Rule Measures Tradeoff Using
Game-Theoretic Rough Sets
Yan Zhang
JingTao Yao
Department of Computer Science
University of Regina, CANADA
[email protected], [email protected]
Dec 2012
Outline
Introduction
Concepts and Background
Formulating Competitions Between Measures
Example
Conclusion
Outline
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Introduction
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Concepts and background
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Formulating competition between measures with GTRS
I
Example of using measures in GTRS
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Conclusion and future research
Y. Zhang and J. T. Yao
Rule Measures Tradeoff Using Game-Theoretic Rough Sets
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Outline
Introduction
Concepts and Background
Formulating Competitions Between Measures
Example
Conclusion
Research problem
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Classification rules can be induced from positive and negative
regions based on rough sets theory.
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Large boundary region limits the practical applicability of rough
sets.
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Probabilistic rough sets provide a solution to define the more
applicable regions by weakening the strict conditions.
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Determining effective probabilistic thresholds remains a major
challenge. Multiple criteria or aspects may be considered for
obtaining effective thresholds
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Rule Measures Tradeoff Using Game-Theoretic Rough Sets
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Outline
Introduction
Concepts and Background
Formulating Competitions Between Measures
Example
Conclusion
Approach
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Game-Theoretic Rough Rets (GTRS): use game mechanism to
exploit the relationships between multiple measures and
thresholds to obtain effective rough set regions.
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Multiple measures are set as players in the game, which depend
on the application requirements.
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How to select suitable players is the fundamental for game
formulation.
GTRS
rules
Players Strategies Payoff functions
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Outline
Introduction
Concepts and Background
Formulating Competitions Between Measures
Example
Conclusion
Game-Theoretic Rough Sets (GTRS)
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was proposed by J.T. Yao and J.P. Herbert on 2008;
I
aims to achieve suitable probabilistic thresholds that satisfies two
or more criteria by formulating game among multi criteria;
I
can be applied in rules mining, feature selection, classification,
uncertainty analysis, etc.
G = {O, S, P}
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I
I
I
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O: a set of players, O = {o1 , o2 , ..., on }
S: a set of strategies S = {S1 , S2 , ..., Sn } and Si : a set of possible
actions for player oi , i.e. Si = {ai1 , ai2 , ..., aimi }
P: the set of payoff functions, P = {u1 , u2 , ..., un }
payoff table and equilibrium
o1
Y. Zhang and J. T. Yao
a11
a12
......
a21
u1 (a11 , a21 ), u2 (a11 , a21 )
u1 (a12 , a21 ), u2 (a12 , a21 )
......
o2
a22
u1 (a11 , a22 ), u2 (a11 , a22 )
u1 (a12 , a22 ), u2 (a12 , a22 )
......
Rule Measures Tradeoff Using Game-Theoretic Rough Sets
.....
......
......
......
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Outline
Introduction
Concepts and Background
Formulating Competitions Between Measures
Example
Conclusion
Multiple Criteria and Thresholds
Utilitiesvforv Utilitiesvforv
criterionvC1 criterionvC2
Y. Zhang and J. T. Yao
0.7
0.6
0.8
0.7
0.9
0.5
0.4
0.9
0.3
0.8
Dilemma
v-vRankingvofvC1vvsvC2
v-vwhichvpairvtovselect
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Outline
Introduction
Concepts and Background
Formulating Competitions Between Measures
Example
Conclusion
Game-Theoretic Rough Set Approach
Obtaining probabilistic thresholds with GTRS. The effective
threshold pair is determined with game-theoretic equilibrium analysis.
C2
......
C1
......
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Rule Measures Tradeoff Using Game-Theoretic Rough Sets
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Outline
Introduction
Concepts and Background
Formulating Competitions Between Measures
Example
Conclusion
Immediate decision rules
I
Decision table S = (U, At = C ∪ D, {Va |a ∈ At}, {Ia |a ∈ At}), C
is the set of condition attributes, D is the set of decision attributes
and D = {D1 , D2 , ..., Dn };
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For [x] ∈ POSEC (EDi ), we can get immediate decision rules
des([x]) −→ des(Di );
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With the increase of positive region, more rules can be induced
but with lower accuracy;
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Many different measures can be used to evaluate immediate
decision rules set (IDRS).
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Outline
Introduction
Concepts and Background
Formulating Competitions Between Measures
Example
Conclusion
Different formulating approaches with GTRS
Game formulation
probabilistic
thresholds
classification
measures
properties of
rough sets
Players
Strategies
threshold (α)
Sα = {α(1 − c1 ), α(1 − c2 ), α(1 − c3 )}
threshold (β )
Sβ = {β (1 + c1 ), β (1 + c2 ), β (1 + c3 )}
accuracy (φ )
Sφ = {↓ RP , ↑ RN , ↑ RB }
precision (ψ)
Sψ = {↓ RP , ↑ RN , ↑ RB }
accuracy (a)
Sa = {(α, β ), (α(1 − c%), β ), (α, β (1 +
c%)), (α(1 − c%), β (1 + c%)}
generality (g)
Sg = {(α, β ), (α(1 − c%), β ), (α, β (1 +
c%)), (α(1 − c%), β (1 + c%)}
SI = {α ↓, β ↑, α ↓ β ↑}
uncertainty of
regions
uncertainty of POS
and NEG (I)
uncertainty of BND (D)
positive rules
evaluation
confidence (con)
Scon = {α, α(1 − 5%), α(1 − 10%)}
coverage (cov)
Scov = {α, α(1 − 5%), α(1 − 10%)}
Y. Zhang and J. T. Yao
SD = {α ↓, β ↑, α ↓ β ↑}
Rule Measures Tradeoff Using Game-Theoretic Rough Sets
Payoff functions
|POS|‘ −|POS|
α×(1−ci )
|NEG|‘ −|NEG|
u(βi ) = β ×(c −1)
i
|apr(α,β ) (A)|
uφ = |apr
(α,β ) (A)|
|apr(α,β ) (A)|
uψ =
|A|
correct# classified by POS&NEG
total# classified by POS&NEG
u(αi ) =
ua =
ug =
total# classified by POS&NEG
uI =
|U|
(1−4P (α,β ))+(1−4N (α,β ))
2
uD = 1 − 4B (α, β )
# correctly classified by PRS
ucon = # totally classified by PRS C
C
# correctly classified by PRSC
ucov =
|C|
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Outline
Introduction
Concepts and Background
Formulating Competitions Between Measures
Example
Conclusion
Framework
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Players: O = {m1 , m2 }
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Strategies: S = {S1 , S2 }
for player mi , Si = {ai1 , ..., aik } and aij = fij (α, β )
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Payoff functions: P = {u1 , u2 }
u1 (αij , βij ), here (αij , βij ) = F(f1i (α, β ), f2j (α, β ))
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Payoff table
m1
Y. Zhang and J. T. Yao
a11 = f11 (α, β )
a12 = f12 (α, β )
......
a21 = f21 (α, β )
u1 (α11 , β11 ), u2 (α11 , β11 )
u1 (α21 , β21 ), u2 (α21 , β21 )
......
m2
a22 = f22 (α, β )
u1 (α12 , β12 ), u2 (α12 , β12 )
u1 (α21 , β22 ), u2 (α22 , β22 )
......
Rule Measures Tradeoff Using Game-Theoretic Rough Sets
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......
......
......
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Outline
Introduction
Concepts and Background
Formulating Competitions Between Measures
Example
Conclusion
Measures
There are many different measures for rules evaluation, and they can
be selected as game players and compete against each other in GTRS
to get suitable thresholds.
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Accuracy
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Confidence
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Coverage
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Generality
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Certainty
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Support
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Rule Measures Tradeoff Using Game-Theoretic Rough Sets
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Outline
Introduction
Concepts and Background
Formulating Competitions Between Measures
Example
Conclusion
Game formulation
P(Xi )
P(D1 |Xi )
P(D2 |Xi )
P(D3 |Xi )
0.034
1
0
0
0.067
1
0
0
0.098
0
1
0
0.121
0
0
1
0.139
0.42
0.26
0.32
0.157
0.25
0.55
0.2
0.146
0.11
0.26
0.63
0.117
0.23
0.72
0.05
0.088
0.91
0.02
0.07
0.033
0.06
0.05
0.89
Objective: to determine the thresholds (α, β ) which can make the
confidence of IDRS greater than 0.8 and generality of IDRS greater
than 0.7.
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Y. Zhang and J. T. Yao
Players: confidence (γ) and generality (τ), O = {γ, τ}
)
Strategies: S = {Sγ , Sτ }, aij = fij (α, β ) = α − (j−1)(α−β
10
Sγ = {a11 , a12 , a13 , a14 } = {(1, 0), (0.9, 0), (0.8, 0), (0.7, 0)}
Sτ = {a21 , a22 , a23 , a24 } = {(1, 0), (0.9, 0), (0.8, 0), (0.7, 0)}
Payoff functions: P = {uγ , uτ }
f (α,β )+f2j (α,β )
f (α,β )+f2j (α,β )
uγ (αij , βij ) = γ( 1i
), uτ (αij , βij ) = τ( 1i
)
2
2
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Outline
Introduction
Concepts and Background
Formulating Competitions Between Measures
Example
Conclusion
Payoff table
a11 = (α, β )
confidence(γ)
a12 = (0.9α, β )
a13 = (0.8α, β )
a14 = (0.7α, β )
a21
= (α, β )
γ(α, β ),
τ(α, β )
γ(0.95α, β ),
τ(0.95α, β )
γ(0.90α, β ),
τ(0.90α, β )
γ(0.85α, β ),
τ(0.85α, β )
generality(τ)
a
22 = (0.9α, β )
a23
= (0.8α, β )
γ(0.95α, β ),
γ(0.9α, β ),
τ(0.95α, β )
τ(0.9α, β )
γ(0.90α, β ),
γ(0.85α, β ),
τ(0.85α,
β)
τ(0.90α, β )
γ(0.85α, β ),
γ(0.8α,
β
),
τ(0.85α,
β)
τ(0.8α, β )
γ(0.8α, β ),
γ(0.75α,
β
),
τ(0.8α, β )
τ(0.75α, β )
a
24 = (0.7α, β )
γ(0.85α, β ),
τ(0.85α,
β)
γ(0.8α, β ),
τ(0.8α, β )
γ(0.75α, β ),
τ(0.75α,
β)
γ(0.7α, β ),
τ(0.7α, β )
τ
γ
a11
a12
a13
a14
a21
(1, 0)
(0.95, 0)
(0.9, 0)
(0.85, 0)
a22
(0.95, 0)
(0.9, 0)
(0.85, 0)
(0.8, 0)
a23
(0.9, 0)
(0.85, 0)
(0.8, 0)
(0.75, 0)
a24
(0.85, 0)
(0.8, 0)
(0.75, 0)
(0.7, 0)
τ
γ
a11
a12
a13
a14
a21
< 1, 0.32 >
< 1, 0.32 >
< 0.981, 0.408 >
< 0.973, 0.441 >
a22
< 1, 0.32 >
< 0.981, 0.408 >
< 0.973, 0.441 >
< 0.973, 0.441 >
a23
< 0.981, 0.408 >
< 0.973, 0.441 >
< 0.973, 0.441 >
< 0.973, 0.441 >
a24
<0.973,0.441>
< 0.973, 0.441 >
< 0.973, 0.441 >
< 0.921, 0.558 >
(a11 , a24 ) is the equilibrium, but < 0.973, 0.441 > does not satisfy the
stop condition, so the whole process should be repeated by setting
(0.85, 0) as initial thresholds.
Y. Zhang and J. T. Yao
Rule Measures Tradeoff Using Game-Theoretic Rough Sets
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Outline
Introduction
Concepts and Background
Formulating Competitions Between Measures
Example
Conclusion
Repetitive thresholds modification
)
(α, β ) = (0.85, 0) and aij = fij (α, β ) = α − (j−1)(α−β
10
τ
γ
a11
a12
a13
a14
a21
(0.85, 0)
(0.8075, 0)
(0.765, 0)
(0.7225, 0)
a22
(0.8075, 0)
(0.765, 0)
(0.7225, 0)
(0.68, 0)
a23
(0.765, 0)
(0.7225, 0)
(0.68, 0)
(0.6375, 0)
a24
(0.7225, 0)
(0.68, 0)
(0.6375, 0)
(0.595, 0)
τ
γ
a11
a12
a13
a14
a21
< 0.9738, 0.4410 >
< 0.9738, 0.4410 >
< 0.9738, 0.4413 >
< 0.9738, 0.4413 >
a22
< 0.9738, 0.4410 >
< 0.9738, 0.4410 >
< 0.9738, 0.4413 >
< 0.9206, 0.5580 >
a23
< 0.9738, 0.4410 >
< 0.9738, 0.4410 >
< 0.9206, 0.5580 >
< 0.9206, 0.5580 >
a24
< 0.9738, 0.4410 >
< 0.9206, 0.5580 >
< 0.9206, 0.5580 >
<0.8603,0.7040>
When (α, β ) = (0.595, 0), the confidence of IDRS is greater than 0.8
and the generality of IDRS is greater than 0.7. So we can get qualified
rules from POS(0.595,0) (D).
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Outline
Introduction
Concepts and Background
Formulating Competitions Between Measures
Example
Conclusion
Conclusion and Future Research
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We present GTRS for formulating competition between
measures.
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We discuss possible measures for evaluating a set of immediate
decision rules and analyze their properties.
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More games can be formulated by using these measures, and the
result in this study may enhance our understanding and the
applicability of GTRS.
The future research will focus on:
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Y. Zhang and J. T. Yao
more general GTRS model;
more formulation approaches of GTRS;
the algorithm and implementation of GTRS.
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