AAAA NewNewNewNewMethodMethodMethodMethod ofofofof AscertainingAscertainingAscertainingAscertaining AttributeAttributeAttributeAttribute WeightWeightWeightWeight BasedBasedBasedBased ononononRoughRoughRoughRough SetsSetsSetsSets TheorTheorTheorTheoryyyy

BAO Xin-zhong LIU Ying-wenSchool of Economics and Management School of Economics and Management

University of Science and Technology Beijing University of Science and Technology BeijingBeijing 10083, China Beijing 10083, China

baoxz@manage.ustb.edu.cn

Abstract----------------AttributeAttributeAttributeAttribute importanceimportanceimportanceimportance representedrepresentedrepresentedrepresented bybybybyroughroughroughrough setsetsetset isisisis studiedstudiedstudiedstudied deeply.deeply.deeply.deeply. AimingAimingAimingAiming atatatat thethethetheshortcomingsshortcomingsshortcomingsshortcomings ofofofof existingexistingexistingexisting methodsmethodsmethodsmethods ofofofof attributeattributeattributeattributeweightweightweightweight assignmentassignmentassignmentassignment basedbasedbasedbased onononon roughroughroughrough set,set,set,set, bybybybyconsideringconsideringconsideringconsidering bothbothbothboth thethethethe overalloveralloveralloverall significancesignificancesignificancesignificance degreedegreedegreedegreeandandandand thethethethe individualindividualindividualindividual significancesignificancesignificancesignificance degreedegreedegreedegree ofofofof eacheacheacheachattributeattributeattributeattribute inininin thethethethe conditionalconditionalconditionalconditional attributes,attributes,attributes,attributes, aaaa newnewnewnewmethodmethodmethodmethod ofofofof attributeattributeattributeattribute weightweightweightweight assignmentassignmentassignmentassignment basedbasedbasedbased ononononroughroughroughrough setsetsetset isisisis proposed,proposed,proposed,proposed, whichwhichwhichwhich cancancancan improveimproveimproveimprove thethethetheabilityabilityabilityability ofofofof generalizationgeneralizationgeneralizationgeneralization andandandand interpretation.interpretation.interpretation.interpretation.

KeyKeyKeyKey words:words:words:words: weight;weight;weight;weight; roughroughroughrough sets;sets;sets;sets; attributeattributeattributeattributesignificancesignificancesignificancesignificance degreedegreedegreedegree;;;; decisiondecisiondecisiondecision tabletabletabletable

I. PREFACEWeight is vitally important in the process of

judging and decision making, which reflects thestatus and function of every attribute. Methods ofattribute weight assignment that are normallyused include expert evaluation method, fuzzyStatistics method, method of ranking by dualcomparison and so on. The disadvantage of thesemethods is that attribute weight assignment islimited by experts’ experience. Now, someacademicians did researches on methods ofattribute weight assignment based on rough set.

Rough sets theorem was raised by ProfessorPawlak from Poland Warsaw Institute ofTechnology in 1980s, which is a theoreticalmethod researching on the expression, study andgeneralization of incomplete and uncertainknowledge and data. Since it does not need priorinformation, rough set has been appliedsuccessfully in various fields such as expertsystem, machine study and mode identifying [2-4]. Literature [5] provides the method ofconditional attribute weight assignment indecision table, which has been quoted in differentareas by many academicians [6-8]. This thesisanalyzed the shortcomings of existing methods ofattribute weight assignment based on rough set,proposed a new method of attribute weightassignment based on rough set and proved itsrationality.

II. ANALYSIS OF THE EXISTING METHODS OFATTRIBUTE WEIGHT ASSIGNMENT BASED ON

ROUGH SETThe significance degree of attribute (indicator)

in the conditional attributes varies from oneattribute to another, so we should ascertain thesignificance degree of each attribute at first.While applying the rough sets theory, we take outan attribute and then consider how theclassification will change without this attribute. Ifthe classification changes relatively large afterremoving the attribute, we can conclude that thestrength of this attribute is large and the attributeis very important; otherwise, the strength of thisattribute is small and the attribute is of littleimportance.[4-5]. According to this feature,literature [5] defines the importance of attribute(indicator) as follows.

Dependence on knowledge:

DefinitionDefinitionDefinitionDefinition 1111[1]: In decision table ,( , , , , )S UCDV f=

the dependence of decision attribute D( )B Dγ

to conditional attribute sets (indicators) B C⊆

is defined as follows:( ) | ( ) | / | |B BD POS D Uγ =

DefinitionDefinitionDefinitionDefinition 2222[7]: In decision table ,( , , , , )S UCDV f=

the significance degree of conditional attribute(indicator) is defined as follows:c

.{ }( ) ( ) ( )C C cSig c D Dg g -= -

The weight of conditional attribute0( )W c

(indicator) is defined as :c0

( )( )( )

a C

Sig cW cSig a

Î

=å

Note: The definition illustrates that the

greater is, the more important the( )Sig cconditional attribute (indicator) is, so as tocthe weight of the attribute.

2009 International Conference on Electronic Commerce and Business Intelligence

978-0-7695-3661-3/09 $25.00 © 2009 IEEE

DOI 10.1109/ECBI.2009.129

507

2009 International Conference on Electronic Commerce and Business Intelligence

978-0-7695-3661-3/09 $25.00 © 2009 IEEE

DOI 10.1109/ECBI.2009.129

507

In order to explain the shortage of thedefinition better, take an example as follows.

Example 1: The decision table 1

Table 1: The decision table 1

From the decision table 1, we can easily get:U/{a,b,c}={{x1,x2},{x3},{x4},{x5},{x6,x7},

{x8,x9}, {x10,x11}}U/D={{x1,x5,x6,x8,x11},{x2,x3,x4,x7,x9,x1

0}}Therefore：

POS(a,b,c)(D)={x3,x4,x5};POS(a,b)(D)={x3,x4,x5}POS(b,c)(D)={x3,x4,x5};POS(a,c)(D)={x3,x4,x5}.

As a result：

.{ , , } { , , }( ) | ( ) | / | | 3 / 11a b c a b cD POS D Ug = =

Similar that:

.{ , } { , } { , }( ) ( ) ( ) 3 / 11a b a c b cD D Dg g g= = =

Then: ;{ , , } { , }( ) ( ) ( ) 0a b c b cSig a D Dg g= - =

;{ , , } { , }( ) ( ) ( ) 0a b c a cSig b D Dg g= - =

.{ , , } { , }( ) ( ) ( ) 0a b c a bSig c D Dg g= - =

From the example above, we can see that theweight of conditional attributes can not becalculated accurately in that event. Conditionalattributes (a, b, and c) are unnecessary in thedecision Table 1 in respect to simple attribute.Aiming at solving this problem, literature [7]simplified the decision table and figured out theimportance of the attributes. In this way, theimportance of at least one attribute is not zero,and then we can figured out the weight with theformula of ascertaining weight in literature [5].Although this method can guarantee that at leastone weight not to be zero, it ignores the practicalsignificance of the attributes which occupies noweight and the reduced ones. In order to solvethis problem, we offer a new method ofascertaining weight.

III. Analysis of significance degree ofattribute weight based on rough sets

a b c dx1 0 1 2 0x2 0 1 2 1x3 1 0 1 1x4 0 0 1 1x5 1 3 1 0x6 2 1 2 0x7 2 1 2 1x8 2 1 1 0x9 2 1 1 1x10 0 2 2 1x11 0 2 2 0

Definition 3: In decision table ,( , , , , )S UCDV f=

compared to the conditional attribute sets( ), the conditionalC 1 2/ { , , , }mU C C C C= ⋯

entropy of decision attribute sets( | )I D C

( ) is defined as follows:D 1 2/ { , , , }kU D D D D= ⋯

2

21 1

| | | || |( | ) 1

| | | || |

m kj i j ii

i j i i

D C D CCI D CC CU= =

∩ ∩⎛ ⎞= −⎜ ⎟

⎝ ⎠∑ ∑

Definition4: In decisiontable ,for , denote( , , , , )S U C DV f= , ( )P Q C D∀ ⊆ ∪

, ,if1 2/ { , ,..., }hU P P P P= 1 2/ { , ,..., }SU Q Q Q Q=

, such that ,then/ /i jP U P Q U Q∀ ∈ ⇒ ∃ ∈ i jP Q⊆

called as the refinement of , denoted/U Q /U P

by / /U P U Q°

Theorem 1: In decision table ,( , , , , )S U C DV f=

for , there was .( )Q P C D∀ ⊆ ⊆ ∪ / /U P U Q°

Proof:

Denoted , ,1 2/ { , , , }hU P P P P= ⋯ 1 2/ { , , , }sU Q Q Q Q= ⋯ ∀

,due to ,then[ ] /i PP x U P= ∈ Q P⊆

,|{][ PayxP pi ∈∀== jQaxfayf ⊆= )},(),(

.)},(),(,|{][ axfayfQayx Q =∈∀==

According to the randomicity of , we caniP

get ./ /U P U Q°

Theorem 2: In domain of discourse , letU

,1 2/ { , , , }tU B B B B= ⋯

1 2/ { , , , }kU D D D D= ⋯

,1 2 1 1 1 1/ { , , , , , , , , , , }p p q q t p qU B B B B B B B B B B− + − +′ = ∪⋯ ⋯ ⋯

then .( | ) ( | )I D B I D B′ ≥

The equality holds if and only if:|)||(||| qjqpj BDBBD ∩−∩

=0(j=1,2,...,k)|)||(|||0 qjqpj BDBBD ∩−∩∧=

Proof:( | ) ( | )I D B I D B′ −

2

21

| | | ( ) | | ( ) |1

| | | || |

kP q j P q j P q

j P q P q

B B D B B D B BB B B BU =

⎛ ⎞∪ ∩ ∪ ∩ ∪= −⎜ ⎟⎜ ⎟∪ ∪⎝ ⎠

∑22

2 21 1

| | | | | | | | | || |1 1

| | | | | | | || | | |

k kj P j P q j q j qP

j jP P q q

D B D B B D B D BBB B B BU U= =

⎛ ⎞∩ ∩ ∩ ∩⎛ ⎞− − − −⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

∑ ∑

21

1 (| | | |)(| | | | | |)| |

k

j P j q P q j P j qj

D B D B B B D B D BU =

= ∩ + ∩ ∪ − ∩ − ∩∑

|)||(|||||

11

2 pjppj

k

jBDBBD

U∩−∩− ∑

=

508508

|)||(|||||

11

2 qjqqj

k

jBDBBD

U∩−∩− ∑

=

Let ,| |, | |, | |, | |j j P j j q P qx D B y D B x B y B= ∩ = ∩ = =

then ,and0, 0j jx x y y≥ ≥ ≥ ≥ ( | ) ( | )I D B I D B′ −

21

1 ( )( )| |

k

j j j jj

x y x y x yU =

= + + − −∑

2 21 1

1 1( ) ( )| | | |

k k

j j j jj jx x x y y y

U U= =

− − − −∑ ∑

21

1 ( ( ) ( ) ) 0| |

k

j j j jj

x y y x x yU =

= − + − ≥∑

（for ）0, 0j jx x y y≥ ≥ ≥ ≥

The equality holds if and only if :

For , ,{1,2, , }j k∀ ∈ ⋯ ( ) 0 ( ) 0j j j jx y y x x y− = ∧ − =

| |(| | | |) 0 | |(| | | |) 0j P q j q j q p j pD B B D B D B B D B∩ − ∩ = ∧ ∩ − ∩ =

Theorem 3: In decision table S=(U,C,D,V,f),, , thenB P C∀ ⊆ ⊆ 1 2/ { , , , }kU D D D D= ⋯

.( | ) ( | )I D P I D B≤

The equality holds if and only if:

For , if ,, /i jX X U P∀ ∈ ( ) /i jX X Y U B∪ ⊆ ∈

there were :|)||(||| jrjir XDXXD ∩−∩

|)||(|||0 jrjir XDXXD ∩−∩∧==0 (r=1,2,…,k)

Proof: Let , ,1 2/ { , , , }sU P P P P= ⋯ 1 2/ { , , , }tU B B B B= ⋯according to the probative process of theorem 1,

we can easily get that: for any ,there/iB U B∈

exists , which causes1 2, , , /j j jhP P P U P∈⋯

.1 2i j j jhB P P P= ∪ ∪ ∪⋯

According to theorem 2, we can easily getthat: .( | ) ( | )I D P I D B≤

The equality holds if and only if:

For , if ,, /i jX X U P∀ ∈ ( ) /i jX X Y U B∪ ⊆ ∈

there were

: |)||(||| jrjir XDXXD ∩−∩

|)||(|||0 jrjir XDXXD ∩−∩∧==0(r=1,2,…,k)Note: The theorem illustrates that the

information entropy of the conditional attributessets is monotonically descending. So we candefine the importance of attributes as follows:

Definition5: In decision table S=(U,C,D,V,f),for , the significance of attribute (index)c C∀ ∈

is defined as ：c

.( ) ( | ( { })) ( | )NewSig c I D C c I D C= − −

Theorem 4: In decision table S=(U,C,D,V,f),

for , NewSig(c)=0, then Sig(c)=0.c C∀ ∈

Note: Theorem 4 shows that if thesignificance degree of a certain attribute is zeroaccording to the definition of new attribute'ssignificance degree, then it is also zero with theoriginal attribute's significance degree. But theconverse proposition of theorem 4 does not hold,so we illustrate this with the decision table 1.

According to the foregoing calculation indecision table 1, we can get that : Sig(a)=Sig(b)=Sig(c)=0.

In decision table 1:U/{a,b,c}={{x1,x2},{x3},{x4},{x5},{x6,.x7},{x

8,x9},{x10,x11}}U/D={{x1,x5,x6,x8,x11},{x2,x3,x4,x7,x9,x10}},then:

2

21 1

| | | || |( | ) 1

| | | || |

m kj i j ii

i j i i

D C D CCI D CC CU= =

∩ ∩⎛ ⎞= −⎜ ⎟

⎝ ⎠∑ ∑ 11

121=

U/{a,b}={{x1,x2},{x3},{x4},{x5},{x6,.x7,x8,x9},{x10,x11}}( | ( { }))I D C c−

+⎟⎠⎞

⎜⎝⎛ −+−⎟

⎠⎞

⎜⎝⎛= )

211(

21)

211(

21

112 2

⎟⎠⎞

⎜⎝⎛ −+−⎟

⎠⎞

⎜⎝⎛ )

111(

11)

101(

10

111 2

+ +⎟⎠⎞

⎜⎝⎛ −+−⎟

⎠⎞

⎜⎝⎛ )

111(

11)

101(

10

111 2

⎟⎠⎞

⎜⎝⎛ −+−⎟

⎠⎞

⎜⎝⎛ )

101(

10)

111(

11

111 2

+⎟⎠⎞

⎜⎝⎛ −+−⎟

⎠⎞

⎜⎝⎛ )

421(

42)

421(

42

114 2

⎟⎠⎞

⎜⎝⎛ −+−⎟

⎠⎞

⎜⎝⎛+ )

211(

21)

211(

21

112 2

.15121

=

Thus:( ) ( |( { })) ( | ) 15/121 11/121 4/121 0Sig c I D C c I D C= − − = − = >

Similar that: ,( ) 4 /121 0Sig b = >

.( ) 6 /121 0Sig a = >

The example shows that the significance ofeach conditional attribute is not zero according tothe new significance degree. This example andtheorem 4 show that the new definition ofattribute’s significance degree is more reasonable.But when we calculating the importance degreeof attributes in whichever decision table with thenew definition ,we cannot guarantee that everyattribute’s significance degree is note zero .Inorder to solve this problem, we give a newdefinition of attribute weight as follows.

IV.A new method of attribute weight assignmentbased on rough set

509509

Definition 6:In decision table S=(U,C,D,V,f),,then conditional attribute (index) c’sc C∀ ∈

weight W(c) is defined as ：

.

( ) ( | { })( ){ ( ) ( | { })}

a C

NewSig c I D cW cNewSig a I D a

∈

+=

+∑Note: In this definition, NewSig(c) represents

conditional attribute c’s importance degree in thewhole set, and I(D|{c}) represents conditionalattribute c’ s importance degree in the system.

NewSig(c) represents the variation degree ofattribute set after removing attribute c, so it iseasy to cause some conditional attributes to beunnecessary when the number of attributes in thedecision table is large. The attribute’ssignificance degree is zero when applying themethod of weight assignment in literature [7].Weight represents the certain attribute’srelatively importance degree in the whole and theattribute’s objective weight should not be zero.This definition introduced the definition ofweight into the combination of NewSig(c)andI(D|{c}). On one hand, it is the overallinvestigation of the significance degree ofattribute in the whole set and in itself. On theother hand, it makes every attribute’s weight notto be zero and therefore makes weightassignment to be more reasonable.

Theorem 5: In decision table S=(U,C,D,V,f),

, there at least exists , suchc C∀ ∈ ,i jx x U∈

that and ,( , } ( , }i jf x c f x c= ( , } ( , }i jf x D f x D≠

then W(c)>0.Proof: Let { | ( , } ( , }i iD x x U f x D f x D= ∈ ∧ =

,{ | ( , } ( , }j jD x x U f x D f x D= ∈ ∧ =

due to , then and .( , } ( , }i jf x D f x D≠ i jD D≠ ,i i j jx D x D∈ ∈

Due to , then ,( , } ( , }i jf x c f x c= { }[ ]i j cx x∈

thus .Due to ,{ }[ ]i i j cx D x∈ ∩ i i i jx D x D∈ ∧ ∉

then , therefore{ } { }| [ ] | | [ ] |i j c j cD x x∩ <

.

2{ } { } { }2

{ } { }

| [ ] | | [ ] | | [ ] |1 0

| [ ] | | [ ] || |j c i j c i j c

j c j c

x D x D xx xU

⎛ ⎞∩ ∩− >⎜ ⎟⎜ ⎟

⎝ ⎠Then I(D|{c})>0, thus W(c)>0..According to the upper probative process, in a

decision table which contains at least twodecision variable: for any conditional attribute c,there exist more than two objects based on cwhich cannot be distinguished, but their decisionattribute values are different. That is, more thantwo objects’ values based on c are the same, butdifferent in their decision attribute. It shows that,in that decision table, we cannot get assured

decision result with a single attribute c. Underthis condition, all results of attribute weight arenot zero according to our method of attributeweight assignment based on rough set.

V. Conclusion

Shortcomings of existing methods of attributeweight assignment based on rough set have beendiscussed. The significant degree of rough setattributes has been analyzed. A new method ofattribute weight assignment based on rough sethas been proposed and justified. The new method,by considering both the overall significancedegree and the individual significance degree ofeach attribute in the conditional attributes,improves the ability of generalization andinterpretation.

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