Learning rules from incomplete training examples by rough sets

Tzung-Pei Hong, Li-Huei Tseng, Shyue-Liang Wang

Expert Systems with Applications 22(2002) 285-293

2006. 5. 17(Wed)

Introduction

• deal with the problem of producing a set of certain and possible rules from incomplete data sets

• propose a new learning approach based on rough sets– derive rules from incomplete data sets

– estimate the missing values in the learning process

• Unknown values are first assumed to be any possible values and are gradually redefined according to the incomplete lower & upper approximations

• The examples and the approximations interact on each other to derive certain and possible rules and to estimate appropriate unknown values.

Review of the rough set theory

• Example 1.

– indiscernibility relation and belong to the same equivalence class for SP

– equivalence partition for singleton attributes

– lower approximation of X

– upper approximation of X

:& )4()1( ObjObj

class set : BP

possible values : {Low(L), Normal(N), High(H)}

},,,{ )7()2()1( ObjObjObjU },,{ DPSPA

}},}{,,}{,{{}/{ )4()1()7()6()3()5()2( ObjObjObjObjObjObjObjSPU }},}{,}{,,{{}/{ )6()5()7()4()3()2()1( ObjObjObjObjObjObjObjDPU

:})(,|{)(* XxBUxxXB :})(and|{)(* XxBUxxXB

Definitions

• incomplete equivalence classes– each object is represented as a tuple (obj, symbol)

• symbol : certain(c) or uncertain(u)

– If an object has a certain value for attribute ,

then is put in the equivalence class for ;

otherwise, is put in each equivalence class of attribute

– above definition(for single attributes) can easily be extended to attribute subsets

– The set of incomplete equivalence classes for subset B is referred to as B-elementary set

)(iobj )(ijv jA

),( )( cobj i )(ijv

),( )( uobj ijA

• Example 3.

• the incomplete elementary set of attribute SP

• the incomplete elementary set of attribute DP

for SP

e.g.

),( )1( cobj

),( )5( uobj

)},,)(,)(,{()},,)(,)(,)(,{{(}/{ )9()5()2()9()5()6()3( uObjuObjcObjuObjuObjcObjcObjSPU )}},)(,)(,)(,)(,)(,{( )9()5()8()7()4()1( uObjuObjcObjcObjcObjcObj

),)(,{()},,)(,)(,{{(}/{ )5()3()7()9()1( cObjcObjuObjcObjcObjDPU )}},)(,)(,{()},,)(,)(,( )7()4()2()7()8()6( uObjcObjcObjuObjcObjcObj

• represents the incomplete equivalence classes in which exists

)( )7(objDP)7(obj

• Example 4. – assume

– incomplete lower approximation for attribute SP on X

– incomplete upper approximation for attribute SP on X

},,,{ )9()6()5()2( ObjObjObjObjX

A rough set based approach to simultaneously estimate missing values and derive rules

• proposed learning algorithm

An example

• Step 1. partition

• Step 2.– the incomplete elementary set of attribute SP

– the incomplete elementary set of attribute DP

Obj SP DP BP

1 L N N

2 H L H

3 N H N

4 L L L

5 * H H

6 N H H

7 L * L

8 L H N

9 * N H

• Step 3. q =1

• Step 4.– incomplete lower approximation

• Step 5.– each uncertain object is checked for change to

certain objects. e.g. in

– the incomplete elementary set of attribute SP

Obj SP DP BP

1 L N N

2 H L H

3 N H N

4 L L L

5 * H H

6 N H H

7 L * L

8 L H N

9 * N H

)(* HXSP

),(and),(),(and),( )9()5()9()5( cObjcObjuObjuObj

Obj SP DP BP

1 L N N

2 H L H

3 N H N

4 L L L

5 H H H

6 N H H

7 L * L

8 L H N

9 H N H

• Step 6. q = q+1 = 2, and Steps 4-6 are repeated– incomplete elementary set of attributes {SP,DP}

– incomplete lower approximations of {SP,DP}Obj SP DP BP

1 L N N

2 H L H

3 N H N

4 L L L

5 H H H

6 N H H

7 L L L

8 L H N

9 H N H

Obj SP DP BP

1 L N N

2 H L H

3 N H N

4 L L L

5 H H H

6 N H H

7 L * L

8 L H N

9 H N H

),(),( )7()7( cObjuObj

– incomplete elementary set of attributes {SP,DP}

– incomplete elementary set of attributes DP

Obj SP DP BP

1 L N N

2 H L H

3 N H N

4 L L L

5 H H H

6 N H H

7 L L L

8 L H N

9 H N H

Conclusion and future work

• The proposed approach is different from others in that it can derive rules and estimate the missing values at the same time.

• The incomplete lower and upper approximations was defined

• The interaction between data and approximations helps derive certain and possible rules from incomplete data sets and estimate appropriate unknown values