- Decision-theoretic rough sets
-
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
and
region parameters. Like rough sets, the lower and upper approximations of a set are used.
Contents
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
be a finite set of
possible actions and let
be a finite set of s 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 (conditional risk) 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 (P, N, B):
- P: If
and
, decide POS(
);
- N: If
and
, decide NEG(
);
- B: If
, decide BND(
);
where,
,
,
.
The
,
, and
values define the three different regions, giving us an associated risk for classifying an object. When
beta" border="0">, we get
gamma > \beta" border="0"> and can simplify (P, N, B) into (P1, N1, B1):
- P1: If
, decide POS(
);
- N1: If
, decide NEG(
);
- B1: If
, decide BND(
).
When
, we can simplify the rules (P-B) into (P2-B2), which divide the regions based solely on
:
- P2: If
alpha" border="0">, decide POS(
);
- N2: If
, decide NEG(
);
- B2: If
, decide BND(
).
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
- ^ Yao, Y.Y.; Wong, S.K.M. and 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
Categories: - P: If
Wikimedia Foundation. 2010.