association action rules b y zbigniew w. ras 1,5 agnieszka dardzinska 2 li-shiang tsay 3 hanna...

Association Action RulesAssociation Action Rules

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Zbigniew W. RasZbigniew W. Ras1,51,5 Agnieszka DardzinskaAgnieszka Dardzinska22

Li-Shiang Tsay3 Hanna Wasyluk4

1)1) University of North Carolina, Charlotte, NC, USAUniversity of North Carolina, Charlotte, NC, USA2)2) Bialystok Technical University, Bialystok, Poland Bialystok Technical University, Bialystok, Poland 3)3) North Carolina A&T State Univ., Greensboro, USANorth Carolina A&T State Univ., Greensboro, USA4)4) Medical Center of Postgraduate Education, Warsaw, PolandMedical Center of Postgraduate Education, Warsaw, Poland5)5) Polish-Japanese Institute of Information Technology, Warsaw, PolandPolish-Japanese Institute of Information Technology, Warsaw, Poland

Example

XX a a bb cc dd

xx11 aa11 bb11 cc11 dd11






xx77 aa22 bb11 cc2 2 dd22


(a, a2) (b, b1 → b2)(c, c2)(d, d1 → d2)

Information System Information System SS

r=[(a, a2)(b, b1→b2)] → (d, d1→d2)

action rule

Dom (r) = {a, b, d}

Generating Frequent Action Sets (Apriori)

S = (X, A , V) – information systemλ1 – minimum support

ta - atomic action term, where NS(ta) = [Y1, Y2] and a A.

ta – FREQUENT, ifcard(Y1) ≥ λ1 and card(Y2) ≥ λ1


S=(X, A, V) – information systemλ1 – minimum support

ta - an atomic action set, where NS(ta) = [Y1, Y2] and a A.

1. Merging Step: Merge pairs (t1, t2) of frequent k-element action sets into (k + 1)-element candidate action set if all elements in t1 and t2 are the same except the last elements.

Example: If (a, a1 a2).(b, b1 b2), (a, a1 a2).(c, c2 c1) are frequent, then (a, a1 a2).(b, b1 b2).(c, c2 c1) is a candidate action set. It is frequent if c b and its support is not smaller than λ1.




1. Merging Step: Merge pairs (t1, t2) of frequent k-element action sets into (k + 1)-element candidate action set if all elements in t1 and t2 are the same except the last elements.

2. Pruning Step: Delete each (k + 1)-element candidate action set t if

either some k-element subset of t is not frequent or t is not frequent.Example: If (a, a1 a2).(b, b1 b2).(c, c2 c1) is a candidate

action set, then check if (b, b1 b2).(c, c2 c1), (a, a1 a2).(c, c2 c1),

(a, a1 a2).(b, b1 b2) are all frequent.




1. Merging Step: Merge pairs (t1, t2) of frequent k-element action sets into (k + 1)-element candidate action set if all elements in t1 and t2 are the same except the last elements.2. Pruning Step: Delete each (k + 1)-element candidate action set t if

either t is not an action set or some k-element subset of t is not a frequent k-element action set.

If t is (k + 1)-element candidate action set, all attributes listed in t are different,NS(t) = [Y1, Y2], and Card(Y1) ≥ λ1 and Card(Y2)

≥ λ1

then t is a frequent (k + 1)-element action set.

Generating Association Action Rules

S=(X, A, V) – information systemλ1 – minimum support & λ2 – minimum confidence

Definition: t – is a frequent action set in S, if t is frequent k-element action set in S, for some k.

Notation:

[t - t1] - action set containing all atomic action sets listed in t but not listed in t1.

AARS(λ1, λ2) - set of association action rules in S satisfying both thresholds λ1, λ2 for minimum support and minimum confidence.

Generating Association Action Rules

S=(X, A, V) – information systemλ1 – minimum support & λ2 – minimum confidence

Definition: t – is a frequent action set in S, if t is frequent k-element action set in S, for some k.

Notation:

[t - t1] - action set containing all atomic action sets listed in t but not listed in t1.

AARS(λ1, λ2) - set of association action rules in S satisfying both thresholds λ1, λ2 for minimum support and minimum confidence.

Construction:

t - frequent action set in S and t1 is its subset.

Any action rule r = [(t-t1)→t1] is an association action rule in AARS(λ1, λ2), if conf(r) ≥ λ2.

Association Action Rules, Example

XX a a bb cc dd









Stable: a, c

λ1=2, λ2 =4/9

Frequent Atomic Action Sets:

(a, a1) – support 2

(a, a2) – support 6

(b, b1) – support 4

(b, b2) – support 4

(b, b1→b2) – support 4

(b, b2→b1) – support 4

(c, c1) – support 5

(c, c2) – support 3

(d, d1) – support 4

(d, d2) – support 4

(d, d1→d2) – support 4

(d, d2→d1) – support 4


XX a a bb cc dd









Stable: a, c

λ1=2, λ2 =4/9

Frequent Action Sets:

(a, a1) (b, b1) – support 1 not frequent

(a, a1) (b, b2) – support 1 not frequent

(a, a1) (b, b1→b2) – support 1 not frequent

(a, a1) (b, b2→b1) – support 1 not frequent

(a, a1) (c, c1) – support 1 not frequent

(a, a1) (c, c2) – support 1 not frequent

(a, a1) (d, d1) – support 2

(a, a1) (d, d2) – support 0 not frequent

(a, a1) (d, d1→d2) – support 0 not frequent

(a, a1) (d, d2→d1) – support 0 not frequent

6. Association Action Rules, Example

XX a a bb cc dd









Stable: a, c

λ1=2, λ2 =4/9

Frequent Action Sets:

(a, a2) (b, b1) – support 3

(a, a2) (b, b2) – support 3

(a, a2) (b, b1→b2) – support 3

(a, a2) (b, b2→b1) – support 3

(a, a2) (c, c1) – support 4

(a, a2) (c, c2) – support 2



(a, a2) (d, d1→d2) – support 2

(a, a2) (d, d2→d1) – support 2


XX a a bb cc dd









Stable: a, c

λ1=2, λ2 =4/9

Frequent Action Sets:………….

…….……

…….……

(a, a2) (b, b1→b2) (c, c1) (d, d1→d2) –

- support 2

Association action rules can be constructed from frequent action sets.

We can construct association action rule:

[(a, a2)·(b, b1→b2)]→[(c, c1)·(d, d1→d2)]

Confidence: 4/9

Simple Association Action Rules

(a, a1→ a2) - atomic action set

cost((a, a1→ a2)) - cost of action expecting to change value of

attribute a from a1 to a2 .t1 = (a, a1→a2), t2 = (b, b1 → b2) - two atomic action sets

t1, t2 are positively correlated if changes t1, t2 support each other

Change t1 implies change t2 and t2 implies change t1.


Definition:

Let t = t1t2…tm is a frequent action set, where each ti - atomic action set.

Let T = {t1, t2,…, tm} and: ti~tj iff ti and tj are positively correlated.

Equivalence relation, partitions T into equivalence classes (T = T1 T2 … Tk)


Definition:



Now:In each equivalence class Ti, an atomic action set a(Ti) of the lowest cost is identified.

The cost of t is defined as: cost(t) =∑{cost(a(Ti)): 1≤ i ≤ k}


Definition:



Now:The cost of t is defined as: cost(t) =∑{cost(a(Ti)): 1≤ i ≤

k}

a(T1) a(T2) … a(Tk) → [t – {a(Ti): 1 i k}] - simple association action rule.


Definition:



Now:The cost of t is defined as: cost(t) =∑{cost(a(Ti)): 1≤ i ≤

k}

r = [ a(T1) a(T2) … a(Tk) → [t – {a(Ti): 1 i k}] ] - simple association action rule.

The cost of r is defined as the cost of a(T1) a(T2) … a(Tk)


Algorithm generating simple association action rules:

User gives three threshold values:

λ1 - minimum support, λ2 - minimum confidence, λ3 - maximum cost.

Strategy similar to Apriori [atomic action sets are ordered with respect to cost increase]

Thank You

Questions?

association action rules b y zbigniew w. ras 1,5 agnieszka dardzinska 2 li-shiang tsay 3 hanna...

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