Decision-theoretic rough sets

In the mathematical theory of decisions, decision-theoretic rough sets (DTRS) is a probabilistic extension of rough set classification. First created in 1990 by Dr. Yiyu Yao,[1] the extension makes use of loss functions to derive and region parameters. Like rough sets, the lower and upper approximations of a set are used.

Definitions

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The following contains the basic principles of decision-theoretic rough sets.

Conditional risk

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Using the Bayesian decision procedure, the decision-theoretic rough set (DTRS) approach allows for minimum-risk decision making based on observed evidence. Let   be a finite set of   possible actions and let   be a finite set of   states.   is calculated as the conditional probability of an object   being in state   given the object description  .   denotes the loss, or cost, for performing action   when the state is  . The expected loss (conditional risk) associated with taking action   is given by:

 

Object classification with the approximation operators can be fitted into the Bayesian decision framework. The set of actions is given by  , where  ,  , and   represent the three actions in classifying an object into POS( ), NEG( ), and BND( ) respectively. To indicate whether an element is in   or not in  , the set of states is given by  . Let   denote the loss incurred by taking action   when an object belongs to  , and let   denote the loss incurred by take the same action when the object belongs to  .

Loss functions

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Let   denote the loss function for classifying an object in   into the POS region,   denote the loss function for classifying an object in   into the BND region, and let   denote the loss function for classifying an object in   into the NEG region. A loss function   denotes the loss of classifying an object that does not belong to   into the regions specified by  .

Taking individual can be associated with the expected loss  actions and can be expressed as:

 
 
 

where  ,  , and  ,  , or  .

Minimum-risk decision rules

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If we consider the loss functions   and  , the following decision rules are formulated (P, N, B):

  • P: If   and  , decide POS( );
  • N: If   and  , decide NEG( );
  • B: If  , decide BND( );

where,

 
 
 

The  ,  , and   values define the three different regions, giving us an associated risk for classifying an object. When  , we get   and can simplify (P, N, B) into (P1, N1, B1):

  • P1: If  , decide POS( );
  • N1: If  , decide NEG( );
  • B1: If  , decide BND( ).

When  , we can simplify the rules (P-B) into (P2-B2), which divide the regions based solely on  :

  • P2: If  , decide POS( );
  • N2: If  , decide NEG( );
  • B2: If  , decide BND( ).

Data mining, feature selection, information retrieval, and classifications are just some of the applications in which the DTRS approach has been successfully used.

See also

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References

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  1. ^ Yao, Y.Y.; Wong, S.K.M.; Lingras, P. (1990). "A decision-theoretic rough set model". Methodologies for Intelligent Systems, 5, Proceedings of the 5th International Symposium on Methodologies for Intelligent Systems. Knoxville, Tennessee, USA: North-Holland: 17–25.
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