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Action Rules Discovery Systems: DEAR1, DEAR2, ARED, ….. by Zbigniew W. Ra ś. Y = {x 2 , x 4 } Z = {x 1 ,x 2 ,x 3 ,x 4 ,x 5 ,x 7 }. LERS. (a, a 1 ) (a, a 2 ) (b, b 1 ) (b,b 2 ) ……….. (d,d 1 ) (d,d 2 ). atomic terms. Decision System S. r = [[(a, a 2 )*(b, b 1 )] → (d, d 1 )] - PowerPoint PPT Presentation
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Action Rules Discovery Systems: Action Rules Discovery Systems: DEAR1, DEAR2, ARED, …..DEAR1, DEAR2, ARED, …..
byby
Zbigniew Zbigniew W. W. RaRaśś
LERS
XX a a bb cc dd
xx11 aa11 bb11 cc11 dd11
xx22 aa22 bb11 cc22 dd11
xx33 aa22 bb22 cc22 dd11
xx44 aa22 bb11 cc11 dd11
xx55 aa22 bb33 cc22 dd11
xx66 aa11 bb11 cc22 dd22
xx77 aa11 bb22 cc2 2 dd11
xx88 aa11 bb22 cc11 dd33
(a, a1) (a, a2)(b, b1)(b,b2)………..(d,d1)(d,d2)
Decision System Decision System SSatomic terms
r = [[(a, a2)*(b, b1)] → (d, d1)]
w Y → w Z
rule
Support:
Confidence:
)(
)()(
Ycard
ZYcardrconf
)()sup( ZYcardr
Y = {x2, x4}Z = {x1,x2,x3,x4,x5,x7}
sup(r) = 2conf(r) = 2/2 = 1
a b c e f d
2 1 1 7 8 1
2 5 4 6 8 1
1 1 4 9 4 2
1 4 5 8 7 2
2 1 5 2 8 3
2 1 4 2 8 3
1 2 4 7 1 2
2 1 1 6 8 1
3 2 4 6 8 2
3 3 5 7 4 2
3 3 5 6 2 3
2 5 4 6 8 3
b c e f d
1 1 7 8 1
5 4 6 8 1
1 5 2 8 3
1 4 2 8 3
1 1 6 8 1
5 4 6 8 3
b c e f d
2 4 6 8 2
3 5 7 4 2
3 5 6 2 3
Splitting the node using the stable attributeDom(a) = {1,2,3} & Dom(b) = {1,2,3,4,5}
All objects have the same decision value, so this sub-table is not analyzed any further
None of the objects contain the desired class “1”, so this sub-table stops splitting any further
a = 1 a =
2
a = 3
b c e f d
1 9 4 4 2
4 5 8 7 2
2 4 7 1 2
c e d
1 7 1
5 2 3
4 2 3
1 6 1
c e d
4 6 1
4 6 3
b = 1 b = 5
All the flexible values are the same for both objects , therefore this sub-table is not analyzed any further
Partition decision table S Stable:{ a, b}
Flexible: {c, e, f}
Reclassification direction:
2 1 or 3 1
All objects have the same value 8 for attribute f, so it is crossed out from the sub-table ( this condition is used for stable attributes as well)
T1
T2
T3
T4
T5
Action Rules Discovery (Preprocessing)
Table: Set of rules R with supporting objects
Figure of (d, H)-tree T1
Figure of (d, L)-tree T2
Objects a b c dx1, x2, x3, x4 0 L
x1, x3 0 L
x2, x4 2 L
x2, x4 1 L
x5, x6 3 L
x7, x8 2 1 H
x7, x8 1 2 H
Objects a b cx1, x2, x3, x4 0
x1, x3 0
x2, x4 2
x2, x4 1
x5, x6 3
Objects b c
x1, x3 0
x2, x4 2
x2, x4 1
x5, x6 3
Objects b
x2, x4 2
x5, x6 3
c = 1c = ? c = 0
Objects b c
x1, x2, x3, x4
Objects b
x1, x3
a = 0
Objects b
x2, x4
a = ?
Objects a b cx7, x8 2 1
x7, x8 1 2
Objects b c
x7, x8 1
a = 2
Objects b c
x7, x8 1 2
a = ?
Stable Attribute: {a, c}
Flexible Attribute: b
Decision Attribute: d
T1 T2
T3T4
(T3, T1) : (a = 2) (b, 21) ( d, L H)
(a = 2) (b, 31) ( d, L H)
Objects b
x7, x8 1
c = ? c = 2Objects b
x7, x8 1
c = ?Objects b
x1, x2, x3, x4
T5
T6
System DEAR1 System DEAR1
Objects a b c d
r1 x1, x2, x3, x4 0 L
r2 x1, x3 0 L
r3 x2, x4 2 L
r4 x2, x4 1 L
r5 x5, x6 3 L
r6 x7, x8 2 1 H
r7 x7, x8 1 2 H
Objects a b c dx1, x2, x3, x4 0 L
x1, x3 0 L
x2, x4 2 L
x2, x4 1 L
x5, x6 3 L
x7, x8 2 1 H
x7, x8 1 2 H
Stable Attribute: b
Flexible Attribute: {a, c}
Decision Attribute: d
Objects a c dx1, x2, x3, x4 0 L
x1, x3 0 L
x2, x4 L
x2, x4 1 L
b = 2
Objects a c dx1, x2, x3, x4 0 L
x1, x3 0 L
x2, x4 1 L
x5, x6 L
b = 3
Objects a c dx1, x2, x3, x4 0 L
x1, x3 0 L
x2, x4 1 L
x7, x8 2 H
x7, x8 2 H
b = 1
Objects a cx1, x2, x3, x4 0
x1, x3 0
x2, x4 1
Objects a cx7, x8 2
x7, x8 2
d = L d = H
Set of rules R with supporting objects
(b = 1) (a, 02) ( d, L H)(b = 1) (c, 02) ( d, L H)(b = 1) (c, 12) ( d, L H)
System DEAR2
Cost of Action Rule
Action rule r:
[(b1, v1→ w1) (b2, v2→ w2) … ( bp, vp→ wp)](x) (d, k1→ k2)(x)
The cost of r in S:
costS(r) = {S(vi , wi) : 1 i p}
Action rule r is feasible in S, if costS(r) < S(k1 , k2).
For any feasible action rule r, the cost of the conditional
part of r is lower than the cost of its decision part.
Example:
r = [(b1, v1 → w1) … (bj, vj → wj) … ( bp, vp → wp)](x)
(d, k1 → k2)(x)
In RS[(bj, vj → wj)] we find
r1 = [(bj1, vj1 → wj1) (bj2, vj2 → wj2) … ( bjq, vjq → wjq)](x)
(bj, vj → wj)(x)
Then, we can compose r with r1 and the same replace
term (bj, vj → wj) by term from the left hand side of r1:
[(b1, v1 → w1) … [(bj1, vj1 → wj1) (bj2, vj2 → wj2) …
( bjq, vjq → wjq)] … ( bp, vp → wp)](x) (d, k1 → k2)(x)
Cost of Action Rule
ARED
XX a a bb cc dd
xx11 aa11 bb11 cc11 dd11
xx22 aa22 bb11 cc22 dd11
xx33 aa22 bb22 cc22 dd11
xx44 aa22 bb11 cc11 dd11
xx55 aa22 bb33 cc22 dd11
xx66 aa11 bb11 cc22 dd22
xx77 aa11 bb22 cc2 2 dd11
xx88 aa11 bb22 cc11 dd33
(a, a1 → a1) (a, a2 → a2)(b, b1 → b1)(b, b2 → b2) ………..(d, d1 → d1)(d, d2 → d2)
Decision System Decision System SS atomic action terms
r=[(a, a2 → a2)*(b, b1 → b1)] → (d, d1 → d1)
(w, w) (Y, Y ) → (w,w) (Z, Z)
action rule
Support:
Confidence:
)(
)()(
Ycard
ZYcardrconf
)()sup( ZYcardr
Y = {x2, x4}Z = {x1,x2,x3,x4,x5,x7}
sup(r) = 2conf(r) = 2/2 = 1
ARED
XX a a bb cc dd
xx11 aa11 bb11 cc11 dd11
xx22 aa22 bb11 cc22 dd11
xx33 aa22 bb22 cc22 dd11
xx44 aa22 bb11 cc11 dd11
xx55 aa22 bb33 cc22 dd11
xx66 aa11 bb11 cc22 dd22
xx77 aa11 bb22 cc2 2 dd11
xx88 aa11 bb22 cc11 dd33
(a, a1 → a1) (a, a2 → a2)(b, b1 → b1)(b, b2 → b2) ………..(d, d1 → d1)(d, d2 → d2)
Decision System Decision System SS atomic action terms
r=[(a, a2 → a1)*(b, b1 → b1)] → (d, d1 → d2)
(w1, w2) (Y1, Y 2) → (w1,w2) (Z1, Z2)
action rule
Support:
Confidence:
)(
)()(
Ycard
ZYcardrconf
)()sup( ZYcardr
Y = {x2, x4}Z = {x1,x2,x3,x4,x5,x7}
sup(r) = ?conf(r) = ?
Y=(Y1,Y2), Z=(Z1,Z2)w = (w1,w2)
ARED
XX a a bb cc dd
xx11 aa11 bb11 cc11 dd11
xx22 aa22 bb11 cc22 dd11
xx33 aa22 bb22 cc22 dd11
xx44 aa22 bb11 cc11 dd11
xx55 aa22 bb33 cc22 dd11
xx66 aa11 bb11 cc22 dd22
xx77 aa11 bb22 cc2 2 dd11
xx88 aa11 bb22 cc11 dd33
(a, a1 → a1) (a, a1 → a2)(b, b1 → b2)(b, b2 → b2) ………..(d, d1 → d1)(d, d2 → d2)
Decision System Decision System SSatomic action
terms
r=[(a, a2 → a1)*(b, b1 → b1)] → (d, d1 → d2)
(Y1, Y 2) (Z1, Z2)
action rule
Support:
Confidence: ???)(
)()(
1
11
Ycard
ZYcardrconf
)()sup( 11 ZYcardr Y1 = {x2, x4}Z1 = {x1,x2,x3,x4,x5,x7}Y2 = {x1, x6}Z2 = { x6}
sup(r) = 2conf(r) = 2/2 = 1
Y1 → Z1, Y2 → Z2
ARED
XX a a bb cc dd
xx11 aa11 bb11 cc11 dd11
xx22 aa22 bb11 cc22 dd11
xx33 aa22 bb22 cc22 dd11
xx44 aa22 bb11 cc11 dd11
xx55 aa22 bb33 cc22 dd11
xx66 aa11 bb11 cc22 dd22
xx77 aa11 bb22 cc2 2 dd11
xx88 aa11 bb22 cc11 dd33
(a, a1 → a1) (a, a1 → a2)(b, b1 → b2)(b, b2 → b2) ………..(d, d1 → d1)(d, d2 → d2)
Decision System Decision System SSatomic terms
r=[(a, a2 → a1)*(b, b1 → b1)] → (d, d1 → d2)
(Y1, Y 2) (Z1, Z2)
rule
)(
)(*
)(
)()(
2
2
1
1 21
Ycard
ZYcard
Ycard
ZYcardrconf
???)()sup( 11 ZYcardr Y1 = {x2, x4}Z1 = {x1,x2,x3,x4,x5,x7}Y2 = {x1, x6}Z2 = { x6}
sup(r) = 2conf(r) = 2/2 = 1
ARED
XX a a bb cc dd
xx11 aa11 bb11 cc11 dd11
xx22 aa22 bb11 cc22 dd11
xx33 aa22 bb22 cc22 dd11
xx44 aa22 bb11 cc11 dd11
xx55 aa22 bb33 cc22 dd11
xx66 aa11 bb11 cc22 dd22
xx77 aa11 bb22 cc2 2 dd11
xx88 aa11 bb22 cc11 dd33
(a, a1 → a1) (a, a1 → a2)(b, b1 → b2)(b, b2 → b2) ………..(d, d1 → d1)(d, d2 → d2)
Decision System Decision System SSatomic terms
r=[(a, a2 → a1)*(b, b1 → b1)] → (d, d1 → d2)
(Y1, Y 2) (Z1, Z2)
rule
)(
)(*
)(
)()(
2
2
1
1 21
Ycard
ZYcard
Ycard
ZYcardrconf
)}(),(min{)sup( 2211 ZYcardZYcardr Y1 = {x2, x4}Z1 = {x1,x2,x3,x4,x5,x7}Y2 = {x1, x6}Z2 = { x6}
sup(r) = 2conf(r) = 2/2 = 1
ARED
Meaning of (d,d1 d2) in S:
NS(d,d1 d2)=[{x1,x2, x3, x4, x5, x7}, {x6}]
XX a a bb cc dd
xx11 aa11 bb11 cc11 dd11
xx22 aa22 bb11 cc22 dd11
xx33 aa22 bb22 cc22 dd11
xx44 aa22 bb11 cc11 dd11
xx55 aa22 bb33 cc22 dd11
xx66 aa11 bb11 cc22 dd22
xx77 aa11 bb22 cc2 2 dd11
xx88 aa11 bb22 cc11 dd33
stable attribute
flexible attributes
Object reclassification from class d1 to d2 λ1=2, λ2=1/4
Atomic classification terms:
(b,b1b1), (b,b2b2), (b,b3b3)(a,a1a2), (a,a1a1), (a,a2a2), (a,a2a1) (c,c1c2), (c,c2c1), (c,c1c1), (c,c2c2)
λ1 - minimum support, λ2 - minimum confidence
ARED
XX a a bb cc dd
xx11 aa11 bb11 cc11 dd11
xx22 aa22 bb11 cc22 dd11
xx33 aa22 bb22 cc22 dd11
xx44 aa22 bb11 cc11 dd11
xx55 aa22 bb33 cc22 dd11
xx66 aa11 bb11 cc22 dd22
xx77 aa11 bb22 cc2 2 dd11
xx88 aa11 bb22 cc11 dd33
stable attribute
flexible attributes
Object reclassification from class d1 to d2 λ1=2, λ2=1/4
Notation: t1=(b,b1b1), t2=(b,b2b2), t3=(b,b3b3),
t4=(a,a1a2), t5=(a,a1a1), t6=(a,a2a2), t7=(a,a2a1),
t8=(c,c1c2), t9=(c,c2c1), t10=(c,c1c1),t11=(c,c2c2),
t12 = (d,d1 d2).
λ1 - minimum support, λ2 - minimum confidence
For decision attribute in S:NS(d,d1 d2)=[{x1,x2, x3, x4, x5, x7}, {x6}]
Object reclassification from class d1 to d2 λ1=2, λ2=1/4
For classification attribute in S:
NS(t1) = NS(b,b1b1) = [{x1,x2, x4, x6}, {x1,x2, x4, x6}] NS(t2) = NS(b,b2b2) = [{x3,x7, x8}, {x3,x7, x8}]
NS(t3) = NS(b,b3b3) = [{x5}, {x5}]
NS(t4) = NS (a,a1a2) = [{x1,x6, x7, x8}, {x2,x3, x4, x5}]
XX a a bb cc dd
xx11 aa11 bb11 cc11 dd11
xx22 aa22 bb11 cc22 dd11
xx33 aa22 bb22 cc22 dd11
xx44 aa22 bb11 cc11 dd11
xx55 aa22 bb33 cc22 dd11
xx66 aa11 bb11 cc22 dd22
xx77 aa11 bb22 cc2 2 dd11
xx88 aa11 bb22 cc11 dd33
Not marked λ1=3
Mark “-” λ2=0
Mark “-” λ1=1
Mark “-” λ2=0
)()sup( 11 ZYcardr
For decision attribute in S:NS(d,d1 d2)=[{x1,x2, x3, x4, x5, x7}, {x6}]
Object reclassification from class d1 to d2 λ1=2, λ2=1/4
For classification attribute in S:
XX a a bb cc dd
xx11 aa11 bb11 cc11 dd11
xx22 aa22 bb11 cc22 dd11
xx33 aa22 bb22 cc22 dd11
xx44 aa22 bb11 cc11 dd11
xx55 aa22 bb33 cc22 dd11
xx66 aa11 bb11 cc22 dd22
xx77 aa11 bb22 cc2 2 dd11
xx88 aa11 bb22 cc11 dd33
Not marked λ1=2
Mark “-” λ2= 0
Mark “+” λ1=4,λ2=1/4
NS(t5) = NS(a,a1a1) = [{x1,x6, x7, x8}, {x1,x6, x7, x8}] NS(t6)= NS(a,a2a2) = [{x2,x3, x4, x5}, {x2,x3, x4, x5}] NS(t7)= NS(a,a2a1) = [{x2,x3, x4, x5}, {x1,x6, x7, x8}]
For decision attribute in S:
NS(t12)=[{x1,x2, x3, x4, x5, x7}, {x6}]
Object reclassification from class d1 to d2
λ1=2, λ2=1/4
For classification attribute in S:
NS(t1)=[{x1,x2, x4, x6}, {x1,x2, x4, x6}]
Not marked λ1=3
NS(t2)=[{x3,x7, x8}, {x3,x7, x8}] Marked “-” λ2=0
NS(t3)=[{x5}, {x5}] Marked “-” λ1=1
NS(t4)=[{x1,x6, x7, x8}, {x2,x3, x4, x5}]
Marked “-” λ2=0
NS(t5)=[{x1,x6, x7, x8}, {x1,x6, x7, x8}]
Not marked λ1=2
NS(t6)=[{x2,x3, x4, x5}, {x2,x3, x4, x5}]
Marked “-” λ2=0
Mark “+” λ1=4, λ2=1/4NS(t7)=[{x2,x3, x4, x5}, {x1,x6, x7, x8}]
r = [t7 t1]
XX a a bb cc dd
xx11 aa11 bb11 cc11 dd11
xx22 aa22 bb11 cc22 dd11
xx33 aa22 bb22 cc22 dd11
xx44 aa22 bb11 cc11 dd11
xx55 aa22 bb33 cc22 dd11
xx66 aa11 bb11 cc22 dd22
xx77 aa11 bb22 cc2 2 dd11
xx88 aa11 bb22 cc11 dd33
For decision attribute in S:
NS(t12)=[{x1,x2, x3, x4, x5, x7}, {x6}]
Object reclassification from class d1 to d2
λ1=2, λ2=1/4
For classification attribute in S:
NS(t8)= NS(c,c1c2) = [{x1,x4, x8}, {x2, x3, x5, x6, x7}]
Not marked
Marked “-”
NS(t10) = NS(c,c1c1) = [{x1, x4, x8}, {x1, x4, x8}] Marked “-”
NS(t11) = NS (c,c2c2)= [{x2, x3, x5, x6, x7}, {x2, x3, x5, x6, x7}]
Not marked
conf = 2/3 *1/5 <λ2
NS(t9) = NS(c,c2c1) = [{x2, x3, x5, x6, x7}, {x1, x4, x8}]
XX a a bb cc dd
xx11 aa11 bb11 cc11 dd11
xx22 aa22 bb11 cc22 dd11
xx33 aa22 bb22 cc22 dd11
xx44 aa22 bb11 cc11 dd11
xx55 aa22 bb33 cc22 dd11
xx66 aa11 bb11 cc22 dd22
xx77 aa11 bb22 cc2 2 dd11
xx88 aa11 bb22 cc11 dd33
For decision attribute in S:
NS(t12)=[{x1,x2, x3, x4, x5, x7}, {x6}]
Object reclassification from class d1 to d2 λ1=2, λ2=1/4
For classification attribute in S:
Marked “+”
NS(t1*t11)=[{x2, x6}, {x2, x6}] Marked “-”, λ1=1
NS(t5*t8)=[{x1, x8}, {x6, x7}] Marked “-”, λ1=1
Rule r = [t1*t8→t12], conf = 1/2 ≥ λ2, sup=2 ≥ λ1
Now action terms of length 2 from unmarked action terms of length 1
NS(t1*t5)=[{x1, x6}, {x1, x6}] Marked “-”, λ1=1
NS(t1*t8)=[{x1, x4}, {x2, x6}]
NS(t5*t11)=[{x6, x7}, {x6, x7}] Marked “-”, λ1=1
NS(t8*t11)=[Ø, {x2, x3, x5, x6, x7}] Marked “-”
NS(t1)=[{x1,x2, x4, x6}, {x1,x2, x4, x6}] , NS(t5)=[{x1,x6, x7, x8}, {x1,x6, x7, x8}],
NS(t8)=[{x1,x4, x8}, {x2, x3, x5, x6, x7}], NS(t11)= [{x2, x3, x5, x6, x7}, {x2, x3, x5, x6, x7}].
ARED Algorithm
For decision attribute in S:
NS(t12)=[{x1,x2, x3, x4, x5, x7}, {x6}]
Object reclassification from class d1 to d2 λ1=2, λ2=1/4
For classification attribute in S:
Action rules:
[[(b,b1→b1)*(c,c1→c2)] → (d, d1→d2)]
[[(a,a2→a1] → (d, d1→d2)]
XX a a bb cc dd
xx11 aa11 bb11 cc11 dd11
xx22 aa22 bb11 cc22 dd11
xx33 aa22 bb22 cc22 dd11
xx44 aa22 bb11 cc11 dd11
xx55 aa22 bb33 cc22 dd11
xx66 aa11 bb11 cc22 dd22
xx77 aa11 bb22 cc2 2 dd11
xx88 aa11 bb22 cc11 dd33
Thank You
Questions?