lecture 21 rule discovery strategies lers & erid
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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