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 imlicational association rules -double imlicational association rules equivalency association rules - equivalency association rules rules with F-property
4 Hájek, P. - Havránek T.: Mechanising Hypothesis Formation – Mathematical Foundations for a General Theory. Berlin – Heidelberg - New York, Springer-Verlag, 1978, 396 pp, 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 pp 68 – Literature
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 * is implicational quantifier M ab cd M’M’ a’b’ c’d’ 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 a’ a b’ b: Implication quantifiers – examples (1) Founded implication: p,B (a,b,c,d) = 1 iff Founded 2b - implication: p,B (a,b,c,d) = 1 iff
8 Lower critical implication for 0 < p 1, 0 0.5: ! p; (a,b,c,d) = 1 iff a’ a b’ b: Implication quantifiers – examples (2) The rule ! p; corresponds to the statistical test (on the level ) of the null hypothesis H 0 : P( | ) p against the alternative one H 1 : P( | ) > p. Here P( | ) is the conditional probability of the validity of under the condition .
9 Deduction rules (1) M E E Aab A cd M EFEF (E F) Aa’b’ A c’d’ 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 Is the deduction rule correct? Yes, the deduction rule is correct.
10 Deduction rules (2) M E E Aab A cd M EFEF (E F) Aa’b’ A c’d’ 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 Is the deduction rule correct? Yes, the deduction rule is correct.
11 Deduction rules (3) Additional correct deduction rules (prove it home): * implication quantifier: iff ??? Question:
12 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, F are 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 Deduction rules – two notions
13 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 implication quantifiers Deduction rules – two notions Implicational quantifier is interesting :
14 Correct Deduction Rules is the correct deduction rule iff 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: A B E A (E B) and A ( E B) A B E are tautologies is correct because of
16 implication quantifier: if *(a, b, c, d) = 1 and 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) Table of Critical Frequencies
17 Table of maximal b b a
18 Class of - double implication 4ft quantifiers M Y YY Xab XX cd M’M’ Y YY Xa’b’ XX c’d’ 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 Class of - equivalence 4ft quantifiers M Y YY Xab XX cd M’M’ Y YY Xa’b’ XX c’d’ 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 1)If (a,b,c,d) = 1 and b c – 1 0 then (a,b+1,c-1,d) = 1 2)If (a,b,c,d) = 1 and c b – 1 0 then (a,b -1,c+1,d) = 1 If is symmetrical and has 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