# 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, 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.