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Modeling, Representing and Querying the Uncertainty of Moving Objects in Spatio-temporal Databases

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Title:
Modeling, Representing and Querying the Uncertainty of Moving Objects in Spatio-temporal Databases
Creator:
Liu, Hechen
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
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Language:
english
Physical Description:
1 online resource (185 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Computer Engineering
Computer and Information Science and Engineering
Committee Chair:
Schneider, Markus
Committee Members:
Xia, Ye
Kahveci, Tamer
Dobra, Alin
Matyas, Corene J.
Graduation Date:
12/15/2012

Subjects

Subjects / Keywords:
Balloons ( jstor )
Cardinal points ( jstor )
Connected regions ( jstor )
Data types ( jstor )
Databases ( jstor )
Hurricanes ( jstor )
Mining ( jstor )
Modeling ( jstor )
Pendants ( jstor )
Trajectories ( jstor )
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
gis -- modeling -- movingobjects -- uncertainty
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Computer Engineering thesis, Ph.D.

Notes

Abstract:
Moving objects such as moving points, moving lines and moving regions are widely observed in the real world. They describe the continuous changing of locations and geometries over time and  involve many fields such as location-based services (LBS), artificial intelligence and transportation management, etc. As the advance of geographical information systems (GIS), researchers have been studying moving objects in the context of moving object databases. This dissertation studies the spatio-temporal uncertainty problem in moving objects. Spatio-temporal uncertainty is an intrinsic feature of moving objects due to the inability of precisely capturing their continuously changing locations over time. The uncertainty feature of moving objects exists in two scenarios. The first scenario is about moving objects in the past. Since the exact trajectories of moving objects cannot be captured by devices such as GPS sensors all the time, the locations of a moving object when it is not being tracked is unknown and the uncertainty exists. Queries like whether two cars could possibly meet during their past movements can not be answered  without knowing the exact movement of both moving objects. In the second scenario, the uncertainty also exists in the movement in the future. Lacking the observations of future movement, the locations of moving objects in the future can only be predicted. The query such as whether an airplane will enter hurricane Katrina in two days in the future could be of interest. This dissertation performs a study on the spatio-temporal uncertainty problem of moving objects in the database context. The author provides solutions to the problem in three stages. In the first stage, the author provides several abstract models to represent historical and future moving objects which take care of the uncertainty feature. The author proposes three uncertainty models, the pendant model representing the moving objects with uncertainty in the past, the balloon model which represents both historical and future uncertain movements, and a model using data mining approach to predict future locations of moving objects. Operations and predicates which can be used to query moving objects are defined under these models. In the second stage, the author designs discrete representations for the uncertainty models proposed in the first stage, including data structures for moving objects and efficient algorithms for operations and predicates. The final goal of this research is to provide a solution to the spatio-temporal uncertainty problem in the database context. Therefore, in the third stage, the author implement the uncertain moving object data types, operations and predicates as software packages and integrate them into extensible database systems and query languages. ( en )
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2012.
Local:
Adviser: Schneider, Markus.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2014-12-31
Statement of Responsibility:
by Hechen Liu.

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Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Embargo Date:
12/31/2014
Resource Identifier:
870531697 ( OCLC )
Classification:
LD1780 2012 ( lcc )

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MODELING,REPRESENTINGANDQUERYINGTHEUNCERTAINTYOFMOVINGOBJECTSINSPATIO-TEMPORALDATABASESByHECHENLIUADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012HechenLiu 2

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Tomyparents 3

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ACKNOWLEDGMENTS Iwouldliketogivemygreatestappreciationtomyadvisor,Dr.MarkusSchneider.ThankyouforyourpatientguidanceforyearswhichhelpmeestablishedthemostinterestingresearchtopicinSpatialDatabasesandGIS,andpublishedover10researchpaperswithmeinconferencesallovertheworld.Ithankmydissertationcommittee,Dr.AlinDobra,Dr.YeXia,Dr.TamerKahveciandDr.CoreneMatyasforyouadviceonmyresearch.IwouldliketogivemyspecialthankstomymentorsinMicrosoftResearchAsiaduringmyinternship,Dr.YuZhengandDr.XingXie,fromwhomIhavelearnedalotonhowtoconverttheresearchresultstoindustryproducts.Thetopicwehaveworkedtogethercomposespartofthisdissertation.IthankDr.Shen-ShyangHo,assistantprofessorinNanyangTechnologicalUniversity,Singapore,fortheadviceonourjointproject.Ithankmylab-matesandco-authorsDr.TaoChen,Dr.GaneshViswanathan,Dr.WenjieYuan,Dr.ViruKanjilal,LinQiandAnshuRanjanforthebrainstormandhelponmyresearch.Finally,IthankmydearestfriendsinUniversityofFloridaforbeingtogetherwithmeintheseyearsandprovidinghelptomylifeandstudy,includingJieXiang,ChenYang,LiangliangJiang,XiaozhenShen,HongjieDong,ChuanSun,TaoLi,XiaokeQin,JieFan,XinyueFan,HengxingTan,HuafengJin,MeizhuLiu,XuelianXiao,YuchenXie,JianminChen,LixiaChen,BoLi,Iek-HengChu,OuZhang,LiXie,YanDeng,JipengTan,KunLiandotherswhosenamescannotallbelisted.Iwillcherishthefriendshipwithyouforever.Inparticular,IthankZhuoHuang,myboyfriendforthelove,encouragementandsupportonmystudy,myfuturecareerandmylife. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 13 CHAPTER 1INTRODUCTION ................................... 15 1.1Motivation .................................... 15 1.2SolutionsandApproaches ........................... 17 1.3OrganizationoftheDocument ......................... 19 2RELATEDWORK .................................. 20 2.1ModelsofMovingObjects ........................... 20 2.1.1ModelsofMovingObjectTrajectories ................. 20 2.1.2ModelsofSpatio-TemporalUncertainty ................ 22 2.1.3ModelsofSpatialRelationshipsandSpatio-temporalRelationships 26 2.2ImplementationofMovingObjectsandQueriesinDatabases ....... 27 2.2.1MovingObjectImplementationandIndexes ............. 28 2.2.2QueryingMovingObjectsinDatabases ............... 28 3ABSTRACTMODELSOFUNCERTAINTYINHISTORICALANDPREDICTIVEMOVINGOBJECTS ................................. 31 3.1PendantModel:RepresentingHistoricalMovingObjectswithUncertainty 31 3.1.1Motivation:TheUncertaintyofMovingObjectsinthePast ..... 32 3.1.2RepresentingHistoricalMovingObjectswithUncertainty ...... 36 3.1.3OperationsonHistoricalMovingObjectswithUncertainty ..... 39 3.1.4Spatio-temporalUncertainPredicates ................ 43 3.1.4.1OverviewofSpatio-temporalUncertainPredicates .... 44 3.1.4.2PredicatesbetweenUncertainMovingPoints ....... 45 3.1.4.3PredicatesbetweenanUncertainMovingPointandanUncertainMovingRegion .................. 49 3.1.4.4PredicatesbetweenUncertainMovingRegions ...... 53 3.1.5Spatio-TemporalUncertainQueryLanguage ............. 56 3.1.6UncertainCardinalDirectionDevelopmentPredicates ....... 59 3.1.6.1CardinalDirectionsbetweenStaticPoints ......... 59 3.1.6.2TheDevelopmentofCardinalDirectionsbetweenTwoMovingPoints ........................ 60 3.1.7UncertainTopologicalChangesofComplexMovingRegions .... 66 5

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3.1.7.1ComplexRegions ...................... 66 3.1.7.2SnapshotRepresentationofMovingRegions ....... 71 3.1.7.3EvaluatingtheTopologicalChangesinaComplexMovingRegion ............................ 72 3.2BalloonModel:RepresentingHistoricalandPredictiveMovingObjects .. 77 3.2.1TheNatureofMovingObjects ..................... 77 3.2.1.1DissimilarityMeasurement ................. 78 3.2.1.2TheContinuityPropertyofMovingObjects ........ 80 3.2.1.3ModelingHistoricalMovingObjects ............ 87 3.2.1.4ModelingFutureMovingObjectswithUncertainty .... 89 3.2.2BalloonDataTypes:RepresentingHistoricalandPredictiveMovingObjectswithUncertainty ........................ 94 3.2.3OperationsonBalloonDataTypes .................. 97 3.2.3.1OperationsonPredictiveMovingObjects ......... 97 3.2.3.2OperationsonBalloonObjects ............... 101 3.2.4Spatio-TemporalPredicates ...................... 104 3.2.4.1Spatio-TemporalPredicatesonBalloonModel ...... 104 3.2.4.2ModelingPotentialPredicatesintheFuture ........ 110 4REPRESENTATIONOFHISTORICALANDPREDICTIVEMOVINGOBJECTSWITHUNCERTAINTY ................................ 113 4.1RepresentingMovingObjects ......................... 113 4.1.1SliceRepresentationofMovingObjects ............... 113 4.1.2RepresentingUncertainMovingObjects ............... 115 4.2AlgorithmsofMovingObjectswithUncertainty ............... 117 4.2.1AlgorithmsofComputingCardinalDirectionDevelopments ..... 117 4.2.2AlgorithmsofSpatio-temporalUncertainPredicates ........ 118 4.3InferringFutureLocationsofMovingObjectsfromSimilarTrajectories .. 121 4.3.1Overview ................................ 121 4.3.2MiningUncertainTrajectories ..................... 122 4.3.3RouteGeneration ............................ 124 4.4SimilarityMeasurementofMovingObjectTrajectories ........... 126 4.4.1Overview ................................ 126 4.4.2PreliminaryandSystemFramework ................. 128 4.4.2.1Preliminary .......................... 128 4.4.2.2SystemFramework ..................... 130 4.4.3MiningSimilarTrajectories ....................... 130 4.4.3.1Sub-trajectoryPartitioning .................. 131 4.4.3.2GeographicDistance .................... 133 4.4.3.3SemanticSimilarity ..................... 137 5IMPLEMENTATIONOFMOVINGOBJECTSWITHUNCERTAINTY ....... 142 5.1ReviewofANovelApproachonImplementingComplexSpatialDataTypes ...................................... 142 6

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5.1.1IBLOB:StoreComplexSpatialObjectsUsingIntelligentLargeObjects ................................. 143 5.1.2TypeStructureSpecication(TSS) .................. 147 5.2ImplementationofUncertainMovingObjectDataTypesusingIBLOBandTSS ..................................... 149 5.2.1ImplementationofthePendantModel ................ 149 5.2.2ImplementationoftheBalloonModel ................. 151 6SYSTEM,QUERIESANDEXPERIMENTS .................... 153 6.1QueryingHistoricalMovingObjects:CardinalDirectionDevelopmentfromDataofNationalHurricaneCenter(NHC) ............... 153 6.1.1Environment ............................... 153 6.1.2EntireCardinalDirectionDevelopmentQuery ............ 154 6.1.3ExistentialQuery ............................ 156 6.1.4Top-kQuery ............................... 156 6.2ARouteDiscoverySystemforPredictiveMovingObjects ......... 158 6.2.1OverviewoftheSystem ........................ 158 6.2.2SystemImplementation ........................ 159 6.2.3ExperimentalEvaluation ........................ 160 6.3BalloonSystem:QueryHistoricalandPredictiveMovingObjects ..... 162 6.3.1SupportofSpatio-temporalUncertainQueries ............ 162 6.3.2DemoSystemofHistoricalandPredictiveQueries ......... 163 6.3.3ExperimentalStudy ........................... 165 6.3.3.1StatisticsoftheDataset ................... 166 6.3.3.2Accuracyanalysis ...................... 167 6.3.3.3RuntimeAnalysis ...................... 168 6.3.3.4Discussion .......................... 168 6.4ImplementationandExperimentsofInferringFutureLocationsfromSimilarTrajectories ................................... 170 6.4.1SettingsandDataPreprocessing ................... 171 6.4.2EffectsEvaluation ............................ 171 7CONCLUSIONS ................................... 174 REFERENCES ....................................... 177 BIOGRAPHICALSKETCH ................................ 185 7

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LISTOFTABLES Table page 3-1OperationsonthePendantmodel ......................... 40 3-2ComponentsofanSTUPexpression ........................ 44 3-3STUPpredicatesunderthependantmodel .................... 46 3-4Operationsonhistoricalandpredictivemovingobjects. ............. 98 3-5Operationsonballoonobjectsandmovingballoonobjects ............ 103 3-6Assigningnamingprexestopairwisecombinationsofinteractions. ...... 108 3-7Numberofballoonpredicatesbetweenballoon pp,balloon pr,andballoon rrobjects. ........................................ 110 3-8Inferringthetypesofinteractionbetweenactualobjects ............. 111 4-1NotationsofParameters ............................... 122 4-2Notations ....................................... 131 5-1GrammarOfTSS ................................... 148 5-2TSSGrammarRepresentationofRegion ..................... 148 5-3TSSGrammarRepresentationofUncertainMovingPoints ............ 150 5-4TSSGrammarRepresentationofUncertainMovingRegions .......... 151 5-5TSSGrammarRepresentationofBalloon prObjects ............... 152 6-1SummaryofDatasets ................................ 160 6-2Spatio-temporalqueriessupportedbythesystem ................. 164 6-3StatisticsofDatasets ................................. 167 6-4SummaryofDataset ................................. 171 8

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LISTOFFIGURES Figure page 3-1Thelinearmovementbetweentwoconsecutiveobservations,andthepartialmovementbetween[ts,te] .............................. 33 3-2Apendantmovementandapartialpendantmovement .............. 34 3-3AumPointmovementwhichconsistsofasequenceofcertainanduncertainmovement ....................................... 38 3-4Uncertainmovementofacircleregionandapolygonregion ........... 39 3-5TheatInstantandtemporalSelectoperations ................... 43 3-6Thedomainsofaninstantpredicateandanintervalpredicate .......... 45 3-7Thedifferencebetweenaninstantpredicateandanintervalpredicate ..... 46 3-8Instantpredicatesofmovingpointswithuncertainty ............... 48 3-9Examplesofspatio-temporaluncertainpredicatesbetweentwomovingpoints 49 3-10Examplesofspatio-temporaluncertainpredicatesbetweenamovingpointandamovingregion ................................. 50 3-11Examplesofspatio-temporaluncertainpredicatesbetweenamovingpointandamovingregion ................................. 53 3-12Examplesofspatio-temporaluncertainpredicatesbetweentwomovingregions 54 3-13Examplesofspatio-temporaluncertainpredicatesbetweentwomovingregions 56 3-14Examplesoftwomovingpointswithchangedandunchangeddirectionsovertime .......................................... 61 3-15Thestatetransitiondiagramofallcardinaldirections ............... 63 3-16Anexampleofacardinaldirectiondevelopment .................. 64 3-17Adisconnectedregion,aregionwithdanglingpointsandlineswhichisnotclose,andanunboundedregion .......................... 68 3-18Sixdifferentshapesofregionobjects ........................ 70 3-19TwosnapshotsO1andO2ofamovingregionR .................. 72 3-20Thestatetransitiondiagramrepresentingvalidtopologicalchangesofamovingregion ......................................... 73 9

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3-21Dissimilarityfunctiondenedontypesofinteger,real,point,points,lineandregion ......................................... 80 3-22Singlecomponentmovingobjects ......................... 84 3-23Examplesofphi-continuousfunctions ....................... 84 3-24Examplesofdiscontinuityimplyingvalidtopologicalchanges .......... 86 3-25Event--discontinuity ................................. 87 3-26Themovementofamovingobjectinthefuture12-hourperiod ......... 90 3-27Differentcondencedistributionfunctions ..................... 92 3-28Thetimeinstancet0,thehistoricalandpredictivetimeintervals ......... 95 3-29Sixvalidballoondatatypes ............................. 96 3-30Examplesofpredictionsatatimeinstantunderdifferentuncertaintymodels .. 101 3-31Possiblerelationshipsbetweenpartsofballoonobjects ............. 106 3-32Afuturecrossingsituationbetweenaballoon ppobjectPandaballoon probjectR ........................................ 107 4-1Examplesoftheslicerepresentations ....................... 115 4-2Sliceunitrepresentationofmovingobjectswithuncertainty ........... 116 4-3Algorithmsthatcomputethetime-synchronizedintervalrenementfortwomovingpoints,andcardinaldirectiondevelopments ............... 119 4-4ThealgorithmtestUnitIntersectiontodetermineswhethertwoslicesintersect 120 4-5Thealgorithmofpossibly encounter ........................ 120 4-6Anexampleofthetrajectoryindexingstructure .................. 123 4-7Examplesofcorrelatedtrajectories ......................... 124 4-8Anexampleofconstructingaconnectedregion. ................. 124 4-9Edgeinferenceandremoveredundantedge ................... 125 4-10Querytransformation,routing,androuterenement ............... 125 4-11Trajectoriesformedbyusercheck-insequencesfromFoursquarec ....... 127 4-12Examplesofarawtrajectoryandasemantictrajectory .............. 129 4-13Systemframeworkforinferringfuturelocations .................. 131 10

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4-14Anexamplethatawholetrajectorymaynotworkinidentifyingclustersandusingturnstodetectthepointtopartitionsub-trajectories ............ 132 4-15Differentsituationsconsideredwhenmeasuringthesimilarity .......... 134 4-16Centersofmassofdifferenttrajectories ...................... 135 4-17Thealgorithmofrevisedlongestcommonsubsequencetodeterminesthesemanticsimilarityratiobetweentwotrajectories ................. 139 5-1Aregionobjectasanexampleofacomplex,structuredapplicationobject ... 145 5-2Thelayeredarchitecture,theintegratedarchitectureandtheiBLOBsolution .. 145 5-3Thestructureindexesinsidearegionwhichconsistsofnfacesandnoffsets 146 5-4AnexampleofsequenceindexesinsideaniBLOB ................ 146 5-5ThehierarchicaliBLOBrepresentationofaregionobject ............. 147 5-6ThehierarchicaliBLOBrepresentationofaregionobject ............. 147 5-7Sliceunitrepresentationofmovingobjectswithuncertainty ........... 149 5-8TSSDesignofUncertainMovingPointsinthePendantModel .......... 150 5-9TSSDesignofUncertainMovingRegionsinthePendantModel ........ 151 5-10TSSDesignofUncertainMovingObjectintheBalloonModel .......... 152 6-1ThetrajectoriesofhurricanesPHILIPPEANDRITA. ............... 154 6-2Overviewofthesystemframework ......................... 158 6-3Userinterface ..................................... 159 6-4Trajectoryqualityanduser'scontribution ...................... 161 6-5RoutablegraphconstructionontheareaofNYC ................. 161 6-6EffectandeffeciencyoftheMiningUncertaintyTrajectory(MUT)algorithm .. 162 6-7Ademosystemofspatio-temporaluncertainqueries ............... 164 6-8Visualizationofhistoricalqueries .......................... 165 6-918-hourpredictionmadeatAug-25-200518:00:00and6hourslater ...... 166 6-1072-hourstreamingpredictionwithupdatesevery6hours ............ 166 6-11Accuracyanalysisintermsofhitrate ........................ 168 6-12Predictionofallhurricanesfrom2005to2010 ................... 169 11

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6-13Inaccuratepredictioncostbysuddenchangeofdirections ............ 170 6-14Accuracyanalysisintermsoferrordistancesandtheruntime .......... 170 6-15Numberofvisitstodifferentcategories ....................... 172 6-16Resultsofon-lineprediction ............................. 173 12

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyMODELING,REPRESENTINGANDQUERYINGTHEUNCERTAINTYOFMOVINGOBJECTSINSPATIO-TEMPORALDATABASESByHechenLiuDecember2012Chair:MarkusSchneiderMajor:ComputerEngineeringMovingobjectssuchasmovingpoints,movinglinesandmovingregionsarewidelyobservedintherealworld.Theydescribethecontinuouschangingoflocationsandgeometriesovertimeandinvolvemanyeldssuchaslocation-basedservices(LBS),articialintelligenceandtransportationmanagement,etc.Astheadvanceofgeographicalinformationsystems(GIS),researchershavebeenstudyingmovingobjectsinthecontextofmovingobjectdatabases.Thisdissertationstudiesthespatio-temporaluncertaintyprobleminmovingobjects.Spatio-temporaluncertaintyisanintrinsicfeatureofmovingobjectsduetotheinabilityofpreciselycapturingtheircontinuouslychanginglocationsovertime.Theuncertaintyfeatureofmovingobjectsexistsintwoscenarios.Therstscenarioisaboutmovingobjectsinthepast.SincetheexacttrajectoriesofmovingobjectscannotbecapturedbydevicessuchasGPSsensorsallthetime,thelocationsofamovingobjectwhenitisnotbeingtrackedisunknownandtheuncertaintyexists.Querieslikewhethertwocarscouldpossiblymeetduringtheirpastmovementscannotbeansweredwithoutknowingtheexactmovementofbothmovingobjects.Inthesecondscenario,theuncertaintyalsoexistsinthemovementinthefuture.Lackingtheobservationsoffuturemovement,thelocationsofmovingobjectsinthefuturecanonlybepredicted.ThequerysuchaswhetheranairplanewillenterhurricaneKatrinaintwodaysinthefuturecouldbeofinterest. 13

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Thisdissertationperformsastudyonthespatio-temporaluncertaintyproblemofmovingobjectsinthedatabasecontext.Theauthorprovidessolutionstotheprobleminthreestages.Intherststage,theauthorprovidesseveralabstractmodelstorepresenthistoricalandfuturemovingobjectswhichtakecareoftheuncertaintyfeature.Theauthorproposesthreeuncertaintymodels,thependantmodelrepresentingthemovingobjectswithuncertaintyinthepast,theballoonmodelwhichrepresentsbothhistoricalandfutureuncertainmovements,andamodelusingdataminingapproachtopredictfuturelocationsofmovingobjects.Operationsandpredicateswhichcanbeusedtoquerymovingobjectsaredenedunderthesemodels.Inthesecondstage,theauthordesignsdiscreterepresentationsfortheuncertaintymodelsproposedintherststage,includingdatastructuresformovingobjectsandefcientalgorithmsforoperationsandpredicates.Thenalgoalofthisresearchistoprovideasolutiontothespatio-temporaluncertaintyprobleminthedatabasecontext.Therefore,inthethirdstage,theauthorimplementtheuncertainmovingobjectdatatypes,operationsandpredicatesassoftwarepackagesandintegratethemintoextensibledatabasesystemsandquerylanguages. 14

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CHAPTER1INTRODUCTION 1.1MotivationInrecentyears,astheadvanceofmobiledevicesandthetechnologyofglobalpositioningsystems(GPS),therehavebeenemergingapplicationssuchaslocation-basedservices(LBS),mobilesensornetworks,etc.Theyallinvolvethestudyofmovingob-jects.Movingobjectssuchasmovingpoints,movinglinesandmovingregionshaveacommonpropertythattheirlocationsorgeometriesarechangingcontinuouslyovertime.Therehasbeenanintensivestudyonmovingobjectsinthepastdecadeintermsofmovingobjectdatabases[ 28 29 ].Amovingobjectcanberepresentedbyalistof(time,latitude,longitude)tuples,whichformsatrajectory.Previousresearchershaveperformedthestudyontrajectoriestoanalyzethemovementsofmovingobjects.Amovementfunction,oramotionvectorcanbedetectedbyextractingthethemovementpatternofthemovingobjectfromitstrajectory[ 73 ].Forexample,giventwoconsecutiveGPSpoints(t1,x1,y1)and(t2,x2,y2)ofacardrivingonthehighwaywhoselocationsarecapturedattimet1andt2respectively,wecanassumethatthemovementbetweent1andt2islinear.Thereforealinearfunctiondescribesthemovementofthiscarcanbederivedandwecanobtainthelocationofthiscaratanyinstantbetweent1andt2byapplyingthefunctiontothemovementofthecar.However,intheseperiodswhenamovingobjectisnotbeingtracked,orthemovingpatterncannotbeeasilydetected,thelocationofthismovingobjectisnotdeterministic.Themostrecentlocationcapturedbythedevicesmaynotreectthereallocationofthemovingobject,andthustheun-certaintyexists.Uncertaintyisaninherentfeatureofmovingobjectsduetotheinabilityofcapturingtheexactlocationsalloverthetime.Whenuncertaintyexists,thepossiblelocationofamovingobjectisnotasinglepointbutcanbeanywherewithinanuncertainarea.Severalapproacheshavebeenproposedtomanagetheuncertaintyinmoving 15

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objectdatabases,includingthe3Dcylindermodel[ 83 ][ 80 ],space-timeprismmodel[ 62 ][ 31 ][ 18 ][ 57 ],etc.Therearetwomajorscenarioswheretheuncertaintymayexist.Therstscenarioisaboutthemovingobjectinthepastwhichtrajectoriescannotbetrackedprecisely.Forexample,atatimeinstancewhenacarisnotbeinglocatedbyGPSdevices,itspossiblelocationcanbeanywhereinanuncertainarea.Theuncertaintyoflocationsbringsuncertaintytomanyotheraspectsofmovingobjects.Oneaspectreferstothetopologicalrelationships.Atopologicalrelationship,suchasmeet,disjointorinside,characterizestherelativepositionbetweentwospatialobjects.Inthespatio-temporalcontext,topologicalrelationshipsbetweenmovingobjectsarenotconstantbutaredeveloping.Forexample,anairplaneisdisjointwithahurricaneatthebeginning,andlateritiesapproachingthehurricane,andnallylocatesinsidethehurricane.Thisdevelopingrelationshipisnamedasenter[ 21 ].DuetotheinabilityoftrackingthecontinuousmovementofmovingobjectsallthetimeduetotheimprecisionoftheGPSdevices,themovementsbetweenthesetimeinstantsarealwaysuncertain,thereforethedevelopingtopologicalrelationshipsbetweenmovingobjectscannotbeinferredprecisely.Thesecondscenariowheretheuncertaintyexistsisthemovementofmovingobjectsinthefuture.Duetolimitedobservationsoffuturemovement,thelocationsofmovingobjectsinthefuturecanonlybepredicted.Forexample,whenscientistsstudythemovementofhurricanestopreventsignicantdamage,theymaybeinterestedinsuchakindofquery,TellmeallhurricanesthatwillpossiblyenterFloridainthenextmonth?SincehurricanescanmovefreelyinthetwodimensionalEuclideanspaceandarenotlimitedbyconstraintssuchasroadnetworks,theirmovementsinthefutureareunknown.Thereforethelocationsofahurricaneinthefutureinuncertainwithoutanyspecicknowledge.However,ifweobtaintheknowledgerelatedtothepatternofahurricane'smovement,suchasitsmovementfunctionatthecurrentmoment,the 16

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locationofthishurricaneatatimeinstanceinthenearfuturecouldbepredictable.QuerieslikeWhatistheprobabilitythatHurricaneKatrinawilltraverseFloridainthenext5days?couldbeanswered.Thegoalofthisresearchistoprovideapproachestorepresentandqueryhistoricalandfuturemovingobjectsinthedatabasecontextwhichtakescareoftheuncertaintyproperty.Thestorywillstartfromtheabstractionofmovingobjectsintherealworld,thendesignaconcreterepresentationincludingdatastructuresandalgorithmsforvariouskindofmovingobjectdatatypessuchasmovingpoints,movinglinesandmovingregions.Andthenalstepistheimplementationofthemovingobjectswithuncertaintyinextensibledatabasesystems. 1.2SolutionsandApproachesThisresearchprovidesolutionstotheprobleminthreestages.Intherststage,theauthorprovidesthreeabstractmodelstorepresenthistoricalandfuturemovingobjectswhichtakecareoftheuncertaintyfeature.Therstabstractmodeliscalledthependantmodel,whichrepresentsthetemporalevolutionofmovingobjectsinanuncertainenvironmentinadatabasescontext.Thismodelisbasedonthewell-knownspace-timeprismmodel.Inthesimplestcase,weassumethatthemovementbetweentwoobservationsarelinear.Inanuncertainenvironment,however,becauseofthemaximumspeedconstraint,thepossiblelocationsbetweentwoobservationscannotexceedadouble-conevolume.Amovingobjectwithuncertaintyisrepresentedbyaanecklacewhichcomposedbylinearcomponents(string)anduncertaincomponents(pendant).Asanimportantpartofthismodel,wedeneasetofspatio-temporaloperationsaswellasbooleanpredicatesnamedasspatio-temporaluncertaintypredicates(STUP)whichdescribethedevelopmentsoftopologicalrelationshipsbetweenmovingobjects.Thebenetofintroducingthesepredicatesisthattheycanbeusedasselectionandjoinconditionsindatabasequerylanguages. 17

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Thesecondabstractmodeliscalledtheballoonmodel,whichrepresentsbothhistoricalandpredictivefuturemovementswhichemphasisonthefuturepartofmovingobjects.Thetermballoonisgeneratedfromthemetaphorofaballoontorepresentamovingobjectinthe2D+timespacewhoselifetimelastsfromthepasttothefuture.Thetailoftheballoonrepresentsthehistoricalpartwhichisalreadyknown,andthebodyoftheballoonrepresentsthefuturepartofthemovementwhichisuncertain.Theballoonmodelisappliedspecicallytohandletheuncertainmovementsofmovingobjectswhichlastfromthepasttothenearfuture.Insteadofrestrictingtheuncertaintymovementofamovingobjecttoacylinderoracone,thefuturepartoftheballoonmodeldoesnothaveamaximumspeedconstraintandthuscanbeappliedtomoremovementsingeneral.Operationsandpredicateswhichcanbeusedtoquerymovingobjectsaredenedunderthesemodels.Thethirdmodelisadataminingapproachwhichperformsthepredictionofmovingobjectsinthefuturebyminingfromalargenumberofhistoricaluncertaintrajectories.WiththeadvancesinGPStechnology,thelocationsofmovingobjectsovertimecanbeeasilyobtainedandtrajectoriesareavailable.Sincetrajectoriesareoftengeneratedatalowfrequencyduetotheconsiderationofenergysaving,theroutepassingtwoconsecutivesamplingpointsbecomeuncertain.Whilesuchtrajectoriesimplyrichknowledgeaboutthemobilityofmovingobjects,theyarelessusefulindividually.However,combiningotheruncertaintrajectoriesanddetectcommonmovingpatterns,peoplecaninfertherealpathofthesemovingobjects.Thismodelenablesroutediscoveringthroughminingalargenumberofuncertaintrajectories.Inthesecondstage,theauthordesignsdiscreterepresentationsforhistoricalandfuturemovingobjectwithuncertaintyundertheabstractmodelsoftherststage.Datastructuresofvariouskindsofmovingobjectdatatypesandalgorithmsforoperationsandpredicatesaredesigned.Theauthorintroducestheimplementationconceptsofthedatastructures,operationsandpredicatesofthependantmodelandballoonmodel, 18

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respectively.Inthisstage,theauthoralsointroducestheimplementationofaroutediscoverysystemwhichimplementsthethirdmodelintherststage.Thethirdstageistheintegrationoftheuncertainmovingobjectdatatypes,operationsandpredicatesintoextensibledatabasesystemsandquerylanguages.Inthisdocument,theauthorpresentsthesystemsandsoftwarepackagesoftheproposedabstractmodelsthathavebeenaccomplished.Extensiveexperimentalstudyareperformed,whichshowstheeffectivenessandefciencyoftheapporaches. 1.3OrganizationoftheDocumentTherestofthisdocumentisorganizedasfollows.Chapter 2 discussestherelatedworkinmovingobjectsandspatio-temporaluncertainty.Chapter 3 introducesabstractmodelsofuncertainhistoricalandpredictivefuturemovingobjects.Chapter 4 discussesthediscreterepresentationofuncertainmovingobjectsbasedontheabstractmodels.Chapter 5 presentstheimplementationofmovingobjectswithuncertainty.Chapter 6 discussestheimplementedsystemforuncertainmovingobjects,queryresultsandexperiments.Chapter 7 drawsconclusions. 19

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CHAPTER2RELATEDWORKInthischapter,wesummarizetherelatedworkonmovingobjectdatabasesrelatedtoourapproachinthisdissertation.Weclassifythemintotwocategories.WerstreviewtheworkonmodelsofmovingobjectsinSection 2.1 ,includingthemodelsofmovingobjecttrajectories,themodelsofthespatio-temporaluncertaintyandmodelsofspatialandspatio-temporalrelationships.ThenwereviewtheimplementationmethodsofmovingobjectsinSection 2.2 includingindexingandqueryingapproaches. 2.1ModelsofMovingObjectsTherehasbeenintensivestudyonmovingobjectsintermsofmovingobjectdatabase[ 29 ][ 28 ],includingalotofinterestingandimportantapplications.Themostwidelystudiedtopicaboutmovingobjectsishowtomodeltrajectoriesofmovingobjects.WerstreviewmodelsonmovingobjecttrajectoriesinSection 2.1.1 .Astheactualrepresentationofmovingobjectscannotcapturetheexactmovementallthetime,i.e.,wedonotknowthelocationofamovingobjectwhenitisnotbeingobservedandtheuncertaintyexist.Andthisdissertationwillbefocusingonmodelingandqueryingtheuncertaintyofmovingobjects.Therehasbeenalargenumberofmodelsdealingwiththespatio-temporaluncertainty.WewillreviewthesemodelsandcomparethemwithoursinSection 2.1.2 .Anotherinterestingtopicinmovingobjectsistostudythespatialrelationships,suchasthecardinaldirectionrelationshipsandthetopologicalrelationshipsbetweenmovingobjects.WewillreviewtheliteraturesofthistopicinSection 2.1.3 2.1.1ModelsofMovingObjectTrajectoriesThereareseveralapproachesproposedtorepresentmovingobjecttrajectories.Anapproachrepresentsthepastmovementofmovingobjectsasafunctionfromtimetospace[ 29 ][ 28 ].TheMOSTmodelrepresentsthepositionofamovingobjectatatimeinstanceasafunctionofitsmotionvector[ 73 ]. 20

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Unliketheaboveapproacheswhichrepresentmovingobjecttrajectoriesascontinuousfunctions,someotherapproachesrepresentatrajectoryasalistofspatio-temporalpoints.Anapproachrepresentsthecontinuousmovementofamovingobjectasalistofpositions[ 85 ].Itstatesthatanobject'spositionatsometimetisgivenbyx(t)=(x1(t),x2(t),...,xn(t)),whereitisassumedthatthetimestarenotbeforethecurrenttime.[ 45 ]representsatrajectoryasasequenceofmulti-dimensionalpoints.ItisdenotedasTRi=p1p2p3...pjpleni(1inumtra).Here,pjisad-dimensionalpoint.Thelengthleniofatrajectorycanbedifferentfromthoseofothertrajectories.[ 95 ]representsataxi'strajectoryasasequenceofGPSpoints,whereeachpointconsistsofalatitude,alongitudeandatime-stamp.Whiletheaboveapproachesmainlyfocusonmovingpointobjects,someresearchersnoticethatnotonlythelocation,butalsothegeometryofmovingobjectsareofimportance.Acomprehensivesetofmovingobjectdatatypesisintroducedin[ 23 ].Thesedatatypesincludesnotonlymovingpointsbutalsomovinglinesandmovingregions.Foranarbitrarynon-temporaldatatype,itscorrespondingtemporaldatatypeisprovidedbyatypeconstructor()whichisafunctiontypethatmapsfromthetemporaldomaintimeto,thatis,()=time!.Byapplyingthetypeconstructortothespatialdatatypespoint,line,andregion,weobtainthecorrespondingspatio-temporaldatatypesnamedmpointmlineandmregionrespectively,shownasfollows,mpoint=(point)=time!pointmline=(line)=time!linemregion=(region)=time!regionAfewapproachesfocusonthestudyofmovingregionswhichareasarechangingovertime.Amethoddescribinghowtoconstructmovingregionsfromsnapshotobservationsisdiscussedin[ 79 ].Thedetectionoftopologicalchangesofmoving 21

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regionswiththehelpofsensornetworksisintroducedin[ 37 ][ 38 ][ 39 ].Theydetecttopologicalchangingofarealobjectssuchasregionmergingandregionsplitting.Theauthorherselfhasproposedamodelforrepresentingtopologicalchangesofcomplexregionsin[ 48 ][ 51 ].Withtheincreasingcapabilityofstoringlargeamountdataincomputeranddatabasesystems,researchersofmovingobjectsrecentlybecomemoreinterestinminingusefulinformationfromlargeamountofmovingobjectdata.Asystematicintroductionontrajectorydataminingisshownin[ 58 ].Aframeworkofdetectingmovingpatternsfromanalyzingalargedatasetoftrajectoriesisintroducedin[ 27 ].TherstmethodthatminesthemostpopulartravelroutesfromGPStrajectoriesisintroducedin[ 102 ].Amethodthatsearchesforanexistingtrajectoryfromatrajectorydatasetaccordingtoasetofpointlocationsisproposedin[ 11 ].Adataminingapproachwhichinfersfastestroutesfromlow-samplingratetaxitrajectoriesareintroducedin[ 95 ][ 94 ].Astheprocessofminingmovingobjecttrajectoriesneedsalotofcomputing,[ 104 ]providesacomprehensiveoverviewofthegeneralconcepts,techniques,andapplicationsoftrajectorycomputing. 2.1.2ModelsofSpatio-TemporalUncertaintySeveralmodelshavebeenproposedtodiscusstheuncertaintyinmovingobjects.[ 87 ][ 88 ]addresstheproblemsofqueryingmovingobjectswithfrequentupdatesindatabases.Theyproposeaninformationcostmodelthatcapturesuncertainty,deviation,andcommunicationtosolvetheproblem.The3Dcylindermodelrepresentsthetrajectoriesofmovingobjectsasavolumein3D(2DEuclidean+time)space[ 83 ][ 80 ].Itstatesthatallthepossiblelocationsofamovingobjectwithuncertaintyarewithinacircleareacalledtheuncertaintyregion.Therefore,withtimepassingby,themovementcanbetreatedasacylinderin3Dspace.Themodelisbasedontheassumptionthatthedegreeofuncertaintyofamovingobjectremainsconstantduringaperiodoftime.However,thisisnotalwaystrueinreality.Acomparableapproach,the 22

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space-timeprism(beads)model[ 30 ][ 62 ][ 31 ][ 18 ][ 57 ]assumesthattheuncertaintyofamovingobjectgrowslargerwhenitleavesfartherfromobservation.Forexample,amovingobjectisobservedattimet1andt2,thelargestuncertaintymayhappenattimearound(t1+t2)=2andminimumuncertaintyappearswhenitapproachest1ort2.Therefore,thespace-timeprismmodelrepresentstheuncertainmovementofamovingobjectastheunionoftwohalf-conesinthe3Dspace.Giventheoriginandthedestinationaswellasthemaximumspeedofamovingobject,allpossibletrajectoriesofthismovingobjectareboundedbythebead.Accordingtothegeometricpropertyofconesandcylinders,thismodelismuchmoreefcientthanthe3Dcylindermodelsinceitreducestheuncertaintytoonethird.Inrecentyears,thespace-timeprismmodelhasdevelopedalot.Samequeriesin[ 83 ]and[ 80 ]havebeendiscussedagainundertheprismmodelandimprovementsareseenin[ 81 ].Approachesthatusetheprismtosolvetheuncertaintyinroadnetworksarediscussedin[ 42 ][ 43 ].Anapproachusingtheprismtosolvethealibiqueriesisintroducedin[ 40 ].Arecentpaperextendstheclassicalspace-timeprismbyimposinganupperboundonthemovingobject'sacceleration[ 41 ].Theauthor'spreviousownworkin[ 49 ]proposesthenameofthependantmodelforthersttime.Themodelusesthespace-timeprismtorepresenttheuncertainpartofthemovementandcontainsthecertainpartofthemovementaswell.Themostimportantcontributionofthatworkisthatitdenesalistofspatio-temporaluncertaintypredicates,whichenablesthequeryontheuncertaintyofmovingobjectsinthepast.Incomparisonto3Dcylindermodelandbeadsmodelwhichassumethatallpossiblemovementsarewithinanuncertainvolume,anotherclassofresearchpapersonspatio-temporaluncertaintydonotspecifyaparticularvolumeshape,butintroducingtheprobabilitytheorytodescribetheuncertainty.Chenget.alproposetheuncertainregionconceptandusingprobabilitydensityfunction(pdf)todescribetheuncertaintydistributioninsideanuncertainregion,andsolvestheprobabilisticrangequeries(PRQ)andprobabilisticnearest-neighborqueries(PNNQ)[ 13 ][ 12 ].[ 82 ]discussesthe 23

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continuousprobabilisticnearestneighborsqueryproblemunderthecylindermodel.Anapproachusingprobabilisticmodelstosolveuncertainrangequeriesinroadnetworkenvironmentsisintroducedin[ 97 ].Afewapproacheshavebeenproposedtopredictthefuturelocationsofmovingobjects.Themoststraightforwardpredictionmethodistoassumethatthemovingobjectsmoveslinearly[ 78 ][ 61 ].Animprovedmethodthatmakespredictiononnon-linearmotionfunctionisintroducedin[ 77 ].Anovelresearchproposesahybridmethodwhichmakesthepredictionbothonexistingmotionfunctionsaswellasthepatterninformation[ 35 ].Themethodcannotonlypredictthenearfuturelocations,butcanalsopredictthelocationswhenthequerytimeisfarawayfromthecurrenttime.Anapproachthatreducestheuncertaintyinthefutureandmakepredictiononmovingobjectsinroadnetworksarediscussedin[ 36 ].Anindexingmethodonuncertaintymovingobjectswiththeprobabilisticmodelisproposedin[ 96 ],whichsolvesthePRQandPNNQmoreefciently.Alltheaboveapproachesmainlyfocusonpredictingthefuturelocationbypastgeographicpropertiesofmovingobjectsinthepast.Anotherclassofpredictionapproachesarebasedonmeasuringthesimilarityofthemovingobjecttrajectoriesandclusteringthesimilartrajectories,thentherepresentativetrajectorycanbetakenasthebasisforpredictingthefuturelocationsofmovingobjectswhichbelongtothecluster.ExamplesoftheclusteringmethodsinspatialdatabaseareDBSCAN[ 69 ]andOPTICS[ 2 ],whicharedensity-basedapproachdetectingclustersofarbitraryshapes.Apartition-and-groupalgorithmcalledTraClasstoclustersimilartrajectoriesisdiscussedin[ 45 ]and[ 44 ].Itdenesthreemeasurement,i.e.,perpendiculardistance,paralleldistanceandangledistancetoevaluatethesimilarityoftwomovingobjecttrajectories.[ 84 ]ndsimilartrajectoriesfromalongestcommonsubsequencemethod(LCSS).Whiletheabovemodelsmakepredictionbasedongeographicfeatures,recentapproachesintroducethesemantictagstohelpprediction.[ 53 ]statesthatphysical 24

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locationsarenotusefulformostusers,whereassemanticlocationsarecriticaltodetermineuser'sactivities.Theauthorsproposeamethodthatautomaticallyderivessemanticlocationsfromuser'strace.Thetermsemantictrajectoriesisproposed[ 1 ][ 90 ][ 4 ].In[ 1 ]theauthorsclaimthatmeaningfulpatternsfordecisionmakingprocessesinrealapplicationscannotbeextractedfromrawtrajectorydata,thussemanticsareintroduced.Theyconsidertrajectoriesasasetofstopsandmoves,wherestopsaretheimportantplacesfortheapplication,andmovesaretransitionsbetweenconsecutivestops.Anapproachwhichminesthesimilarityofusers'trajectorybasedonlocationhistoriesisproposedin[ 47 ].Itdenesastaypointastheplacewhereauserstaysforawhile.Astaypointthereforecarriesaparticularsemanticmeaning,suchastheplaceauserworksorlives,orarestauranthevisits.Theauthorsintroduceahierachical-graph-basedmeasurement(HGSM)todetectthesimilaritybetweenusers.[ 103 ]minesinterestinglocationsfromhistoricaluserstrajectoriesandperformsrecommendations.Tondthesimilaritybetweentwotrajectories,thestaypointsequencesarecomparedandthesimilarityiscalculatedintermsofTFIDFvalue[ 101 ].Arecentapproachwhichdetectssimilarityinsemantictrajectoriesisproposedin[ 92 ].Theapproachusesthelongestcommonsubsequence(LCSS)algorithmtondthesimilaritymainlyonthesemantics.Acontinuouswork[ 91 ]addsgeographicfeaturetomeasurethesimilarityandmakeprediction.Itintroducestwomeasurements,i.e.SemanticScoreandGeographicScore.However,thisapproachrstltersthetrajectoriesbysemanticsimilarityandthendetectthegeographicsimilarity,thereforetwotrajectorieswhicharefarfromeachotherinlocationmighthaveaveryhighsimilarityscore,iftheirsemanticmeaningsareveryclosetoeachother.Thedifferenceofourapproachisthatwecomparegeographicsimilarityatthebeginningandthenmeasurethesemanticsimilarity.Anapproachoftheauthors'ownndsfrequenttravelpatternsbyminingfromuncertainhistoricaltrajectoriesofmovingobjectsand 25

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inferstheirmostpossibleroutes[ 52 ].Asimilarapproach,whichsolvesthesameprobleminroadnetworkenvironmentisdiscussedin[ 98 ].Theauthor'sgrouprstproposetheballoonmodelin[ 65 ].Thismodelisanovelapproachusinganabstract3Dshapeotherthanconeorcylindertorepresenttheuncertaintyofmovingobjectinthefuture.Acontinuousworkemphasisontherepresentationoftheballoonisdiscussedin[ 50 ].However,thesepapershavenotstudiedthecontinuouspropertyofmovingobjectsandgivenacorrectformaldenitiononthemovingfunctions,andhavenotdiscussedcomplexmovingobjects.Inthisarticle,wewillgiveaformalandcomprehensivesetofdenitionsonmovingobjectsinuncertainenvironments. 2.1.3ModelsofSpatialRelationshipsandSpatio-temporalRelationshipsAnimportanteldinstudyingmovingobjectsisthestudyofspatio-temporalrelationships.Spatialrelationshipsincludetopologicalrelationshipsandcardinaldirectionrelationships.Atopologicalrelationshipisaqualitativespatialrelationshipwhichdescribeshowtheboundaryandinteriorbetweentwospatialobjectsintersect.Examplesoftopologicalrelationshipsaremeet,inside,overlat,etc.Thetopologicalrelationshipisintroducedandformallydenedin[ 16 ][ 20 ].Thewell-knownmodelof9-intersectionmatrixareusedtoperformthereasoningaboutthetopologicalrelationships[ 17 ].Astudyonthedeformationsofspatialobjectssuchastranslation,rotation,expansionetc.arestudied,andformaldenitionsofsuchdeformationsaregivenin[ 19 ].Agraphoftransitionsisdepictedandthenexttopologicalrelationshipcanbepredictedfromthegraph.Whiletheaboveapproachesmainlystudythetopologicalrelationshipsbetweensimplespatialobjects,somelaterresearchhasbeenperformedonthetopologicalrelationshipsbetweencomplexspatialobjects[ 70 ][ 71 ][ 56 ][ 55 ].[ 15 ]discusseshowtodescribetopologicalrelationshipsfromusersperspective.Theauthorlatershowsthatthemethodcanalsobeappliedtospatialobjectswithcomplexfeatures[ 14 ].Arecentapproachintroducestheconceptoftopologicalfeaturevectorstomodel 26

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topologicalrelationshipsbetweencomplexobjects[ 66 ][ 67 ].Topologicalrelationshipsbetweenthreedimensionalspatialobjectsarediscussedin[ 9 ].Spatialdatatypessuchaspoints,linesandregionsandtheiroperationsofspatialdatatypescanconstructcorrespondingmovingdatatypesandoperationsbyaliftingprocess[ 28 ].Whenliftingtopologicalrelationshipsbetweenspatialobjectstomovingobjects,wecanmodeltherelativepositionsbetweenmovingobjectsovertime.Asetofbinarypredicatessuchasenters,crossareinvitedtomodelsuchrelationships,calledspatio-temporalpredicates,whichareintroducedin[ 21 ].ThepurposeofintroducingsuchrelationshipsintobinarypredicatesistousethemeasierinquerylanguagessuchasSQLindatabases[ 24 ].Aspecialkindoftopologicalrelationshipbetweenspatialobjectsisthecardinaldirectionrelationship.[ 26 ]introducesanalgebraicmethodtoformalizethemeaningofcardinaldirections.[ 74 ]focusesonthecompositionoperatorfortwocardinaldirectionrelations.Afamilyofexpressivemodelsforqualitativespatialreasoningwithdirectionsisproposedin[ 75 ],whichisbasedonthecognitiveplausiblecone-basedmodel.AnewapproachcalledtheOIMmodelwhichrepresentsthecardinaldirectionbetweencomplexregionsisproposedin[ 54 ].Sofar,therehavebeenveryfewresearchoncardinaldirectionrelationshipsbetweenmovingobjects.Theauthor'sgrouphaveproposedanovelapproachoncomputingthecardinaldirectiondevelopmentsbetweenmovingobjectsin[ 7 ][ 8 ]. 2.2ImplementationofMovingObjectsandQueriesinDatabasesIntheprevioussection,wehavereviewedtherelatedworkonmodelingmovingobjects.Inthissection,wewillreviewthemethodsofimplementingmovingobjectsindatabasesandqueryingmovingobjects.Therearetwoimportantissues,therstisthedesignofdatastructuresformovingobjects,indexingmethodsandapplications.Thesecondissueisthequerylanguageofmovingobjectsandtheintegrationtodatabases. 27

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WewillreviewtheliteraturesoftherstissueinSection 2.2.1 ,andtheliteraturesofthesecondissueinSection 2.2.2 2.2.1MovingObjectImplementationandIndexesAswehavementionedabove,movingobjectmodelsincludingthedatatypesandoperationsattheabstractlevelhavebeenintroducedin[ 28 ].Acontinuousresearchofthismodel,whichistheimplementationmethodofthesedatatypesandalgorithmsareprovidedin[ 25 ][ 46 ].Theimplementationusesaslicerepresentationmethod,inwhichmovingobjecttrajectoriesarerepresentedbyalistofunits.Operationsonmovingobjectssuchasintersection,unionanddifferencearealsoimplementedusingplane-sweepalgorithms.SECONDOisanextensibledatabasesystemwhichimplementedvariesspatialdatatypesandspatio-temporaldatatypesalongwithalotofoperations.Itisoneoftherstdatabasesystemprototypesthatcanhandlemovingobjects[ 72 ].Itimplementsmovingpoints,movinglinesandmovingregionsaccordingtotheslicerepresentationmethod.Animportanteldinqueryingmovingobjectsishowtoindexmovingobjecttrajectoriessothatsearchcanbeperformedefciently.Thereareanumberofindexingmethodsofmovingobjectsinordertoretrievemovingobjectsefcientlyindatabases[ 85 ][ 63 ][ 64 ].AnR-treebasedtechniquefortheindexingofthecurrentpositionsofobjectswhichhavenotreportedtheirpositionwithinaspecieddurationoftimeisdiscussedin[ 68 ].Jensenet.alpointoutthatR-treebasedindexingmethodmaycostoverheadduetonodesplitting,andtheyproposeanewindexingmethodbasedonB+-treewhichoutperformstheR-treebasedTPR-treeforbothsingleandconcurrentaccessscenarios[ 34 ]. 2.2.2QueryingMovingObjectsinDatabasesMostoftheapproachesofmodelingmovingobjectsaimtoimplementedtheoperationsindatabasesandperformdifferentqueries,sothatapplicationscanbebuiltontopoftheseapproaches.Afewapproachesareimplementedonextensible 28

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databases.[ 86 ]introducesaDOMINOprojectwhichbuildsanenvelopeofrepresentingandqueryingmovingobjectsontopofexistingDBMSs,sothatitcananswerquerieslikeretrievethefreecabsthatarecurrentlywithin1mileof33N.MichiganAve.,Chicago.Themethodconsiderseveralimportantissuessuchaslocationmodeling,linguisticissues,indexing,uncertainty,dynamicattributesandquerylanguages.FinallyitintroducesaFutureTemporalLogic(FTL)languageforqueryandtriggerspecicationsinmovingobjectsdatabases.ThelanguageisnaturalandintuitivetouseinformulatingMODqueries,anditusesbothspatialoperators(inside)andtemporaloperators(until,eventually).AnexamplequeryofFTLwhichasksRetrievethepairsofobjectsoandnsuchthatthedistancebetweenoandnstayswithin5milesuntiltheybothenterthepolygonPisshownasfollows, RETRIEVEo,nFROMMoving-ObjectsWHEREbegin-time(DIST(o,n)<=5)<=nowandend-time(DIST(o,n)<=5)>=begin-time(INSIDE(o,P)AINSIDE(n,P)).WenoticethatinordertousethisFTLlanguage,notonlythenewdatatypesbutalsoanewinterpretermustbeimplemented.Incontrast,inourownresearch,wetakeadvantageofexistingDBMSlikeOracle,andimplementedoursoftwarelibraryontopofthem.Manyclassicalcomputationalgeometryproblemsbecomeinterestingundertheenvironmentsofmovingobjects,andgeneratealistofusefulqueries.Forexample,thequeriesofnearestneighbors[ 59 ]andreversenearestneighbors(RNN)queries[ 3 ].Theformertypeofqueryreturnskobjectsnearesttoaqueryobjectforeachtimepointduringatimeinterval,whilethelatterreturnstheobjectsthathaveaspeciedqueryobjectasoneoftheirkclosestneighbors,againforeachtimepointduringatimeinterval.Somerecentresearchonk-nearestneighborsqueries(kNN)usingdifferent 29

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indexingmethodsarediscussedin[ 76 ][ 33 ][ 89 ][ 93 ].[ 33 ]discussescontinuouskNNquerieswithfrequentupdatesindatabases.[ 89 ]proposesamethodcalledSEA-CNNwhichachievesbothefciencyandscalabilityinthepresenceofasetofconcurrentqueries.[ 93 ]proposestwoefcientandscalablealgorithmsusinggridindexes,andtheyndthatitoutperformstheR-treebasedindexesalot. 30

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CHAPTER3ABSTRACTMODELSOFUNCERTAINTYINHISTORICALANDPREDICTIVEMOVINGOBJECTSInthischapter,weintroducethreemodelsrepresentingthespatio-temporaluncertaintyofmovingobjects.Section 3.1 introducesthependantmodelwhichrepresentsthehistoricalmovingobjectswithuncertainty.Section 3.2 introducestheballoonmodelwhichisabletorepresentbothhistoricalandfuturemovementsofmovingobjectsinuncertainenvironments.Section 4.3 introducesadataminingapproachwhichisabletopredicttheroutesofmovingobjectscollectivelyminingfrommassiveuncertaintrajectories.Section 4.4 discussesamethodoninferringfuturelocationsfromdetectingsimilartrajectories. 3.1PendantModel:RepresentingHistoricalMovingObjectswithUncertaintyInthissection,weproposeanewmodelcalledthependantmodeltorepresenttheuncertainmovingobjects.Similarwiththespace-timeprismmodel,weassumethatthedegreeofuncertaintyofamovingobjectchangesovertime.However,unlikethespace-timeprismmodel,thispendantmodelisanintegratedandseamlessmodelwhichcombinesbothknownmovementsanduncertainmovements.Further,itformallydenesspatio-temporaluncertaintypredicates(STUP)thatexpressthetopologicalrelationshipsbetweenuncertainmovingobjects.Thisisimportantinqueryingmovingobjectswithuncertaintyinthedatabasecontextsincetheycanbeusedasselectionconditionsindatabasequerylanguages.Asanimportantpartofthemodel,weformallydeneoperationsrelatedtoretrievingandmanipulatinguncertaindata,andSTUPssuchaspossibly meet at,possibly enter,anddenitely cross,etc.,whicharedenedonthebasisoftheoperations.Queriesrelatedtotheuncertaintyinthetopologicalrelationshipsbetweenmovingobjectscanthenbeanswered.Section 3.1.1 discussesthespatio-temporaluncertaintyprobleminhistoricalmovingobjects.Section 3.1.2 introducesourpendantmodelofmovingobjectswithuncertainty.Section 3.1.3 denesasetofoperationsonthependantmodel. 31

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Section 3.1.4 introducesspatio-temporaluncertaintypredicates(STUP).Section 3.1.5 discusseshowSTUPcanbeintegratedtodatabasequerylanguages. 3.1.1Motivation:TheUncertaintyofMovingObjectsinthePastWestartourdiscussionbyreviewingtheclassicspace-timeuncertaintyproblem.Assumethatacellphoneuseriswalkingonthestreetandherlocationsarerecordedbythecellphonecompanyasatrajectory,denotedbyasequenceof(time,latitude,longitude)records.Aninterestingquestioniswhereherlocationsarewhensheisnotbeingobserved.Assumethatthepersonhasamaximumspeed,denotedbyvmax,andwedenotetwoconsecutiveobservationsasthelocationsofp1(x1,y1)attimet1,andp2(x2,y2)attimet2.Ourgoalistondallherpossiblelocationsatanytimetbetweentwoconsecutiveobservationtimest1andt2.Beforesolvingtheproblem,werstgivethedenitionofthespatio-temporalobservationswementionedabovewhichwillbefrequentlyusedinthelatersections.Weallknowthepointdatatypewhichisrepresentedbyapairof(x,y)coordinatesinthe2DEuclideanspace.Now,weextendthisdatatimeinour2D+timespace,andgivethedenitionofspatio-temporalpointasfollows. Denition3.1.1.1(spatio-temporalpoint). Aspatio-temporalpoint,denotedbystPoint,withtheformof(t,x,y),istheobservationofthelocation(x,y)ofanobjectattheparticulartimeinstancet,wheret2timeandx,y2R2.Toanswertheabovequestion,wemustconsidertwodifferentsituations.Therstsituationisquitestraightforward.Assumethatthelengthoftheinterval[t1,t2]isverysmall,wecanapproximatethemovementbetweentwoobservationsasastraightlinesegment.Evenif[t1,t2]isnotsmall,butweknowthatthepersonistravelingtowardaknowndirection,onafreewayforexample,thenhermovementcanstillberepresentedbylinearapproximation.Therefore,wetakethemetaphorstringtorepresentsuchkindofmovement,incomparisontothewordpendantwhichwillbeintroducedlater.Animportantfactisthatthemovementbetweenthetwoobservationsatt1andt2follows 32

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(a)(b)Figure3-1. Thelinearmovementbetweentwoconsecutiveobservations,andthepartialmovementbetween[ts,te] thelinearpattern,however,duringanyintervalswithin[t1,t2],themovementshouldalsofollowthesamepattern.Assumethatwehaveastartingtimeinstancetst1andanendingtimeinstancetet2,thenthemovementbetween[ts,te]overlapsthemovementbetween[t1,t2].Adegeneratecaseisthatwhents=t1,andte=t2,themovementbetweents,teistheentirestringmovement.Therefore,themovementduringanintervalbetweentwoconsecutivespatio-temporalpointscanbedescribedinDenition 3.1.1.2 Denition3.1.1.2(stringmovement). Giventwospatio-temporalpointsp1=(t1,x1,y1)andp2(t2,x2,y2)astwoconsecutiveobservationsofamovingpoint,andaninterval[ts,te]wheretst1andtet2,andthemovementbetweent1andt2isknowntobelinear.Themovementfunctionbetween[ts,te]isapartialmovementofthemovementbetween[t1,t2],andcanbedenedasfollows, string(p1,p2,ts,te)=f(t,x,y)jt2[ts,te],x=x1+(x2)]TJ /F6 7.97 Tf 6.59 0 Td[(x1)(t)]TJ /F6 7.97 Tf 6.59 0 Td[(t1) t2)]TJ /F6 7.97 Tf 6.58 0 Td[(t1,y=y1+(y2)]TJ /F6 7.97 Tf 6.59 0 Td[(y1)(t)]TJ /F6 7.97 Tf 6.59 0 Td[(t1) t2)]TJ /F6 7.97 Tf 6.58 0 Td[(t1gThestringmovementisillustratedinFigure 3-1 .Figure 3-1 ashowstheentirelinearmovementbetween[t1,t2],andFigure 3-1 showsthepartiallinearmovementbetween[ts,te],where[ts,te]iswithin[t1,t2].Inthesecondsituation,ifthelengthoftheinterval[t1,t2]isrelativelylarge,orthedirectionofthemovementisnotpredictable,thenthecellphoneuser(themovingpointobject)doesnotactuallytaketheshortestpath.Thenherpossiblelocation(x,y)attime 33

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(a)(b)Figure3-2. Apendantmovementandapartialpendantmovement tisboundedbyhermaximumvelocitymultiplyingthetraveltime.Astimepassingby,allpossiblemovementsovertimeareboundedbya3Dvolume.Sincetheactualdistancethemovingobjecttravelscannotexceedthemaximumdistanceitcantravel,themovementin3Disactuallyacone-shapedvolume,whichisanalogoustothependantofanecklace.Similartothestringmovement,ifwewanttotrackthemovementduringanintervalwithintwospatio-temporalpoints,wewillgetapartialpendant.Therefore,wegivethedenitionofpendantwhichdescribesthemovementorpartofthemovementdenedonanuncertaininterval. Denition3.1.1.3(pendantmovement). Giventwoconsecutivespatio-temporalpointsp1=(t1,x1,y1)andp2=(t2,x2,y2)ofamovingpointwhosemaximumvelocityisvmax,andaninterval[ts,te]wheretst1andtet2,theuncertainmovementbetweentsandteisdenedasfollows, pendant(p1,p2,ts,te,vmax)=f(t,x,y)jt2[ts,te],p (x)]TJ /F5 11.955 Tf 11.96 0 Td[(x1)2+(y)]TJ /F5 11.955 Tf 11.95 0 Td[(y1)2(t)]TJ /F5 11.955 Tf 11.95 0 Td[(t1)vmax,p (x)]TJ /F5 11.955 Tf 11.96 0 Td[(x2)2+(y)]TJ /F5 11.955 Tf 11.95 0 Td[(y2)2(t2)]TJ /F5 11.955 Tf 11.95 0 Td[(t)vmaxgInaspecialcase,whents=t1,andte=t2,theabovedenitiondescribesallthepossiblemovementsbetweent1andt2.TheentirependantandpartialpendantmovementareillustratedinFigure 3-2 aandFigure 3-2 brespectively. 34

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Wendthatthependantparthasanimportantproperty.Ifweprojectallpossiblelocationstothe2DEuclideanplane,thelocationsareboundedbyanellipsewithp1(x1,y1)andp2(x2,y2)astwofocuses.Thisisstatedinthefollowinglemma. Lemma3.1.1.1(ellipseproperty). Ifamovingpointwithamaximumvelocityvmaxtravelsfromtwoconsecutivespatio-temporalpointsp1(t1,x1,y1)top2(t2,x2,y2),allpossiblelocationsitcantravelduringthisperiodisboundedbyanellipsewhichcanberepresentedby,(2x)]TJ /F5 11.955 Tf 11.96 0 Td[(x1)]TJ /F5 11.955 Tf 11.96 0 Td[(x2)2 v2max(t2)]TJ /F5 11.955 Tf 11.95 0 Td[(t1)2+(2y)]TJ /F5 11.955 Tf 11.95 0 Td[(y1)]TJ /F5 11.955 Tf 11.95 0 Td[(y2)2 v2max(t2)]TJ /F5 11.955 Tf 11.96 0 Td[(t1)2)]TJ /F4 11.955 Tf 11.96 0 Td[((x2)]TJ /F5 11.955 Tf 11.95 0 Td[(x)2)]TJ /F4 11.955 Tf 11.95 0 Td[((y2)]TJ /F5 11.955 Tf 11.96 0 Td[(y)2=1 Proof. Sincethemaximumspeedofthemovingpointisvmax,themaximumdistanceitcantravelisvmax(t2)]TJ /F5 11.955 Tf 12.43 0 Td[(t1),whichequalstwicethemajoraxis.TheshortestdistanceittravelsistheEuclideandistancebetweenp1andp2,i.e.,p (x2)]TJ /F5 11.955 Tf 11.95 0 Td[(x1)2+(y2)]TJ /F5 11.955 Tf 11.95 0 Td[(y1)2,whichequalsthedistancebetweentwofoci.Thenwetranslatetheorigintothemid-pointofp1p2,and(x,y)istranslatedto(x)]TJ /F6 7.97 Tf 13.15 4.88 Td[(x1+x2 2,y)]TJ /F6 7.97 Tf 13.15 5.03 Td[(y1+y2 2).Thereforewehave,4(x)]TJ /F6 7.97 Tf 13.15 4.88 Td[(x1+x2 2)2 v2max(t2)]TJ /F5 11.955 Tf 11.95 0 Td[(t1)2+4(y)]TJ /F6 7.97 Tf 13.15 5.03 Td[(y1+y2 2)2 v2max(t2)]TJ /F5 11.955 Tf 11.96 0 Td[(t1)2)]TJ /F4 11.955 Tf 11.96 0 Td[((x2)]TJ /F5 11.955 Tf 11.96 0 Td[(x1)2)]TJ /F4 11.955 Tf 11.96 0 Td[((y2)]TJ /F5 11.955 Tf 11.96 0 Td[(y1)2=1Tosimplify,weget,(2x)]TJ /F5 11.955 Tf 11.96 0 Td[(x1)]TJ /F5 11.955 Tf 11.96 0 Td[(x2)2 v2max(t2)]TJ /F5 11.955 Tf 11.95 0 Td[(t1)2+(2y)]TJ /F5 11.955 Tf 11.95 0 Td[(y1)]TJ /F5 11.955 Tf 11.95 0 Td[(y2)2 v2max(t2)]TJ /F5 11.955 Tf 11.96 0 Td[(t1)2)]TJ /F4 11.955 Tf 11.96 0 Td[((x2)]TJ /F5 11.955 Tf 11.95 0 Td[(x)2)]TJ /F4 11.955 Tf 11.95 0 Td[((y2)]TJ /F5 11.955 Tf 11.96 0 Td[(y)2=1 Lemma 3.1.1.1 showsaveryimportantproperty:withtheconstraintofmaximumvelocity,wecanalwaysndtheboundaryofthefarthestdistancethemovingobjectcantravel.Thisfeatureisessentialindeterminingtheuncertaintopologicalrelationshipsbetweenmovingobjects.FromLemma 3.1.1.1 wegivethefollowingtheorem. 35

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Theorem3.1.1.1. Ifamovingpointwithamaximumvelocityvmaxtravelsfromp1(t1,x1,y1)top2(t2,x2,y2)during[t1,t2],allpossiblemovementsovertimeareboundedbythevolume,whichsatisesp (x)]TJ /F5 11.955 Tf 11.96 0 Td[(x1)2+(y)]TJ /F5 11.955 Tf 11.96 0 Td[(y1)2(t)]TJ /F5 11.955 Tf 11.96 0 Td[(t1)vmax,andp (x)]TJ /F5 11.955 Tf 11.96 0 Td[(x2)2+(y)]TJ /F5 11.955 Tf 11.96 0 Td[(y2)2(t2)]TJ /F5 11.955 Tf 11.95 0 Td[(t)vmax. Proof. TheproofisshownbyDenition 3.2.1.9 andLemma 3.1.1.1 AsshowninFigure 3-2 ,theprojectionofthependantintothe2DEuclideanspaceisanellipse.Themovementrepresentedbyadashedlineistheshortestpathofthemovingpoint.Theothermovementwhichrepresentedbythesolidlineshowsthatthemovingpointtravelswiththemaximumspeedallthetime,andthusittravelsthemaximumdistance,i.e.,followingtheboundaryofthependant.Therefore,amovementwithuncertaintycanbecomposedbyasequenceoflinearmotioncurvesrepresentingallpartsofmovementwhichareknown,togetherwithasequenceofuncertainvolumes.Inthenextsubsection,wedenetherealmovementformally. 3.1.2RepresentingHistoricalMovingObjectswithUncertaintyInmostscenariosinreality,themovementofamovingobjectoftenconsistsofbothuncertainmovementsandcertainmovementstogether.Forexample,themovementofanairplaneistrackedbyaradaratatimeperiod,whichcanbeconsideredcertain.However,atalatertime,thesignalislost,thenthepositionoftheairplaneisuncertain.TheseexactlycorrespondtothetwosituationswementionedinSection 3.1.1 .Therefore,inourpendantmodel,amovementconsistsofbothcertainpartsanduncertainpartsofmovementtogether.OnthebasisoftheDenition 3.1.1.2 andDenition 3.2.1.9 ,wegiveourdenitionofthecombinedmovementasfollows.Inthisdenition,weusethefunctionrestrictionconceptinmathematics.Assumewehaveafunctionf:X!Y,therestrictionfjAmeansthatAX,andfisonlydenedonA. 36

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Denition3.1.2.1(movingpointwithuncertainty). Givenalistofspatio-temporalpointsSTPList=andIS=f1,2,...,ng,witht1
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(a)(b)Figure3-3. AumPointmovementwhichconsistsofasequenceofcertainanduncertainmovement Ifweextendthemovingpointtoageneralmovingobject,i.e.,amovingregionwithanarbitraryshape,wecangiveasimilardenitionasshowninDenition 3.1.2.3 .Similarly,werstgivethedenitionofthespatio-temporalregionobject. Denition3.1.2.2(spatio-temporalregion). Aspatio-temporalregion,withtheformofr(t,s),istheobservationoftheshapesofobjectrattheparticulartimeinstancet,wheret2timeandr2region,withthefollowingcondition,8p(t,x,y),if(x,y)2s,thenwesayp2r.Now,wegivethedenitionofamovingregionwithuncertainty. Denition3.1.2.3(movingregionwithuncertainty). Givenalistofspatio-temporalregionsSTRListandIS=f1,2,...,ng,witht1
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(a)(b)Figure3-4. Uncertainmovementofacircleregionandapolygonregion umRegion=ff:time!regionj(i)dom(f)=If[t1,t2],...,[tn)]TJ /F7 7.97 Tf 6.58 0 Td[(1,tn]g(ii)8[tk,tk+1]2C\I:9ri(ti,si),ri+1(ti+1,si+1)2STRListand[tk,tk+1][ti,ti+1],thenfj[tk,tk+1]=Spi2ri,pi+12ri+1string(pi,pi+1,tk,tk+1)(iii)8[tk,tk+1]2U\I:9ri(ti,si),ri+1(ti+1,si+1)2STRListand[tk,tk+1][ti,ti+1],thenfj[tk,tk+1]=Spi2ri,pi+12ri+1pendant(pi,pi+1,tk,tk+1,vmaxi)gIntheabovedenition,Conditionii)showsthatintheintervalsotherthantheuncertainpart,theareamoveslinearly,whichistheunionofthestringmovementofallmovingpointsinit.Conditioniii)showsthatatanytimeinstanceintheuncertainintervals,allpossiblelocationsofamovingregionisdeterminedbytheunionofthependantofallmovingpointsinsidethemovingregion.ThemovementsofuncertainregionsareshowninFigure 3-4 3.1.3OperationsonHistoricalMovingObjectswithUncertaintyAfterintroducingourpendantmodel,weintroducetheoperationsthatwillbeperformedonthemodel.Theyareimportantsincetheywillbeintegratedintodatabasesandusedasfunctionsofretrievingandmanipulatingdata.Intherestofthissection,weintroducesomeimportantoperationsofourpendantmodel.Theywillbeusefulin 39

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helpingusdenespatio-temporaluncertainpredicatesinthenextsection.WelistalltheoperationswhichwillbedenedinthissectioninTable 3.2.3 .Thenwegivetheformaldenitionsoftheseoperations. Table3-1. OperationsonthePendantmodel createUmPoint:Listperiods!umPointcreateUmRegion:Listperiods!umRegiongetLifetime:umPoint!periodsumRegion!periodsgetUncertainIntervals:umPoint!periodsatInstant:umPointinstant!pointregionumRegioninstant!regiontemporalSelect:umPointperiods!umPointumRegionperiods!umRegion Nowwegiveformaldenitionsofeachoftheaboveoperations.Werstdenetheoperationsofcreatinganuncertainmovingobject,asshowninDenition 3.1.3.1 andDenition 3.1.3.2 Denition3.1.3.1(createUmPoint). Givenalistofspatio-temporalpointsSTPList=<(t1,x1,y1),...,(ti,xi,yi),...,(tn,xn,yn)>,IS=f1,2,...,ngCISIS,andC=f[ti,ti+1]ji2CISgdenotesthecertainintervals.ThencreateUmpoint(stPoint[],periods)willconstructtheuncertainmovingpointobjectaccordingtoDenition 3.1.2.1 Denition3.1.3.2(createUmRegion). Givenalistofspatio-temporalregionsSTRList=<(t1,r1),...,(ti,ri),...,(tn,rn)>IS=f1,2,...,ngCISIS,andC=f[ti,ti+1]ji2CISgdenotesthecertainintervals.ThencreateUmRegion(STRList,periods)willconstructtheuncertainmovingregionobjectaccordingtoDenition 3.1.2.3 Denition3.1.3.3(getLifetime). Givenanuncertainmovingpointump2umPoint.Letdom(ump)=[ts,te],theoperationget lifetimewillreturntheintervalinwhichthismovingpointexists.Itisdenedasfollows, getLifetime(ump)=[ts,te],ts,te2time 40

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Denition3.1.3.4(getUncertainInterval). Givenanuncertainmovingpointump2umPoint,dom(ump)=I=f[t1,tn]g.letCIbeasetofcertainintervals.theoperationgetUncertainIntervalwillreturnalluncertaintyintervals, getUncertainInterval(ump)=I)]TJ /F5 11.955 Tf 11.96 0 Td[(UNowweintroduceaveryimportantoperationatInstant.Whenamovingobjectisuncertain,itspossiblelocationatatimeinstantisnotdeterministicbutiswithinanuncertainarea.WeintroducetheatInstantoperationwhichwillreturnthepossiblelocationsofanuncertainmovingobject.Sinceitcanbeappliedtobothmovingpointsandmovingregions,thisisanoverloadingfunction.Intherstsituation,ittakesanuncertainmovingpointandatimeinstantasinput,andreturnsthepossiblelocationofthismovingpointatthattime.Inthesecondsituation,ittakesanuncertainmovingregionandatimeinstant,andreturnsthepossiblelocationofthemovingregionatthattime.TheoperationisillustratedinFigure 3-5 a. Denition3.1.3.5(atInstant). Givenanuncertainmovingpointump2umPointandatimeinstancet,andletCI=dom(ump)denotetheunionofcertainintervals,andU=I)]TJ /F5 11.955 Tf 11.95 0 Td[(Cdenotestheuncertainintervals, atInstant(ump,t)=(p,r)(i)p2point,r2region(ii)8t2U:p=8t2C:r=Givenanuncertainmovingpointumr2umRegion,andatimeinstancet,theatInstantoperationisdenedas, atInstant(umr,t)=regionTheabovedenitionshowsthatbecauseoftheuncertaintyfeature,thepossiblelocationsofamovingpointatatimeinstantcanbeapointoraregion.Ifthetimeinstantisintheuncertaininterval,thenthemovingpointisinthependant,i.e.,theresultisinanuncertainarea.Otherwise,thetimeinstantisinthecertaininterval,andtheresultofthe 41

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movingpointisacertainpointwhichcanbecalculatedfromthefunctionrepresentingthestringpartofthemovement.However,foramovingregionobject,theresultoftheatInstantoperationwillalwaysbearegion,nomatterornotthetimeinstantiswithintheuncertainintervals.IncontrasttotheatInstantoperationwhichreturnsthepossiblelocationsofamovingobjectataninstant,weintroduceanotheroperationtemporalSelectwhichretrievestheuncertainmovementduringaperiodoftime.Thisoperationisanoverloadingfunction,similartotheatInstantoperation.Intherstcase,ittakesanuncertainmovingpointandatimeintervalasinput,andreturnspartialmovementoftheuncertainmovingpoint.Inthesecondcase,ittakesanuncertainmovingregionandatimeinstance,andreturnspartialmovementoftheuncertainmovingregion.TheoperationisillustratedinFigure 3-5 b. Denition3.1.3.6(temporalSelect). Givenanuncertainmovingpointump2umPoint,whichisconstructedfromaspatiotemporalpointlistSTPList=anddom(ump)=f[t1,t2],...,[tn,tn)]TJ /F7 7.97 Tf 6.58 0 Td[(1]g.LetCdenotethecertainintervalsofump,andUdenotetheuncertainintervalsofump.GivenanintervalIdom(ump),thenatemporalselectionoperationonumpwithrespecttointervalIisdenedas, temporalSelect(ump,I)=f,withthefollowingconditions(i)f2umPoint(ii)8[tl,tr]2I\C:9pi,pi+12STPListfj[tl,tr]=string(pi,pi+1,tl,tr)(iii)8[tl,tr]2I\U:9pi,pi+12STPListfj[tl,tr]=pendant(pi,pi+1,tl,tr,vmaxi)Wecandenethetemporalselectoperationonaumregionobjectsimilarly.Givenanuncertainmovingpointumr2umRegion,whichisconstructedfromaspatiotemporalregionlistSTRList=anddom(ump)= 42

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(a)(b)Figure3-5. TheatInstantandtemporalSelectoperations f[t1,t2],...,[tn,tn)]TJ /F7 7.97 Tf 6.59 0 Td[(1]g.GivenatimeintervalIdom(umr),thenatemporalselectionoperationonumrwithrespecttointervalIisdenedas, temporalSelect(umr,I)=f,withthefollowingconditions(i)f2umRegion(ii)8[tl,tr]2I\C:9ri,ri+12STRListfj[tl,tr]=Spi2ri,pi+12ri+1string(pi,pi+1,tl,tr)(iii)8[tl,tr]2I\U:9ri,ri+12STRListfj[tl,tr]=Spi2ri,pi+12ri+1pendant(pi,pi+1,tl,tr,vmaxi) 3.1.4Spatio-temporalUncertainPredicatesAnimportantresearchtopicofmovingobjectisthetopologicalrelationshipbetweenmovingobjects.Thetopologicalrelationshipsbetweenmovingobjectswithuncertaintyhavebeenrarelystudied.Peoplearealwaysinterestedinwhethertwomovingobjectscouldpossiblymeetduringsomeperiod.Inthissection,weintroduceaconceptcalledspatio-temporaluncertainpredicates(STUP).AnSTUPisabinarypredicateswhichresultiseithertrueoffalse.Itdescribestherelativepositionbetweenamovingobjectandastaticobject,ortwomovingobjects.AnexampleofSTUPispossiblyCross,whichdescribeswhetheramovingobjectcancrossastaticregion,forexample,itcanhelpdeterminewhetherhurricaneKatrinawillcrossthestateofFlorida.Inthissection,weintroducethespatio-temporaluncertainpredicatesunderourpendant 43

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Table3-2. ComponentsofanSTUPexpression CategoryOperators Logicaloperator:,9,8,^,_Setoperator\,[,2,2Dpredicatedisjoint,meet,overlap,covers,coveredBy,equals,inside,containsPendantoperatoratInstant,temporalSelect,getLifetimeUncertaintydegreedenitely,possibly model.InSection 3.1.4.1 ,wegiveanoverviewofSTUPs.WeclassifytheSTUPsintothreecategories,i.e.,thepredicatesbetweenmovingpointobjects,discussedinSection 3.1.4.2 ,andpredicatesbetweenamovingpointobjectandamovingregionobject,asdiscussedinSection 3.1.4.3 ,andtheSTUPbetweenmovingregionobjects,asdiscussedinSection 3.1.4.4 3.1.4.1OverviewofSpatio-temporalUncertainPredicatesInthepast,topologicalpredicateswhichdescribetherelationshipbetweenspatialobjectshavebeenwellstudied.8topologicalpredicatesbetweentworegionobjectshavebeendenedusingthe9-intersectionMatrix[ 17 ],whicharedisjoint,meet,overlap,covers,coveredBy,equal,inside,containsrespectively.Forexample,inside(A,B)meansthattheareaofregionAislocatedentirelyinregionB[ 66 ].WhenausersubmitsaqueryFindallnationalparksthatareinsideofstateofFlorida,thispredicatewillbeusedasselectionconditionsinthewhereclauseofaSQLquery,andalltheresultsthatwillmakethepredicatetobetruewillbereturned.Thisisaneasyandconvenientwaytoperformdatabasequeries.Similarly,wedeneourspatio-temporaluncertaintypredicateswhichwillenableuserstoquerytherelationshipbetweenmovingobjectsintheuncertaintyenvironmenteasily.Aspatio-temporaluncertainpredicate(STUP)isabooleanexpressionthatconsistsof2Dtopologicalpredicates,mathnotationsandoperationsunderourpendantmodel.AnSTUPexpressioncontainsthefollowingcomponentsshowninTable 3-2 ,Theeighttopologicalpredicatesdescribetherelationshipbetweentworegions,andtheyformthebasisofourspatio-temporaluncertainpredicates.Weusethelogic 44

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(a)(b)Figure3-6. Thedomainsofaninstantpredicateandanintervalpredicate operatorsandsetoperatorstoconnecttermsandformtheexpressions.Wenametherelationshipbetweenthematatimeinstanceasaninstantpredicate,andtherelationshipswhichlastsforaperiodasaintervalpredicate,shownasfollows.Aninstantpredicatebetweentwomovingobjectswithuncertaintyisafunction,time!B,where,2fumPoint,umRegiong.Anintervalpredicatebetweentwomovingobjectswithuncertaintyisafunction,interval!B,where,2fumPoint,umRegiong.Figure 3-6 aandFigure 3-6 bshowthedomainofaninstantpredicateandthedomainofanintervalpredicaterespectively.Intable 3-3 wedetectallspatio-temporaluncertaintypredicatesunderourpendantmodel,includinginstantpredicatesandintervalpredicates.Wewilldenethemformallyintherestofthissection.Figure 3-7 illustratesthedifferencebetweeninstantpredicatesandintervalpredicates.Withoutconsideringtheuncertaintyaspect,aninstantpredicate,suchasinside,showsthetopologicalrelationshipbetweenthesetwomovingobjectsataparticulartimeinstance,asshowninFigure 3-7 a.Incontrast,anintervalpredicate,suchascross,showstheevolvingtopologicalrelationshipbetweentwomovingobjectswithinaninterval,asshowninFigure 3-7 b.Ifweconsidertheuncertaintyaspect,theinstantpredicatebetweenamovingpointandaregioncanbepossibly inside,asshowninFigure 3-7 c,however,theintervalpredicatepossiblyCrossisdenedonaninterval[t1,t3],asshowninFigure 3-7 d. 3.1.4.2PredicatesbetweenUncertainMovingPointsWerstexamtheinstantpredicatesbetweentwomovingpoints.Withouttheuncertaintyaspect,thetopologicalrelationshipbetweentwomovingpointsatatime 45

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(a)(b) (c)(d)Figure3-7. Thedifferencebetweenaninstantpredicateandanintervalpredicate Table3-3. STUPpredicatesunderthependantmodel CategoriesumPointumPointumPointumRegionumRegionumRegion InstantdisjointAtdisjointAtdisjointAtPredicatepossiblyMeetAtpossiblyInsideAtpossiblyOverlapAtdenitelyMeetAtdenitelyInsideAtdenitelyInsideAt/denitelyCoverAt IntervaldenitelyEncounterdenitelyEnterdenitelyEnterPredicatepossiblyEncounterpossiblyEnterpossiblyEnterdenitelyLeavedenitelySweeppossiblyLeavepossiblySweepdenitelyCrossdenitelyLeavepossiblyCrosspossiblyLeavedenitelyCrosspossiblyCross instantisconsideredasthetopologicalrelationshipbetweentwostaticpoints,thusitiseitherdisjointormeet.However,whenintroducingtheuncertainty,atatimeinstanttherearethreepossiblerelationships.Theycanbedisjoint,ordenitelymeet,orpossiblymeet.Thereforewehavethefollowingthreeinstantpredicatesbetweenmovingpoints. Denition3.1.4.1(disjointat). Giventwomovingpointsp,q2umPointandatimeinstantt.ThepredicatedisjointAtisdenedas, disjointAt(p,q,t):=t2(getLifetime(p)\getLifetime(q))^disjoint(atInstant(p,t),atInstant(q,t)) 46

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Inthisdenition,weshowthatinordertomakedisjointAt(p,q,t)tobetrue,therstconditionisthatthequerytimetmustbelongtothelifetimeofbothmovingpoints.Then,wewillapplyatInstantoperationtobothobjectsandgettwospatialobjects.Thenwewillcheckwhetherthesetwoobjectsaredisjointwitheachother.Similarly,wedenedenitelyMeetAtandpossiblyMeetAtoperationsasfollows. Denition3.1.4.2(denitelymeetat). Giventwomovingpointsp,q2umPointandatimeinstantt.ThepredicatedenitelyMeetAtisdenedasfollows, denitelyMeetAt(p,q,t):=t2(getLifetime(p)\getLifetime(q))^atInstant(p,t)2point^equals(atInstant(p,t),atInstant(q,t)) Denition3.1.4.3(possiblymeetat). Giventwomovingpointsp,q2umPointandatimeinstantt.ThepredicatepossiblyMeetAtisdenedasfollows, possiblyMeetAt(p,q,t):=t2(getLifetime(p)\getLifetime(q))^:disjointAt(p,q,t)^:denitelyMeetAt(p,q,t)FromDenition 3.1.4.2 toDenition 3.1.4.3 ,welearnthatatopologicalrelationshipcanbedescribeddifferentlywhentheuncertaintyaspectisintroduced.Sinceeachmovingpointhasanuncertaintyregionatthatparticulartimeinstance,ifthetheiruncertaintyregionsdonotoverlap,thetwomovingpointswillnothaveachancetomeeteachotheratthattime.ThenwehavethedisjointAtpredicatetobetrue.However,iftheatInstantoperationontwoobjectsresulttwopointsthatareofthesameposition,theywilldenitelymeetatthistimeinstance.Inothercases,atthisparticulartimeinstant,theresultsoftheatInstantoperationonthesetwomovingobjectspatiallyoverlap,thentherelationshipbetweenthesetwomovingpointsispossiblyMeetAt.Figure 3-8 illustratestheabovethreepredicates.Theabovedenitionsonlyintroducepredicatesofmovingobjectwithuncertaintyataparticulartimeinstanceintheirlifetime,however,wearealsointerestedintheevolvingrelationshipbetweenmovingobjectswhichcanlastforaperiodoftime. 47

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(a)(b)(c)Figure3-8. Instantpredicatesofmovingpointswithuncertainty Forexample,amovingpointatthebeginningisdisjointwithanothermovingpoint,butitispossiblethatatalatertime,themovingpointsmeetataparticularlocation.Wecandescribesuchkindofevolvingrelationshipasencounter.Ifintroducingtheuncertaintyaspect,thisrelationshipcanfurtherbecometwodifferentpredicates,whicharepossiblyEncounteranddenitelyEncounter.TheserelationshipscanalsobeimplementedasbinarypredicatessothattheycanbeusedinSQLqueries.Wehavenamedsuchkindofrelationshipsasintervalpredicates.Intherestofthissubsection,wedenesomepredicatesthatdescribetheevolvingtopologicalrelationshipbetweenmovingobjectsoveraperiodoftime.WerstgivethedenitionoftheintervalpredicatesdenitelyEncounterandpossiblyEncounter. Denition3.1.4.4. LetpI:=temporalSelect(p,I),qI:=temporalSelect(q,I), denitelyEncounter(p,q,I):=I(getLifetime(p)\getLifetime(q))^9t1,t2,t32I^t1
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disjointAtrespectively.Similarly,wehavethepossiblyEncounterpredicatesbetweentwomovingpoints,denedasfollows, Denition3.1.4.5. LetpI:=temporalSelect(p,I),qI:=temporalSelect(q,I), possiblyEncounter(p,q,I):=I(getLifetime(p)\getLifetime(q))^9t1,t2,t32I^t1
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(a)(b)(c)Figure3-10. Examplesofspatio-temporaluncertainpredicatesbetweenamovingpointandamovingregion hurricaneKatrina.Now,wedenetheinstantpredicateswhichrepresenttopologicalrelationshipsbetweenamovingpointandamovingregionobject. Denition3.1.4.6. Giventhemovementofamovingpointp2unmpoint,amovingregionr2region,andatimeinstantt2time,thedisjointAtpredicateisdenedas, disjointAt(p,r,t):=t2getLifetime(p)\getLifetime(r)^disjoint(atInstant(p,t),atInstant(r,t)) Denition3.1.4.7. Giventhemovementofamovingpointp2unmpoint,amovingregionr2region,andatimeinstantt2time,thedenitelyInsideAtpredicateisdenedas, denitelyInsideAt(p,r,t):=t2getLifetime(p)\getLifetime(r)^inside(atInstant(p,t),atInstant(r,t)) Denition3.1.4.8. Giventhemovementofamovingpointp2unmpoint,amovingregionR2region,andatimeinstantt2time,thepossiblyInsideAtpredicateisdenedas, possiblyInsideAt(p,R,t):=t2getLifetime(p)\getLifetime(r)^:disjointAt(p,r,t)^:denitelyInsideAt(p,r,t)Theabovedenitionformalizethreeinstantuncertaintypredicatesbetweenamovingpointobjectandamovingregionobject.Figure 3-10 a-cshowtheabovethreepredicates,wherethecirclerepresentstheuncertainregionofamovingpointatthegiventimeinstanceandthepolygonrepresentstheresultoftheatInstantoperationonamovingregion.Nowwedenetheuncertaintypredicateswhichinvolvemovingregionobjects. 50

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Denition3.1.4.9. Giventheuncertainmovementofamovingpointp2umPoint,amovingregionr2umRegion,andatimeinstantt2time.LetpI:=temporalSelect(p,I),rI:=temporalSelect(r,I), denitelyEnter(p,r,I):=I(getLifetime(p)\getLifetime(r))^9t1,t22I^t1
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possiblyLeave(p,r,I):=I(getLifetime(p)\getLifetime(r))^9t1,t22I^t1
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(a)(b)Figure3-11. Examplesofspatio-temporaluncertainpredicatesbetweenamovingpointandamovingregion 3.1.4.4PredicatesbetweenUncertainMovingRegionsInthissection,weintroducethepredicatesbetweentwomovingregions.Itcanhelp,forexample,determineallthestatesthathavebeentraversedbyhurricaneKatrina.Werstdenetheinstantpredicatesbetweentwomovingregionobjects. Denition3.1.4.15. Giventwomovingregionsr,s2umRegion,andatimeinstantt2time,thedisjointAtpredicateisdenedas, disjointAt(r,s,t):=t2getLifetime(r)\getLifetime(s)^disjoint(atInstant(r,t),atInstant(s,t)) Denition3.1.4.16. Giventwomovingregionsr,s2umRegion,andatimeinstantt2time,thedenitelyInsideAtpredicateisdenedas, denitelyInsideAt(r,s,t):=t2getLifetime(r)\getLifetime(s)^inside(atInstant(r,t),atInstant(s,t))ThepredicatedenitelyCoverAtisdenedas, denitelyCoverAt(r,s,t):=denitelyInsideAt(s,r,t) Denition3.1.4.17. Giventwomovingregionsr,s2umRegion,andatimeinstantt2time,thepossiblyInsideAtpredicateisdenedas, possiblyInsideAt(r,s,t):=t2getLifetime(r)\getLifetime(s)^:disjointAt(r,s,t)^:denitelyInsideAt(r,s,t)^:denitelyCoverAt(r,s,t) 53

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(a)(b)(c)Figure3-12. Examplesofspatio-temporaluncertainpredicatesbetweentwomovingregions Theabovedenitionsformalizeinstantuncertaintypredicatesbetweentwomovingregionobjects.Figure 3-12 a-cshowtheabovethreepredicates.Nowweintroducetheintervalpredicatesbetweentwomovingregions. Denition3.1.4.18. Giventwomovingregionsr,s2umRegion,andatimeinstantt2time.LetrI:=temporalSelect(r,I),sI:=temporalSelect(s,I), denitelyEnter(r,s,I):=I(getLifetime(r)\getLifetime(s))^9t1,t22I^t1
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denitelyLeave(r,s,I):=I(getLifetime(r)\getLifetime(s))^9t1,t22I^t1
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ThepredicatesofpossiblyCrossandpossiblyEnterbetweentwomovingregionsareillustratedinFigure 3-13 (a)(b)Figure3-13. Examplesofspatio-temporaluncertainpredicatesbetweentwomovingregions 3.1.5Spatio-TemporalUncertainQueryLanguageInthissection,wediscusshowtointegrateSTUPswehavedenedintheprevioussectionintodatabasequeries.CurrentdatabasequerylanguagessuchasSQLarenotabletoanswertemporalqueriesbecausetheydonotsupporttemporaloperators.ThisproblemcouldbesolvedbyimplementingtheSTUPsasoperatorsinqueries.Thus,weareabletoextendtheSQLlanguagetoamorecomprehensivequerylanguage,namedasspatio-temporaluncertaintyquerylanguage(STUQL).TheSTUQLlanguageextendsSQLandsupportsthespatio-temporaluncertaintyoperationsintermsofSTUPs.Intherestofthissectionweshowsomeexamplesofqueryingthespatio-temporaluncertaintyindatabasesusingSTUQL.FirstwewillintroducethenewdatatypesweneedinourMovingObjectDatabase(MOD).Besidestheprimitivedatatypessuchasinteger,string,etc.,wewillimplementthemovingobjectdatatypesinourpendantmodel.Herewehavetwonewdatatypesumpointandumregion,representingtheumPointandumRegionobjectswehavedenedinSection 3.1.2 respectively. 56

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Assumethatwewanttodetectwhethertwopersonshasapossibilitytomeetduringaperiodoftime,andwehavethefollowingschemaofpersonsinthedatabase, persons(id:integer,name:string,trajectory:umpoint)Hereumpointistheuncertainmovingpointdatatype.ThequeryFindallpersonsthatmaypossiblybecomethewitnessofthecriminalTrudyduringtheperiodfrom10amto12pmcanbeansweredbytheSTULqueryasfollows, SELECTp1.idFROMpersonsp1,personsp2WHEREpossibly_encounter(p1.trajectory,p2.trajectory,10:00:00,12:00:00)ANDp2.name=`Trudy'Intheabovequery,weimplementthepossiblyEncounterpredicateasafunctionthatcanbeintegratedintotheSQLqueryintheWHEREclause.NowwegiveanexampleofSQLlikequeryonthepredicatesbetweenanuncertainmovingpointandastaticregion.Rememberthatinourmodel,astaticregionistreatedasaspecialcaseofamovingregion.Thereforewehavetheareaofanairportasanuncertainmovingregion.Assumethatwehavethefollowingschemas, airplanes(id:string,flight:unmpoint)airports(name:string,area:umregion)ThequeryFindallplanesthathavepossiblyenteredtheLosAngelesairportfrom2:00pmto2:30pmcanbewrittenasfollows, SELECTairplanes.idFROMairplanes,airportsWHEREpossibly_enter(airplanes.flight,airports.area,14:00:00,14:30:00)ANDairports.name=`LAX'; 57

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Wearealsoabletoqueryonthetopologicalrelationshipbetweenanuncertainmovingpointandanuncertainmovingregion.Anexampleofamovingregionwithuncertaintyisthehurricane.Assumethatwehavethefollowingschema, hurricanes(name:string,extent:umregion)ThequeryFindallplanesthathavepossiblycrossedtheextentofhurricaneKatrinabetweenAug24toAug25,2005canbewrittenasfollows, SELECTa.idFROMairplanesa,hurricaneshWHEREpossibly_cross(a.flight,h.extent,2005-08-24,2005-08-25)ANDh.name=`Katrina' 58

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3.1.6UncertainCardinalDirectionDevelopmentPredicatesInthesamewayasmovingobjectscanchangetheirlocationovertime,thespatialrelationshipsbetweenthemcanchangeovertime.Animportantclassofspatialrelationshipsarecardinaldirectionslikenorthandsoutheast.InspatialdatabasesandGIS,theycharacterizetherelativedirectionalpositionbetweenstaticobjectsinspaceandarefrequentlyusedasselectionandjoincriteriainspatialqueries.Transferredtoaspatiotemporalcontext,thesimultaneouslocationchangeofdifferentmovingobjectscanimplyatemporalevolutionoftheirdirectionalrelationships,calleddevelopment.Nowweintroducetheconceptofcardinaldirectiondevelopment,whichisaspecialkindofspatio-temporaluncertainpredicatedescribingthedynamiccardinaldirectionsbetweentwomovingobjects.InSection 3.1.6.1 ,werstreviewthedenitionsforcardinaldirectionsbetweenstaticpointswithouttheconsiderationoftime.Then,inSection 3.1.6.2 ,wemodelthetemporalevolutionofthecardinaldirectionsbetweentwomovingpointsasacardinaldirectiondevelopment. 3.1.6.1CardinalDirectionsbetweenStaticPointsTheapproachthatisusuallytakenfordeningcardinaldirectionsbetweentwostaticpointsintheEuclideanplaneistodividetheplaneintopartitionsusingthetwopoints.Onepopularpartitionmethodistheprojection-basedmethodthatuseslinesorthogonaltothex-andy-coordinateaxestomakepartitions[ 26 60 ].Thepointthatisusedtocreatethepartitionsiscalledthereferencepoint,andtheotherpointiscalledthetargetpoint.Thedirectionalrelationbetweentwopointsisthendeterminedbythepartitionthatthetargetobjectisin,withrespecttothereferenceobject.LetPointsdenotethesetofstaticpointobjects,andletp,q2Pointsbetwostaticpointobjects,wherepisthetargetpointandqisthereferencepoint.Atotalof9mutuallyexclusivecardinaldirectionsarepossiblebetweenpandq.LetCDdenotethesetof9cardinaldirections,thenCD=fnorthwest(NW),restrictednorth(N),northeast(NE),restrictedwest 59

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(W),sameposition(SP),restrictedeast(E),southwest(SW),restrictedsouth(S),southeast(SE)g.Further,letXandYbefunctionsthatreturnthexandycoordinateofapointobjectrespectively.Thecardinaldirectiondir(p,q)2CDbetweenpandqisthereforedenedas dir(p,q)=8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:NWifX(p)Y(q)NifX(p)=X(q)^Y(p)>Y(q)NEifX(p)>X(q)^Y(p)>Y(q)WifX(p)X(q)^Y(p)=Y(q)SWifX(p)X(q)^Y(p)
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(a)(b)Figure3-14. Examplesoftwomovingpointswithchangedandunchangeddirectionsovertime andtoanswerthequestionwhetherthereexistsatimeinstancet(t1><>>:trueif8t2I:dir(A(t),B(t))=dfalseotherwise 61

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ForagiventimeintervalI,wemakethefollowingobservations:(i)ifthereexistsacardinaldirectiond2CDsuchthatholds(A,B,I,d)=true,thenwesayAandBhaveauniquecardinaldirectiononthetimeintervalI;(ii)ifbothAandBaredenedonI,andthepredicateholdsreturnsfalseforall9basiccardinaldirectionsinCD,thenwesaythatAandBhaveadevelopingcardinaldirectionrelationshipoverI;(iii)ifneitherAnorBisdenedonI,wesaythecardinaldirectionbetweenAandBisnotdenedonI.Therefore,wecanonlydeterminecardinaldirectionsbetweenAandBduringintervalsonwhichtheyaredened.Foranintervalwherethereisnouniquebasiccardinaldirectionthatholdsovertheentireperiod,wesplititintoseveralsub-intervalssuchthatoneachsub-intervalonlyauniquecardinaldirectionholds.Further,ifweregarddifferentcardinaldirectionsthatholdoverdifferentsub-intervalsascardinaldirectionstates,thedevelopmentofthecardinaldirectionsreferstoasequenceoftransitionsbetweenthesestates.Forexample,AmovingfromNWtoWtoSWofBisadevelopmentofcardinaldirectionsbetweentwomovingpointsAandB.However,notalltransitionsarepossiblebetweenanytwostates.Figure 3-15 showsallpossibletransitionsbetweendifferentstates.Forexample,ifthecardinaldirectionbetweentwomovingpointsAandBhasbeenNWsofar,theniftimechanges,thecardinaldirectionmightstaythesameasNW,orchangetoeitherN,W,orSP.ItisnotpossiblethatAmovesdirectlytothesouth(S)ofBwithoutcrossinganyotherdirections.Thisstatetransitiondiagramimpliesthatonlydevelopmentsthatinvolvevalidtransitionsarepossiblebetweentwomovingpoints,e.g,AmovesfromNWtoWtoSWofB.DevelopmentsthatinvolveinvalidtransitionslikeAmovingfromNWtoSofBarenotpossibleandthusnotallowed.LetthepredicateisValidTrans:CDCD!booltaketwocardinaldirectionsasinput,andyieldtrueifthetransitionbetweenthemisvalid.Then,forexample,isValidTrans(NW,W)=truewhileisValidTrans(NW,S)=false.Nowwecandenethedevelopmentofthecardinal 62

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Figure3-15. Thestatetransitiondiagramofallcardinaldirections directionsbetweentwomovingpointsAandBonanygiventimeintervalIonwhichAandBarebothdened. Denition3.1.6.2. GiventwomovingpointsA,B2MPointsandatimeintervalIonwhichbothAandBaredened.AssumetheorderingofanytwointervalsI1=[tb1,te1]andI2=[tb2,te2]isdenedaste1tb2,I1I2.Letthesymbol.representthetransitionfromonecardinaldirectionstatetoanother.ThenthedevelopmentofcardinaldirectionsbetweenAandBonintervalI,denotedasdev(A,B,I)canbedenedas: dev(A,B,I)=d1.d2.....dnifthefollowingconditionshold: (i)n2N(ii)81in:di2CD(iii)81in)]TJ /F4 11.955 Tf 11.96 0 Td[(1:di6=di+1,isValidTrans(di,di+1)=true(iv)9I1,I2,...,In:(a)81in:Iiisatimeinterval,holds(A,B,Ii,di)=true(b)81in)]TJ /F4 11.955 Tf 11.96 0 Td[(1:IiIi+1,(c)n[i=1Ii=I 63

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Figure3-16. Anexampleofacardinaldirectiondevelopment InDenition 3.1.6.2 ,wesplitthegiventimeintervalintoasequenceofnon-overlappingsub-intervals.Thedevelopmentdevrepresentsthetransitionofcardinaldirectionsoverthesesub-intervals.Condition(iv)(a)ensuresthatauniquecardinaldirectionbetweenAandBholdsoneachsub-interval,andcondition(iv)(c)ensuresthatallsub-intervalstogetherformafulldecompositionofthegivenintervalI.Further,accordingtocondition(iii),onlyvalidtransitionsareallowedbetweentwocardinaldirectionsthatholdonadjacentsub-intervals.AnexampleofsuchadevelopmentcanbederivedfromFigure 3-16 ,whereAmovesfromlocationa1tolocationa2andBdoesnotmoveduringthetimeintervalI=[t1,t2].ThedevelopmentofcardinaldirectionsbetweenAandBduringIisthereforedev(A,B,I)=NW.N.NE.N.NW.W.SW.Itdescribesthatfromtimet1totimet2,AstartsinNWofB,crossesNandreachesNEofB,thenitturnsaroundandcrossesNagain,andreturnstoNWofB.Finally,AcrossesWofBandendsupintheSWofB.Nowwearereadytodenecardinaldirectiondevelopmentsbetweentwomovingpointsduringtheirentirelifetime.Theideaistorstndouttheircommonlifetimeinter-vals,onwhichbothAandBaredened.ThenweapplythedevfunctiontodeterminethedevelopmentofcardinaldirectionsbetweenAandBduringeachcommonlifetime 64

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interval.Finally,wecomposethecardinaldirectiondevelopmentsondifferentcommonlifetimeintervalsanddeneitasthedevelopmentofcardinaldirectionsbetweenthetwomovingpointsAandB.Werstndoutthecommonlifetimeintervalsfortwomovingpoints. Denition3.1.6.3. GiventwomovingpointsA,B2MPoints,letLTA=IA1,IA2,...,IAm,LTB=IB1,IB2,...,IBnbetwolifetimeintervalsequencesofAandBrespectivelysuchthatIAi
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Denition 3.1.7.10 generalizesthedevelopmentofcardinaldirectionsbetweentwomovingpointsfromagivenintervaltotheirentirelifetime.ThecardinaldirectiondevelopmentbetweenAandBinFigure 3-16 isthereforeDEV(A,B)=NW.N.NE.N.NW.W.SW.?.SW.?.SE.E.NE. 3.1.7UncertainTopologicalChangesofComplexMovingRegionsInprevioussections,wehaveaddressedtheproblemofuncertaintyinmovingobjects.Mostoftheexampleswehaveshowndealwiththeuncertaintyinmovingpoints.Nowwecometothediscussionofmovingregions.Inthissection,wediscusstheproblemoftopologicalchangesofmovingregions.Amovingregionwhoselocationandextentchangeovertimecanundergoseveraltopologicalchangessuchasthesplittingofaregionortheformationofahole.Thestudyofthiskindofchangesisimportantinmanyapplications,e.g.,forthetopologycontrolofwirelesssensornetworksandtheprocessingofanimationimagesinmultimediaapplications.Sinceweoftenlacktheabilityofcapturingthelocation,extent,andshapechangesofamovingregionduringitslifespan,uncertaintyexists.Weproposeanapproachpursuingathree-phasestrategytodeterminethetopologicalchangesofacomplexmovingregionrepresentedbyasequenceofsnapshots.Westartthediscussionfromregionobjects.Section 3.1.7.1 formallydiscussesspatialregionobjectsandtheirproperties,anddepictstheconceptandtherepresentationofmovingregionobjects.Section 3.1.7.2 introducesoursnapshotrepresentationofmovingregion.Section 3.1.7.3 presentsthethree-phasealgorithmsforevaluatingthetopologicalchangesinacomplexmovingregion. 3.1.7.1ComplexRegionsAmovingobjectisdescribedbyafunctionfromtimetoitscorrespondingspatialobject[ 28 ].Similarly,amovingregionobjectshowsamappingfromtimetoare-gionobject.Thus,beforewediscusswhatisamovingregion,werstdiscussitscorrespondingspatialdatatype,i.e.region. 66

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Inordertogivetheformaldenitionofamovingregion,weintroducesomebasicconceptsofregions,forexample,theconnectivity,closedpropertyandboundedproperty,whichareneededinthelaterdenitions.Ourdenitionsarebasedonpointsettheoryandpointsettopologyandourpreviousresearchoncomplexregionobjects[?],whereregionsareembeddedintothetwo-dimensionalEuclideanspaceR2andmodeledasinnitepointsets.Inthesimplestcase,aregioniscomposedbyasingleconnectedcomponent,whichiscalledasimpleregion.Thepointsetinthe2DEuclideanspacerepresentingasimpleregionobjectshouldbeconnected,closedandbounded.Thisthreecharacteristicsareformallydescribedinthefollowingthreedenitions. Denition3.1.7.1(Connectivity). LetXR2,andX,X)]TJ /F3 11.955 Tf 7.08 -4.33 Td[(,and@Xdenotetheinterior,exteriorandboundaryofX;let XdenotetheclosureofXand X=@X[X.TwosetsX,YR2aresaidtobeseparatedif,andonlyifX\ Y== X\Y.AsetXR2isconnectedif,andonlyif,@Y,ZX,sothat (i)X6=,Y6=(ii)X=Y[Z(iii)YandZareseparated.Denition 3.1.7.1 showsthatifaregionisconnected,itshouldnotequaltotheunionoftwononemptyseparatedsets.AcounterexampleisshowninFigure 3-17 a,whereR=R1[R2,R16=,andR26=.However,sinceR1andR2areseparated,thesituationinthisgureviolatestheconnectivityproperty. Denition3.1.7.2(ClosedProperty). LetXR2.Xissaidtoberegularclosedif,andonlyif,X= X.Denition 3.1.7.2 showsthataregularclosedregionisainnitepointsetthatremovesallthegeometricanomalies.Theinterioroperationeliminatesdanglingpoints,danglinglinesandboundaryparts.Theclosureoperatoraddstheboundaryand 67

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(a)(b)(c)Figure3-17. Adisconnectedregion,aregionwithdanglingpointsandlineswhichisnotclose,andanunboundedregion eliminatescutsandpuncturesbysupplementingpoints.AnexampleofaregionwhichisnotclosedisshowninFigure 3-17 b,wherewecanseedanglingpointsandlines,aswellascuts. Denition3.1.7.3(BoundedProperty). LetXR2,p=(x1,y1)2X,q=(x2,y2)2X,andd(p,q)=p (x1)]TJ /F5 11.955 Tf 11.95 0 Td[(x2)2+(y1)]TJ /F5 11.955 Tf 11.95 0 Td[(y2)2,Xissaidtobebounded,if 8p,q2X:9r2R+,suchthatd(p,q)
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Denition3.1.7.5(SimpleRegion). Thesimpleregiondatatype,denotedbySRisdenedas, SR=fRregionj(i)Risconnected(ii)Risregularclosed(iii)RisboundedgThereareeighttopologicalrelationshipsbetweentwosimpleregions,introducedin[ 16 ],whicharedisjoint,meet,overlap,covers,coveredBy,equal,contains,andinside.Theywillbeusedinthedenitionsofothertypesofregionobjectsintherestofthissection.Inmanycases,aregionobjectmaybeclosedandbounded,butnotnecessaryconnected.Therearetwodifferentsituations.Intherstsituation,aregionmaybecomposedbyseveralseparatedcomponents.Thesecondsituationisthataregionhasoneormoreholesinsideit,violatingtheconnectivity.Considertherstsituation,wedenethespecictypemulti-regionasfollows, Denition3.1.7.6(Multi-region). Themulti-regiondatatype,denotedbyMR,isdenedas, MR=fRregion,R=R1[R2...[Ri...[Rnj(i)81in:Ri2SR(ii)81ijn:disjoint(Ri,Rj)gAnexampleofmulti-regionisshowninFigure 4-8 (b).Theotherfactthatcanviolatetheconnectivityofaregionobjectistheexistenceofholes.Thuswehavetwomoretypesofregions:aregionwithoneholeinsideit,oraregionwithmultipleholes.Wegivethedenitionsofthesetwotypesofregionsasfollows, Denition3.1.7.7(SimpleRegionwithOneHole). Thesimpleregionwithoneholedatatype,denotedbySRH,isdenedas, SRH=fRregion,R=R0)]TJ /F5 11.955 Tf 11.96 0 Td[(R1j(i)R0SR,R1SR(ii)contains(R0,R1)g 69

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(a)(b)(c) (d)(e)(f)Figure3-18. Sixdifferentshapesofregionobjects Denition3.1.7.8(SimpleRegionwithMultipleHoles). Thesimpleregionwithmultipleholesdatatype,denotedbySMH,isdenedas, SMH=fRregion,R=R0)]TJ /F16 11.955 Tf 11.96 8.97 Td[(Sni=1j(i)81in:Ri2SR(ii)1in:contains(R0,Ri)(iii)1ijn:disjoint(Ri,Rj)gThediagramsofsimpleregionwithoneholeandsimpleregionwithmultipleholesareshowninFigure 4-8 (c)and(d)respectively.Next,wegiveourdenitionoftheregionwiththemostcomplicatedproperties. Denition3.1.7.9(ComplexRegion). Thecomplexregiondatatype,denotedbyCR,isdenedas, CR=fR2region,R=R1[R2...[Ri...[Rnj(i)81in:Ri2SR,orRi2SRH,orRi2SMR(ii)81ijn:disjoint(Ri,Rj)gTherefore,accordingtoDenition 3.1.7.5 3.1.7.9 ,weobtainvedifferentgeometriesforaregionobject,whicharesimpleregion,multi-region,simpleregionwithahole,simpleregionwithmultipleholesandcomplexregion.Inaddition,weconsideranemptyregionasaspecialcaseofregionobjects,asillustratedinFigure 4-8 (f)thenwehavethesixthpossibleshapesofaregionobject. 70

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3.1.7.2SnapshotRepresentationofMovingRegionsInordertodetectthetopologicaldevelopmentinamovingregion,therstimportanttaskistorepresentthemovingregionproperly.However,thistaskischallenging.Becauseamovingregioncontinuouslychangeitslocationsandshape,wearenotabletotrackthecontinuousdeformationofthatmovingregionatalltimesduetotheshortcomingsofthetrackingdevices.Forexample,indetectingaforestre,sincesensorsusuallytakesmeasurementatdiscretetimes,wegetthereportfromthesensorsatdiscretetimeinstances.Whenstudyingwhetherthereisahurricane,weanalyzethepicturesfromthesatelliteswhicharecapturedeveryfewhours.Thus,ourideaistorepresentamovingobjectasasequenceofsnapshots.Inthispaper,wecallasnapshotasanobservation.Thesnapshotbasedapproachhasbeenwidelyacceptedbyresearchersinmanyelds.Incomputer-basedanimations,preciseimagesarecapturedlessfrequentlyandnamedasI-frames,andinterpolationisperformedtollthegapbetweentwoI-frames.Similarly,inourmodel,werepresentamovingregionatdifferenttimeinstancesandinterpretthetransitionsinbetween.Atdifferenttimeinstantswecan,forexample,obtaintheobservationsthatamovingregionobjectisasimpleregion,amulti-regionwithoutholes,orasimpleregionwithholes.Ourmodelwillbeabletocharacterizethebasictopologicalchangesbetweentwoconsecutiveobservationssuchasthesplittingofaregionortheformationofahole.Figure 3-19 (a)and(b)representtwoobservationsofacomplexmovingregioncapturedattimet1andt2respectively.Wecanndthatthisregionobjectmovesrightanddown,andthereisamergebetweenthelargestregionwiththeregionattherightabovecorner.Also,thereisaholeappearingattheregioninthecenter.However,theinterpretationisintuitivewhichlacksaformalexplanation.Forsuchacomplexmovingregioncontainingmultiplecomponents,wecannottellpreciselywhichcomponentbeforethechangecorrespondstowhichcomponentafterthechange.Thereforeitisdifcult 71

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toformallydeterminefromtwoconsecutivesnapshots,withouthumanintuitionand/orbackgroundinformation,whetheraspotofredisappears,orwhetheritmergeswithanotherspotofre.Wesolvethisproblembyprovidingathree-phasestrategy,whichisabletouniquelyinterpretthetopologicalchangebetweentwoconsecutivesnapshotsinthenexttwosections. (a)(b)Figure3-19. TwosnapshotsO1andO2ofamovingregionR 3.1.7.3EvaluatingtheTopologicalChangesinaComplexMovingRegionAmovingregionmayhavedifferentshapesatdifferentinstants,leadingtotopologicalchanges.However,becauseofthecontinuityproperty,topologicalchangescannothappenbetweeneverypairofstates.Forexample,adirectchangefromasimpleregiontoacomplexregionisimpossible.Instead,theremustbeotherintermediatestatesbetweenthem.Therefore,weintroducetheStateTransitionDiagram,whichshowsthevalidityoftransitionsbetweendifferentstatesofamovingregion.ThestatetransitiondiagramrepresentingalldirecttopologicalchangesisshowninFigure 3-20 .Anarrowbetweentwostatesshowsthatthereexistsadirecttopologicalchangebetweenthesetwostates.Iftherearenoarrowsbetweentwostates,itmeansthatadirecttopologicalchangebetweenthemarenotvalid.Therecanbemorethanonepossibletopologicalchangesbetweentwosamestates.Forexample,fromasimpleregiontoasimpleregionwithholes,twotopologicalchangesmayhappen:eitheraholeisformedinsidetheregion,ortheregiontouchesitselfandformshole.Thesetwotopologicalchangesarenamedasholeformandregionself-touchrespectively. 72

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Figure3-20. Thestatetransitiondiagramrepresentingvalidtopologicalchangesofamovingregion Ifwegroupallthesixstatesofamovingregionatdifferenttimeinstancestogether,thenwehaveaStateSet.LetEM,SR,SRH,SMH,MR,CRdenoteempty,simpleregion,simpleregionwithahole,simpleregionwithmultipleholes,multi-region,andcomplexregionrespectively,thenStateSet=fEM,SR,SRH,SMH,MR,CRg.LetSrepresentanyoneofthe6statesintheStateSet.Nowwegivetheformaldenitionsof11basictopologicalchangesthatareshowninFigure 3-20 Denition3.1.7.10(BasicTopologicalChange). Abasictopologicalchangeofamovingregiondi=S1!S2,isatransitionprocessbetweentwostatesS1,S22StateSet,wheret(S1)
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d0(topologypreserve):S)166(!Sd1(regionappear):EM)166(!SRSRH)166(!CRSR)166(!MRSMH)166(!CRd2(regiondisappear):SR)166(!EMCR)166(!SRHMR)166(!SRCR)166(!SMHd3(holeform):SR)166(!RIMR)166(!CRd4(holell):SRH)166(!SRCR)166(!MRd5(regionsplit):SR)166(!MRSRH)166(!CRSMH)166(!CRd6(regionmerge):MR)166(!SRCR)166(!SMHCR)166(!SRHd7(regionself-touch):SR)166(!SRHMR)166(!CRd8(ringsplit):SRH)166(!SRCR)166(!MRd9(holesplit):SRH)166(!SMHd10(holemerge):SMH)166(!SRH Denition3.1.7.11(TopologyPreserve). AbasictopologicalchangeS1!S2,iscalledtopologypreserve,denotedbyd0,ifS1,S22StateSetandS1=S2. Denition3.1.7.12(RegionAppear). AbasictopologicalchangeS1!S2,iscalledregionappear,denotedbyd1,if9rS2,r*S1andrisasimpleregion. Denition3.1.7.13(RegionDisappear). AbasictopologicalchangeS1!S2,iscalledregiondisappear,denotedbyd2,ifthechangeS2!S3isregionappear. Denition3.1.7.14(HoleForm). AbasictopologicalchangeS1!S2,iscalledholeform,denotedbyd3,if9R0=S1)]TJ /F5 11.955 Tf 11.95 0 Td[(S2,whereR0isasimpleregion,andcontains(S2, R0).InDenition 3.1.7.14 R0denotestheclosureofR0. Denition3.1.7.15(HoleFill). AbasictopologicalchangeS1!S2,iscalledholell,denotedbyd4,ifthechangeS2!S1isholeform. Denition3.1.7.16(RegionMerge). AbasictopologicalchangeS1!S2,iscalledregionmerge,denotedbyd5,ifgivenR0S1,R1[R2S2,andR1andR2areseparated,whereR0,R1,R2aresimpleregions,9Rx=R0)]TJ /F4 11.955 Tf 10.83 0 Td[((R1[R2),sothatR1[R2[Rxisconnected. Denition3.1.7.17(RegionSplit). AbasictopologicalchangeS1!S2,iscalledregionsplit,denotedbyd6,ifthechangeS2!S1isregionmerge. 74

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Denition3.1.7.18(RegionSelf-touch). AbasictopologicalchangeS1!S2,iscalledregionself-touch,denotedbyd7,ifgivenR0S1,R1S2,whereR0isasimpleregionandR1isasimpleregionwithonehole,9Rx=R1)]TJ /F5 11.955 Tf 11.95 0 Td[(R0,sothatR0[Rxisconnected. Denition3.1.7.19(RingSplit). AbasictopologicalchangeS1!S2,iscalledringsplit,denotedbyd8,ifthechangeS2!S1isregionself-touch. Denition3.1.7.20(HoleSplit). AbasictopologicalchangeS1!S2,iscalledholesplit,denotedbyd9,ifgivenR0S1,R1S2,whereR0isasimpleregionwithonehole,andR1isasimpleregionwithmultipleholes,9Rx= R0)]TJ /F5 11.955 Tf 12.61 0 Td[(R0,andRy= R1)]TJ /F5 11.955 Tf 12.62 0 Td[(R1,sothatRx)166(!Rxisregionsplit. Denition3.1.7.21(HoleMerge). AbasictopologicalchangeS1!S2,iscalledholemerge,denotedbyd10,ifthechangeS2!S1isholesplit.Intheabovedenitions,d0isaspecialcasethatthetopologyremainsthesameafterthechange,i.e.,thestatesbetweenthechangeandthestateafterthechangearethesame.Inotherwords,d0isnotastricttopologicalchange,butatopologypreservingchange.Suchchangesincludepositionchanging,growingorshrinking.Forexample,aspotofforestlespreadsandbecomesalargerareacatchingres,thisisatopologypreservingchange.Abasictopologicalchangecanappearmorethanonceinthediagram,forexample,regionappear(d1)canhappenfromemptytoasimpleregion,orfromasimpleregionwithoneholetoacomplexregion,orfromasimpleregionwithmultipleholestoacomplexregion,etc.Also,betweentwostates,therecanbemorethanonetopologicalchanges.Ifasimpleregionchangestoamulti-regionwithoutholes,eitheraregion-split(d3)oraregionappear(d7)mayhappen.Withtimepassingby,amovingregionmayhaveexperiencedasequenceofbasictopologicalchanges.Wecallthisatopologicaldevelopment.Nowwegivetheformaldenitiononthetopologicaldevelopmentofamovingregionduringaperiodoftime. 75

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Denition3.1.7.22. Letdenotetheoperatorthatconnectingtwobasictopologicalchangeswhichhappenconsecutively,thenthetopologicaldevelopmentofacomplexmovingregionC,isdenedas dev(C)=D1D2...Dnwiththefollowingconditions, (i)n2N(ii)81in:Di2fd0,d1,...,d10g(iii)LetDi=Si1)166(!Si2,andDj=Sj1)167(!Sj281ijn)]TJ /F4 11.955 Tf 11.96 0 Td[(1:t(Si2)t(Sj1)InDenition 3.1.7.22 ,Condition(i)showsthatthetopologicaldevelopmentofamovingregioniscomposedbyasequenceofdirecttopologicalchanges,denotedbyDi,wherethenumberofdirecttopologicalchangesisnite.Condition(ii)showsthattheunitcomposingthetopologicaldevelopmentisfromthe11basictopologicalchangeswehavedenedin 3.1.7.10 .Condition(iii)showsthatforanyconsecutivedirecttopologicalchanges,theoneinfrontofthenotationhappensbeforetheoneafterthenotation. 76

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3.2BalloonModel:RepresentingHistoricalandPredictiveMovingObjectsUncertaintyisaninherentfeatureofmovingobjectsduetotheinabilityofcapturingtheexactlocationsofmovingobjects.Previousapproacheswhichstudytheuncertaintyinmovingobjectmainlyfocusontheuncertaintyofmovingobjectsinthepast.However,themostcommonscenariowheretheuncertaintyexistsisthemovementsinthefuture.Becauselackingoftheobservationsoffuturemovement,thelocationsofmovingobjectsinthefuturecanonlybepredicted.Inthissection,westudytheuncertaintyinthefuturemovementsofmovingobjects.WeproposeanabstracttypesystemcalledBalloonModelthatisabletorepresentthehistoricalandfuturemovementsofmovingobjectsinuncertainenvironments.Asmostofthepreviousresearchersmainlyfocusonthemovementofasinglemovingpoint,whilemovingobjectsinrealityarecomplex,ourmodelcandealwiththerepresentationofacomplexmovingobjectwhichcancontainseveralmovingcomponents.Weintroducetheballoonmodeldatatypesandprovideformaldenitionsofthedatatypeswhichrepresentthemovingpointsandmovingregionsrespectively.Weintroduceacomprehensiveofoperationsonthemovingobjectswithuncertaintyunderthisballoonmodelandshowhowtheycanbeusedindatabasesqueries.Thismodelprovidesintegratedandseamlesssupportforbothhistoricalandpredictedmovementsofmovingobjectsinuncertainenvironments.Section 3.2.1 formalizethedenitionofmovingobjectsandtheirproperties.Sectionsubsec:mfmodiscussestheuncertaintyprobleminfuturemovingobjects.Section 3.2.2 proposesourballoonmodelrepresentingthemovingpointobjectsaswellasmovingregionobjectswithuncertainty.Section 3.2.3 discussestheoperationsontheballoonmodelwhichcanbefurtherintegratedintodatabases. 3.2.1TheNatureofMovingObjectsAnimportantaspectinthestudyofmovingobjectsistogiveaproperdescriptionofthemovementfunctionitself.Previousresearcheroftendenemovementasafunctionoftime,however,thisdenitiondoesnotconsiderthefactofcontinuity,i.e.,whether 77

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instantaneousjumpordenitiongapshouldbeallowed.Continuityisanintrinsicfeatureofmovingobjects.Sofar,thereisrarelyresearchonmovingobjectsthathaveformalizethecontinuityproperty.Inthissection,wespecifythecharacteristicfeaturesofmovingobjectsanddealwiththeproblemofhandlingthepastmovementinadatabasecontext.InSection 3.2.1.1 ,wegivethedenitionofthedissimilaritymeasurementwhichwillbeusedtodenethecontinuousmovements.Insection 3.2.1.2 ,wediscussthecontinuitypropertyandgiveaformaldenitionofcontinuousmovement.InSection 3.2.1.3 ,wedenethehistoricalmovingobjectsonthebasisofthecontinuousmovement.InSection 3.2.1.4 wedenefuturemovingobjectswhichtakestheuncertaintyaspectintoconsideration. 3.2.1.1DissimilarityMeasurementMovingobjectswhoselocationsandgeometryareevolvingovertimeoftenshowacontinuousfeature.Forexample,theextendofaforestregrowssmoothly,andaninstantaneousjumpofthecenterofahurricanecannothappen.Thiscontinuityfeatureisnotonlyshowninspatio-temporalobjects,butcanbeshowninothertemporalobjectssuchastemporalrealobjectsaswell.Forexample,thedynamicchangingofthetemperaturealsoshowsthisproperty.Intuitively,wecandescribethecontinuityfeatureasslighttimedifferencewillleadtoslightlychangeintheresult.However,howtomeasurethechanges?Itisnecessarytodenethechangeasquantitativemeasurements.Inmathematics,acontinuityfunctionisdenedasafunctionwhosesmallchangesintheinputwillresultinsmallchangesintheoutput.Thereforeitisnecessaryforustodenehowthechangesbetweentwospatialobjectssuchastwopoints,twolinesandtworegions,aswellastwonon-spatialobjects,forexample,twobooleans,twointegersandtworealscanbecharacterizedandplottedinto2Deuclideanplane.Thereforeweproposeamethodbelowtomeasurethedissimilarityoftwoobjectsofthesametype.Letdenotethedatatypewherethedissimilaritybetweentwoobjectscanbemeasured,andcanbeeitherspatialornon-spatialtype.Letdist(p,q)denote 78

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theEuclideandistancebetweentwosinglepointspandq,andletdist(p,Q)bethedistancefromptotheclosestpointinapointobjectQ.Letlength(L)returnthelengthofalineobjectL,andletarea(R)returntheareaofaregionobjectR.Further,letthenotation?denoteanemptyvalueoranundenedobject.Wedenethismeasurementasdissimilarityfunction,denotedby,asshowninDenition 3.2.1.1 .Theadvantageofintroducingthisdenitionisthatnomatterwhatinputdatatypesare,wecanalwaysgetaresultwhichisinnumericformat.Thisisimportantforustodenethecontinuitypropertyinthefuture. Denition3.2.1.1. Foranytype2fbool,int,string,real,point,line,regiong,adissimilaritymeasure:!Risdenedasfollows: (i)8x,y22fbool,int,stringg,x6=?,y6=?:(x,y)=8>><>>:0ifx=y1otherwise(ii)8x,y2=real,x6=?,y6=?:(x,y)=jx)]TJ /F5 11.955 Tf 11.95 0 Td[(yj(iii)8p,q2=point,x6=?,y6=?:(p,q)=dist(p,q)(iv)8P,Q2=points,p6=?,q6=?:(P,Q)=Xp2PnQdist(p,Q)+Xp2QnPdist(p,P)(v)8L1,L22=line,L16=?,L26=?:(L1,L2)=length(L1nL2)+length(L2nL1)(vi)8R1,R22=region,R16=?,R26=?:(R1,R2)=area(R1nR2)+area(R2nR1)InDenition 3.2.1.1 ,(i)showsthatwhenmeasuringthedissimilarityondiscretetypessuchasboolean,integerandstring,theresultchangesabruptly,asshowninFigure 3-21 a.However,ifwemeasurethedissimilarityoncontinuoustypes,theresultchangessmoothly,asshowninFigure 3-21 b.(iii)denesthatthedissimilaritybetweentwopointsintheeuclideanspaceistheeuclideandistancebetweenthem.(iv)showsthattomeasurethedissimilaritybetweentwopointsets,wewillsumupthedistanceofeachpointinonesetrespecttotheotherset,andviceversa,andaddthetwosumstogether.(iii)isaspecialcaseof(iv).(v)and(vi)aretheotherversionof(iii),wherepointsarenotlongerdiscrete,butcontinuous,andthedissimilaritiesaremeasuredby 79

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(a)(b)(c) (d)(e)(f)Figure3-21. Dissimilarityfunctiondenedontypesofinteger,real,point,points,lineandregion thelengthofthedifferenceoftwolines,andtheareaofthedifferenceoftworegions,respectively. 3.2.1.2TheContinuityPropertyofMovingObjectsInSection 3.2.1.1 ,wehavedenedthedissimilarityfunctionondifferentinputformats.Itwillbeusedlaterinthissectiontodenethecontinuousmovement.Beforewegivetheformaldenitionofacontinuousmovementfunction.Nowweintroducesomeotherimportantconcepts.Sincethemovementofmovingobjectsofteninvolvethechangeoflocationsandgeometriesovertime,modelingsuchchangesrequiresaconceptoftimeandspace.WedenethedatatypetimeasaspecialclassofsetRrepresentingrealnumbers.Next,wedeneR2asthe2DEuclideanspace,andbeanyspatialornon-spatialdatatypeinR2.Thenwecandeneatemporaldatatypeasafunctionoftime,asourpreviousworkdoes[ 21 22 28 ].()=f:time!Intheabovedenition,isaconstructorwhichliftanon-temporaldatatypetoatemporaldatatype.canbeappliedtobothspatialornon-spatialdatatypes.Therefore 80

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canbeallkindsofdatatypeswehaveintroducedinSection 3.2.1.1 .Forinstance,if=real,then()denoteamovingrealobjects,whichcanrepresentthecurveoftemperaturechanges.If=boolean,itcanrepresentthedynamicchangingofbooleanvalues.If2fpoint,line,regiong,then()canyieldspatio-temporaldatatypeswhichcanrepresentmovingpoints,movinglinesandmovingregions.However,thisfunctiondescriptionoftemporalobjectshassomeproblem.First,atemporalfunctionshouldbedescribedasapartialfunction,i.e.,theremustbeaconstraintspecifyingwhenthefunctionisdened.Forexample,atemporalobjectcanappear,disappearandreappear.Atthetimetwhenthetemporalobjectdoesnotexist,wewillhavef(t)=?.Therefore,anadditionalconstraintshouldbeaddedtotheabovetemporalobjectdenition,dom(f)=ft2timejf(t)6=?gFurther,iffisundenedatt,wehavef(t)=?.Thereforeitallowsustomodeldifferentsituationswhetherthetemporalobjectsappearordisappearattimet.Then,thenextproblemappearsthatwemustconsiderhowtomodeltheboundarypointsoftheappearanceordisappearanceofmovingobjects.Forexample,whentheobjectjustappearattimet,itisdenedattheupperpartoft,butnotatthelowerpartoft.Thereforewemustconsidertheone-sidedortwo-sidedlimitproblem.Nowwespecifytheconceptoflimitofatemporalfunctionfatatimeinstantt.Aone-sidedortwo-sidedlimitcanonlybedenedifaone-sidedortwo-sidedtimeintervalbelongstodom(f).Therefore,inDenition 3.2.1.2 ,wespecifytwopredicatesdenedatthebottomoft,anddenedatthetopoft,denotedbydfbanddft,whichcheckwhetherfisdenedattfromthebottomandfromthetoprespectively. Denition3.2.1.2. Let2fbool,int,string,real,point,line,regiong,f2()=time!,andt2time. (i)dfb(f,t):=92R^>080<<:f(t)]TJ /F8 11.955 Tf 11.95 0 Td[()6=?(ii)dft(f,t):=92R^>080<<:f(t+)6=? 81

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InDenition 3.2.1.2 (i),wesaythatfisdenedatthebottomoft,ifthereexistasmallpositivevaluesothatforanypositiverealnumberwhichissmallerthen,f(t)]TJ /F8 11.955 Tf 12.7 0 Td[()dened.Ifsuchdoesnotexist,wethendfb(f,t)isfalse.Similarly,wehavedft(f,t)inDenition 3.2.1.2 (ii).BasedonDenition 3.2.1.2 ,wedenethelimitofatemporalfunctionfatatimeinstanttinitsdomain. Denition3.2.1.3(Limit). Let2fbool,int,string,real,point,line,regiong,f2()=time!,andt2time. (i)lim!0f(t)]TJ /F8 11.955 Tf 11.96 0 Td[()=L,ifandonlyif,dfb(f,t)^82R,>092R,>0,80<<:(f(t)]TJ /F8 11.955 Tf 11.95 0 Td[(),L)<(ii)lim!0f(t+)=Lif,andonlyif,dft(f,t)^82R,>092R,>0,80<<:(f(t+),L)<(iii)lim!0f(t)=Lif,andonlyif,lim!0f(t)]TJ /F8 11.955 Tf 11.96 0 Td[()=lim!0f(t+)=LIntheabovedenition,(i)denesthelimitofffromthebottomoft.(ii)denesthelimitofffromthetopoft.(iii)denesthelimitoffatt.Thisdenitionshowsthatfdoesnothavetobeexactlydenedattandthatthelimitspecicationsrequiretheexistenceeitheranupperlimitoralowerlimit.Basedontheconceptoflimit,wearenowabletoapproachadenitionofcontinuityforatemporalobjectatatimeinstant.Fortemporalobjectsbasedonanon-spatialdatatype,thenotionofcontinuityisquitestandard.Theseobjectshavethefeaturethat,ateachtimeinstantoftheirdomain,weobtainasinglevalue(likeasinglestringvalueorasingleintegervalue)thatevolvesovertime.Fortemporalobjectsbasedonthethreediscretetypesboolean,integer,andstring,wecanimmediatelyconcludethattherearenocontinuouschangessinceasmoothtransitionovertimeisimpossibleondiscretedata.Weoftenseeaconstantvaluewhichlastsforaperiodoftime,followsbyastepwisechangeatalatertimeinstant.Forrealnumbers,weapplytheclassicaldenitionofreal-valuedcontinuousfunctions.Ifatemporalobjectfbasedonanynon-spatialdatatypehas 82

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ajumpdiscontinuityatatimeinstantt,weassumethatthefunctionvaluef(t)=lim!0f(t+).Thus,f(t)6=lim!0f(t)]TJ /F8 11.955 Tf 11.96 0 Td[().Wemustpayattentionthatateachtimeinstantofitsdomain,amovingobjectmayincludemultiplesimplespatialvalues.Apointobjectmayconsistofseveralsinglepoints,alineobjectmayincludeseveralblocks,andaregionobjectmayincorporateseveralfaces.Similarly,amovingobjectmaycontainseveralmovingcomponents.Themultiplesimplevaluesofamovingobjectatatimeinstantmaymovesimultaneouslyovertime,stayseparatefromeachother,interact,coincide,merge,split,partiallystoptoexist,orpartiallystarttoexist.Wegivethefollowingdenitionstodiscusssuchkindsofmovements.InDenition 3.2.1.4 ,wespecifytheimportantconceptofcontinuityatatimeinstantforamovingobject.ItrestsonthelimitconceptofDenition 3.2.1.3 Denition3.2.1.4. Let2fpoint,line,regiong,f2()=time!,t2time,andf(t)6=?.Then (i)fis-continuousfromthebottomattif,andonlyif,lim!0f(t)]TJ /F8 11.955 Tf 11.96 0 Td[()=f(t)(ii)fis-continuousfromthetopattif,andonlyif,lim!0f(t+)=f(t)(iii)fis-continuousattif,andonlyif,lim!0f(t)=f(t)(iv)fis-discontinuousattif,andonlyif,fisnot-continuousattToexplainDenition 3.2.1.4 ,let'srstconsiderthesimplestcasethatamovingobjectonlyhasonecomponent,asshowninFigure 3-22 aandFigure 3-22 b.Theyshowamovingpointobjectandamovingregionobjectrespectively.Theyare-continuousatanytimeinstanceoftheopeninterval,i.e.,t2(t1,t2).Theyare-continuousatthetopattimet1,and-continuousatthebottomattimet2.Nowweconsideracomplexmovingobjectwhichhasmultiplecomponents.Figure 3-23 ashowsthatamovingpointcanconsistofmorethanonesimultaneousmovingcomponents.Figure 3-23 bshowsthattwomovingpointscanmergeintoonesinglemovingpoint.Figure 3-23 cshowsthatamovingpointcansplitintotwodifferentcomponentsastimepassing.Alloftheabovesituationssatisfytheproperty 83

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(a)(b)Figure3-22. Singlecomponentmovingobjects (a)(b)(c)Figure3-23. Examplesofphi-continuousfunctions of-continuousbetweentheopeninterval(t1,t2)wehavedenedinDenition 3.2.1.4 .Andtheyalsoshowa-discontinuouspropertyatendpointsoftheinterval.Therefore,ourdenitioniscorrectunderbothsinglemovingobjectsaswellascomplexmovingobjects.Weallowdiscontinuityatendpointsofintervalsbecauseitoftenshowsatopologicalchange.Inmostofthecases,adiscontinuityoftendescribesameaningfultemporalbehavior.Forexample,theappearanceofacomponentofacomplexmovingregionmayindicatethatthemovingregionissplitintotwoparts.Therefore,suchsituationsshouldbeallowed.Figure 3-24 aandFigure 3-24 billustratesuchspecialsituations.ThemovementinFigure 3-24 ais-continuousfromthetopbutnotfromthebottomatt1andt2,sincethemovementchangesfromemptyobjecttooneobjectatt1,andfromoneobjecttotwoobjectsattimet2.Similarly,itis-continuousfromthebottombutnotfromthetopattimeinstancesoft3andt4,becauseatbothtimeinstances,object 84

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disappearancehappens.Figure 3-24 showsa-discontinuousattimet4,becausefisneither-continuousatthetopoft4,noris-continuousatthebottomoft4.Therefore,weintroducetheconceptofevent--continuousmovement,itallowsvalidtopologicalchangessuchasappearsplit,merge,etc.inthemovementofamovingobject.Figure 3-24 crepresentsamovingobjectwhichisalternatelycontinuousanddiscontinuousondisjointtimeintervals,i.e.,itisrepresentedbypartialfunctionsoftime.Figure 3-24 dillustratesanexampleofaninstantlyappearingmovingobjectwithisolatedpoints.Weallowthiskindofdiscontinuitysinceitismeaningfulinsomespecialsituations.Themainpurposeofallowthissituationconsistsindesiredclosurepropertiesofspatio-temporaloperations.Forexample,iftwomovingpointsintersectatasinglepointattimet,wewanttomodelthereturnedpartasamovingpointobjectaswell,thenwemustthinkofawaytorepresentthisisolatedpoint.Thisensurestheclosurepropertyofalltheoperationsunderthismodel.WealsopermitthemovementinFigure 3-24 dandFigure 3-24 easvalidmovements.Thereasonisthatinstantaneousjumpcanbeinterpretedasthedisappearanceofonemovementandtheappearanceofanothermovementatthesametimeinstance.WedenoteallthesituationsfromFigure 3-24 atoFigure 3-24 gasevent--discontinuousmovement.Asituationthatwedonotallowinourmodelisthespatio-temporaloutlierasshowninFigure 3-24 f.Itisgivenbyatemporalfunctionthatdoesnotrepresentarealisticmovementsinceintuitivelyitdeviatesfromitsgeneralrouteandreturnstoitforatimeinstantonly.Thefollowingdenitionprovidestheevent--discontinuitydescriptionaswehavediscussedabove. Denition3.2.1.5. Let2fpoint,line,regiong,f2()=time!,t2time,andf(t)6=?.Further,letl=lim!0f(t)]TJ /F8 11.955 Tf 12.7 0 Td[()denotethelimitoffatthebottomoftifitexists,andletu=lim!0f(t+)denotethelimitoffatthetopoftifitexists.Thenfisevent--discontinuousattifoneofthefollowingconditionsholds: 85

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(a)(b)(c) (d)(e)(f)Figure3-24. Examplesofdiscontinuityimplyingvalidtopologicalchanges (i):dfb(f,t)^:dft(f,t)(ii):dfb(f,t)^dft(f,t))uf(t)(iii)dfb(f,t)^:dft(f,t))lf(t)(iv)dfb(f,t)^dft(f,t)^u6=l)u[lf(t)(v)dfb(f,t)^dft(f,t)^u=l)uf(t)InDenition 3.2.1.5 ,(i)meansthatfisisolatedatt.(ii)and(iii)preventaspatialoutlieratanendpointtofatimeintervalofthedomainoff.Inbothcases,thelimitsmustequalfunctionvalueatt.Ifaspatialoutlieroccursinthemiddleofatimeinterval,wehavetodistinguishtwocases.Ifthelimitsfromthetopandfromthebottomaredifferent,theymustbepartoforequaltothefunctionvalueattimet(iv).Ifthelimitsareequal,thecommonlimitmustbeproperlycontainedinf(t)sinceequalitywouldmean-continuityattincontrasttoourassumption(v).Theabove5differentcasesofevent--discontinuityisshowninFigure 3-25 ae. 86

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(a)(b)(c)(d)(e)Figure3-25. Event--discontinuity 3.2.1.3ModelingHistoricalMovingObjectsBasedonDenitions 3.2.1.4 and 3.2.1.5 ,wearenowableinDenition 3.2.1.6 tospecifythedesiredpropertiesofthetypeconstructorforrepresentingvalidmovingobjects.Thenotations[a,b]and(a,b)representclosedandopenintervalsrespectivelywithendpointsaandb. Denition3.2.1.6. Let2fpoint,line,regiongand()=time!.Werestricttocontainonlytemporalfunctionsf2()thatfulllthefollowingconditions: (i)9n2N:dom(f)=Sni=1[t2i)]TJ /F7 7.97 Tf 6.59 0 Td[(1,t2i](ii)81in:t2i)]TJ /F7 7.97 Tf 6.58 0 Td[(1t2i(iii)81i
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timeinterval.Thereasonisthatamovingobjectcomponenthastobeisolatedifitonlyexistsforatimeinstant.(vi)requires-continuitywithintimeintervals.(vii)allowsevent--discontinuitytoappearattimeintervalendpointsthereforewecanrepresentvalidtopologicalchangesofmovingobjectsproperly.Now,wewillusethetypeconstructorandallthetypesandconceptsderivedfromitinthesenseofDenition 3.2.1.6 .Acquiringknowledgeofthehistoricallocationsandmovement(trajectories,routes)ofmovingobjectsisimportantformanyanalysistasksinordertolearnfromthepast.Forexample,hurricaneresearchbenetsfromtheobservationofformerhurricanesinordertolearntheirmovingpatternsandtopredicttheirlocationsinthefuture.Bystudyingthepast,remanagementisabletoidentifycriticalareashavingahighprobabilityofareoutbreakandtoanalyzethespread,merge,andsplitofresovertime.Ourmodelingofhistoricalmovementassumesfullknowledgeaboutthepastlocationsandextentofmovingobjectsintheirtimedomains(i.e.,whentheyaredened).Byusingpartialtemporalfunctions,lackingknowledgeisexpressedbytimeintervalswhensuchfunctionsareundened.Denition 3.2.1.7 extendstheapproachin[ 22 28 ]andmodelsdatatypesformovingobjectsinthepastbyatypeconstructoronthebasisofthetypeconstructorsuchthat()()for2fpoint,line,regiong. Denition3.2.1.7. Let2fpoint,line,regiong,andlet()=time!representthemovementfunctiondenedinDenition 3.2.1.6 .Wedenethehistoricalmovingobjectsas, ()=ff2()j8t2dom(f):tnowgwherethefollowingdenitions (i)hmpoint=(point)(ii)hmline=(line)(iii)hmregion=(region) 88

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describehistoricaldatatypesandarecalledashistoricalmovingpoints,historicalmovinglines,andhistoricalmovingregionsrespectively.Duetotheirprecisespecication,thesetypesreplaceandextendthedatatypesmpoint,mline,andmregiondiscussedinprevioussections. 3.2.1.4ModelingFutureMovingObjectswithUncertaintyWehavereviewedourpreviousworkofthependantmodelonthehistoricaluncertaintyofmovingobjects.Intherestofthissection,wediscusshowtomodeltheuncertaintyofmovingobjectsinthefuture.Predictingthefuturelocationsofmovingobjectsisofgreatimportanceinmanyapplicationsandiscalledlocationmanagement.Forexample,topredictthefuturelocationsofhurricanesandthegrowingofforestresareusefulindisastermanagement.Unlikemovementsinthepastforwhichweassumetohavepreciseknowledgeevenfortheuncertaintyaspectinthepastmovement,futurepredictionsinvolvetheinherentfeatureofuncertaintywithregardtothefuturelocationsorextentofmovingobjects.Fromadatabaseperspective,representingthisuncertaintyfeatureinvolvetwoissues.Therstissueishowtopredictfuturespatialevolutionanddealswiththedevelopmentofpredictionmethods.Acommonprobleminthepredictionofmovingobjectsisthatthepredictionmethodsareoftendomainspecic.Forexample,thepredictionofthemovementofhurricanesrequirestheknowledgeinmeteorology.Thesecondissueishowtointegratethedomainspecicpredictionmethodsintoourmodelandapplyourmodeltorealapplications.Sinceweaimatprovidingageneralpurposesolutiontodifferentapplicationsoffuturemovementprediction,wethereforethinkthatthesecondissueshouldbesupportedbythedatabasesystembutnottherstissue.Thismeansthattheapplicationdomainsshoulddeveloppredictionmodelsoutsideofthedatabasesystem.However,itisnecessaryforadatabasesystemtoprovidesolutionstorepresent,storeandquerypredictedspatio-temporaldata.Therefore,intherestofthis 89

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(a)(b)Figure3-26. Themovementofamovingobjectinthefuture12-hourperiod article,wefocusonthedatamodelingaspectofthefuturepredictionsofmovingobjectsandhowthistypeofdatacanberepresentedandqueriedinthedatabase.Weleavethetaskofpredictiontotheapplicationdomains.Westartfromanexampleinordertounderstandtheuncertainfeatureofmovingobjectsinthefuture.Forexample,thepossiblepositionsoftheeyeofahurricaneat12hoursfromnowcanbeanywherewithinaregion,asshowninFigure 3-22 b.Similarly,ifweareinterestedinthepossiblepositionsinacertainperiodinthefuture,forexample,fromthepresenttimenowto12hoursinthefuture,thentheactualpositioncanbeanywherewithinapredictedvolumeifweconsidertimeasthethirddimension.Figure 3-26 ashowsthepossiblepositionsofamovingpointatthepresenttime,4hourslater,8hourslater,and12hourslaterrespectively.Ifweintegrateallthepossiblelocationsovertime,wewillgeta3Dvolumewhoseshapelookslikeaballoon,asshowninFigure 3-26 b.Atanytimeinstanceinthefuture,thepossiblelocationsarewithinanarea,thereforeitcanberepresentedbyourspatio-temporaltype(region)aswehaveintroducedinSection 3.2.1 .Intheaboveexample,wehaveonlyconsideredthefuturemovementofamovingpointobject.However,amovingobjectcanhaveanextent.Forexample,aregionobjectsuchasahurricane,thefuturepredictionofitsextentisalwaysaregion.Therefore,over 90

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aperiodoftimeinthefuture,thetemporalevolutionofthisregioncanberepresentedbyamovingregionobjectoftype(region).Wehavebeenabletorepresenttheuncertaintyofthemovingobjectataspecictimeinstanceasaregion,however,howtheuncertaintyisdistributedinthisregion?Forexample,itmightbeinterestedtoasktheprobabilitythatamovingpointwillbeinsideasub-regionofitsuncertainregion.Iftheuncertaintyisuniformlydistributedintheuncertainregion,thentheprobabilitythatthemovingpointwillbeinsidethesub-regionequalstheareaofthesub-regiondividedbytheareaoftheentireuncertainregion.Iftheuncertaintyisnotuniformlydistributedintheuncertainarea,thenwewillcalculatetheprobabilityaccordingtootherdistributionfunctionoftheuncertainty.Therefore,itisnecessarytointroducetheconcepttodescribehowtheuncertaintyisdistributedamongtheuncertainregion.Wecallthisconceptcondence. Denition3.2.1.8. Let2fpoint,line,regiong,andletcbeafunctionc:R2![0,1],thenc(x,y)iscalledthecondencedistributionfunction,andC()=![0,1]iscalledthecondenceof,whichdenotestheprobabilitythatwillbetheareaofallpotentiallocationsofaspatialobject,withthefollowingconditions, (i)ifp=(x,y)2,=point,C(p)=c(x,y)2[0,1](ii)ifo2,2fpoint,regiong8p=(x,y)2o,c(x,y)=0C(o)=RR(x,y)2oc(x,y)dxdyIntheabovedenition,wecanseethatthecondencedistributionfunctiondescribeshowtheuncertaintyisdistributedamongthepossibleareaofaspatialobject.Condition(i)showsthatiftheuncertaintyisdistributedamongdiscretepoints,thenthecondenceofsuchapointisavaluebetween[0,1].Condition(ii)showsthatifwewanttogetthecondenceofaregion,thenthecondencedistributionfunctioniscontinuous.Thereforethecondenceofanypointinsidetheregioniszero,andthecondenceoftheregionistheintegralofthecondenceofallpoints. 91

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(a)(b)Figure3-27. Differentcondencedistributionfunctions Condencedistributionfunctionscandescribedifferentkindsofuncertainty,whichareapplicationdependent.Forexample,acondencedistributionfunctioncanbeauniformdistributedfunction,inwhichcasethatallpointsintheuncertaintyareahavetheequalopportunitytobethepotentialpositionofamovingpoint,asshowninFigure 3-27 a.Inotherapplicationdomain,thecondencecanbeaGaussiandistribution,inwhichcasethecenteroftheareahasmorechancetobethepotentiallocationofamovingobject,asshowninFigure 3-27 b.Toapplythisconceptofcondencedistributionfunctiontoamovingobjectforrepresentingfuturepredictionsovertime,wecanusetheconceptoftemporallifting,introducedin[ 21 28 ],toliftourdenitionofcondencedistributionfunctiontoamovingcondencedistributionfunction.Thedenitionisshownasfollows. Denition3.2.1.9. Letmc:time!(R2![0,1])bethetemporalver-sionofacondencedistributionfunctionon.Further,werequiretheexistenceofafunctionmgeo:(time!(![0,1]))!(time!)suchthatmgeo(mc)=f(t,dom(mc(t)))jt2dom(mc)g2(),accordingtoDenition 3.2.1.6 .Thenmciscalledthemovingcondencedistributionfunctionwithrespectto.Let'()=(C())=time!C()=time!(![0,1])bethetypeconstructorthatcontainsallmovingcondencedistributionfunctionsmcwithrespecttosuchthatmgeo:()!().Wedenote, 92

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(i)pmpoint='(point)(ii)pmline='(line)(iii)pmregion='(region)aspredictivemovingpoints,predictivemovinglinesandpredictivemovingregionsrespectively.Theabovedenitionenablesapplicationstodescribeapredictivemovingobjectinthesensethatitsgeometryisgivenasamovingobjectoftype()andretrievablebythefunctionmgeoandthatitsuncertaintyisrepresentedbyacondencevaluebetween[0,1]foreachpointbelongingtothepredictivemovingobject.Toillustratetheconceptswehavepresented,weconsidertheexampleofahurricane.Wecanmodelthepotentialpositionsoftheeyeofthehurricaneusing(region)object,asshowninFigure 3-26 b.Byapplyingthemovingcondencedistributionfunctiononthisobject,wecanobtainanewkindofobjectwhichrepresentsthepotentialfuturepositionswithuncertainty.Thesurfaceoftheuncertaintyregionateachtimeisassociatedwithacondencedistributionfunction,asshowninFigure 3-27 aandFigure 3-27 b.AsshowninDenition 3.2.1.9 ,predictivespatio-temporaldatatypesareonlydenedforfuturepredictionsofmovingobjects,thereforewehavet>now.Theydonotmakeanyreferenceorassumptiononthehistoricaldevelopmentrepresentedbyhistoricalspatio-temporaldatatypesofmovingobjects.However,wecansettheinstancenoweithertothecurrenttimeoratimeinstanceinthepast.Forexample,anobjectoftypepmregioncanbeusedtorepresentthefuturepredictionofeitherahistoricalmovingpoint,ahistoricalmovinglineorahistoricalmovingregion.Nowtheproblemcomeshowtorepresentamovingobjectwhosemovementlastsfromthepasttothefuture?Inthenextsection,wewilldenesuchdatatypesthatcombinethehistoricalandpredictivemovements. 93

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3.2.2BalloonDataTypes:RepresentingHistoricalandPredictiveMovingObjectswithUncertaintyInthissection,wecombineourobservationsandmodelingapproachesofhistoricalandpredictivemovingobjectswehavepresentedinprevioussections.Weintroducetheballoondatatypestorepresentsuchkindofmovingobjects.Eachballoondataobjectconsistsofahistoricalmovingobjectpartandapredictivemovingobjectpart.Thetermballoonisusedasametaphortodescribethehistoricalandfuturepartsofamovingobjectwithrespecttoaspecictimeinstancet0whenthismovingobjectisbeingobserved.However,t0doesnotnecessarytobethetimeatthecurrentmoment.Itcanbeanytimebeforethepresenttime,i.e.,t0now.t0cannotbeatimegreaterthannow.Therefore,t0representthelateststateofthemovementinourknowledge.Theperiodbeforet0(inclusive)isthedenedtimeofthehistoricalmovementandcorrespondstothestringoftheballoon.Theperiodaftert0(exclusive)isthedenedtimeofthefuturepartofthemovementandcorrespondstothebodyoftheballoon.Wetakethehurricanestudyasanexample.Thecenterofahurricaneisusuallyillustratedasashapethatresemblesaballoononthewebsites.Thepastmovementofthecenterofthehurricanecanbeseenasamovementalongalineoraroutewhichresemblesthetailoftheballoon.Thepositionofthehurricanecenteratafuturetimeinstancecanbeanywherewithinanareaofuncertainty.Therefore,thefuturepredictionoftheeyecanbeseenasamovingregionofuncertaintythatresemblesthebodyofaballoon.Itcorrespondstoapredictivemovingregion.Eachballoonobjectbohasitsownlatestknownstateataspecictimeinstantt0.Therefore,thedomainofaballoontypecanbedividedintotwoparts.Therstparthtime=ft2timejtt0gisthedomainofthehistoricalmovementofbo.Thesecondpartftime=ft2timejt>t0gisthedomainofthefuturemovementofbo.Figure 3-28 illustratesthedenitiondomainofthehistoricalmovementandthpredictivefuturemovementofaballoonobject. 94

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Figure3-28. Thetimeinstancet0,thehistoricalandpredictivetimeintervals Now,wegivethedenitionofballoondatatypesbyintroducinganewtypeconstructortocombinethehistoricalandfuturemovementsintoageneraldatatypewhichcontainsbothpartstogether.Theideaistodenethisnewdatatypebasedontheavailabletypeconstructorsand'wehaveintroducedbefore.Theresultobjectisapairofahistoricalmovingobjectandafuturemovingobjectwithcertainconstraints. Denition3.2.2.1. Let,2fpoint,line,regiong,andletdimbeafunctionthatreturnsthedimensionofaspatialdatatype.Thatis,dim(point)=0,dim(line)=1,anddim(region)=2.Wedenethetypeconstructoras (,)=fbo=(p,f)2()()j(i)dim()dim()(ii)t0=max(dom(p))(iii)dom(p)(,t0]dom(f)(t0,1)gThenwegeneratethefollowingballoondatatypes balloon pp=(point,point)balloon pl=(point,line)balloon pr=(point,region)balloon ll=(line,line)balloon lr=(line,region)balloon rr=(region,region)Aswehavediscussedinprevioussections,theunderlyingspatialdatatypesforconstructingthehistoricalspatiotemporaldatatype()andforconstructingthepredictivespatio-temporaldatatype'()canbedifferent.Therefore,theconstructorgetstwopossiblydifferentspatialdatatypesasoperands.Aswehavediscussedbefore,notallcombinationsoftypesandarevalidmovements.Condition(i)states 95

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thatthedimensionofthefuturedatatypeshouldnotbelessthanthedimensionofthehistoricaldatatype.Thisreectsthefeatureoftheuncertaintythattheuncertaintygrowswithtime.Anexampleisthatthepossiblelocationsoftheeyeofahurricanecouldbeapoint,alineoraregion.However,theoppositecasedoesnothold.Forexample,ifweobtainthepossiblelocationsofahurricaneasaregionatatimeinstanceinthefuture,itspossiblelocationsafterthattimeinstantcouldnotbeapoint,whichresultsadimensionelapse.Condition(ii)statesthatthetimeinstanceofthelatestknownstateofaballoonobjectmustbeequaltothelasttimeinstantofthedomainofitshistoricalmovingobject.Condition(iii)statesthatthedomainsofthehistoricalandpredictivemovingobjectsofaballoonobjectmustbedisjointandbeinthecorrecttimeperiod.Theabove6balloondatatypesareillustratedinFigure 3-29 atoFigure 3-29 frespectively. (a)(b)(c) (d)(e)(f)Figure3-29. Sixvalidballoondatatypes 96

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3.2.3OperationsonBalloonDataTypesWehaveintroducedourballoonmodelwhichcontainsseveralcombinationsofhistoricalandfuturecombinationsofdatatypes.Weprovideallthesedatatypesandcorrespondingoperationsonthem.Weuse()todenotethehistoricalpartofmovingobjects,anduse'()todenotefuturemovements.Sinceinourpreviousworkin[ 28 ]and[ 49 ],wehavealreadyintroducedadatatypeorientedmodelforhistoricalmovingobjectswithacomprehensivesetofoperations,wecompletelyintegratethispartintoourballoonmodel.Theonlydifferenceisthatweapplytheconstructorinfrontofthespatialdatatypestoconstructourhistoricalmovingdatatypes.Therefore,wedonotrepeattheirdenitionsinthispaper.Instead,wejustlistthiscomprehensivesetofoperationsintheleftcolumnofTable 3.2.3 .Theonlyonethingwewanttomentionisaboutthemeaningofmin(,).Let,2fpoint,lineregiong.Weusetheoperatorasanoverloadingoperatortocomparethedimensionoftwospatialdatatypesandweletpoint<>:if=_
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Table3-4. Operationsonhistoricalandpredictivemovingobjects. TemporallyLiftedOperationsApplicationtoHistoricalMovementsApplicationtoFuturePredictions intersection()()!(min(,))'()'()!(min(,))union,minus()()!()'()'()!()crossings(line)(line)!(point)'(line)'(line)!(point)touch points(region)(line)!(point)'(region)'(line)!(point)common border(region)(region)!(line)'(region)'(region)!(line)no components()!(int)'()!(int)length(line)!(real)'(line)!(real)area(region)!(real)'(region)!(real)perimeter(region)!(real)'(region)!(real)distance()()!(real)'()'()!(real)direction(point)(point)!(real)'(point)'(point)!(real) ProjectiontoDomainorRangeApplicationtoHistoricalMovementsApplicationtoFuturePredictions deftime()!periods'()!periodslocations(point)!point'(point)!pointtrajectory(point)!line'(point)!linetraversed(line)!region'(line)!regiontraversed(region)!region'(region)!regionroutes(point)!line'(point)!lineinstintime()!timeinfutime()!timevalintime()!infutime()! InteractionwithDomainorRangeApplicationtoHistoricalMovementsApplicationtoFuturePredictions atinstant()time!intime()'()time!infutime()atperiods()periods!()'()periods!'()initial,nal()!intime()'()!infutime()present()time!bool'()time!boolpresent()periods!bool'()periods!boolat()!(min(,))'()!(min(,))passes()!bool'()!boolwhen()(!bool)!()'()(!bool)!'()mconfN/A'()!MC()confN/A'()time!C()point confN/A'()pointtime!realpointset confN/A'()time!real RateofChangeApplicationtoHistoricalMovementsApplicationtoFuturePredictions derivative(real)!(real)'(real)!(real)speed,mdirection(point)!(real)'(point)!(real)velocity(point)!(point)'(point)!(real) Letthetypeinfutime()=C()instantrepresentthestateofapredictionataninstantintime.Wecandecomposethisdatatypeusingthreeoperationsinst,valandconf.Inthehistoricalmovement,theoperationsofatinstant,initialandnalwillreturntheintime()datatype.Similarly,inthefuturemovementcontext,theseoperationswillreturnainfutime().Now,weintroducetheoperationsthatarerelatedtotheuncertaintyinthefuture. 98

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Theoperationmconfisappliedtothefuturemovementofamovingobject,andwillreturnacondencevaluebetween[0,1]overtime.Ithasthesignaturemconf:'()!MC(),whereMC()correspondstothemovingcondenceconceptwehaveintroducedinDenition 3.2.1.9 Denition3.2.3.1. Let2fpoint,line,regiongand'betheconstructoroffuturemovingobjects.Letpmo2'()Theoperationmconf(pmo)iscalledthemovingcondenceofthefuturemovementofpmoandisdenedas,mconf(pmo)=f(t,C)jt2R,C2[0,1]gIfwewanttoknowthecondenceofpmoinaparticulartimeinstanceinthefuture,wewillusetheconfoperation.Ithasthesignatureconf:'()time!C(),whereC()correspondstothecondenceconceptinDenition 3.2.1.8 .Wegivethefollowingdenitionoftheconfoperation. Denition3.2.3.2. Let2fpoint,line,regiongand'betheconstructoroffuturemovingobjects.Letpmo2'(),andt2time.Theoperationconf(pmo,t)iscalledthemovingcondenceofthefuturemovementofpmoandisdenedas,conf(pmo,t)=C,C2[0,1]Nowwegivethedenitionsoftwootheroperationsrelatedtothecondenceoffuturemovements,namelypoint confandpointset conf.Thepoint confoperationisusedtodeterminetheprobabilityofoccurrencethatapointwillbeinsidearegionataspecicinstanceoftime,whichisavaluebetween[0,1].Ithasthesignature'()pointtime!real.Thedenitionofpoint confisasfollows. Denition3.2.3.3. Let2fpoint,line,regiongand'betheconstructoroffuturemovingobjects.Letpmo2'(),p2pointandt2time.Theoperationpoint conf(pmo,p,t)iscalledthemovingcondenceofthefuturemovementofpmo.Ithasthesignature'()pointtime!real,andisdenedas,point conf(pmo,p,t)=C,C2[0,1] 99

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Thispoint confoperationwillgivetheresultofthecondencethatamovingobjectwillmeetapointatafuturetimeinstance.Similarly,ifwewanttoobtainthecondencethatthismovingobjectwillbeinsideaparticularregion,orlyingonaline,wecanapplythepointset conf,whichdeterminestheprobabilitythatapointsetwillbethepossiblefuturelocationsofamovingobject.Thepointset confoperationisdenedasfollows. Denition3.2.3.4. Let2fpoint,line,regiongand'betheconstructoroffuturemovingobjects.Letpmo2'(),r2,and2fpoint,line,regiong,andt2time.Theoperationpointset conf(pmo,r,t)iscalledthemovingcondenceofthefuturemovementofpmo.Ithasthesignature'()time!real,andisdenedas,pointset conf(pmo,r,t)=CwhereCsatises, (i)C2[0,1](ii)Letpmo=f(t),c(x,y)bethecondencedistributionfunction8(x,y)2f(t)ifc(x,y)iscontinuous,C=RR(x,y)2rc(x,y)dxdyifc(x,y)isdiscrete,C=0Fromtheabovetwodenition,wecanseethattheresultofoperationspoint confandpointset confaredependentonthecondencedistributionfunction.Ifacondencedistributionfunctioniscontinuous,thenthepoint confresultwillbezeroforanypoint.However,ifthecondencedistributionfunctionisdiscrete,thentheresultcanbegreaterthanzero.Theresultofpointset confiscalculatedthroughintegralofallcondencevaluesamongallpoints.Itisobviousthatforasinglepoint,theareaorlengthiszero,andtheintegralresultiszero.Thereforethecondencevalueatasinglepointiszeroifthecondencedistributionfunctioniscontinuous.Tobetterillustratetheconceptsofpoint confandpointset conf,weFigure 3-30 asanexample.Figure 3-30 ashowsapredictionfunctionwithadiscretecondencedistribution.Thecondencevalueofpointsp,q,rare0.25,0.5and0.25respectively.Thismeansthatthesethreepointsaretheonlypossiblelocationsofthemovingpointinthefuture,thereforethesumofthecondencevalueatthesethreepointsequalsto1. 100

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However,thepossiblelocationsofamovingpointinthefuturecouldalsolieinapointset,suchasalineoraregion.Figure 3-30 bshowsacontinuouscondencedistributionfunction,whichisaGaussiandistributiononaline.Theentireareaenclosedbythecondencedistributionfunction(cdf)curveandthelinesegmentisthetotalprobabilitythatthemovingpointwilllieinthesegment,whichequalsto1.Thecondencethatthemovingpointwillliebetweensegmentpqequalstotheintegralofthecondenceatallpointsonpq,whichisshownbytheshadedarea.Thereforethevalueisbetween(0,1).Similarly,Figure 3-30 cshowsacondencedistributionfunctionwhichsatisesauniformdistributiononacirclearea.ThecondencethatthemovingpointlieonareaAequalstothecylindervolumeasshowninthediagram.FromFigure 3-30 bandFigure 3-30 cwecaneasilyunderstandwhythepoint confvaluecanbezeroatasinglepointinacontinuouscondencedistributionfunction,becausetheareaorthelengthwewanttomakeintegraliszero. (a)(b)(c)Figure3-30. Examplesofpredictionsatatimeinstantunderdifferentuncertaintymodels 3.2.3.2OperationsonBalloonObjectsWehaveintroducedsomeimportantnewoperationsonthepredictivefuturemovingobjects.Nowweintroducenewoperationsontheballoonobjectwhichcanbeappliedtobothhistoricalpartandthefuturepart. 101

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Amajoradvantageofintroducingtheballoondatatypeisthatwecanapplymostoftheexistingoperationsonthehistoricalmovementandfuturemovementtotheentiremovingobject.Theseoperationsincludeprojectionoperationssuchasdeftime,location,trajectory,traversed,interactionoperationssuchaspresent,passes,liftedoperationssuchaslength,area,perimeter,distanceanddirection,etc.Thesemanticoftheseoperationscanbeexpressedastheunionbetweentheresultsofapplyingtheoperationtobothhistoricalandfuturecomponentsoftheballoonobjects.However,someoperationswhichhasinteractionwithtimemustbehandledcarefully.Forexample,theoperationsofatinstant,initialandnalmustberstdecomposedintotwopartsofhistoricalmovementandfuturemovement,andthentheoperationonthecorrespondingdomaincouldbeapplied.Nowwedenetwonewoperationspast projandfuture projwhichreturnthehistoricalmovementandfuturemovementrespectively. Denition3.2.3.5. Thepast projoperationwillreturnthehistoricalpartofmovementandhasthesignature(,)!()past proj((,))=ff:time!jdom(f)nowgTheabovedenitionshowsthatwhenwemakethepast projoperation,wemakeacutontheinstancenow,andextractthehistoricalmovementfromtheentiremovingobject.Similarly,wegivethefollowingdenitionofthefutureprojectionoperation. Denition3.2.3.6. Thefuture projoperationwillreturnthepredictivefuturepartofmovementandhasthesignature(,)!'()future proj((,)=ff:time!jdom(f)>nowgAfterintroducingtheabovetwodenitions,weareabletoredenetheoperationswhichinteractwithtime.Nowwedenetheatinstantoperationsonaballoonobject. Denition3.2.3.7. Theatinstantoperationisappliedtothelifetimeofaballoonobjectbo.Ithasthesignature(,)time!intime()infutime().Itisdenedas 102

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Table3-5. Operationsonballoonobjectsandmovingballoonobjects TemporaryLiftedOperationsApplicationtoBalloonDataTypes no components(,)!(int)length(line,line)!(real)area(region,region)!(real)perimeter(region,region)!(real)distance(,)!(real)distance(1,1)(2,2)!(real)direction(point,point)point!(real)direction(point,point)(point,point)!(real) ProjectiontoDomainorRangeApplicationtoBalloonDataTypes deftime(,)!periodslocations(point,point)!pointtrajectory(point,point)!linetraversed(line,line)!regiontraversed(line,region)!regiontraversed(region,region)!regionroutes(point,point)!lineinst(,)!timeval(,)! InteractionwithDomainorRangeApplicationtoBalloonDataTypes past proj(,)!()future proj(,)!'()atinstant(,)time!intime()infutime()atperiods(,)periods!(,)initial,nal(,)!intime()infutime()present(,)instant!boolpresent(,)periods!boolpasses(,)!bool RateofChangeApplicationtoBalloonDataTypes turn,velocity(point,point)!(real) atinstant(bo,t)=fp,fj(i)p2(),f2'()(ii)max(())now,min('()>now(iii)iftnow,f=(iv)ift>now,p=gTheabovedenitionshowsthattheatinstantoperationcanbeappliedtoanytimeinstanceduringthelifetimeofthemovingobject.Theresultofthisoperationiscomposedbytwoparts,thehistoricalpartandthefuturepart(Condition(i)and(ii)).Thereasonwhywerepresenttheresultinthiswayisbecausewewanttomaketheoperationtypecompatible,i.e.,thereturntypemustbeidenticaleverytimewhentheoperationiscalled.Therefore,Condition(iii)and(iv)showthatiftheinputtimeisless 103

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thanorequaltonow,thefuturepartoftheresultisempty,andiftheinputtimeisgreaterthanthecurrenttime,thehistoricalpartoftheresultisempty.Operationsinitialandnalarespecialcasesofatinstant,wherethetimeoperatorissettothersttimeinstantandthelasttimeinstantinthelifetimeofthemovingobject. 3.2.4Spatio-TemporalPredicatesInthissection,wediscussthespatio-temporalpredicatesundertheballoonmodel.Aspatio-temporalpredicatesdescribesthedevelopmentofrelationshipsbetweenmovingobjects[ 21 ].Itisafunctionfromspatio-temporalobjectstoafactwhichcanbeeithertrueorfalse,andiscomposedbyspatialrelationshipsovertime.Anexampleofaspatio-temporalpredicateisthecrossrelationshipbetweenanairplaneandahurricane.Therelationshipsbetweentheairplaneandthehurricaneovertimearedisjoint,touch,inside,touch,disjoint,andthecrosspredicateiscomposedbytheabovepredicates.However,suchpredicatesonlydescribethedevelopingrelationshipswhichhavehappenedforsure.Todescribethespatio-temporalrelationshipsinthefuture,weneedtoconsiderthecondenceintothepredicates.Asuncertaintyexistsinthefuture,thecrossrelationshipbetweenanairplaneandahurricanehasapossibility.Therefore,wemustconsidertwoimportantissuesinmodelingfuturespatio-temporalrelationships.Oneisthespatio-temporalrelationshipbetweenthemovinggeometries,andtheotheristhequanticationofthechancethattherewillbeaninteractionbetweenthetwoobjectsinthefuture.Wediscussthesetwoissuesseparatelysothatwecanpresentthemodelinthesimplestform.InSection 3.2.4.1 wedenespatio-temporalpredicatesbetweenballoonobjects.InSection 3.2.4.2 weprovideourreasoningaboutthepotentialfutureinteractionbetweentheactualobjects. 3.2.4.1Spatio-TemporalPredicatesonBalloonModelInthissection,weexplainthemethodfordeningspatio-temporalpredicatesontheballoonmodel.Werstdescribeourgeneralmechanism.Laterwediscusshow 104

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aballoonpredicatecanbespeciedusingtraditionalSTPs.Thenwedeterminethecanonicalcollectionofballoonpredicates.Wedenetheballoonpredicatesbymakinguseofexistingdenitionsoftraditionalspatio-temporalpredicates(STPs).Withthisapproach,wecanbenetfromboththeoreticalandimplementationadvantagessuchthattheformalismandimplementationofballoonpredicatescanmakeuseoftheexistingworkfortraditionalmovingobjectdatamodels.Thegeneralmethodweproposecharacterizesballoonpredicatesbasedontheideathatastwospatialobjectsmoveovertime,therelationshipbetweenthemmayalsodevelopsovertime.Byspecifyingthischangingrelationshipasapredicate,wecanaskabooleanqueryofwhetherornotsuchachangingrelationshipoccurs.Thus,wecandeneaballoonpredicateasafunctionfromaballoondatatypetoabooleanvalue.Wegivethedenitionofthespatio-temporalpredicatesonballoondatatypeasfollows. Denition1. Aballoonpredicateisafunctionoftheform(1,1)(2,2)!boolfor1,1,2,22fpoint,line,regiong.Thechangeofrelationshipovertimebetweentwoballoonobjectsindicatesthatthereisasequenceofrelationshipsovertime.Thissuggeststhataballoonpredicatecanalsobemodeledasadevelopmentofspatio-temporalpredicates.Sinceaballoonobjectconsistsofahistorypartandapredictionpart,thespecicationofaballoonpredicatemusttakeintoaccounttherelationshipsbetweenbothparts.Therefore,werstexplorehowtheserelationshipscanbemodeled.Eachballoonobjecthasadenedcurrentstateatitscurrentinstanttcwhichseparatesthehistorypartandthepredictionpart.BetweentwoballoonobjectsA=(Ah,Ap)andB=(Bh,Bp),A'scurrentinstantmayeitherbeearlier,atthesametime,orlaterthanB'scurrentinstant.Ineachofthesescenarios,certainsequencesofspatio-temporalrelationshipsarepossiblebetweenthepartsofAandB.Here,weareonlyinterestedintherelationshipsbetweenapartofAandanotherpartofBwhosetemporaldomainsoverlapsinceundersuchconditions 105

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(a)(b)(c)Figure3-31. Possiblerelationshipsbetweenpartsofballoonobjects thetwopartsmaybedenedonthesameperiodoftime.Figure 3-31 illustratesallthepossiblerelatedpairsforeachscenariobetweenpartsofAandB.Althoughtherearefourpossibletypesofrelationshipsbetweenallpartsoftwoballoonobjects,itturnsoutthatinanycase,thereareatmostthreetypesofrelationshipsthatmayexistbetweenpartsofanytwoballoonobjects.Theseincludehistory/history,history/predictionorprediction/history,andprediction/predictionrelationships.Thehistory/predictionandprediction/historyrelationshipscannotexistatthesametimeduetothetemporalcompositionbetweenthehistoryandpredictionpartsofaballoonobject.Hereastaticobjectcanbetreatedasaspecialballoonobject.AnexampleishurricaneKatrinaandthestateofFlorida,wherewetreatedFloridaasamovingregionwhoseshaperemainsthesameallthetime.Thereforewecanndhistory/history,history/predictionandprediction/predictionrelationshipsbetweenthestateofFloridaandKatrina.Fromobservation,wecanndthatalltherelationshipsbetweenthepartsoftwoballoonobjectsthatmayexistinascenarioformadevelopmentsuchthattheentirerelationshipbetweenthetwoballoonobjectscanbeseenasacombinationoftheserelationshipsbetweentheirparts.Forexample,consideranairplanerepresentedbyaballoon ppobjectP=(Ph,Pp)andahurricanerepresentedbyaballoon probjectR=(Rh,Rp)(Figure 3-32 ).Inthehistoricalpart,PhasbeendisjointfromR.However,thepredictedrouteofPcrossesthepredictedfutureofR.TherelationshipbetweenPandRcanbedescribedasadevelopmentofuncertainspatio-temporalpredicatesovertime,i.e.,disjoint.meet.inside.meet.disjoint. 106

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However,thesespatialandspatio-temporalpredicatesmayrepresentrelationshipsbetweendifferentpartsoftheballoonobjects.Forinstance,therstdisjointpredicateisactuallyatemporalcompositionofthreedifferenttypesofdisjointednessbetweenthecorrespondingpartsofPandR,i.e.,thehistoricalpartofPandR,thehistoricalpartofPandpredictivepartofR,andthepredictivepartsofPandR.Therestofthepredicatesrepresentrelationshipsbetweenthepredictionpartsofbothobjects.Hence,wecanexpandtheoriginalsequenceasdisjoint(Ph,Rh).disjoint(Ph,Rp).disjoint(Pp,Rp).meet(Pp,Rp).inside(Pp,Rp).meet(Pp,Rp).disjoint(Pp,Rp).Inthissequence,thesubsequencedisjoint(Pp,Rp).meet(Pp,Rp).inside(Pp,Rp).meet(Pp,Rp).disjoint(Pp,Rp)canberepresentedbyanSTPcross(Pp,Rp)asanewspatio-temporalpredicate.Thus,wehavedisjoint(Ph,Rh).disjoint(Ph,Rp).cross(Pp,Rp).Therefore,weareleftwithasequenceofthreeSTPseachappliedtodifferentcombinationpairsofpartsoftheballoonobjects.ThisexampleillustratesthatballoonpredicatescanbeappropriatelymodeledbysequencesofthreeSTPsbetweentherelatedpartsoftheobjects.Hence,wecanspecifyballoonpredicatesbasedonthetraditionalSTPsinthedenitionasfollows: Denition2. LetPandRbetwoballoonobjectsoftype(1,1)and(2,2)respectively.AballoonpredicatebetweenPandRisatemporalcompositionoftraditionalspatio-temporalpredicates:stp((1),(2)).(stp((1),(2))jstp((1),(2))).stp((1),(2)). Figure3-32. Afuturecrossingsituationbetweenaballoon ppobjectPandaballoon probjectR 107

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AnSTPbetweentwomovingobjectsismeaningfulifandonlyifthereexistsaperiodoftimeforwhichbothobjectsaredened.Hence,eachelementoftheabovesequenceismeaningfulonlyiftheeachcorrespondingpartisdened.Thepredicateoftherstelementinthesequencerepresentsaninteractionthatdidoccur.Therstandsecondalternativepredicatesofthesecondelementinthesequencerepresentsaninteractionthatmayhaveoccurred.Thesepredicateoptionsreecttheconstraintwedescribedabove,whichstatesthatthetwopredicatescannotexistatthesametime.Thepredicateofthethirdelementinthesequencedenotesaninteractionthatprobablywilloccur.Thus,thesecondandthirdelementsindicatewhetherthereisapossibilitythataninteractionwilloccurwhereastherstelementtellsexactlywhetherornotaninteractionhasoccurred.Thecombinationsofmultipleoftheseinteractionsrepresentsamorecomplexrelationshipbetweenballoonobjects.Forexample,aninteractionthatdidoccurinthepastandprobablywilloccurinthefuturecanindicatethatthereisachancethatitprobablyalwaysoccurs.Table 3-6 showsanexampleofassigningameaningfulprextothenameforeachpairwisecombinationbetweentheseinteractions.Other Table3-6. Assigningnamingprexestopairwisecombinationsofinteractions. didmayhaveprobablywill did-mayhavebeenprobablyalways maymayhavebeen-probablywillhave probablywillprobablyalwaysprobablywillhavecombinationswithlargernumberofinteractionsalsoexist,butitisusuallynotobvioustonametheserelationships.Herearesomeexamplesofballoonpredicates:did cross:=cross((1),(2))probably will cross:=cross((1),(2))may have been disjoint:=disjoint((1),(2)).disjoint((1),(2))probably always inside:=inside((1),(2)).inside((1),(2))Aswehavedenedamodelforballoonpredicatesabove,thenwecansearchforacanonicalcollectionofballoonpredicates.Thedenitionofballoonpredicatesimpliesthatthecanonicalcollectionofballoonpredicatescanbeexpressedintermsofthe 108

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canonicalcollectionoftraditionalSTPs,whichisprovidedin[ 21 ].Anotherimportantfactorthataffectsthecanonicalcollectioniswhetherdependenciesexistbetweenthethreeelementsofthesequence.Morespecically,weneedtoinvestigatewhethertheexistenceofaSTPasanelementofthesequencecanpreventorrestrictanotherSTPfromrepresentinganotherelementofthesequence.Wecanprovethatthisisnottrue.In[ 21 ],thedependencybetweenSTPsisexpressedusingadevelopmentgraph.ThisgraphdescribesallthepossibledevelopmentsofSTPswhichcorrespondtocontinuoustopologicalchangesofmovingobjects.Forexample,ifamovingpointentersamovingregion,itmustbedisjointfromtheregionrst,thenmeetstheregionandintheendinsidetheregion.ThisconstraintreliesonthecontinuityofmovingobjectsasweintroducedinSection 3.2.1 .Althoughthehistorypartandthepredictionpartofaballoonobjectcannottemporallyoverlapeachother,theymightbeseparatedbyaperiodofunknownmovement.Further,therecanalsobeperiodsofunknownmovementwithinthehistoryorthepredictionpartofaballoonobject.Duetothepossiblediscontinuityofballoonobjects,wecanimplythateachelementofthepredicatesequenceisindependentofeachother.Thus,allthecombinationsoftheSTPsinvolvedarepossible.ThismeansthatthecanonicalcollectionofballoonpredicatescanbedeterminedsolelybasedonthecanonicalcollectionsofthetraditionalSTPsinvolved.Asprovidedin[ 21 ],thereare13distincttemporalevolutionsbetweentwomovingpointswithoutrepetitions,28betweenamovingpointandamovingregion,and2,198betweentwomovingregions.Withthisinformation,wecandetermine,forexample,thenumberofdistinct,non-repetitiveballoonpredicatesbetweentwoballoon ppobjectstobe13(13+13)13=4,394.EachofthethreepartsofthemultiplicationrepresentsthenumberofdistinctSTPsforeachelementofthesequence.Similarly,wecandeterminethenumberofballoonpredicatesbetweenalltypecombinationsofballoon pp,balloon pr,andballoon rrasshowninTable 3-7 below. 109

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Table3-7. Numberofballoonpredicatesbetweenballoon pp,balloon pr,andballoon rrobjects. balloon ppballoon prballoon rr balloon pp4,39414,92443,904 balloon pr14,9241,600,144136,996,944 balloon rr43,904136,996,94421,237,972,784 3.2.4.2ModelingPotentialPredicatesintheFutureIntheSection 3.2.4.1 ,wehavemodeledballoonpredicatesbasedontraditionalSTPs.Thisallowsustodistinguishrelationshipsinvolvingfuturepredictionsasuncertainrelationshipswithrespecttothemovingobjectsthemselves.Unlikerelationshipsbetweenthepastmovementhistorieswhichindicatedisjointorinteractionrelationshipsthathaddenitelyoccurredbetweenthemovingobjects,uncertainrelationshipsonlyindicatetheexistenceofachancewhetherthemovingobjectswillinteractwitheachother.Asthepossiblelocationofamovingobjectisnotdeterministicbutcanbeanywherewithinanarea.Thusinthissection,wewillstudyhowthischanceoffutureinteractionbetweentheactualmovingobjectscanbequantiedbasedonthegivenrelationshipoftheirpredictions.Recallthatthefuturepredictionofaballoonobjectrepresentsthesetofallpotentialfuturepositionsorextentsofthemovingobject.Thismeansthatanon-interactionrelationshipwiththisfuturepredictioncomponentguaranteesanon-interactionrelationshipwiththeactualobjectinthefuture.However,aninteractionrelationshipwiththisfuturepredictioncomponentcanonlysignifyapotentialinteractionwiththeactualobjectinthefuture.Forexample,iftherouteofashipdoesnotintersectthefuturepredictionofahurricane,thismeansthatthereisnochancethattheshipwillencounterthehurricaneinthefuture.However,iftheroutecrossesthehurricane'sfutureprediction,anumberofpossibilitiescanhappen.Theshipwilleitherencounteroravoidthehurricane.Therearetwointerestingquestionswhichinvolvetheuncertaintythatweneedtoinvestigate:(1)Whatarethedifferenttypesofpossibleinteractions 110

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betweentheactualobjectsinthefuturegivenaninteractionbetweentheirfuturepredictions?and(2)Howmuchofachancethattheobjectswillinteractinthefuture?Theproblemoftherstquestionissimilartotheproblemofinferringthesetofpotentialtopologicalrelationshipsbetweentwomovingobjectsinourpendantmodel[ 49 ].Atanyinstantofaprediction,amovingobjectcanbeanywherewithinitsprediction.Thisallowsalargenumberofpossiblecongurationoftheobjectwithinitsprediction,morespecically,withinanydivisiblepartoftheinteriorofitsprediction.Thismeansthatforaninteractionbetweentwopredictionswheretheinteriorsofthepredictionsintersect,allpossibletypesofinteractionarepossiblebetweentheactualobjects.Ontheotherhand,iftheinteriorsofthepredictionsdonotintersectbuttheirboundariesintersect,theactualmovingobjectscaneitherinteractbysharingtheirboundariesorbedisjoint.Finally,ifthepredictionsaredisjoint,thisimpliesthattheactualmovingobjectswillbedisjointaswell.Table 3-8 summarizestheseinteractioninferences. Table3-8. Inferringthetypesofinteractionbetweenactualobjects PredictionInteractionsPossibleObjectInteractions interiorintersectionanyinteractionpossible boundaryintersectionboundaryintersection,disjoint disjointdisjoint Toanswerthesecondquestion,weshouldconsidereachtypeofpredictioninteractions.Fordisjointpredictions,itisguaranteedthattheobjectwillbedisjoint.Thusthechanceofinteractioninthiscaseis0.Forpredictionswithboundaryintersection,thechanceoftheactualobjectssharingtheirboundariesatthisintersectionisproportionaltotheproductofthepoint-setcondencevaluesoftheintersectionwithrespecttoeachobject.Thisquantityisaninnitelysmallpositivenumberapproaching0sincethedimensionoftheboundaryintersectionisalwayssmallerthanthedimensionoftheprediction.Butthereisstillapossibilitythattheboundaryintersectioninteractioncanoccurbetweentheactualobjects.Similarly,inthecaseofpredictionswithinterior 111

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intersection,thechancethattheactualobjectswillinteractatthisintersectionisproportionaltotheproductofthepoint-setcondencevaluesoftheintersectionofeachprediction.However,thisquantityhereisameaningfulquantitysinceeachofthepoint-setcondencevaluesisameaningfulvalue.Itisimportanttonotethatthisquantitydoesnotindicatetheprobabilityoftheinteractionbetweentheactualobjects,butmerelyrepresentstheprobabilityofbothobjectsbeingintheintersection.However,itisreasonabletosaythatthehighertheprobabilityofbothobjectsbeingintheintersection,thehigherthechancethattheywillinteractwithoneanother.Weusetheoperationinteraction potentialforthispurpose.Theresultofthisoperationisoftype(real)indicatingthetemporallydependentvalueofthechancethattheobjectswillbeintheproximity(intersection)whereinteractionispossible.Todeterminewhetherthereisapossibilityofinteractionthusdistinguishingtheboundaryintersectioncasefromthedisjointcase,weusethepredicateoperationinteraction possible.Byusingthecombinationoftheseoperationstogetherwiththebinarypredicateoperation,onecanobtaintheuncertaintyinformationoffutureinteractionsbetweenmovingobjects. 112

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CHAPTER4REPRESENTATIONOFHISTORICALANDPREDICTIVEMOVINGOBJECTSWITHUNCERTAINTYInthischapter,weintroducethediscreterepresentationofmovingobjectswithuncertaintyaccordingtothemodelsintheprevioussection.Section 4.1 introducesthedatastructuresofmovingobjectsunderthepedantmodel.Section 4.2 discussesthealgorithmsforoperationsandpredicatesonthependantmodel.Section 4.3 introducesamethodonminingfromuncertaintrajectories.Section 4.4 introducesanovelapproachonsimilaritymeasurementofmovingobjecttrajectories. 4.1RepresentingMovingObjectsSofar,wehaveintroducedhowtomodelmovingobjects,suchasfunctionsfromtimetospace,ora3D(2D+time)volume.However,thesemethodsareonlyattheabstractlevel.Inthissection,weintroducethediscretemethodofrepresentingmovingobjects,whichcanbeimplementedbyprograminglanguagesandfurtherintegratedintoextensibledatabases.Werstintroducetheslicerepresentationformovingobjectswithoutuncertaintywhichsolvestheproblemofcomputingcardinaldirections.Thenweextendthesliceconcepttorepresentmovingobjectswithuncertainty. 4.1.1SliceRepresentationofMovingObjectsSincewetakethespecicationofthemovingpointdatatypein[ 23 ][ 25 ]asourbasis,werstreviewtherepresentationofthemovingpointdatatype.Accordingtothedenition,themovingpointdatetypedescribesthetemporaldevelopmentofacomplexpointobjectwhichmaybeapointcloud.However,wehereonlyconsiderthesimplemovingpointthatinvolvesexactlyonesinglepoint.Aslicerepresenta-tiontechniqueisemployedtorepresentamovingpointobject.Thebasicideaistodecomposeitstemporaldevelopmentintofragmentscalledslices,wherewithineachslicethisdevelopmentisdescribedbyasimplelinearfunction.Asliceofasinglemovingpointiscalledaupoint,whichisapairofvalues(interval,unit-function).Theintervalvaluedenesthetimeintervalforwhichtheunitisvalid;theunit-function 113

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valuecontainsarecord(x0,x1,y0,y1)ofcoefcientsrepresentingthelinearfunctionf(t)=(x0+x1t,y0+y1t),wheretisatimevariable.Suchfunctionsdescribealinearlymovingpoint.Thetimeintervalsofanytwodistinctsliceunitsaredisjoint;henceunitscanbetotallyorderedbytime.Moreformally,letAbeasinglemovingpointrepresentation,interval=timetime,real4=realrealrealreal,andupoint=intervalreal4.ThenAcanberepresentedasanarrayofsliceunitsorderedbytime,thatis,A=h(I1,c1),(I2,c2),...,(In,cn)iwherefor1inholdsthatIi2intervalandci2real4containsthecoefcientsofalinearunitfunctionfi.Further,werequirethatIi
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(a)(b)Figure4-1. Examplesoftheslicerepresentations Therstsetofoperationsisprovidedformanipulatingmovingpoints.Theget rst sliceoperationretrievestherstsliceunitinaslicesequenceofamovingpoint,andsetsthecurrentpositionto1.Theget next sliceoperationreturnsthenextsliceunitofthecurrentpositioninthesequenceandincrementsthecurrentposition.Thepredicateend of sequenceyieldstrueifthecurrentpositionexceedstheendoftheslicesequence.Theoperationcreate newcreatesanemptyMPointobjectwithanemptyslicesequence.Finally,theoperationadd sliceaddsasliceunittotheendoftheslicesequenceofamovingpoint.Thesecondsetofoperationsisprovidedforaccessingelementsinasliceunit.Theoperationget intervalreturnsthetimeintervalofasliceunit.Theoperationget unit functionreturnsarecordthatrepresentsthelinearfunctionofasliceunit.Thecreate sliceoperationcreatesasliceunitbasedontheprovidedtimeintervalandthelinearfunction. 4.1.2RepresentingUncertainMovingObjectsSimilarly,weareabletorepresenttheuncertainmovingobjectsofourpendantmodelbasedontheslicerepresentation.InSection 3.1 ,wehavestatedthateachmovingobjectinthependantmodelisrepresentedbyasetofpartialfunctionsonaunionofintervals,andeachintervalhasitsownmovingpattern,i.e.,eitheraknownmovementrepresentedbyalinearlyfunctionoranunknownmovementrepresentedbyadouble-conevolume.Thus,weare 115

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abletofragmenttheentiremovementasasetofslices.Asliceisthesmallestunitofevaluatingthespatio-temporaluncertaintypredicate,whichiseitheralinesegmentoradouble-conevolume.ThesliceunitrepresentationofmovingobjectswithuncertaintyisillustratedinFigure 5-7 ,wheremovingobjectAisrepresentedbywithelementsorderedbytime,andBisrepresentedby.Weevaluatethepredicatebetweentwoentiremovingobjectsbyevaluatingwhetheranintersectionexistsbetweenapairofslices.Therearethreesituations:1.Bothslicesarelinesegments;2.onesliceisalinesegmentwhiletheotherisadouble-cone;3.bothslicesaredouble-conevolumes.Therstsituationisthesimplestoneinthatwecanrepresenttwolinesegmentsbyequationsinthe3Dplaneandcomputewhethertheyintersectatacommonpoint,denotedbycomPoint(seg1,seg2).Forsituation2and3,sinceitiscumbersometocalculatetheintersectionin3Dvolumes,weintroducethemethodtotesttheintersectiononlyinsometimeinstants,whicharecalledcriticalinstants.Alinesegmenthastwocriticalinstants:thestartingtimeandendingtimerespectively.Astraightdouble-conevolumehas3criticalinstants:theinstantsatthebottomapex,thebaseandthetopapex.Anobliquedouble-conevolumehas4criticalinstants:thebottomapex,thelowerbasepoint,theupperbasepointandthetopapex.Figure 5-7 (b)illustratescriticalinstantsonthethreetypesofvolumes. (a)(b)Figure4-2. Sliceunitrepresentationofmovingobjectswithuncertainty 116

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4.2AlgorithmsofMovingObjectswithUncertainty 4.2.1AlgorithmsofComputingCardinalDirectionDevelopmentsNow,weintroducea3-phaseapproachofcomputingthecardinaldirectiondevelopmentsbetweenmovingobjects.Therstphaseiscalledthetime-synchronizedintervalrenementphase.Sinceasliceisthesmallestunitintheslicerepresentationofmovingpoints,werstconsidertheproblemofcomputingcardinaldirectionsbetweentwomovingpointslices.Accordingtoourdenitionsin[ 7 ]thecardinaldirectionsonlymakesensewhenthesametimeintervalsareconsideredforbothmovingpoints.However,matching,i.e.,equal,sliceintervalscanusuallynotbefoundinbothmovingpoints.Forexample,inFigure 4-1 ,thesliceintervalIA1=[t2,t4]ofAdoesnotmatchanyofthesliceintervalsofB.AlthoughthesliceintervalIB1=[t1,t3]ofBoverlapswithIA1,italsocoversasub-interval[t1,t2]thatisnotpartofIA1,whichmakesthetwoslicesdenedinIA1andIB1incomparable.Thus,inordertocomputethecardinaldirectionsbetweentwomovingpointslices,atime-synchronizedintervalrenementforbothmovingpointsisnecessary.Weintroducealinearalgorithminterval syncforsynchronizingtheintervalsofbothmovingpoints.Theinputofthealgorithmconsistsoftwoslicesequencesmp1andmp2thatrepresentthetwooriginalmovingpoints,andtwoemptylistsnmp1andnmp2thatareusedtostorethetwonewintervalrenedmovingpoints.Thealgorithmperformsaparallelscanofthetwooriginalslicesequences,andcomputestheintersectionsbetweenthetimeintervalsfromtwomovingpoints.Onceanintervalintersectioniscaptured,twonewslicesassociatedwiththeintervalintersectionarecreatedforbothmovingpointsandareaddedtothenewslicesequencesofthetwomovingpoints.LetI=[t1,t2]andI`=[t1`,t2`]denotetwotimeintervals,andletlower thandenotethepredicatethatcheckstherelationshipbetweentwointervals.Thenwehavelower than(I,I`)=trueifandonlyift2
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thatcomputestheintersectionoftwotimeintervals,whichreturns;ifnointersectionexists.Wepresentthecorrespondingalgorithminterval syncinFigure 4-3 .Asaresultofthealgorithm,weobtaintwonewslicesequencesforthetwomovingpointsinwhichbothoperandobjectsaresynchronizedinthesensethatforeachunitintherstmovingpointthereexistsamatchingunitinthesecondmovingpointwiththesameunitintervalandviceversa.Forexample,afterthetime-synchronizedintervalrenement,thetwoslicerepresentationsofthemovingpointsAandBinFigure 4-1 becomeA=(I1,cA1),(I2,cA1),(I3,cA2),(I4,cA2)andB=(I1,cB1),(I2,cB2),(I3,cB2),(I4,cB3),wherethecAiwithi2f1,2gcontainthecoefcientsofthelinearunitfunctionsfAi,thecBiwithi2f1,2,3gcontainthecoefcientsofthelinearunitfunctionsfBi,andI1=intersection(IA1,IB1)=[t2,t3],I2=intersection(IA1,IB2)=[t3,t4],I3=intersection(IA2,IB2)=[t4,t5],andI4=intersection(IA2,IB3)=[t5,t6]. 4.2.2AlgorithmsofSpatio-temporalUncertainPredicatesWedesignanalgorithmtocomputewhethertwoslicecouldintersectbytestingwhethertheyintersectatcriticalinstants,shownbyFigure 4-4 .Becauseweonlyexamtheintersectionatafewnumberofcriticalinstants,thecomplexityofthisalgorithmisconstant.Thepossibly encounterpredicatecanthenbedeterminedbyexaminingtheintersectionbetweenpairsofslices.ThealgorithmisshowninFigure 4-5 .Assumethattherstmovingobjecthasmslicesandthesecondhasnslices,thisalgorithmwillexamm+ntimesofunitIntersection.SincethecomplexityofunitIntersectionisO(1),thetotalcomplexitytodeterminepossibly encounterisO(m+n). 118

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methodinterval sync(mp1,mp2,nmp1,nmp2)s1 get rst slice(mp1)s2 get rst slice(mp2)whilenotend of sequence(mp1)andnotend of sequence(mp2)doi1 get interval(s1)i2 get interval(s2)i intersection(i1,i2)ifi6=;thenf1 get unit function(s1)f2 get unit function(s2)ns1 create slice(i,f1)ns2 create slice(i,f2)add slice(nmp1,ns1)add slice(nmp2,ns2)endififlower than(i1,i2)thens1 get next slice(mp1)elses2 get next slice(mp2)endifendwhileend 1 methodcompute dir dev(sl1,sl2)2 dev list emptylist3 s1 get rst slice(sl1)4 s2 get rst slice(sl2)5 slice dir list compute slice dir(s1,s2)6 append(dev list,slice dir list)7 whilenotend of sequence(sl1)8 andnotend of sequence(sl2)do9 (b,e) get interval(s1)10 s1 get next slice(sl1)11 s2 get next slice(sl2)12 (b new,e new) get interval(s1)13 ife
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algorithmunitIntersect(sliceS1,sliceS2) 1 intersect false 2 S empty//sequenceofinstants 3 m num of critical instants(S1) 4 n num of critical instants(S2) 5 whilefimandjng 6 iftime[i]
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4.3InferringFutureLocationsofMovingObjectsfromSimilarTrajectoriesMovingobjectsinthephysicalworldusuallygeneratemanyuncertaintrajectoriesforsomereasonssuchastheconsiderationofenergyconsumption,leavingtheroutepassingtwoconsecutivesamplingpointsunknown.Whilesuchtrajectoriesimplyrichknowledgeaboutthemobilityofmovingobjects,theyarelessusefulindividually.Thissectionintroducesanapproachofpredictingtheroutesofmovingobjectscollectivelyminingfrommassiveuncertaintrajectoriesfollowingaparadigmofuncertain+uncertain!certain.Thisapproachrstbuildsaroutablegraphfromuncertaintrajectories,andthenanswersauser'sonlinequery(asequenceofpointlocations)bysearchingtop-kroutesonthegraph. 4.3.1OverviewWiththeadvancesinlocation-acquisitiontechnology(e.g.,GPSservices),thestudyofmovingobjectshasbeenexperiencingpopularity[ 29 ][ 100 ],[ 99 ].Amovingobjectsuchasavehicleorapersoncanberepresentedbyasequenceoflocationswithincrementontime,oratrajectory.Forexample,asequenceofplacesofinterest(POI)atravelervisits,themigrationofbirds,andthemovementofhurricanescanallberepresentedbytrajectories.Obtainingthesetrajectorieswillbeusefulindiscoveringknowledgefrommovingobjects.However,trajectoriesareoftengeneratedatalowfrequencyduetotheconsiderationofenergysavingorotherapplicationfeatures.Forexample,atravelerwithasmartphonecannottakeageo-taggedphotoevery10seconds;asensortrackingahurricanecannotreportthelocationeverysecond.AssumethatatravelerinNewYorkCityhasvisited5POIsincludingtheStatueofLiberty,ChinaTown,TimesSquare,CentralPark,andtheMetropolitanMuseumofArt,butonlytakenphotosattheStatueofLibertyandCentralPark.Therealpathofthistravelerisuncertain.However,anotherpersonwhohasalsovisitedthesamePOIsmayhavephotosatTimesSquareandtheMetropolitanMuseumofArt.Combiningtheiruncertaintrajectories,wecaninfertheirrealpaths,i.e.uncertain+uncertain!certain. 121

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Thegoalofthisresearchistointroduceanonlinesystemthatenablesroutediscoveringthroughminingalargenumberofuncertaintrajectories.Wecollectthedatasetsofuncertaintytrajectoriesfromtwosources,travelers'check-insandtaxitrajectories.Inrecentyears,emergingsocialnetworkwebsiteswithphotosharingandcheck-infunctionsmakethetrajectoriesoftravelersavailable:apersonwithasmartphonecancreateatraveltiporcaptureageo-taggedphotoatanytimeanduploadittoasocialnetworksuchasFoursquarec.Wehavecollectedmorethan425,000check-insofthreemonthsinNewYorkCity,anddetectedover73,000uncertaintrajectories.Wehavealsocollectedover15,000taxitrajectoriesinBeijingwiththehelpoftheGPSsensorsembeddedintaxis.Aroutablegraphontopofthecityareaisbuiltwherethetripplanningisperformed.Whenauserinputsasequenceofquerylocations,thesystemwillsearchallpossibleroutestraversingthemonthegraph.Thesystemwillscoretheseroutesandreporttop-koptimalroutes. 4.3.2MiningUncertainTrajectoriesInthissubsection,weintroduceourmininguncertaintytrajectoriesapproach.Table 4.3.2 showsthenotationsoftheparametersthatwillbeneededinthemodel,aswehavementioned. Table4-1. NotationsofParameters NotationsDescription glgridlength srsamplingrateofrawtrajectories temporalconstraintbetween[0,1] Cconnectionsupport )]TJ /F19 9.963 Tf 57.89 0 Td[(transitiontimeofasub-trajectory krankingofoptimaltrajectory Werstdivideageographicalareaintoasetofdisjointgridcells.Acellisasquareandisdenotedby(i,j)indicatingthecellid.AtrajectoryisindexedbytheorderofthegridcellsittraversesinFigure 4-6 .c(g)denotesthenumberofdistincttrajectoriestraversinggridcellg.Thetrajectoriesinagridcellareorderedbyavariablemtraina 122

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Figure4-6. Anexampleofthetrajectoryindexingstructure descendingorder,wheremtradenotesthemedianofc(g)ofallcellstraversedbyatrajectorytra. Denition4.3.2.1. Giventwogridsg1=(x1,y1)andg2=(x2,y2),thegridsg1andg2aresaidtobespatialcloseifjx)]TJ /F5 11.955 Tf 11.96 0 Td[(x1j1andjy)]TJ /F5 11.955 Tf 11.95 0 Td[(y2j1.Thus,acellisspatial-closeto8cellssurroundingit. Denition4.3.2.2(correlatedtrajectories). Wesaythattwosub-trajectories(segmentsoftrajectories)arecorrelated,ifthefollowingconditionshold:(i)Theratioofthedifferenceofthetransitiontimestothemaximumtransitiontimeofthetwosub-trajectoriesislessthanathreshold,where, jt1)]TJ /F7 7.97 Tf 6.59 0 Td[(t2 maxft1,t2g(ii)Eitherthesourcegridcelloftwosub-trajectoriesarespatial-closeandtheyhavethesamesinkgridcell(Rule1),ortheyhavethesamesourcegridcellandtheirsinkgridcellsarespatial-close(Rule2).Rule1andRule2areshowninFigure 4-7 Denition4.3.2.3(neighbors). Wesaythattwogridcellsg1andg2areneighbors,org1Ng2,ifthefollowingconditionshold.(i)g1andg2arespatial-close;(ii)theconnectionsupportCofg1andg2isgreaterthanorequaltothegivenconnectionsupportC0speciedbytheuser. 123

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Figure4-7. Examplesofcorrelatedtrajectories Figure4-8. Anexampleofconstructingaconnectedregion. Denition4.3.2.4(connectedregion). AsetofgridsGiscalledaconnectedregion,ifforanygridcellg2G,wecanalwaysndg02G,sothatgNg0issatised.Then,wecanobtainasetofconnectedregions,asshowninFigure 4-8 .Edgescarryinginformationsuchasthedirection,theconnectionsupportandthetransitiontimewillbeadded.Internaledgeswillbeaddedwithinaregion,andexternaledgeswillbeaddedbetweendifferentregions,asshowninFigure 4-9 a.Iftherearemultipleedgesbetweensameregions,edgeswithconnectionsupportof0orlongertransitiontimeswillberemoved( 4-9 b).Thentheroutablegraphisbuilt. 4.3.3RouteGenerationThelaststepofthisapproachisroutegeneration.Givenasequenceofquerylocationsandatimespan,wesearchqualiedroutesontheroutablegraph.Werstmapallquerypointstogridcellsonthegraph.Ifaquerypointisnotinanygrid,wemapittogridsthatareclosesttoit.Itispossiblethataquerypointismappedtomore 124

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(a)(b)Figure4-9. Edgeinferenceandremoveredundantedge (a)(b)(c)Figure4-10. Querytransformation,routing,androuterenement thanonegridcell(Figure 4-10 a).Wedenearoutescorefunctionwhichconsidersrouteswithhigherconnectionsupportasbetterroutes.Wendtop-klocalroutesbasedonanA-likealgorithmbetweenanytwoconsecutivegrids[ 5 ].Thenwesearchtop-kglobalroutesbyabranch-and-boundsearchapproach(Figure 4-10 b).Whensearchinglocalroutesbetweentwogridcellswhicharelocatedindifferentregions,weproposeatwo-layerroutingalgorithm.Wedeterminetheorderoftheregionstoreducethesearchspace.Byutilizingalowerboundoftransitiontimesbetweenanytworegions,wegenerateregionsequenceswithrespecttothegivengridcells,andsearchbysequentiallytraversingtheseregions.Intheend,werenetop-kroutesbyndingthesegmentsfromhistoricaltrajectories.Weadoptlinearregressiononthepointsetineachgridtoderiveasegment,andconcatenatethesegments(Figure 4-10 c). 125

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4.4SimilarityMeasurementofMovingObjectTrajectories 4.4.1OverviewInmovingobjectapplicationssuchaslocation-basedservice,hurricaneresearchandtransportationmanagement,thefuturelocationsofmovingobjectsarealwaysinterestingsincetheyarehelpfulinsolvingproblemssuchascrisespreventionandlocationrecommendationinnowadayssocialnetworks.Thefuturelocationsarealwaysuncertain.Previousresearchershavestudiedalotonthepredictionoffuturelocationsofmovingobjects.Mostofthemestimatethefuturelocationsofmovingobjectsbymotionfunctionswhichcanbederivedfromthehistoricaltrajectoryofamovingobject.Asthemovingobjectschangetheirlocationssmoothly,wecanassumethatthelocationscapturedatrecenttimescanreectthemovingpatternoftheobjectatthecurrentmoment,andthereforewecandetectthecurrentlocationofthemovingobjectfromthemotioncurvewend.However,thisisnotalwaystruesincethedirectionsofamovingobjectscanchangeveryoften.Assumethatapersonisdrivingtoworkwithaconstantspeed,thenwecanpredictthatinthenext5minutes,heisstillatsomewhereonthewayfromhishometohisofce.However,wemaynotknowthatheactuallywillgotoacafeteriarsttohavebreakfastbeforehegoestotheofce.Theninthenextfewminutes,hemaymakeaturnatanintersectionandleavethewayonwhichheisdriving,arriveatthecafeteriaandhavebreakfast,andthencomebacktothewaytoworkagain.However,ifweknowfromthehistoricaltrajectoriesthatthepersonfollowthesamerouteseveraltimes,orotherpeoplewhoalsohavethesimilarbehavior,wecanmakethepredictionmoreeasily.Thiswillrequireamethodtondthesimilaritybetweentrajectoriesofpeople.Therefore,ourfocusshouldbendingthesimilaritybetweentrajectoriesinsteadofmerelystudyingtheirrecentmovements.Frommanyobservationswendthatsimilartrajectoriesshouldsatisfysomeparticularrequirement,forexample,theyshouldbecloseenoughtoeachotherintheEuclideanspace,andtheyevenshouldhavesimilardirection.Thenachallenging 126

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problemcomes,howtomeasurethecloseness?Further,similaritybetweenhuman'strajectoriesnotonlylieingeometrybutalsootherinformationsuchassemanticmeanings[ 92 ][ 1 ].Howtomeasurethesemanticsimilarity? (a)(b)Figure4-11. Trajectoriesformedbyusercheck-insequencesfromFoursquarec Inthispart,weproposeapredictionmethodonthefuturelocationsofmovingobjects.Thepredictionmethodisbasedonbothgeographicfeatureaswellassemanticpropertiesofmovingobjecttrajectories.Wedenegeographicsimilarityaswellassemanticsimilarityonthetrajectories.Thenwegiveasounddenitiontocombinebothsimilaritymeasurementstogetheranddenesanoveldistancefunction.Weadoptawellknowndensity-basedclusteringmethod[ 69 ][ 44 ]todetecttheclustersofsimilartrajectories.Afterthatwegeneraterepresentativeonesforeachclusteraccordingtotheresultofthesimilarity.Intheend,givenoneorfewlocationsofatrajectoryinthepastanditsdestination,weareabletopredictitsfuturelocationbasedontherepresentativetrajectory. 127

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4.4.2PreliminaryandSystemFrameworkInthissection,wegivepreliminaryconceptsofourresearch.Werstgivesomedenitionsonthetermsthatwillbeusedinfurtherdiscussionintherestofthepaper.Thenweshowtheframeworkofoursystem. 4.4.2.1Preliminary Denition4.4.2.1(Rawtrajectory). Arawtrajectoryisalistofspatio-temporalpoints,i.e.,<(x1,y1,t1),...,(xi,yi,ti),...,(xn,yn,tn)>,wherexi,yi,tirepresentlatitude,longitudeandtimerespectively.Arawtrajectorysatisestheconditionthatthepointsareorderedbytime,i.e.8i
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temporalinformation,whichmeansthatthemappingprocessisunique,i.e.,itdoesnotdependonwhenaplaceisvisitedandhowlongitisstayed.Thisisanadvantagethatwecanndsimilaritybetweentwopeopleeveniftheyvisitaplaceatdifferenttimesandwithdifferenttravelspeeds.Addingsuchsemantictagstoatrajectory,itisnolongeramerelypointsequencesbutcontainsricherinformation.Therefore,wegiveournewdenitiononsemantictrajectoryasfollows, Denition4.4.2.3(Semantictrajectory). Asemantictrajectoryisalistofpoints<(x1,y1,t1,S1),...,(xi,yi,ti,Si),...,(xn,yn,tn,Sn)>,withthefollowingconditions, i)n2N,i2f1,2,...,ngii)81
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4.4.2.2SystemFrameworkByintroducingtheconceptofrawtrajectoriesandsemantictrajectories,ourpurposeistotakeadvantagesofthespatio-temporalpropertyaswellasthesemanticstohelpuspredictthefuturelocationsofamovingobject.TheframeworkofoursystemisshowninFigure 4-13 .Oursystemconsistsofthreephases.Inthedatapreprocessingphase,weextractrawtrajectoryfromusers'visitingsequences,i.e.,alargenumberofGPSpoints.Forexample,inourrealsystem,wecollectusers'check-indatafromFoursquarec.Wedetectthetravelsequencesofeachuser,andanumberofrawtrajectoriesaregenerated.WiththevenueinformationcollectedfromtheInternetincludingGoogleMapsc,BingTMMapsandFoursquarec,weareabletoperformthesemanticmappingoneachlocationthatexistsinrawtrajectories.Therefore,eachlocationismappedtoasemanticsetandanumberofsemantictrajectoriesareobtained.Inthesimilartrajectoryminingphase,wedenethegeographicandsemanticsimilaritiesrespectively,andcalculatethescoresbetweenalltrajectories.Thenanumberofclustersaregeneratedaccordingtothesimilaritymeasurement.Afterthatweareabletondtherepresentativetrajectoriesforeachcluster.Inthelocationpredictionphase,giventhehistoricalpartofatrajectory,wewilldiscoverrepresentativetrajectoriesfromtheclusterswhichcanrepresentthemovingpatternofthequeriedtrajectorybest.Wepredictthefuturelocationsofthistrajectoryfromtherepresentativetrajectorieswediscovered.Afterastepofrenement,wewillgetourpredictionresult.Thersttwomodulesareoff-linemodules,whereweperformthedataminingprocess.Thethirdmoduleistheon-linemodule,wherewecanreturntherequiredanswerinstantly.Inthefollowingsections,weshowindetailhowourapproachisprocessed. 4.4.3MiningSimilarTrajectoriesInthissection,wedenethesimilaritybetweentrajectories,whichwillbethefoundationofourapproach.Weconsiderintwoaspects,thegeographicsimilarityandsemanticsimilarity.Wedenethemintermsofdistance,i.e.,thelessthedistanceis, 130

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Figure4-13. Systemframeworkforinferringfuturelocations Table4-2. Notations SymbolDescription Segmentbearing 'Trajectoryturn Atemporalconstraint sDisplacement ktrakLengthoftrajectorytra jtrajNumberofGPSpointsintra dist(p,q)Euclideandistancebetweenp,q cos(tra1,tra2)Cosinesimilarityoftra1,tra2 Aweightofthesemanticsimilarity themoresimilartwotrajectoreisare.InTable 4-2 welistasetofnotationsthatwillbeneededinthissection. 4.4.3.1Sub-trajectoryPartitioningFirstofallwedenethesimilaritymeasurementbetweentrajectoriesingeometry.Acommonprobleminmeasuringthesimilarityoftrajectoriesandtrajectoryclusteringisthehandlingofsub-trajectories[ 45 ].GivenatrajectorywhichconsistsofalistofGPSpointstra=<(x1,y1,t1),(x2,y2,t2),...,(xn,yn,tn)>,asub-trajectorytrakisapartiallistoftra,andtrak=<(xk1,yk1,tk1),...,(xkm,ykm,tkm)>,where1k1
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Inourcontext,wehavesub-trajectoriesofarawtrajectory,aswellassub-trajectoriesofasemantictrajectory.Ifasub-trajectorycontainsonlytwoGPSpoints,thenitisatrajectorysegment.Aproblemcostbynotidentifyingsub-trajectoriesisshowninFigure 4-14 a.Assumethatwehaveatrajectorytra1=,wherea1,a2,...arealistofGPSpoints,andtra2=,wendthatpartoftra1andtra2areverysimilar.However,ifwecomparetheentiretrajectoryoftra1withtra2,it'shardtotellwhethertra1andtra2aresimilar,becausetra1hasasuddenturnatpointa3.Wecallsuchpointaturningpoint.However,ifwepartitiontra1intotwosub-trajectoriesastra11=andtra12=,weareabletomeasurethesimilaritybetweentra11andtra2.Therefore,animportanttaskistodetecttheturningpointsofatrajectoryandpartitionitintoalistofsub-trajectories. (a)(b)Figure4-14. Anexamplethatawholetrajectorymaynotworkinidentifyingclustersandusingturnstodetectthepointtopartitionsub-trajectories Aturningpointisdetectedbygivenaspecicparameter'whichshowsthedegreeofturnsmadebyatrajectory.Wedeneiastheabsolutebearing(directiontonorth)ofthei-thsegment.Giventhelatitudeandlongitudeofaspatialpoint,itsbearingcanbedetermined.Let'idenotesthedegreethatatrajectoryturnsatthei-thpoint,then'=i)]TJ /F8 11.955 Tf 12.14 0 Td[(i)]TJ /F7 7.97 Tf 6.59 0 Td[(1,asshowninFigure 4-14 b.Thenwecanselectarangeof',sothatiftheabsolutevalueofthedegreeofaturnisgreaterthan',thenwesplitthetrajectoryatthisturningpointintotwosub-trajectories.Inoneextremecase,ifwechoose'tobe0, 132

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theneachsegmentwillbepartitioned.Intheothercase,ifwechoose'tobe,thennotrajectorieswillbesplit.Anothercriteriathatwewanttotaketopartitionatrajectoryintoseveralsub-trajectoriesisatemporalconstraint.Manytrajectory-clusteringalgorithmsdonotconsidertheeffectontime,i.e.,theyonlyconsiderthesimilaritybetweentheshapesoftrajectories.However,wethinkthattimeisanimportantfeatureinthemovingpatternofamovingobject.Forexample,aperson'strajectorymayconsistoftwoparts:intherstpart,hewalkswiththespeedof4km=h,whileinthesecondparthetakesataxiandtravelswiththespeedof50km=h.Sincethesetwopartsshowdifferentmovingpatterns,wewillsplitthem.Thismayalsoreectsomespecicreasons.Forexample,insomeplacestheremightbeabridgeoratunnel,whichcanonlypassedbycar.Ifsuchplacesbelongtothetrajectoriesofsomeusers,thentheymustbeextractedandconsidereddifferentlywithotherlocations.Thereforewedeneaspeedratioofthei-thsegmentiandatemporalconstraintas,i=vi viwhereviistheaveragespeedofthemovingobjectatthei-thsegment,andviistheaveragespeedsofar.Ifiexceedthethresholdweset,thenwewillpartitionthetrajectoryatthei-thpoint. 4.4.3.2GeographicDistanceAfterpartitioningtrajectoriesintosub-trajectories,weareabletodenethesimilaritybetweensub-trajectories.Inordertoformthedenition,wenowdiscussafewwellphysicalmetricsunderoursub-trajectorycontext.Whenmeasuringthesimilaritybetweenspatialobjects,anintuitivewayistomeasurehowcloseindistancetheyaretoeachother.Amethodistoconsiderthesetwospatialobjectsastwopointsets,andndtheminimumdistancebetweenthetwopointsets.AsshowninFigure 4-15 .However,thisisnot 133

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(a)(b)(c)Figure4-15. Differentsituationsconsideredwhenmeasuringthesimilarity acorrectassumptionwhenitappliestothesimilarityoftrajectories.Forexample,iftwotrajectoriesoverlap,thenthedistancewillbezero,andthesetwotrajectorieswillbeconsideredverysimilar,whichisnotaccurate.ShownbyFigure 4-15 a,theblueandredtrajectoriesalthoughintersect,theirdirectionsarequitedifferent,therefore,weneedtoconsidertheirdirectionsinthesimilaritymeasurement.Further,wemusttakeintoconsiderationthediferenceofthelengthsbetweentrajectories.Ashorttrajectorywillnotbesimilartoalengtrajectory,asshowninFigure 4-15 c.Withthecarefulconsiderationofalltheaboveissues,weintroducethefollowingconceptswhicharenecessaryforustodenethetrajectorysimilarity. Centerofmass.Werstdiscussthecenterofmassconcept.Thecenterofmassofanarbitrary2Dshape(x,y)isthegeometriccenterofthisshape.Weusectr(tra)todenotethecenterofmassofthetrajectorytra.Itiscalculatedby,ctr(tra)=(x,y)=(Rxf(x)dx Rf(x)dx,Ryf(y)dy Rf(y)dy)wheref(x)andf(y)denotesthedensitydistributiononx-coordinateandy-coordinate.Assumethatwehaveasub-trajectory(x1,y1,t1),(x2,y2,t2),...(xn,yn,tn)>,thenxiscalculatedasx=Pn)]TJ /F7 7.97 Tf 6.59 0 Td[(1i=1Rxi+1xixf(x)dx Pni=1Rxi+1xif(x)d(x) 134

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Sinceweassumeauniformdistributionofdensity,f(x)=f(y)=1.Thenthecenterofmassis,(x,y)=(Pn)]TJ /F7 7.97 Tf 6.59 0 Td[(1i=1(x2i+1)]TJ /F5 11.955 Tf 11.96 0 Td[(x2i) 2Pni=1(xi+1)]TJ /F5 11.955 Tf 11.96 0 Td[(xi),Pn)]TJ /F7 7.97 Tf 6.58 0 Td[(1i=1(y2i+1)]TJ /F5 11.955 Tf 11.95 0 Td[(y2i) 2Pni=1(yi+1)]TJ /F5 11.955 Tf 11.96 0 Td[(yi))Aspecialcaseistondthecenterofasegment,i.e.,asub-trajectoryjustcontainstwoGPSpoints(x1,y1),(x2,y2).Thecenteris(x1+x2 2,y1+y2 2),whichisinaccordancewiththeaboveequation.Itisobviousthatthecenterofmassofatrajectorymayormaynotlieonthetrajectory,asshownbyFigure 4-16 aandb.Therefore,wecancalculatetheEuclideandistancebetweenthemassofcentersoftwotrajectoriestra1andtra2asdist(ctr(tra1),ctr(tra2)).WenamethistermctrDist(tra1,tra2). Displacement.Thedisplacementofatrajectorytraistheshortestdistancefromitsorigintoitsdestination.AnexampleisshowninFigure 4-16 c.Weintroducethisconceptbecauseitshowsashortcutofatrajectory.Theshortcutalwaysleadstotherightdestinationandisusefulinidentifyingrepresentativetrajectories.Weusethetermstodenotethedisplacement. (a)(b) (c)(d)Figure4-16. Centersofmassofdifferenttrajectories Cosinesimilarity.Thecosinesimilaritybetweentwovectorisameasureofthecosineoftheanglebetweenthem,andthevalueisbetween[-1,1].Twovectorsare 135

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thesame,theircosinesimilarityvalueis1.Iftheyhavethesamevaluebutpointtotheoppositedirection,thiscosinesimilarityis-1.Hereweuseittomeasurethesimilaritybetweenthedirectionoftwosub-trajectories.Wecalculatethecosinesimilaritybetweenthedisplacementsoftwotrajectoriesandaddittooursimilaritymeasurement.Alargervalueshowsahighersimilarity,incontrasttotheeuclideandistance,thereforeweaddthismetricasannegativeterm.Lets1,s2denotethedisplacementoftrajectorytra1andtra2respectively,thecosinesimilarityisdenedas,cos(s1,s2)=s1s2 ks1kks2k (4)Nowweareablegivethedenitionofthegeometricdistancebetweentwosub-trajectories. Denition4.4.3.1(Geographicdistance). Lettra1andtra2denotetwotrajectories.Lets1ands2denotethedisplacementoftra1andtra2respectively,andktrakdenotethelengthoftra.Thegeographicdistancebetweentra1andtra2isdenedas,geoDist(tra1,tra2)=ctrDist(tra1,tra2)+ctrDist(tra1,tra2)jktra1k)-223(ktra2kj max(ktra1,ktra2k))]TJ /F4 11.955 Tf 11.95 0 Td[(avg(ks1k,ks2k)cos(s1,s2) (4)Intheabovedenition,thersttermmeasuresthedistancebetweenthecentersofmass.Thesecondtermmeasuresthedifferencebetweenthelengthofthetrajectories.AswehaveshowninFigure 4-15 c,thelengthsofthetrajectorieseffecttheirsimilarity,sothatweconsidertherationofthedifferencetothemaximumlengthoftwotrajectories.Thersttwotermsbothincreasethedistancebetweentwotrajectories. 136

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Whilethethirdtermisthecosinesimilaritytimestheaveragelengthoftwotrajectories,whichreducethedistancebetweentrajectories,i.e.,similartrajectorieshavealargercosinesimilarityvalue.Thereforethistermisnegative.Anadvantageofthisdenitionisthatthedistancefunctionissymmetric,i.e.,thedistancefromtra1totra2isthesamefromthedistancefromtra2totra1.Weshowthispropertyinthefollowinglemma. Lemma4.4.3.1. ThedistancefunctiondenedinDenition 4.4.3.1 issymmetric,whichsatises,geoDist(tra1,tra2)=geoDist(tra2,tra1) Proof. ThersttermofEquation(2)istheeuclideandistancebetweentwopoints,i.e.,thecenterofmassoftwotrajectories,whichissymmetric.Thesecondtermisthersttermtimesaratio,andthedenominateoftheratioisthelargervaluebetweenthetwo,whichisalwaysthesame,whilethenumeratoristheabsolutevalueofthedifferenceoftwonumbersandwillnotchange.Thusthesecondtermissymmetric.ThethirdtermisaratioasdenedinEquation(1),thenumeratoristhedotoftwovectors,whichiscommutable,andthedenominatorwillalwaysreturnthesamevaluenomatterwhichvectorcomesrst.ThereforethedistancefunctiondenedinEquation(2)issymmetric. Afterdeningsuchadistancefunction,ourgoalistondclustersoftrajectoriesintermsofTra=ftra1,...,trang,wherendenotesthenumberoftrajectoriesinacluster,andarepresentativetrajectoryiforeachclusterwhichcansolvetheproblemargminij6=iXjgeoDist(trai,traj). 4.4.3.3SemanticSimilarityNowwediscussthesemanticsimilaritybetweentrajectories.Weadoptthewellknownlongestcommonsubsequencealgorithmtondthecommonpartoftwo 137

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trajectories.Similarmethodhasappearedin[ 84 ][ 92 ].However,[ 92 ]denesthesemanticsimilaritybetweentwotrajectoriesastworatios,i.e.,thesimilarityofthersttrajectorytothesecondtrajectoryisdifferentfromthesimilarityofthesecondtrajectorytothersttrajectory.Weconsiderthatthesemanticsimilarityshouldbesymmetric,whichisinaccordancetothegeographicsimilarity.Wedeneasemanticratioasfollows. Denition4.4.3.2(Semanticsimilarity). Thesemanticratiobetweentwotrajectoriesmeasuresadegreeofsemanticsimilaritybetweenthem,andisdenedassemRatio(tra1,tra2)=LCSS(tra1,tra2) min(jtra1j,jtra2j) (4)whereLCSS(tra1,tra2)isdenedby LCSS(tra1,tra2)=8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:0,ifjtra1j=0,orjtra2j=0max(LCSS(H(tra1),H(tra2))+1,LCSS(H(tra1),tra2),LCSS(tra1,H(tra2)),otherwise(4)whereH(tra1)denotetheheadoftra1,andjtra1jdenotethenumberofGPSpointsintra1.NoticethatherewedifferentiateL1-normtoL2-normwehaveusedinprevioussections.WeusejtrajtorepresentthenumberofGPSpointsintra,howeverweusektraktodenotethelengthofthetrajectory.WeuseH(tra)todenotetheheadoftra,whichisthesub-trajectorypriortothevisitingofthecurrentpoint.Forexample,assumethesemanticpartoftra1isWork,Food,Shop,Entertainment,theheadwillbeWork, 138

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Food,Shop.WeusethewellknowndynamicprogrammingapproachtocalculatetheLCSSoftwotrajectories,asshowninFigure 4-17 AlgorithmRevisedlongestcommonsubsequence Input:twosemantictrajectoriestra1,tra2Output:thesemanticsimilarityratiobetweentra1andtra21m jtra1j,n jtra2j,2InitializeM[m][n]//matrixtostoretheLCSS3fori 1tom4M[i][0] 05forj 1ton6M[0][j] 07fori 1tom8forj 1ton9max len =M[i)]TJ /F4 11.955 Tf 11.96 0 Td[(1][j)]TJ /F4 11.955 Tf 11.96 0 Td[(1]+110ifmax lenM[i][j)]TJ /F4 11.955 Tf 11.95 0 Td[(1]12max len M[i)]TJ /F4 11.955 Tf 11.95 0 Td[(1][j]13elseifmax len
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andthedenominatoralwaysreturntheminimumlengthbetweenthetwo,thenthesemRatioisalwayssymmetric. Fromtheabovealgorithmweareabletoobtainthesimilaritymeasurementbetweenthesemanticpartoftrajectories.AswehavealreadyknownhowtomeasurethegeographicdistanceinSection 4.4.3.2 ,thenextimportantthingis:howcanwecombinebothmeasurementtogether?IfwetakealookatDenition 4.4.3.1 ,wemayndthatthisisameasurementofthelength.However,whenwelookintoDenition 4.4.3.2 weobservethatitisaratiobetween[0,1].Anideaistotakethesecondmeasurementasamultiplicator.However,thereisanotherdifferencebetweenthesetwomeasurementsthattheformerisadistancefunction,i.e.,thehigherthedistance,thelessthesimilarity.Howeverthelattershowsahigherscorewhentwotrajectoriesaresimilar.Thusweshoulddividethesecondfactorfromtherst.Thereforewegivethefollowingdenition. Denition4.4.3.3. Thetotaldistanceoftwotrajectoriestra1andtra2measuresthesimilaritybetweenthemconsideringbothgeographicandsementicfeatures,andisdenedastotalDist(tra1,tra2)=geoDist(tra1,tra2)1 1+semRatio(tra1,tra2) (4)where01Hereweintroduceaparametertoadjusttheportionthathowmuchthesemanticfeaturescanaffectthesimilarity.Asincreases,thehighertheportionis,asitwillshrinkthevalueofthenaldistanceandmaketwotrajectoriesmoresimilar.Ifwesetaszero,thedistanceismeasuredmerelybasedongeographicdistance.Therefore,ourgoalbecomestondrepresentativetrajectoriesthatcanminimizethesumofthenaldistancesamongalltrajectoriesinacluster,i.e. 140

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argminij6=iXjtotalDist(trai,traj)WeshowthatthedenitioninEquation(5)issymmetricbythefollowingtheorem. Theorem4.4.3.1. ThetotaldistancefunctionmeasuringthesimilaritybetweentwotrajectoriesinEquation(5)issymmetric,i.e.,totalDist(tra1,tra2)=totalDist(tra2,tra1) Proof. FromLemma 4.4.3.1 andLemma 4.4.3.2 wehaveprovedthatthegeographicdistanceandthesemanticratioarebothsymmetric,thereforeinEquation(5)wewillalwaysgetthesamevaluenomatterwhichtrajectorycomesrst. 141

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CHAPTER5IMPLEMENTATIONOFMOVINGOBJECTSWITHUNCERTAINTYAfterintroducingtheuncertaintyofmovingobjectsatanabstractlevelinChapter 3 ,andtheimplementationconceptandalgorithmintroducedinChapter4,wenowexploreindetailshowtheuncertainmovingobjectmodelscanbeintegratedinourmovingobjectdatabases.Therearetwomaintasksinvolvedintheintegrationprocess,theimplementationofthedatatypesofuncertainmovingobjects,theoperationsandpredicatesimplementedasfunctionsofthesedatatypes,andtheintegrationofoperationsandpredicatesinSQLlanguagewhichcanbesupportedbycommercialdatabaseslikeOracle.Wedescribethersttask,i.e.,theimplementationofuncertainmovingobjectdatatypesandfunctionsinthischapter.ThenwewillshowqueryinguncertainmovingobjectsinSQLlanguagesinourmovingdatabasesandtheextensiveexperimentsinChapter 6 .InSection 5.1 werstreviewanoveldatatypecallediBLOB(intelligentbinarylargeobject)designedbyresearchersinrecentdayswhichisimplementedindatabasesystemsandcanhandlecomplexspatialobjectsandsupportoperationssuchasretrieval,insertandupdatefunctionsinanefcientmanner.Thenwedescribeauser-friendlygeneralframeworktocaptureandvalidatethestructureofapplicationobjectsontopofiBLOB,sothatuserscanbuildtheirdatatypeseasierandfaster.InSection 5.2 wedescribehowouruncertainmovingobjectsareimplementedusingiBLOBdatatypesandTSSframework. 5.1ReviewofANovelApproachonImplementingComplexSpatialDataTypesSincemovingobjectdatacontainsspatio-temporalinformationandareoftencomplexinstructuresandlargeinsizes.AswehaveintroducedinChapter 4 ,weusetheslicestructuretorepresenttheapproximationofmovingobjects,itisimportantthathowtheseslicesareconstructedandstoredinrealcomputeranddatabasesystems.Achallengingproblemishowtohandlelargeandcomplexdatacorrectlyandefciently.Inthischapter,werstdescribeagenericframeworkthatisbenecialtothe 142

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implementationoflarge,structuredandcomplexobjects,forexample,movingobjectsinourcontext.Inthischapter,werstreviewanovelapproachnamedintelligentbinarylargeobject(iBLOB)proposedin[ 6 ]inSection 5.1.1 .ThenwedescribeageneralizedconceptualframeworktocaptureandvalidatethestructureofapplicationobjectsbymeansofatypestructurespecicationinSection 5.1.2 5.1.1IBLOB:StoreComplexSpatialObjectsUsingIntelligentLargeObjectsInrecentyears,manynewemergingapplicationsincludingbio-informatics,multimedia,andgeospatialtechnologieshavenecessitatedthehandlingofcomplexapplicationobjectsthatarehighlystructured,large,andofvariablelengths,forexample,biologicalsequencedata(DNA)andsatelliteimages.CurrentapproachesoftenhandlethesecomplexdatausinglesystemformatssuchasHDF,NetCDF,XML.However,someoftheseapproachesareveryapplicationspecicanddonotprovideproperlevelsofdataabstractionfortheusers.Binarylargeobjects(BLOBs)aretheonlymeanstostoresuchobjects.However,BLOBsrepresentthemaslow-level,binarystringsanddonotpreservetheirstructure.Anovelapproachnamedintelligentbinarylargeobject(iBLOB)whichcansolvealltheseproblemshasbeenproposedin[ 6 ].IBLOBisinventedfortheefcientandhigh-levelstorage,retrieval,andupdateofhierarchicallystructuredcomplexobjectsindatabases.ThestructurestorescomplexobjectsbyutilizingtheunstructuredstoragecapabilitiesofDBMSandprovidecomponent-wiseaccess.Inthissense,theyserveasacommunicationbridgebetweenthehigh-levelabstracttypesystemandthelow-levelbinarystorage.Thisframeworkisbasedontwoorthogonalconceptscalledstructuredindexandsequenceindex.AstructuredindexfacilitatesthepreservationofthestructuralcompositionofapplicationobjectsinunstructuredBLOBstorage.AsequenceindexisamechanismthatpermitsfullsupportofrandomupdatesinaBLOBenvironment.Figure 5-1 showsacomplexregionexample,wheretheregionconsistsofthreeconnectedcomponent(face),andeachfacemayormaynothavecycles.Applications 143

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thatdealwithregionsmightbeinterestedinnumericoperationsthatcomputethearea,theperimeteraswellasgeometricoperationssuchasintersection,union,anddifferenceoftworegions.Theimplementationoftheseoperationsrequireseasyaccesstounitcomponentsofthecomplexregionobjectsuchasfacesorsegments.IfwewanttostoreandquerysuchobjectsintraditionalDBMS,weeitherusethelayeredarchitectureasshowninFigure 5-2 a,whichaddsalotofoverheadinthemiddlewareandcannotbenetfromtheadvantageofDBMSitself.orusethebuilt-indatatypeBLOB(binarylargeobject)tostorethecomplexregionasshowninFigure 5-2 b.TheproblemofusingBLOBisthattheentireBLOBhastobeloadedintomainmemoryeachtimeforprocessingpurposes.WecannotretrieveasinglecomponentwithoutloadingtheentireBLOBintomainmemory.TheiBLOBstructureisshowninFigure 5-2 c.Thismethodproposesagenericstoragestructure,theimplementationofwhichisbasedontheBLOBtypeandwhichmaintainshierarchicalinformation.ItisintelligentbecauseunlikeBLOBs,itunderstandsthestructureoftheobjectstoredandsupportsfastaccess,insertionandupdatetocomponentsatanylevelintheobjecthierarchy.OntopoftheiBLOBstructure,thetypestructurespecication(TSS)intheframeworkprovidesanabstractviewoftheapplicationobjectswhichhidestheimplementationdetailsoftheunderlyingcomplexdatatypes.IBLOBensuresagenericstoragesolutionforanykindsofstructuredapplicationobjects,andenabletheimplementationofthehigh-levelinterfacesprovidedbythetypestructurespecication.Therefore,thetypestructurespecicationandiBLOBtogetherenableaneasyimplementationforabstractdatatypes.TheiBLOBframeworkconsistsoftwomainsectionscalledthestructureindexandthesequenceindex.Thestructureindexpreservesthephysicalstructureofapplicationobjectsinunstructuredstoragespace,asshowninFigure 5-3 .Thesequenceindexpreservesthelogicalstructureofapplicationobjects,asshowninFigure 5-4 .Thestructindexisusedasareferencetoaccessthehierarchyofthedata.Themechanismisnotintendedtoenforceconstraintsonthedatawithinit;thus,ithas 144

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Figure5-1. Aregionobjectasanexampleofacomplex,structuredapplicationobject Figure5-2. Thelayeredarchitecture,theintegratedarchitectureandtheiBLOBsolution noknowledgeofthesemanticsofthedatauponwhichitisimposed.Thisconceptconsidershierarchicallystructuredobjectsasconsistingofanumberofvariable-lengthsub-objectswhereeachsub-objectcaneitherbeastructuredobjectorabaseobject.Withineachstructuredobject,itssub-objectsresideinsequentiallynumberedslots.Theleavesofthestructurehierarchycontainbaseobjects.AnexampleofthehierarchyisshowninFigure 5-5 .IfweusethetraditionalBLOBobjecttostorethecomplexregion,wehavetoloadandsequentiallytraversetheentireBLOBuntilthedesiredfacewouldbefound.Incontrast,thehierarchyenablesustorecursivelyndthecomponentbygoingdowntotheleavesfromtherootwithoutloadingtheentireobject. 145

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Figure5-3. Thestructureindexesinsidearegionwhichconsistsofnfacesandnoffsets Figure5-4. AnexampleofsequenceindexesinsideaniBLOB Structuredlargeobjectsrequiretheabilitytoupdatesub-objectswithinastructure.Specically,theyrequirerandomupdateswhichincludeinsertion,deletionandtheabilitytoreplacedatawithnewdataofarbitrarysize.Givenalargeregionobject,updatingitforachangeinaface,acycleorsegmentbecomesverycostlywhenstoredinBLOBs.Thus,itisdesirabletoupdateonlythepartofthestructurethatneedsupdating.Introducingthesequenceindexconceptenablestherandomreadanddataappendoperationsmoreefcient.TheideaistophysicallystorenewdataattheendofaBLOBandprovideanindexthatpreservesthelogicallycorrectorderofdata.Besidesstructureindex,asequenceindexconceptispresented,whichisbasedontherandomreadanddataappendoperationssupportedbyBLOBsExtracapabilitiesprovidedbyhigherlevelBLOBsareafurtherimprovementandserveforoptimizationpurposes.ThesequenceindexconceptisbasedontheideaofphysicallystoringnewdataattheendofaBLOBandprovidinganindexthatpreservesthelogicallycorrectorderofdata.Exampleofinsertion,deletionandupdatingwithsequenceindexisshowninFigure 5-6 a,bandcrespectively. 146

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Figure5-5. ThehierarchicaliBLOBrepresentationofaregionobject (a) (b) (b)Figure5-6. ThehierarchicaliBLOBrepresentationofaregionobject Inthenextsubsection,wewillbrieyreviewhowiBLOBcanbeusedtoconstructcomplexdatatypeswiththehelpoftheTSSframework. 5.1.2TypeStructureSpecication(TSS)WehavereviewedtheiBLOBstructurewhichcanpreservethehierarchyofacomplexobject,nowwedescribehowtoconnectthisstructuretotheapplicationleveldatatypes.Thehierarchyofastructuredobjectcanalwaysberepresentedasatree.Inthetreerepresentation,therootnoderepresentsthestructuredobjectitself,andeachchildnoderepresentsasub-object.Asub-objectcanfurtherhavea 147

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structure,whichisrepresentedinasub-treerootedwiththatsub-objectnode.Inatreerepresentation,eachleafnodeisabaseobjectwhileinternalnodesrepresentstructuredobjects.Atreerepresentationisausefultooltodescribehierarchicalinformationataconceptuallevel.However,togiveamoreprecisedescriptionandtomakeitunderstandabletocomputers,aformalspecicationwouldbemoreappropriate.Thetypestructurespecicationprovidesanalternativeofthetreerepresentationfordescribingthehierarchicalstructureofapplicationobjects.Structureexpressionsdenethehierarchyofastructuredobject.Astructureexpressioniscomposedofstructuretags(TAGs)andstructuretaglists(TAGLISTs).Astructuretag(TAG)providesthedeclarationforasinglecomponentofastructuredobject,whereasastructuretaglist(TAGLIST)providesthedeclarationforalistofcomponentsthathavethesamestructure.Thefollowingshowsthegrammarofthestructureexpression. Table5-1. GrammarOfTSS TerminalSetSf:=,h,i,,[,],SO,BO,:gExpression::=TAG:=hTAGjTAGLISTi+;TAGLIST::=TAG[]TAG::=hNAME:TYPEiTYPE::=hSOjBOiNAME::=IDENTIFIER Withstructureexpressions,thetypesystemimplementercanrecursivelydenethestructureofstructuredsub-objectsuntilnostructuredsub-objectsareleftundened.Alistofstructureexpressionsthenformsaspecication.AdenitionoftheregiondatatypeusingTSScanbeshowninTable 5.1.2 Table5-2. TSSGrammarRepresentationofRegion hregion:SOi:=hregionLabel:BOihface:SOi[];hface:SOi:=hfaceLabel:BOihouterCycle:SOihholeCycle:SOi[];houterCycle:SOi:=hsegment:BOi[];hholeCycle:SOi:=hsegment:BOi[]; 148

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5.2ImplementationofUncertainMovingObjectDataTypesusingIBLOBandTSSAfterreviewingtheiBLOBandTSSconceptintheprevioussection,wenowintroducehowtousethemtoimplementourownuncertainmovingobjectdatatypes.Section 5.2.1 presentstheimplementationofthependantmodel,andSection 5.2.2 presentstheimplementationoftheballoonmodel. 5.2.1ImplementationofthePendantModelRecallthatwehaveintroducedtheslicerepresentationofourPendantmodelinSection 4.1.2 ,asillustratedinFigure 5-7 Figure5-7. Sliceunitrepresentationofmovingobjectswithuncertainty Basedontheslicerepresentation,thetreestructureoftheuncertainmovingpointinthependantmodelisshowninFigure 5-9 .Anuncertainmovingpointconsistsofanumberofunits,eithercertain(string)oruncertain(pendant).Herewenamethecomposedunitpendant,nomatterifitiscertain(actuallystring)oruncertain.TheisCertaineldwillindicatethis.Ifitistrue,thenweconstructastringunit.Ifnot,weconstructapendantunit.Endpointsaretheobservationsofmovingpoint.Theyareoftenrepresentedby(t,x,y)GPSpointsintheapplications.Forexample,theobservedlocationsofhurricanecentersfromNHC,ortheobservedtrajectorypointsofacar.Animportantpropertymaxvelocitywillbestoredhere.Maxvelocityisnotglobalinformation.Indifferentintervals,maxvelocitycanvary.Iftheunitisuncertain,thenthis 149

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Figure5-8. TSSDesignofUncertainMovingPointsinthePendantModel indicatesthemaximumvelocityofthemovingpointatthisinterval,andcanbehelpedtocalculatetheuncertainvolume(cone)andtheprojectionuncertainarea(ellipse)forfurtheroperations.Ifitiscertain,thanmaxvelocityisjustageneralpropertyofthemovingobject.Wecantreatpoint2Dasabaseobject,orwecanalsotreatitasastructuralobject,andletxcoordinateandycoordinatebebaseobjects.WetransferittotheTSSgrammarexpression,asshowninTable 5.2.1 Table5-3. TSSGrammarRepresentationofUncertainMovingPoints umpoint::=pendant+lifetimeboundingBoxnumOfPendants:Ipendant::=endpoint:poi2D+intervalboundingBoxmaxVelocity:DisCertain:Blifetime::=interval+boundingBox::=lbpt:poi2Drtpt:poi2Dinterval::=startTime:DendTime:DopenStatus:I ThedifferencebetweenuncertainmovingregionwiththeuncertainmovingpointisthatthecomposedunitofumregionareendRegionsinsteadofendPoints.AnendRegionisanobservedregionofthismovingregionataparticulartime.Inthismodel,wewanttondallpossiblelocationsofamovingregion,thereforeweareonlyinterestedintheoutercycleofmovingregions,anddonotconsiderthecasethataregionmayhaveholes. 150

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Figure5-9. TSSDesignofUncertainMovingRegionsinthePendantModel Table5-4. TSSGrammarRepresentationofUncertainMovingRegions umregion::=pendant+lifetimeboundingBoxnumOfPendants:Ipendant::=endregion+intervalboundingboxmaxVelocity:DisCertain:Blifetime::=interval+boundingBox::=leftBottom:poi2DrightTop:poi2Dinterval::=startTime:DendTime:DopenStatus:Iendregion::=outercycleouterCycle::=segment:poi2D+boundingBoxnumberOfOSegment:I 5.2.2ImplementationoftheBalloonModelInthissection,wepresenttheimplementationoftheballoon prdatatype,whichcapturesthemostwidelyseennatureinuncertainmovingobjects.Thehistoricalpartofasuchkindofobjectisamovingpoint,whileitevolvesamovingregionastheuncertaintygrowsinthefuture.ThetreestructurerepresentationofsuchanobjectisseeninFigure 5-10 .TheTSSgrammarspecicationisshowninTable 5-5 .Inthenextsection,wewillshowtheperformanceresultsoftheimplementationoftheuncertainmovingobjectdatatypesthroughthequeries. 151

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Figure5-10. TSSDesignofUncertainMovingObjectintheBalloonModel Table5-5. TSSGrammarRepresentationofBalloon prObjects balloon pr::=slice+lifetimeboundingboxnumOfPendants:Islice::=endPoint+uncertainRegion+intervalboundingboxmaxVelocity:DisCertain:Blifetime::=interval+boundingBox::=leftBottom:poi2DrightTop:poi2Dendpoint::=poi2DuncertainRegion::=outercycleinterval::=startTime:DendTime:DopenStatus:IouterCycle::=segment:poi2D+boundingboxnumberOfOSegment:I 152

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CHAPTER6SYSTEM,QUERIESANDEXPERIMENTSInthischapter,weintroducethesystemsimplementedforuncertainmovingobjects,theintegrationofSQLqueriesintoourmovingobjectdatabasesandextensiveexperimentsofthesequeries. 6.1QueryingHistoricalMovingObjects:CardinalDirectionDevelopmentfromDataofNationalHurricaneCenter(NHC)Inthissection,weimplementthealgorithmsofcomputingthecardinaldirectiondevelopmentsbetweentwomovingpoints,whichhasbeenproposedinSection 4.2.1 ,andperformsomeexperimentstoevaluateseveralimportantqueriesoncardinaldirectiondevelopments.OurexperimentsareperformedonrealworldhistoricalhurricanedatafromNationalHurricaneCenter(NHC)ofyear2005.AlltheoriginaldatacanbedownloadedfromthewebsiteofNHC[ 32 ]. 6.1.1EnvironmentThehurricanedataarerecordedinASCIIformatandstoredin.HURDATles,representingHurricaneBestTrackFiles.Eachdatasourceleisgeneratedbysensorsontheground,whichkeeptrackingaparticularhurricaneeverysixhoursperday,i.e.at00:00,06:00,12:00and18:00UTCtimerespectively.Thenameofaparticularhurricaneisincludedintheheaderofan.HURDATle,followedbytheentriesofrealhurricanedata.Eachdataentryinthelecontainsthe6hourlycenterlocationscomposedbylatitudeandlongitudeintenthsofdegrees,andintensitiesincludingmaximum1-minutesurfacewindspeedsandminimumcentralpressures.Sinceweareonlyinterestedinqueryingthechangeofcardinaldirections,wejuststorethespatio-temporalattributes,e.g.,latitude,longitudeandtime,withoutwindspeedorairpressure.Therefore,weareabletoconstructourmovingpointdatatypebasedonthespatio-temporalrelateddataintheoriginalle,wherethelatitudecorrespondstoanxvalueandthelongitudecorrespondtoayvalueatthetimeinstancestinourcoordinatesystem. 153

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Figure6-1. ThetrajectoriesofhurricanesPHILIPPEANDRITA. AccordingtotheNHChurricanedatale,wedetectthetrajectoriesof28hurricanesthathavebeenactiveonAtlanticOceanintheyear2005,andstorethemintothetabletest movingwiththefollowingdesign, test_moving(id:integer,name:string,track:mpoint)Thetrackattributerepresentsthetrajectoryofahurricanecenterandisstoredasacomplexdatatypemovingpoint,whichisspatio-temporaldatatypewedenedontopoftheprimitivedatatypessupportedbyOracledatabase.Thenweimplementthealgorithmofcomputingthecardinaldirectiondevelopmentbetweentwomovingpointsasafunction.Therefore,weabletoanswerthreetypesofimportantqueries,whicharedescribedinthefollowingthreesubsections. 6.1.2EntireCardinalDirectionDevelopmentQueryForthersttypeofqueries,giventwomovingpoints,weareabletondtheentirecardinaldirectiondevelopmentbetweenthem.Forexample,weareinterestedinthequery,Q1:FindthecardinaldirectiondevelopmentbetweenPHILIPPEandRITA?Intheabovequery,RITAisthereferencemovingpointandPHILLIPPEisthetargetmovingpoint.Wehavetheirtrajectoriesstoredinthedatabaseinmpointformatintabletest moving.Then,wewritethefollowingSQLquerytondtheanswer: SELECTm1.name,m2.name,mdir(m1.track,m2.track), 154

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FROMtest_movingm1,test_movingm2WHEREm1.name=`PHILIPPE'ANDm2.name=`RITA';Whenwesubmitthequery,thedatabasewillrstretrievethetrajectoriesofPHILLIPEandRITAfromtabletest moving.Whenwegetthetrajectorydata,itisrstconvertedthedatatoKMLlesothatthetrajectoriescanbeseenonthemapusingGoogleMapscAPI,asshowninFigure 6-1 .ThebluelinepassingtheGulfofMexicorepresentsthetrajectoryofPHILLIPPE,andtheredlinepassingtheAtlanticOceanrepresentsthetrajectoryofRITA.However,fromthediagram,wecanonlyndaroughrelativepositionbetweenthesetwohurricanesduringtheirlifetimesbutcannottelltheexactcardinaldirectiondevelopment.Sincethemdirfunctionimplementsouralgorithminterval syncandcompute dir devproposedinSection 4.2.1 ,itisabletoreturnanorderedlistcontainingallcardinaldirectionsinconsecutivetimeintervals.Afterwesubmitthequery,weobtainthefollowingresult, NAMENAME----------------------------------------MDIR(M1.TRACK,M2.TRACK)-------------------------------------------------------------PHILIPPERITA->undefined[2005091712,2005091800)->NW[2005091800,2005092212)->W[2005092212,2005092212]->SW(2005092212,2005092402)->W[2005092402,2005092402]->NW(2005092402,2005092406)->undefined[2005092406,2005092606)Theresultisalistofcardinaldirectionsintimeintervalswithascendingorder.Theentirelifetimeofthetwohurricanesisfrom2005-09-1712:00:00to2005-09-2606:00:00UTC.Therstcardinaldirectionbetweenthemisundened,whichmeansthatonlyoneofthehurricanesexistinthetimeinterval[2005-09-1712:00:00,2005-09-1800:00:00).Thenthecardinaldirectionchangestonorthwestinthenexttimeinterval 155

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[2005-09-1800:00:00,2005-09-2212:00:00),etc.Thus,thecardinaldirectiondevelopmentbetweenPHILIPPEandRITAcanberepresentedas,DEV(PHILIPPE,RITA)=?NWWSWWNW? 6.1.3ExistentialQueryThesecondtypeofqueriesistheexistentialpredicate.Forexample,aninterestingquerycanbe,Q2:FindallhurricanesthathaveexistedtotheSWofhurricaneDELTA?Wewritethefollowingquery, SELECTm1.name,m2.namefromtest_movingm1,test_movingm2whereexistsw(m1.track,m2.track)=1ANDm1.name=`DELTA'Afterwesubmitthequery,wegettheanswer, NAMENAME----------------------------------------DELTAGAMMADELTAEPSILONFromtheresultwecanndthattherearetwohurricanesGAMMAandEPSILONrespectively,whichhaveexistedtothesouthwestofthehurricaneDELTA. 6.1.4Top-kQueryThethirdimportantquerywesupportinthesystemisthetop-kquery.Giventwohurricanes,weareabletondthetop-kcardinaldirectionsbetweenthem.Forexample,weareinterestedinthequery,Q3:Findtop3cardinaldirectionsbetweenMARIAwithothertracksWewritethefollowingdatabasequery, SELECTm1.name,m2.name,topKDir(m1.track,m2.track,3)FROMtest_movingm1,test_movingm2WHEREm1.name=`MARIA'ANDm1.name<>m2.name 156

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ANDtopKDir(m1.track,m2.track,3)<>`'ThetopKDir(m1.track,m2.track,3)<>`'expressionshowsthatwewanttondthetop3cardinaldirectionsbetweenthegivenhurricaneandotherhurricanesexcludingtheundeneddirection.Afterwesubmitthequery,wegetthefollowinganswer, NAMENAME----------------------------------------TOPKDIR(M1.TRACK,M2.TRACK,3)-----------------------------------------MARIALEENW(0.137179)NE(0.00128205)N(0.001)MARIANATESW(0.571429)MARIAOPHELIASW(0.348837)Fromtheresultwecandetectthreeotherhurricanesthathavenon-emptycardinaldirectionswithrespecttoMARIA.ThetopthreedirectionbetweenMARIAandLEEareNW,NEandN,withtheprobabilityof13.71%,0.128%and0.1%respectively.Theremainingprobabilityisassociatedtotheundeneddirection,thusitisnotshown.TheresultofMARIAandNATEonlycontainsonecardinaldirectionSW,withtheprobabilityof57.14%,sinceitistheonlynon-emptycardinaldirectionexistingbetweenthesetwohurricanes.ThesameexplanationcanbemadetotheresultbetweenMARIAandOPHELIA. 157

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Figure6-2. Overviewofthesystemframework 6.2ARouteDiscoverySystemforPredictiveMovingObjects 6.2.1OverviewoftheSystemTheframeworkofthesystemisshowninFigure 6-2 .Thesystemconsistsoftwocomponents:graphconstructionandtripplanning.Thegraphconstructionpartisanofinemodule.Havingalargedatasetofrawgeo-taggedphotosandtravelers'tipsasinput,werstgroupthembyusersandgetasetofuncertaintytrajectories.Thenwepartitionthecity'sspaceintoasetofdisjointgrids,andindexthetrajectories.Asetofconnectedgridcellswillformaconnectedregion.Edgesconnectingcellswillbeinferred.Eachedgecarriessomeimportantinformation,suchasthemovingdirection,itssupport(numberoftrajectoriestraversedthisedge)andthetransitiontime.Theregionsandedgesformaroutablegraphthatwillbestored.Thesecondstageisperformedonline.WhenauserinputsasetofPOIsandatimespan,thesystemwillndtop-kroughroutesonthebasisoftheroutablegraph.Aroughroutecontainingalistofgridswillberstgenerated.Intheendwerenetheroutesandgeneratethetripplan.Thetripplanningsystemisbuiltasawebsite.Userscouldsubmitaqueryonlineandgettheresponsequicklyinlessthanonesecond.TheuserinterfaceofthesystemisshowninFigure 6-3 a.HereweapplythetripplanningsystemtotheareaoftheNew 158

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Figure6-3. Userinterface YorkCity.Therightsideofthewebsiteshowsthemenuofthesystem(Figure 6-3 c).Beforesubmittingqueries,parametersshouldbesetproperly.ThemeaningoftheseparameterswillbeintroducedinSectionII.Ausercouldinteractwiththesystembyperformingactionsonthemap.Thereddotsonthemaprepresenttop-100pointsofinterest(wehavemuchmorePOIsinthedatabase).WhentheusermovesthemouseonaPOI,thedetailswillbedisplayed,includingitsname,addressandaphoto.ThetravelercanchoosethisPOIasoneofhis/herdestinationsbyclickSelectthisPOI(Figure 6-3 b).Ausercanalsoarbitrarilychooseanon-POIlocationonthemapbyright-clickingtheposition.AusercanselectuptofourPOIs(Figure 6-3 d),andclicktheQuerybuttonunderRouting(Figure 6-3 c).Withinasecond,thesystemwillalertthetravelerwhethertop-kroutesarefound.Theusercanvieweachofthetop-kpaths.Thetop-1pathofthequeryisshowninFigure 6-3 e.Thebluesegmentsshowtherealtrajectorysegmentsfromtherawdata.Theblackdashlinesaretheinferredlinesfromtheroutingalgorithm 6.2.2SystemImplementationTogeneratetrajectoriesoftravelers,wemineallusers'tipsdata(atipisashortmessagedescribingtheuser'sexperienceataPOI,likeIamattheApplestoreanditissocrowded!)inNewYorkCityfromFoursquarecbetweenMay.2008andJun. 159

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2011.Atrajectoryisdetectedbyasequenceofcheck-insperdayperuser.Theusershouldcheckinatleast3timesperdayinordertoformatrajectory.Thus,userswhoonlycheckinrandomlyareconsideredasnoiseandareremoved.WealsocollecttaxitrajectoriesinBeijingwiththehelpofGPSsensorsembeddedineachtaxi.TherawdatawecollectaresummarizedinTable 6-1 Table6-1. SummaryofDatasets DataSocialNetworkDataTaxiData POIs206,194 Users49,0233,531 Check-insequences425,5582,989,165 Trajectories73,08815,098 Contributedusers10,3373,531 AtrajectorywithmorenumbersofPOIspotentiallyhasmoreknowledgethanatrajectorywithlessnumberofPOIs.ThusthetrajectorieswithmorePOIsareconsideredtohavegoodquality.Amongalltheuncertaintytrajectorieswedetected,mostofthem(8089)havelessthan5POIs,1750trajectoriescontains6to10POIs(Figure 6-4 a).Alargenumberoftravelers(5323)contributeonlyonetrajectory(Figure 6-4 b).Therefore,mostofthedatacomesfromalargenumberofdifferentusers,whichreectstherealworld.Withtheuncertaintrajectorydataset,webuildtheroutablegraph(Figure 6-5 a),withgridlength200meters,=0.3,andt=1houronrawtrajectories.Differentcolorsrepresentdifferentregions.TheinternaledgesandexternaledgesareshowninFigure 6-5 b. 6.2.3ExperimentalEvaluationTomeasuretheeffectandefciencyofourapproach,weapplyittolargerealdatasetsandstudytheperformance.WechoosethetaxitrajectoriesinBeijingbecause:1)Ataxitrajectorycontainsalargenumberofpoints,thuswecansetarawtaxitrajectoryasagroundtruthandsampledtrajectoriesasuncertaintrajectories;2)takingtaxisisalsoanimportantwaywhenatravelermakesatriptoanewcity.Wechoosethedatasetwhichcontains15,098rawtrajectoriesfrom3531users.Werstndthe 160

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(a)(b)Figure6-4. Trajectoryqualityanduser'scontribution (a)(b)Figure6-5. RoutablegraphconstructionontheareaofNYC groundtruth.Given2to4querylocations,weselectrawtrajectoriesthathavetraversedthesequerylocationsandrankthem.Trajectorieswhichtraversemoresegmentsthataretraversedmorefrequentlywillreceivehigherranking.Wechoosethetop-1rawtrajectoryasthegroundtruth.Weintroduceameasurementcalledlength-normalizeddynamictimewarpingdistance(NDTW)betweentwotrajectories,whichismodiedfromdynamictimewarpingdistance(DTW),NDTW(tra1,tra2)=DTW(tra1,tra2) length(tra1) 161

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LowerNDTWindicatesmorepreciselyinferredroutes.WecomparetheNDTWofourmininguncertaintytrajectoriesapproach(denotedbyMUT)withtheMPRapproachin[ 10 ]ontheinferredrouteswithsamplingratesof3and5minutesrespectively.Thedistancebetweentwoquerypointsaredeterminedbyti.e.,thetransitiontimebetweenthetwoquerypoints.Theexperimentresult(Figure 6-6 a)showsthatourapproachwillndmorepreciseroutes.Wealsoevaluatethequerytime(Figure 6-6 b)whenthenumbersofquerypointsare2,3and4respectively. (a)(b)Figure6-6. EffectandeffeciencyoftheMiningUncertaintyTrajectory(MUT)algorithm 6.3BalloonSystem:QueryHistoricalandPredictiveMovingObjectsInthissection,weintroducethesystemofourballoonmodel.Section 6.3.1 showsexamplequerieswesupport.Section 6.3.2 showsthedemosystemofthequeryresults.Section 6.3.3 showstheexperimentresultsoftheprediction.Andintheend,Section 6.3.3.4 discussestheresults. 6.3.1SupportofSpatio-temporalUncertainQueriesWeimplementedsomeimportantspatio-temporaluncertainoperationsandpredicatesintheballoonmodelandintegratedintoOracledatabases.Assumewehavethefollowingschemasofmovingobjects. hurricanes(id:varchar(100),name:varchar(100),eye:balloon_pr)states(name:char(2),extent:region) 162

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Twoexamplesofqueriesthesystemcansupportisshownasfollows,Q1:WhatareawillpotentiallybeaffectedbytheeyeofhurricaneKatrinaat12hoursfromnow? SELECTatinstant(future_proj(eye)),now()+12h)FROMhurricanesWHEREname='Katrina';Q2:FindallstatesthatwillpossiblytraversedbyKatrinabetween25Aug2007and27Aug2007. SELECTs.nameFROMhurricanesh,statessWHEREpossibly_enter(h.eye,s.extent,interval(25-AUG-2005,27-AUG-2005)Andh.name='Katrina';Q1isthequeryfortheuncertainarea.Q2isanextensionofQ1,whichisaspatio-temporalquery.Q1isanimportantquerysincetheresultsarehighlydependentonthepredictionmodel.Inthefollowingpart,wewillshowsomeexperimentresultsaboutQ1. 6.3.2DemoSystemofHistoricalandPredictiveQueriesInthissection,weshowthedemonstrationsystemwehaveimplementedforourhistoricalandpredictivequeriesonmovingobjects.Figure 6-7 illustratestheuserinterfaceofthesystem.ThesystemisimplementedusingASP.NETandBingTMMapsAPI.Therearethreecomponents:usermenu,map,andresultpanel.Theleftmostpartistheusermenu,whereuserscaninputandsubmitqueries.Themiddlepartisthemapvisualization,whereweuseBingTMMapsAPItoshowthequeryresults.Therightmostpartshowstext-basedresultslikethewindspeed,winddirection,etc.ThequerieswesupportarelistedinTable 6-2 .Figure 6-8 illustratessomeexamplequerieswesupported.Figure 6-8 ashowsthevisualizationofthetrajectoryquery.ThequeryasksthetrajectoryofhurricaneKatrina. 163

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Figure6-7. Ademosystemofspatio-temporaluncertainqueries Table6-2. Spatio-temporalqueriessupportedbythesystem TimeCategoryName HistoricalDomainandRangeTrajectoryBufferQueryQueriesInstantQueriesAtInstanceVelocityAtRelationshipsCardinaldirectionsSpatio-temporalpredicates PredictiveClassicationHurricanecategoriesQueriesMaxVelocityGreaterThanUncertainQueriesUncertainAreaSTUPsStreamingprediction Figure 6-8 bshowstheoutboundbufferquery.Thequeryasksthedangerousareawithin10kilometersaroundhurricaneKatrina.Ithasthesignature.hmpointreal!regionThemovementandthedangerousdistancearetheinputparameters,andapolygonshowingthebufferedregionistheresult.Figure 6-14 showstheresultsofthepredictionqueries.WerstadopttheMOSTmodel[ 73 ],wherethepredictionisperformedbyusinglinearfunctions.Thelinear 164

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(a)(b)Figure6-8. Visualizationofhistoricalqueries functionhasthefollowingassumption:thelocationofanobjectl(t)attimetiscalculatedby,l(t)=l(t0)+v0(t)]TJ /F5 11.955 Tf 11.96 0 Td[(t0)Wheret0isthelasttime-stampthatobjectoissuedandupdate,l(t0)isthelocationoftheobjectattimet0,andv0denotesthemostrecentvelocitythatisavailable,whichistreatedasaconstanthere.Bothlandv0ared-dimensionalvectors.Anupdateisnecessarywhenv0changes.Figure 6-14 ashowsthevisualizationresultofthe18-hourpredictionofHurricaneKatrinaatAug.25th2005.Figure 6-14 bshowsthepredictionat6hourslater.Asthecurrentvelocityisupdated,ourresultisalsoupdated.Figure 6-10 showsacontinuous72-hourstreamingpredictionwithupdatesofthemostcurrentlocation. 6.3.3ExperimentalStudyWeperformextensiveexperimentsonthequeryofuncertainareaprediction.Inthissection,weshowtheexperimentalresults. 165

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(a)(b)Figure6-9. 18-hourpredictionmadeatAug-25-200518:00:00and6hourslater Figure6-10. 72-hourstreamingpredictionwithupdatesevery6hours 6.3.3.1StatisticsoftheDatasetWerstshowsomestatisticsofourdatasets.WeusedtheNHCbesttrackdatafromyear1851to2010.SomestatisticsofthetrajectoriesareshowninTable 6.3.3.1 166

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Table6-3. StatisticsofDatasets FieldValue Totalnumberoftrajectories1446 Sizeof.txtformat1.1MB Sizeofxmlformat5.2MB SizeofiBLOBformat2.6MB Averageduration153hours Averagelength3698km 6.3.3.2AccuracyanalysisWeanalysistheaccuracyofthepredictionusingMOSTmodelintermsofhitrate,i.e.,iftheactuallocationisinsidetheuncertainareaofthepredictedlocation,itisconsideredasahit,otherwise,itisamiss.Incomparison,weimplementanotherpredictionapproach,thequadraticfunction,andcomparetheiraccuracy.DifferentfromthelinearfunctionintheMOSTmodel,thequadraticfunctionintroducesanacceleratorvariable.Itworksasfollows.l(t)=l(t0)+v0(t)]TJ /F5 11.955 Tf 11.96 0 Td[(t0)+1=2a0(t)]TJ /F5 11.955 Tf 11.96 0 Td[(t0)2Wherea0istheaccelerationvectorofv0.Thismethodcaptureslinearmotionasaspecialcase.Itcanhandletheproblemsofdirectionandspeedchanginginamovement.Wecomparethepredictionaccuracyfrombothmethods.TheresultareshowninFigure 6-11 .Wecanndthatthetwocurvesareveryclosetoeachother,andtheaccuracyisoptimal.Thisisbecauseweallowanuncertainareaintheprediction.Forexample,ifweproducethelocationofahurricaneintwodays,theerrorofseveralhundredskilometersistolerable.InordertondthereasonfortheinaccuratepredictionundertheMOSTmodel,weshowthe48-predictionresultofeachhurricanefromyear2005toyear2010inthebarcharofFigure 6-12 .Thebluepartofthebarshowsaccurateprediction,whiletheredpartofthebarshowsinaccurateprediction.Wendthathurricaneswithid1385and1405havelessaccuracy.WevisualizetheirshapesinFigure 6-13 aandbrespectively.Wecaninferthattheinaccurateresultscomefromthearbitrarychangingofshapes.ThisisreasonablesinceboththeMOSTmodelandthequadraticfunctioncan 167

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Figure6-11. Accuracyanalysisintermsofhitrate onlymodelnearfuturepredictionandaresensitivetothemostrecentlocations.Theycannothandlethesuddenchangeofspeedsanddirections.However,ourresearchistoprovideageneralapproachandinterfaceforqueryingmovingobject.Wewillnotbefocusingonthedomainexpertknowledgesuchastheatmospherepropertiesinpredictinghurricanes. 6.3.3.3RuntimeAnalysisFigure 6-14 bshowstheresultsoftheruntimeanalysisontheuncertainareapredictionofhurricanes.Weruntheexperimentsthreetimesandchoosetheaveragetime.Wendthatbothmethodscanmakethepredictionveryfast,asthedatasizeissmall.ThequadraticmodeltakesmoretimethantheMOSTmodel. 6.3.3.4DiscussionFromtheaboveexperiments,wecanndthatcurrentpredictionmethodscanonlypredictnearfuturemovements.Withtheincrementofpredictiontime,theaccuracyof 168

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Figure6-12. Predictionofallhurricanesfrom2005to2010 thepredictiondropsandtheerrordistanceincreases.ThelinearfunctionofMOSTisthemoststraightforwardpredictionmethod,whichassumesthatthespeedanddirectiondoesnotchangeinthenearfuture.Thequadraticfunctiontakestheaccelerationintoconsideration.Thereforeitcancapturethechangeofthedirectionandspeed.However,fromrealexperiments,wendthatthequadraticfunctiondoesnotperformbetterthanthelinearfunctioninthepredictionaccuracy.Thisisbecausethemovementsofhurricanesaremorecomplexthanlinearorquadraticmotionmodels.Wemusttakemoreinputssuchasatmosphericvariables.Thequadraticfunctioncostmoretimethanthelinearfunctionsinceitneedstocomputetheaccelerator. 169

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(a)(b)Figure6-13. Inaccuratepredictioncostbysuddenchangeofdirections (a)(b)Figure6-14. Accuracyanalysisintermsoferrordistancesandtheruntime 6.4ImplementationandExperimentsofInferringFutureLocationsfromSimilarTrajectoriesInthissection,weconductseveralexperimentstoevaluateourproposedapproachofthispaper.AlltheexperimentsareimplementedinC#andevaluatedonanIntelCore(TM)i52.53GHZCPUrunningWindows7operatingsystemwith4GBmainmemory.Intherestofthissection,werstintroducethesettingsoftheexperiment 170

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Table6-4. SummaryofDataset RawDataAmountTrajectoriesStatistics Venues206194Totaltrajectories10564 Users49023Avg.length18.19km Check-ins425588Avg.duration202.8min Activeusers6979Avg.#check-ins5 anddatapreprocessing.Thenweshowtheeffectsandefciencyofouralgorithmsrespectively. 6.4.1SettingsandDataPreprocessingAllthedataweusedinexperimentsareusertrajectorydataintherealworld.Wecrawledusercheck-insequencesinNewYorkCityfromFoursquarec.TheinformationofthedatasourceissummarizedinTable 6-4 .Eachcheck-inrecordshowstheuser,thevenue,thetimeandthetipsleftbytheuser.Byorderingthecheck-insequencesofauserovertime,weareabletoformtrajectoriesofeachuser.Wenoticethatalthoughtherearealargenumberofcheck-insandusers,onlyasmallportioncanformtrajectories.First,manyregisteredusersarenotactive,theycheckinrandomly,aboutjustonceortwice.Second,someuserscheckinverysparsely.Ifauserjustchecksinonceaday,itisnothelpfulforustodetectthemovingpatternofthisuser.Thereforewerequirethateachusermustcheckinatleastthreetimesduringoneday.Thereforewelterouttherandomcheck-ins.Thenweobtainedanumberof10564rawtrajectories.Wendthatthedataisquitesparse,i.e.,auserchecksinaboutevery3hoursonaverage.Fromallcheck-ins,weperformthesemanticmappingandobtainoursemantictrajectorydataset.Wedetect8categoriesand231sub-categories.Thedistributionofthe8maincategoriesamongallthecheck-inrecordsareshowninFigure 6-15 6.4.2EffectsEvaluationWeusealltrajectoriesastrainingdata,off-linecalculatedallclusters.TworesultsofpredictionisshowninFigure 6-16 aandbrespectively.Toobtainthisresult,weset 171

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Figure6-15. Numberofvisitstodifferentcategories trajectoryturn'(wementionedinSection 4.4.3.1 )tobe=4,thetemporalconstraintas20,andas1. 172

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(a) (b)Figure6-16. Resultsofon-lineprediction 173

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CHAPTER7CONCLUSIONSMovingobjectsarespatialobjects,whoselocationsorgeometrieschangewithtime.Movingobjectsareoftenrepresentedbytrajectories.Atrajectoryisasequenceoftime-stampedlocations.Inrealworldapplications,duetosomespecicissuessuchastosavetheenergyconsumption,theobservationsofmovingobjectsarenotavailableallthetime.Thereforetheuncertaintyexists.Theuncertaintyisaninherentfeatureofmovingobjectsresultingfromtheinabilityoftrackingthecontinuousmotionfunctions,orlackingtheknowledgeaboutthefuturemovements.Solvingthespatio-temporaluncertaintyproblemisimportantinmanyapplicationssuchastrafcmanagement,recommendationroutesinlocationbasedsystems,andhurricanesearchandprediction.Theuncertaintyexistsinbothhistoricalmovementsandfuturemovementsofmovingobjects.Thepurposeofthisresearchistobuildandintegrateacomprehensivecomponentinextensibledatabasemanagementsystemsforrepresentingandqueryingtheuncertaintyofmovingobjects.Thesystemconsistsofvariousdatatypesformovingobjectswithuncertainty,andimportantoperationsandpredicatessuchastheuncertainarea,topologicalrelationships,andthedirectionalrelationshipsbetweenmovingobject,aswellastheintegrationoftheseoperationsandpredicatesintoSQLtosupportqueriesinextensibledatabases.Toachievethisgoal,wehavedevelopedsolutionsatthreemainlevelsofabstraction.Startingfromtheabstractlevel,weproposeseveralmodelswhichproperlyrepresenttheuncertaintyofinhistoricalmovementsaswellasfuturemovements.Weproposethependantmodeltorepresentthehistoricalmovingobjectswithuncertainty.Thependantmodelisanimprovementofthewellknownspace-timeprismmodel.Giventwoobservationsatthestartandtheendofamovement,allpossiblelocationsbetweenthesetwoobservationsarewithinatwo-coneshapedvolume.Thereforethe 174

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uncertainareaatanytimeinstancebetweenthesetwoobservationscanbecalculatedthroughmathematicalformulas.Wehavemodeledthisfactformallyandintroducedspatio-temporaluncertainpredicates(STUP).Thebenetofusingsuchpredicatesistoenablethequeryindatabaseseasily.Thenweproposetheballoonmodeltorepresenttheintegrationofbothhistoricalandfuturemovementswithuncertainty.Thehistoricalmovementisrepresentedbythestringoftheballoon,whilethefuturemovementisrepresentedbythebodyoftheballoonastheuncertaintygrows.InbothmodelswedesignthedatatypesandacomprehensivesetofoperationsandpredicateswhichcanbeintegratedinSQLs.Inthesecondlevel,wedesignarepresentationmethodforuncertainmovingobjectdatatypesweproposedintherstlevel,theslicerepresentation.Theideaistoprovidetherepresentationwhichcancapturethechangesofmovementsthroughunitcomponents.Wedesignalgorithmsofvariousoperationsandpredicatesbasedonthisslicerepresentation.Forexample,weusedtheplane-sweepalgorithmtocalculatetheintersectionwhichtakestheadvantageoftheslicerepresentation.Inthislevel,wealsointroducearesearchonminingfromuncertaintrajectoriesanddiscoverroutes.Weproposenovelmethodstopre-processuncertaintrajectoriesandusegridindex.Weconstructrouteablegraphontopofthegridindex.Giveninterestinglocations,weprovideanalgorithmtondtherouteswhichconnecttheselocations.Therefore,theycanbeadoptinarouterecommendationsystem.Whilemostoftheresearchmainlyfocusonthegeographicpropertyofmovingobjects,wendinterestinthesemanticpropertiesofmovingobjecttrajectories.Atrajectoryofapersonoftencontainssemantictagssuchasshop,school,restaurantetc.andcanreectthebehaviorofthepeople.Whenseveralpeoplevisitsameplaces,wecanndcommonbehaviorofthem,whichishelpfulforustorecommendlocationsinthefuture.Thereforeweproposeanovelmethodsonmeasuringthesimilaritybetweensemantictrajectories.Wetakebothgeographicandsemanticinformationtomeasurethesimilarity. 175

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Thethirdlevelistheimplementationlevel.Weimplementedthealgorithms,operationsandpredicates,andintegratethemintoextensibledatabasesystemsandquerylanguages.WeadoptarecentresearchiBLOBwhichcanstoreandquerycomplexspatialdataefciently.WeusetheiBLOBstructuretostoreourmovingobjectdata.Wealsoprovidedemonstrationsystemsforqueryinguncertainmovingobjectswhichisbuiltontopoftheextensibledatabases.On-linequeriescanbeperformedonthesystems.Weperformextensiveexperimentalanalysistotesttheeffectivenessandefciencyofthealgorithmsinouruncertaintymodels. 176

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BIOGRAPHICALSKETCH HechenLiuwasborninHarbin,HeilongjiangProvince,China,in1984.ShereceivedherB.S.degreeinInformationSystemsinHarbinInstituteofTechnology,Chinain2007.ShestartedhergraduatestudyintheDepartmentofComputerandInformationScienceandEngineering,UniversityofFloridainAugust2007,underthesupervisionofDr.MarkusSchneider.Herresearchinterestsarespatialdatabasesandspatio-temporaldatabases.SheworkedasaninternintheWebSearchandMininggroupofMicrosoftResearchAsia(MSRA)inSummer2011. 185