Mining Preferences from Superior and Inferior

Examples

KDD’08

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Outline

Introduction Problem Definition Greedy Method

Term-Based Algorithm Condition-Based Algorithm

Experimental Result Conclusion

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Introduction

In a multidimensional space where the user preferences on some categorical attribute are unknown.

Example

ID Price Age Developer

a 1600 2 X

b 2400 1 Y

c 3000 5 Z

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Cont.

Superior Example (S) For each superior example o, according to the

customer preference there does not exist another realty o’ which is as good as o in every aspect, and is better than o in at least one aspect.

Inferior Example (Q) For each inferior example o, according to the

customer preference there exist at least one realty o’ which is as good as o in every aspect, and is better than o in at least one aspect.

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Problem Definition

D = such that : Determined attribute. : Undetermined attribute.

SPS ( Satisfying Preference Set)

d: The dimensionality of Dd’: The number of determined attribute

D UD D =D UD D DD

UD

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'+1{ ,..., } is called a SPS ifd dR

' '(1 - ') is a perference on attribute d i d ii d d D

' 1( , ,..., )D d d

Example

A: Determined attribute B,C: Undetermined attribute S = { } Q ={ }

Object

A B C

o1 a1 b1 c1

o2 a1 b1 c2

o3 a2 b1 c2

o4 a2 b2 c2

o5 a2 b3 c2

1 3,o o 2o

1 2Cc c

1 2Aa a

1 2Cc c

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Cont.

Problem 1 (SPS Existence) Given a set of superior examples S and a set of

inferior examples Q , determine whether there exists at least SPS R with respect to S and Q.

Problem (Minimal SPS) For a set of superior example S and a set of

inferior examples Q, find a SPS with respect to S and Q such that is minimized. R is called a minimal SPS.

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'+1{ ,..., }d dR ' 1 | ( ,... ) | d dE

Cont.

Theorem (Multidimensional Preference) In space , let

be a preference on attribute Di and .

Then

1 2 ... dD D D D (1 )i i d 1( ,..., )d

1 1

| ( ) | ( | ( ) | | | ) - | |d d

i i ii i

E E D D

| |iD is the number of distinct values in attribute Di

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Term-Based AlgorithmObject-

idD1 D2 D3 D4 Label

o1 1 5 a3 b3

o2 1 6 a2 b1 Inferior

o3 1 6 a2 b3

o4 2 2 a1 b1

o5 2 5 a2 b2 Inferior

o6 3 1 a4 b3

o7 3 4 a2 b2 Inferior

o8 6 1 a5 b1 Inferior

o9 6 1 a5 b3

o10 6 2 a1 b1 Inferior

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Cont.

Inferior P(q) Condition Cq (p)

o2o1

o3

o5o1

o4

o7o4

o6

o8o6

o9

2 3 3 4 1( ) ( )OC O b b 2 1 3 3 2 3 4 1( ) ( ) ( )OC O a a b b

5 1 3 3 2 3 4 2( ) ( ) ( )OC O a a b b

5 4 1 3 2 1 4 2( ) ( ) ( )OC O a a b b

7 4 1 3 2 1 4 2( ) ( ) ( )OC O a a b b

7 6 4 3 2 3 4 2( ) ( ) ( )OC O a a b b

8 6 4 3 5 3 4 1( ) ( ) ( )OC O a a b b

8 9 3 4 1( ) ( )OC O b b 10

Cont.

Complexity increment CI

If is selected on D3

3 4{ , }R 3 4| | 5,| | 3D D

3 1 3 2 4{ } and a a

1 1

| ( ) | ( | ( ) | | | ) - | |d d

i i ii i

E E D D

| ( ) | (1 5) (0 3) 5 3 3E R

3 3 1a a

3| ( ) | 3E

| ( ) | (3 5) (0 3) 5 3 9E R

3 3 1( ) 9 3CI a a

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Cont. Inferior Example Coverage

Cov(t) is the number of interior examples newly satisfied if t is selected.

Inferior P(q) Condition Cq (p)

o2o1

o3

o5o1

o4

o7o4

o6

o8o6

o9

2 3 3 4 1( ) ( )OC O b b 2 1 3 3 2 3 4 1( ) ( ) ( )OC O a a b b

5 1 3 3 2 3 4 2( ) ( ) ( )OC O a a b b

5 4 1 3 2 1 4 2( ) ( ) ( )OC O a a b b

7 4 1 3 2 1 4 2( ) ( ) ( )OC O a a b b

7 6 4 3 2 3 4 2( ) ( ) ( )OC O a a b b

8 6 4 3 5 3 4 1( ) ( ) ( )OC O a a b b

8 9 3 4 1( ) ( )OC O b b

3 4 1( )Cov b b max{0.5,1} max{0.5,1} 2

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Cont.

Term\Iteration 1 2 3

D3

1/3 1/4* \

1/3 1/8 1/8

1/6 1/8 1/8

1/6 \ \

D4

1/5 1/10 2/12*

2/5* \ \

1/5 1/5 1/6

1 3 2a a

3 3 2a a

4 3 2a a

4 3 5a a

1 4 2b b

3 4 1b b

3 4 2b b

( )( )

( )

Cov tscore t

CI t

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Condition-Based Algorithm

Inferior Condition 1 2 3

o2o1 3/9 \ \

o3 2/5* \ \

o5o1 2/9 1.5/10* \

o4 2/9 2/16

o7o4 2/9 2/16 1/12

o6 1.5/9 1.5/10 1/5*

o8o6 2.5/9 \ \

o9 2/5 \ \14

Experimental Results

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Conclusion

Mining Preference

Greedy Method: Score (t) ?

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