Continual Neighborhood Tracking for Moving Objects
Yoshiharu IshikawaHiroyuki KitagawaTooru Kawashima
University of Tsukuba, Japan{ishikawa,kitagawa}@is.tsukuba.ac.jp
Using Adaptive Distances
Organization
Background and Overview Our Approach Experiments Query Processing with Spatial Indexes Incremental Query Update Conclusions and Future Work
Background
Progress of Digital Cartography Development of GPS Technologies Wide Use of PDA and Hand-held Devices
New Types of Information Services: Providing neighborhood information to moving objects (people with PDAs, cars with navigation systems) considering their locations and trajectories
Motivating Example (1)
Neighborhood Query:A user at point x wantsto find nearby gas stations
Typical Approach:retrieve gas stationswith their distances lessthan 200 meters from x
x
A spatial query based onthe Euclidean distance
Motivating Example (2)
A
What’s Wrong? If we know user’s past and future trajectories,we can provide moreappropriate information
past trajectory
future trajectory
Our Idea (1)
A
Use of an ellipsoid region to represent a neighborhood query
An ellipsoid region is computed based on the past/future trajectories
A neighborhood query is specified as a spatial query with an ellipsoid distance
Our Idea (2)Neighborhood InfoRetrieval System
start pointdestination
start point
destination
initial queryparameters
: Data objects: sampled estimated positions of the moving object Sample positions are
taken by unit-time basis At each sample
position, a spatial query is generated
The system perform queries continuously
Problems and Solutions How can we generate appropriate spatial
queries? Introduction of influence model of trajectory points Proposal of query derivation models
How about efficiency? Use of spatial indexes for efficient query processing Low-cost query update procedure for continuous
queries
Organization
Background and Overview Our Approach
Influence model of trajectory points Query derivation model
Experiments Query Processing with Spatial Indexes Incremental Query Update Conclusions and Future Work
Representation of LocationInformation (1) Object locations are represented by d-D vectors
Tidii xxx ],...,[ 1 d : no. of dimensions
1x
2x
1x
x
x
2x
x
current point
start point
destination
Representation of LocationInformation (2) Locations of a moving object:
Assumption:past/future trajectory points are given in unit-time basis
t : current time t : estimated arrival time
1t : departure time
Influence Model of Trajectory Points (1)
currentposition
We usually set high importance on current neighborhood points
Influence Model of Trajectory Points (2)
currentposition
A user may be interested in near future neighborhood where he or she will arrive soon
Influence Model of Trajectory Points (3) The influence model sets the highest weight “1” on
location information at time t = + ( unit times after the current time )
The influence values decay exponentially towards past and future with parameters and , respectively
time
Influence Value
1
τ+στ+σ+1
τ+σ+2τ+σ- 1
τ+σ- 2
)...,,(
)...,,1()(
t
tt
t
t
1x
2x
1x
x
x
2x
x
current point
start point
destination
Influence value for each point when = 1
Influence Model of Trajectory Points (4)
’-1
3x
1x
’-2
highest weight pointsince = 1
Organization
Background and Overview Our Approach
Influence model of trajectory points Query derivation model
Experiments Query Processing with Spatial Indexes Incremental Query Update Conclusions and Future Work
Query Derivation Model Neighborhood queries for moving objects are issued to a spatial database A spatial query is fixed specifying
query center q two models (cur, avg)
distance function D three models (EU, OV, HB)
query task range query and k-nn query
q
D
Derivation of Query Centers (1) Model cur: set the point with the highest
importance to the query center
xq
x x
1xcurrentposition
x
query center q
Derivation of Query Centers (2) Model avg: weighted average based on influence
values
1
1
)(
)(
t
t t
t
t
xq
x x
1xcurrentposition query center q
Setting of parameters and changes the query center
Query Derivation Model Neighborhood queries for moving objects are issued to a spatial database A spatial query is fixed specifying
query center q two models (cur, avg)
distance function D three models (EU, OV, HB)
query task range query and k-nn query
q
D
Distance Function Derivation Models (1)
Model EU: Euclid distance-based model
)()()( qxqxqx TD ,2
Pros- simple and intuitive- easy to compute
Cons- do not consider past/future- trajectory information
q
q
Ellipsoid Distance
)()(),(2 qxqxqx ATD
Appropriate setting of the distancematrix A allows flexibletuning of distances
We derive an appropriate matrixA using past/future trajectoryinformation
q
Distance Function Derivation Models (2) Model OV: ellipsoid distance-based model
1
)())((mint
tT
tt qxqx AMA
11))(det( CCM d
1
)(t
Tiit qxqxC
q
derive a distance matrix M that reflects the sample pointdistribution nearby the query point [19]
q
C is the weighted covariance matrix
Distance Function Derivation Models (3) Model OV: ellipsoid distance-based model
pros allows retrieval along the trajectory since the derived
distance is an extended version of the Mahalanobis distance [8, 20]
cons: not robust compared to the Euclidean distance When an object is moving along a straight line or stay
ing in some place, the matrix C becomes an ill-conditioned matrix: therefore, we cannot derive the distance matrix M!
Model HB: hybrid model integrates the benefits of EU and OV models
Distance Function Derivation Models (4)
I
I
C
CC )1(~
11 )~
())(det( CCM d
10 I : unit matrix
C~
becomes an regular matrix
regularization
Query Derivation Model Neighborhood queries for moving objects are issued to a spatial database A spatial query is fixed specifying
query center q two models (cur, avg)
distance function D three models (EU, OV, HB)
query task range query and k-nn query
q
D
Query Task (1)
Range Query: At each point, retrieve objects within distance
q
ε
Query Task (2) k-Nearest Neighbor
Query: At each point retrieve nearest k objects
q
when k = 3
Organization
Background and Overview Our Approach Experiments Query Processing with Spatial Indexes Incremental Query Update Conclusions and Future Work
Experiment 1: Observation of Behaviors
Query generation example for the trajectory (blue line)
Target points are shown in green points
Queries are generated based on the hybrid model
x
Experiment 1 (1) Comparison of Euclidean distance and ellipsoid
distance
Experiment 1 (2) Set the “near future” point as query center
initial parametersσ= 0 , μ=0.5ν=0.5, λ= 1.0
modified parameters σ= 5 , μ=0.4ν=0.4, λ= 1.0
x
y
Experiment 1 (3) Set high weights on future trajectory
initial parametersσ= 0 , μ=0.4ν=0.4, λ= 1.0
refined parametersσ= 0 , μ=0.4ν=0.9, λ= 1.0
x
Experiment 1 (4)
x
Use of the regularization parameter
initial parametersσ= 0 , μ=0.4ν=0.4, λ= 1.0
refined parametersσ= 0 , μ=0.4ν=0.4, λ= 0.7
Experiment 2: Simulation Based on Trace Data (1)
Car driving trace data is used to compute queries
Experiment 2: Simulation Based on Trace Data (2)
Each isosurface represents the query generated at the point
Organization
Background and Overview Our Approach Experiments Query Processing with Spatial Indexes Incremental Query Update Conclusions and Future Work
Query Processing Based on Spatial Indexes
Most of spatial indexes do not support ellipsoid distance-based queries
We extend the approach of Seidl & Kriegel [30] to support ellipsoid distance-based queries with conventional spatial indexes
Assumptions: only three generic retrieval functions are supported by the underlying spatial indexes
Generic Retrieval Functions (1)
rect_search(r): retrieve objects within the specified rectangle region r r
Generic Retrieval Functions (2)
dist_search(q, ): retrieve objects within distance from q using the Euclidean distance
q
Generic Retrieval Functions (3)
knn_search(q, k): retrieve nearest k objects from the query center q using the Euclidean distance
q
Minimal Bounding Box (MBB) for Ellipsoid Isosurface [30]
MBB that tightly encloses the ellipsoid ellip(M, q, )
1 iiiq M 1 iiiq M
1 jjjq M
1 jjjq M
j-th dimension
i-th dimension
1iiM : (i, i) element of
the inverse of Mq
ellip(M, q, )
Minimal Bounding Sphere (MBS) for Ellipsoid Isosuraface [30]
MBS that tightly encloses the ellipsoid ellip(M, q, )
min
min : the smallest eigenvalue of Mq
ellip(M, q, )
Query Processing (1)
Range query processing with MBB approximation
q
Query Processing (1)
q
Range query processing with MBB approximation
Query Processing (2)
k-NN query (k = 3)
q
Query Processing (2)
k-NN query (k = 3) 1. Perform k-NN query based on the Euclidean distance
2. Derive an ellipsoid that tightly encloses k-NN objects3. Perform a range query with MBS (or MBB) that tightly encloses the ellipsoid region4. Select nearest k objects from the retrieved objects using the ellipsoid distance
q
Experiment: Retrieval I/O Cost with Spatial Indexes (1)
I/O cost evaluation using R-tree (GiST) Target dataset (green points): 39,22
6 crossroad points of Maryland County in U.S.
Query: 62 blue points along the road
I/O costs are compared for sequential scan ellipsoid distance query with the su
pport of spatial indexes k-NN (k = 1, 10, …, 150) results ar
e shown
Experiment: Retrieval I/O Cost with Spatial Indexes (2)
Average page I/O cost per query
Organization
Background and Overview Our Approach Examples Query Processing with Spatial Indexes Incremental Query Update Conclusions and Future Work
Query Update
In each query point, a slightly different query is generated The query center and the distance function will change
Naïve update strategy Derive the query center and the distance function from
scratch The generation cost is quite large
It requires calculation from past/future trajectory information
Can we update queries incrementally? Answer: Yes, but periodic reorganization is required
Incremental Query Update (1)
Basic Idea Decompose statistics used to generate a query into past
part and future part At each update, make “one step shift” from the future
part to the past part Exponential decay factors allow a simple and efficient
procedure
Incremental Query Update (2)
Example: Incremental update of query center for model avg Decompose x|as
|w
ssx
-
τ|
tt
t
tt
t
xs
xs
'
1
1
|
|
Incremental Query Update (3)
Then update using the following formulas
We can make an incremental update for covariance matrix (C) in a similar manner
1
111
11
11
|
|||
|1
|
||
w
-
ssx
xss
xss
Incremental Query Update (4)
Incremental query update procedure allows constant update cost for fixed dimensionality d
Bad news: two problems A moving object may reach early or late to the next point.
Moreover, it may change the estimated route. A number of incremental updates will result in incorrect
query generation since the proposed incremental update procedure amplifies the noise.
Practical update procedure Use incremental update procedure for short period and
recalculate statistics periodically
Organization
Background and Overview Our Approach Examples Query Processing with Spatial Indexes Incremental Query Update Conclusions and Future Work
Conclusions
Generation of Neighborhood Tracking Queries Based on Adaptive Distances (Ellipsoid Distances) Introduction of Influence Decay Model of Trajectory
Points Proposal of Spatial Query Generation Models
Efficient Query Evaluation with Spatial Indexes Query Update Method for Continual Query
Processing
Future Work
Development of parameter set-up method that considers query workloads and query tasks
Use of previous query results (cached results) for efficient continual query processing
Development of Prototype System
Prototype System
Under developmenton top ofArcView GIS
Support ofdynamic location feedingfrom GPS
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