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.

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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.

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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

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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.

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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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