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 ${\displaystyle \textstyle A=\{a_{1},\ldots ,a_{m}\}}$ be a finite set of $\textstyle m$ possible actions and let ${\displaystyle \textstyle \Omega =\{w_{1},\ldots ,w_{s}\}}$ be a finite set of $s$ states. ${\displaystyle \textstyle P(w_{j}\mid [x])}$ is calculated as the conditional probability of an object $\textstyle x$ being in state ${\displaystyle \textstyle w_{j}}$ given the object description $\textstyle [x]$. ${\displaystyle \textstyle \lambda (a_{i}\mid w_{j})}$ denotes the loss, or cost, for performing action $\textstyle a_{i}$ when the state is ${\displaystyle \textstyle w_{j}}$. The expected loss (conditional risk) associated with taking action $\textstyle a_{i}$ is given by:

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

Object classification with the approximation operators can be fitted into the Bayesian decision framework. The set of actions is given by ${\displaystyle \textstyle A=\{a_{P},a_{N},a_{B}\}}$, where ${\displaystyle \textstyle a_{P}}$, ${\displaystyle \textstyle a_{N}}$, and ${\displaystyle \textstyle a_{B}}$ 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 ${\displaystyle \textstyle \Omega =\{A,A^{c}\}}$. Let ${\displaystyle \textstyle \lambda (a_{\diamond }\mid A)}$ denote the loss incurred by taking action ${\displaystyle \textstyle a_{\diamond }}$ when an object belongs to $\textstyle A$, and let ${\displaystyle \textstyle \lambda (a_{\diamond }\mid A^{c})}$ denote the loss incurred by take the same action when the object belongs to ${\displaystyle \textstyle A^{c}}$.

Loss functions

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

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

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

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

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

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

Minimum-risk decision rules

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

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

where,

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

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

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

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

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

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

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

See also

• Rough sets
• Granular computing
• Soft computing
• Fuzzy set theory

References

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.

External links

• The International Rough Set Society
• The Decision-theoretic Rough Set Portal