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.