spatial range querying for gaussian-based imprecise query objects yoshiharu ishikawa, yuichi iijima...

Spatial Range Querying forGaussian-Based Imprecise

Query Objects

Yoshiharu Ishikawa, Yuichi IijimaNagoya University

Jeffrey Xu YuThe Chinese University of Hong Kong

Outline

• Background and Problem Formulation• Related Work• Query Processing Strategies• Experimental Results• Conclusions

2

3

Imprecise Location Information

• Sensor Environments– Frequent updates may not be possible

• GPS-based positioning consumes batteries

• Robotics– Localization using sensing and

movement histories– Probabilistic approach has

vagueness• Privacy

– Location Anonymity

4

Location-based Range Queries

• Location-based Range Queries– Example: Find hotels located within 2 km from

Yuyuan Garden– Traditional problem in spatial databases

• Efficient query processing using spatial indices• Extensible to multi-dimensional cases (e.g., image

retrieval)

• What happen if the location of query object is uncertain? q

5

Probabilistic Range Query (PRQ) (1)

• Assumptions– Location of query object q is

specified as a Gaussian distribution– Target data: static points

• Gaussian Distribution

– Σ: Covariance matrix

)()(

21exp

||)2(1)( 1

2/12/ qxqxx ΣΣ

tdqp

6


• Probabilistic Range Query (PRQ)

• Find objects such that the probabilities thattheir distances from qare less than δ are greater than θ

q

})Pr(,|{),,( 22 oxΟooqPRQ

7

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

-4-2

0 2

4

0 0.02 0.04 0.06 0.08

• Is distance between q and p within ?


ppdf of q (Gaussian distribution)

Numericalintegraiton is required

8

Naïve Approach for Query Processing

• Exchanging roles– Pr[p is within from q] = Pr[q is within from p]

• Naïve approach– For each object p,

integrate pdf forsphere region R

– R : sphere withcenter p andradius δ

– If the result it is qualified

• Quite costly!

q

pR

Outline


9

Related Work

• Query processing methods for uncertain (location) data– Cheng, Prabhakar, et al. (SIGMOD’03, VLDB’04, …)– Tao et al. (VLDB’05, TODS’07)– Parker, Subrahmanian, et al. (TKDE’07, ‘09)– Consider arbitrary PDFs or uniform PDFs– Target objects may be uncertain

• Research related to Gaussian distribution– Gauss-tree [Böhm et al., ICDE’06]– Target objects are based on Gaussian

distributions

10

Outline


11

Outline of Query Processing

• Generic query processing strategy consists of three phases1. Index-Based Search: Retrieve all candidate

objects using spatial index (R-tree)2. Filtering: Using several conditions, some

candidates are pruned3. Probability Computation: Perform numerical

integration (Monte Carlo method) to evaluate exact probability

• Phase 3 dominates processing cost– Filtering (phase 2) is important for efficiency

12

Query Processing Strategies

• Three strategies1. Rectilinear-Region-Based Approach (RR)2. Oblique-Region-Based Approach (OR)3. Bounding-Function-Based Approach (BF)

• Combination of strategies is also possible

13

Rectilinear-Region-Based (RR) (1)

• Use the concept of -region– Similar concepts are used in query processing for

uncertain spatial databases -region: Ellipsoidal region for which the result of

the integration becomes 1 – 2 :

• The ellipsoidal region

is the -region

21 )()(21)(

r qt

dpqxqx

xxΣ

21)()( r

t

qxqx Σ

15

Rectilinear-Region-Based (RR) (2)• Query processing

– Given a query, -region is computed: it is suffice if we have rθ-table for “normal” Gaussian pdf

• “Normal” Gaussian: = I, q = 0• Given , it returns appropriate rθ

– Derive MBR for -region and perform Minkowski Sum– Retrieve candidates then perform numerical integration

q q

-region

a

b

c

16

Rectilinear-Region-Based (RR) (3)

• Geometry of bounding box

where (Σ)ii is the (i, i) entry of Σ

q

wj

wj

wi wi

xi

xj

iii

ii rw

)(Σ

• Use of oblique rectangle– Query processing based on axis transformation– Not effective for phase 1 (index-based search): Only used

for filtering (phase 2)

17

Oblique-Region-Based (OR) (1)

q

a

b

cb

c

q

a

• Step 1: Rotate candidate objects– Based on the result of eigenvalue decomposition of Σ-1

• Step 2: Check whether each object is inside of the rectangle

– λi: Eigenvalue of Σ-1 for i-th dimension18

Oblique-Region-Based (OR) (2)

q

2/1)( jr

2/1)( ir

19

Bounding-Function-Based (BF) (1)

• Basic idea– Covariance matrix = I (“normal” Gaussian pdf)– Isosurface of pdf has a spherical shape

• Approach– Let be the radius for

which the integration result is

– If dist(q, p) ≤ then p satisfies the condition

– Construct a table that gives (, ) beforehand

q

Pr <

Pr >

Pr =

20


• General case– isosurface has an ellipsoidal shape

• Approach– Use of upper- and

lower-bounding functions for pdf

• They have spherericalisosurfaces

• Derived from covariance matrix

q

21

Bounding Functions

• Original Gaussian pdf

• Upper- and lower-bounding functions

2

2/12/

2

2/12/

2exp

||)2(1)(

2exp

||)2(1)(

qxx

qxx

dq

dq

p

p

T T

)()()( xxx qqq ppp T

Isosurfacehas asphericalshape

)()(

21exp

||)2(1)( 1

2/12/ qxqxx

tdqp

holds

TT

Note: λT = min{λ i}λ = max{λ i}

T

T ( ): Radius with which the integration result of upper- (lower-) bounding function is

22


q

xy

z

T

T

T

T

T

Prob. (integrationresult) =

22

Upper-boundingfunction

Lower-bounding function

Originalpdf


• Theoretical result– Let ST be a spherical region with radius and

its center relative to the origin is βT, and assume that ST satisfies the following equation:

– Using table that gives (, ) , we can get βT:

– Then we can get

23

2/12/norm ||)()( Σd

Sdp

TT

xxx

T

TTT )||)(,( 2/12/d

T

TT

24


• Step 1: Use of R-tree– {b, c, d } are retrieved as

candidates• Step 2: Filtering using T

– b is deleted• Step 2’: Filtering using

– We can determine d as an answer withoutnumerical integration

• Step 3: Numerical integration– Performed on {c}

q

T

T

a

b

c

d T

Outline

• Background and Problem Formulation• Related Work• Query Processing Strategies• Experimental Results• and Conclusions

25

26

Experiments on 2D Data (1)

• Map of Long Beach, CA– Normalized into [0, 1000] [0, 1000]

732327Σ

• 50,747 entries• Indexed by R-tree• Covariance matrix

• γ : Scaling parameter• Default: γ = 10

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Example Query

• Find objects within distance δ = 50 with probability threshold θ = 1%

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• Numerical integration dominates the total cost• R-tree-based search is negligible• ALL is the most effective strategy

δ = 25θ = 0.01γ = 1

γ = 1000

50

100

150

200

RR BF RR+BF RR+OR BF+OR ALL

γ = 1

γ = 10

γ = 100


• Filtering regions (δ = 25, θ = 0.01, γ = 10)

29

RROR

BF (upper)

BF (lower)

IIntegration region for ALL

y

x


• Filtering regions for different uncertainty setting (δ = 25, θ = 0.01)

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γ = 1 :Nearly exact γ = 10 :

Medium uncertaintyγ = 100 :Uncertain


• Motivating Scenario: Example-Based Image Retrieval– User specifies

sample images– Image retrieval

system estimates his interest as a Gaussian distribution

31


• Data set: Corel Image Features data set– From UCI KDD Archive– Color Moments data– 68,040 9D vectors– Euclidean-distance based similarity

• Experimental Scenario: Pseudo-Feedback– Select a random query object, then retrieve k-

NN query (k = 20) as sample images– Derive the covariance matrix from samples

32IΣΣ κ~ : Sample covariance matrix

κ : Normalization parameterΣ~


• Parameters– δ = 0.7: For exact case, it retrieves 15.3 objects – θ = 40%

• Number of candidates (ANS: answer objs)

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0

500

1000

1500

2000

2500

3000

3500

4000

RR BF RR+BF RR+OR BF+OR ALL ANS

3713

3216

2468

1905 19981699

3.9

Too many candidates to retrieve only 3.9 objects!


• Reason: Curse of dimensionality• Plot shows existence probability for pnorm for

different radii and dimensions

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Location of query object is too vague: In medium dimension, it is quite apart from its distribution center on averageExample: For 9D case, the probability that query object is within distance two is only 9%

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

Prob

abili

ty o

f Exi

sten

ce

Radius

2D

3D

5D

9D

15D

Outline


35

Conclusions

• Spatial range query processing methods for imprecise query objects– Location of query object is represented by

Gaussian distribution– Three strategies and their combinations– Reduction of numerical integration is important– Problem is difficult for medium- and high-

dimensional data• Our related work

– Probabilistic Nearest Neighbor Queries (MDM’09)

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Spatial Range Querying forGaussian-Based Imprecise

Query Objects

Yoshiharu Ishikawa, Yuichi IijimaNagoya University

Jeffrey Xu YuThe Chinese University of Hong Kong

spatial range querying for gaussian-based imprecise query objects yoshiharu ishikawa, yuichi iijima...

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