Lecture 2 1

Rule discovery strategiesLERS & ERID

Lecture 2 2

Input data is represented as a decision table.

In the decision table examples are described by values of attributes and characterized by a value of a decision.

All examples with the same value of the decision belong to the same concept.

This system looks for regularities in the decision table.

System LERS (Learning from Examples based on Rough Sets)

Lecture 2 3

System LERS (Learning from Examples based on Rough Sets)

- The first implementation of LERS was done by John

S. Dean and Douglas J. Sikora in 1988.

- Other important steps were:

-adding two modules of LEM (Learning from

Examples Module): module LEM1, module LEM2,

-improvements in the basic algorithm,

-implementation, and the fundamental

implementation.

Lecture 2 4

has two main options of rule induction, which are:

1. a basic algorithm, invoked by selecting Induce Rules from the menu Induce Rule Set (LEM 2).

This algorithm works on the level of attribute-value pairs. A local covering for each of the concepts is computed

System LERS (Learning from Examples based on Rough Sets)

Lecture 2 5

has two main options of rule induction, which are:

2. the option Induce Rules Using Priorities on Concept Level, of the menu Induce Rule Set working on entire attributes (LEM 1).

System LERS (Learning from Examples based on Rough Sets)

Lecture 2 6

Algorithm (LEM 1)

Let be the information system.),,( VAXS

X a b c d f

x1 0 0 0 1 0

x2 0 1 1 1 1

x3 0 0 0 1 0

x4 0 1 1 1 1

x5 1 1 0 1 2

x6 1 1 0 1 2

x7 2 2 2 0 3

x8 2 2 2 0 3

},,,,,,,{ 87654321 xxxxxxxxX

Lecture 2 7

Algorithm (LEM 1)

Let be the information system.),,( VAXS

},,,,,,,{ 87654321 xxxxxxxxX

Classification attributes

X a b c d f

x1 0 0 0 1 0

x2 0 1 1 1 1

x3 0 0 0 1 0

x4 0 1 1 1 1

x5 1 1 0 1 2

x6 1 1 0 1 2

x7 2 2 2 0 3

x8 2 2 2 0 3

Lecture 2 8

Algorithm (LEM 1)

Let be the information system.),,( VAXS

},,,,,,,{ 87654321 xxxxxxxxX

Decision attribute

X a b c d f

x1 0 0 0 1 0

x2 0 1 1 1 1

x3 0 0 0 1 0

x4 0 1 1 1 1

x5 1 1 0 1 2

x6 1 1 0 1 2

x7 2 2 2 0 3

x8 2 2 2 0 3

Lecture 2 9

Algorithm (LEM 1)

Let be the information system.),,( VAXS

},,,,,,,{ 87654321 xxxxxxxxX

The partitions of X, generated by single attributes are:

}},{},,,,{},,{{}*{ 87654231 xxxxxxxxb

}},{},,{},,,,{{}*{ 87426531 xxxxxxxxc

}},{},,,,,,{{}*{ 87654321 xxxxxxxxd

}},{},,{},,{},,{{}*{ 87654231 xxxxxxxxf

Let C be the set containing of one attribute {f}:

X a b c d f

x1 0 0 0 1 0

x2 0 1 1 1 1

x3 0 0 0 1 0

x4 0 1 1 1 1

x5 1 1 0 1 2

x6 1 1 0 1 2

x7 2 2 2 0 3

x8 2 2 2 0 3

}},{},,{},,,,{{}*{ 87654321 xxxxxxxxa

Lecture 2 10

Algorithm (LEM 1)

Let be the information system.),,( VAXS

},,,,,,,{ 87654321 xxxxxxxxX

The partitions of X, generated by single attributes are:

}},{},,,,{},,{{}*{ 87654231 xxxxxxxxb

}},{},,{},,,,{{}*{ 87426531 xxxxxxxxc

}},{},,,,,,{{}*{ 87654321 xxxxxxxxd

Let C be the set containing of one attribute {f}:

None of the sets is a subset of {f}*

X a b c d f

x1 0 0 0 1 0

x2 0 1 1 1 1

x3 0 0 0 1 0

x4 0 1 1 1 1

x5 1 1 0 1 2

x6 1 1 0 1 2

x7 2 2 2 0 3

x8 2 2 2 0 3

}},{},,{},,{},,{{}*{ 87654231 xxxxxxxxf

}},{},,{},,,,{{}*{ 87654321 xxxxxxxxa

Lecture 2 11

Algorithm (LEM 1)

Let be the information system.),,( VAXS

X a b c d f

x1 0 0 0 1 0

x2 0 1 1 1 1

x3 0 0 0 1 0

x4 0 1 1 1 1

x5 1 1 0 1 2

x6 1 1 0 1 2

x7 2 2 2 0 3

x8 2 2 2 0 3

},,,,,,,{ 87654321 xxxxxxxxX

}},{},,{},,{},,{{}*{ 87654231 xxxxxxxxf

*}{}},{},,{},,{},,{{}*,{ 87654231 fxxxxxxxxba *}{}},{},,{},,{},,{{}*,{ 87654231 fxxxxxxxxca

*}{}},{},,{},,,,{{}*,{ 87654321 axxxxxxxxda *}{}},{},,{},,{},,{{}*,{ 87654231 fxxxxxxxxcb

*}{}},{},,{},,{},,{{}*,{ 87654231 fxxxxxxxxdb

*}{}},{},,{},,,,{{}*,{ 87426531 cxxxxxxxxdc

forming two item sets:

Lecture 2 12

Algorithm (LEM 1)

Let be the information system.),,( VAXS

X a b c d f

x1 0 0 0 1 0

x2 0 1 1 1 1

x3 0 0 0 1 0

x4 0 1 1 1 1

x5 1 1 0 1 2

x6 1 1 0 1 2

x7 2 2 2 0 3

x8 2 2 2 0 3

},,,,,,,{ 87654321 xxxxxxxxX

}},{},,{},,{},,{{}*{ 87654231 xxxxxxxxf

*}{}},{},,{},,{},,{{}*,{ 87654231 fxxxxxxxxba *}{}},{},,{},,{},,{{}*,{ 87654231 fxxxxxxxxca

*}{}},{},,{},,,,{{}*,{ 87654321 axxxxxxxxda *}{}},{},,{},,{},,{{}*,{ 87654231 fxxxxxxxxcb

*}{}},{},,{},,{},,{{}*,{ 87654231 fxxxxxxxxdb

*}{}},{},,{},,,,{{}*,{ 87426531 cxxxxxxxxdc

marked

Lecture 2 13

Algorithm (LEM 1)

Let be the information system.),,( VAXS

X a b c d f

x1 0 0 0 1 0

x2 0 1 1 1 1

x3 0 0 0 1 0

x4 0 1 1 1 1

x5 1 1 0 1 2

x6 1 1 0 1 2

x7 2 2 2 0 3

x8 2 2 2 0 3

},,,,,,,{ 87654321 xxxxxxxxX

}},{},,{},,{},,{{}*{ 87654231 xxxxxxxxf

*}{}},{},,{},,{},,{{}*,{ 87654231 fxxxxxxxxba *}{}},{},,{},,{},,{{}*,{ 87654231 fxxxxxxxxca

*}{}},{},,{},,,,{{}*,{ 87654321 axxxxxxxxda *}{}},{},,{},,{},,{{}*,{ 87654231 fxxxxxxxxcb

*}{}},{},,{},,{},,{{}*,{ 87654231 fxxxxxxxxdb

*}{}},{},,{},,,,{{}*,{ 87426531 cxxxxxxxxdc

marked, but not covering of f

Lecture 2 14

Algorithm (LEM 1)

Let be the information system.),,( VAXS

X a b c d f

x1 0 0 0 1 0

x2 0 1 1 1 1

x3 0 0 0 1 0

x4 0 1 1 1 1

x5 1 1 0 1 2

x6 1 1 0 1 2

x7 2 2 2 0 3

x8 2 2 2 0 3

},,,,,,,{ 87654321 xxxxxxxxX

*}{}},{},,{},,{},,{{}*,{ 87654231 fxxxxxxxxba *}{}},{},,{},,{},,{{}*,{ 87654231 fxxxxxxxxca

*}{}},{},,{},,{},,{{}*,{ 87654231 fxxxxxxxxcb *}{}},{},,{},,{},,{{}*,{ 87654231 fxxxxxxxxdb

The coverings of C are:},{ ba },{ ca },{ cb },{ db

All of the sets are marked!

Lecture 2 15

How to find rules from coverings ?

Lecture 2 16

Algorithm (LEM 1)

Let be the information system.),,( VAXS

X a b c d f

x1 0 0 0 1 0

x2 0 1 1 1 1

x3 0 0 0 1 0

x4 0 1 1 1 1

x5 1 1 0 1 2

x6 1 1 0 1 2

x7 2 2 2 0 3

x8 2 2 2 0 3

Covering {a,b}

},,,{)*0,( 4321 xxxxa

*)2,(},{)*1,( 65 fxxa

*)3,(},{)*2,( 87 fxxa

*)0,(},{)*0,( 31 fxxb

},,,{)*1,( 6542 xxxxb

*)3,(},{)*2,( 87 fxxb

marked }},{},,{},,{},,{{}*{ 87654231 xxxxxxxxf

Lecture 2 17

Algorithm (LEM 1)

Let be the information system.),,( VAXS

X a b c d f

x1 0 0 0 1 0

x2 0 1 1 1 1

x3 0 0 0 1 0

x4 0 1 1 1 1

x5 1 1 0 1 2

x6 1 1 0 1 2

x7 2 2 2 0 3

x8 2 2 2 0 3

Covering {a,b}

},,,{)*0,( 4321 xxxxa

*)2,(},{)*1,( 65 fxxa

*)3,(},{)*2,( 87 fxxa

*)0,(},{)*0,( 31 fxxb

},,,{)*1,( 6542 xxxxb

*)3,(},{)*2,( 87 fxxb

marked*)1,(},{))*1,()0,(( 42 fxxba

}},{},,{},,{},,{{}*{ 87654231 xxxxxxxxf

Lecture 2 18

Algorithm (LEM 1)

Let be the information system.),,( VAXS

X a b c d f

x1 0 0 0 1 0

x2 0 1 1 1 1

x3 0 0 0 1 0

x4 0 1 1 1 1

x5 1 1 0 1 2

x6 1 1 0 1 2

x7 2 2 2 0 3

x8 2 2 2 0 3

Covering {a,b}

*)2,(},{)*1,( 65 fxxa *)3,(},{)*2,( 87 fxxa *)0,(},{)*0,( 31 fxxb

*)3,(},{)*2,( 87 fxxb *)1,(},{))*1,()0,(( 42 fxxba

Certain rules, obtained from marked items:

)2,()1,( fa )3,()2,( fa )0,()0,( fb

)3,()2,( fb

)1,()1,()0,( fba

Lecture 2 19

Algorithm (LEM 1)

Let be the information system.),,( VAXS

X a b c d f

x1 0 0 0 1 0

x2 0 1 1 1 1

x3 0 0 0 1 0

x4 0 1 1 1 1

x5 1 1 0 1 2

x6 1 1 0 1 2

x7 2 2 2 0 3

x8 2 2 2 0 3

Covering {a,b}

*)2,(},{)*1,( 65 fxxa *)3,(},{)*2,( 87 fxxa *)0,(},{)*0,( 31 fxxb

*)3,(},{)*2,( 87 fxxb *)1,(},{))*1,()0,(( 42 fxxba

Possible rules, obtained from non-marked items:

)0,()0,( fa )1,()0,( fa )1,()1,( fb

)2,()1,( fb

with confidence ½

with confidence ½

with confidence ½

with confidence ½

Lecture 2 20

New Rule Discovery Method for Incomplete IS

New strategy for discovering rules from incomplete information

systems We allow to use not only sets of attribute values as values of an object but also we allow to assign a weight to each value in such set.

)},(),,{( 31

32 brownblue

Lecture 2 21

New Rule Discovery Method for Incomplete IS

New strategy for discovering rules from incomplete information

systems We allow to use not only sets of attribute values as values of an object but also we allow to assign a weight to each value in such set.

)},(),,{( 31

32 brownblue

the confidence that object x has blue eyes is 2/3, whereas the confidence that x has brown

eyes is 1/2

Lecture 2 22

Incomplete Information System is a triple (X, A, V) where:

• X is a nonempty, finite set of objects,• A is a nonempty, finite set of attributes,• is a set of values of

attributes,

where Va is a set of values of attribute a, for any

We assume that for each attribute and

}:{ AaVV a Aa

XxAa

Definition 2.2 2

}1])[(:),{()( )()( iaixaxaii pVaJiJipaxa

Null value assigned to an object is interpreted as all possible values of an attribute with equal confidence assigned to all of them.

Lecture 2 23

x8

x7

x6

x5

x4

x3

x2

x1

edcbaX

),( 32

2a),( 3

11b

1e

1c 1d),( 2

11e

),( 21

2e),( 3

11a ),( 3

21b

),( 31

2b),( 4

12a

),( 43

3a ),( 32

2b 2d

1a 2b),( 2

11c

),( 21

3c 2d 3e

3a 2c 1d),( 3

21e

),( 31

2e),( 3

21a

),( 31

2a1b 2c 1e

2a 2b 3c 2d),( 3

12e

),( 32

3e

2a),( 4

11b

),( 43

2b),( 3

11c

),( 32

2c 2d 2e

3a 2b 1c 1d 3e

Extract rules from S describing attribute e in terms of attributes

{a,b,c,d} ( following a

strategy similar to LERS )

Lecture 2 24

Goal: Describe e in terms of {a,b,c,d}

x8

x7

x6

x5

x4

x3

x2

x1

edcbaX

),( 32

2a),( 3

11b

1e

1c 1d),( 2

11e

),( 21

2e),( 3

11a ),( 3

21b

),( 31

2b

),( 41

2a),( 4

33a ),( 3

22b 2d

1a 2b),( 2

11c

),( 21

3c 2d 3e

3a 2c 1d),( 3

21e

),( 31

2e

),( 32

1a),( 3

12a

1b 2c 1e

2a 2b 3c 2d),( 3

12e

),( 32

3e

2a),( 4

11b

),( 43

2b),( 3

11c

),( 32

2c 2d 2e

3a 2b 1c 1d 3e

Algorithm ERID (Extracting Rules from partially Incomplete

Information System(Database))

)},(),1,(),,{(* 32

5331

11 xxxa

)}1,(),1,(),,(),,(),,{(* 7631

541

232

12 xxxxxa

)}1,(),1,(),,{(* 8443

23 xxxa

)},(),1,(),,(),,(),,{(* 41

7521

431

231

11 xxxxxb

)}1,(),,(

),1,(),,(),1,(),,(),,{(*

843

7

621

4332

231

12

xx

xxxxxb

Lecture 2 25

x8

x7

x6

x5

x4

x3

x2

x1

edcbaX

),( 32

2a),( 3

11b

1e

1c 1d),( 2

11e

),( 21

2e),( 3

11a ),( 3

21b

),( 31

2b

),( 41

2a),( 4

33a ),( 3

22b 2d

1a 2b),( 2

11c

),( 21

3c 2d 3e

3a 2c 1d),( 3

21e

),( 31

2e

),( 32

1a),( 3

12a

1b 2c 1e

2a 2b 3c 2d),( 3

12e

),( 32

3e

2a),( 4

11b

),( 43

2b),( 3

11c

),( 32

2c 2d 2e

3a 2b 1c 1d 3e

)}1,(),,(),,(),,(),1,{(* 831

721

331

211 xxxxxc

)},(),1,(),1,(),,{(* 32

25431

22 xxxxc

)}1,(),,(),,{(* 621

331

23 xxxc

)}1,(),,(),1,(),1,{(* 821

5411 xxxxd

)}1,(),1,(),,(),1,(),1,{(* 7621

5322 xxxxxd

Algorithm ERID for Extracting Rules from partially Incomplete Information System

Goal: Describe e in terms of {a,b,c,d}

Lecture 2 26

Algorithm ERID for Extracting Rules from partially Incomplete Information System

Goal: Describe e in terms of {a,b,c,d}

x8

x7

x6

x5

x4

x3

x2

x1

edcbaX

),( 32

2a),( 3

11b

1e

1c 1d),( 2

11e

),( 21

2e),( 3

11a ),( 3

21b

),( 31

2b

),( 41

2a),( 4

33a ),( 3

22b 2d

1a 2b),( 2

11c

),( 21

3c 2d 3e

3a 2c 1d),( 3

21e

),( 31

2e

),( 32

1a),( 3

12a

1b 2c 1e

2a 2b 3c 2d),( 3

12e

),( 32

3e

2a),( 4

11b

),( 43

2b),( 3

11c

),( 32

2c 2d 2e

3a 2b 1c 1d 3e

For the values of the decision attribute we have:

)}1,(),,(),1,(),,{(* 532

4221

11 xxxxe

)}1,(),,(),,(),,{(* 731

631

421

12 xxxxe

)}1,(),,(),1,{(* 832

633 xxxe

Lecture 2 27

x8

x7

x6

x5

x4

x3

x2

x1

edcbaX

),( 32

2a),( 3

11b

1e

1c 1d),( 2

11e

),( 21

2e),( 3

11a ),( 3

21b

),( 31

2b

),( 41

2a),( 4

33a ),( 3

22b 2d

1a 2b),( 2

11c

),( 21

3c 2d 3e

3a 2c 1d),( 3

21e

),( 31

2e

),( 32

1a),( 3

12a

1b 2c 1e

2a 2b 3c 2d),( 3

12e

),( 32

3e

2a),( 4

11b

),( 43

2b),( 3

11c

),( 32

2c 2d 2e

3a 2b 1c 1d 3e

2. Check the relationship “ ”

between values of classification

attributes {a,b,c,d} and values

of decision attribute e

Algorithm ERID for Extracting Rules from partially Incomplete Information System

Goal: Describe e in terms of {a,b,c,d}

Lecture 2 28

x8

x7

x6

x5

x4

x3

x2

x1

edcbaX

),( 32

2a),( 3

11b

1e

1c 1d),( 2

11e

),( 21

2e),( 3

11a ),( 3

21b

),( 31

2b

),( 41

2a),( 4

33a ),( 3

22b 2d

1a 2b),( 2

11c

),( 21

3c 2d 3e

3a 2c 1d),( 3

21e

),( 31

2e

),( 32

1a),( 3

12a

1b 2c 1e

2a 2b 3c 2d),( 3

12e

),( 32

3e

2a),( 4

11b

),( 43

2b),( 3

11c

),( 32

2c 2d 2e

3a 2b 1c 1d 3e

Niiii pxc )},{(* Njjjj qye )},{(*Let , .

and confidence of the rule are above some threshold values.

We say that:

** ji ec iff support

ji ec

Algorithm ERID for Extracting Rules from partially Incomplete Information System

Goal: Describe e in terms of {a,b,c,d}

Lecture 2 29

Algorithm ERID for Extracting Rules from partially Incomplete Information System

Goal: Describe e in terms of {a,b,c,d}

x8

x7

x6

x5

x4

x3

x2

x1

edcbaX

),( 32

2a),( 3

11b

1e

1c 1d),( 2

11e

),( 21

2e),( 3

11a ),( 3

21b

),( 31

2b

),( 41

2a),( 4

33a ),( 3

22b 2d

1a 2b),( 2

11c

),( 21

3c 2d 3e

3a 2c 1d),( 3

21e

),( 31

2e

),( 32

1a),( 3

12a

1b 2c 1e

2a 2b 3c 2d),( 3

12e

),( 32

3e

2a),( 4

11b

),( 43

2b),( 3

11c

),( 32

2c 2d 2e

3a 2b 1c 1d 3e

Niiii pxc )},{(* Njjjj qye )},{(*Let , .

and confidence of the rule are above some threshold values.

We say that:** ji ec iff support

ji ec

How to define support and confidence

of a rule ?ji ec

Lecture 2 30

Definition of Support and Confidence (by example)

To define support and confidence of the rule a1 e3 we compute: )},(),1,(),,{(*

32

5331

11 xxxa

10110)sup( 32

31

31 ea

)}1,(),,(),1,{(* 832

633 xxxe

2

1

)sup(

)sup()(

1

3131

a

eaeaconf

21)sup( 32

31

1 a

Support of the rule:

Support of the term a1:

Confidence of the rule:

Lecture 2 31

Extracting Rules from partially Incomplete Information System (Algorithm ERID(λ1, λ2))

x8

x7

x6

x5

x4

x3

x2

x1

edcbaX

),( 32

2a),( 3

11b

1e

1c 1d),( 2

11e

),( 21

2e),( 3

11a ),( 3

21b

),( 31

2b

),( 41

2a),( 4

33a ),( 3

22b 2d

1a 2b),( 2

11c

),( 21

3c 2d 3e

3a 2c 1d),( 3

21e

),( 31

2e

),( 32

1a),( 3

12a

1b 2c 1e

2a 2b 3c 2d),( 3

12e

),( 32

3e

2a),( 4

11b

),( 43

2b),( 3

11c

),( 32

2c 2d 2e

3a 2b 1c 1d 3e

** 11 ea )1(sup 65 - marked negative

** 21 ea )1(sup 61

** 31 ea )11(sup

- marked positive

)5.0( conf

Thresholds (provided by user):

Minimal support (λ1 = 1)

Minimal confidence (λ2 = ½)

- marked negative

Goal: Describe e in terms of {a,b,c,d}

Lecture 2 32

x8

x7

x6

x5

x4

x3

x2

x1

edcbaX

),( 32

2a),( 3

11b

1e

1c 1d),( 2

11e

),( 21

2e),( 3

11a ),( 3

21b

),( 31

2b

),( 41

2a),( 4

33a ),( 3

22b 2d

1a 2b),( 2

11c

),( 21

3c 2d 3e

3a 2c 1d),( 3

21e

),( 31

2e

),( 32

1a),( 3

12a

1b 2c 1e

2a 2b 3c 2d),( 3

12e

),( 32

3e

2a),( 4

11b

),( 43

2b),( 3

11c

),( 32

2c 2d 2e

3a 2b 1c 1d 3e

**

33ea )11(sup but )36.0( conf

**

12eb )1(sup 6

7 but )22.0( conf

**

22eb )1(sup 12

17 but )27.0( conf**

31ec )1(sup 2

3 but )47.0( conf**

22ec )11(sup but )33.0( conf

**

11ed )1(sup 3

5 but )48.0( conf**

31ed )11(sup but )28.0( conf

**

12ed )1(sup 2

3 but )33.0( conf

**

32ed )1(sup 3

5 but )37.0( conf

Extracting Rules from partially Incomplete Information System (Algorithm ERID(λ1, λ2))

Lecture 2 33

x8

x7

x6

x5

x4

x3

x2

x1

edcbaX

),( 32

2a),( 3

11b

1e

1c 1d),( 2

11e

),( 21

2e),( 3

11a ),( 3

21b

),( 31

2b

),( 41

2a),( 4

33a ),( 3

22b 2d

1a 2b),( 2

11c

),( 21

3c 2d 3e

3a 2c 1d),( 3

21e

),( 31

2e

),( 32

1a),( 3

12a

1b 2c 1e

2a 2b 3c 2d),( 3

12e

),( 32

3e

2a),( 4

11b

),( 43

2b),( 3

11c

),( 32

2c 2d 2e

3a 2b 1c 1d 3e

)11(sup but )36.0( conf

)1(sup 67 but )22.0( conf

)1(sup 1217 but )27.0( conf

)1(sup 23 but )47.0( conf

)11(sup but )33.0( conf

)1(sup 35 but )48.0( conf

)11(sup but )28.0( conf

)1(sup 23 but )33.0( conf

)1(sup 35 but )37.0( conf

They all are not marked

**

33ea

**

12eb

**

22eb

**

31ec

**

22ec

**

11ed

**

31ed

**

12ed

**

32ed

Extracting Rules from partially Incomplete Information System (Algorithm ERID(λ1, λ2))

Lecture 2 34

x8

x7

x6

x5

x4

x3

x2

x1

edcbaX

),( 32

2a),( 3

11b

1e

1c 1d),( 2

11e

),( 21

2e),( 3

11a ),( 3

21b

),( 31

2b

),( 41

2a),( 4

33a ),( 3

22b 2d

1a 2b),( 2

11c

),( 21

3c 2d 3e

3a 2c 1d),( 3

21e

),( 31

2e

),( 32

1a),( 3

12a

1b 2c 1e

2a 2b 3c 2d),( 3

12e

),( 32

3e

2a),( 4

11b

),( 43

2b),( 3

11c

),( 32

2c 2d 2e

3a 2b 1c 1d 3e

*)*( 313 eca )11(sup and )8.0( conf

*)*( 313 eda )11(sup and )5.0( conf

*)*( 323 eda )10(sup *)*( 122 edb )1(sup 3

2 *)*( 222 ecb )1(sup 2

1

Extracting Rules from partially Incomplete Information System (Algorithm ERID(λ1, λ2))

Lecture 2 35

Extracting Rules from partially Incomplete Information System (Algorithm ERID(λ1, λ2))

x8

x7

x6

x5

x4

x3

x2

x1

edcbaX

),( 32

2a),( 3

11b

1e

1c 1d),( 2

11e

),( 21

2e),( 3

11a ),( 3

21b

),( 31

2b

),( 41

2a),( 4

33a ),( 3

22b 2d

1a 2b),( 2

11c

),( 21

3c 2d 3e

3a 2c 1d),( 3

21e

),( 31

2e

),( 32

1a),( 3

12a

1b 2c 1e

2a 2b 3c 2d),( 3

12e

),( 32

3e

2a),( 4

11b

),( 43

2b),( 3

11c

),( 32

2c 2d 2e

3a 2b 1c 1d 3e

)11(sup and )8.0( conf

)11(sup and )5.0( conf

)10(sup

)1(sup 32

)1(sup 21

They all are marked positive

*)*( 313 eca

*)*( 313 eda

*)*( 323 eda

*)*( 122 edb

*)*( 222 ecb

Lecture 2 36

Extracting Rules from partially Incomplete Information System (Algorithm ERID(λ1, λ2))

x8

x7

x6

x5

x4

x3

x2

x1

edcbaX

),( 32

2a),( 3

11b

1e

1c 1d),( 2

11e

),( 21

2e),( 3

11a ),( 3

21b

),( 31

2b

),( 41

2a),( 4

33a ),( 3

22b 2d

1a 2b),( 2

11c

),( 21

3c 2d 3e

3a 2c 1d),( 3

21e

),( 31

2e

),( 32

1a),( 3

12a

1b 2c 1e

2a 2b 3c 2d),( 3

12e

),( 32

3e

2a),( 4

11b

),( 43

2b),( 3

11c

),( 32

2c 2d 2e

3a 2b 1c 1d 3e

)11(sup and )8.0( conf

)11(sup and )5.0( conf

)10(sup

)1(sup 32

)1(sup 21

They all are marked positive

They all are marked negative

*)*( 313 eca

*)*( 313 eda

*)*( 323 eda

*)*( 122 edb

*)*( 222 ecb