Decision-theoretic rough sets


In the mathematical theory of decisions, decision-theoretic rough sets 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 ' and ' 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 approach allows for minimum-risk decision making based on observed evidence. Let be a finite set of
possible actions and let be a finite set of states. is
calculated as the conditional probability of an object being in state given the object description
. denotes the loss, or cost, for performing action when the state is.
The expected loss associated with taking action is given
by:
Object classification with the approximation operators can be fitted into the Bayesian decision framework. The
set of actions is given by, where,, and represent the three
actions in classifying an object into POS, NEG, and BND respectively. To indicate whether an
element is in or not in, the set of states is given by. Let
denote the loss incurred by taking action when an object belongs to
, and let denote the loss incurred by take the same action when the object
belongs to.

Loss functions

Let denote the loss function for classifying an object in into the POS region, denote the loss function for classifying an object in into the BND region, and let denote the loss function for classifying an object in into the NEG region. A loss function denotes the loss of classifying an object that does not belong to into the regions specified by.
Taking individual can be associated with the expected loss actions and can be expressed as:
where,, and,, or.

Minimum-risk decision rules

If we consider the loss functions and, the following decision rules are formulated :
where,
The,, and values define the three different regions, giving us an associated risk for classifying an object. When, we get and can simplify into :
When, we can simplify the rules into, which divide the regions based solely on :
Data mining, feature selection, information retrieval, and classifications are just some of the applications in which the DTRS approach has been successfully used.