Databases Reference
In-Depth Information
In addition we sometimes use
r
=
a
+
b
,
k
=
a
+
c
and
n
=
a
+
b
+
c
+
d
,see
also Table 1. We use the abbreviation “4ft” instead of the expression “fourfold
table”. The notion
4ft table
is used for all possible tables 4
ft
(
ϕ
,
ψ
,
M
).
We write
(
a,b,c,d
)=1
if the condition corresponding to the 4ft-quantifier
≈
≈
is satisfied for the
quadruple
a,b,c,d
of integer non-negative numbers, otherwise we write
≈
(
a,b,c,d
)=0.
The
(
a,b,c,d
) defined for all 4ft tables can be
informally called
associated function of 4-ft quantifier
{
0
,
1
}
- valued function
≈
≈
. The precise definition
of associated function of 4ft-quantifier is given in [2].
We write
Val
(
ϕ
≈
ψ,
M
) = 1 if the association rule
ϕ
≈
ψ
is true in data
matrix
)=0.
Various kinds of dependencies of the Boolean attributes
ϕ
and
ψ
can be
expressed by suitable 4ft-quantifiers. Some examples follow (see also [9]).
The 4ft-quantifier
M
, otherwise we write
Val
(
ϕ
≈
ψ,
M
⇒
p,Base
of
founded implication
for 0
<p
≤
1and
Base >
0 [2] is defined by the condition
a
a
+
b
≥
p
∧
a
≥
Base
.
This means that at least 100
p
per cent of objects satisfying
ϕ
satisfy also
ψ
and that there are at least
Base
objects of
M
satisfying both
ϕ
and
ψ
.
!
p,α,Base
The 4ft-quantifier
⇒
of
lower critical implication
for 0
<p
≤
1,
0
<α<
0
.
5and
Base >
0 [2] is defined by the condition
a
+
b
i
p
i
(1
a
+
b
p
)
a
+
b−i
−
≤
α
∧
a
≥
Base
.
i
=
a
This corresponds to the statistical test (on the level
α
) of the null hypothesis
H
0
:
P
(
ψ
|
ϕ
)
≤
p
against the alternative one
H
1
:
P
(
ψ
|
ϕ
)
>p
. Here
P
(
ψ
|
ϕ
)is
the conditional probability of the validity of
ψ
under the condition
ϕ
.
The 4ft-quantifier
⇔
p,Base
of
founded double implication
for 0
<p
≤
1
and
Base >
0 [4] is defined by the condition
a
a
+
b
+
c
≥
p
∧
a
≥
Base
.
This means that at least 100
p
per cent of objects satisfying
ϕ
or
ψ
satisfy
both
ϕ
and
ψ
and that there are at least
Base
objects of
M
satisfying both
ϕ
and
ψ
.
The 4ft-quantifier
!
p,α,Base
⇔
of
lower critical double implication
for 0
<
p
≤
1, 0
<α<
0
.
5and
Base >
0 [4] is defined by the condition
a
+
b
+
c
i
p
i
(1
a
+
b
+
c
p
)
a
+
b
+
c−i
−
≤
α
∧
a
≥
Base
.
i
=
a
