Extracting Mobility Statistics from Indexed Spatio-Temporal Datasets

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Extracting Mobility Statistics from Indexed Spatio-Temporal Datasets. Yoshiharu Ishikawa Yuichi Tsukamoto Hiroyuki Kitagawa University of Tsukuba. Outline. Background and objectives Markov transition probability Indexing method for moving trajectories Proposed methods naïve algorithm - PowerPoint PPT Presentation

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  • Extracting Mobility Statistics from Indexed Spatio-Temporal DatasetsYoshiharu IshikawaYuichi TsukamotoHiroyuki KitagawaUniversity of Tsukuba

  • OutlineBackground and objectivesMarkov transition probabilityIndexing method for moving trajectoriesProposed methodsnave algorithmCSP-based algorithmExperimental resultsConclusions

  • BackgroundMoving object databasesstores and manages information on a huge number of moving objectssupports queries on moving trajectories and/or moving statusResearch issuesspatio-temporal indexesextraction of statistics (e.g., selectivities)Statics in spatio-temporal databasesused for query optimizationalso useful in mobility analysis

  • Our ApproachObjective: extracting mobility statistics from spatio-temporal databasesTarget: trajectory data indexed using R-treesStatistics to be extractedMarkov transition probabilitytarget space is decomposed in cellsestimating transition probabilities between cells using the indexed trajectory dataFeaturessearch problem is formalized as constraint satisfaction problem (CSP) efficient processing using R-trees

  • OutlineBackground and objectivesMarkov transition probabilityIndexing method for moving trajectoriesProposed methodsnave algorithmCSP-based algorithmExperimental resultsConclusions

  • Markov Transition Probability (1)Assumption: target space is decomposed in cellsExample 1: What is the estimated probability that an object currently in cell c0 moves in cell c1 in a unit time later?

    First-order Markov transition probability Pr(c1|c0)c1c0

  • Markov Transition Probability (2)Example 2: What is the probability that an object which moves from c0 to cell c1 in a unit time moves to cell c2 in the next unit time?

    Second-order transition probability Pr(c2|c0, c1) Extension to order-n Markov transition probability Pr(cn|c0, , cn-1) is easyc0

  • Markov Transition ProbabilityConventional technique in traffic data analysisUpton & Fingleton, 1989 [13]Special kind of association rulesprobability corresponds to the confidence factordifference: existence of orderUsagetrajectory estimationestimates where a moving object moves to in the next periodsimulation of movement statusgiven status of moving objects at t = , we can estimate the change of the status at t = + 1, + 2,

  • AssumptionsMovement patterns obeys stationary processmovement tendency does not change as time passesCell decompositioneach cell is a rectanglecell size is arbitrary: non-uniform decomposition is allowedcell decomposition can be specified dynamicallyUnit time lengthunit time can be specified as arbitrary length (e.g., one minuite, 10 minuites, )but a unit time length should be a multiple of sampling time length

  • Formalization of Probability (1)Target data: trajectory data from t = 0 to t = T Definition of first-order Markov transition probability

    objs(ci, t): set of objects which were in cell ci at t denominator: no. of objects which were in cell c0 at arbitrary t (0 t T 1) numerator: no. of objects each of which contained in denominator and moved cell c1 at t + 1

  • Formalization of Probability (2)Definition of order-n Markov Transition Probability

    denominator: no. of objects each of which was in cell c0 at t (0 t T 1), in cell c1 at t + 1, , and in cell cn 1 at t + n 1numerator: no. of objects each of which is contained in Dominator and moved cell cn at t + n

  • Generalized Transition Probability Estimation Problem (1)Derives transition probability according to the specified cell sets at once Given n + 1 cell sets

    for each of arbitrary cell combinations

    output Pr(cn|c0,,cn-1)

  • Generalized Transition Probability Estimation Problem (2)Example: Given C0 = {c0, c1}, C1 = {c1, c2}, C2 = {c1, c2, c3}, estimate second-order probabilities

    Algorithm outputs 12 probabilities Pr(c1|c0, c1), Pr(c2|c0, c1), , Pr(c3|c1, c2)

  • OutlineBackground and objectivesMarkov transition probabilityIndexing method for moving trajectoriesProposed methodsnave algorithmCSP-based algorithmExperimental resultsConclusions

  • Indexing Methods for TrajectoriesR-tree-based approach is assumedPoint-based representation: trajectories is represented as a set of points(d+1)-dimension R-tree is used (e.g., 3D R-tree)incorporating temporal dimension

  • (d +1)-D R-tree-based RepresentationSampling-based representationAB

  • OutlineBackground and objectivesMarkov transition probabilityIndexing method for moving trajectory dataProposed methodsnave algorithmCSP-based algorithmExperimental resultsConclusions

  • Nave Algorithm (1)Based on the definition of the Markov transition probabilityExample: Estimating Pr(c2|c0, c1) Determine objs(c0, ) and objs(c1, + 1) using the R-treeobjs(ci, t): the set of objects which were in cell ci at time tTake intersection of two sets; the cardinality of the intersection is added to Scount If the intersection is not empty objs(c2, + 2) is determined using the R-treeTake intersection of objs(c0, ), objs(c1, + 1) , objs(c2, + 2); the cardinality of the result is added to QcountThis process is repeated for each (0 T n) Calculate Pr(c2|c0, c1) based on Scount, Qcount No. of search on R-tree is proportional to T

  • Nave Algorithm (2)Example: estimation of

  • OutlineBackground and objectivesMarkov transition probabilityIndexing method for moving trajectoriesProposed methodsnave algorithmCSP-based algorithmExperimental resultsConclusions

  • Basic Idea (1)Estimation of Pr(cn|c0, , cn-1) based on three steps:Count the no. of objects which were in c0, , cn-1 at each unit time using an R-treeCount the no. of objects which were in c0, , cnat each unit time using an R-treeCompute Pr(cn|c0, , cn-1) by [result of step 2] / [result of step 1]Benefitsstep 1 & 2 can be processed using the same algorithmalgorithm for step 1 is given by setting n n 1requires only two searches on R-tree

  • Basic Idea (2) 1 2 3 4 5 6 7 8 (= T )xcellc2Example: estimation of Pr(c2|c0, c1)cellc1cellc0Step 1: count objectswhich moved from c0 to c1 within aunit timeScount = 2Step 2: count objectsthat moved asc0 , c1, c2 at eachunit timeQcount = 1Pr(c2|c0, c1) = Step 3: computeprobability

  • Counting Using R-tree (1)How can we compute no. of objects which were in c0, , cn at each unit time?Idea: the problem is formalized as a constraint satisfaction problem (CSP)An object satisfying the constraint fulfills the following constraints for some it was in cell c0 at t = it was in cell c1 at t = + 1 it was in cell cn at t = + nSearch objects that satisfy all n + 1 constraints

  • Counting Using R-tree (2)Effective use of R-tree is necessaryWe extend the CSP solution search method using R-trees (Papadias et al, VLDB98) [7]considers spatial constraintsExample: find all spatial objects x, y, z that satisfy overlap(x, y) and north(y, z)search CSP solutions from the root to leavesUse of pruning and backtracksReduce search space using constraintsenumerates all solutions with one R-tree access

  • Example of Counting (1)abcrootccFor C0 = {c1}, C1 = {c1, c2},C2={c2}, derive probabilities for (C0, C1, C2) Derive two probabilities at once Pr(c2|c1, c1): the probability that an object which have moved as c1c1 next moves to c2 Pr(c2|c1, c2)

  • Example of Counting (2)rootabc1 2 3 4 5 6R-treeabcrootc1c2

  • Pruning Method (1)Pruning condition 1:Movement between two R-tree nodes which do not temporary consecutive is impossibleCandidates can be deletedExample: movement such as a b and b c are allowed movement a c is impossible

  • Pruning Method (2)cell c1Pruning condition 2:Trajectory is not containedin the target cellExample: When we are counting for c1 c1, we should consider only nodes that overlaps with c1

  • Pruning Method (3)distancebetweenMBRsPruning condition 3:If [max distance an object can move] < [distance between MBRs] then an object cannot move from a node to next node

  • Query Processing Examplepruningpruningtree level= 1pruningtreelevel=0Targets:c1 c1 c2c1 c2 c2

  • OutlineBackground and objectivesMarkov transition probabilityIndexing method for moving trajectory dataProposed methodsNave algorithmCSP-based algorithmExperimental resultsConclusions

  • Dataset (1)Generated using the moving object simulator made by Brinkoff [1]Simulates car movement situation on actual city road networkOldenburg city, Germany (about 2.5km x 2.8km)no. of initial moving objects: 55 objects are created in a minuteon average 100 objects are moving in the map at a timedata is generated for T = 1000 minutes120K points are stored in 3-D R-tree

  • Dataset (2)Example forestimating using 3 x 3 cells

  • Experimental Result (1)Map is decomposed into 30 x 30 cellsFirst-order Markov transition probabilitiesRandomly 3 x 3 cells are selected

    Graph1

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    Nave

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    Graph2

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    T (minute)

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    Sheet1

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    T=10030*303*3 10