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 EFEF  (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 EFEF  (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 YY Xab XX cd M’M’ Y YY Xa’b’ XX 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 YY Xab XX cd M’M’ Y YY Xa’b’ XX 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