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August 30, 2004 STDBM 2004 at Toronto

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 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

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 index

ed trajectory data

Features search problem is formalized as constraint satisfaction problem

(CSP) efficient processing using R-trees

Our Approach




Markov Transition Probability (1) Assumption: target space is decomposed in cells Example 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)t =τ

A

t =τ+1

A

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 easy

t =τ

A

t =τ+1

A

t =τ+2

A

c1c0

c2

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 stationary process

movement tendency does not change as time passes Cell decomposition

each cell is a rectangle cell size is arbitrary: non-uniform decomposition is all

owed cell decomposition can be specified dynamically

Unit time length unit time can be specified as arbitrary length (e.g., on

e minuite, 10 minuites, …) but a unit time length should be a multiple of samplin

g 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

1

00

1

010

01

|),(objs|

|)1,(objs),(objs|)|Pr( T

t

T

t

tc

tctccc

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

y

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 i

n Dominator and moved cell cn at t + n

1

0

10

1

00

10

|),(objs|

|),(objs|),,|Pr( T

ti

ni

T

ti

ni

nn

itc

itcccc

Generalized Transition Probability Estimation Problem (1)

Given n + 1 cell sets

for each of arbitrary cell combinations

output Pr(cn|c0,…,cn-1)

Derives transition probability according to the specified cell sets at once

},,,{,},,,{ ||,1,||,01,00 0 CnnnnC ccCccC

,),,( 00 nn CCcc

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)

c0 c1 c2

c3




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

０　 1 2 3 4 5 6 7 8 (=T)

x

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

Sampling-based representation

A

B

root

a b c

1 2 3 4 5 6

０　 1 2 3 4 5 6 7 8 (=T)

x

１

２４

５３６

a

b

c

root

Outline Background and objectives Markov transition probability Indexing method for moving trajectory data Proposed methods



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 cell ci at 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 t

he R-tree Take intersection of objs(c0, ), objs(c1, + 1) , objs(c2, + 2); the car

dinality of the result is added to Qcount This 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

Naïve Algorithm (2) 102 ,|Pr ccc

０　 1 2 3 4 5 6 7 8 (=T)

x

cell c1

Example: estimation of

Qcount += 1

No. of searchon R-treeis proportionalto T

Output =　 Qcount　　　 Scount

Scount += 1 Scount += 1

cell c0

cell c2




Basic Idea (1)Estimation of Pr(cn|c0, …, cn-1) based on three steps:

1. Count the no. of objects which were in c0, …, cn-1 at each unit time using an R-tree

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

3. 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)

０　 1 2 3 4 5 6 7 8 (= T )

x

cellc2

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

cellc1

cellc0

Step 1: count objectswhich moved from c0 to c1 within aunit time

Scount = 2

Step 2: count objectsthat moved asc0 , c1, c2 at eachunit time

Qcount = 1Pr(c2|c0, c1) = ―――――

Step 3: computeprobability

Counting Using R-tree (1) How can we compute no. of objects which were i

n c0, …, cn at each unit time? Idea: the problem is formalized as a constraint s

atisfaction problem (CSP) An object satisfying the constraint fulfills the follo

wing 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 = + 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 u

sing 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)

０　　 1 　 2 3 4 　 5 6 7 8 (=T)

x

１

２

４

５

３６

a

b

c

root

c １

c ２

For 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)

root

a b c

1 2 3 4 5 6

R-tree

０　　 1 　 2 3 4 　 5 6 7 8 (=T)

x

１

２

４

５

３６

a

b

c

root

c1

c2

Pruning Method (1)

Pruning condition 1:Movement between two R-tree nodes which do not temporary consecutive is impossible

Candidates can be deleted

０　 1 2 3 4 5 6 7 8 (=T)

x

a

cb

Example: - movement such as a b and b c are allowed- movement a c is impossible

Pruning Method (2)

０　 1 2 3 4 5 6 7 8 (=T)

x

cell c1

Pruning condition 2:Trajectory is not containedin the target cell

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

Pruning Method (3)

０　 1 2 3 4 5 6 7 8 (=T)

x

2

1

distancebetweenMBRs

Pruning condition 3:If [max distance an objectcan move] < [distance betweenMBRs] then an object cannotmove from a node to next node

Query Processing Example

cell c1

cell c2

cell c1

cell c2

treelevel= 2

cell c1

cell c2

x

t

root root root

pruning

a

bc

pruning

1

2

treelevel= 1

pruning

treelevel=0

backtrack

An object thatmoved asc1 c1 c2

is found andcounted

There is no objects thatmoved asc1 c1 c2

c1 c2 c2

Targets:c1 c1 c2

c1 c2 c2

Outline Background and objectives Markov transition probability Indexing method for moving trajectory data Proposed methods

Naïve algorithm CSP-based algorithm


Dataset (1) Generated using the moving object simulator ma

de 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 ti

me 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

Example forestimating using 3 x 3 cells

0 0.183 0.04

0.081 0.348 0.10

0.08 0.01 0.02

Experimental Result (1) Map is decomposed into 30 x 30 cells First-order Markov transition probabilities Randomly 3 x 3 cells are selected

00.10.20.30.40.50.60.70.80.9

1

T=10

0

T=20

0

T=30

0

T=40

0

T=50

0

T=60

0

T=70

0

T=80

0

T=90

0

T=10

00

T (minute)

Ella

psed

Tim

e (s

econ

d) NaïveCSP

Experimental Result (2) Estimation of second-order transition probabilities Other parameters are same to the former case

0

1

2

3

4

5

6

7

8

T=10

0

T=20

0

T=30

0

T=40

0

T=50

0

T=60

0

T=70

0

T=80

0

T=90

0

T=10

00

T (minute)

Ella

psed

Tim

e (s

econ

d)

NaïveCSP

Experimental Result (3) Estimation of third-order transition probabilities Other parameters are similar to the former case

0

20

40

60

80

100

120

T=10

0

T=20

0

T=30

0

T=40

0

T=50

0

T=60

0

T=70

0

T=80

0

T=90

0

T=10

00

T (minute)

Ella

psed

Tim

e (s

econ

d) NaïveCSP

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

0

5

10

15

20

25

T=10

0

T=20

0

T=30

0

T=40

0

T=50

0

T=60

0

T=70

0

T=80

0

T=90

0

T=10

00

T (minute)

Ella

psed

Tim

e (s

econ

d)

NaïveCSP

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