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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

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Extracting Mobility Statistics from Indexed Spatio-Temporal Datasets

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Extracting mobility statistics from indexed spatio temporal datasets

Extracting Mobility Statistics from Indexed Spatio-Temporal Datasets

Yoshiharu Ishikawa

Yuichi Tsukamoto

Hiroyuki Kitagawa

University of Tsukuba

STDBM 2004 at Toronto


Outline

Outline

  • Background and objectives

  • Markov transition probability

  • Indexing method for moving trajectories

  • Proposed methods

    • naïve algorithm

    • CSP-based algorithm

  • Experimental results

  • Conclusions


Background

Background

  • Moving object databases

    • stores and manages information on a huge number of moving objects

    • supports queries on moving trajectories and/or moving status

  • Research issues

    • spatio-temporal indexes

    • extraction of statistics (e.g., selectivities)

  • Statics in spatio-temporal databases

    • used for query optimization

    • also useful in mobility analysis


Our approach

Our Approach

  • Objective: extracting mobility statistics from spatio-temporal databases

  • Target: trajectory data indexed using R-trees

  • Statistics to be extracted:Markov transition probability

    • target space is decomposed in cells

    • estimating transition probabilities between cells using the indexed trajectory data

  • Features

    • search problem is formalized as constraint satisfaction problem (CSP)

    • efficient processing usingR-trees


Outline1

Outline

  • Background and objectives

  • Markov transition probability

  • Indexing method for moving trajectories

  • Proposed methods

    • naïve algorithm

    • CSP-based algorithm

  • Experimental results

  • Conclusions


Markov transition probability 1

A

A

t =τ

t =τ+1

Markov Transition Probability (1)

  • Assumption: target space is decomposed in cells

  • Example 1: What is the estimated probability that an object currently in cell c0moves in cell c1in a unit time later?

  • First-orderMarkov transition probability Pr(c1|c0)

c0

c1


Markov transition probability 2

A

A

A

c1

c2

t =τ

t =τ+1

t =τ+2

Markov Transition Probability (2)

  • Example 2: What is the probability that an object which moves from c0to cell c1in a unit time moves to cell c2in the next unit time?

  • Second-order transition probability Pr(c2|c0, c1)

  • Extension toorder-n Markov transitionprobability Pr(cn|c0, …, cn-1) is easy

c0


Markov transition probability

Markov Transition Probability

  • Conventional technique in traffic data analysis

    • Upton & Fingleton, 1989 [13]

  • Special kind of association rules

    • probability corresponds to the confidence factor

    • difference: existence of order

  • Usage

    • trajectory estimation

      • estimates where a moving object moves to in the next period

    • simulation of movement status

      • given status of moving objects at t = , we can estimate the change of the status at t =  + 1,  + 2, …


Assumptions

Assumptions

  • Movement patterns obeys stationaryprocess

    • movement tendency does not change as time passes

  • Cell decomposition

    • each cell is a rectangle

    • cell size is arbitrary: non-uniform decomposition is allowed

    • cell decomposition can be specified dynamically

  • Unit time length

    • unit 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

Formalization of Probability (1)

  • Target data: trajectory data fromt = 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

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 1

    • numerator: no. of objects each of which is contained in Dominator and moved cell cn at t + n


Generalized transition probability estimation problem 1

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

  • outputPr(cn|c0,…,cn-1)


Generalized transition probability estimation problem 2

c0

c1

c2

c3

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)


Outline2

Outline

  • Background and objectives

  • Markov transition probability

  • Indexing method for moving trajectories

  • Proposed methods

    • naïve algorithm

    • CSP-based algorithm

  • Experimental results

  • Conclusions


Indexing methods for trajectories

Indexing Methods for Trajectories

  • R-tree-based approach is assumed

  • Point-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 representation

x

x

root

b

a

c

root

0  1 2 3 4 5 6 7 8

(=T)

0  1 2 3 4 5 6 7 8

(=T)

a

b

c

1

2

3

4

5

6

(d +1)-D R-tree-based Representation

B

A

Sampling-based representation


Outline3

Outline

  • Background and objectives

  • Markov transition probability

  • Indexing method for moving trajectory data

  • Proposed methods

    • naïve algorithm

    • CSP-based algorithm

  • Experimental results

  • Conclusions


Na ve algorithm 1

Naïve Algorithm (1)

  • Based on the definition of the Markov transition probability

  • Example: Estimating Pr(c2|c0, c1)

    • Determine objs(c0, ) and objs(c1,  + 1) using the R-tree

      • objs(ci, t): the set of objects which were in cellciat time t

    • Take 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-tree

    • Take intersection of objs(c0, ), objs(c1,  + 1) , objs(c2,  + 2);the cardinality of the result is added toQcount

    • This process is repeated for each  (0 ≤≤T – n)

    • CalculatePr(c2|c0, c1) based on Scount, Qcount

  • No. of search on R-tree is proportional to T


Na ve algorithm 2

cell c0

cell c1

Output =

Qcount

Scount

x

Qcount += 1

cell c2

0  1 2 3 4 5 6 7 8

(=T)

No. of search

on R-tree

is proportional

to T

Scount += 1

Scount += 1

Naïve Algorithm (2)

Example: estimation of


Outline4

Outline

  • Background and objectives

  • Markov transition probability

  • Indexing method for moving trajectories

  • Proposed methods

    • naïve algorithm

    • CSP-based algorithm

  • Experimental results

  • Conclusions


Basic idea 1

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-tree

    • Count the no. of objects which were in c0, …, cnat each unit time using an R-tree

    • Compute Pr(cn|c0, …, cn-1) by [result of step 2] / [result of step 1]

  • Benefits

    • step 1 & 2 can be processed using the same algorithm

      • algorithm for step 1 is given by setting n → n – 1

    • requires only two searches on R-tree


Basic idea 2

Basic Idea (2)

Example: estimation of Pr(c2|c0, c1)

x

Step 1: count objects

which moved from

c0 toc1within a

unit time

cell

c2

Step 2: count objects

that moved as

c0 , c1, c2 at each

unit time

cell

c1

Step 3: compute

probability

cell

c0

Qcount = 1

Pr(c2|c0, c1) = ―――――

Scount = 2

0  1 2 3 4 5 6 7 8 (= T)


Counting using r tree 1

Counting Using R-tree (1)

  • How can we compute no. of objects which were in c0, …, cnat 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 cellc0at t = 

    • it was in cellc1at t =  + 1

    • it was in cellcnat t =  + n

  • Search objects that satisfy all n + 1 constraints


  • Counting using r tree 2

    Counting Using R-tree (2)

    • Effective use of R-tree is necessary

    • We extend the CSP solution search method using R-trees(Papadias et al, VLDB’98) [7]

      • considers spatial constraints

        • Example: find all spatial objects x, y, z that satisfy overlap(x, y) and north(y, z)

      • search CSP solutions from the root to leaves

        • Use of pruning and backtracks

        • Reduce search space using constraints

      • enumerates all solutions with one R-tree access


    Example of counting 1

    x

    0   1 2 3 4 5 6 7 8

    (=T)

    Example of Counting (1)

    root

    ForC0 = {c1}, C1 = {c1, c2},

    C2={c2}, derive

    probabilities for(C0, C1, C2)

    b

    • Derive two probabilities at once

    • Pr(c2|c1, c1): the probability that an objectwhich have moved as c1c1 next moves to c2

    • Pr(c2|c1, c2)

    c2

    a

    c

    c1


    Example of counting 2

    x

    0   1 2 3 4 5 6 7 8

    (=T)

    Example of Counting (2)

    root

    R-tree

    b

    root

    c2

    a

    a

    b

    c

    c

    c1

    1 2 3 4 5 6


    Pruning method 1

    x

    c

    b

    a

    0  1 2 3 4 5 6 7 8

    (=T)

    Pruning Method (1)

    Pruning condition 1:

    Movement between two R-tree nodes which do not temporary consecutive is impossible

    Candidates can be deleted

    Example:

    • movement such as ab and bc are allowed

    • movement ac is impossible


    Pruning method 2

    x

    0  1 2 3 4 5 6 7 8

    (=T)

    Pruning Method (2)

    Pruning condition 2:

    Trajectory is not contained

    in the target cell

    Example: When we are counting for c1 c1, we should consider only nodesthat overlaps with c1

    cellc1


    Pruning method 3

    x

    1

    2

    0  1 2 3 4 5 6 7 8

    (=T)

    Pruning Method (3)

    Pruning condition 3:

    If [max distance an objectcan move] < [distance betweenMBRs] then an object cannotmove from a node to next node

    distance

    between

    MBRs


    Query processing example

    Query Processing Example

    x

    treelevel

    = 2

    root

    root

    root

    cell c2

    cell c2

    cell c2

    a

    c

    b

    cell c1

    cell c1

    cell c1

    t

    1

    2

    There is no

    objects that

    moved as

    c1 c1 c2

    c1 c2 c2

    backtrack

    An object that

    moved as

    c1 c1 c2

    is found and

    counted

    Targets:

    c1 c1 c2

    c1 c2 c2

    pruning

    pruning

    treelevel

    = 1

    pruning

    tree

    level

    =0


    Outline5

    Outline

    • Background and objectives

    • Markov transition probability

    • Indexing method for moving trajectory data

    • Proposed methods

      • Naïve algorithm

      • CSP-based algorithm

    • Experimental results

    • Conclusions


    Dataset 1

    Dataset (1)

    • Generated using the moving object simulator made by Brinkoff [1]

    • Simulates car movement situation on actual city road network

      • Oldenburg city, Germany (about 2.5km x 2.8km)

      • no. of initial moving objects: 5

      • 5 objects are created in a minute

      • on average 100 objects are moving in the map at a time

      • data is generated for T = 1000 minutes

      • 120K points are stored in 3-D R-tree


    Dataset 2

    c0   c3    c6

    c1   c4    c7

    c2   c5    c8

    0 0.183 0.04

    0.081 0.348 0.10

    0.08 0.01 0.02

    Dataset (2)

    Example for

    estimating using 3 x 3 cells


    Experimental result 1

    Experimental Result (1)

    • Map is decomposed into 30 x 30 cells

    • First-order Markov transition probabilities

    • Randomly 3 x 3 cells are selected


    Experimental result 2

    Experimental Result (2)

    • Estimation of second-order transition probabilities

    • Other parameters are same to the former case


    Experimental result 3

    Experimental Result (3)

    • Estimation of third-order transition probabilities

    • Other parameters are similar to the former case


    Experimental result 4

    Experimental Result (4)

    • The case when CSP-based approach is not effective

      • Target space is decomposed into 20 x 20 cells

      • Estimation of second-order transition probabilities

    Since cell decomposition is coarse, the pruning cannot reduce candidates


    Conclusions and future work

    Conclusions and Future Work

    • Conclusions

      • mobility statistics based on Markov transition probability

      • proposals of two algorithms

        • naïve approach

        • CSP-based approach

      • CSP-based approach effectively utilizes R-tree structure

    • Future Work

      • adaptive cell decompositions

      • extension to non-stationary Markov transitions


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