Classifying Attributes with Gametheoretic Rough Sets
Nouman Azam and JingTao Yao
Department of Computer Science
University of Regina
CANADA S4S 0A2
azam200n@cs.uregina.ca jtyao@cs.uregina.ca
http://www.cs.uregina.ca/~azam200n http://www.cs.uregina.ca/~jtyao
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
Visualizing Rough Sets
• Let
• Lower approximation.
• Upper approximation.
• Positive Region.
• Boundary.
• Negative region.
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
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 gametheoretic process among multiple criteria.
Game-theoretic Rough Set Model
Utilities for Criterion C1
(α1, β1)
0.5
(α2, β2)
0.7
(α3, β3)
0.9
Rankings based on C1
(α4, β4)
0.6
(α5, β5)
0.3
(α6, β6)
0.2
1
2
3
4
5
6
Game-theoretic Rough Set Model
(α1, β1)
Utilities for Criterion C2
Rankings based on C2
0.7
1
(α2, β2)
0.1
• Dilemma:
(α3, β3)
0.5
– Ranking of C1 vs C2
(α4, β4)
0.6
– Which pair to select
(α5, β5)
0.8
(α6, β6)
0.3
2
3
4
5
6
Game Theory for Solving Dilemma
• Game theory is a core subject in decision
sciences.
– Prisoners Dilemma.
• A classical example in Game Theory.
A Game-theoretic Rough Set Approach
• Obtaining Probabilistic threshold with GTRS.
– An (α, β) pair is determined with game-theoretic
equilibrium analysis.
C1
C2
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.
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.
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.
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.
Players and Strategies
• 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
J T Yao
Incorporating Game Theory in Feature Selection for TC
13
Payoff Functions
• Let
represents a particular measure.
– The value of
be given as,
corresponding to an attribute A, may
• 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,
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.
Attribute Classification Algorithm
A Demonstrative Example
• Core = {e}
• Reduct = {a,c,e}
• Non-reduct = {b,d,f}
The Measures in the Game
• Conditional Entropy.
• Attribute Dependency.
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}.
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.
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.