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![Page 1: Classifying Attributes with Game- theoretic Rough Sets Nouman Azam and JingTao Yao Department of Computer Science University of Regina CANADA S4S 0A2 azam200n@cs.uregina.cajtyao@cs.uregina.ca.](https://reader033.fdocuments.us/reader033/viewer/2022051416/56649ea05503460f94ba2ca3/html5/thumbnails/1.jpg)
Classifying Attributes with Game-theoretic Rough Sets
Nouman Azam and JingTao Yao
Department of Computer Science University of ReginaCANADA S4S 0A2
[email protected] [email protected]://www.cs.uregina.ca/~azam200n http://www.cs.uregina.ca/~jtyao
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Rough Sets
• Sets derived from imperfect, imprecise, and incomplete data may not be able to be precisely defined.
• Sets have to be approximated.
• Rough sets introduces a pair of sets for such approximation.– Lower approximation– Upper approximation
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Visualizing Rough Sets• Let• Lower approximation.
• Upper approximation.
• Positive Region.
• Boundary.
• Negative region.
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Probabilistic Rough Sets• Defines the approximations in terms of conditional
probabilities.– Introduces a pair of threshold denoted as (α, β) to determine
the rough set approximations and regions
– Lower approximation
– Upper approximation
– The three Regions are defined as
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A Key Issue in Probabilistic Rough Sets
• Two extreme cases.– Pawlak Model: (α, β) = (1,0)
• Large boundary. Not suitable in practical applications.
– Two-way Decision Model: α = β• No boundary: Forced to make decisions even in cases of
insufficient information.
• Determining Effective Probabilistic thresholds.
• The GTRS model.– Finds effective values of thresholds with a game-
theoretic process among multiple criteria.
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Game-theoretic Rough Set ModelUtilities for Criterion C1
0.5
0.7
0.9
0.6
0.3
0.2
(α1, β1)
(α2, β2)
(α6, β6)
(α3, β3)
(α4, β4)
(α5, β5)
Rankings based on C1
1 2 3 4 5 6
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Game-theoretic Rough Set ModelUtilities for Criterion C2
0.7
0.1
0.5
0.6
0.8
0.3
(α1, β1)
(α2, β2)
(α6, β6)
(α3, β3)
(α4, β4)
(α5, β5)
Rankings based on C2
1 2 3 4 5 6• Dilemma:
– Ranking of C1 vs C2– Which pair to select
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Game Theory for Solving Dilemma
• Game theory is a core subject in decision sciences.– Prisoners Dilemma.
• A classical example in Game Theory.
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A Game-theoretic Rough Set Approach
• Obtaining Probabilistic threshold with GTRS.– An (α, β) pair is determined with game-theoretic
equilibrium analysis.
C1
C2
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Attribute Types in Rough Sets• Reduct.
– A minimal set of attribute set having the same classification ability as the entire attribute set.
– Generally there may exist multiple reducts.
• Core attribute.– An attribute appearing in every reduct.
• Reduct attribute.– An attribute appearing in at least one reduct.
• Non-reduct attribute.– An attribute that does not appear in any reduct.
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Limitations of Existing Methods
• For classifying attributes we need to find most, if not all, reducts.
• Existing methods for finding multiple reducts.– Commonly involve an iterative process.– Each iteration involves a sub-iterative process for
searching a single reduct.– An attribute may be processed multiple times in
different iterations of these methods.
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A GTRS Based Approach
• We try to find an additional mechanism for classifying attributes.– Processing each attribute once to avoid extensive
computations.
• A GTRS based solution– Interpreting the classification of a feature as a
decision problem within a game.
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Attribute Classification with GTRS
• Formulating problems with GTRS model requires to,– Identify the players.– Identify the strategies of players.– Determine the payoff functions.– Implement a competition.
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J T Yao Incorporating Game Theory in Feature Selection for TC 14
• Players were selected as measures of an attribute importance. – Each measure analyzes an attribute for its
importance.– A case of two player game was considered.
• Two strategies were formulated for each player.– Accepting an attribute, denoted as – Rejecting an attribute, denoted as
Players and Strategies
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Payoff Functions
• Let represents a particular measure.– The value of corresponding to an attribute A, may
be given as,
• Notation for a payoff function.– Payoff of measure , performing action j, given
action k of his opponent is denoted as,
• The payoff functions of a player in four different situations of a game are calculated as,
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Obtaining Attribute Classification• The game may result in three possible
outcomes.– Both players choose to select– One of the players choose to select– None of the players choose to select.
• Attribute classification: An attribute is considered as, – core, when both players choose to select.– reduct, when one of the players choose to select.– Non-reduct, when none of the players select.
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Attribute Classification Algorithm
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A Demonstrative Example
• Core = {e}• Reduct = {a,c,e}• Non-reduct = {b,d,f}
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The Measures in the Game• Conditional Entropy.
• Attribute Dependency.
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Payoff Tables
• The bold cell represents Nash equilibrium.– None of the players can achieve a higher payoff
given their opponents chosen action.– The attribute is classified as core, since both
measures choose to select, i.e. core = {e}.
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Payoff Tables (Cont.)
• The actions of players classify the above attributes as reduct attributes.
• Equilibrium analysis for attribute b, d, f suggest their classification as non-reduct attributes.
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Conclusion
• Limitations of existing approaches.– Extensive computation due to multiple processing of
individual attributes.
• GTRS based method.– Interprets the classification of attributes as a game
among multiple measures of attribute importance.
• Importance of the method.– Each attribute is processed only once in obtaining
the classification of attributes.