Effectively Indexing Uncertain Moving Objects for Predictive Queries

School of ComputingNational University of Singapore

Department of Computer ScienceAalborg University

Meihui Zhang, Su Chen, Christian S. Jensen, Beng Chin Ooi, Zhenjie Zhang

Outline

• Introduction• Uncertain Moving Object Model• Movement Inference Model• Indexing and Query Processing

– Index structure– Probabilistic Range Query– k-PNN Query

• Experiments• Conclusion

Outline

• Introduction• Uncertain Moving Object Model• Movement Inference Model• Indexing and Query Processing

– Index structure– Probabilistic Range Query– k-PNN Query

• Experiments• Conclusion

Introduction• Trends and applications

– Positioning Systems, Wireless Communication, etc.– Intelligent Transport Systems, Location-Based Services, etc.

Motivation• Existing work on moving object management

Assumption: deterministic movement

• Real world– Limited accuracy– Complex and

stochastic movement– …

-1 0 1-1

0

1

velocities

speed on x-axis (km/min)

spee

d on

y-a

xis (

km/m

in)

School bus in Athens metropolitan

Motivation• Problem

– Information gap between real movement and deterministic models

• Solution– Introduce an uncertainty model

Our contributions• Uncertain moving object model

– Take into account the uncertainties of both location and velocity

• Movement inference model– Infer the location distribution at t

• Ease of integration into existing index structures– Indexing– Query processing

Outline

• Introduction• Uncertain Moving Object Model• Movement Inference Model• Indexing and Query Processing

– Index structure– Probabilistic Range Query– k-PNN Query

• Experiments• Conclusion

Uncertain Moving Object Model• Discrete timestamps• 2-D moving objects

• Uncertain moving object representation– Distributions – Instead of exact values

Distribution Representation• Distribution

– Domain discretization– Probability assigned to each cell– Uniform distribution assumption in each cell

0.7 0.3

0.1

0.2

0.2

0.5

0 0.25 0.5 0.75 1

0.25

0.5

0.75

1

-0.2 -0.1 0 0.1 0.2

-0.1

0

0.1

0.2

Location distribution Velocity distribution

Cells with non-zero probabilities

Outline

• Introduction• Uncertain Moving Object Model• Movement Inference Model• Indexing and Query Processing

– Index structure– Probabilistic Range Query– k-PNN Query

• Experiments• Conclusion

Movement Inference Model• Location distribution prediction

– given the location and velocity distributions of oi , update time tu , query time t

– derive a new location distribution for object oi at near-future time t

• Solutions– Rectangle inference– Monte Carlo simulation

Rectangle Inference

0.7 0.3

0.5

0 0.25 0.5 0.75 1

0.25

0.5

0.75

1

-0.2 -0.1 0 0.1 0.2

-0.1

0

0.1

0.2

Location distribution Velocity distribution

0.1

0.2

0.2

0.5

Rectangle Inference

0.7 0.3

0.5

0 0.25 0.5 0.75 1

0.25

0.5

0.75

1

-0.2 -0.1 0 0.1 0.2

-0.1

0

0.1

0.2

Location distribution Velocity distribution

Rectangle Inference

0 0.25 0.5 0.75

0.25

0.5

0.75

1tu

0 0.25 0.5 0.75

0.25

0.5

0.75

1tu+ 1

0 0.25 0.5 0.75

0.25

0.5

0.75

1tu+ 2

0.5

1 1 1

IR1

0.35IR2

0.15

0.25

0.6

0.60.250.25

0.6

0.70.35

IR3

0.245IR4

0.105IR5

0.105IR6

0.045

Monte Carlo Simulation• Randomized method to simulate the motion

– Error rate , confidence , simulation number N– For each simulation

• Initial step: selects a random location • following steps: pick up a velocity• final step: returns location

– Estimate the location distribution with the simulation results

Outline

• Introduction• Uncertain Moving Object Model• Movement Inference Model• Indexing and Query Processing

– Index structure– Probabilistic Range Query– k-PNN Query

• Experiments• Conclusion

Bx-Tree• Use B+-tree to index moving objects• Space filling curve

– 2-D location 1-D key value– 2-D range query Several 1-D range queries

Indexing Uncertain Moving Objects• Index structure

– Time domain partition– Two sub-trees– Reference time– Sub-trees roll over

– Index each location cell with non-zero probabilityalong with the probability and velocity distribution info

Indexing Uncertain Moving Objects• Index update

– Insertion• Identify the sub-tree in which to insert• Infer the location distribution at tref

• Insert the spatial cells with non-zero probabilities

– Deletion• Locate the record in data file• Identify the sub-tree• Infer the location distribution• Delete from the index

Velocity-Based Partitioning• Uncertain larger query expansion

• Tighten velocity bound s.t. decrease query expansion

• Velocity Minimal Bounding Rectangle (VMBR)

• Partition each sub-tree into K logical sub-trees– reduce VMBRs recorded at each sub-tree root 0.7 0.3

-0.2 -0.1 0 0.1 0.2

-0.1

0

0.1

0.2

0.2 0.8

Query Processing• Probabilistic Range Query

– Given a spatial range R, a query time t, and a threshold θ, the probabilistic range query returns all uncertain moving objects falling into R with probability no smaller than θ at time t

• Top-k Probabilistic NN Query(k-PNN)– Given a query location q and a query time t, the k-PNN

query returns k uncertain moving objects with the highest probabilities of being the nearest neighbor of q

Probabilistic Range Query• Growing step

– Issue range query on index– Construct a candidate object list

• Verification step– Rectangle inference works as a filter– Monte Carlo Simulation verifies

k-PNN Query• Issue a series of circular region range queries• Maintain

– lower bound– upper bound– accumulated probability

• Terminate– kth highest lower bound > any other upper bound– accumulated probability = 1

q

k-PNN Query

q

PC(o1,q,3) = 0.8

• PC(oi,q,r)Probability of oi belonging to circle centered at q with radius r

• PR(oi,q,r1,r2) (r1 < r2) Probability of oi belonging to ring centered at q with radius between r1 and r2

o1

PR(o1,q,,2) = 0.2 PR(o1,q,2,3) = 0.6

k-PNN Query

q

iteration object acci lowi upiiteration object acci lowi upi

iteration 2

iteration object acci lowi upi

iteration 2

o1 0.2 0.2 1

iteration object acci lowi upi

iteration 2

o1 0.2 0.2 1

o2 0 0 0.8

iteration object acci lowi upi

iteration 2

o1 0.2 0.2 1

o2 0 0 0.8

o3 0 0 0.8

iteration object acci lowi upi

iteration 2

o1 0.2 0.2 1

o2 0 0 0.8

o3 0 0 0.8

iteration 3

o1 0.8 0.416 0.488

o2 0.6 0.108 0.18

o3 0.1 0.008 0.08

iteration object acci lowi upi

iteration 2

o1 0.2 0.2 1

o2 0 0 0.8

o3 0 0 0.8

iteration 3

o1 0.8 0.416 0.488

o2 0.6 0.108 0.18

o3 0.1 0.008 0.08

iteration 4

o1 1 0.42 0.42

o2 0.9 0.108 0.108

o3 0.8 0.008 0.008

PC(o1,q,2) = 0.2PC(o2,q,2) = 0

0.2 1 10.2 + 0.8 1 10 0.8 10 + 0.8 1 1

PR(o1,q,3,4) = 0.2PC(o1,q,3) = 0.2 + 0.6

0.2 + 0.6 0.4 0.90.416 + 0.2 0.4 0.9

1-PC(o1,q,2) = 0.81-PC(o1,q,3) = 0.2PR(o1,q,,2) = 0.2

o1

PR(o1,q,2,3) = 0.6

PR(o2,q,2,3) = 0.6

1-PC(o2,q,2) = 1 1-PC(o3,q,2) = 1

o3

PR(o3,q,2,3) = 0.1

1-PC(o2,q,3) = 0.4 1-PC(o3,q,3) = 0.9

o2

PR(o2,q,3,4) = 0.3 PR(o3,q,3,4) = 0.7

Outline

• Introduction• Uncertain Moving Object Model• Movement Inference Model• Indexing and Query Processing

– Index structure– Probabilistic Range Query– k-PNN Query

• Experiments• Conclusion

Experiments• Synthetic data

– Uniformly distribute locations– Randomly select directions and speeds– Model uncertainty by Gaussian distribution

• Performance study– Certain model vs. uncertain model– Efficiency tests

Certain Model vs. Uncertain Model• Certain model

– Simple linear motion function– Average velocity and location

• Measurement– Recall – Precision

Certain Model vs. Uncertain Model• Varying probability threshold

0.1 0.15 0.2 0.25 0.30

0.2

0.4

0.6

0.8

1Certain ModelUncertain Model

threshold

Reca

ll

0.1 0.15 0.2 0.25 0.30

0.2

0.4

0.6

0.8

1Certain ModelUncertain Model

thershold

Prec

ision

(a) Recall (b) Precision

Efficiency Tests• Range query size

– NP-tree: index w./o. velocity partition– VP-tree: index w. velocity partition

1 1.5 2 2.5 30

100

200

300

400

500

600

700

800

NP-TreeVP-Tree

Query Size (km)

avg

I/O

cos

t

1 1.5 2 2.5 30

10

20

30

40

50

NP-TreeVP-Tree

Query Size (km)

avg

CPU

cost

(ms)

(a) I/O cost (b) CPU cost

Efficiency Tests• Range query time

– NP-tree: index w./o. velocity partition– VP-tree: index w. velocity partition

10 20 30 40 500

100

200

300

400

500

600

700

800

NP-TreeVP-Tree

Query Time (sec)

avg

I/O

cos

t

10 20 30 40 500

10

20

30

40

50

60

NP-TreeVP-Tree

Query Time (sec)

avg

CPU

cost

(ms)

(a) I/O cost (b) CPU cost

Conclusion• Inferring current/near-future uncertain locations from

past uncertain velocity and location information

• Indexing the uncertain moving objects by means of an adapted Bx-tree

• Processing probabilistic range and nearest neighbor queries

Thank You!