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=\$ be a finite set of $\textstyle m$
possible actions and let $\textstyle \Omega=\$ be a finite set of $s$ states. \textstyle P(w_j\mid is
calculated as the conditional probability of an object $\textstyle x$ being in state $\textstyle w_j$ given the object description
\textstyle /math>. $\textstyle \lambda(a_i\mid w_j)$ 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 = \sum_^s \lambda(a_i\mid w_j)P(w_j\mid .

Object classification with the approximation operators can be fitted into the Bayesian decision framework. The
set of actions is given by $\textstyle A=\$, 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=\$. 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 $\textstyle \lambda_$ denote the loss function for classifying an object in $\textstyle A$ into the POS region, $\textstyle \lambda_$ denote the loss function for classifying an object in $\textstyle A$ into the BND region, and let $\textstyle \lambda_$ denote the loss function for classifying an object in $\textstyle A$ into the NEG region. A loss function $\textstyle \lambda_$ 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 actions and can be expressed as:

: \textstyle R(a_P\mid = \lambda_P(A\mid + \lambda_P(A^c\mid ,

: \textstyle R(a_N\mid = \lambda_P(A\mid + \lambda_P(A^c\mid ,

: \textstyle R(a_B\mid = \lambda_P(A\mid + \lambda_P(A^c\mid ,

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

Minimum-risk decision rules

If we consider the loss functions $\textstyle \lambda_ \leq \lambda_ < \lambda_$ and $\textstyle \lambda_ \leq \lambda_ < \lambda_$, the following decision rules are formulated (''P'', ''N'', ''B''):

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

where,

: $\alpha = \frac,$

: $\gamma = \frac,$

: $\beta = \frac.$

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 (''P''1, ''N''1, ''B''1):

* P1: If \textstyle P(A\mid \geq \alpha, decide POS($\textstyle A$);
* N1: If \textstyle P(A\mid \leq \beta, decide NEG($\textstyle A$);
* B1: If \textstyle \beta < P(A\mid < \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 > \alpha, decide POS($\textstyle A$);
* N2: If \textstyle P(A\mid < \alpha, decide NEG($\textstyle A$);
* B2: If \textstyle P(A\mid = \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
* Fuzzy set theory

References

External links

The International Rough Set Society

The Decision-theoretic Rough Set Portal

{{DEFAULTSORT:Decision-Theoretic Rough Sets
Decision theory