Decision-theoretic rough sets explained

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,^[1] the extension makes use of loss functions to derive

style\alpha

and

style\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

styleA=\{a_1,\ldots,a_m\}

be a finite set of

stylem

possible actions and let

style\Omega=\{w_1,\ldots,w_s\}

be a finite set of

states.

styleP(w_j\mid[x])

iscalculated as the conditional probability of an object

stylex

being in state

stylew_j

given the object description

style[x]

styleλ(a_i\midw_j)

denotes the loss, or cost, for performing action

stylea_i

when the state is

stylew_j

.The expected loss (conditional risk) associated with taking action

stylea_i

is givenby:

R(a_i\mid[x])=

	s
\sum
	j=1

λ(a_i\midw_j)P(w_j\mid[x]).

Object classification with the approximation operators can be fitted into the Bayesian decision framework. Theset of actions is given by

styleA=\{a_P,a_N,a_B\}

, where

stylea_P

stylea_N

, and

stylea_B

represent the threeactions in classifying an object into POS(

styleA

), NEG(

styleA

), and BND(

styleA

) respectively. To indicate whether anelement is in

styleA

or not in

styleA

, the set of states is given by

style\Omega=\{A,A^c\}

. Let

styleλ(a_\diamond\midA)

denote the loss incurred by taking action

stylea_\diamond

when an object belongs to

styleA

, and let

styleλ(a_\diamond\midA^c)

denote the loss incurred by take the same action when the objectbelongs to

styleA^c

Loss functions

Let

styleλ_PP

denote the loss function for classifying an object in

styleA

into the POS region,

styleλ_BP

denote the loss function for classifying an object in

styleA

into the BND region, and let

styleλ_NP

denote the loss function for classifying an object in

styleA

into the NEG region. A loss function

styleλ_\diamond

denotes the loss of classifying an object that does not belong to

styleA

into the regions specified by

style\diamond

Taking individual can be associated with the expected loss

styleR(a_{\diamond\mid[x])}

actions and can be expressed as:

styleR(a_P\mid[x])=λ_PPP(A\mid[x])+λ_PNP(A^c\mid[x]),

styleR(a_N\mid[x])=λ_NPP(A\mid[x])+λ_NNP(A^c\mid[x]),

styleR(a_B\mid[x])=λ_BPP(A\mid[x])+λ_BNP(A^c\mid[x]),

where

styleλ_\diamond=λ(a_\diamond\midA)

styleλ_\diamond=λ(a_\diamond\midA^c)

, and

style\diamond=P

styleN

, or

styleB

Minimum-risk decision rules

If we consider the loss functions

styleλ_PP\leqλ_BP<λ_NP

and

styleλ_NN\leqλ_BN<λ_PN

, the following decision rules are formulated (P, N, B):

P: If

styleP(A\mid[x])\geq\gamma

and

styleP(A\mid[x])\geq\alpha

, decide POS(

styleA

);

N: If

styleP(A\mid[x])\leq\beta

and

styleP(A\mid[x])\leq\gamma

, decide NEG(

styleA

);

B: If

style\beta\leqP(A\mid[x])\leq\alpha

, decide BND(

styleA

);

where,

\alpha=

	λ_PN-λ_BN
	(λ_BP-λ_BN)-(λ_PP-λ_PN)

\gamma=

	λ_PN-λ_NN
	(λ_NP-λ_NN)-(λ_PP-λ_PN)

\beta=

	λ_BN-λ_NN
	(λ_NP-λ_NN)-(λ_BP-λ_BN)

The

style\alpha

style\beta

, and

style\gamma

values define the three different regions, giving us an associated risk for classifying an object. When

style\alpha>\beta

, we get

style\alpha>\gamma>\beta

and can simplify (P, N, B) into (P1, N1, B1):

P1: If

styleP(A\mid[x])\geq\alpha

, decide POS(

styleA

);

N1: If

styleP(A\mid[x])\leq\beta

, decide NEG(

styleA

);

B1: If

style\beta<P(A\mid[x])<\alpha

, decide BND(

styleA

When

style\alpha=\beta=\gamma

, we can simplify the rules (P-B) into (P2-B2), which divide the regions based solely on

style\alpha

P2: If

styleP(A\mid[x])>\alpha

, decide POS(

styleA

);

N2: If

styleP(A\mid[x])<\alpha

, decide NEG(

styleA

);

B2: If

styleP(A\mid[x])=\alpha

, decide BND(

styleA

Data mining, feature selection, information retrieval, and classifications are just some of the applications in which the DTRS approach has been successfully used.

References

Yao. Y.Y.. Wong, S.K.M. . 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. North-Holland. Knoxville, Tennessee, USA. 17–25.

Decision-theoretic rough sets explained

Definitions

Conditional risk

Loss functions

Minimum-risk decision rules

See also

References

External links