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 $\textstyle \alpha$ and $\textstyle \beta$ region parameters. Like rough sets, the lower and upper approximations of a set are used.

Definitions

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

Conditional risk

Using the Bayesian decision procedure, the decision-theoretic rough set (DTRS) approach allows for minimum-risk decision making based on observed evidence. Let $\textstyle A=\{a_{1},\ldots ,a_{m}\}$ be a finite set of $\textstyle m$ possible actions and let $\textstyle \Omega =\{w_{1},\ldots ,w_{s}\}$ be a finite set of $s$ states. $\textstyle P(w_{j}\mid [x])$ is calculated as the conditional probability of an object $\textstyle x$ being in state $\textstyle w_{j}$ given the object description $\textstyle [x]$ . $\textstyle \lambda (a_{i}\mid w_{j})$ denotes the loss, or cost, for performing action $\textstyle a_{i}$ when the state is $\textstyle w_{j}$ . The expected loss (conditional risk) associated with taking action $\textstyle a_{i}$ is given by:

R(a_{i}\mid [x])=\sum _{j=1}^{s}\lambda (a_{i}\mid w_{j})P(w_{j}\mid [x]).

Object classification with the approximation operators can be fitted into the Bayesian decision framework. The set of actions is given by $\textstyle A=\{a_{P},a_{N},a_{B}\}$ , where $\textstyle a_{P}$ , $\textstyle a_{N}$ , and $\textstyle a_{B}$ represent the three actions in classifying an object into POS( $\textstyle A$ ), NEG( $\textstyle A$ ), and BND( $\textstyle A$ ) respectively. To indicate whether an element is in $\textstyle A$ or not in $\textstyle A$ , the set of states is given by $\textstyle \Omega =\{A,A^{c}\}$ . Let $\textstyle \lambda (a_{\diamond }\mid A)$ denote the loss incurred by taking action $\textstyle a_{\diamond }$ when an object belongs to $\textstyle A$ , and let $\textstyle \lambda (a_{\diamond }\mid A^{c})$ denote the loss incurred by take the same action when the object belongs to $\textstyle A^{c}$ .

Loss functions

Let $\textstyle \lambda _{PP}$ denote the loss function for classifying an object in $\textstyle A$ into the POS region, $\textstyle \lambda _{BP}$ denote the loss function for classifying an object in $\textstyle A$ into the BND region, and let $\textstyle \lambda _{NP}$ denote the loss function for classifying an object in $\textstyle A$ into the NEG region. A loss function $\textstyle \lambda _{\diamond N}$ denotes the loss of classifying an object that does not belong to $\textstyle A$ into the regions specified by $\textstyle \diamond$ .

Taking individual can be associated with the expected loss $\textstyle R(a_{\diamond }\mid [x])$ actions and can be expressed as:

\textstyle R(a_{P}\mid [x])=\lambda _{PP}P(A\mid [x])+\lambda _{PN}P(A^{c}\mid [x]),

\textstyle R(a_{N}\mid [x])=\lambda _{NP}P(A\mid [x])+\lambda _{NN}P(A^{c}\mid [x]),

\textstyle R(a_{B}\mid [x])=\lambda _{BP}P(A\mid [x])+\lambda _{BN}P(A^{c}\mid [x]),

where $\textstyle \lambda _{\diamond P}=\lambda (a_{\diamond }\mid A)$ , $\textstyle \lambda _{\diamond N}=\lambda (a_{\diamond }\mid A^{c})$ , and $\textstyle \diamond =P$ , $\textstyle N$ , or $\textstyle B$ .

Minimum-risk decision rules

If we consider the loss functions $\textstyle \lambda _{PP}\leq \lambda _{BP}<\lambda _{NP}$ and $\textstyle \lambda _{NN}\leq \lambda _{BN}<\lambda _{PN}$ , the following decision rules are formulated (P, N, B):

P: If $\textstyle P(A\mid [x])\geq \gamma$ and $\textstyle P(A\mid [x])\geq \alpha$ , decide POS( $\textstyle A$ );
N: If $\textstyle P(A\mid [x])\leq \beta$ and $\textstyle P(A\mid [x])\leq \gamma$ , decide NEG( $\textstyle A$ );
B: If $\textstyle \beta \leq P(A\mid [x])\leq \alpha$ , decide BND( $\textstyle A$ );

where,

\alpha ={\frac {\lambda _{PN}-\lambda _{BN}}{(\lambda _{BP}-\lambda _{BN})-(\lambda _{PP}-\lambda _{PN})}},

\gamma ={\frac {\lambda _{PN}-\lambda _{NN}}{(\lambda _{NP}-\lambda _{NN})-(\lambda _{PP}-\lambda _{PN})}},

\beta ={\frac {\lambda _{BN}-\lambda _{NN}}{(\lambda _{NP}-\lambda _{NN})-(\lambda _{BP}-\lambda _{BN})}}.

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

P1: If $\textstyle P(A\mid [x])\geq \alpha$ , decide POS( $\textstyle A$ );
N1: If $\textstyle P(A\mid [x])\leq \beta$ , decide NEG( $\textstyle A$ );
B1: If $\textstyle \beta <P(A\mid [x])<\alpha$ , decide BND( $\textstyle A$ ).

When $\textstyle \alpha =\beta =\gamma$ , we can simplify the rules (P-B) into (P2-B2), which divide the regions based solely on $\textstyle \alpha$ :

P2: If $\textstyle P(A\mid [x])>\alpha$ , decide POS( $\textstyle A$ );
N2: If $\textstyle P(A\mid [x])<\alpha$ , decide NEG( $\textstyle A$ );
B2: If $\textstyle P(A\mid [x])=\alpha$ , decide BND( $\textstyle A$ ).

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

References

↑ 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.

External links

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