselves.
3. Stableornot:Inastablenetworkthereadingofthevalueofeachsensoris
availableatany time (i.e.Thesensorcanbe readatany time).Whenthe
network is not stable the valueof some sensors cannot be read at some
pointsbecause;theyarenotavailable.Exampleofthesesensorsarethefirst
respondersindisasterscenarios,whenthecommunicationbackboneisnot
stable to communicate with each first responders (i.e. sensor node) at
disasterlocations.
4. Real-time or not: The collected data from the sensors at the real-time
networksisnotdifferentthantheothernetworks,howeverbecauseofthe
natureof thenetwork and the limited time for processing thedata, data
collection using selected sensors in a real time network cannot be as
8
accurateas theothernetworks.Also, thecollecteddata fromthesensors
withhighlatencyisnotasvaluableastheothersensorswithlowerlatency.
1.4. Sensorselection
Sensorselectionistheprocessofselectingasubsetofsensorsforevaluationofan
objectivefunction.Therearetwomodelsforsensorselection:
1. Selectionofthesensorsbycentralserver[53,3,15,106,105,131,24,79,
26,132,54,11,111,25,40,112,108,102,133,2,39,125,101,94,63,88,
138,51,56]
2. Selectionofthesensorsbythemselves(Selfactivationofthesensors)[42,
57,7,109,118,9,17,80,37,31,20]
Sometimesthecommunicationcostisthemajorbottleneckforthenetwork[30,
29,32,45,126],especiallywhenthedataisbeingreadbythecentralserverina
multi-hopmethod, in this case self-activationof sensors canhelp to reduce the
communicationcost.Insomeapproaches[119,103,127,84],sensornodesrunthe
sameprediction function thathappens in theserveron theirpreviousvalues to
obtainanestimateoftheircurrentreadingandsendthereadingtotheserveronly
iftheestimatedvalueisfarfromtheactualreading.
1.4.1.Selectiongoal
Thedataofthenetworkcanbeusedforavarietyofapplications.Thisdatacanbe
rawdatacollectedfromallthesensors,oritmightbesomeprocesseddatathat
9
can be the result of selection and prediction of sensors. Sensor selection is
equivalent to feature selection in data mining process. In which the selected
sensorsaretheselectedfeatures.Theclasslabelcanbeanobjectivefunctionor
the value of the remaining sensors (that can be viewed as a specific objective
function).
Sometimesthegoalofsensorselection istoconservetheenergyofthesensors
[53],Inthiscase,eithercommunicatingthedataorreadingoftheenvironmental
phenomena is energy consuming, and this energy is expensive/impossible to
provide.Also,insomecasestherechargeablesourceofenergyhaslimitationsfor
the number of recharges (e.g. the rechargeable battery has limited numbers of
recharges)andnotallthesensorsshouldbeactiveatthesametime.
Sometimes the goal of the sensor selection is to conserve the bandwidth by
reducingthesizeofdata[42,83,71,116].Inthiscase,itispossibletousesomesort
ofdatacompressiontechniquetoreducethesizeofdata[77,75,120,85,110,129,
114],butanothergoodmethod,especiallywhenthedatahavelowcompression
rate,istousesensorselectiontoselectonlythemostvaluabledata.
Thelimitationsofthecentralserverfordataprocessingisanotherissue[76,135,
93,1].Inlargenetworks,thecentralservermaynothaveenoughprocessingpower
toprocessallthecollecteddataspeciallywhenthesystemisreal-time.Inthiscase
sensorselectionisrecommended.
Insomemodelsofsensorselection,thegoalistomaximizethelifetimeofallthe
sensors [74, 28, 86, 98, 52, 89, 78, 10, 95], Instead of minimizing the energy
10
consumptionof each individual sensor. In this case it is possible tomake some
sensors active in each iteration and try to minimize the overall battery
consumption,thiswaythesensorwiththeminimumbatteryhashigherweightfor
selection.
1.4.2. Sensorplacementversussensorselection
Thesensorplacementproblem[27,73,123,58,14,22,23,43]isatypeofsensor
selectioninwhichtheobjectivefunctionwillbeapproximatedoverthesimulated
data (i.e. Sensors values). The goal of sensor selection in the sensor placement
problemistoselectthebestlocationfordeploymentofthesensors.Usuallythe
goalofthesensorplacementistoefficientlymaximizethecoverage[27,73,123,
58,50].
1.5.Sensorprediction
Thetopicofpredictioninthenetworkmightbeviewedfromdifferentapproaches;
insomeliterature,predictionreferstothepredictionoverenergyconsumptionof
thesensors[67,59].However,inthissurvey,thepredictiontaskisrelatedtothe
collecteddata(eitherpredictingthevalueofeachsensor,oranobjectivefunction).
Thesensorpredictiontaskisadataminingtaskforpredictingtheunknownvalues
ofthesensornodes[137,42,3,25,94].Usuallythisprocessresultsincreationofa
prediction model using the training set that will be evaluated, for validation
purpose,onthetestset.Theattributesetofpredictiontasksarethevaluesofthose
sensors that have been selected using the sensor selection algorithms (Or the
11
sensornodesthatareavailableforread[70,128,97,99,87])andtheclasslabels
aretheremainingsensors.Sometimesthegoalofsensorselectionisnotpredicting
the valueof theother sensors, insteadweare interested in thepredictionof a
specific objective function (e.g. predicting the average), in this case the sensor
selectiontaskisutilizedtoproducethebestpredictionvalueofthefunctioninstead
ofpredictingthevalueofeachindividualsensorinthenetwork.
1.6.Motivation
There are many approaches to problems in the area of sensor selection and
prediction.Mathematicalcomparisonoftheseapproachesdoesnotgiveusagood
depthofknowledgeofhowatypicalresearchcanbeaccomplishedintheareaof
sensor selection and prediction. Different approaches use different calculation
methodstoselectand/orpredict.Thesemethodsmayhavedifferentapplications
indifferentscenarios.Forexample,asensorselectionmethodmightbegoodfor
preserving energy in one network and it might be good in preventing the
equipment fromwearing out in the other. For this reason, comparing different
methods of selection might be inappropriate. Instead of comparing different
methods, we prefer to compare different works of arts in the area of sensor
selectionandprediction.
Ourapproachinthissurveyistolookatthreedifferentpaperstoseehowatypical
sensorselectionandpredictionresearchcanbeaccomplished.Papersdiscussing
theproblemofsensorselectionandpredictionhavedifferentpointsofviewand
togethertheyarecomplementary.Eachpaperhasitsownuniquepointofviewto
12
thesensorselectionandpredictionproblemthatexplainshowtypicalresearchin
theareacanbeaccomplished.
The firstpaper (i.e. SensorSelection forMinimizingWorst-CasePredictionError
[25]) explains an offline model for sensor selection, this paper is especially
interestingbecausethegoalof itssensorselectionalgorithmsarenottopredict
thevalueofindividualsensorsandinsteaditwantstopredictafunctionoverthe
valueofall sensors (e.g. theaverage function). Ithasauniquepointof view to
sensorselection.
Thesecondpaper(i.e.Onlinedistributedsensorselection[42])explainsanonline
distributed algorithm for sensor selection. The viewpoint of this publication to
solve sensor selection and prediction is completely different; it starts from an
offline central selection and prediction to online single and multiple sensor
selection and prediction and then online distributed single andmultiple sensor
selectionandprediction.Thiswayitiseasiertounderstandthedistributedmodel
andcompareitwiththeothermodels.Thealgorithmsofthispaperonlyneedthe
objectivefunctionbesubmodularandtheydonotspecifywhattheformulationis.
Thethirdpaper(i.e.Ondynamicdata-drivenselectionofsensorstreams[3])has
specificallydefinedwhatistheformulationofobjectivefunctionandhowtosolve
thatformulation.Themodelofthethirdpaperisonline(dynamic)anditdiscusses
aboutthetopicoflagcorrelationbetweensensors.
Thisselectionisveryappropriatebecausetheviewofthereviewedpapersbecome
narrower andmore specific from the first one up to the third one and exactly
13
specifiesifsomeonewantstodoresearchinthesensornetworkarea,whatarethe
different depth of views. Beside that, almost all the variety of sensor
selection/predictionmodelsandalgorithmshasbeenusedintherelevantcontexts.
2. SensorSelectionforMinimizingWorst-CasePredictionError
2.1. Introduction
This paper explains an offline sensor selection andpredictionmodel inwhich a
streamofsensordataisusedtocreateanofflinemodelandthismodelisusedfor
predicting the value of specific aggregation functions. This method of sensor
selection and prediction is especially useful in real time systems (i.e.when the
processing time is limited) or in large networks, where it becomes difficult to
update themodelor createanewoneat run time. Thepaper reviewed in this
sectioncreatesamodelforminimizingtheworst-casepredictionerror.
In the following section, after notations and discussion about the target
aggregationfunctions(i.e.theaggregationfunctionsthatareusedinthispaper),
theadversarybasedsensorselectionandpredictionapproachisdiscussedanda
generalization of that approach for a specific class of aggregation functions is
explained, then the selection methods that are used for target aggregation
functionsandtherelatedexperimental resultsareexplained.Finally,wediscuss
thecritiquetothemethodsoftheliterature.
14
2.2. Notations
Inthispaper,eachsensorisdenotedby𝑥(,thesetofallsensorsby𝐴andtheset
ofselectedsensorsby𝑆.Thecostofselectingeachsensoristhesame(Unitcost).
Theaggregationfunction𝑓acceptsavectorofsensorsvalues𝑉andproducesone
number𝑓(𝑉) = 𝑈,thedistancebetweenthisnumberandtheestimatedvalueof
theaggregationfunction(𝑓’)overtheselectedsetiscalledpredictionerror(𝐸).The
numberofsensorsinthevectorislessthanorequaltothebudget𝐵.Thevalueof
sensor𝑥( isdenotedby𝑣( andthedistancebetweentwosensors isdenotedby
𝑑(𝑥(, 𝑥7) which satisfies non-negativity, symmetry and triangle inequality. The
distancebetweentwosensorscannotbesmallerthanthedifferencebetweentheir
reading values at any time (i.e. 𝑣( − 𝑣7 ≤ 𝑑(𝑥(, 𝑥7)) and it is the maximum
differencebetweenreadingvalueofthemintheirstreams.Thismaylooklikean
invalidassumptionforamajorityofnetworks,butif 𝑣( − 𝑣7 ≤ 𝜑 𝑑 𝑥(, 𝑥7 and
if 𝜑 is an increasing concave function, then we can assume 𝑑ʹ 𝑥(, 𝑥7 =
𝜑 𝑑 𝑥(, 𝑥7 and the settings are still valid. Themaximum difference between
readingsoftwosensorsintheirstreamsiscalledthedistancebetweenthosetwo
sensors.Theselectionprocessusesthedistancebetweenthesensorsratherthan
theirvalues.
Thepaperfocusesonoptimizingthreedifferentaggregationfunctions:Max,Min
andAveragethatarerespectivelyminimum,maximumandtheaveragevalueofall
sensors.Unlikethemostpublications,there isnosensorvalueprediction inthis
paper;insteadthefocusisonpredictingthevalueofanobjectivefunction(i.e.Max,
MinorAverage)usingalimitednumberofselectedsensors.
15
2.3.Adversary-basedselectionandpredictionapproach
Supposethedistancematrix(i.e.amatrixofalldistancesbetweenanytwosensors)
isgiventoanadversary.Theadversaryhasfullcontrolofwhatwouldbethevalue
of the sensors. The algorithm selects the candidate sensors by looking at the
distancematrix.Aftertheselectionprocessisfinished,theadversarywhoknows
what is the aggregate function (i.e. Knowswhat is the result of the prediction)
assignsvalidvalues(Avalidvalueassignmentisanassignmentthatthedifference
between thevalueofany twosensors isatmostequal to theirdistance) forall
sensorssuchthatitmaximizesthedistancebetweenthepredictionofthesensors
andtheirrealvalue.
Thebestvaluesthatanadversarycanselecttoproduceaworst-casescenarioin
generalistoselectzeroforthevaluesoftheselectedsetsandfornot-selectedset
assignthemaximumorminimumpossiblevaluestoincreasethedistancebetween
theselectedsetandtheothersensorsandincreasethepredictionerror(Proofisin
thepaper).
Figure 4: The choice of adversary for unselected set to increase the prediction error of average
However,forthisargumenttobevalidtheaggregationfunctionshouldhaveaa
specificfeaturethatiscalledita0-centeredaggregationfunction(discussedinthe
nextpart).Then,if𝑑 𝑥(, 𝑆 = min7∈?
𝑑(𝑥(, 𝑥7)theworst-casepredictionerrorforthe
16
average aggregation functionwill become:𝐸 𝑆 = @A 𝑑(𝑥(, 𝑆)BC∉? , for themax
𝐸 𝑆 = @EmaxBC∉?
𝑑(𝑥(, 𝑆)andforthemin𝐸(𝑆) = @EminBC∉?
𝑑(𝑥(, 𝑆).Theproofofthese
equationsislinkedtothedecisionoftheadversarythatafterselectingset𝑆itcan
decidetoextendthevaluesoftheremainingsettoeitheroftwodirections(assign
thesmallestpossiblevaluesforeachremainingsensororassignthelargestpossible
valuesforeachremainingsensor),thiscausesthemax,minoraveragefunctions
(thatare0-centered)toproducelargererrors.
2.3.1 Generalizedresults
Themodelofthispapercanbeusedforany0-centeredaggregationfunctions.A
functioniscalled0-centered,ifithasthefollowingspecifications:
1. Itismonotonicallynon-decreasinginallitsvariablesandhaswell-defined
firstandsecondpartialderivatives.
2. Thepartialderivativesaresymmetricaround0ineachvariable.Inother
words,foreach𝑥( and𝑥7,wehaveHHIC𝑓(𝑉) = H
HIC𝑓(𝑉7→KIL).𝑉7→KIL isa
vectorwithentriesequalto𝑉exceptthe𝑗’thentrywhichisequalto−𝑣7.
3. Forall𝑥( ,HE
HEIC𝑓 𝑉 ≤ 0whenever𝑣( > 0,and H
E
HEIC𝑓(𝑉) ≥ 0whenever
𝑣( < 0.
4. Partialderivativesaremaximizednear0,inthesensethat HHIC𝑓(0(→IC) ≥
HHIC𝑓(𝑉) for all𝑥( .0(→IC is a vectorwith zero entries except 𝑖’th entry
whichis𝑣(.
17
If𝑓 is0-centered,thenforanysubset𝑆ofsensors,𝐸(𝑆) = 𝐸(𝑆, 0(?)),the0(?)
meanstheassignmentofvalue0sensorsin𝑆.Thatis,theadversarycanmaximize
the prediction error by showing the value 0 at all sampled locations (Selected
sensors). Furthermore, the worst-case error is then exactly 𝐸 𝑆 = @E(𝑓(𝛥) −
𝑓(−𝛥)), where 𝛥 is the vector of distances from 𝑆, i.e., 𝛥( = 𝑑(𝑥(, 𝑆) =
minBL∈?
𝑑(𝑥(, 𝑥7).
2.4. Sensorselection
Theselectionalgorithmsinthispaperareofflinealgorithms.Thismeansthatthe
objectivefunctionshouldbeavailabletobeabletofindthecandidatesensorsfor
selectionalsoitisstaticbecausethesinglemodelwillbecreatedusingthedistance
matrix and the model will never change after that. The selection method is a
centralizedmethod.
Thepapersuggeststwomethodsofsensorselectionforaggregationfunctions.
1. 𝐾-medianalgorithm:Thealgorithmstartsbychainingaarbitrarysubsetof
sensors𝑆Tandparameter𝑝 ≤ 𝐵.Ateachiteration,ittriestominimizethe
predictionerrorbyreplacingatmost𝑚 ≤ 𝑝sensorsintheselectedsetby
sensors in unselected set. It continues until reduction in error is not
significant and returns set𝑆W after 𝑡 iterations. It is proven that this local
search algorithmhas3 + 2/𝑝 + 𝜀 approximation ratio (i.e., the algorithm
outputsaset𝑆suchthatthetotaldistanceofallnodesfrom𝑆 iswithina
3 + 2/𝑝 + 𝜀 factor of the total distance from the best set 𝑆 ∗). The
experimentsinthispaperhaveused𝑝 = 2thatis4 + 𝜀approximationratio.
18
2. 𝐾-centeralgorithm:Thisalgorithmstartswithonesensor𝑥(andtriestoadd
𝑥7 to𝑆suchthat𝑑(𝑥(, 𝑆)ismaximum.Itcontinuestheprocessuntil𝐾sensors
hasbeenselected.Thisalgorithmhasthe2+𝜀approximationratio.
Inthispaper𝐾-centeralgorithmisusedforpredictingthemaximumorminimum
and𝐾-medianforaverageaggregationfunction.
2.5.Experimentalresults
Multipledatasetshavebeenselectedtoevaluatetheresultsonthereal-worlddata.
Todosothetimeseriesvaluesofmultiplesensorsisusedondata.Thetrainingset
includesasampleofthevaluesoverthetimeserieswhichincludesthevaluesof
sensors for multiple times. Using these sample points, the maximum distance
betweeneachtwosensorsisestimated.Thisdistanceisusedtoselectaportionof
sensors(inthispaper10%ofthesensors)topredicttheaggregationfunctions.We
consideroneattributeeachfromthetwodatasets: lightfromthefirstdataset,
andhumidityfromtheseconddataset.Totesttheperformanceofthealgorithms
they have compared themwith the run time of the algorithmover50 random
samples.Theresultsofrunningthealgorithmsover50samplesaredisplayedinthe
followingtables:
Scheme AverageError MinimumError5-mediansamples 10.1 -5randomsamples 18.1 12.710randomsamples 13.5 10.115randomsamples 9.7 7.7Table 1: Predicting Average over light measurement
19
Scheme AverageError MinimumError5-centersamples 14.9 -5randomsamples 23.7 17.210randomsamples 19.8 15.115randomsamples 14.7 10.7Table 2: Predicting maximum over light measurement
Scheme AverageError MinimumError3-mediansamples 2.2 -3randomsamples 2.6 1.76randomsamples 2.0 1.39randomsamples 1.1 0.47Table 3: Predicting average over humidity sensors
Asyoucansee,topredicttheaverageandmaximum,smartselectionof5sensors
canbecomparedwithrandomselectionof15sensors.Butforthehumiditythe
resultsarenotthatinteresting.
2.6.Discussionand critiques
Themethods of this paper are not utilized to take complete advantage of the
dependenciesamongdatastreamsanditsimplyignoresthosedependencies;the
paper justmentioned the sensorsare correlatedbecauseof theirdistance.This
modelmightbeagoodmodel if it isdifficult tocalculate thedependenciesbut
usually it is easy to find a rough estimate of dependencies; this means the
applicationofthispapertotherealworldsensorselectionisnotrecommended.
The algorithmof this paper cannot be applied in any dynamic network (Where
dependencies between sensors change by time). However, themain goal is to
optimizefortheworst-casescenario.
20
Theotherissueabouttheresultsofthispaperisitdoesnotlookverypromisingon
somenetworkslikehumidity.Asyouseeusingthemethodsofthispapertheerror
is2.2 howevera randomsampleproducesvery closeerror2.6.And the lowest
errorover50samplesis1.7.Ontheotherhand,oneofthemostimportantbenefits
ofthistypeofsensorselectionisitssimilaritytosensorplacementmethods.This
meanshavingtheobjectivefunctionandasetofcandidate’slocationsitiseasyto
use thealgorithmsof thispaper to select thebestpossible locations for sensor
deployment.Alsothemethodsofthispaperareveryfastandsuitableforthereal-
timenetworkswithlargesensorsetandcomplexdependencies.
In the conclusion of the paper the authors recommend using an asymmetric
approachforsensorselection.Inthispaperthedefaultassumptionistofindout
whatisthemaximumdistancebetweentwosensors𝑥( and𝑥7 andusethatdistance
asalimitationbetweenthemaximumvalueof𝑥( and𝑥7 so𝑣( < 𝑑(𝑥(, 𝑥7) + 𝑣7 and
𝑣7 < 𝑑(𝑥(, 𝑥7) + 𝑣( but,sometimesitisbettertohavetwodistancesforexample
afterrunningthealgorithmwefindoutthatintheworstcase𝑥( = 𝑑 + 𝑥7 and𝑥7 =
3𝑑 + 𝑥(, in this case we can say 𝑑(𝑥(, 𝑥7) = 𝑑 and 𝑑(𝑥7, 𝑥() = 3𝑑. The new
distance function has differentmeaning than the old one𝑑(𝑥(, 𝑥7) = 𝑑 means
distancebetweenIandjwhenIismaximizedandjisminimized.Thismightbean
improvement to themethodsof thispaper.Despite theseproblems,oneof the
advantagesofthismethodofsensorselectionisitsapplicationinsensorplacement.
Thismeansbyhavingtheknowledgeaboutarrangementofthesensorsandtheir
distanceweselectthebestlocationstoplacesensors.Itisusefulbecauseusinga
roughestimateofthedistancebetweensensorsreadingandwithoutanyreading
informationitispossibletoselectthebestsensorslocationstoplacethesensors.
21
3. OnlineDistributedSensorSelection
3.1.Introduction
This paper discusses the distributed online sensor selection problem, when a
centralizedsensorselectionalgorithmisnotefficientinpracticeandtheobjective
function is not known in advance. Thepaperdoes a smooth transition froman
offlinemodeltoadistributedonlinemodelinwhichthecentralizedserverisnot
involved in theselectionprocess.Thepaperhasan interestingdiscussionabout
multiplesensorselectionstrategies.
In the following sections, after the notation, the simple offline sensor selection
problemand theonline sensor selectionproblemare explained.After that, the
distributedsingleandmulti sensorselectionareexplainedandaspecialcaseof
distributedsensorselectioninwhichthesensorscanonlycommunicatewiththe
centralserver(i.e.theycannotcommunicatewitheachother)isexplained.
3.2.Notation
Eachsensorisdenotedby𝑥(,thesetofallsensorsby𝐴( 𝐴 = 𝑛)andthesetof
selected sensorsby𝑆. The costof selectingeach sensor inuniquecostand the
budgetofselectionis𝐵thattotalnumberofselectedsensorscannotexceedthis
number. The weight of sensor 𝑥( is denoted by𝑤( (That can be seen as the
importance of that sensor for prediction task) and its activation probability is
denoted by 𝑝( (i.e. the probability that the sensor activates itself), also, the
probabilityofselecting𝑥( is𝑝ʹ(.Letter𝑓isusedfortheobjectivefunction,which
22
acceptsasetofselectedsensors𝑆andproducesonenumber𝑓(𝑆) = 𝑅,whichis
therewardofselectedsensors.Oneoftheassumptionofthispaperistheobjective
functionissub-modular,itmeansthatselectinganewsensorincorporatessmaller
increaseinthevalueoftotalrewardthanthesumoftherewardofthissensorand
thevalueofthetotalrewardbeforethisselection.Thevaluesofsensorsarenot
mentionedinthispaper;thepaperonlydiscussestheamountofincreasetothe
objective function value by adding sensor 𝑥( to the set of currently selected
sensors,butitdoesnotexplainhowtheobjectivefunctioniscalculated.
3.3.TheOfflinesensorselection
Theofflinesensorselectionproblemisthesimplestcase.Undertheofflinemodel
the objective function 𝑓 is known in advanced (i.e. we know how much the
selection of each individual sensor can increase the sensing quality objective
function). The information about the objective function can be obtain from
differentsourceslikedomainknowledgeordatafromapilotdeployment.
Intheofflinemodelasimplegreedyalgorithmproducesgoodapproximations;the
algorithmisa𝐵stepsensorselection.Ateachstep,thisalgorithmselectsasensor
thatmaximizesthetotalutility:
𝑥( = argmax 𝑓 𝑆jKk ∪ 𝑥7BL∈m\?op@
𝑆j = 𝑆jKk ∪ {𝑥(}
Thisgreedysolutionwillresultinatleastaconstantfractionoftheoptimal:
23
𝑓(𝑆s) ≥ 1 − 1 𝑒 max|?|vs
𝑓(𝑆)
Thisgreedysolutionisespeciallyusefulinsensorplacement.
3.4.Theonlinesensorselectionproblem
In the onlinemodel, there is no information about the objective function. The
onlinemodelhasanextraparameter𝑇whichdeterminesthenumberofiterations
thatsensorselectionprocessisrepeatedanditlearnstheobjectivefunctioninan
onlinemanner.
Thesolutionoftheonlinealgorithmswillbecomparedtothebeststrategythat
obtains reward 𝑚𝑎𝑥?⊆m:|?|vs 𝑓W(𝑆)yWzk . The difference between the optimal
rewardandobtainedrewardiscalledregret.Astrategyiscallednoregretstrategy
if itsaverage regret tends to zerowhen𝑇 → ∞ (Theaverage regret is the total
regretofallturnsoverthenumberofturns).
Even if the objective function is known in advance, it may change during the
iterations,theofflineselectionignoresthischangebuttheonlinesensorselection
adaptsitselfwiththischangeandconvergestobetterresults.
3.5.Centralizedonlinesinglesensorselection
Themainideabehindthecentralizedonlinesinglesensorselectionistoconvert
theproblemtoamulti-armedbandit(MAB)problem.UsingMABalgorithm,one
sensorwillbeselectedineachiteration.Thereadingoftheselectedsensorwillbe
24
processedbythecentralservertoproduceafeedback(therewardofthisselection)
forthenextiteration.Thisway,ineachiteration,abettersensorwillbeselected
untilitconvergestotheselectionofthebestsensorineachiteration.
The problem is called multi-armed bandit [139] because, each sensor can be
consideredasanarmforamulti-armedbanditmachineandselectionofthatarm
hassomerandompayoff(reward)basedonsomeprobabilitydistribution𝑝(,the
goalistofindthearmthathasthemaximumpayoff.𝐸𝑋𝑃3isoneofthealgorithms
that can solve the multi-armed bandit problem.𝐸𝑋𝑃3 stands for Exponential-
weightalgorithmforExplorationandExploitation.Itworksasfollows:
1. Given 𝛾 ∈ [0, 1], initialize the weights of each sensor𝑤((1) = 1 for 𝑖 =
1,… , 𝑛.
2. Ineachround𝑡:
I. Set𝑝( 𝑡 = 1 − 𝛾 𝑤((𝑡) 𝑤7(𝑡)�7zk + 𝛾 𝑁foreach𝑖.
II. Selecttheonesensor𝑥7 randomlyaccordingtothedistribution𝑝at
𝑡’thiteration.
III. Observereward𝑓W(𝑥7)(Thevalueofthesensingquality).
IV. Settheweightofthesensor𝑥7 to:𝑤7 𝑡 + 1 = 𝑤7 𝑡 𝑒(���(BL))/(�L�)
𝛾iscalledtheexplorationprobability,ifitiscloseto1thenthealgorithmexplores
otherwiseitexploitsbasedontheweight.Itisproventhatifsuitableparameters
are selected, then the relative regret𝑅y of𝐸𝑋𝑃3 becomes𝑂( 𝑇𝑛 ln 𝑛) so the
averageregret𝑅y/𝑇tendstothezero.
25
3.6.Centralizedonlinemultiplesensorselection
Theproblemofmultiplesensorselectionisitsscalability(i.e.therunningtimeof
algorithmgrowsexponentiallybythebudgetorthesizeofthenetwork),because,
mappingtheproblemtoamulti-armedbanditproblemwillincreasethenumber
ofarmsexponentiallybythenumberofsensors(i.e. �s -banditproblem).Inthis
case, the average regret becomes 𝑂 𝑛s � 𝑇 ln 𝑛 . However, if the utility
functionissub-modularthenthearmshavedependentvaluesandtheproblemcan
beconvertedto𝐵, 𝑛-armedbanditproblemwithasmallloseintheaccuracy.
M.StreeterandD.Golovin[113]usedthe𝑂𝐺��(Walgorithmtosolvethisproblem.
Themainideabehind𝑂𝐺��(Wistoconverttheofflinegreedyselectionofsensor𝑥(
totheonlinemulti-armedbanditproblem𝑀𝐴𝐵7.So𝑂𝐺��(W isanalgorithmthat
usesmultiplemulti-armedbanditalgorithms(Thenumberof𝑀𝐴𝐵sisequaltothe
budget𝐵) in parallel to do the sensor selection. Each𝑀𝐴𝐵7 is responsible for
addingthenextsensortothesetofalreadyselectedsensorsbytheprevious𝑀𝐴𝐵
algorithms.Ineachiterationallthese𝑀𝐴𝐵algorithmsworktogetherandeachof
themtriestooptimizeitsownselectionofsensor.
1. Initialize 𝐵 multi-armed bandit algorithms 𝑀𝐴𝐵k,… ,𝑀𝐴𝐵s for each
candidatesensorforselection.
2. Foreachround𝑡 ∈ 𝑇:
I. Foreach𝑖 ∈ 𝐵inparallel𝑀𝐴𝐵( selectssensor𝑥WC (Theassumptionis
notwoalgorithmsselectthesamesensor).
26
II. Foreach𝑖 ≤ 𝐵inparallelfeedback𝑓W(𝑥W@, … , 𝑥WC)–𝑓W(𝑥W@, … , 𝑥WCp@)
to𝑀𝐴𝐵( which is the relative increase of the utility function (the
amountof increasetotheutility functionbyadding𝑥WCtothesetof
selectedsensors).
Thisalgorithmhasa(1 − 1/𝑒)-regretboundof𝑂(𝐵𝑅)[114](Assumingeach𝑀𝐴𝐵(
hasexpectedregretatmost𝑅andthereare𝐵algorithms).Inthispaper𝐸𝑋𝑃3is
selectedfor𝑀𝐴𝐵(Sothefeedbackistherewardofselectionthatwillbeusedto
adjusttheweightofsensor𝑥WC for𝑀𝐴𝐵().Thus,theabovealgorithmhasno-(1 −
1/𝑒)-regret(i.e.𝑅 = 𝑂 𝑇𝑛 ln 𝑛 and𝐵 ≤ 𝑛then limy→∞
s y� ���y
= 0).
3.7.Distributedalgorithmforonlinesensorselection
Unlikecentralizedalgorithms,distributedalgorithmsdonotneedcentralserverfor
selectionand they cando the selectionby communicatingwitheachother and
activatingthemselves.DistributedOnlineGreedy 𝐷𝑂𝐺 isanefficientalgorithm
forthedistributedonlinesensorselectionthathasthreesimpleassumptions:
1. Eachsensor isable tocompute itscontribution to theutilityof thesetof
selectedsensors.
2. Eachsensorcanbroadcasttoallothersensors.
3. Thesensorshavecalibratedclocksandunique,linearlyorderedidentifiers.
Indistributedsensorselection,eachsensorhasanactivationprobability𝑝( (i.e.in
eachiterationitactivateswithprobability𝑝()andthecentralserverisnotinvolved
intheselectionprocess.Afterasetsofsensorshasbeenselectedtheycalculate
27
theobjectivefunctionandeachofthembroadcastsafeedbacktoallsensorsinthe
networkforthenextiteration.
3.7.1.Onlinedistributedsinglesensorselection
Toselectasinglesensorinadistributedmannertherearetwocommonstrategies:
Thefirststrategyistouseanaïvedistributedsamplingscheme;Usingthismethod,
asinglesensorwillproducearandomnumber𝑢andbroadcaststhisnumbertothe
othersensors, thesensor𝑥( forwhich 𝑝77�( < 𝑢 ≤ 𝑝77v( willbeactivated.
Theproblemof thismethod is thatall thesensorsneedtokeeptrackofall the
activationprobability𝑝( (Theactivationprobabilitieswillchangeineachiteration
andeachsensorneedtokeepanupdatedversion),memorywisethisisnotagood
strategy. On the other hand, each sensormay keep track of its own activation
probabilityandthenthesensorsstartsendingtheiractivationprobabilityaccording
to their 𝐼𝐷 and they stop once the sum is greater than 𝑢. In this case, 𝜃(𝑛)
messagesneedtobesentwhichisimpractical.
Thesecondstrategyisdistributedmultinomialsampling;Thismethodneeds𝑂(1)
messagesandaconstantamountoflocalstoragehoweverinthismethodthereis
a probability of activating no sensor or more than one sensor so it should be
somehowmanaged.
Supposetheinputdistributionforsampling is𝑝, thismeanseachsensortriesto
automaticallyactivatebasedonitsactivationprobability𝑝(.Let𝑝’betheresulting
distributionand𝑝ʹ( betheprobabilitythatsensor𝑥( isactivatedandselected.One
method is to let the sensors activate themselves automatically based on their
28
probability𝑝. Ifmore thanone sensor is activatedornothing is activated, then
repeat the experiment. The problem of this type of sampling is the difference
betweeninputdistributionandtheoutputdistribution.Forexample,ifthereare
twosensors𝑝k = 𝜀,𝑝� = 1 − 𝜀then,fortheasmall𝜀thisalgorithmyields𝑝kʹ =
𝜀� 1 − 2𝜀 + 2𝜀� = 𝑂 𝜀� ,sothefirstsensorwillbeseverelyunderrepresented.
Thesecondmethodistostartsamplingandifthemorethanonesensorisselected
then try to select one of them uniformly at random. This method is also not
promising, suppose there are three sensors with the following probability
distribution;𝑝k = 0.1,𝑝� = 0.3,𝑝� = 0.6,thentheoutputdistributionbecomes:
𝑃’k = 0.1×0.7×0.4 + 0.1×0.3×0.4×0.5 + 0.1×0.7×0.6×0.5
+ 0.1×0.3×0.6×0.33 = 0.061
𝑃’� = 0.9×0.3×0.4 + 0.1×0.3×0.4×0.5 + 0.9×0.3×0.6×0.5
+ 0.1×0.3×0.6×0.33 = 0.201
𝑃’� = 0.9×0.7×0.6 + 0.9×0.3×0.6×0.5 + 0.1×0.7×0.6×0.5
+ 0.1×0.3×0.6×0.33 = 0.486
𝑃’T = 0.9×0.7×0.4 = 0.252
Theratioofinputdistributionandoutputdistributionwillbedifferentfordifferent
sensors: 𝑝k/𝑝’k ≅ 1.64, 𝑝�/𝑝’� ≅ 1.49, 𝑝�/𝑝’� ≅ 1.23. And as you see the
probabilityofselecting𝑥�is8times𝑥kbuttheinputprobabilitywas6times.
29
Thethirdmethodistoassumetheactivationprobabilityofeachsensor𝑥( is𝑘(/𝑁.
Thenconverteachsensor𝑥( to𝑘( fakesensors𝑥(7 eachwithactivationprobability
of 1/𝑁. After that, for each sensor 𝑥( take a sample of binomial distribution
𝐵𝑖𝑛𝑜𝑚𝑖𝑎𝑙 𝑁. 𝑝( ,k�
, if more than one fake sensor is activated, activate the
relatedsensor.Finally,ifmorethanonesensorisactivated,trytoactivateoneof
thembasedontheratioofitsfakesensorsoverthetotalnumberoffakesensors,
this helps to maintain the activation probability ratio. If𝑁 → ∞ the binomial
distributionwillbeconvertedtoaPoissondistribution(i.e.𝑃𝑜𝑖𝑠𝑠𝑜𝑛 𝑝( ).
Also,itispossibletomultiplytheactivationprobabilityonsomevariable∝∈ [1, 𝑛]
andsamplefrom𝑃𝑜𝑖𝑠𝑠𝑜𝑛 𝛼𝑝I andtheratiosstaysthesame;wecallthismethod
PoissonMultinomialSampling(𝑃𝑀𝑆).Thisway,theprobabilityofactivatingmore
than0sensorswillincrease.Theexpectednumberofmessages𝐶thatwillbesend
bythismethodbecomes:
𝔼 𝐶 = Pr 𝑋( ≥ 1 =(
(1 − 𝑒K¤�C)(
≤ 𝛼𝑝((
= 𝛼
In the broadcast model, running 𝐸𝑋𝑃3 using 𝑃𝑀𝑆 Protocol with 𝛼 = 1, and
rerunningtheprotocolwhenevernothingisselected,yieldsexactlythesameregret
boundasstandard𝐸𝑋𝑃3,andineachroundatmost ¥¥–k
+ 2 ≈ 3.582messages
arebroadcastedinexpectation.
30
3.7.2.Thedistributedonlinegreedy(DOG)algorithm
To develop themultiple sensor selection,𝐵 distributed𝐸𝑋𝑃3 sensor selection
algorithms𝑀𝐴𝐵( are used. Using this method each sensor 𝑥( has 𝐵 weights
𝑤(k, …𝑤(s and𝐵 normalizing constants𝑍(k, …𝑍(s (for𝐵 multi-armed bandit),
eachnormalizingconstantequalstothetotalweightofallsensorsforeach𝑀𝐴𝐵(
(𝑖 = 1… 𝐵).Bothweightsandnormalizingconstantsarestoredinthesensoritself.
Usingthe𝑃𝑀𝑆protocolappliedtothedistribution(1 − 𝛾)𝑤(7/𝑍(7 + 𝛾/𝑛select𝑘
sensors such that, each𝑀𝐴𝐵7 algorithm tries to maximize the value of 𝑓W(𝑆 ∪
{𝑥(}) − 𝑓W(𝑆)byselectingthebestsinglesensor𝑥(.Eachselectedsensorcomputes
itsrewardanditsnewweight𝑤ʹ(7 andsendsthedifferencebetweenitsoldweight
andnewweighttoalltheothersensorstoupdatetheirnormalizingconstants.This
way,thereisnoneedtokeepstrackofallweights(Eachsensorneedstokeepits
ownweight and the total weights for each𝑀𝐴𝐵7 algorithm). This algorithm in
expectation needs 𝑂(𝐵) messages in each selection round. The comparison
betweentheofflinealgorithmandDOGalgorithmisshowninfigure1.Wehave
selected5sensorsusingexplorationprobability𝛾 = 0.01.
Figure 5: Experimental results on [T] Temperature data, [R] precipitation data
31
3.7.3.Lazyrenormalizationanddistributed𝑬𝑿𝑷𝟑(Singlesensorselection)
Sometimesthesensorscannotcommunicatewitheachother,alsocalculatingthe
payoffmaynotbepossibleforindividualsensors(Computationallyexpensive).In
thiscase,thereshouldbeacentralservertocontrolthecommunicationsandthe
modeliscalledthestarnetworkmodel.
The𝐸𝑋𝑃3algorithmneedstomaintainadistributionoveractionsandupdatethis
distributionineachround.In𝐸𝑋𝑃3,eachsensorneedstostoreitsownweightand
thesumoftotalweights in itsmemory. Inthelazyrenormalization,eachsensor
needs to store the same information; however, because the sensors cannot
communicatewitheachother,onceasensorhasbeenselected,itcannotsendits
feedback (i.e. difference between its newweight and old weight) to the other
sensorsdirectly. Inthestarmodel,eachsensorcanonlycommunicatewiththe
server.Initiallyeachsensorhastheweightof𝑤( = 1,andnormalizingconstantof
𝑍( = 𝑛.Theserverstoresonlythenormalizingconstant𝑍,and𝑍( getsupdatedto
𝑍 when the sensor communicates with the server; this way, there is no
communicationoverhead.
To do a single sensor selection using lazy renormalizationmethod, each sensor
samplesbasedon𝑞(~𝑃𝑜𝑖𝑠𝑠𝑜𝑛(𝑝().Ifthesampledvalueisgreaterthan1,then,it
communicates with the server and send its sampled value 𝑞(. The server in
response,sendsthenormalizingconstant𝑍(i.e.Updatedweightofallsensors)and
updatedweight𝑤( backtothesensor(Thisiswherethenormalizingconstantof
thesensorbecomesupdated.Asensorwithoutdatednormalizingconstantsuses
the old normalizing constant until its sampling value is greater than 1 (i.e. it
32
communicateswithserver).Then,asinglesensoramongtheactivatedsensorswill
beselectedbytheserver.Usingthismethod,thesumofactivationprobabilityof
allsensorsbecomesmorethanone(Becausesomenormalizingconstantsarenot
updated)andthereisoveractivationofsensors,however,itsignificantlyreduces
thecommunicationcost.Thenumberofsensoractivationusingthismethodis∝
+(𝑒 − 1)inexpectationand𝑂(∝ +𝑙𝑜𝑔𝑛)withhighprobabilityandthenumber
ofmessagesisatmosttwicetheactivation.
In the star model the sensors do not need to know anything about objective
functionandtheycanbeactivatedinadistributedmanner,howevertheserveris
responsibleforcommunicatingwithalltheactivatedsensorsandselectingoneof
thembasedonitsprobability.
3.7.4.LazyDOG
The 𝑙𝑎𝑧𝑦𝐷𝑂𝐺 uses the same 𝑂𝐺��(W algorithm but 𝑃𝑀𝑆 algorithm uses lazy
renormalization scheme for its𝐸𝑋𝑃3 algorithm. For 𝑙𝑎𝑧𝑦𝐷𝑂𝐺, the number of
activated sensors in each round is “𝐵𝑙𝑛𝑛 + 𝑂(𝐵)” in expectation and
𝑂(𝐵𝑙𝑜𝑔𝑛)withhighprobabilityandthenumberofmessagesistwicethenumber
ofactivation.
3.7.5.ObservationDependent-DistributedOnlineGreedy(OD-DOG)
Inmanyapplications,wewouldliketoperformwelloniterationswith“atypical”
objectivefunctions.Forexample,inanoutbreakdetectionapplication,wewould
liketogetaverygooddataonroundswithsignificantevents.Inthissituationwe
33
prefer𝐷𝑂𝐺algorithmtohavesomeoveractivationofsensorstoletthebestsensor
tobeactivatedineachiterationtoobtainthemaximumpayoff.
𝑂𝐷 − 𝐷𝑂𝐺assignsathreshold𝑇tothesensorsandtriestoactivatethesensors
basedonboththresholdandprobabilitythatcausesoveractivationofsensors.In
each roundof𝑂𝐷 − 𝐷𝑂𝐺 for each𝑀𝐴𝐵 each sensor𝑥( calculates its estimate
payoff𝜋( ≅ 𝑓W 𝑥( by using some local information from the environment and
activatesif𝜋( ≥ 𝑇.
In 𝑂𝐷 − 𝐷𝑂𝐺, one of the important problems is to decide what is the best
threshold to activate the sensors. To answer to this question, assume that
threshold𝑇 is selected and our goal is to somehow evaluate how good is this
threshold.Todothat,wespecifyarewardfunctionforactivatedsensors.Asyou
know,theactivationofsensorsisbasedonbothprobabilityandthreshold,results
inoveractivation,so,tocalculatetherewardofthreshold𝑇,weneedtoassignan
activationcost𝑐forthesensorsthatactivatebasedontheirthreshold𝜋 ≥ 𝑇.
To specify the reward function, Suppose set𝐷 of sensors are activated on this
iteration for a specific𝑀𝑂𝐵7. If the marginal benefit of selecting sensor 𝑥( be
𝜋( = 𝑓W 𝑆 ∪ {𝑥(} − 𝑓W 𝑆 ,thentherewardofthisactivatedsensorbecome:
𝜓( 𝑇 =𝑐( − max 𝜋( − 𝑚𝑎𝑥 𝜋 ´\BC , 0 𝑖𝑓𝜋µ < 𝑇max 𝜋( − 𝑚𝑎𝑥 𝜋 ´\BC , 0 − 𝑐(𝑖𝑓𝜋µ ≥ 𝑇
Asyousee,ifweonlyusethresholdforsensorselection,thepayoffofanactivated
sensor(𝜋 ≥ 𝑇)istheimprovementoverthebestpayoffamongallotheractivated
34
sensorsminusitscost.Therewardofthreshold𝑇j isthesumoftherewardsofall
activatedsensor.
Now,supposethereisathresholdset{𝑇( ∶ 𝑖 ∈ [1, … ,𝑚]},oneofthemethodsto
select the best threshold is to use a randomized weighted majority algorithm
overtime. This algorithmmaintainsweights𝑤 𝑇( = exp 𝜂. 𝜓WºW»¼ 𝑇( for each
threshold 𝑇(, where 𝜂 > 0 is a learning parameter and𝜓WºW»¼ 𝑇( is the total
cumulativerewardforplaying𝑇( ineveryroundsofar.Ineachiteration,foreach
𝑀𝐴𝐵7,thealgorithmselectsathreshold𝑇( withprobabilityof𝑤 𝑇( / 𝑤 𝑇77∈[½]
andthenupdatesthe𝑤 𝑇( usingthetotalobtainedreward.Thisalgorithmcan
optimize theselectionof thresholdovertimeand lets theperformanceof𝑂𝐷 −
𝐷𝑂𝐺behigherthan𝐷𝑂𝐺,butitneedstheactivationcosttobeselectedcorrectly.
Also,𝑊𝑀𝑅 algorithm can bemodified towork for the star network setting to
obtainabetterresult.Infigure6youseetheresultofrunning𝑂𝐷 − 𝐷𝑂𝐺onwater
distributionnetworkdata.Thealgorithmselects10sensorsoutof12537sensors
and optimizes its selection overtime. The results drastically are improved
comparedto𝐷𝑂𝐺.
Figure 6: Experimental results on [W] water distribution network data
35
3.8.Discussionandcritiques
One of the interesting approaches in this paper is its generalization over the
objectivefunction.Unliketheotherreviewsofoursurveythatspecifythetypeof
objectivefunction,thisworkonlyassumestheobjectivefunctionissub-modular
and it can be anything, it can even change during the sensor selection process
whichisanadvantage.Alsohavinglazyrenormalizationversionofsensorselection
besidetheoriginaldistributedversionmakesthealgorithmsmoreuseable;itmight
be even possible to use combination of bothmethods for a network (e.g. Use
𝐿𝑎𝑧𝑦𝐷𝑂𝐺unlessyouneedmoreaccuratepredictions).
Oneoftheproblemsofthepaperisthelazyrenormalizationmethod.Theproblem
is thesensornormalizationparametercanbeupdatedonly if it isactivatedand
communicateswiththeserver.However,itispossibletheserversendsabroadcast
messagetoall thesensors, insteadofreachingeachsensor individually,sothey
knowtheupdatedvaluerightaway,butinthepaperitismentionedthattoremove
thecommunicationoverheadtheparameterofeachsensorwillbeupdatedonlyif
itdirectlycommunicateswiththeserver.
Finally,itisnevermentionedthathowthesetofcandidatethresholdsshouldbe
selectedinitiallyandhowtodecidewhatisthebestvalueforactivationcostusing
OD-DOGalgorithmanditslooklikeitshouldbeselectedbytryanderrorwhichis
notagoodmethodforhighlydynamicnetworks.
36
4. OnDynamicData-DrivenSelectionofSensorStreams
4.1.Introduction
This paper covers an online sensor selection problem where dependencies
betweensensorsdynamicallychangeovertimeandamodelthatisworkingforthe
selectionofthesensorsattime𝑇kmaynotworkforthethesamenetworkattime
𝑇�,𝑇� > 𝑇k.Theselectionmodelofthispaperissimilartothemodeldiscussedin
section3(i.e.Onlinedistributedsensorselection)butitisbasedonawell-defined
centralized algorithm (Central server calculates the utility function and decides
whichsensorsshouldbeselected).
Themaingoalofthesensorselection inthiswork istopreservethebattery.To
accomplishthisgoal,aftertheselectiontaskisfinished,thevalueoftheselected
sensorswillbereadinmultipleconsequentturns,thatarecalledactiveturns,and
usedtopredictthevalueofunselectedsensorsinthoseturns.Aftersequenceof
activeturns,thevalueofallsensorswillbereadinaturnthatiscalledapassive
turn.Basedonthereadingofthemostrecentpassiveturns,theactivesensorset
ofthenextintervalwillbeselected.Theartofthispaperistofindawaytoswitch
betweentheactiveandpassiveselectionmodesinsomespecificupdateintervals
(numberofactiveturnsbeforeapassiveturn)tofindtheoptimizedselectionof
sensors for each interval of data collection. This way it preserves the power
requirementbyactivatingonlyafewsensorsformultipleturnsandthentryingto
maintain(update)theactivesetusingthecollecteddataofthepassiveturns.In
figure7youseetheexampleofthismethodofthesensorselectionwithupdate
intervalof4.
37
The rest of this review after the notation includes discussion about the sensor
selectionprocessandhowthesensorpredictiontaskworksandhowtooptimize
thesensorselectionprocessintermsofthespeedandpredictionaccuracy.Finally,
the experimental result and critiques about the methods of this paper are
discussed.
Figure 7: Example of the concept of passive and active turns; in this figure each row represents one iteration of data collection and each column represents a sensor. In this case the budget of sensor selection is 5 and the update interval is 4.
38
4.2. Notation
Eachsensor isdenotedby𝑥(, thesetofallsensorsby𝐴andthesetofselected
sensorsby𝑆.Thevalueofsensor𝑥( isrepresentedby𝑣( andthecostis𝑐( whichis
thepowerrequirementofthatsensor.Ourobjectivefunctionis𝑓whichacceptsa
setofsensors𝑆withtotalcostof𝐵(Thebudget)andproducesonenumber𝑓 𝑆 =
𝐸,theaverageerrorofthatset(Thepaperdefinestheobjectivefunctionandtries
tofindasolutionforthat).
Inaddition,theupdateinterval𝑅,isthenumberofactiveturnsbeforeapassive
turn,windowsize𝑊,isthenumberofpassiveturnsusedintheregressionmodel,
andthemaximumlag𝑚𝑎𝑥𝑙𝑎𝑔,determineswhatisthemaximumlagbetweenthe
readingofanytwosensors.
4.3. Localregressionclustering
Inthispart,theproblemofpredictingthevalueoftheunselectedsensorsissolved.
Atanymomentintime,eachstream(Valuesofaspecificsensorovertime)belongs
toaparticularcluster(Setofstreams).Theseclustersmayevolveovertimeasthe
underlying stream patterns change. The approach of the paper is a𝑘-medoids
based partitioning approach in which the data is clustered around 𝑘-medoids.
Thesemedoidsareusedastherepresentativesoftheclustersandourprediction
willbearoundthem.
Supposewewanttopredictsensor𝑥7 from𝑥((Whichisoneofthemedoids)and
wehavethevalueforbothsensorsforthelast𝑤passiveturns.Wecall𝑤thesize
ofthewindow.Ifthereissomelag𝑟forpredicting𝑥7 from𝑥( thenwecanwrite
thefollowing:
39
𝑣7 𝑘 = 𝑎. 𝑣( 𝑘 − 𝑟 + 𝑏∀𝑘 ∈ 𝑟 + 1… 𝑤
Sothegoalistoselectgoodconstants𝑎and𝑏suchthatitminimizestheerror𝑒
whichcanbedefinedasthefollowing:
𝑒 𝑥(, 𝑥7, 𝑟 = 𝑎. 𝑣( 𝑘 − 𝑟 + 𝑏–𝑣7𝑘�
Â
jzÃÄk
Supposetheoptimalvaluesfor𝑎and𝑏are𝑎∗and𝑏∗,thentheoptimalvaluefor𝑒
becomes𝑒∗,andforthelag𝑟theaverageerrorofpredictionbecomes:
𝐸∗ 𝑥(, 𝑥7, 𝑟 =𝑒∗ 𝑥(, 𝑥7, 𝑟𝑤 − 𝑟
The distance function 𝐷 𝑥(, 𝑥7 between sensors 𝑥( and 𝑥7 is defined as the
minimumvalueof𝐸∗ 𝑥(, 𝑥7, 𝑟 overallpossiblevaluesofthelag𝑟 ∈ [0,𝑚𝑎𝑥𝑙𝑎𝑔]
(Notethat𝑚𝑎𝑥𝑙𝑎𝑔isavariablethatshouldbeselectedcarefullytooptimizethe
tradeoffbetweentheerrorandtherunningtime).
𝐷 𝑥(, 𝑥7 = 𝑚𝑖𝑛Ã∈ T,½»B¼»Å 𝐸∗ 𝑥(, 𝑥7, 𝑟
Usingunivariate regression analysisweuse theoptimal valueof𝑎∗ and𝑏∗ and
optimalvalueforthelag𝑟∗topredictsensor𝑥7 fromitsclosestsensor𝑥(∗.Wecan
usethispredictionmodelforthenextsectiontoselecttheoptimalsetofsensors
asourmedoids,thestrategyistoselectrandommedoidsatthebeginningandthen
graduallytochangethem.
4.4. Sensorselectionprocess
Todothesensorselectiontask,thefirstpriorityistodefinetheobjectivefunction
(asyouseethisreviewismoredetailedthanthereviewinsection3andit’sbased
40
onawelldefinedobjectivefunctionformulation).Supposewehavethevalueofall
sensorsforlast𝑤passiveiterations,𝑒𝑟𝑟 𝑆, 𝑥7 𝑡 ,𝑀 istheerrorofpredictingthe
sensor𝑥7 fromtheselectedset𝑆usingmodel𝑀onawindowofsize𝑤atturn𝑡(In
univariate regression analysis it becomes 𝐷W 𝑥(∗, 𝑥7 such that
𝑥(∗ ∈ 𝑆is the closest sensor to 𝑥7 ∉ 𝑆). Suppose𝑦(𝑆, 𝑥7(𝑡),𝑀) is the estimated
valueofsensor𝑥7 at turn𝑡 (Explained intheprevioussectionfortheunivariate
case)thentofindtheerroronawindowofsize𝑤wehave:
𝑒𝑟𝑟(𝑆, 𝑥7(𝑡),𝑀) = |𝑦(𝑆, 𝑥7(𝑡 − 𝑟),𝑀)–𝑣7(𝑡 − 𝑟)|ÂKk
ÃzT
Thusthesensorselectionproblemcanbedefinedas:
Byhavingthemodel𝑀and|𝐴| = 𝑛,determineasubsetofsensors𝑆tofindthe
optimumvalueof𝑓 𝑆,𝑀 inwhich:
𝑓(𝑆,𝑀) = ¥Ãà ?,BL W ,Æ�K ?(∉? and 𝑐((∉? ≤ 𝐵
Byhavingthisobjectivefunction,wecaneasilydothesensorselectiontask,the
taskismaintaining(Updating)theactivesetaftereachpassiveiteration.
MaintainActiveSet (Sensor Streams: [𝟏 … 𝐧], Power Constraint: 𝐁)
S = Randomly sampled set of sensors with aggregate power requirement less than B
Repeat
Add that sensor to S which leads to maximum decrease in prediction error
While S violates power constraints
Drop the sensor from S which leads to the least increase of the prediction error
Until S did not change in the last iteration.
Algorithm 1: Maintaining active set of sensors
41
4.5. SpeedingupwithRankingVariations
Finding the sensor thatminimizes the error is a little bit slow because in each
iterationofthemaintainactiveset,weshouldgothroughallsensorstofindthe
onethatminimizestheobjectivefunction.Oneofthetechniquestoovercomethis
problem is to go through all the sensors in the first iteration, then rank all the
sensors based on their regression predictability and for the next iteration go
throughthesensorsequentiallybasedontheirregressionpredictabilityandstop
wheneverthepredictionvaluedoesn’tchangeanymore.Thismethodmaynotbe
asaccurateastheoriginalmethodbutitisaveryefficientintermsofrunningtime.
4.6. LeveragingLocal-RegressionClustersforPrediction
Sofarwehaveonlytalkedabouttheunivariateregressionmodels.However,ifwe
havemore computing power thenwe can predict sensor𝑥7 not only fromone
sensor,butfromalltheselectedsensors.Thiswaytheaccuracywillincreasebut
thespeeddecreases.Toimprovethetimeefficiency,wecanusetherankinglink
togetherwiththismethod.
4.7. Experimentalresults
Twobaselineshavebeenselected:sampledunivariateandsampledmultivariate
whichrespectivelydoesunivariateregressionanalysisandmultivariateregression
analysisona randomsampleof the sensors. Forpowerefficient sampling (PES)
threemodesareused:PESunivariate,PESmultivariateandPESmultivariateusing
a ranking list (𝑅𝐿). Experiments onmultiple datasets shows a small difference
betweentheaccuracyofthemultivariatepowerefficientsamplingusingaranking
list.Without a ranking list, however, running time usingmultivariate PES using
42
ranking listwasthesameasunivariatePESandbetter thanregularmultivariate
PES.Alsointermofpredictability,multivariatesensorpredictionwasalwaysbetter
thantheunivariateone.Thesesamplingmethodsaretestedondifferentwindow
sizes,maximumlagsandupdateintervals.Alsotheresultsaretestedondifferent
budgets(powerconstraints).Asexpectedbyincreasingthemaximumlagorbudget
the error reduces and by increasing the maximum update interval the error
increases.Forthewindowsize,theresultwasveryinterestingbecausebymaking
thewindowsizetoolargeitwillsmoothouttheresultanditincreasestheerror,
bymaking it toosmall it cannotcaptureall correlationsand theerror increases
again.Sotheparametershouldbeselectedverycarefully.Finally,therunningtime
analysisshowssignificantimprovementforusingrankinglistinprediction.
Figure 8: Error Plot on Power Constraint; as you see the relative error of all algorithms is reducing by increasing the
power constraint (Budget).
Figure 9: Error Plot on Window Size. Too big and too small window size are increasing the error.
43
Figure 10: Error Plot on Update Interval; increase in the update interval will increase the error.
Figure 11: Error Plot on Maximum Lag. Increase in the lag correlation will increase the error.
Figure 12: Running Time (Data Set intel-humidity). PES multivariate is substantially slower than the other methods, however, using ranking list it works similar to the PES univariate.
4.8. DiscussionandCritique
Thispaperhasmoredetailthanthepaperofsectionthree.Itexplainswhatshould
beformulationoftheobjectivefunctionandwhatisthegoalofsensorselection
(i.e.Toreducethepowerconsumption).Inthispapertheobjectivefunctionisnot
knowninadvancedandaftereachpassiveturn,itdoesaregressionanalysistofind
44
thebestsetofsensorsthathavethebestpredictability.Thissetmaychangefor
each interval. Two methods of regression analysis have been explained:
multivariate andunivariate.Using a ranking listwas amethodwhich helped to
reducetherunningtimeofthebothmethods.
There are some issues in this paper: First of all, the settings are not optimal,
becausetheupdateintervalisalwaysafixednumberanditisnotdependenton
the predictability of the sensors. A better strategy is tomake it variable: if the
predictabilityisbetterthansomethreshold,thenincreasethesizeoftheupdate
interval. This produces a lower number of passive turns and less power
consumption.Amorecritical issueof thispaper is thedefinitionof theoptimal
model.Thepaperassignsaselectionbudgetwhichistheaveragesumofpower
requirement inactivesetsanditdoesnottalkaboutthepassiveturns.Abetter
objectiveistoassignthebudgetastheaveragepowerconsumptionofeachturn.
Thiswaythegoalwillchangefromonlyselectinganactivesettoselectinganactive
setandupdateintervaltogether.
Anotherissueofthepaperisthatitneverexplainshowtomaketheparameters
optimal.Thismagnifiesanotherbigproblem;inthispapereveryparameterisused
fortheentiresensorsetbutitshouldbelikethat(thefirstissue).Forexample,if
youuseabigvaluefor𝑚𝑎𝑥𝑙𝑎𝑔parameterthenthespeeddecreasesandifyouuse
smallvalueaccuracydecreases.Thesolutionistofindoutforwhichpair𝑚𝑎𝑥𝑙𝑎𝑔
shouldbehighandforwhichsmallandusecorrespondingvalues.Thesamecanbe
trueforthewindowsize.
Finally,thispaper,onlytakescareofthepowerconsumption,thenetworklifeis
notimportant.Onestrategyistoincreasethecostofselectionofthesensorswith
45
the lowenergy to be able tomaintain the network for a longer periodof time
(howeverthismayreducetheaccuracyofprediction).
5. Summeryandconclusion
Inthissurveywehaveinvestigatedthemethodofsensorselectionandprediction
in the context of three papers. Each paper approached to the problem from a
differentpointofview.Thestaticmodelofsensornetworkhasbeendiscussedin
the first paper andpart of the secondpaper (Greedy sensor selection) and the
dynamic model has been discussed in the second and the third paper. The
centralizedsensorselectionmodelwasmentionedinallpapersbutthedistributed
versionwas only in the second paper. Stable network is notmentioned in any
papershoweverthemodelofthesecondpaperfordistributedgreedyselectionis
afaulttolerancemodelandcanacceptunstableenvironment.Afastmodelisthe
onethatcanbeusedfortherealtimesystems.Allthemodelsofthreepaperscan
beusedforsuitablereal-timenetworks.Forexample,inanetworkwithverylow
temporalvariancethemodelof thefirstpaperwhich isastaticmodelmightbe
useful.Inanetworkwithhightemporalvariancethemodelofthethirdpaperusing
smallupdateintervalmightbeuseful.Ifthereisnocentralserverthenthemodels
ofthesecondpapermightbeuseful.
Fromanotherpointofvieweachpaperexplainstheprobleminadifferentway;the
firstpaperlooksfromaverygeneralviewtotheproblemandtriestodothesensor
selectionusingaspecifiedobjectivefunctionwhichisamathematicalfunctionover
thevalueofallthesensors.Forpredictionpartittriestopredictthevalueofthis
function.Inthesecondpaperwhatistheobjectivefunctionisnotspecifiedandthe
goalofsensorselectionispredictionoverthevalueofunknownsensorsandinthe
46
thirdpapertheobjective function isdefinedandchangesover timeandhowto
calculate theobjective functionandhow topredict thevalueofeach individual
sensoriscompletelyexplained.
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