Classes of association rules short overview

1. Classes of association rules short overview Jan Rauch, Department of Knowledge and Information Engineering University of Economics, Prague

2. Classes of association rules – overview • Introduction, classes of rules and quantifiers • Implicational quantifiers • Deduction rules for implicational quantifiers • Tables of critical frequencies for implicational quantifiers •  -double implication 4ft quantifiers •  - equivalence 4ft quantifiers • 4ft quantifiers with F-property

3. Classes of association rules – Introduction • Simple intuitive definition • Each class contains both simple association rules and comlex association rules corresponding to statistical hypothesis tests • Important both theoretical and practical properties • Examples: • imlicational association rules • double imlicationalassociation rules • -double imlicational association rules • equivalency association rules •  - equivalency association rules • rules with F-property

4. Literature Hájek, P. - Havránek T.: Mechanising Hypothesis Formation – Mathematical Foundations for a General Theory. Berlin – Heidelberg - New York, Springer-Verlag, 1978, 396 pp, http://www.cs.cas.cz/~hajek/guhabook/ Rauch, J.: Logic of Association Rules. Applied Intelligence, 2005, No. 22, 9-28 Rauch, J.: Classes of Association Rules, An Overview. In: LIN, T.Y. Ying, X.(Ed.): Foundation of Semantic Oriented Data and Web Mining. Proceedings of an ICDM 2005 Workshop, IEEE Houston 2005. pp 68 – 74. http://www.cs.sjsu.edu/faculty/tylin/ICDM05/proceeding.pdf

5. Classes of 4ft-quantifiers Association rule    belongs to the class  of association rules if and only if the 4ft-quantifier belongs to the class  of 4ft-quantifiers Examples: • association rule    is implicational iff  is implicational • association rule    is -double implicational iff  is -double implicational • association rule    is - equivalency iff  is - equivalency

6. M M’      a b  a’ b’  c d  c’ d’ * is implicational quantifier M’ is better from the point of view of implication: a’ a  b’ b If *(a, b, c, d) = 1 and a’  a  b’  b then *(a’, b’, c’, d’) = 1 Truth Preservation Condition for implicational quantifiers: TPC :a’ a  b’ b * is implicational: If *(a, b, c, d) = 1 and TPC then *(a’, b’, c’, d’) = 1

7. Implication quantifiers – examples (1) Founded implication: p,B (a,b,c,d) = 1 iff a’ a b’ b: Founded 2b - implication: p,B (a,b,c,d) = 1 iff

8. Implication quantifiers – examples (2) Lower critical implication for0 < p  1, 0  0.5: !p; (a,b,c,d) = 1 iff The rule  !p; corresponds to the statistical test (on the level )of the null hypothesis H0: P( |  ) p againstthe alternative one H1: P( |  ) > p. Here P( |  ) is the conditional probability of thevalidity of  under the condition . a’ a b’ b:

9. M M EF E  E (EF) A a b A a’ b’  A c d  A c’ d’ Deduction rules (1) Is the deduction rule correct? we see: a’ a b’ b and TPC thus if 0.9,50(a,b,c,d) = 1 then also 0.9, 50(a,b,c,d) = 1 Yes, the deduction rule is correct.

10. M M EF E  E (EF) A a b A a’ b’  A c d  A c’ d’ Deduction rules (2) Is the deduction rule correct? we see: a’ a b’ b and it is TPC and thus if !0.95,0.05(a,b,c,d) = 1 then also !0.95, 0.05(a,b,c,d) = 1 Yes, the deduction rule is correct.

11. Deduction rules (3) Additional correct deduction rules (prove it home): Question: * implication quantifier: iff ???

12. Deduction rules – two notions Associated propositional formula () associated to Boolean attribute : Rule p,B e.g. A  B C p,B D E  F A, B, C, B, D, E, Fare Boolean attributes ( ): Boolean attributes  propositional variables () = A  B  C () = D E  F A, B, C, D, E, F are propositional variables, we can decide if () is a tautology

13. Deduction rules – two notions Implicational quantifier  is interesting: Iis a – dependent , b – dependent and(0,0,c,d) = 0 is a - dependent if exists a, a’, b, c, d : (a,b,c,d) (a’, b, c, d) 0.9, 50, !0.9, 0.05 are interesting implicationquantifiers

14. Correct Deduction Rules is thecorrect deduction ruleiff 1) or 2) are satisfied: 1) both (X) (Y) (X’) (Y’) and (X’) (Y’) (X) (Y) are tautologies 2) (X)  (Y) is a tautology

15. Correct Deduction Rules Example: is correct because of A  B EA  (EB)and A ( E B)  A  BE are tautologies

16. Table of Critical Frequencies implication quantifier: if*(a, b, c, d) = 1and a’ a b’ b then*(a’, b’, c’, d’) = 1 *is c, d independent, thus *(a, b) instead of*(a, b, c, d) Table of maximal b for *:Tb*(a) = min {e|*(a, e) = 0} *(a, b)= 1 iff b < Tb* (a)

17. Table of maximal b b a

18. M’ M Y Y Y Y X X a a’ b’ b X X c c’ d’ d Class of  -double implication 4ft quantifiers True Preservation Condition: a’ a b’ + c’  b + c example: X p Y a/(a + b + c) p TCF: Tb*(a) = min{b+c|*(a, b, c) = 0} *(a, b, c)= 1 iff b + c < Tb* (a) is correct iff ...

19. M’ M Y Y Y Y X X a a’ b’ b X X c c’ d’ d Class of  - equivalence 4ft quantifiers True Preservation Condition: a’ + d’  a + d  b’ + c’  b + c example:X p Y (a + d)/(a+b+c+d)  p TCF: Tb*(F)=min {b+c | *(a,b,c,d)=0  a+d=F} *(a, b,c,d)= 1 iff b +c < Tb*(a + d) is correct iff ...

20. 4ft quantifiers with F-property  has the F-property if it satisfies If (a,b,c,d) = 1 and b c – 1  0 then(a,b+1,c-1,d)= 1 If (a,b,c,d) = 1 and c b – 1  0 then (a,b -1,c+1,d)= 1 If is symmetrical andhas the F-property then there is a function T(a,d,n) such that for a+b+c+d = n is  (a,b,c,d) = 1 iff| b-c | T(a,d,n) Fisher’s quantifier and 2 quantifier have the F-property

21. AA - quantifier has F-property