Databases Reference
In-Depth Information
3 Classes of Association Rules
3.1 Truth Preservation Condition and Implicational Quantifiers
Classes of association rules are defined by classes of 4ft quantifiers. The as-
sociation rule
ϕ
≈
ψ
belongs to the
class of implicational association rules
if
the 4ft quantifier
belongs to the
class of implicational quantifiers
. We also
say that the association rule
ϕ
≈
≈
ψ
is an
implicational rule
and that the 4ft
quantifier
is an
implicational quantifier
. This is the same for other classes
of association rules.
There are various important classes of 4ft quantifiers defined by
truth
preservation conditions
[12]. We say that class
≈
of 4ft-quantifiers is de-
fined by truth preservation condition
TPC
C
if there is a Boolean condi-
tion
TPC
C
(
a,b,c,d,a
,b
,c
,d
) concerning two fourfold contingency tables
C
a
,b
,c
,d
a,b,c,d
and
such that the following is true:
4ft quantifier
≈
belongs to the class
C
if and only if
TPC
C
(
a,b,c,d,a
,b
,c
,d
)
≈
(
a,b,c,d
)=1
∧
implies
(
a
,b
,c
,d
)=1
≈
a
,b
,c
,d
for all 4ft tables
.
The class of implicational quantifiers was defined in [2] by the
truth preser-
vation condition TPC
⇒
for implicational quantifiers
.Itis
a,b,c,d
and
TPC
⇒
=
a
≥
b
a
∧
≤
b
.
It means that the 4ft quantifier
≈
is
implicational
if
a
≥
b
≈
(
a,b,c,d
)=1
∧
a
∧
≤
b
implies
(
a
,b
,c
,d
)=1
≈
for all 4ft tables
a,b,c,d
and
a
,b
,c
,d
.
The truth preservation condition
TPC
⇒
for implicational quantifiers (i.e.
a
≥
b
≤
a
,b
,c
,d
a
∧
b
) means that the fourfold table
is
“better from
the point of view of implication”
than the fourfold table
a,b,c,d
(i-better
according to [2]). If
a,b,c,d
is the fourfold table of
ϕ
and
ψ
in data matrix
M
,
then the sentence “
better from the point of view of implication
”means:indata
matrix
a
,b
,c
,d
M
and if
is the fourfold table of
ϕ
and
ψ
in data matrix
M
there are more rows satisfying both
ϕ
and
ψ
than in data matrix
M
M
there are fewer rows satisfying
ϕ
and not satisfying
ψ
than in
and in
M
.
Thus if it is
a
≥
b
≤
a
∧
b
then it is reasonable to expect that if the
implicational association rule
ϕ
≈
ψ
(i.e. the rule expressing the implication
by
≈
) is true in the data matrix
M
then this rule is also true in data matrix
