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

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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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