Post on 17-Jan-2016
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
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) 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
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 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
53 6
a
b
c
root
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 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
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)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
3 6
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
3 6
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
Experimental results Conclusions
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