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### Continual Neighborhood Tracking for Moving Objects

Using Adaptive Distances

Yoshiharu Ishikawa

Hiroyuki Kitagawa

Tooru Kawashima

University of Tsukuba, Japan

{ishikawa,kitagawa}@is.tsukuba.ac.jp

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 wants

to find nearby gas stations

Typical Approach:

retrieve gas stations

with their distances less

than 200 meters from x

x

A spatial query based on

the Euclidean distance

future trajectory

Motivating Example (2)What’s Wrong?

If we know user’s past and future trajectories,

we can provide more

appropriate information

A

Our Idea (1)

- 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

A

At each sample position, a spatial query is generated

initial query

parameters

start point

destination

: sampled estimated

positions of the

moving object

destination

start point

Our Idea (2)Neighborhood Info

Retrieval System

: Data objects

- Sample positions are taken by unit-time basis

- 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

: no. of dimensions

Representation of LocationInformation (2)

- Locations of a moving object:

: departure time

: current time

: estimated arrival time

destination

current point

start point

Assumption:

past/future trajectory points are given in unit-time basis

position

Influence Model of Trajectory Points (1)- We usually set high importance on current neighborhood points

position

Influence Model of Trajectory Points (2)- 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 = + s(s unit times after the current time )
- The influence values decay exponentially towards past and future with parameters m and n, respectively

Influence Value

time

τ+σ－2

τ+σ

τ+σ+2

τ+σ－1

τ+σ+1

Influence Model of Trajectory Points (4)

- Influence value for each point when s = 1

nt’-1

nt’-2

m

m2

1

n

destination

mt

mt+1

current point

n2

highest weight point

since s = 1

start point

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

q

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

Derivation of Query Centers (1)

- Model cur: set the point with the highest importance to the query center

current

position

Derivation of Query Centers (2)

- Model avg: weighted average based on influence values

Setting of parameters m and n

changes the query center

current

position

query center q

q

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

Distance Function Derivation Models (1)

- Model EU: Euclid distance-based model

- Pros
- - simple and intuitive
- - easy to compute
- Cons
- - do not consider past/future
- - trajectory information

Ellipsoid Distance

Appropriate setting of the distance

matrix A allows flexible

tuning of distances

We derive an appropriate matrix

A using past/future trajectory

information

Distance Function Derivation Models (2)

- Model OV: ellipsoid distance-based model

derive a distance matrix M that reflects the sample point

distribution nearby the query point [19]

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 staying in some place, the matrix C becomes an ill-conditioned matrix: therefore, we cannot derive the distance matrix M!

: unit matrix

Distance Function Derivation Models (4)

- Model HB: hybrid model
- integrates the benefits of EU and OV models

becomes an regular matrix

q

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

Query Task (1)

- Range Query: At each point, retrieve objects within distance e

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

modified parameters

σ= 5 , μ=0.4

ν=0.4, λ= 1.0

Experiment 1 (2)- Set the “near future” point as query center

initial parameters

σ= 0 , μ=0.5

ν=0.5, λ= 1.0

x

σ= 0 , μ=0.4

ν=0.9, λ= 1.0

Experiment 1 (3)- Set high weights on future trajectory

initial parameters

σ= 0 , μ=0.4

ν=0.4, λ= 1.0

x

σ= 0 , μ=0.4

ν=0.4, λ= 0.7

Experiment 1 (4)- Use of the regularization parameter l

initial parameters

σ= 0 , μ=0.4

ν=0.4, λ= 1.0

x

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

Generic Retrieval Functions (2)

- dist_search(q, ):retrieve objects within distance e from q using the Euclidean distance

Generic Retrieval Functions (3)

- knn_search(q, k): retrieve nearest k objects from the query center q using the Euclidean distance

Minimal Bounding Box (MBB) for Ellipsoid Isosurface [30]

- MBB that tightly encloses the ellipsoid ellip(M, q, e)

ellip(M, q, e)

j-th dimension

: (i, i) element of

the inverse of M

i-th dimension

Minimal Bounding Sphere (MBS) for Ellipsoid Isosuraface [30]

- MBS that tightly encloses the ellipsoid ellip(M, q, e)

ellip(M, q, e)

: the smallest

eigenvalue of M

Query Processing (2)

- k-NN query (k = 3)

Query Processing (2)

- k-NN query (k = 3)

- Perform k-NN query
- based on the Euclidean
- distance

2. Derive an ellipsoid that

tightly encloses k-NN

objects

3. Perform a range query

with MBS (or MBB) that

tightly encloses the

ellipsoid region

4. Select nearest k objects

from the retrieved objects

using the ellipsoid distance

Experiment: Retrieval I/O Cost with Spatial Indexes (1)

- I/O cost evaluation using R-tree (GiST)
- Target dataset (green points): 39,226 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 support of spatial indexes
- k-NN (k = 1, 10, …, 150) results are 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|t as

Incremental Query Update (3)

- Then update using the following formulas
- We can make an incremental update for covariance matrix (C) in a similar manner

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

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