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

Classes of association rules
short overview
Jan Rauch,
Department of Knowledge and Information
Engineering University of Economics, Prague
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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
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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
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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
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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
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* is implicational quantifier
M


M’



a
c
b
d

a’
c’
b’
d’


M’ is better from the point of view of implication:
a’  a  b’  b
If *(a, b, c, d) = 1 and
*(a’, b’, c’, d’) = 1
a’  a  b’  b
then
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
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Implication quantifiers – examples (1)
Founded implication: p,B (a,b,c,d) = 1 iff
a’  a  b’  b:
a
 p  a  Base
ab
a´
a´
a


 p  a´ a  Base
a´b´ a´b a  b
Founded 2b - implication: p,B (a,b,c,d) = 1 iff
a
 p  a  Base
a  2b
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Implication quantifiers – examples (2)
Lower critical implication for 0 < p  1, 0    0.5:
!p; (a,b,c,d) = 1
a+b
iff

(a+b)!
i!( a+bi)!
* pi *(1  p) abi  
i a
The rule  !p;  corresponds to the statistical test (on the level ) of the null
hypothesis H0: P( |  )  p against the alternative one H1: P( |  ) > p.
Here P( |  ) is the conditional probability of the validity of  under the
condition .
a +b
' '
a’  a  b’  b:

ia
(a '+b ')!
i!( a '+b 'i )!
* p * (1  p)
i
a ' b 'i

a +b

i a
(a+b) !
i!( a+bi )!
* p i * (1  p) abi  
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Deduction rules (1)
Is the deduction rule
M
A
A
E
a
c
A 0.9,50 E
A 0.9,50 E  F
E
b
d
we see:
correct?
M EF (EF)
a’
b’
A
c’
d’
A
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
A 0.9,50 E
A 0.9,50 E  F
is correct.
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Deduction rules (2)
A !0.95,0.05 E
Is the deduction rule
A !0.95,0.05 E  F
M
A
A
E
a
c
E
b
d
we see:
correct?
M EF (EF)
a’
b’
A
c’
d’
A
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
A !0.95,0.05 E
A !0.95,0.05 E  F
is correct.
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Deduction rules (3)
Additional correct deduction rules (prove it home):
A  B 0.9,50 E
A 0.9,50 E  B
A  B !0.95,0.05 E
A !0.95,0.05 E  B
Question:
* implication quantifier:
X*Y
X'  * Y '
iff ???
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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, 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
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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 implication quantifiers
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Correct Deduction Rules
X*Y
X'  * Y'
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
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Correct Deduction Rules
Example:
A  B !0.95,0.05 E
A !0.95,0.05 E  B
A  B  E  A  (E  B)
is correct because of
and
A  ( E   B)  A  B  E
are tautologies
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Table of Critical Frequencies
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)
 p, (a,b)  1
!
a +b
iff

i a
!p, (a,b)  1 iff
(a+b) !
i!( a+b i )!
* p i * (1  p) a b i  
b  Tb! p , (a )
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Table of maximal b
b
a +b

ia
(a + b) !
i
a bi
*
p
*
(
1

p
)

i! ( a + b  i )!
a+b

(a+b)!
i!( a+bi)!
* pi *(1  p) abi  
i a
a
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Class of  - double implication 4ft quantifiers
M
X
X
Y
a
c
M’
X
X
Y
b
d
True Preservation Condition:
example: X p Y
Y
a’
c’
Y
b’
d’
a’  a  b’ + c’  b + c
a/(a + b + c)  p
TCF: Tb*(a) = min{b+c| *(a, b, c) = 0}
*(a, b, c)= 1 iff b + c < Tb* (a)
X *Y
X'  * Y'
is correct iff ...
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Class of  - equivalence 4ft quantifiers
M
X
X
Y
a
c
M’
X
X
Y
b
d
True Preservation Condition:
example: X p Y
TCF:
Y
a’
c’
Y
b’
d’
a’ + d’  a + d  b’ + c’  b + c
(a + d)/(a+b+c+d)  p
Tb*(F) = min {b+c | *(a,b,c,d)=0  a+d=F}
*(a, b,c,d)= 1 iff b + c < Tb*(a + d)
X *Y
X'  * Y'
is correct iff ...
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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
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AA - quantifier has F-property
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