Numerical characterizations of covering rough sets

based on evidence theory Chen Degang, Zhang Xiao

Department of Mathematics and Physics, North China Electric Power University, Beijing, 102206, P. R. China

Outline: 1. Introduction 2. Basic notions related to covering rough sets 3. Belief function and plausibility function of covering rough sets 4. Numerical characterizations of attribute reduction of covering information system 5. Numerical characterizations of attribute reduction of covering decision systems 6. Conclusions

What is covering rough sets?Covering rough sets are improvements of traditional rough sets by considering cover of universe instead of partition.

1. Introduction

Why do we need covering rough sets?Partition and equivalence relation are too restrictive to many applications. One response to this argument is to extend an equivalence relation to more general relations such as similarity relation [21], tolerance relation [4,20] or others [22,25,26]. Another response is to consider a cover instead of partition and obtain covering rough sets [1-3,5,9,16,28,29-32].

The existing study on covering rough setsZakowski employed coverings of universe to establish the generalized rough sets [28]. Bonikowski et al. [1] studied the structures of covers. Mordeson [9] examined the relationship between the approximations of sets defined with respect to covers and some axioms satisfied by traditional rough sets. Chen et al. [5] discussed the covering rough set under the framework of a complete completely distributive lattice. Zhu and Wang [29-32] compared three kinds of generalized rough sets to deal with vagueness and granularity in information system.

Chen et al.[6] began to develop definition and methods of attribute reduction with covering rough sets. In [6] the intersection of coverings was defined and the discernibility matrix was employed to compute all reducts. Their study established a theoretical foundation for attribute reduction of covering decision systems.

1. Introduction

Among these work on covering rough sets, less effort has been concentrated on developing measures for covering rough sets up to now. As well known, in traditional rough set theory different kinds of measures are proposed to reveal numerical characterizations of rough sets and applied to develop algorithms of finding reducts. This fact motivates our idea in this paper to develop measures to characterize covering rough sets numerically.

As pointed in [23,24], there is closed connection between rough set theory and evidence theory. This connection further motivates us to set up connection between covering rough sets and evidence theory, i.e., to characterize approximations and attribute reductions in covering rough sets by employing measures in evidence theory.

1. IntroductionTwo facts motivate our idea

2. Basic notions related to covering rough setsWe recall the basic concepts related to covering rough sets [6]. Definition 2.1. Let be a universe, and a family of subsets of . is called a covering of if none elements in is empty and .

U C UC U C UC

1 2{ , , , }nC C CC Ux U { : , }x j j jC C C x C C ( ) { : }xCov C x U C

U C

Definition 2.2. Let be a covering of . For every , let , then

is also a covering of . We call it the induced covering of .

.

Definition 2.3. Let { : 1, , }i i m C be a family of coverings of U

. For every x U , let { : ( )}x ix ix iC C Cov C , then ( ) { : }xCov x U is also a covering of

U

. We call it the induced covering of .U

For every , the lower and the upper approximations of with respect to are defined as follows

X U X

( ) { : }x xX X ( ) { : }x xX X

The positive region . ( ) ( )Pos X X

In this section, we first discuss the property of lower approximation and propose a new upper approximation for covering rough sets. Lemma 3.1. Let ( , )U be a covering information system and

be a family of coverings of U . For and , we have { : 1,..., }i i m CΔ x U X U

( ) { : } { : }x x xX X x X

In this paper, we define a new upper approximation of to the induced cover of as developed in terms of an induced cover, it certainly can be defined for arbitrary covering. Furthermore, we have the following conclusions.

. Here X with respect

*( ) { : }xX x X * is

3. Belief function and plausibility function of covering rough sets

3. Belief function and plausibility function of covering rough sets

Theorem 3.2. Suppose is a family of coverings of { : 1,..., }i i m CΔ

, the covering lower approximation U and upper approximation *have the following properties:

( )X X *( )X X(1L) (Contraction) (1U) (Extension)(2) (Duality) (Duality) (3L) (Normality) (3U) (Normality)(4L) (Co-normality) (4U) (Co-normality)(5L) (Multiplication)(5U) (Addition)(6L) (Monotone) (6U) (Monotone)(7L) (Idempotency)(7U) (Idempotency)

*(~ ) ~ ( )X X *(~ ) ~ ( )X X ( ) *( ) ( )U U *( )U U

( ) ( ) ( )X Y X Y * * *( ) ( ) ( )X Y X Y

( ) ( )X Y X Y * *( ) ( )X Y X Y ( ( )) ( )X X * * *( ( )) ( )X X

3. Belief function and plausibility function of covering rough sets

Theorem 3.3. Let be a covering information system, , for any , denote , .Then and are belief and plausibility functions on respectively, and the corresponding mass distribution is

here defined as ,

and .

( , )U 1 2{ , , , }nU x x x X U

( ) ( )Bel X X n *( ) ( )Pl X X n

Bel Pl U

1( ), ,( )

0,

ii

fXm X n

otherwise

Δ : ( )f U Cov Δ ( )i if x

1( ) { : ( ) }i k k if x f x

4. Numerical characterizations of attribute reduction of covering information system

The reduct of covering information systems is the minimal subset of

that preserves the induced c

overing .

( )Cov

Theorem 4.1. Let be a covering information system and be

a family of coverings, , , ,

then is a reduct of iff

, and for any nonempty subset , .

( , )S U

1 2{ , , , }nU x x x 1 2( ) { , , , }nCov PP S

1

1 ( ) 1n

ii i

Bel

P P P

1

1 ( ) 1n

ii i

Bel

P

Theorem 4.2. Let be a covering information system and be

a family of coverings, , , ,

, then is a reduct of iff

, and for any nonempty subset , .

( , )S U

1 2{ , , , }nU x x x 1 2( ) { , , , }nCov PP S

P P

1

1 ( )n

ii i

Pl M

Δ

1

1 ( )n

ii i

Pl M

P

1

1 ( )n

ii i

Pl M

P'

4. Numerical characterizations of attribute reduction of covering information system

4. Numerical characterizations of attribute reduction of covering information system

From Theorem 4.1 and 4.2 we conclude that the purpose of attribute reduction in covering information systems is to find a minimal subset which preserves or . In Theorem 4.3 may not hold since may not always hold. Generally we always have even is a basic granule, and this is one difference between covering rough sets and traditional rough sets since every basic granule equals to its lower and upper approximations in traditional rough sets.Now we define the significance of a covering in in a covering information system.

P1

( ) 1n

i ii

Bel

P1

1 ( )n

ii i

Pl M

P

1M ( )i i Δ

( )i i Δ i

4. Numerical characterizations of attribute reduction of covering information system

Definition 4.3. Let be a covering information system.

, we define the significance of the covering by

.

( , )S U

1 2{ , , , }nU x x x C Δ

{ }1 1

( ) ( ) ( )n n

i i i ii i

Sig Bel Bel

CCΔ Δ Δ

Theorem4.4. Let be a covering information system. For

every , is indispensable in in iff .

( , )S U

C Δ D ( ) 0Sig CΔC

Theorem 4.5. .( ) { : ( ) 0}Core Sig C CΔΔ Δ

Definition 4.6. Let be a covering information system.

, for every covering , we define the

significance of the covering relative to by

.

( , )S U

1 2{ , , , }nU x x x (CoreC Δ)

C ( )Core

( ( ) { } ( )1 1

( ) ( ) ( )n n

Core Core i i Core i ii i

Sig Bel Bel

CC Δ) Δ Δ

4. Numerical characterizations of attribute reduction of covering information system

Algorithm 1. Acquire the core and the reduct for a covering information system.

( )Core

C Δ { }1 1

( ) ( ) ( )n n

i i i ii i

Sig Bel Bel

CCΔ Δ Δ

ΔC ( ) 0Sig CΔ( )Core

( ) 0Sig CΔ( ) ( ) { }Core Core CΔ Δ

( )1

( ) 1n

Core i ii

Bel

( )Core

(1) let ;

(2) for each , calculate

(3) if for every , , then , go to step (6);

(4) If , then let ;

(5) if then return , else go to step (6);

(6) let ;

(7) for each , calculate ;

(8) if , then ;

(9) if then stop and output as a reduct , else

go back to step (7).

( )CoreP

{ } C Δ P ( )Sig CP

{ } CP P

1

( ) 1n

i ii

Bel

P P

{ }( ) max ( )i iSig Sig CC CP Δ-P P

Let and , the time complexity of Algorithm 1 is . By Algorithm 1, we can acquire not only the core but also a proper reduct.

m U n 2 2( )O m n

5. Numerical characterizations of attribute reduction of covering decision systems

Similar to attribute reduction of decision systems in traditional rough sets, attribute reduction of covering decision systems aims to find the minimal set of conditional attributes to preserve the positive region of decision attribute [6].

Lemma 5.1. Let be a covering decision system,

, then we have .

( , , { })S U D d

1 2/ { , , , }rU D D D D 1

( ) ( )r

jj

Bel D Pos D U

Theorem 5.2. Let be a covering decision system, , , , then is a relative reduct of iff , and for any nonempty subset ,

.

( , , { })S U D d

1 2{ , , , }nU x x x P1 2/ { , , , }rU D D D D P

S1 1

( ) ( )r r

j jj j

Bel D Bel D

P P P

1 1

( ) ( )r r

j jj j

Bel D Bel D

P P

5. Numerical characterizations of attribute reduction of covering decision systems

Definition 5.3. Let be a covering decision system.

, for every , we define the significance of the covering relative to in by

( , , { })S U D d

1 2/ { , , , }rU D D D D C ΔC D

{ }1 1

( ) ( ) ( )r r

D j jj i

Sig Bel D Bel D

CCΔ Δ Δ

Theorem 5.4. Let be a covering decision system.

For every , is indispensable relative to in iff .

( , , { })S U D d

C Δ C D ( ) 0DSig CΔ

The covering decision systems can be divided into consistent covering decision systems and inconsistent covering decision systems[6]. Generally speaking, for covering decision systems, we only consider to find a minimal subset of to preserve the sum of belief functions of all decision classes.

By Theorem 5.4 and the definition of , we have the following result.Theorem 5.5. .

( )DCore

( ) { : ( ) 0}D DCore Sig C CΔΔ Δ

Definition 5.6. Let be a covering decision system, . For every but , we define the relative significance of the covering to by .

( , , { })S U D d

1 2/ { , , , }rU D D D D ( )i DCoreC C ΔC ( )DCore

( ) ( ) { } ( )1 1

( ) ( ) ( )D D D

r r

Core Core j Core jj i

Sig Bel D Bel D

CC Δ Δ Δ

5. Numerical characterizations of attribute reduction of covering decision systems

Algorithm 2. Acquire the core and the reduction in a covering decision system.

( )Core

C Δ { }1 1

( ) ( ) ( )r r

D j jj i

Sig Bel D Bel D

CCΔ Δ Δ

ΔC ( ) 0DSig CΔ ( )DCore

( ) 0Sig CΔ D ( ) ( ) { }D DCore Core CΔ Δ

( )1 1

( ) ( )r r

Core j jj j

Bel D Bel D

( )DCore

(1) let ;

(2) for each , calculate ;

(3) if for every , , then , go to step (6);

(4) If , then ;

(5) if then return , else go to

step (6);

(6) let ;

(7) for each , calculate ;

(8) if , then ;

(9) if then stop and output as a reduct ,

else go back to step (7).

( )DCoreP

{ } C Δ P { }1 1

( ) ( ) ( )r r

j jj i

Sig Bel D Bel D

CC P P P

{ }( ) max ( )i iSig Sig CC CP Δ-P P { } CP P

1 1

( ) ( )r r

j jj j

Bel D Bel D

P P

Let and , the time complexity of Algorithm 2 is . By Algorithm 2, we can acquire not only the core but also a proper reduct.

m U n 2 2( )O m n

The covering rough set theory is a generalization of traditional rough set theory characterized by covers instead of partitions. Since there is closed connection between rough set theory and evidence theory, we try to propose belief and plausibility functions to characterize covering rough sets. The relationships between attribute reduction and belief (plausibility) function are analyzed in covering information and decision systems respectively. We give the concepts of significance and relative significance of coverings to find reducts in covering information and decision systems. In addition, the relevant algorithms are designed. In a word, we develop a numerical method to find reducts by employing the belief function in covering information (decision) systems. Our future work will concentrate on developing other measures for covering rough sets based on belief and plausibility functions in this paper.

6. Conclusions

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