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