Decision-theoretic rough sets
Encyclopedia
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
and 
region parameters. Like rough sets, the lower and upper approximations of a set are used.
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 (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
.
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 
.
and 
, the following decision rules are formulated (P, N, B):
where,
,
,
.
The
, 
, and 
values define the three different regions, giving us an associated risk for classifying an object. When 
, we get 
and can simplify (P, N, B) into (P1, N1, B1):
When
, we can simplify the rules (P-B) into (P2-B2), 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.
Rough set
In computer science, a rough set, first described by a Polish computer scientist Zdzisław I. Pawlak, is a formal approximation of a crisp set in terms of a pair of sets which give the lower and the upper approximation of the original 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.Conditional Risk
Using the Bayesian decision procedure, the decision-theoretic rough set (DTRS) approach allows for minimumrisk decision making based on observed evidence. Let
be a finite set of possible actions and let
be a finite set of states. iscalculated 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 givenby:
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 threeactions in classifying an object into POS(
), NEG(), and BND() respectively. To indicate whether anelement 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 objectbelongs 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 , we get 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
, decide POS(
); - N2: If
, decide NEG(
); - B2: If
, decide BND(
).
Data mining
Data mining
Data mining , a relatively young and interdisciplinary field of computer science is the process of discovering new patterns from large data sets involving methods at the intersection of artificial intelligence, machine learning, statistics and database systems...
, feature selection
Feature selection
In machine learning and statistics, feature selection, also known as variable selection, feature reduction, attribute selection or variable subset selection, is the technique of selecting a subset of relevant features for building robust learning models...
, information retrieval
Information retrieval
Information retrieval is the area of study concerned with searching for documents, for information within documents, and for metadata about documents, as well as that of searching structured storage, relational databases, and the World Wide Web...
, and classifications are just some of the applications in which the DTRS approach has been successfully used.