action rules discovery systems: dear1, dear2, ared, ….. by zbigniew w. raś

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






xx77 aa11 bb22 cc2 2 dd11


(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


x1, x3 0 L

x2, x4 1 L

x5, x6 L

b = 3


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









(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










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










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











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











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









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









stable attribute

flexible attributes


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}]


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









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}]



XX a a bb cc dd









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


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










NS(t12)=[{x1,x2, x3, x4, x5, x7}, {x6}]

Object reclassification from class d1 to d2

λ1=2, λ2=1/4


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










NS(t12)=[{x1,x2, x3, x4, x5, x7}, {x6}]



Marked “+”

NS(t1*t11)=[{x2, x6}, {x2, x6}] 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*t8)=[{x1, x4}, {x2, x6}]


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


NS(t12)=[{x1,x2, x3, x4, x5, x7}, {x6}]



Action rules:

[[(b,b1→b1)*(c,c1→c2)] → (d, d1→d2)]

[[(a,a2→a1] → (d, d1→d2)]

XX a a bb cc dd









Thank You

Questions?

action rules discovery systems: dear1, dear2, ared, ….. by zbigniew w. raś

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