Cheng, Chen, Chen, Xie Evaluating Probability Threshold k- Nearest-Neighbor Queries over Uncertain...

Cheng, Chen, Chen, Xie

Evaluating Probability Threshold k-Nearest-Neighbor Queries over Uncertain Data

Reynold Cheng (University of Hong Kong)Lei Chen (Hong Kong University of Science &Tech)Jinchuan Chen (Hong Kong Polytechnic University)Xike Xie (University of Hong Kong)

International Conference on Extending Database Technology 2009


Agenda

1. Introduction 2. Problem Definition 3. Basic Solution 4. Efficient Solution 5. Results


Data Uncertainty

Inherent in various applicationsLocation-based services (e.g., using GPS,

RFID) [TDRP98, SSDBM99]Natural habitat monitoring with sensor

networks [VLDB04a]Biomedical and biometric databases[ICDE06,

ICDE07]


Attribute Uncertainty Model [TDRP98,ISSD99,VLDB04b]

pdf

y

PreciseLocation

x

yL

CloakedLocation

x

y

probabilitydensity function

U

(pdf)

Uncertainty region

We represent an uncertainty pdf as a histogram


k-NN Queries

k-NN Query over Precise Data- application in LBS [VLDB03]- natural habitat monitoring system [VLDB04a]- network traffic analysis [ICDCS07]- pattern matching in CAM [VLDB04c]

k-NN over Uncertain Objects- [VLDB08a] ranks the probability each object is the NN of the query point.- [ICDE07a] use expected distance and does not discuss the probability.


Agenda



Probability Threshold k-Nearest-Neighbor Query (T-k-PNN)

INPUT

1. A query point q, parameter k, threshold T

2. A set of n objects with uncertainty regions and pdfs

OUTPUT A number of k-subset

p(S) is the qualification probability of the k-subset S})(|||{ TSpkSDSS

}...,{ 21 nOOOD


Example of a k-PNN query (k=3)

{O1, O2 , O3}

{O1, O2 , O4}

O2

O3

O1

O4

O5

O6

O7

O8

q


Example of a k-PNN query (k=3)

O2

O3

O1

O4

O5

O6

q

{O1, O2, O3} {O1, O2, O4}

…

{O6, O7, O8}

5683 C

k-bound

2063 C

{O1, O2, O3} {O1, O2, O4}

…

{O4, O5, O6}

O7

O8


k-bound Filtering (k=3)

O2

O3

O1

O4

O5

O6

q

k-bound

O7

O8

f1

f2

f3

fk (k-bound): is the k-th minimum maximum distance

Since min(r7)> f3, O7 can not be 3-NN of q. Because there are always 3 objects with distances smaller than f3.We apply k-bound filtering on an index (e.g. R-tree) to prune unqualified objects.


Agenda



Basic solution for a T-k-PNN query (k=3,T=0.1)

3-subset QP

{O1, O2, O3} 0.2

{O1, O2, O4} 0.1

{O1, O3, O4} 0.1

{O2, O3, O4} 0.1

0.05{O2, O3, O5}

0.05{O1, O3, O5}

……

0.05{O1, O2, O5}

0.05{O1, O2, O5}

0.1{O2, O3, O4}

0.1{O1, O3, O4}

0.1{O1, O2, O4}

0.2{O1, O2, O3}

QP3-subset

O2

O3

O1

O4

O5

O6

q

k-bound

T=0.1

Exact QP is expensive to compute!

Too many k-subsets!

Step1: k-bound filteringStep2: QP CalculationStep3: Accept S, if qp(S)≥T

Symbol Meaning

ri |oi − q|

di(r) pdf of ri (distance pdf)

Di(r) cdf of ri (distance cdf)


Agenda



Efficient Solution Framework (GVR)

Lower bound

Upper bound3.

4. Refinement

k-subset Generation

k-subset VerificationAndRefinement

k-subsets

rejectedk-subsets

acceptedk-subsets

Candidate Objects

1. k-bound Filtering

2. Probabilistic Candidate Selection

k-subsets GenerationVerificationRefinement


Probabilistic Candidates Selection

so kii

frScp )Pr()(

)()( ScpSqp

O2

O3

O1

O4

O5

O6

q

k-bound

0.1

0.2

0.5

Cutoff Probability of Oi : Pr(ri ≤fk)

SSScpScp '),()(

S1={O4, O5,O6}cp(S1)=0.5*0.2*0.1 = 0.01S2={O4, O5}

cp(S2)=0.5*0.2 = 0.1Given T=0.2, if cp(S2) < T, then qp(S1)<cp(S1)<T.S1 can be pruned.


Probabilistic Candidates Selection

0.5{O4}

0.2{O5}

0.1{O6}

1{O3}

1{O2}

1{O1}

CP1-subset

0.2{O2, O3, O5}

0.2{O1, O3, O5}

0.1{O1, O4, O5}

0.5{O2, O3, O4}

0.1{O2, O4, O5}

0.1{O3, O4, O5}

0.5{O1, O3, O4}

0.2{O1, O2, O5}

0.5{O1, O2, O4}

1{O1, O2, O3}

CP3-subset

1{O2,O3}

0.5{O2,O4}

0.2{O2,O5}

0.5{O3,O4}

0.2{O3,O5}

0.2{O1,O5}

0.1{O4,O5}

0.5{O1,O4}

1{O1,O3}

1{O1,O2}

CP2-subset

T=0.2, k=3


Storage Efficient Compression

1{O2,O3}

0.5{O2,O4}

0.2{O2,O5}

0.5{O3,O4}

0.2{O3,O5}

0.2{O1,O5}

0.5{O1,O4}

1{O1,O3}

1{O1,O2}

CP2-subset

Subsets are sorted in descending order of their CPs.

{O3,O5}

{O2,O5}

{O1,O5}

Size-2 Set

Original subsets

Compressed subsets

Store the common prefix of the subsetsAnd the last element of the subset that has the minimum product of cutoff probability greater than T


Storage Efficient Compression

0.5{O4}

0.2{O5}

0.1{O6}

1{O3}

1{O2}

1{O1}

CP1-subset

0.2{O2, O3, O5}

0.2{O1, O3, O5}

0.1{O1, O4, O5}

0.5{O2, O3, O4}

0.1{O2, O4, O5}

0.1{O3, O4, O5}

0.5{O1, O3, O4}

0.2{O1, O2, O5}

0.5{O1, O2, O4}

1{O1, O2, O3}

CP3-subset

1{O2,O3}

0.5{O2,O4}

0.2{O2,O5}

0.5{O3,O4}

0.2{O3,O5}

0.2{O1,O5}

0.1{O4,O5}

0.5{O1,O4}

1{O1,O3}

1{O1,O2}

CP2-subset

{O4}

{O5}

{O3}

{O2}

{O1}

Size-1 Set

{O3,O5}

{O2,O5}

{O1,O5}

Size-2 Set Size-3 Set

{O1,O2,O5}

{O1,O3,O5}

{O2,O3,O5}

T=0,2, k=3


O3

Seeds Pruning

O1

O2

q

O4

k=3

f1

f2f3

min(r4) > f2 > f1

Seeds: o1, o2, o3

If o4 belongs to a 3-nn set S, o1 and o2 must also belong to S.

r4 > r2 r4 > r1

min(r4)

For example, we can prune the set {o1,o3,o4}, according to the above rule.

max(r1) =f1 max(r2) =f2 max(r3) =f3

No CP calculation is needed.

Can prune more candidate k-sets


Verifiers: Upper and Lower Bounds (T=0.2)

Candidates k-subsets

(After PCS)

0

1

S1 1

0

0.190.19

0.6

0.10.5 ?

0.4

0.540.14

0.15

0.180.03

Verifier Incremental Refinement

Classifier

1

1

0

S2

1 S3

0

1


Verification and Refinement

0.3 0.3 0.1

0.3 0.7

0.30.7

r1

r2

r3

0.3 0.5 0.8

0.3 1

0.9

10.7

P1 P2 P3 P4

D1(e4)

e2 e3 e4e1

P1 P2 P3

r1

r2

r3

q

e5

f2

0.1

P4

0.2

Partitions Stair-Case Model

Divide the range [min(r1), fk] into a series of partitions.

Extended from the probabilistic verifiers in [ICDE08b]

Build a data structure, i.e. stair-case model, to store the distance cdf of each object.

Derive the lower and upper bounds of a k-set’s QP based on the stair-case model.

Reject (Accept) a k-set once its QP must be lower (larger) than the threshold.


Agenda



Experiment Setup

Uncertain Object DB

Long Beach (53k)(http://www.census.gov/geo/www/tiger/)

Uncertainty pdf Uniform (default)Gaussian (represented by histograms)

Threshold (T) 0.1

k 6


1. k-bound Filtering


2. Performance of GVR


3. k-subset Generation


4. Verification and Refinement


5. Time Analysis


6. Gaussian Distribution


Conclusion

We proposed an efficient evaluation framework for T-k-PNN query

We proposed various techniques:- k-bound to filter away those unqualified objects- PCS to reduce the number of k-subsets- verification/refinement methods to avoid exact calculation

Future Work- extend the techniques to other queries


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