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Extracting Mobility Statistics from Indexed Spatio-Temporal Datasets

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### Extracting Mobility Statistics from Indexed Spatio-Temporal Datasets

Outline

Yoshiharu Ishikawa

Yuichi Tsukamoto

Hiroyuki Kitagawa

University of Tsukuba

STDBM 2004 at Toronto

Outline

- Background and objectives
- Markov transition probability
- Indexing method for moving trajectories
- Proposed methods
- naïve algorithm
- CSP-based algorithm
- Experimental results
- Conclusions

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

- 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

Outline

- Background and objectives
- Markov transition probability
- Indexing method for moving trajectories
- Proposed methods
- naïve algorithm
- CSP-based algorithm
- Experimental results
- Conclusions

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

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

- 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

- 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)

- 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)

- 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)

- 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)

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)

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

- 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

x

x

root

b

１

５

６

３

a

４

c

２

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 RepresentationB

A

Sampling-based representation

Outline

- Background and objectives
- Markov transition probability
- Indexing method for moving trajectory data
- Proposed methods
- naïve algorithm
- CSP-based algorithm
- Experimental results
- Conclusions

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

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

- Background and objectives
- Markov transition probability
- Indexing method for moving trajectories
- Proposed methods
- naïve algorithm
- CSP-based algorithm
- Experimental results
- Conclusions

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)

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)

- 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)

- 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

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

１

５

６

３

- Derive two probabilities at once
- Pr(c2|c1, c1): the probability that an objectwhich have moved as c1c1 next moves to c2
- Pr(c2|c1, c2)

c２

a

４

c

２

c１

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 ab and bc are allowed
- movement ac is impossible

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

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

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

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)

- 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

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)

- Map is decomposed into 30 x 30 cells
- First-order Markov transition probabilities
- Randomly 3 x 3 cells are selected

Experimental Result (2)

- Estimation of second-order transition probabilities
- Other parameters are same to the former case

Experimental Result (3)

- Estimation of third-order transition probabilities
- Other parameters are similar to the former case

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
- 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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