overview of rough sets

Rough

Set

Overview of Rough Sets

Rough Sets (Theoretical Aspects of Reasoning about Data)by Zdzislaw Pawlak

2

Contents

1. Introduction 2. Basic concepts of Rough Sets

information system equivalence relation / equivalence class / indiscernibility relation set approximation (Lower & Upper Approximations)

- accuracy of Approximation- extension of the definition of approximation of sets

dispensable & indispensable- reducts and core

dispensable & indispensable attributes- independent- relative reduct & relative core

dependency in knowledge- partial dependency of attribute (knowledge)- significance of attributes

discernibility Matrix

3. Example decision Table dissimilarity Analysis

3

Introduction

Rough Set theory

by Zdzislaw Pawlak in the early 1980’s

use : AI, information processing, data mining, KDD etc.

ex) feature selection, feature extraction, data reduction,

decision rule generation and pattern extraction (association rules) etc.

theory for dealing with information with uncertainties.

reasoning from imprecision data.

more specifically, discovering relationships in data.

The idea of the rough set consists of the approximation of a set(X) by a p

air of sets, called the low and the upper approximation of this set(X)

4

Information System

Knowledge Representation System ( KR-system , KRS )

Information systems I = < U, Ω>

a finite set U of objects , U={x1, x2, .., xn} ( universe )

a finite set Ω of attributes , Ω = {q1, q2, .., qm} ={ C, d }

C : set of condition attribute d : decision attribute

ex) I = < U, {a, c, d}>

UU a c dx1 1 4 yes

x2 1 1 no

x3 2 2 no

x4 2 2 Yes

x5 3 3 no

x6 1 3 yes

x7 3 3 no

5

Equivalence Relation

An equivalence relation R on a set U is defined as

i.e. a collection R of ordered pairs of elements of U, satisfying certain properties.

1. Reflexive: xRx for all x in U ,

2. Symmetric: xRy implies yRx for all x, y in U

3. Transitive: xRy and yRz imply xRz for all x, y, z in U

))}()((|),{(R ycxcUUyx jjc j

6

Equivalence Class

R : any subset of attributes( )

If R is an equivalence relation over U,

then U/R is the family of all equivalence classes of R

: equivalence class in R containing an element

ex) an subset of attribute ‘R1={a}’ is equivalence relation

the family of all equivalence classes of {a}

: U/ R1 ={{ 1, 2, 6}{3, 4}{5, 7}}

equivalence class

:

※ A family of equivalence relation over U will be call a knowledge base over U

UU a c d1 1 4 yes

2 1 1 no

3 2 2 no

4 2 2 Yes

5 3 3 no

6 1 3 yes

7 3 3 no

R

Rx][ Ux

}6,2,1{]1[1R }4,3{]3[

1R }7,5{]5[

1R

7

Indiscernibility relation

If and ,

then is also an equivalence relation, ( IND(R) )

and will be called an indiscernibility relation over R ※ : intersection of all equivalence relations belonging to R

equivalence class of the equivalence relation IND(R)

:

U/IND(R) : the family of all equivalence classes of IND(R)

ex) an subset of attribute ‘R={a, c}’

the family of all equivalence classes of {a}

: U/{a}={{ 1, 2, 6}{3, 4}{5, 7}}

the family of all equivalence classes of {b}

: U/{c}={{ 2}{3, 4}{5, 6, 7}{1}}

the family of all equivalence classes of IND(R)

: U/IND(R) ={{2}{6}{1} {3,4}{5,7}}=U/{a,c}

UU a c d

1 1 4 yes

2 1 1 no

3 2 2 no

4 2 2 Yes

5 3 3 no

6 1 3 yes

7 3 3 no

R)R( ][][ xx IND

R

R R

R

8

I = <U, Ω> = <U, {a, c}>

U is a set (called the universe) Ω is an equivalence relation on U (called an indiscernibility relation).

U is partitioned by Ω into equivalence classes,

elements within an equivalence class are indistinguishable in I.

An equivalence relation induces a partitioning of the universe.

The partitions can be used to build new subsets of the universe.

※ equivalence classes of IND(R) are called basic categories (concepts) of knowledge R

※ even union of R-basic categories will be called R-category

UU a c1 1 4

2 1 1

3 2 2

4 2 2

6 1 3

5 3 3

7 3 3

9

Set Approximation

Given I = <U, Ω> Let and R : equivalence relation We can approximate X using only the information contained in R

by constructing the R-lower( ) and R-upper( ) approximations of X,

where

or

X is R-definable (or crisp) if and only if ( i.e X is the union of some R-basic categories, called R-definable set, R-exact set)

X is R-undefinable (rough) with respect to R if and only if ( called R-inexact, R-rough)

UX

XR XR

}.][|{ XxxXR R

},][|{ XxxXR R

}.:/{ XYRUYXR

}:/{ XYRUYXR

XRXR

XRXR

)I(INDR

10

R-positive region of X : R-borderline region of X : R-negative region of X :

XRXRXBNR )(

XRUXNEGR )(

XRXPOSR )(

U

U/R R : subset of attributes

set X

XRXR ∴ X is R-definable

U/R

U

set X

∴ X is R-rough (undefinable)XR XR

11

EX) I = <U, Ω>, let R={a, c} , X={x | d(x) = yes}={1, 4, 6}

► approximate set X using only the information contained in R

the family of all equivalence classes of IND(R)

: U/IND(R) = U/R = {{1}{ 2}{6} {3,4}{5,7}}

R-lower approximations of X

:

R-lower approximations of X

:

※ The set X is R-rough since the boundary region is not empty

UU a c d

1 1 4 yes

2 1 1 no

3 2 2 no

4 2 2 Yes

5 3 3 no

6 1 3 yes

7 3 3 no}4,3,6,1{}][|{R XxxX B

}6,1{}][|{R XxxX B

}4,3{)(R XBN }7,5,2{)(R XNEG}6,1{)(R XPOS

12

Lower & Upper Approximations

yes

yes/no

no

{x1, x6}

{x3, x4}

{x2, x5,x7}

XR

XR

13

Accuracy of Approximation

accuracy measure αR(X)

: the degree of completeness of our knowledge R about the set X

If , the R-borderline region of X is empty

and the set X is R-definable (i.e X is crisp with respect to R).

If , the set X has some non-empty R-borderline region

and X is R-undefinable (i.e X is rough with respect to R).

ex) let R={a, c} , X={x | d(x) = yes}={1, 4, 6}

Rcard

RcardXR )( .X.10 R

1)( XR

1)( XR

}4,3,6,1{}][|{R XxxX B

}6,1{}][|{R XxxX B

5.04

2)(R

Rcard

RcardX

14

R-roughness of X

: the degree of incompleteness of knowledge R about the set X

ex) let R={a, c} , X={x | d(x) = yes}={1, 4, 6}

Y={x | d(x) = no}={2, 3, 6, 7}

U/IND(R) = U/R = {{1}{ 2}{6} {3,4}{5,7}}

)(1)( XX RR

5.05.01)(1)(R XX R

}4,3,6,1{}][|{R R XxxX

}6,1{}][|{R R XxxX

5.04/2)(R X

}6,2{}][|{R R YxxY

}7,5,6,4,3,2{}][|{R R YxxY

67.06/4)(1)( RR YY 33.06/2)(R Y

15

Extension of the definition of approximation of sets

F={X1, X2, ..., Xn} : a family of non-empty sets and

=> R-lower approximation of the family F : R-upper approximation of the family F :

},,{ 21 nXRXRXRFR

},,,{ 11 nXRXRXRFR

ex) R={a, c}

F={X, Y}={{1,4,6}{2,3,5,7}} , X={x | d(x) = yes},

Y={x | d(x) = no}

U/IND(R) = U/R = {{1}{ 2}{6} {3,4}{5,7}}

}7,6,5,2,1{}}7,5,2}{6,1{{},{ YRXRFR

}7,6,5,4,3,2,1{}}7,6,5,4,3,2}{6.4.3.1{{},{ YRXRFR

UF

UU a c d

1 1 4 yes

2 1 1 no

3 2 2 no

4 2 2 Yes

5 3 3 no

6 1 3 yes

7 3 3 no

16

the accuracy of approximation of F: the percentage of possible correct decisions when classifying objects employing the knowledge R

the quality of approximation of F : the percentage of objects which can be correctly classified to classes of F

employing the knowledge R

i

iR

XRcard

XRcardF )(

Ucard

XRcardF i

R)(

7/5)( FR2/1)64/()32()( FR

ex) R={a, c}

F={X, Y}={{1,4,6}{2,3,6,7}} , X={x | d(x) = yes}, Y={x | d(x) = no}

}7,6,5,2,1{}}7,5,2}{6,1{{},{ YRXRFR

}7,6,5,4,3,2,1{}}7,6,5,4,3,2}{6.4.3.1{{},{ YRXRFR

17

Dispensable & Indispensable

Let R be a family of equivalence relations let if IND(R) = IND(R-{a}), then a is dispensable in R if IND(R) ≠ IND(R-{a}), then a is indispensable in R

the family R is independent if each is indispensable in R ; otherwise R is dependent

ex) R={a, c}

U/{a}={{ 1, 2, 6}{3, 4}{5, 7}}

U/{b}={{ 2}{3, 4}{5, 6, 7}{1}}

U/IND(R) ={{1}{ 2}{6} {3,4}{5,7}}=U/{a,b}

∴ a, b : indispensable in R

∴ R is independent (∵ U/IR ≠ U/{b}, U/IR ≠ U/{a})

Ra

Ra

UU a c1 1 4

2 1 1

3 2 2

4 2 2

5 3 3

6 1 3

7 3 3

18

Core & Reduct

the set of all indispensable relation in R => the core of R , ( CORE(R) ) is a reduct of R if Q is independent and IND(Q) = IND(R) , ( RED(R) )RQ

)R()R( REDCORE

ex) a family of equivalence relations R={P, Q, R}

U/P ={{1,4,5}{2,8}{3}{6,7}}

U/Q ={{1,3,5}{6}{2,4,7,8}}

U/R ={{1,5}{6}{2,7,8}{3,4}}

U/{P,Q}={{1,5}{4}{{2,8}{3}{6}{7}}

U/{P,R}={{1,5}{4}{2,8}{3}{6}{7}}

U/{Q,R}={{1,5}{3}{6}{2,7,8}{4}}

U/R={{1,5}{6}{2,8}{3}{4}{7}}

U/{P,Q}= U/R =>R is dispensable in R

U/{P,R} }= U/R => Q is dispensable in R

U/{Q,R} }≠ U/R => P is indispensable in R

∴DORE(R) ={P}

∴RED(R) = {P,Q} and {P,R}

( U/{∵ P,Q}≠U/{P} , U/{P,Q}≠U/{Q}

U/{P,R}≠U/{P} , U/{P,R}≠U/{R} )

※ a reduct of knowledge is its essential part.

※ a core is in a certain sense its most important part.

19

Dispensable & Indispensable AttributesLet R and D be families of equivalence relation over U,

if , then the attribute a is dispensable in I ,

if , then the attribute a is indispensable in I ,

The R-positive region of D :

.Ra

XPOSUX

D/R R)D(

)D()D( }){R(R aPOSPOS )D()D( }){R(R aPOSPOS

20

ex) R={a, c} D={d}

U/{a}={{ 1, 2, 6}{3, 4}{5, 7}}

U/{c}={{ 2}{3, 4}{5, 6, 7}{1}}

U/D={{1,4,6}{2,3,5,7}}

U/IND(R) ={{1}{ 2}{6} {3,4}{5,7}}=U/{a,c}

UU a c d

1 1 4 yes

2 1 1 no

3 2 2 no

4 2 2 Yes

5 3 3 no

6 1 3 yes

7 3 3 no

}7,6,5,2,1{}}7,5,2}{6,,1{{R)D(D/

R

XPOSUX

}2,1{}}2}{1{{)D(}){R( aPOS

}7,5{}}7,5{{)D(}){R( cPOS

)D()D( }){R(R aPOSPOS

)D()D( }){R(R cPOSPOS

=> the relation ‘a’ is indispensable in R (‘a’ is indispensable attribute)

=> the relation ‘c’ is indispensable in R (‘c’ is indispensable attribute)

21

Independent

If every c in R is D-indispensable, then we say that R is D-independent

(or R is independent with respect to D)

ex) R={a, c} D={d}

∴ R is D-independent ( , ))D()D( }){R(R aPOSPOS )D()D( }){R(R cPOSPOS

UU a c d

1 1 4 yes

2 1 1 no

3 2 2 no

4 2 2 Yes

5 3 3 no

6 1 3 yes

7 3 3 no

22

Relative Reduct & Relative Core

The set of all D-indispensable elementary relation in R will be called the D-core of R, and will be denoted as CORED(R)

※ a core is in a certain sense its most important part.

The set of attributes is called a reduct of R,

if C is the D-independent subfamily of R and

=> C is a reduct of R ( REDD(R) )

※ a reduct of knowledge is its essential part.

※ REDD(R) is the family of all D-reducts of R

ex) R={a, c} D={d}

CORED(R) ={a, c} REDD(R) ={a, c}

RC

).D()D( RPOSPOSC

)()( RREDRCORED

23

An Example of Reducts & Core

U Headache Muscle pain

Temp. Flu

U1 Yes Yes Normal No U2 Yes Yes High Yes U3 Yes Yes Very-high Yes U4 No Yes Normal No U5 No No High No U6 No Yes Very-high Yes

U={U1, U2, U3, U4, U5, U6} =let {1,2,3,4,5,6}

Ω={headache, Muscle pan, Temp, Flu}={a, b, c, d}

condition R={a, b, c}, decision D={d}

U/{a}={{1,2,3}{4,5,6}}

U/{b}={{1,2,3,4,6}{5}}

U/{c}={{1,4}{2,5}{3,6}}

U/{a,b}={1,2,3}{4,6}{5}}

U/{a,c}={{1}{2}{3}{4}{5}{6}}

U/{b,c}={{1,4}{2}{3,6}{5}}

U/R={{1}{4}{2}{5}{3}{6}}

U/D={{1,4,5}{2,3,6}}

POSR(D)={{1,4,5}{2,3,6}}={1,2,3,4,5,6}

POSR-{a}(D)={{1,4,5}{2,3,6}}={1,2,3,4,5,6}

POSR-{b}(D)={{{1,4,5}{2,3,6}}={1,2,3,4,5,6}

POSR-{c}(D)={{5}}={5}

• relation ‘a’, ‘b’ is dispensable• relation ‘c’ is indispensable

=> D-core of R =CORED(R)={c}

to find reducts of R={a, b, c}

• {a, c} is D-independent and POS{a, c}(D)=POSR(D)

(∵POS{a}(D)={} ≠POS{a, c}(D)

POS{c}(D)={1,4,3,6} ≠POS{a, c}(D) )

• {b, c} is D-independent and POS{b, c}(D)=POSR(D) => {a, c} {b, c} is the D-reduct of R

POSR-{ab}(D)={{1,4}{3,6}}={1,4,3,6}POSR-{ac}(D)={{5}}={5}POSR-{bc}(D)={}

24


Temp. Flu


U Musclepain

Temp. Flu

Ｕ1,U4 Yes Normal No

U2 Yes High YesU3,U6 Yes Very-high YesU5 No High No

U Headache Temp. Flu

U1 Yes Norlmal NoU2 Yes High YesU3 Yes Very-high YesU4 No Normal NoU5 No High NoU6 No Very-high Yes

Reduct1 = {Muscle-pain,Temp.}

Reduct2 = {Headache, Temp.}CORE = {Headache, Temp} ∩ {Muscle Pain, Temp}

　　 = {Temp}

25

Dependency in knowledge

Given knowledge P, Q U/P={{1,5}{2,8}{3}{4}{6}{7}} U/Q={{1,5}{2,7,8}{3,4,6}}

If , then Q depends on P (P Q)⇒)Q()P( INDIND

26

Partial Dependency of knowledge

I=<U, Ω> and Knowledge Q depends in a degree k (0≤k≤1 ) from knowledge P

(P⇒k Q)

ex) U/Q={{1}{2,7}{3,6}{4}{5,8}}

U/P={{1,5}{2,8}{3}{4}{6}{7}}

POSP(Q) = {3,4,6,7}

the degree of dependency between Q and P

: (P⇒0.5 Q )

If k = 1 we say that Q depends totally on P. If k < 1 we say that Q depends partially (in a degree k) on P.

QP,

Ucard

POScardk

)Q()Q( P

P

5.08

4)Q()Q( P

P Ucard

POSk

27

Significance of attributes

ex) R={a, b, c}, decision D={d}

U/{a}={{1,2,3}{4,5,6}}

U/{b}={{1,2,3,4,6}{5}}

U/{c}={{1,4}{2,5}{3,6}}

U/{a,b}={1,2,3}{4,6}{5}}

U/{a,c}={{1}{2}{3}{4}{5}{6}}

U/{b,c}={{1,4}{2}{3,6}{5}}

U/R={{1}{4}{2}{5}{3}{6}}

U/D={{1,4,5}{2,3,6}}

POSR(D)={{1,4,5}{2,3,6}}={1,2,3,4,5,6}

POSR-{a}(D)={{1,4,5}{2,3,6}}={1,2,3,4,5,6}

POSR-{b}(D)={{{1,4,5}{2,3,6}}={1,2,3,4,5,6}

POSR-{c}(D)={{5}}={5}

011)()( {b}-RR DD significance of attribute ‘b’ :

83.06/56/11)()( {c}-RR DD significance of attribute ‘c’ :

∴ the attribute c is most significant, since it most changes the positive region of U/IND(D)

011)()( {a}-RR DD significance of attribute ‘a’ :

28

Discernibility Matrix

Let I = (U, Ω) be a decision table,

with U={x1, x2, .., xn}

C={a, b, c} : condition attribute set , D={d} : decision attribute set

By a discernibility matrix of I, denoted M(I)={mij}n×n

mij is the set of all the condition attributes that classify objects xi and xj into different classes.

U1 U2 U3 U4 U5 U6U1U2 cU3 c -U4 - a, c a, c

U5 - a, b a, b, c -

U6 a, c - - c b, c

- : same equivalence classes of the relation IND(d)

< Decision Table >


Temp. Flu


(a) (b) (d)(c)< Discernibility Matrix >

29

Compute value cores and value reducts from the M(I)

the core can be defined now as the set of all single element entries of the discernibility matrix,

is the reduct of R, if B is the minimal subset of R such that

for any nonempty entry c ( ) in M(I)

},),(:{)( jisomeforamRaRCORE ij

RB

cB c

U1 U2 U3 U4 U5

U2 c

U3 c -

U4 - a, c a, c

U5 - a, b a, b, c -

U6 a, c - - c b, c

d-CORE(R) d-reducts : {a, c} {b, c}

CHAPTER 6. Decision Tables

31

• Proposition 6.2Each decision table can be uniquely decomposed

into two decision tables and such that in and in , where and

– compute the dependency between condition and decision attributes

– decompose the table into two subtables

),,,( DCAUT ),,,( 11 DCAUT

),,,( 22 DCAUT DC 1 1T DC 0 2T)(1 DPOSU C

)(/2 )(

DINDUXC XBNU

32

• Example 1.

U a b c d e

1 1 0 2 2 0

2 0 1 1 1 2

3 2 0 0 1 1

4 1 1 0 2 2

5 1 0 2 0 1

6 2 2 0 1 1

7 2 1 1 1 2

8 0 1 1 0 1

condition

attribute

decision attribute

U a b c d e

3 2 0 0 1 1

4 1 1 0 2 2

6 2 2 0 1 1

7 2 1 1 1 2

U a b c d e

1 1 0 2 2 0

2 0 1 1 1 2

5 1 0 2 0 1

8 0 1 1 0 1

Table 2

Table 3

Table 1

• Table 2 is consistent, Table 3 is totally inconsistent

→ All decision rules in Table 2 are consistent

All decision rules in Table 3 are inconsistent

33

• simplification of decision tables : reduction of condition attributes• steps

1) Computation of reducts of condition attributes which is equivalent to elimination of some column from the decision tables

2) Elimination of duplicate rows

3) Elimination of superfluous values of attributes

34

• Example 2

U a b c d e

1 1 0 0 1 1

2 1 0 0 0 1

3 0 0 0 0 0

4 1 1 0 1 0

5 1 1 0 2 2

6 2 1 0 2 2

7 2 2 2 2 2

U a b d e

1 1 0 1 1

2 1 0 0 1

3 0 0 0 0

4 1 1 1 0

5 1 1 2 2

6 2 1 2 2

7 2 2 2 2

condition

attribute

decision attribute

e-dispensable condition attribute is c.

let R={a, b, c, d}, D={e}

CORED(R) ={a, b, d}

REDD(R) ={a, b, d}

remove column c

35

• we have to reduce superfluous values of condition attributes in every decision rules

→ compute the core values1. In the 1st decision rules

• the core of the family of sets

• the core value is

U a b d e

1 1 0 1 1

2 1 0 0 1

3 0 0 0 0

4 1 1 1 0

5 1 1 2 2

6 2 1 2 2

7 2 2 2 2

}}4,1{},3,2,1{},5,4,2,1{{}]1[,]1[,]1{[ dbaF

dbadba ]1[]1[]1[]1[ },,{ }1{}4,1{}3,2,1{}5,4,2,1{

}2,1{]1[,1)1(,0)1(,1)1( edba

}1{}4,1{}3,2,1{]1[]1[ db

}4,1{}4,1{}5,4,2,1{]1[]1[ da

}2,1{}3,2,1{}5,4,2,1{]1[]1[ ba

0)1( b

36

2. In the 2nd decision rules• the core of the family of sets


3. In the 3rd decision rules• the core of the family of sets


U a b d e

1 1 0 1 1

2 1 0 0 1

3 0 0 0 0

4 1 1 1 0

5 1 1 2 2

6 2 1 2 2

7 2 2 2 2

}}3,2{},3,2,1{},5,4,2,1{{}]1[,]1[,]1{[ dbaF

}2{]1[]1[]1[]1[ },,{ dbadba

}2,1{]1[,0)1(,0)1(,1)1( edba

}2,1{]1[]1[},2{]1[]1[},3,2{]1[]1[ badadb

1)1( a

}}3,2{},3,2,1{},3{{}]0[,]0[,]0{[ dbaF

}3{]0[]0[]0[]0[ },,{ dbadba

}4,3{]0[,0)0(,0)0(,0)0( edba

}3{]0[]0[},3{]0[]0[},3,2{]0[]0[ badadb

0)0( a

37

4. In the 4th decision rules• the core of the family of sets

• the core value :



U a b d e

1 1 0 1 1

2 1 0 0 1

3 0 0 0 0

4 1 1 1 0

5 1 1 2 2

6 2 1 2 2

7 2 2 2 2

}}4,1{},6,5,4{},5,4,2,1{{}]0[,]0[,]0{[ dbaF

}4{]0[]0[]0[]0[ },,{ dbadba

}4,3{]1[,1)0(,1)0(,1)0( edba

}5,4{]0[]0[},4,1{]0[]0[},4{]0[]0[ badadb

1)0(,1)0( db

}}7,6,5{},6,5,4{},5,4,2,1{{}]2[,]2[,]2{[ dbaF

}5{]2[]2[]2[]2[ },,{ dbadba

}7,6,5{]2[,2)2(,1)2(,1)2( edba

}5,4{]2[]2[},5{]2[]2[},6,5{]2[]2[ badadb

2)2( d

38


• the core value : not exist


• the core value : not exist

U a b d e

1 1 0 1 1

2 1 0 0 1

3 0 0 0 0

4 1 1 1 0

5 1 1 2 2

6 2 1 2 2

7 2 2 2 2

}}7,6,5{},6,5,4{},7,6{{}]2[,]2[,]2{[ dbaF

}6{]2[]2[]2[]2[ },,{ dbadba

}7,6,5{]2[,2)2(,1)2(,2)2( edba

}6{]2[]2[},7,6{]2[]2[},6,5{]2[]2[ badadb

}}7,6,5{},7{},7,6{{}]2[,]2[,]2{[ dbaF

}7{]2[]2[]2[]2[ },,{ dbadba

}7,6,5{]2[,2)2(,2)2(,2)2( edba

}7{]2[]2[},7,6{]2[]2[},7{]2[]2[ badadb

U a b d e

1 - 0 - 1

2 1 - - 1

3 0 - - 0

4 - 1 1 0

5 - - 2 2

6 - - - 2

7 - - - 2

39

• to compute value reducts– let’s compute value reducts for the ~1. 1st decision rules of the decision table

– 2 value reducts1. 2.

– Intersection of reducts : → core value

}2,1{]1[}1{}4,1{}3,2,1{]1[]1[ edb

eda ]1[}4,1{}4,1{}5,4,2,1{]1[]1[

edebea ]1[}4,1{]1[,]1[}3,2,1{]1[,]1[}5,4,2,1{]1[

1)1(and0)1( db0)1(and1)1( ba

0)1( b

eba ]1[}2,1{}3,2,1{}5,4,2,1{]1[]1[

40

2. 2nd decision rules of the decision table

– 2 value reducts : – Intersection of reducts : → core value

3. 3rd decision rules of the decision table

– 1 value reduct : – Intersection of reducts : → core value

}2,1{]1[}3,2{]1[]1[ edb

ebaeda ]1[}2,1{]1[]1[,]1[}2{]1[]1[

edebea ]1[}3,2{]1[,]1[}3,2,1{]1[,]1[}5,4,2,1{]1[

0)1(and1)1(or0)1(and1)1( bada

0)1( a

}4,3{]0[}3,2{]0[]0[ edb

ebaeda ]0[}3{]0[]0[,]0[}3{]0[]0[

edebea ]0[}3,2{]0[,]0[}3,2,1{]0[,]0[}3{]0[

0)0( a0)0( a

41

4. 4th decision rules of the decision table

– 1 value reduct : – Intersection of reducts : → core value


– 1 value reduct :

– Intersection of reducts : → core value

}4,3{]0[}4{]0[]0[ edb

ebaeda ]0[}5,4{]0[]0[,]0[}4,1{]0[]0[

edebea ]0[}4,1{]0[,]0[}3,2,1{]0[,]0[}5,4,2,1{]0[

0)1(and1)1( db

2)2( d

0)1(and1)1( db

edebea ]2[}7,6,5{]2[,]2[}6,5,4{]2[},7,6,5{]0[}5,4,2,1{]2[

2)2( d

}7,6,5{]2[}5{]2[]2[},7,6,5{]2[}6,5{]2[]2[ 22 dadb

}7,6,5{]2[}5,4{]2[]2[ 2 ba

42


– 2 value reducts :

– Intersection of reducts : → core value : not exist

edebea ]2[}7,6,5{]2[,]2[}6,5,4{]2[},7,6,5{]2[}7,6{]2[

2)2(or2)2( da

}7,6,5{]2[}7,6{]2[]2[},7,6,5{]2[}6,5{]2[]2[ edaedb

}7,6,5{]2[}6{]2[]2[ eba

43


– 3 value reducts

– Intersection of reducts : → core value : not exist

– reducts : = 24 solutions to our problem

},7,6,5{]2[}7,6{]2[ ea

2)2(or2)2(or2)2( dba

edeb ]2[}7,6,5{]2[,]2[}7{]2[

U a b d e

1 1 0 Ⅹ 1

1′ Ⅹ 0 1 1

2 1 0 Ⅹ 1

2 ′ 1 Ⅹ 0 1

3 0 Ⅹ Ⅹ 0

4 Ⅹ 1 1 0

5 Ⅹ Ⅹ 2 2

6 Ⅹ Ⅹ 2 2

6 ′ 2 Ⅹ Ⅹ 2

7 Ⅹ Ⅹ 2 2

7 ′ Ⅹ 2 Ⅹ 2

7″ 2 Ⅹ Ⅹ 2

3211122

}7,6,5{]2[}7,6{]2[]2[},7,6,5{]2[}7{]2[]2[ edaedb

}7,6,5{]2[}7{]2[]2[ eba

44

One solution Another solution

U a b d e

1 1 0 Ⅹ 1

2 1 Ⅹ 0 1

3 0 Ⅹ Ⅹ 0

4 Ⅹ 1 1 0

5 Ⅹ Ⅹ 2 2

6 Ⅹ Ⅹ 2 2

7 2 Ⅹ Ⅹ 2

U a b d e

1 1 0 Ⅹ 1

2 1 0 Ⅹ 1

3 0 Ⅹ Ⅹ 0

4 Ⅹ 1 1 0

5 Ⅹ Ⅹ 2 2

6 Ⅹ Ⅹ 2 2

7 Ⅹ Ⅹ 2 2

U a b d e

1,2 1 0 Ⅹ 1

3 0 Ⅹ Ⅹ 0

4 Ⅹ 1 1 0

5,6,7 Ⅹ Ⅹ 2 2

U a b d e

1 1 0 Ⅹ 1

2 0 Ⅹ Ⅹ 0

3 Ⅹ 1 1 0

4 Ⅹ Ⅹ 2 2

identical

enumeration is not essential

minimal solution

45

10.4 Pattern Recognition[ Table10 ] : Digits display unit in a calculator

assumed to represent a characterization of “hand written” digits

UU a b c d e f G0 1 1 1 1 1 1 01 0 1 1 0 0 0 02 1 1 0 1 1 0 13 1 1 1 1 0 0 14 0 1 1 0 0 1 15 1 0 1 1 0 1 16 1 0 1 1 1 1 17 1 1 1 0 0 0 08 1 1 1 1 1 1 19 1 1 1 1 0 1 1

a

b

c

d

e

fg

▶ Out task is to find a minimal description of each digit

and corresponding decision algorithm.

46

compute the core attributesUU b c d e f g0 1 1 1 1 1 01 1 1 0 0 0 02 1 0 1 1 0 13 1 1 1 0 0 14 1 1 0 0 1 15 0 1 1 0 1 16 0 1 1 1 1 17 1 1 0 0 0 08 1 1 1 1 1 19 1 1 1 0 1 1

UU a c d e f g0 1 1 1 1 1 01 0 1 0 0 0 02 1 0 1 1 0 13 1 1 1 0 0 14 0 1 0 0 1 15 1 1 1 0 1 16 1 1 1 1 1 17 1 1 0 0 0 08 1 1 1 1 1 19 1 1 1 0 1 1

[ drop attribute a ]

[ drop attribute b ]

decision rules are inconsistent

Rule1 : b1c1d0e0f0g0 → a0b1c1d0e0f0g0 Rule7 : b1c1d0e0f0g0 → a1b1c1d0e0f0g0

47

UU a b d e f g0 1 1 1 1 1 01 0 1 0 0 0 0

2 1 1 1 1 0 13 1 1 1 0 0 14 0 1 0 0 1 15 1 0 1 0 1 16 1 0 1 1 1 17 1 1 0 0 0 08 1 1 1 1 1 19 1 1 1 0 1 1

UU a b c e f g0 1 1 1 1 1 01 0 1 1 0 0 02 1 1 0 1 0 13 1 1 1 0 0 14 0 1 1 0 1 15 1 0 1 0 1 16 1 0 1 1 1 17 1 1 1 0 0 08 1 1 1 1 1 19 1 1 1 0 1 1

←[drop attribute c]

[drop attribute d]→

UU a b c d f g0 1 1 1 1 1 01 0 1 1 0 0 02 1 1 0 1 0 13 1 1 1 1 0 14 0 1 1 0 1 15 1 0 1 1 1 16 1 0 1 1 1 17 1 1 1 0 0 08 1 1 1 1 1 19 1 1 1 1 1 1

UU a b c d e g0 1 1 1 1 1 01 0 1 1 0 0 02 1 1 0 1 1 13 1 1 1 1 0 14 0 1 1 0 0 15 1 0 1 1 0 16 1 0 1 1 1 17 1 1 1 0 0 08 1 1 1 1 1 19 1 1 1 1 0 1

[drop attribute f]→

←[drop attribute e]

decision rules are

consistent

decision rules are

inconsistent

48

UU a b c d e f0 1 1 1 1 1 11 0 1 1 0 0 02 1 1 0 1 1 03 1 1 1 1 0 04 0 1 1 0 0 15 1 0 1 1 0 16 1 0 1 1 1 17 1 1 1 0 0 08 1 1 1 1 1 19 1 1 1 1 0 1

[drop attribute g]

∴ attribute c, d : dispensable

attribute a, b, e, f, g : indispensable

the set {a, b, e, f, g} : core

sole reducts : {a, b, e, f, g}

decision rules are inconsistent

49

compute reduct • all attribute set : {a, b, c, d, e, f, g}• core : {a, b, e, f, g} • reduct : {a, b, e, f, g}

UU c d a b e f G0 1 1 1 1 1 1 01 1 0 0 1 0 0 02 0 1 1 1 1 0 13 1 1 1 1 0 0 14 1 0 0 1 0 1 15 1 1 1 0 0 1 16 1 1 1 0 1 1 17 1 0 1 1 0 0 08 1 1 1 1 1 1 19 1 1 1 1 0 1 1

50

compute the core values of attributes for table11

UU a b e f G0 1 1 1 1 01 0 1 0 0 02 1 1 1 0 13 1 1 0 0 14 0 1 0 1 15 1 0 0 1 16 1 0 1 1 17 1 1 0 0 08 1 1 1 1 19 1 1 0 1 1

The core value in rule 1 and 4 : a0

The core value in rule 7 and 9 : a1

UU b a e f G0 1 1 1 1 01 1 0 0 0 02 1 1 1 0 13 1 1 0 0 14 1 0 0 1 15 0 1 0 1 16 0 1 1 1 17 1 1 0 0 08 1 1 1 1 19 1 1 0 1 1

The core value in rule 5 and 6 : b0

The core value in rule 8 and 9 : b1

[Table12 : Removing the attribute a ] [Table13 : Removing the attribute b ]

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UU e a b f G0 1 1 1 1 01 0 0 1 0 02 1 1 1 0 13 0 1 1 0 14 0 0 1 1 1

5 0 1 0 1 16 1 1 0 1 17 0 1 1 0 08 1 1 1 1 19 0 1 1 1 1

The core value in rule 2 ,6 and 8 : e0

The core value in rule 3 ,5 and 9 : e1

UU f a b e G0 1 1 1 1 01 0 0 1 0 02 0 1 1 1 13 0 1 1 0 14 1 0 1 0 15 1 1 0 0 16 1 1 0 1 17 0 1 1 0 08 1 1 1 1 19 1 1 1 0 1

[ Table14 : Removing the attribute e ][ Table 15 : Removing the attribute f ]

UU g a b e f0 0 1 1 1 11 0 0 1 0 02 1 1 1 1 03 1 1 1 0 04 1 0 1 0 15 1 1 0 0 16 1 1 0 1 17 0 1 1 0 08 1 1 1 1 19 1 1 1 0 1

The core value in rule 2 and 3 : f0The core value in rule 8 and 9 : f1

[ Table 16 : Removing the attribute g ]

52

core value for all decision rules

UU a b e f G0 _ _ _ 0

1 0 _ _ _

2 _ 1 0 _

3 _ 0 0 1

4 0 _ _ _ _5 _ 0 0 _ _6 _ 0 1 _ _7 1 _ _ _ 08 _ 1 1 1 19 1 1 0 1 _

[Table17]: all core value for table11

• rule 2, 3, 5, 6, 8, 9 are consistent.

• rule 0, 1, 4, 7 are inconsistent.

• to make the rules consistent ⇒ adding proper additional attributes

UU a b e f G0 1 1 1 1 01 0 1 0 0 02 1 1 1 0 13 1 1 0 0 14 0 1 0 1 15 1 0 0 1 16 1 0 1 1 17 1 1 0 0 08 1 1 1 1 19 1 1 0 1 1

[Table11]

UU a b e f g0 x x 1 x 00’ x x x 1 01 0 x x 0 x1’ 0 x x x 02 x x 1 0 x3 x x 0 0 14 0 x x 1 X4’ 0 x x x 15 x 0 0 x x6 x 0 1 x x7 1 x 0 x 07’ 1 x x 0 08 x 1 1 1 19 1 1 0 1 x

[Table 18] All possible value reduct for table11

53

UU a b e f g0 x x 1 x 00’ x x x 1 01 0 x x 0 x1’ 0 x x x 02 x x 1 0 x3 x x 0 0 14 0 x x 1 X4’ 0 x x x 15 x 0 0 x x6 x 0 1 x x7 1 x 0 x 07’ 1 x x 0 08 x 1 1 1 19 1 1 0 1 x

Table18• We have 16=24 minimal decision algorithms.

• One of the possible reduced algorithms

: e1g0 (f1g0) → 0

a0f0 (a0g0) → 1

e1f0 → 2

e0f0g1 → 3

a0f1 (a0g1) → 4

b0e0 → 5

b0e1 → 6

a1e0g0 (a1f0g0) → 7

b1e1f1g1 → 8

a1b1e0f1 → 9

a

b

c

d

e

fg

overview of rough sets

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