mining preferences from superior and inferior examples kdd’08 1
TRANSCRIPT
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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